reworked Probability theory: measures are not type restricted to positive extended reals
1.1 --- a/src/HOL/IsaMakefile Mon Mar 14 14:37:47 2011 +0100
1.2 +++ b/src/HOL/IsaMakefile Mon Mar 14 14:37:49 2011 +0100
1.3 @@ -1168,9 +1168,10 @@
1.4 Multivariate_Analysis/Topology_Euclidean_Space.thy \
1.5 Multivariate_Analysis/document/root.tex \
1.6 Multivariate_Analysis/normarith.ML Library/Glbs.thy \
1.7 - Library/Indicator_Function.thy Library/Inner_Product.thy \
1.8 - Library/Numeral_Type.thy Library/Convex.thy Library/FrechetDeriv.thy \
1.9 - Library/Product_Vector.thy Library/Product_plus.thy
1.10 + Library/Extended_Reals.thy Library/Indicator_Function.thy \
1.11 + Library/Inner_Product.thy Library/Numeral_Type.thy Library/Convex.thy \
1.12 + Library/FrechetDeriv.thy Library/Product_Vector.thy \
1.13 + Library/Product_plus.thy
1.14 @cd Multivariate_Analysis; $(ISABELLE_TOOL) usedir -b -g true $(OUT)/HOL HOL-Multivariate_Analysis
1.15
1.16
1.17 @@ -1185,7 +1186,6 @@
1.18 Probability/ex/Koepf_Duermuth_Countermeasure.thy \
1.19 Probability/Information.thy Probability/Lebesgue_Integration.thy \
1.20 Probability/Lebesgue_Measure.thy Probability/Measure.thy \
1.21 - Probability/Positive_Extended_Real.thy \
1.22 Probability/Probability_Space.thy Probability/Probability.thy \
1.23 Probability/Product_Measure.thy Probability/Radon_Nikodym.thy \
1.24 Probability/ROOT.ML Probability/Sigma_Algebra.thy \
2.1 --- a/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy Mon Mar 14 14:37:47 2011 +0100
2.2 +++ b/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy Mon Mar 14 14:37:49 2011 +0100
2.3 @@ -1028,7 +1028,7 @@
2.4 qed simp
2.5 qed
2.6
2.7 -lemma setsum_of_pextreal:
2.8 +lemma setsum_real_of_extreal:
2.9 assumes "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
2.10 shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)"
2.11 proof -
2.12 @@ -1062,17 +1062,6 @@
2.13 by induct (auto simp: extreal_right_distrib setsum_nonneg)
2.14 qed simp
2.15
2.16 -lemma setsum_real_of_extreal:
2.17 - assumes "\<And>x. x \<in> A \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
2.18 - shows "real (\<Sum>x\<in>A. f x) = (\<Sum>x\<in>A. real (f x))"
2.19 -proof cases
2.20 - assume "finite A" from this assms show ?thesis
2.21 - proof induct
2.22 - case (insert a A) then show ?case
2.23 - by (simp add: real_of_extreal_add setsum_Inf)
2.24 - qed simp
2.25 -qed simp
2.26 -
2.27 lemma sums_extreal_positive:
2.28 fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>i. 0 \<le> f i" shows "f sums (SUP n. \<Sum>i<n. f i)"
2.29 proof -
3.1 --- a/src/HOL/Probability/Borel_Space.thy Mon Mar 14 14:37:47 2011 +0100
3.2 +++ b/src/HOL/Probability/Borel_Space.thy Mon Mar 14 14:37:49 2011 +0100
3.3 @@ -3,13 +3,9 @@
3.4 header {*Borel spaces*}
3.5
3.6 theory Borel_Space
3.7 - imports Sigma_Algebra Positive_Extended_Real Multivariate_Analysis
3.8 + imports Sigma_Algebra Multivariate_Analysis
3.9 begin
3.10
3.11 -lemma LIMSEQ_max:
3.12 - "u ----> (x::real) \<Longrightarrow> (\<lambda>i. max (u i) 0) ----> max x 0"
3.13 - by (fastsimp intro!: LIMSEQ_I dest!: LIMSEQ_D)
3.14 -
3.15 section "Generic Borel spaces"
3.16
3.17 definition "borel = sigma \<lparr> space = UNIV::'a::topological_space set, sets = open\<rparr>"
3.18 @@ -112,26 +108,8 @@
3.19 ultimately show "?I \<in> borel_measurable M" by auto
3.20 qed
3.21
3.22 -lemma borel_measurable_translate:
3.23 - assumes "A \<in> sets borel" and trans: "\<And>B. open B \<Longrightarrow> f -` B \<in> sets borel"
3.24 - shows "f -` A \<in> sets borel"
3.25 -proof -
3.26 - have "A \<in> sigma_sets UNIV open" using assms
3.27 - by (simp add: borel_def sigma_def)
3.28 - thus ?thesis
3.29 - proof (induct rule: sigma_sets.induct)
3.30 - case (Basic a) thus ?case using trans[of a] by (simp add: mem_def)
3.31 - next
3.32 - case (Compl a)
3.33 - moreover have "UNIV \<in> sets borel"
3.34 - using borel.top by simp
3.35 - ultimately show ?case
3.36 - by (auto simp: vimage_Diff borel.Diff)
3.37 - qed (auto simp add: vimage_UN)
3.38 -qed
3.39 -
3.40 lemma (in sigma_algebra) borel_measurable_restricted:
3.41 - fixes f :: "'a \<Rightarrow> 'x\<Colon>{topological_space, semiring_1}" assumes "A \<in> sets M"
3.42 + fixes f :: "'a \<Rightarrow> extreal" assumes "A \<in> sets M"
3.43 shows "f \<in> borel_measurable (restricted_space A) \<longleftrightarrow>
3.44 (\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
3.45 (is "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable M")
3.46 @@ -142,7 +120,7 @@
3.47 show ?thesis unfolding *
3.48 unfolding in_borel_measurable_borel
3.49 proof (simp, safe)
3.50 - fix S :: "'x set" assume "S \<in> sets borel"
3.51 + fix S :: "extreal set" assume "S \<in> sets borel"
3.52 "\<forall>S\<in>sets borel. ?f -` S \<inter> A \<in> op \<inter> A ` sets M"
3.53 then have "?f -` S \<inter> A \<in> op \<inter> A ` sets M" by auto
3.54 then have f: "?f -` S \<inter> A \<in> sets M"
3.55 @@ -161,7 +139,7 @@
3.56 then show ?thesis using f by auto
3.57 qed
3.58 next
3.59 - fix S :: "'x set" assume "S \<in> sets borel"
3.60 + fix S :: "extreal set" assume "S \<in> sets borel"
3.61 "\<forall>S\<in>sets borel. ?f -` S \<inter> space M \<in> sets M"
3.62 then have f: "?f -` S \<inter> space M \<in> sets M" by auto
3.63 then show "?f -` S \<inter> A \<in> op \<inter> A ` sets M"
3.64 @@ -1024,103 +1002,6 @@
3.65 using borel_measurable_euclidean_component
3.66 unfolding nth_conv_component by auto
3.67
3.68 -section "Borel space over the real line with infinity"
3.69 -
3.70 -lemma borel_Real_measurable:
3.71 - "A \<in> sets borel \<Longrightarrow> Real -` A \<in> sets borel"
3.72 -proof (rule borel_measurable_translate)
3.73 - fix B :: "pextreal set" assume "open B"
3.74 - then obtain T x where T: "open T" "Real ` (T \<inter> {0..}) = B - {\<omega>}" and
3.75 - x: "\<omega> \<in> B \<Longrightarrow> 0 \<le> x" "\<omega> \<in> B \<Longrightarrow> {Real x <..} \<subseteq> B"
3.76 - unfolding open_pextreal_def by blast
3.77 - have "Real -` B = Real -` (B - {\<omega>})" by auto
3.78 - also have "\<dots> = Real -` (Real ` (T \<inter> {0..}))" using T by simp
3.79 - also have "\<dots> = (if 0 \<in> T then T \<union> {.. 0} else T \<inter> {0..})"
3.80 - apply (auto simp add: Real_eq_Real image_iff)
3.81 - apply (rule_tac x="max 0 x" in bexI)
3.82 - by (auto simp: max_def)
3.83 - finally show "Real -` B \<in> sets borel"
3.84 - using `open T` by auto
3.85 -qed simp
3.86 -
3.87 -lemma borel_real_measurable:
3.88 - "A \<in> sets borel \<Longrightarrow> (real -` A :: pextreal set) \<in> sets borel"
3.89 -proof (rule borel_measurable_translate)
3.90 - fix B :: "real set" assume "open B"
3.91 - { fix x have "0 < real x \<longleftrightarrow> (\<exists>r>0. x = Real r)" by (cases x) auto }
3.92 - note Ex_less_real = this
3.93 - have *: "real -` B = (if 0 \<in> B then real -` (B \<inter> {0 <..}) \<union> {0, \<omega>} else real -` (B \<inter> {0 <..}))"
3.94 - by (force simp: Ex_less_real)
3.95 -
3.96 - have "open (real -` (B \<inter> {0 <..}) :: pextreal set)"
3.97 - unfolding open_pextreal_def using `open B`
3.98 - by (auto intro!: open_Int exI[of _ "B \<inter> {0 <..}"] simp: image_iff Ex_less_real)
3.99 - then show "(real -` B :: pextreal set) \<in> sets borel" unfolding * by auto
3.100 -qed simp
3.101 -
3.102 -lemma (in sigma_algebra) borel_measurable_Real[intro, simp]:
3.103 - assumes "f \<in> borel_measurable M"
3.104 - shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
3.105 - unfolding in_borel_measurable_borel
3.106 -proof safe
3.107 - fix S :: "pextreal set" assume "S \<in> sets borel"
3.108 - from borel_Real_measurable[OF this]
3.109 - have "(Real \<circ> f) -` S \<inter> space M \<in> sets M"
3.110 - using assms
3.111 - unfolding vimage_compose in_borel_measurable_borel
3.112 - by auto
3.113 - thus "(\<lambda>x. Real (f x)) -` S \<inter> space M \<in> sets M" by (simp add: comp_def)
3.114 -qed
3.115 -
3.116 -lemma (in sigma_algebra) borel_measurable_real[intro, simp]:
3.117 - fixes f :: "'a \<Rightarrow> pextreal"
3.118 - assumes "f \<in> borel_measurable M"
3.119 - shows "(\<lambda>x. real (f x)) \<in> borel_measurable M"
3.120 - unfolding in_borel_measurable_borel
3.121 -proof safe
3.122 - fix S :: "real set" assume "S \<in> sets borel"
3.123 - from borel_real_measurable[OF this]
3.124 - have "(real \<circ> f) -` S \<inter> space M \<in> sets M"
3.125 - using assms
3.126 - unfolding vimage_compose in_borel_measurable_borel
3.127 - by auto
3.128 - thus "(\<lambda>x. real (f x)) -` S \<inter> space M \<in> sets M" by (simp add: comp_def)
3.129 -qed
3.130 -
3.131 -lemma (in sigma_algebra) borel_measurable_Real_eq:
3.132 - assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
3.133 - shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
3.134 -proof
3.135 - have [simp]: "(\<lambda>x. Real (f x)) -` {\<omega>} \<inter> space M = {}"
3.136 - by auto
3.137 - assume "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
3.138 - hence "(\<lambda>x. real (Real (f x))) \<in> borel_measurable M"
3.139 - by (rule borel_measurable_real)
3.140 - moreover have "\<And>x. x \<in> space M \<Longrightarrow> real (Real (f x)) = f x"
3.141 - using assms by auto
3.142 - ultimately show "f \<in> borel_measurable M"
3.143 - by (simp cong: measurable_cong)
3.144 -qed auto
3.145 -
3.146 -lemma (in sigma_algebra) borel_measurable_pextreal_eq_real:
3.147 - "f \<in> borel_measurable M \<longleftrightarrow>
3.148 - ((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<omega>} \<inter> space M \<in> sets M)"
3.149 -proof safe
3.150 - assume "f \<in> borel_measurable M"
3.151 - then show "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<omega>} \<inter> space M \<in> sets M"
3.152 - by (auto intro: borel_measurable_vimage borel_measurable_real)
3.153 -next
3.154 - assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<omega>} \<inter> space M \<in> sets M"
3.155 - have "f -` {\<omega>} \<inter> space M = {x\<in>space M. f x = \<omega>}" by auto
3.156 - with * have **: "{x\<in>space M. f x = \<omega>} \<in> sets M" by simp
3.157 - have f: "f = (\<lambda>x. if f x = \<omega> then \<omega> else Real (real (f x)))"
3.158 - by (simp add: fun_eq_iff Real_real)
3.159 - show "f \<in> borel_measurable M"
3.160 - apply (subst f)
3.161 - apply (rule measurable_If)
3.162 - using * ** by auto
3.163 -qed
3.164 -
3.165 lemma borel_measurable_continuous_on1:
3.166 fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
3.167 assumes "continuous_on UNIV f"
3.168 @@ -1187,206 +1068,213 @@
3.169 using measurable_comp[OF f borel_measurable_borel_log[OF `1 < b`]]
3.170 by (simp add: comp_def)
3.171
3.172 +subsection "Borel space on the extended reals"
3.173 +
3.174 +lemma borel_measurable_extreal_borel:
3.175 + "extreal \<in> borel_measurable borel"
3.176 + unfolding borel_def[where 'a=extreal]
3.177 +proof (rule borel.measurable_sigma)
3.178 + fix X :: "extreal set" assume "X \<in> sets \<lparr>space = UNIV, sets = open \<rparr>"
3.179 + then have "open X" by (auto simp: mem_def)
3.180 + then have "open (extreal -` X \<inter> space borel)"
3.181 + by (simp add: open_extreal_vimage)
3.182 + then show "extreal -` X \<inter> space borel \<in> sets borel" by auto
3.183 +qed auto
3.184 +
3.185 +lemma (in sigma_algebra) borel_measurable_extreal[simp, intro]:
3.186 + assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. extreal (f x)) \<in> borel_measurable M"
3.187 + using measurable_comp[OF f borel_measurable_extreal_borel] unfolding comp_def .
3.188 +
3.189 +lemma borel_measurable_real_of_extreal_borel:
3.190 + "(real :: extreal \<Rightarrow> real) \<in> borel_measurable borel"
3.191 + unfolding borel_def[where 'a=real]
3.192 +proof (rule borel.measurable_sigma)
3.193 + fix B :: "real set" assume "B \<in> sets \<lparr>space = UNIV, sets = open \<rparr>"
3.194 + then have "open B" by (auto simp: mem_def)
3.195 + have *: "extreal -` real -` (B - {0}) = B - {0}" by auto
3.196 + have open_real: "open (real -` (B - {0}) :: extreal set)"
3.197 + unfolding open_extreal_def * using `open B` by auto
3.198 + show "(real -` B \<inter> space borel :: extreal set) \<in> sets borel"
3.199 + proof cases
3.200 + assume "0 \<in> B"
3.201 + then have *: "real -` B = real -` (B - {0}) \<union> {-\<infinity>, \<infinity>, 0}"
3.202 + by (auto simp add: real_of_extreal_eq_0)
3.203 + then show "(real -` B :: extreal set) \<inter> space borel \<in> sets borel"
3.204 + using open_real by auto
3.205 + next
3.206 + assume "0 \<notin> B"
3.207 + then have *: "(real -` B :: extreal set) = real -` (B - {0})"
3.208 + by (auto simp add: real_of_extreal_eq_0)
3.209 + then show "(real -` B :: extreal set) \<inter> space borel \<in> sets borel"
3.210 + using open_real by auto
3.211 + qed
3.212 +qed auto
3.213 +
3.214 +lemma (in sigma_algebra) borel_measurable_real_of_extreal[simp, intro]:
3.215 + assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. real (f x :: extreal)) \<in> borel_measurable M"
3.216 + using measurable_comp[OF f borel_measurable_real_of_extreal_borel] unfolding comp_def .
3.217 +
3.218 +lemma (in sigma_algebra) borel_measurable_extreal_iff:
3.219 + shows "(\<lambda>x. extreal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
3.220 +proof
3.221 + assume "(\<lambda>x. extreal (f x)) \<in> borel_measurable M"
3.222 + from borel_measurable_real_of_extreal[OF this]
3.223 + show "f \<in> borel_measurable M" by auto
3.224 +qed auto
3.225 +
3.226 +lemma (in sigma_algebra) borel_measurable_extreal_iff_real:
3.227 + "f \<in> borel_measurable M \<longleftrightarrow>
3.228 + ((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)"
3.229 +proof safe
3.230 + assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M"
3.231 + have "f -` {\<infinity>} \<inter> space M = {x\<in>space M. f x = \<infinity>}" "f -` {-\<infinity>} \<inter> space M = {x\<in>space M. f x = -\<infinity>}" by auto
3.232 + with * have **: "{x\<in>space M. f x = \<infinity>} \<in> sets M" "{x\<in>space M. f x = -\<infinity>} \<in> sets M" by simp_all
3.233 + let "?f x" = "if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else extreal (real (f x))"
3.234 + have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
3.235 + also have "?f = f" by (auto simp: fun_eq_iff extreal_real)
3.236 + finally show "f \<in> borel_measurable M" .
3.237 +qed (auto intro: measurable_sets borel_measurable_real_of_extreal)
3.238
3.239 lemma (in sigma_algebra) less_eq_ge_measurable:
3.240 fixes f :: "'a \<Rightarrow> 'c::linorder"
3.241 - shows "{x\<in>space M. a < f x} \<in> sets M \<longleftrightarrow> {x\<in>space M. f x \<le> a} \<in> sets M"
3.242 + shows "f -` {a <..} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {..a} \<inter> space M \<in> sets M"
3.243 proof
3.244 - assume "{x\<in>space M. f x \<le> a} \<in> sets M"
3.245 - moreover have "{x\<in>space M. a < f x} = space M - {x\<in>space M. f x \<le> a}" by auto
3.246 - ultimately show "{x\<in>space M. a < f x} \<in> sets M" by auto
3.247 + assume "f -` {a <..} \<inter> space M \<in> sets M"
3.248 + moreover have "f -` {..a} \<inter> space M = space M - f -` {a <..} \<inter> space M" by auto
3.249 + ultimately show "f -` {..a} \<inter> space M \<in> sets M" by auto
3.250 next
3.251 - assume "{x\<in>space M. a < f x} \<in> sets M"
3.252 - moreover have "{x\<in>space M. f x \<le> a} = space M - {x\<in>space M. a < f x}" by auto
3.253 - ultimately show "{x\<in>space M. f x \<le> a} \<in> sets M" by auto
3.254 + assume "f -` {..a} \<inter> space M \<in> sets M"
3.255 + moreover have "f -` {a <..} \<inter> space M = space M - f -` {..a} \<inter> space M" by auto
3.256 + ultimately show "f -` {a <..} \<inter> space M \<in> sets M" by auto
3.257 qed
3.258
3.259 lemma (in sigma_algebra) greater_eq_le_measurable:
3.260 fixes f :: "'a \<Rightarrow> 'c::linorder"
3.261 - shows "{x\<in>space M. f x < a} \<in> sets M \<longleftrightarrow> {x\<in>space M. a \<le> f x} \<in> sets M"
3.262 + shows "f -` {..< a} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {a ..} \<inter> space M \<in> sets M"
3.263 proof
3.264 - assume "{x\<in>space M. a \<le> f x} \<in> sets M"
3.265 - moreover have "{x\<in>space M. f x < a} = space M - {x\<in>space M. a \<le> f x}" by auto
3.266 - ultimately show "{x\<in>space M. f x < a} \<in> sets M" by auto
3.267 + assume "f -` {a ..} \<inter> space M \<in> sets M"
3.268 + moreover have "f -` {..< a} \<inter> space M = space M - f -` {a ..} \<inter> space M" by auto
3.269 + ultimately show "f -` {..< a} \<inter> space M \<in> sets M" by auto
3.270 next
3.271 - assume "{x\<in>space M. f x < a} \<in> sets M"
3.272 - moreover have "{x\<in>space M. a \<le> f x} = space M - {x\<in>space M. f x < a}" by auto
3.273 - ultimately show "{x\<in>space M. a \<le> f x} \<in> sets M" by auto
3.274 + assume "f -` {..< a} \<inter> space M \<in> sets M"
3.275 + moreover have "f -` {a ..} \<inter> space M = space M - f -` {..< a} \<inter> space M" by auto
3.276 + ultimately show "f -` {a ..} \<inter> space M \<in> sets M" by auto
3.277 qed
3.278
3.279 -lemma (in sigma_algebra) less_eq_le_pextreal_measurable:
3.280 - fixes f :: "'a \<Rightarrow> pextreal"
3.281 - shows "(\<forall>a. {x\<in>space M. a < f x} \<in> sets M) \<longleftrightarrow> (\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M)"
3.282 +lemma (in sigma_algebra) borel_measurable_uminus_borel_extreal:
3.283 + "(uminus :: extreal \<Rightarrow> extreal) \<in> borel_measurable borel"
3.284 +proof (subst borel_def, rule borel.measurable_sigma)
3.285 + fix X :: "extreal set" assume "X \<in> sets \<lparr>space = UNIV, sets = open\<rparr>"
3.286 + then have "open X" by (simp add: mem_def)
3.287 + have "uminus -` X = uminus ` X" by (force simp: image_iff)
3.288 + then have "open (uminus -` X)" using `open X` extreal_open_uminus by auto
3.289 + then show "uminus -` X \<inter> space borel \<in> sets borel" by auto
3.290 +qed auto
3.291 +
3.292 +lemma (in sigma_algebra) borel_measurable_uminus_extreal[intro]:
3.293 + assumes "f \<in> borel_measurable M"
3.294 + shows "(\<lambda>x. - f x :: extreal) \<in> borel_measurable M"
3.295 + using measurable_comp[OF assms borel_measurable_uminus_borel_extreal] by (simp add: comp_def)
3.296 +
3.297 +lemma (in sigma_algebra) borel_measurable_uminus_eq_extreal[simp]:
3.298 + "(\<lambda>x. - f x :: extreal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
3.299 proof
3.300 - assume a: "\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M"
3.301 - show "\<forall>a. {x \<in> space M. a < f x} \<in> sets M"
3.302 - proof
3.303 - fix a show "{x \<in> space M. a < f x} \<in> sets M"
3.304 - proof (cases a)
3.305 - case (preal r)
3.306 - have "{x\<in>space M. a < f x} = (\<Union>i. {x\<in>space M. a + inverse (of_nat (Suc i)) \<le> f x})"
3.307 - proof safe
3.308 - fix x assume "a < f x" and [simp]: "x \<in> space M"
3.309 - with ex_pextreal_inverse_of_nat_Suc_less[of "f x - a"]
3.310 - obtain n where "a + inverse (of_nat (Suc n)) < f x"
3.311 - by (cases "f x", auto simp: pextreal_minus_order)
3.312 - then have "a + inverse (of_nat (Suc n)) \<le> f x" by simp
3.313 - then show "x \<in> (\<Union>i. {x \<in> space M. a + inverse (of_nat (Suc i)) \<le> f x})"
3.314 - by auto
3.315 - next
3.316 - fix i x assume [simp]: "x \<in> space M"
3.317 - have "a < a + inverse (of_nat (Suc i))" using preal by auto
3.318 - also assume "a + inverse (of_nat (Suc i)) \<le> f x"
3.319 - finally show "a < f x" .
3.320 - qed
3.321 - with a show ?thesis by auto
3.322 - qed simp
3.323 + assume ?l from borel_measurable_uminus_extreal[OF this] show ?r by simp
3.324 +qed auto
3.325 +
3.326 +lemma (in sigma_algebra) borel_measurable_eq_atMost_extreal:
3.327 + "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
3.328 +proof (intro iffI allI)
3.329 + assume pos[rule_format]: "\<forall>a. f -` {..a} \<inter> space M \<in> sets M"
3.330 + show "f \<in> borel_measurable M"
3.331 + unfolding borel_measurable_extreal_iff_real borel_measurable_iff_le
3.332 + proof (intro conjI allI)
3.333 + fix a :: real
3.334 + { fix x :: extreal assume *: "\<forall>i::nat. real i < x"
3.335 + have "x = \<infinity>"
3.336 + proof (rule extreal_top)
3.337 + fix B from real_arch_lt[of B] guess n ..
3.338 + then have "extreal B < real n" by auto
3.339 + with * show "B \<le> x" by (metis less_trans less_imp_le)
3.340 + qed }
3.341 + then have "f -` {\<infinity>} \<inter> space M = space M - (\<Union>i::nat. f -` {.. real i} \<inter> space M)"
3.342 + by (auto simp: not_le)
3.343 + then show "f -` {\<infinity>} \<inter> space M \<in> sets M" using pos by (auto simp del: UN_simps intro!: Diff)
3.344 + moreover
3.345 + have "{-\<infinity>} = {..-\<infinity>}" by auto
3.346 + then show "f -` {-\<infinity>} \<inter> space M \<in> sets M" using pos by auto
3.347 + moreover have "{x\<in>space M. f x \<le> extreal a} \<in> sets M"
3.348 + using pos[of "extreal a"] by (simp add: vimage_def Int_def conj_commute)
3.349 + moreover have "{w \<in> space M. real (f w) \<le> a} =
3.350 + (if a < 0 then {w \<in> space M. f w \<le> extreal a} - f -` {-\<infinity>} \<inter> space M
3.351 + else {w \<in> space M. f w \<le> extreal a} \<union> (f -` {\<infinity>} \<inter> space M) \<union> (f -` {-\<infinity>} \<inter> space M))" (is "?l = ?r")
3.352 + proof (intro set_eqI) fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases "f x") auto qed
3.353 + ultimately show "{w \<in> space M. real (f w) \<le> a} \<in> sets M" by auto
3.354 qed
3.355 -next
3.356 - assume a': "\<forall>a. {x \<in> space M. a < f x} \<in> sets M"
3.357 - then have a: "\<forall>a. {x \<in> space M. f x \<le> a} \<in> sets M" unfolding less_eq_ge_measurable .
3.358 - show "\<forall>a. {x \<in> space M. a \<le> f x} \<in> sets M" unfolding greater_eq_le_measurable[symmetric]
3.359 - proof
3.360 - fix a show "{x \<in> space M. f x < a} \<in> sets M"
3.361 - proof (cases a)
3.362 - case (preal r)
3.363 - show ?thesis
3.364 - proof cases
3.365 - assume "a = 0" then show ?thesis by simp
3.366 - next
3.367 - assume "a \<noteq> 0"
3.368 - have "{x\<in>space M. f x < a} = (\<Union>i. {x\<in>space M. f x \<le> a - inverse (of_nat (Suc i))})"
3.369 - proof safe
3.370 - fix x assume "f x < a" and [simp]: "x \<in> space M"
3.371 - with ex_pextreal_inverse_of_nat_Suc_less[of "a - f x"]
3.372 - obtain n where "inverse (of_nat (Suc n)) < a - f x"
3.373 - using preal by (cases "f x") auto
3.374 - then have "f x \<le> a - inverse (of_nat (Suc n)) "
3.375 - using preal by (cases "f x") (auto split: split_if_asm)
3.376 - then show "x \<in> (\<Union>i. {x \<in> space M. f x \<le> a - inverse (of_nat (Suc i))})"
3.377 - by auto
3.378 - next
3.379 - fix i x assume [simp]: "x \<in> space M"
3.380 - assume "f x \<le> a - inverse (of_nat (Suc i))"
3.381 - also have "\<dots> < a" using `a \<noteq> 0` preal by auto
3.382 - finally show "f x < a" .
3.383 - qed
3.384 - with a show ?thesis by auto
3.385 - qed
3.386 - next
3.387 - case infinite
3.388 - have "f -` {\<omega>} \<inter> space M = (\<Inter>n. {x\<in>space M. of_nat n < f x})"
3.389 - proof (safe, simp_all, safe)
3.390 - fix x assume *: "\<forall>n::nat. Real (real n) < f x"
3.391 - show "f x = \<omega>" proof (rule ccontr)
3.392 - assume "f x \<noteq> \<omega>"
3.393 - with real_arch_lt[of "real (f x)"] obtain n where "f x < of_nat n"
3.394 - by (auto simp: pextreal_noteq_omega_Ex)
3.395 - with *[THEN spec, of n] show False by auto
3.396 - qed
3.397 - qed
3.398 - with a' have \<omega>: "f -` {\<omega>} \<inter> space M \<in> sets M" by auto
3.399 - moreover have "{x \<in> space M. f x < a} = space M - f -` {\<omega>} \<inter> space M"
3.400 - using infinite by auto
3.401 - ultimately show ?thesis by auto
3.402 - qed
3.403 - qed
3.404 -qed
3.405 +qed (simp add: measurable_sets)
3.406
3.407 -lemma (in sigma_algebra) borel_measurable_pextreal_iff_greater:
3.408 - "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. a < f x} \<in> sets M)"
3.409 -proof safe
3.410 - fix a assume f: "f \<in> borel_measurable M"
3.411 - have "{x\<in>space M. a < f x} = f -` {a <..} \<inter> space M" by auto
3.412 - with f show "{x\<in>space M. a < f x} \<in> sets M"
3.413 - by (auto intro!: measurable_sets)
3.414 -next
3.415 - assume *: "\<forall>a. {x\<in>space M. a < f x} \<in> sets M"
3.416 - hence **: "\<forall>a. {x\<in>space M. f x < a} \<in> sets M"
3.417 - unfolding less_eq_le_pextreal_measurable
3.418 - unfolding greater_eq_le_measurable .
3.419 - show "f \<in> borel_measurable M" unfolding borel_measurable_pextreal_eq_real borel_measurable_iff_greater
3.420 - proof safe
3.421 - have "f -` {\<omega>} \<inter> space M = space M - {x\<in>space M. f x < \<omega>}" by auto
3.422 - then show \<omega>: "f -` {\<omega>} \<inter> space M \<in> sets M" using ** by auto
3.423 - fix a
3.424 - have "{w \<in> space M. a < real (f w)} =
3.425 - (if 0 \<le> a then {w\<in>space M. Real a < f w} - (f -` {\<omega>} \<inter> space M) else space M)"
3.426 - proof (split split_if, safe del: notI)
3.427 - fix x assume "0 \<le> a"
3.428 - { assume "a < real (f x)" then show "Real a < f x" "x \<notin> f -` {\<omega>} \<inter> space M"
3.429 - using `0 \<le> a` by (cases "f x", auto) }
3.430 - { assume "Real a < f x" "x \<notin> f -` {\<omega>}" then show "a < real (f x)"
3.431 - using `0 \<le> a` by (cases "f x", auto) }
3.432 - next
3.433 - fix x assume "\<not> 0 \<le> a" then show "a < real (f x)" by (cases "f x") auto
3.434 - qed
3.435 - then show "{w \<in> space M. a < real (f w)} \<in> sets M"
3.436 - using \<omega> * by (auto intro!: Diff)
3.437 - qed
3.438 -qed
3.439 +lemma (in sigma_algebra) borel_measurable_eq_atLeast_extreal:
3.440 + "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
3.441 +proof
3.442 + assume pos: "\<forall>a. f -` {a..} \<inter> space M \<in> sets M"
3.443 + moreover have "\<And>a. (\<lambda>x. - f x) -` {..a} = f -` {-a ..}"
3.444 + by (auto simp: extreal_uminus_le_reorder)
3.445 + ultimately have "(\<lambda>x. - f x) \<in> borel_measurable M"
3.446 + unfolding borel_measurable_eq_atMost_extreal by auto
3.447 + then show "f \<in> borel_measurable M" by simp
3.448 +qed (simp add: measurable_sets)
3.449
3.450 -lemma (in sigma_algebra) borel_measurable_pextreal_iff_less:
3.451 - "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. f x < a} \<in> sets M)"
3.452 - using borel_measurable_pextreal_iff_greater unfolding less_eq_le_pextreal_measurable greater_eq_le_measurable .
3.453 +lemma (in sigma_algebra) borel_measurable_extreal_iff_less:
3.454 + "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
3.455 + unfolding borel_measurable_eq_atLeast_extreal greater_eq_le_measurable ..
3.456
3.457 -lemma (in sigma_algebra) borel_measurable_pextreal_iff_le:
3.458 - "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. f x \<le> a} \<in> sets M)"
3.459 - using borel_measurable_pextreal_iff_greater unfolding less_eq_ge_measurable .
3.460 +lemma (in sigma_algebra) borel_measurable_extreal_iff_ge:
3.461 + "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
3.462 + unfolding borel_measurable_eq_atMost_extreal less_eq_ge_measurable ..
3.463
3.464 -lemma (in sigma_algebra) borel_measurable_pextreal_iff_ge:
3.465 - "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M)"
3.466 - using borel_measurable_pextreal_iff_greater unfolding less_eq_le_pextreal_measurable .
3.467 -
3.468 -lemma (in sigma_algebra) borel_measurable_pextreal_eq_const:
3.469 - fixes f :: "'a \<Rightarrow> pextreal" assumes "f \<in> borel_measurable M"
3.470 +lemma (in sigma_algebra) borel_measurable_extreal_eq_const:
3.471 + fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M"
3.472 shows "{x\<in>space M. f x = c} \<in> sets M"
3.473 proof -
3.474 have "{x\<in>space M. f x = c} = (f -` {c} \<inter> space M)" by auto
3.475 then show ?thesis using assms by (auto intro!: measurable_sets)
3.476 qed
3.477
3.478 -lemma (in sigma_algebra) borel_measurable_pextreal_neq_const:
3.479 - fixes f :: "'a \<Rightarrow> pextreal"
3.480 - assumes "f \<in> borel_measurable M"
3.481 +lemma (in sigma_algebra) borel_measurable_extreal_neq_const:
3.482 + fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M"
3.483 shows "{x\<in>space M. f x \<noteq> c} \<in> sets M"
3.484 proof -
3.485 have "{x\<in>space M. f x \<noteq> c} = space M - (f -` {c} \<inter> space M)" by auto
3.486 then show ?thesis using assms by (auto intro!: measurable_sets)
3.487 qed
3.488
3.489 -lemma (in sigma_algebra) borel_measurable_pextreal_less[intro,simp]:
3.490 - fixes f g :: "'a \<Rightarrow> pextreal"
3.491 +lemma (in sigma_algebra) borel_measurable_extreal_le[intro,simp]:
3.492 + fixes f g :: "'a \<Rightarrow> extreal"
3.493 + assumes f: "f \<in> borel_measurable M"
3.494 + assumes g: "g \<in> borel_measurable M"
3.495 + shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
3.496 +proof -
3.497 + have "{x \<in> space M. f x \<le> g x} =
3.498 + {x \<in> space M. real (f x) \<le> real (g x)} - (f -` {\<infinity>, -\<infinity>} \<inter> space M \<union> g -` {\<infinity>, -\<infinity>} \<inter> space M) \<union>
3.499 + f -` {-\<infinity>} \<inter> space M \<union> g -` {\<infinity>} \<inter> space M" (is "?l = ?r")
3.500 + proof (intro set_eqI)
3.501 + fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases rule: extreal2_cases[of "f x" "g x"]) auto
3.502 + qed
3.503 + with f g show ?thesis by (auto intro!: Un simp: measurable_sets)
3.504 +qed
3.505 +
3.506 +lemma (in sigma_algebra) borel_measurable_extreal_less[intro,simp]:
3.507 + fixes f :: "'a \<Rightarrow> extreal"
3.508 assumes f: "f \<in> borel_measurable M"
3.509 assumes g: "g \<in> borel_measurable M"
3.510 shows "{x \<in> space M. f x < g x} \<in> sets M"
3.511 proof -
3.512 - have "(\<lambda>x. real (f x)) \<in> borel_measurable M"
3.513 - "(\<lambda>x. real (g x)) \<in> borel_measurable M"
3.514 - using assms by (auto intro!: borel_measurable_real)
3.515 - from borel_measurable_less[OF this]
3.516 - have "{x \<in> space M. real (f x) < real (g x)} \<in> sets M" .
3.517 - moreover have "{x \<in> space M. f x \<noteq> \<omega>} \<in> sets M" using f by (rule borel_measurable_pextreal_neq_const)
3.518 - moreover have "{x \<in> space M. g x = \<omega>} \<in> sets M" using g by (rule borel_measurable_pextreal_eq_const)
3.519 - moreover have "{x \<in> space M. g x \<noteq> \<omega>} \<in> sets M" using g by (rule borel_measurable_pextreal_neq_const)
3.520 - moreover have "{x \<in> space M. f x < g x} = ({x \<in> space M. g x = \<omega>} \<inter> {x \<in> space M. f x \<noteq> \<omega>}) \<union>
3.521 - ({x \<in> space M. g x \<noteq> \<omega>} \<inter> {x \<in> space M. f x \<noteq> \<omega>} \<inter> {x \<in> space M. real (f x) < real (g x)})"
3.522 - by (auto simp: real_of_pextreal_strict_mono_iff)
3.523 - ultimately show ?thesis by auto
3.524 -qed
3.525 -
3.526 -lemma (in sigma_algebra) borel_measurable_pextreal_le[intro,simp]:
3.527 - fixes f :: "'a \<Rightarrow> pextreal"
3.528 - assumes f: "f \<in> borel_measurable M"
3.529 - assumes g: "g \<in> borel_measurable M"
3.530 - shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
3.531 -proof -
3.532 - have "{x \<in> space M. f x \<le> g x} = space M - {x \<in> space M. g x < f x}" by auto
3.533 + have "{x \<in> space M. f x < g x} = space M - {x \<in> space M. g x \<le> f x}" by auto
3.534 then show ?thesis using g f by auto
3.535 qed
3.536
3.537 -lemma (in sigma_algebra) borel_measurable_pextreal_eq[intro,simp]:
3.538 - fixes f :: "'a \<Rightarrow> pextreal"
3.539 +lemma (in sigma_algebra) borel_measurable_extreal_eq[intro,simp]:
3.540 + fixes f :: "'a \<Rightarrow> extreal"
3.541 assumes f: "f \<in> borel_measurable M"
3.542 assumes g: "g \<in> borel_measurable M"
3.543 shows "{w \<in> space M. f w = g w} \<in> sets M"
3.544 @@ -1395,8 +1283,8 @@
3.545 then show ?thesis using g f by auto
3.546 qed
3.547
3.548 -lemma (in sigma_algebra) borel_measurable_pextreal_neq[intro,simp]:
3.549 - fixes f :: "'a \<Rightarrow> pextreal"
3.550 +lemma (in sigma_algebra) borel_measurable_extreal_neq[intro,simp]:
3.551 + fixes f :: "'a \<Rightarrow> extreal"
3.552 assumes f: "f \<in> borel_measurable M"
3.553 assumes g: "g \<in> borel_measurable M"
3.554 shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
3.555 @@ -1405,20 +1293,28 @@
3.556 thus ?thesis using f g by auto
3.557 qed
3.558
3.559 -lemma (in sigma_algebra) borel_measurable_pextreal_add[intro, simp]:
3.560 - fixes f :: "'a \<Rightarrow> pextreal"
3.561 +lemma (in sigma_algebra) split_sets:
3.562 + "{x\<in>space M. P x \<or> Q x} = {x\<in>space M. P x} \<union> {x\<in>space M. Q x}"
3.563 + "{x\<in>space M. P x \<and> Q x} = {x\<in>space M. P x} \<inter> {x\<in>space M. Q x}"
3.564 + by auto
3.565 +
3.566 +lemma (in sigma_algebra) borel_measurable_extreal_add[intro, simp]:
3.567 + fixes f :: "'a \<Rightarrow> extreal"
3.568 assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
3.569 shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
3.570 proof -
3.571 - have *: "(\<lambda>x. f x + g x) =
3.572 - (\<lambda>x. if f x = \<omega> then \<omega> else if g x = \<omega> then \<omega> else Real (real (f x) + real (g x)))"
3.573 - by (auto simp: fun_eq_iff pextreal_noteq_omega_Ex)
3.574 - show ?thesis using assms unfolding *
3.575 - by (auto intro!: measurable_If)
3.576 + { fix x assume "x \<in> space M" then have "f x + g x =
3.577 + (if f x = \<infinity> \<or> g x = \<infinity> then \<infinity>
3.578 + else if f x = -\<infinity> \<or> g x = -\<infinity> then -\<infinity>
3.579 + else extreal (real (f x) + real (g x)))"
3.580 + by (cases rule: extreal2_cases[of "f x" "g x"]) auto }
3.581 + with assms show ?thesis
3.582 + by (auto cong: measurable_cong simp: split_sets
3.583 + intro!: Un measurable_If measurable_sets)
3.584 qed
3.585
3.586 -lemma (in sigma_algebra) borel_measurable_pextreal_setsum[simp, intro]:
3.587 - fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> pextreal"
3.588 +lemma (in sigma_algebra) borel_measurable_extreal_setsum[simp, intro]:
3.589 + fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> extreal"
3.590 assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
3.591 shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
3.592 proof cases
3.593 @@ -1427,20 +1323,49 @@
3.594 by induct auto
3.595 qed (simp add: borel_measurable_const)
3.596
3.597 -lemma (in sigma_algebra) borel_measurable_pextreal_times[intro, simp]:
3.598 - fixes f :: "'a \<Rightarrow> pextreal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
3.599 +lemma abs_extreal_ge0[simp]: "0 \<le> x \<Longrightarrow> \<bar>x :: extreal\<bar> = x"
3.600 + by (cases x) auto
3.601 +
3.602 +lemma abs_extreal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: extreal\<bar> = -x"
3.603 + by (cases x) auto
3.604 +
3.605 +lemma abs_extreal_pos[simp]: "0 \<le> \<bar>x :: extreal\<bar>"
3.606 + by (cases x) auto
3.607 +
3.608 +lemma (in sigma_algebra) borel_measurable_extreal_abs[intro, simp]:
3.609 + fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M"
3.610 + shows "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
3.611 +proof -
3.612 + { fix x have "\<bar>f x\<bar> = (if 0 \<le> f x then f x else - f x)" by auto }
3.613 + then show ?thesis using assms by (auto intro!: measurable_If)
3.614 +qed
3.615 +
3.616 +lemma (in sigma_algebra) borel_measurable_extreal_times[intro, simp]:
3.617 + fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
3.618 shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
3.619 proof -
3.620 + { fix f g :: "'a \<Rightarrow> extreal"
3.621 + assume b: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
3.622 + and pos: "\<And>x. 0 \<le> f x" "\<And>x. 0 \<le> g x"
3.623 + { fix x have *: "f x * g x = (if f x = 0 \<or> g x = 0 then 0
3.624 + else if f x = \<infinity> \<or> g x = \<infinity> then \<infinity>
3.625 + else extreal (real (f x) * real (g x)))"
3.626 + apply (cases rule: extreal2_cases[of "f x" "g x"])
3.627 + using pos[of x] by auto }
3.628 + with b have "(\<lambda>x. f x * g x) \<in> borel_measurable M"
3.629 + by (auto cong: measurable_cong simp: split_sets
3.630 + intro!: Un measurable_If measurable_sets) }
3.631 + note pos_times = this
3.632 have *: "(\<lambda>x. f x * g x) =
3.633 - (\<lambda>x. if f x = 0 then 0 else if g x = 0 then 0 else if f x = \<omega> then \<omega> else if g x = \<omega> then \<omega> else
3.634 - Real (real (f x) * real (g x)))"
3.635 - by (auto simp: fun_eq_iff pextreal_noteq_omega_Ex)
3.636 + (\<lambda>x. if 0 \<le> f x \<and> 0 \<le> g x \<or> f x < 0 \<and> g x < 0 then \<bar>f x\<bar> * \<bar>g x\<bar> else - (\<bar>f x\<bar> * \<bar>g x\<bar>))"
3.637 + by (auto simp: fun_eq_iff)
3.638 show ?thesis using assms unfolding *
3.639 - by (auto intro!: measurable_If)
3.640 + by (intro measurable_If pos_times borel_measurable_uminus_extreal)
3.641 + (auto simp: split_sets intro!: Int)
3.642 qed
3.643
3.644 -lemma (in sigma_algebra) borel_measurable_pextreal_setprod[simp, intro]:
3.645 - fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> pextreal"
3.646 +lemma (in sigma_algebra) borel_measurable_extreal_setprod[simp, intro]:
3.647 + fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> extreal"
3.648 assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
3.649 shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
3.650 proof cases
3.651 @@ -1448,64 +1373,73 @@
3.652 thus ?thesis using assms by induct auto
3.653 qed simp
3.654
3.655 -lemma (in sigma_algebra) borel_measurable_pextreal_min[simp, intro]:
3.656 - fixes f g :: "'a \<Rightarrow> pextreal"
3.657 +lemma (in sigma_algebra) borel_measurable_extreal_min[simp, intro]:
3.658 + fixes f g :: "'a \<Rightarrow> extreal"
3.659 assumes "f \<in> borel_measurable M"
3.660 assumes "g \<in> borel_measurable M"
3.661 shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
3.662 using assms unfolding min_def by (auto intro!: measurable_If)
3.663
3.664 -lemma (in sigma_algebra) borel_measurable_pextreal_max[simp, intro]:
3.665 - fixes f g :: "'a \<Rightarrow> pextreal"
3.666 +lemma (in sigma_algebra) borel_measurable_extreal_max[simp, intro]:
3.667 + fixes f g :: "'a \<Rightarrow> extreal"
3.668 assumes "f \<in> borel_measurable M"
3.669 and "g \<in> borel_measurable M"
3.670 shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
3.671 using assms unfolding max_def by (auto intro!: measurable_If)
3.672
3.673 lemma (in sigma_algebra) borel_measurable_SUP[simp, intro]:
3.674 - fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> pextreal"
3.675 + fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> extreal"
3.676 assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
3.677 shows "(\<lambda>x. SUP i : A. f i x) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M")
3.678 - unfolding borel_measurable_pextreal_iff_greater
3.679 -proof safe
3.680 + unfolding borel_measurable_extreal_iff_ge
3.681 +proof
3.682 fix a
3.683 - have "{x\<in>space M. a < ?sup x} = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"
3.684 + have "?sup -` {a<..} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"
3.685 by (auto simp: less_SUP_iff SUPR_apply)
3.686 - then show "{x\<in>space M. a < ?sup x} \<in> sets M"
3.687 + then show "?sup -` {a<..} \<inter> space M \<in> sets M"
3.688 using assms by auto
3.689 qed
3.690
3.691 lemma (in sigma_algebra) borel_measurable_INF[simp, intro]:
3.692 - fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> pextreal"
3.693 + fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> extreal"
3.694 assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
3.695 shows "(\<lambda>x. INF i : A. f i x) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M")
3.696 - unfolding borel_measurable_pextreal_iff_less
3.697 -proof safe
3.698 + unfolding borel_measurable_extreal_iff_less
3.699 +proof
3.700 fix a
3.701 - have "{x\<in>space M. ?inf x < a} = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"
3.702 + have "?inf -` {..<a} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"
3.703 by (auto simp: INF_less_iff INFI_apply)
3.704 - then show "{x\<in>space M. ?inf x < a} \<in> sets M"
3.705 + then show "?inf -` {..<a} \<inter> space M \<in> sets M"
3.706 using assms by auto
3.707 qed
3.708
3.709 -lemma (in sigma_algebra) borel_measurable_pextreal_diff[simp, intro]:
3.710 - fixes f g :: "'a \<Rightarrow> pextreal"
3.711 +lemma (in sigma_algebra) borel_measurable_liminf[simp, intro]:
3.712 + fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
3.713 + assumes "\<And>i. f i \<in> borel_measurable M"
3.714 + shows "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
3.715 + unfolding liminf_SUPR_INFI using assms by auto
3.716 +
3.717 +lemma (in sigma_algebra) borel_measurable_limsup[simp, intro]:
3.718 + fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
3.719 + assumes "\<And>i. f i \<in> borel_measurable M"
3.720 + shows "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
3.721 + unfolding limsup_INFI_SUPR using assms by auto
3.722 +
3.723 +lemma (in sigma_algebra) borel_measurable_extreal_diff[simp, intro]:
3.724 + fixes f g :: "'a \<Rightarrow> extreal"
3.725 assumes "f \<in> borel_measurable M"
3.726 assumes "g \<in> borel_measurable M"
3.727 shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
3.728 - unfolding borel_measurable_pextreal_iff_greater
3.729 -proof safe
3.730 - fix a
3.731 - have "{x \<in> space M. a < f x - g x} = {x \<in> space M. g x + a < f x}"
3.732 - by (simp add: pextreal_less_minus_iff)
3.733 - then show "{x \<in> space M. a < f x - g x} \<in> sets M"
3.734 - using assms by auto
3.735 -qed
3.736 + unfolding minus_extreal_def using assms by auto
3.737
3.738 lemma (in sigma_algebra) borel_measurable_psuminf[simp, intro]:
3.739 - assumes "\<And>i. f i \<in> borel_measurable M"
3.740 - shows "(\<lambda>x. (\<Sum>\<^isub>\<infinity> i. f i x)) \<in> borel_measurable M"
3.741 - using assms unfolding psuminf_def by auto
3.742 + fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
3.743 + assumes "\<And>i. f i \<in> borel_measurable M" and pos: "\<And>i x. x \<in> space M \<Longrightarrow> 0 \<le> f i x"
3.744 + shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
3.745 + apply (subst measurable_cong)
3.746 + apply (subst suminf_extreal_eq_SUPR)
3.747 + apply (rule pos)
3.748 + using assms by auto
3.749
3.750 section "LIMSEQ is borel measurable"
3.751
3.752 @@ -1515,28 +1449,11 @@
3.753 and u: "\<And>i. u i \<in> borel_measurable M"
3.754 shows "u' \<in> borel_measurable M"
3.755 proof -
3.756 - let "?pu x i" = "max (u i x) 0"
3.757 - let "?nu x i" = "max (- u i x) 0"
3.758 - { fix x assume x: "x \<in> space M"
3.759 - have "(?pu x) ----> max (u' x) 0"
3.760 - "(?nu x) ----> max (- u' x) 0"
3.761 - using u'[OF x] by (auto intro!: LIMSEQ_max LIMSEQ_minus)
3.762 - from LIMSEQ_imp_lim_INF[OF _ this(1)] LIMSEQ_imp_lim_INF[OF _ this(2)]
3.763 - have "(SUP n. INF m. Real (u (n + m) x)) = Real (u' x)"
3.764 - "(SUP n. INF m. Real (- u (n + m) x)) = Real (- u' x)"
3.765 - by (simp_all add: Real_max'[symmetric]) }
3.766 - note eq = this
3.767 - have *: "\<And>x. real (Real (u' x)) - real (Real (- u' x)) = u' x"
3.768 + have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. extreal (u n x)) = extreal (u' x)"
3.769 + using u' by (simp add: lim_imp_Liminf trivial_limit_sequentially lim_extreal)
3.770 + moreover from u have "(\<lambda>x. liminf (\<lambda>n. extreal (u n x))) \<in> borel_measurable M"
3.771 by auto
3.772 - have "(\<lambda>x. SUP n. INF m. Real (u (n + m) x)) \<in> borel_measurable M"
3.773 - "(\<lambda>x. SUP n. INF m. Real (- u (n + m) x)) \<in> borel_measurable M"
3.774 - using u by auto
3.775 - with eq[THEN measurable_cong, of M "\<lambda>x. x" borel]
3.776 - have "(\<lambda>x. Real (u' x)) \<in> borel_measurable M"
3.777 - "(\<lambda>x. Real (- u' x)) \<in> borel_measurable M" by auto
3.778 - note this[THEN borel_measurable_real]
3.779 - from borel_measurable_diff[OF this]
3.780 - show ?thesis unfolding * .
3.781 + ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_extreal_iff)
3.782 qed
3.783
3.784 end
4.1 --- a/src/HOL/Probability/Caratheodory.thy Mon Mar 14 14:37:47 2011 +0100
4.2 +++ b/src/HOL/Probability/Caratheodory.thy Mon Mar 14 14:37:49 2011 +0100
4.3 @@ -1,36 +1,66 @@
4.4 header {*Caratheodory Extension Theorem*}
4.5
4.6 theory Caratheodory
4.7 - imports Sigma_Algebra Positive_Extended_Real
4.8 + imports Sigma_Algebra Extended_Real_Limits
4.9 begin
4.10
4.11 +lemma suminf_extreal_2dimen:
4.12 + fixes f:: "nat \<times> nat \<Rightarrow> extreal"
4.13 + assumes pos: "\<And>p. 0 \<le> f p"
4.14 + assumes "\<And>m. g m = (\<Sum>n. f (m,n))"
4.15 + shows "(\<Sum>i. f (prod_decode i)) = suminf g"
4.16 +proof -
4.17 + have g_def: "g = (\<lambda>m. (\<Sum>n. f (m,n)))"
4.18 + using assms by (simp add: fun_eq_iff)
4.19 + have reindex: "\<And>B. (\<Sum>x\<in>B. f (prod_decode x)) = setsum f (prod_decode ` B)"
4.20 + by (simp add: setsum_reindex[OF inj_prod_decode] comp_def)
4.21 + { fix n
4.22 + let ?M = "\<lambda>f. Suc (Max (f ` prod_decode ` {..<n}))"
4.23 + { fix a b x assume "x < n" and [symmetric]: "(a, b) = prod_decode x"
4.24 + then have "a < ?M fst" "b < ?M snd"
4.25 + by (auto intro!: Max_ge le_imp_less_Suc image_eqI) }
4.26 + then have "setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<?M fst} \<times> {..<?M snd})"
4.27 + by (auto intro!: setsum_mono3 simp: pos)
4.28 + then have "\<exists>a b. setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<a} \<times> {..<b})" by auto }
4.29 + moreover
4.30 + { fix a b
4.31 + let ?M = "prod_decode ` {..<Suc (Max (prod_encode ` ({..<a} \<times> {..<b})))}"
4.32 + { fix a' b' assume "a' < a" "b' < b" then have "(a', b') \<in> ?M"
4.33 + by (auto intro!: Max_ge le_imp_less_Suc image_eqI[where x="prod_encode (a', b')"]) }
4.34 + then have "setsum f ({..<a} \<times> {..<b}) \<le> setsum f ?M"
4.35 + by (auto intro!: setsum_mono3 simp: pos) }
4.36 + ultimately
4.37 + show ?thesis unfolding g_def using pos
4.38 + by (auto intro!: SUPR_eq simp: setsum_cartesian_product reindex le_SUPI2
4.39 + setsum_nonneg suminf_extreal_eq_SUPR SUPR_pair
4.40 + SUPR_extreal_setsum[symmetric] incseq_setsumI setsum_nonneg)
4.41 +qed
4.42 +
4.43 text{*From the Hurd/Coble measure theory development, translated by Lawrence Paulson.*}
4.44
4.45 subsection {* Measure Spaces *}
4.46
4.47 record 'a measure_space = "'a algebra" +
4.48 - measure :: "'a set \<Rightarrow> pextreal"
4.49 + measure :: "'a set \<Rightarrow> extreal"
4.50
4.51 -definition positive where "positive M f \<longleftrightarrow> f {} = (0::pextreal)"
4.52 - -- "Positive is enforced by the type"
4.53 +definition positive where "positive M f \<longleftrightarrow> f {} = (0::extreal) \<and> (\<forall>A\<in>sets M. 0 \<le> f A)"
4.54
4.55 definition additive where "additive M f \<longleftrightarrow>
4.56 (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) = f x + f y)"
4.57
4.58 -definition countably_additive where "countably_additive M f \<longleftrightarrow>
4.59 - (\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow>
4.60 - (\<Sum>\<^isub>\<infinity> n. f (A n)) = f (\<Union>i. A i))"
4.61 +definition countably_additive :: "('a, 'b) algebra_scheme \<Rightarrow> ('a set \<Rightarrow> extreal) \<Rightarrow> bool" where
4.62 + "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow>
4.63 + (\<Sum>i. f (A i)) = f (\<Union>i. A i))"
4.64
4.65 definition increasing where "increasing M f \<longleftrightarrow>
4.66 (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<subseteq> y \<longrightarrow> f x \<le> f y)"
4.67
4.68 definition subadditive where "subadditive M f \<longleftrightarrow>
4.69 - (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} \<longrightarrow>
4.70 - f (x \<union> y) \<le> f x + f y)"
4.71 + (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
4.72
4.73 definition countably_subadditive where "countably_subadditive M f \<longleftrightarrow>
4.74 (\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow>
4.75 - f (\<Union>i. A i) \<le> (\<Sum>\<^isub>\<infinity> n. f (A n)))"
4.76 + (f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))))"
4.77
4.78 definition lambda_system where "lambda_system M f = {l \<in> sets M.
4.79 \<forall>x \<in> sets M. f (l \<inter> x) + f ((space M - l) \<inter> x) = f x}"
4.80 @@ -39,14 +69,19 @@
4.81 positive M f \<and> increasing M f \<and> countably_subadditive M f"
4.82
4.83 definition measure_set where "measure_set M f X = {r.
4.84 - \<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>\<^isub>\<infinity> i. f (A i)) = r}"
4.85 + \<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}"
4.86
4.87 locale measure_space = sigma_algebra M for M :: "('a, 'b) measure_space_scheme" +
4.88 - assumes empty_measure [simp]: "measure M {} = 0"
4.89 + assumes measure_positive: "positive M (measure M)"
4.90 and ca: "countably_additive M (measure M)"
4.91
4.92 abbreviation (in measure_space) "\<mu> \<equiv> measure M"
4.93
4.94 +lemma (in measure_space)
4.95 + shows empty_measure[simp, intro]: "\<mu> {} = 0"
4.96 + and positive_measure[simp, intro!]: "\<And>A. A \<in> sets M \<Longrightarrow> 0 \<le> \<mu> A"
4.97 + using measure_positive unfolding positive_def by auto
4.98 +
4.99 lemma increasingD:
4.100 "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M \<Longrightarrow> f x \<le> f y"
4.101 by (auto simp add: increasing_def)
4.102 @@ -61,39 +96,30 @@
4.103 \<Longrightarrow> f (x \<union> y) = f x + f y"
4.104 by (auto simp add: additive_def)
4.105
4.106 -lemma countably_additiveD:
4.107 - "countably_additive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A
4.108 - \<Longrightarrow> (\<Union>i. A i) \<in> sets M \<Longrightarrow> (\<Sum>\<^isub>\<infinity> n. f (A n)) = f (\<Union>i. A i)"
4.109 - by (simp add: countably_additive_def)
4.110 -
4.111 -lemma countably_subadditiveD:
4.112 - "countably_subadditive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow>
4.113 - (\<Union>i. A i) \<in> sets M \<Longrightarrow> f (\<Union>i. A i) \<le> psuminf (f o A)"
4.114 - by (auto simp add: countably_subadditive_def o_def)
4.115 -
4.116 lemma countably_additiveI:
4.117 - "(\<And>A. range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> sets M
4.118 - \<Longrightarrow> (\<Sum>\<^isub>\<infinity> n. f (A n)) = f (\<Union>i. A i)) \<Longrightarrow> countably_additive M f"
4.119 - by (simp add: countably_additive_def)
4.120 + assumes "\<And>A x. range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> sets M
4.121 + \<Longrightarrow> (\<Sum>i. f (A i)) = f (\<Union>i. A i)"
4.122 + shows "countably_additive M f"
4.123 + using assms by (simp add: countably_additive_def)
4.124
4.125 section "Extend binary sets"
4.126
4.127 lemma LIMSEQ_binaryset:
4.128 assumes f: "f {} = 0"
4.129 - shows "(\<lambda>n. \<Sum>i = 0..<n. f (binaryset A B i)) ----> f A + f B"
4.130 + shows "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) ----> f A + f B"
4.131 proof -
4.132 - have "(\<lambda>n. \<Sum>i = 0..< Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
4.133 + have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
4.134 proof
4.135 fix n
4.136 - show "(\<Sum>i\<Colon>nat = 0\<Colon>nat..<Suc (Suc n). f (binaryset A B i)) = f A + f B"
4.137 + show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B"
4.138 by (induct n) (auto simp add: binaryset_def f)
4.139 qed
4.140 moreover
4.141 have "... ----> f A + f B" by (rule LIMSEQ_const)
4.142 ultimately
4.143 - have "(\<lambda>n. \<Sum>i = 0..< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
4.144 + have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
4.145 by metis
4.146 - hence "(\<lambda>n. \<Sum>i = 0..< n+2. f (binaryset A B i)) ----> f A + f B"
4.147 + hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) ----> f A + f B"
4.148 by simp
4.149 thus ?thesis by (rule LIMSEQ_offset [where k=2])
4.150 qed
4.151 @@ -101,28 +127,13 @@
4.152 lemma binaryset_sums:
4.153 assumes f: "f {} = 0"
4.154 shows "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"
4.155 - by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f])
4.156 + by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan)
4.157
4.158 lemma suminf_binaryset_eq:
4.159 - fixes f :: "'a set \<Rightarrow> real"
4.160 + fixes f :: "'a set \<Rightarrow> 'b::{comm_monoid_add, t2_space}"
4.161 shows "f {} = 0 \<Longrightarrow> (\<Sum>n. f (binaryset A B n)) = f A + f B"
4.162 by (metis binaryset_sums sums_unique)
4.163
4.164 -lemma binaryset_psuminf:
4.165 - assumes "f {} = 0"
4.166 - shows "(\<Sum>\<^isub>\<infinity> n. f (binaryset A B n)) = f A + f B" (is "?suminf = ?sum")
4.167 -proof -
4.168 - have *: "{..<2} = {0, 1::nat}" by auto
4.169 - have "\<forall>n\<ge>2. f (binaryset A B n) = 0"
4.170 - unfolding binaryset_def
4.171 - using assms by auto
4.172 - hence "?suminf = (\<Sum>N<2. f (binaryset A B N))"
4.173 - by (rule psuminf_finite)
4.174 - also have "... = ?sum" unfolding * binaryset_def
4.175 - by simp
4.176 - finally show ?thesis .
4.177 -qed
4.178 -
4.179 subsection {* Lambda Systems *}
4.180
4.181 lemma (in algebra) lambda_system_eq:
4.182 @@ -144,7 +155,7 @@
4.183 by (simp add: lambda_system_def)
4.184
4.185 lemma (in algebra) lambda_system_Compl:
4.186 - fixes f:: "'a set \<Rightarrow> pextreal"
4.187 + fixes f:: "'a set \<Rightarrow> extreal"
4.188 assumes x: "x \<in> lambda_system M f"
4.189 shows "space M - x \<in> lambda_system M f"
4.190 proof -
4.191 @@ -157,7 +168,7 @@
4.192 qed
4.193
4.194 lemma (in algebra) lambda_system_Int:
4.195 - fixes f:: "'a set \<Rightarrow> pextreal"
4.196 + fixes f:: "'a set \<Rightarrow> extreal"
4.197 assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
4.198 shows "x \<inter> y \<in> lambda_system M f"
4.199 proof -
4.200 @@ -191,7 +202,7 @@
4.201 qed
4.202
4.203 lemma (in algebra) lambda_system_Un:
4.204 - fixes f:: "'a set \<Rightarrow> pextreal"
4.205 + fixes f:: "'a set \<Rightarrow> extreal"
4.206 assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
4.207 shows "x \<union> y \<in> lambda_system M f"
4.208 proof -
4.209 @@ -250,53 +261,54 @@
4.210 by (auto simp add: disjoint_family_on_def binaryset_def)
4.211 hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
4.212 (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
4.213 - f (\<Union>i. binaryset x y i) \<le> (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
4.214 - using cs by (simp add: countably_subadditive_def)
4.215 + f (\<Union>i. binaryset x y i) \<le> (\<Sum> n. f (binaryset x y n))"
4.216 + using cs by (auto simp add: countably_subadditive_def)
4.217 hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
4.218 - f (x \<union> y) \<le> (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
4.219 + f (x \<union> y) \<le> (\<Sum> n. f (binaryset x y n))"
4.220 by (simp add: range_binaryset_eq UN_binaryset_eq)
4.221 thus "f (x \<union> y) \<le> f x + f y" using f x y
4.222 - by (auto simp add: Un o_def binaryset_psuminf positive_def)
4.223 + by (auto simp add: Un o_def suminf_binaryset_eq positive_def)
4.224 qed
4.225
4.226 lemma (in algebra) additive_sum:
4.227 fixes A:: "nat \<Rightarrow> 'a set"
4.228 - assumes f: "positive M f" and ad: "additive M f"
4.229 + assumes f: "positive M f" and ad: "additive M f" and "finite S"
4.230 and A: "range A \<subseteq> sets M"
4.231 - and disj: "disjoint_family A"
4.232 - shows "setsum (f \<circ> A) {0..<n} = f (\<Union>i\<in>{0..<n}. A i)"
4.233 -proof (induct n)
4.234 - case 0 show ?case using f by (simp add: positive_def)
4.235 + and disj: "disjoint_family_on A S"
4.236 + shows "(\<Sum>i\<in>S. f (A i)) = f (\<Union>i\<in>S. A i)"
4.237 +using `finite S` disj proof induct
4.238 + case empty show ?case using f by (simp add: positive_def)
4.239 next
4.240 - case (Suc n)
4.241 - have "A n \<inter> (\<Union>i\<in>{0..<n}. A i) = {}" using disj
4.242 - by (auto simp add: disjoint_family_on_def neq_iff) blast
4.243 + case (insert s S)
4.244 + then have "A s \<inter> (\<Union>i\<in>S. A i) = {}"
4.245 + by (auto simp add: disjoint_family_on_def neq_iff)
4.246 moreover
4.247 - have "A n \<in> sets M" using A by blast
4.248 - moreover have "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
4.249 - by (metis A UNION_in_sets atLeast0LessThan)
4.250 + have "A s \<in> sets M" using A by blast
4.251 + moreover have "(\<Union>i\<in>S. A i) \<in> sets M"
4.252 + using A `finite S` by auto
4.253 moreover
4.254 - ultimately have "f (A n \<union> (\<Union>i\<in>{0..<n}. A i)) = f (A n) + f(\<Union>i\<in>{0..<n}. A i)"
4.255 + ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)"
4.256 using ad UNION_in_sets A by (auto simp add: additive_def)
4.257 - with Suc.hyps show ?case using ad
4.258 - by (auto simp add: atLeastLessThanSuc additive_def)
4.259 + with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
4.260 + by (auto simp add: additive_def subset_insertI)
4.261 qed
4.262
4.263 lemma (in algebra) increasing_additive_bound:
4.264 - fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> pextreal"
4.265 + fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> extreal"
4.266 assumes f: "positive M f" and ad: "additive M f"
4.267 and inc: "increasing M f"
4.268 and A: "range A \<subseteq> sets M"
4.269 and disj: "disjoint_family A"
4.270 - shows "psuminf (f \<circ> A) \<le> f (space M)"
4.271 -proof (safe intro!: psuminf_bound)
4.272 + shows "(\<Sum>i. f (A i)) \<le> f (space M)"
4.273 +proof (safe intro!: suminf_bound)
4.274 fix N
4.275 - have "setsum (f \<circ> A) {0..<N} = f (\<Union>i\<in>{0..<N}. A i)"
4.276 - by (rule additive_sum [OF f ad A disj])
4.277 + note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"]
4.278 + have "(\<Sum>i<N. f (A i)) = f (\<Union>i\<in>{..<N}. A i)"
4.279 + by (rule additive_sum [OF f ad _ A]) (auto simp: disj_N)
4.280 also have "... \<le> f (space M)" using space_closed A
4.281 - by (blast intro: increasingD [OF inc] UNION_in_sets top)
4.282 - finally show "setsum (f \<circ> A) {..<N} \<le> f (space M)" by (simp add: atLeast0LessThan)
4.283 -qed
4.284 + by (intro increasingD[OF inc] finite_UN) auto
4.285 + finally show "(\<Sum>i<N. f (A i)) \<le> f (space M)" by simp
4.286 +qed (insert f A, auto simp: positive_def)
4.287
4.288 lemma lambda_system_increasing:
4.289 "increasing M f \<Longrightarrow> increasing (M (|sets := lambda_system M f|)) f"
4.290 @@ -307,7 +319,7 @@
4.291 by (simp add: positive_def lambda_system_def)
4.292
4.293 lemma (in algebra) lambda_system_strong_sum:
4.294 - fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> pextreal"
4.295 + fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> extreal"
4.296 assumes f: "positive M f" and a: "a \<in> sets M"
4.297 and A: "range A \<subseteq> lambda_system M f"
4.298 and disj: "disjoint_family A"
4.299 @@ -331,7 +343,7 @@
4.300 assumes oms: "outer_measure_space M f"
4.301 and A: "range A \<subseteq> lambda_system M f"
4.302 and disj: "disjoint_family A"
4.303 - shows "(\<Union>i. A i) \<in> lambda_system M f \<and> psuminf (f \<circ> A) = f (\<Union>i. A i)"
4.304 + shows "(\<Union>i. A i) \<in> lambda_system M f \<and> (\<Sum>i. f (A i)) = f (\<Union>i. A i)"
4.305 proof -
4.306 have pos: "positive M f" and inc: "increasing M f"
4.307 and csa: "countably_subadditive M f"
4.308 @@ -347,15 +359,16 @@
4.309
4.310 have U_in: "(\<Union>i. A i) \<in> sets M"
4.311 by (metis A'' countable_UN)
4.312 - have U_eq: "f (\<Union>i. A i) = psuminf (f o A)"
4.313 + have U_eq: "f (\<Union>i. A i) = (\<Sum>i. f (A i))"
4.314 proof (rule antisym)
4.315 - show "f (\<Union>i. A i) \<le> psuminf (f \<circ> A)"
4.316 - by (rule countably_subadditiveD [OF csa A'' disj U_in])
4.317 - show "psuminf (f \<circ> A) \<le> f (\<Union>i. A i)"
4.318 - by (rule psuminf_bound, unfold atLeast0LessThan[symmetric])
4.319 - (metis algebra.additive_sum [OF alg_ls] pos disj UN_Un Un_UNIV_right
4.320 - lambda_system_positive lambda_system_additive
4.321 - subset_Un_eq increasingD [OF inc] A' A'' UNION_in_sets U_in)
4.322 + show "f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))"
4.323 + using csa[unfolded countably_subadditive_def] A'' disj U_in by auto
4.324 + have *: "\<And>i. 0 \<le> f (A i)" using pos A'' unfolding positive_def by auto
4.325 + have dis: "\<And>N. disjoint_family_on A {..<N}" by (intro disjoint_family_on_mono[OF _ disj]) auto
4.326 + show "(\<Sum>i. f (A i)) \<le> f (\<Union>i. A i)"
4.327 + using algebra.additive_sum [OF alg_ls lambda_system_positive[OF pos] lambda_system_additive _ A' dis]
4.328 + using A''
4.329 + by (intro suminf_bound[OF _ *]) (auto intro!: increasingD[OF inc] allI countable_UN)
4.330 qed
4.331 {
4.332 fix a
4.333 @@ -373,15 +386,15 @@
4.334 have "a \<inter> (\<Union>i. A i) \<in> sets M"
4.335 by (metis Int U_in a)
4.336 ultimately
4.337 - have "f (a \<inter> (\<Union>i. A i)) \<le> psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)"
4.338 - using countably_subadditiveD [OF csa, of "(\<lambda>i. a \<inter> A i)"]
4.339 + have "f (a \<inter> (\<Union>i. A i)) \<le> (\<Sum>i. f (a \<inter> A i))"
4.340 + using csa[unfolded countably_subadditive_def, rule_format, of "(\<lambda>i. a \<inter> A i)"]
4.341 by (simp add: o_def)
4.342 hence "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le>
4.343 - psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) + f (a - (\<Union>i. A i))"
4.344 + (\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i))"
4.345 by (rule add_right_mono)
4.346 moreover
4.347 - have "psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) + f (a - (\<Union>i. A i)) \<le> f a"
4.348 - proof (safe intro!: psuminf_bound_add)
4.349 + have "(\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
4.350 + proof (intro suminf_bound_add allI)
4.351 fix n
4.352 have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
4.353 by (metis A'' UNION_in_sets)
4.354 @@ -395,8 +408,14 @@
4.355 have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))"
4.356 by (blast intro: increasingD [OF inc] UNION_eq_Union_image
4.357 UNION_in U_in)
4.358 - thus "setsum (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) {..<n} + f (a - (\<Union>i. A i)) \<le> f a"
4.359 + thus "(\<Sum>i<n. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
4.360 by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric])
4.361 + next
4.362 + have "\<And>i. a \<inter> A i \<in> sets M" using A'' by auto
4.363 + then show "\<And>i. 0 \<le> f (a \<inter> A i)" using pos[unfolded positive_def] by auto
4.364 + have "\<And>i. a - (\<Union>i. A i) \<in> sets M" using A'' by auto
4.365 + then have "\<And>i. 0 \<le> f (a - (\<Union>i. A i))" using pos[unfolded positive_def] by auto
4.366 + then show "f (a - (\<Union>i. A i)) \<noteq> -\<infinity>" by auto
4.367 qed
4.368 ultimately show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a"
4.369 by (rule order_trans)
4.370 @@ -443,12 +462,14 @@
4.371 proof (auto simp add: increasing_def)
4.372 fix x y
4.373 assume xy: "x \<in> sets M" "y \<in> sets M" "x \<subseteq> y"
4.374 - have "f x \<le> f x + f (y-x)" ..
4.375 + then have "y - x \<in> sets M" by auto
4.376 + then have "0 \<le> f (y-x)" using posf[unfolded positive_def] by auto
4.377 + then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono) auto
4.378 also have "... = f (x \<union> (y-x))" using addf
4.379 by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
4.380 also have "... = f y"
4.381 by (metis Un_Diff_cancel Un_absorb1 xy(3))
4.382 - finally show "f x \<le> f y" .
4.383 + finally show "f x \<le> f y" by simp
4.384 qed
4.385
4.386 lemma (in algebra) countably_additive_additive:
4.387 @@ -461,27 +482,27 @@
4.388 by (auto simp add: disjoint_family_on_def binaryset_def)
4.389 hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
4.390 (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
4.391 - f (\<Union>i. binaryset x y i) = (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
4.392 + f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))"
4.393 using ca
4.394 by (simp add: countably_additive_def)
4.395 hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
4.396 - f (x \<union> y) = (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
4.397 + f (x \<union> y) = (\<Sum>n. f (binaryset x y n))"
4.398 by (simp add: range_binaryset_eq UN_binaryset_eq)
4.399 thus "f (x \<union> y) = f x + f y" using posf x y
4.400 - by (auto simp add: Un binaryset_psuminf positive_def)
4.401 + by (auto simp add: Un suminf_binaryset_eq positive_def)
4.402 qed
4.403
4.404 lemma inf_measure_nonempty:
4.405 assumes f: "positive M f" and b: "b \<in> sets M" and a: "a \<subseteq> b" "{} \<in> sets M"
4.406 shows "f b \<in> measure_set M f a"
4.407 proof -
4.408 - have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = setsum (f \<circ> (\<lambda>i. {})(0 := b)) {..<1::nat}"
4.409 - by (rule psuminf_finite) (simp add: f[unfolded positive_def])
4.410 + let ?A = "\<lambda>i::nat. (if i = 0 then b else {})"
4.411 + have "(\<Sum>i. f (?A i)) = (\<Sum>i<1::nat. f (?A i))"
4.412 + by (rule suminf_finite) (simp add: f[unfolded positive_def])
4.413 also have "... = f b"
4.414 by simp
4.415 - finally have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = f b" .
4.416 - thus ?thesis using assms
4.417 - by (auto intro!: exI [of _ "(\<lambda>i. {})(0 := b)"]
4.418 + finally show ?thesis using assms
4.419 + by (auto intro!: exI [of _ ?A]
4.420 simp: measure_set_def disjoint_family_on_def split_if_mem2 comp_def)
4.421 qed
4.422
4.423 @@ -489,19 +510,19 @@
4.424 assumes posf: "positive M f" and ca: "countably_additive M f"
4.425 and s: "s \<in> sets M"
4.426 shows "Inf (measure_set M f s) = f s"
4.427 - unfolding Inf_pextreal_def
4.428 + unfolding Inf_extreal_def
4.429 proof (safe intro!: Greatest_equality)
4.430 fix z
4.431 assume z: "z \<in> measure_set M f s"
4.432 from this obtain A where
4.433 A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
4.434 - and "s \<subseteq> (\<Union>x. A x)" and si: "psuminf (f \<circ> A) = z"
4.435 + and "s \<subseteq> (\<Union>x. A x)" and si: "(\<Sum>i. f (A i)) = z"
4.436 by (auto simp add: measure_set_def comp_def)
4.437 hence seq: "s = (\<Union>i. A i \<inter> s)" by blast
4.438 have inc: "increasing M f"
4.439 by (metis additive_increasing ca countably_additive_additive posf)
4.440 - have sums: "psuminf (\<lambda>i. f (A i \<inter> s)) = f (\<Union>i. A i \<inter> s)"
4.441 - proof (rule countably_additiveD [OF ca])
4.442 + have sums: "(\<Sum>i. f (A i \<inter> s)) = f (\<Union>i. A i \<inter> s)"
4.443 + proof (rule ca[unfolded countably_additive_def, rule_format])
4.444 show "range (\<lambda>n. A n \<inter> s) \<subseteq> sets M" using A s
4.445 by blast
4.446 show "disjoint_family (\<lambda>n. A n \<inter> s)" using disj
4.447 @@ -509,12 +530,14 @@
4.448 show "(\<Union>i. A i \<inter> s) \<in> sets M" using A s
4.449 by (metis UN_extend_simps(4) s seq)
4.450 qed
4.451 - hence "f s = psuminf (\<lambda>i. f (A i \<inter> s))"
4.452 + hence "f s = (\<Sum>i. f (A i \<inter> s))"
4.453 using seq [symmetric] by (simp add: sums_iff)
4.454 - also have "... \<le> psuminf (f \<circ> A)"
4.455 - proof (rule psuminf_le)
4.456 - fix n show "f (A n \<inter> s) \<le> (f \<circ> A) n" using A s
4.457 + also have "... \<le> (\<Sum>i. f (A i))"
4.458 + proof (rule suminf_le_pos)
4.459 + fix n show "f (A n \<inter> s) \<le> f (A n)" using A s
4.460 by (force intro: increasingD [OF inc])
4.461 + fix N have "A N \<inter> s \<in> sets M" using A s by auto
4.462 + then show "0 \<le> f (A N \<inter> s)" using posf unfolding positive_def by auto
4.463 qed
4.464 also have "... = z" by (rule si)
4.465 finally show "f s \<le> z" .
4.466 @@ -525,18 +548,40 @@
4.467 by (blast intro: inf_measure_nonempty [of _ f, OF posf s subset_refl])
4.468 qed
4.469
4.470 +lemma measure_set_pos:
4.471 + assumes posf: "positive M f" "r \<in> measure_set M f X"
4.472 + shows "0 \<le> r"
4.473 +proof -
4.474 + obtain A where "range A \<subseteq> sets M" and r: "r = (\<Sum>i. f (A i))"
4.475 + using `r \<in> measure_set M f X` unfolding measure_set_def by auto
4.476 + then show "0 \<le> r" using posf unfolding r positive_def
4.477 + by (intro suminf_0_le) auto
4.478 +qed
4.479 +
4.480 +lemma inf_measure_pos:
4.481 + assumes posf: "positive M f"
4.482 + shows "0 \<le> Inf (measure_set M f X)"
4.483 +proof (rule complete_lattice_class.Inf_greatest)
4.484 + fix r assume "r \<in> measure_set M f X" with posf show "0 \<le> r"
4.485 + by (rule measure_set_pos)
4.486 +qed
4.487 +
4.488 lemma inf_measure_empty:
4.489 - assumes posf: "positive M f" "{} \<in> sets M"
4.490 + assumes posf: "positive M f" and "{} \<in> sets M"
4.491 shows "Inf (measure_set M f {}) = 0"
4.492 proof (rule antisym)
4.493 show "Inf (measure_set M f {}) \<le> 0"
4.494 by (metis complete_lattice_class.Inf_lower `{} \<in> sets M`
4.495 inf_measure_nonempty[OF posf] subset_refl posf[unfolded positive_def])
4.496 -qed simp
4.497 +qed (rule inf_measure_pos[OF posf])
4.498
4.499 lemma (in algebra) inf_measure_positive:
4.500 - "positive M f \<Longrightarrow> positive M (\<lambda>x. Inf (measure_set M f x))"
4.501 - by (simp add: positive_def inf_measure_empty)
4.502 + assumes p: "positive M f" and "{} \<in> sets M"
4.503 + shows "positive M (\<lambda>x. Inf (measure_set M f x))"
4.504 +proof (unfold positive_def, intro conjI ballI)
4.505 + show "Inf (measure_set M f {}) = 0" using inf_measure_empty[OF assms] by auto
4.506 + fix A assume "A \<in> sets M"
4.507 +qed (rule inf_measure_pos[OF p])
4.508
4.509 lemma (in algebra) inf_measure_increasing:
4.510 assumes posf: "positive M f"
4.511 @@ -548,25 +593,25 @@
4.512 apply (clarsimp simp add: measure_set_def, rule_tac x=A in exI, blast)
4.513 done
4.514
4.515 -
4.516 lemma (in algebra) inf_measure_le:
4.517 assumes posf: "positive M f" and inc: "increasing M f"
4.518 - and x: "x \<in> {r . \<exists>A. range A \<subseteq> sets M \<and> s \<subseteq> (\<Union>i. A i) \<and> psuminf (f \<circ> A) = r}"
4.519 + and x: "x \<in> {r . \<exists>A. range A \<subseteq> sets M \<and> s \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}"
4.520 shows "Inf (measure_set M f s) \<le> x"
4.521 proof -
4.522 - from x
4.523 obtain A where A: "range A \<subseteq> sets M" and ss: "s \<subseteq> (\<Union>i. A i)"
4.524 - and xeq: "psuminf (f \<circ> A) = x"
4.525 - by auto
4.526 + and xeq: "(\<Sum>i. f (A i)) = x"
4.527 + using x by auto
4.528 have dA: "range (disjointed A) \<subseteq> sets M"
4.529 by (metis A range_disjointed_sets)
4.530 - have "\<forall>n.(f o disjointed A) n \<le> (f \<circ> A) n" unfolding comp_def
4.531 + have "\<forall>n. f (disjointed A n) \<le> f (A n)"
4.532 by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A comp_def)
4.533 - hence sda: "psuminf (f o disjointed A) \<le> psuminf (f \<circ> A)"
4.534 - by (blast intro: psuminf_le)
4.535 - hence ley: "psuminf (f o disjointed A) \<le> x"
4.536 + moreover have "\<forall>i. 0 \<le> f (disjointed A i)"
4.537 + using posf dA unfolding positive_def by auto
4.538 + ultimately have sda: "(\<Sum>i. f (disjointed A i)) \<le> (\<Sum>i. f (A i))"
4.539 + by (blast intro!: suminf_le_pos)
4.540 + hence ley: "(\<Sum>i. f (disjointed A i)) \<le> x"
4.541 by (metis xeq)
4.542 - hence y: "psuminf (f o disjointed A) \<in> measure_set M f s"
4.543 + hence y: "(\<Sum>i. f (disjointed A i)) \<in> measure_set M f s"
4.544 apply (auto simp add: measure_set_def)
4.545 apply (rule_tac x="disjointed A" in exI)
4.546 apply (simp add: disjoint_family_disjointed UN_disjointed_eq ss dA comp_def)
4.547 @@ -576,13 +621,16 @@
4.548 qed
4.549
4.550 lemma (in algebra) inf_measure_close:
4.551 + fixes e :: extreal
4.552 assumes posf: "positive M f" and e: "0 < e" and ss: "s \<subseteq> (space M)"
4.553 shows "\<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> s \<subseteq> (\<Union>i. A i) \<and>
4.554 - psuminf (f \<circ> A) \<le> Inf (measure_set M f s) + e"
4.555 -proof (cases "Inf (measure_set M f s) = \<omega>")
4.556 + (\<Sum>i. f (A i)) \<le> Inf (measure_set M f s) + e"
4.557 +proof (cases "Inf (measure_set M f s) = \<infinity>")
4.558 case False
4.559 + then have fin: "\<bar>Inf (measure_set M f s)\<bar> \<noteq> \<infinity>"
4.560 + using inf_measure_pos[OF posf, of s] by auto
4.561 obtain l where "l \<in> measure_set M f s" "l \<le> Inf (measure_set M f s) + e"
4.562 - using Inf_close[OF False e] by auto
4.563 + using Inf_extreal_close[OF fin e] by auto
4.564 thus ?thesis
4.565 by (auto intro!: exI[of _ l] simp: measure_set_def comp_def)
4.566 next
4.567 @@ -600,9 +648,8 @@
4.568 shows "countably_subadditive (| space = space M, sets = Pow (space M) |)
4.569 (\<lambda>x. Inf (measure_set M f x))"
4.570 unfolding countably_subadditive_def o_def
4.571 -proof (safe, simp, rule pextreal_le_epsilon)
4.572 - fix A :: "nat \<Rightarrow> 'a set" and e :: pextreal
4.573 -
4.574 +proof (safe, simp, rule extreal_le_epsilon, safe)
4.575 + fix A :: "nat \<Rightarrow> 'a set" and e :: extreal
4.576 let "?outer n" = "Inf (measure_set M f (A n))"
4.577 assume A: "range A \<subseteq> Pow (space M)"
4.578 and disj: "disjoint_family A"
4.579 @@ -610,21 +657,27 @@
4.580 and e: "0 < e"
4.581 hence "\<exists>BB. \<forall>n. range (BB n) \<subseteq> sets M \<and> disjoint_family (BB n) \<and>
4.582 A n \<subseteq> (\<Union>i. BB n i) \<and>
4.583 - psuminf (f o BB n) \<le> ?outer n + e * (1/2)^(Suc n)"
4.584 - apply (safe intro!: choice inf_measure_close [of f, OF posf _])
4.585 - using e sb by (cases e, auto simp add: not_le mult_pos_pos)
4.586 + (\<Sum>i. f (BB n i)) \<le> ?outer n + e * (1/2)^(Suc n)"
4.587 + apply (safe intro!: choice inf_measure_close [of f, OF posf])
4.588 + using e sb by (cases e) (auto simp add: not_le mult_pos_pos one_extreal_def)
4.589 then obtain BB
4.590 where BB: "\<And>n. (range (BB n) \<subseteq> sets M)"
4.591 and disjBB: "\<And>n. disjoint_family (BB n)"
4.592 and sbBB: "\<And>n. A n \<subseteq> (\<Union>i. BB n i)"
4.593 - and BBle: "\<And>n. psuminf (f o BB n) \<le> ?outer n + e * (1/2)^(Suc n)"
4.594 + and BBle: "\<And>n. (\<Sum>i. f (BB n i)) \<le> ?outer n + e * (1/2)^(Suc n)"
4.595 by auto blast
4.596 - have sll: "(\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n)) \<le> psuminf ?outer + e"
4.597 + have sll: "(\<Sum>n. \<Sum>i. (f (BB n i))) \<le> suminf ?outer + e"
4.598 proof -
4.599 - have "(\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n)) \<le> (\<Sum>\<^isub>\<infinity> n. ?outer n + e*(1/2) ^ Suc n)"
4.600 - by (rule psuminf_le[OF BBle])
4.601 - also have "... = psuminf ?outer + e"
4.602 - using psuminf_half_series by simp
4.603 + have sum_eq_1: "(\<Sum>n. e*(1/2) ^ Suc n) = e"
4.604 + using suminf_half_series_extreal e
4.605 + by (simp add: extreal_zero_le_0_iff zero_le_divide_extreal suminf_cmult_extreal)
4.606 + have "\<And>n i. 0 \<le> f (BB n i)" using posf[unfolded positive_def] BB by auto
4.607 + then have "\<And>n. 0 \<le> (\<Sum>i. f (BB n i))" by (rule suminf_0_le)
4.608 + then have "(\<Sum>n. \<Sum>i. (f (BB n i))) \<le> (\<Sum>n. ?outer n + e*(1/2) ^ Suc n)"
4.609 + by (rule suminf_le_pos[OF BBle])
4.610 + also have "... = suminf ?outer + e"
4.611 + using sum_eq_1 inf_measure_pos[OF posf] e
4.612 + by (subst suminf_add_extreal) (auto simp add: extreal_zero_le_0_iff)
4.613 finally show ?thesis .
4.614 qed
4.615 def C \<equiv> "(split BB) o prod_decode"
4.616 @@ -644,23 +697,25 @@
4.617 qed
4.618 have "(f \<circ> C) = (f \<circ> (\<lambda>(x, y). BB x y)) \<circ> prod_decode"
4.619 by (rule ext) (auto simp add: C_def)
4.620 - moreover have "psuminf ... = (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))" using BBle
4.621 - by (force intro!: psuminf_2dimen simp: o_def)
4.622 - ultimately have Csums: "psuminf (f \<circ> C) = (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))" by simp
4.623 - have "Inf (measure_set M f (\<Union>i. A i)) \<le> (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))"
4.624 + moreover have "suminf ... = (\<Sum>n. \<Sum>i. f (BB n i))" using BBle
4.625 + using BB posf[unfolded positive_def]
4.626 + by (force intro!: suminf_extreal_2dimen simp: o_def)
4.627 + ultimately have Csums: "(\<Sum>i. f (C i)) = (\<Sum>n. \<Sum>i. f (BB n i))" by (simp add: o_def)
4.628 + have "Inf (measure_set M f (\<Union>i. A i)) \<le> (\<Sum>n. \<Sum>i. f (BB n i))"
4.629 apply (rule inf_measure_le [OF posf(1) inc], auto)
4.630 apply (rule_tac x="C" in exI)
4.631 apply (auto simp add: C sbC Csums)
4.632 done
4.633 - also have "... \<le> (\<Sum>\<^isub>\<infinity>n. Inf (measure_set M f (A n))) + e" using sll
4.634 + also have "... \<le> (\<Sum>n. Inf (measure_set M f (A n))) + e" using sll
4.635 by blast
4.636 - finally show "Inf (measure_set M f (\<Union>i. A i)) \<le> psuminf ?outer + e" .
4.637 + finally show "Inf (measure_set M f (\<Union>i. A i)) \<le> suminf ?outer + e" .
4.638 qed
4.639
4.640 lemma (in algebra) inf_measure_outer:
4.641 "\<lbrakk> positive M f ; increasing M f \<rbrakk>
4.642 \<Longrightarrow> outer_measure_space \<lparr> space = space M, sets = Pow (space M) \<rparr>
4.643 (\<lambda>x. Inf (measure_set M f x))"
4.644 + using inf_measure_pos[of M f]
4.645 by (simp add: outer_measure_space_def inf_measure_empty
4.646 inf_measure_increasing inf_measure_countably_subadditive positive_def)
4.647
4.648 @@ -680,13 +735,13 @@
4.649 by blast
4.650 have "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
4.651 \<le> Inf (measure_set M f s)"
4.652 - proof (rule pextreal_le_epsilon)
4.653 - fix e :: pextreal
4.654 + proof (rule extreal_le_epsilon, intro allI impI)
4.655 + fix e :: extreal
4.656 assume e: "0 < e"
4.657 from inf_measure_close [of f, OF posf e s]
4.658 obtain A where A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
4.659 and sUN: "s \<subseteq> (\<Union>i. A i)"
4.660 - and l: "psuminf (f \<circ> A) \<le> Inf (measure_set M f s) + e"
4.661 + and l: "(\<Sum>i. f (A i)) \<le> Inf (measure_set M f s) + e"
4.662 by auto
4.663 have [simp]: "!!x. x \<in> sets M \<Longrightarrow>
4.664 (f o (\<lambda>z. z \<inter> (space M - x)) o A) = (f o (\<lambda>z. z - x) o A)"
4.665 @@ -698,9 +753,9 @@
4.666 assume u: "u \<in> sets M"
4.667 have [simp]: "\<And>n. f (A n \<inter> u) \<le> f (A n)"
4.668 by (simp add: increasingD [OF inc] u Int range_subsetD [OF A])
4.669 - have 2: "Inf (measure_set M f (s \<inter> u)) \<le> psuminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A)"
4.670 + have 2: "Inf (measure_set M f (s \<inter> u)) \<le> (\<Sum>i. f (A i \<inter> u))"
4.671 proof (rule complete_lattice_class.Inf_lower)
4.672 - show "psuminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A) \<in> measure_set M f (s \<inter> u)"
4.673 + show "(\<Sum>i. f (A i \<inter> u)) \<in> measure_set M f (s \<inter> u)"
4.674 apply (simp add: measure_set_def)
4.675 apply (rule_tac x="(\<lambda>z. z \<inter> u) o A" in exI)
4.676 apply (auto simp add: disjoint_family_subset [OF disj] o_def)
4.677 @@ -709,15 +764,16 @@
4.678 done
4.679 qed
4.680 } note lesum = this
4.681 - have inf1: "Inf (measure_set M f (s\<inter>x)) \<le> psuminf (f o (\<lambda>z. z\<inter>x) o A)"
4.682 + have [simp]: "\<And>i. A i \<inter> (space M - x) = A i - x" using A sets_into_space by auto
4.683 + have inf1: "Inf (measure_set M f (s\<inter>x)) \<le> (\<Sum>i. f (A i \<inter> x))"
4.684 and inf2: "Inf (measure_set M f (s \<inter> (space M - x)))
4.685 - \<le> psuminf (f o (\<lambda>z. z \<inter> (space M - x)) o A)"
4.686 + \<le> (\<Sum>i. f (A i \<inter> (space M - x)))"
4.687 by (metis Diff lesum top x)+
4.688 hence "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
4.689 - \<le> psuminf (f o (\<lambda>s. s\<inter>x) o A) + psuminf (f o (\<lambda>s. s-x) o A)"
4.690 - by (simp add: x add_mono)
4.691 - also have "... \<le> psuminf (f o A)"
4.692 - by (simp add: x psuminf_add[symmetric] o_def)
4.693 + \<le> (\<Sum>i. f (A i \<inter> x)) + (\<Sum>i. f (A i - x))"
4.694 + by (simp add: add_mono x)
4.695 + also have "... \<le> (\<Sum>i. f (A i))" using posf[unfolded positive_def] A x
4.696 + by (subst suminf_add_extreal[symmetric]) auto
4.697 also have "... \<le> Inf (measure_set M f s) + e"
4.698 by (rule l)
4.699 finally show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
4.700 @@ -732,7 +788,7 @@
4.701 also have "... \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
4.702 apply (rule subadditiveD)
4.703 apply (rule algebra.countably_subadditive_subadditive[OF algebra_Pow])
4.704 - apply (simp add: positive_def inf_measure_empty[OF posf])
4.705 + apply (simp add: positive_def inf_measure_empty[OF posf] inf_measure_pos[OF posf])
4.706 apply (rule inf_measure_countably_subadditive)
4.707 using s by (auto intro!: posf inc)
4.708 finally show ?thesis .
4.709 @@ -751,7 +807,7 @@
4.710
4.711 theorem (in algebra) caratheodory:
4.712 assumes posf: "positive M f" and ca: "countably_additive M f"
4.713 - shows "\<exists>\<mu> :: 'a set \<Rightarrow> pextreal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and>
4.714 + shows "\<exists>\<mu> :: 'a set \<Rightarrow> extreal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and>
4.715 measure_space \<lparr> space = space M, sets = sets (sigma M), measure = \<mu> \<rparr>"
4.716 proof -
4.717 have inc: "increasing M f"
5.1 --- a/src/HOL/Probability/Complete_Measure.thy Mon Mar 14 14:37:47 2011 +0100
5.2 +++ b/src/HOL/Probability/Complete_Measure.thy Mon Mar 14 14:37:49 2011 +0100
5.3 @@ -1,7 +1,6 @@
5.4 -(* Title: HOL/Probability/Complete_Measure.thy
5.5 +(* Title: Complete_Measure.thy
5.6 Author: Robert Himmelmann, Johannes Hoelzl, TU Muenchen
5.7 *)
5.8 -
5.9 theory Complete_Measure
5.10 imports Product_Measure
5.11 begin
5.12 @@ -177,7 +176,8 @@
5.13 proof -
5.14 show "measure_space completion"
5.15 proof
5.16 - show "measure completion {} = 0" by (auto simp: completion_def)
5.17 + show "positive completion (measure completion)"
5.18 + by (auto simp: completion_def positive_def)
5.19 next
5.20 show "countably_additive completion (measure completion)"
5.21 proof (intro countably_additiveI)
5.22 @@ -189,9 +189,9 @@
5.23 using A by (subst (1 2) main_part_null_part_Un) auto
5.24 then show "main_part (A n) \<inter> main_part (A m) = {}" by auto
5.25 qed
5.26 - then have "(\<Sum>\<^isub>\<infinity>n. measure completion (A n)) = \<mu> (\<Union>i. main_part (A i))"
5.27 + then have "(\<Sum>n. measure completion (A n)) = \<mu> (\<Union>i. main_part (A i))"
5.28 unfolding completion_def using A by (auto intro!: measure_countably_additive)
5.29 - then show "(\<Sum>\<^isub>\<infinity>n. measure completion (A n)) = measure completion (UNION UNIV A)"
5.30 + then show "(\<Sum>n. measure completion (A n)) = measure completion (UNION UNIV A)"
5.31 by (simp add: completion_def \<mu>_main_part_UN[OF A(1)])
5.32 qed
5.33 qed
5.34 @@ -251,30 +251,52 @@
5.35 qed
5.36 qed
5.37
5.38 -lemma (in completeable_measure_space) completion_ex_borel_measurable:
5.39 - fixes g :: "'a \<Rightarrow> pextreal"
5.40 - assumes g: "g \<in> borel_measurable completion"
5.41 +lemma (in completeable_measure_space) completion_ex_borel_measurable_pos:
5.42 + fixes g :: "'a \<Rightarrow> extreal"
5.43 + assumes g: "g \<in> borel_measurable completion" and "\<And>x. 0 \<le> g x"
5.44 shows "\<exists>g'\<in>borel_measurable M. (AE x. g x = g' x)"
5.45 proof -
5.46 - from g[THEN completion.borel_measurable_implies_simple_function_sequence]
5.47 - obtain f where "\<And>i. simple_function completion (f i)" "f \<up> g" by auto
5.48 - then have "\<forall>i. \<exists>f'. simple_function M f' \<and> (AE x. f i x = f' x)"
5.49 - using completion_ex_simple_function by auto
5.50 + from g[THEN completion.borel_measurable_implies_simple_function_sequence'] guess f . note f = this
5.51 + from this(1)[THEN completion_ex_simple_function]
5.52 + have "\<forall>i. \<exists>f'. simple_function M f' \<and> (AE x. f i x = f' x)" ..
5.53 from this[THEN choice] obtain f' where
5.54 sf: "\<And>i. simple_function M (f' i)" and
5.55 AE: "\<forall>i. AE x. f i x = f' i x" by auto
5.56 show ?thesis
5.57 proof (intro bexI)
5.58 - from AE[unfolded all_AE_countable]
5.59 + from AE[unfolded AE_all_countable[symmetric]]
5.60 show "AE x. g x = (SUP i. f' i x)" (is "AE x. g x = ?f x")
5.61 proof (elim AE_mp, safe intro!: AE_I2)
5.62 fix x assume eq: "\<forall>i. f i x = f' i x"
5.63 - moreover have "g = SUPR UNIV f" using `f \<up> g` unfolding isoton_def by simp
5.64 - ultimately show "g x = ?f x" by (simp add: SUPR_apply)
5.65 + moreover have "g x = (SUP i. f i x)"
5.66 + unfolding f using `0 \<le> g x` by (auto split: split_max)
5.67 + ultimately show "g x = ?f x" by auto
5.68 qed
5.69 show "?f \<in> borel_measurable M"
5.70 using sf by (auto intro: borel_measurable_simple_function)
5.71 qed
5.72 qed
5.73
5.74 +lemma (in completeable_measure_space) completion_ex_borel_measurable:
5.75 + fixes g :: "'a \<Rightarrow> extreal"
5.76 + assumes g: "g \<in> borel_measurable completion"
5.77 + shows "\<exists>g'\<in>borel_measurable M. (AE x. g x = g' x)"
5.78 +proof -
5.79 + have "(\<lambda>x. max 0 (g x)) \<in> borel_measurable completion" "\<And>x. 0 \<le> max 0 (g x)" using g by auto
5.80 + from completion_ex_borel_measurable_pos[OF this] guess g_pos ..
5.81 + moreover
5.82 + have "(\<lambda>x. max 0 (- g x)) \<in> borel_measurable completion" "\<And>x. 0 \<le> max 0 (- g x)" using g by auto
5.83 + from completion_ex_borel_measurable_pos[OF this] guess g_neg ..
5.84 + ultimately
5.85 + show ?thesis
5.86 + proof (safe intro!: bexI[of _ "\<lambda>x. g_pos x - g_neg x"])
5.87 + show "AE x. max 0 (- g x) = g_neg x \<longrightarrow> max 0 (g x) = g_pos x \<longrightarrow> g x = g_pos x - g_neg x"
5.88 + proof (intro AE_I2 impI)
5.89 + fix x assume g: "max 0 (- g x) = g_neg x" "max 0 (g x) = g_pos x"
5.90 + show "g x = g_pos x - g_neg x" unfolding g[symmetric]
5.91 + by (cases "g x") (auto split: split_max)
5.92 + qed
5.93 + qed auto
5.94 +qed
5.95 +
5.96 end
6.1 --- a/src/HOL/Probability/Information.thy Mon Mar 14 14:37:47 2011 +0100
6.2 +++ b/src/HOL/Probability/Information.thy Mon Mar 14 14:37:49 2011 +0100
6.3 @@ -2,9 +2,12 @@
6.4 imports
6.5 Probability_Space
6.6 "~~/src/HOL/Library/Convex"
6.7 - Lebesgue_Measure
6.8 begin
6.9
6.10 +lemma (in prob_space) not_zero_less_distribution[simp]:
6.11 + "(\<not> 0 < distribution X A) \<longleftrightarrow> distribution X A = 0"
6.12 + using distribution_positive[of X A] by arith
6.13 +
6.14 lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y"
6.15 by (subst log_le_cancel_iff) auto
6.16
6.17 @@ -238,7 +241,7 @@
6.18 have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
6.19 show "(\<Sum>x \<in> space M. log b (real (RN_deriv M \<nu> x)) * real (\<nu> {x})) = ?sum"
6.20 using RN_deriv_finite_measure[OF ms ac]
6.21 - by (auto intro!: setsum_cong simp: field_simps real_of_pextreal_mult[symmetric])
6.22 + by (auto intro!: setsum_cong simp: field_simps)
6.23 qed
6.24
6.25 lemma (in finite_prob_space) KL_divergence_positive_finite:
6.26 @@ -254,7 +257,8 @@
6.27 proof (subst KL_divergence_eq_finite[OF ms ac], safe intro!: log_setsum_divide not_empty)
6.28 show "finite (space M)" using finite_space by simp
6.29 show "1 < b" by fact
6.30 - show "(\<Sum>x\<in>space M. real (\<nu> {x})) = 1" using v.finite_sum_over_space_eq_1 by simp
6.31 + show "(\<Sum>x\<in>space M. real (\<nu> {x})) = 1"
6.32 + using v.finite_sum_over_space_eq_1 by (simp add: v.\<mu>'_def)
6.33
6.34 fix x assume "x \<in> space M"
6.35 then have x: "{x} \<in> sets M" unfolding sets_eq_Pow by auto
6.36 @@ -262,17 +266,19 @@
6.37 then have "\<nu> {x} \<noteq> 0" by auto
6.38 then have "\<mu> {x} \<noteq> 0"
6.39 using ac[unfolded absolutely_continuous_def, THEN bspec, of "{x}"] x by auto
6.40 - thus "0 < prob {x}" using finite_measure[of "{x}"] x by auto }
6.41 - qed auto
6.42 - thus "0 \<le> KL_divergence b M \<nu>" using finite_sum_over_space_eq_1 by simp
6.43 + thus "0 < real (\<mu> {x})" using real_measure[OF x] by auto }
6.44 + show "0 \<le> real (\<mu> {x})" "0 \<le> real (\<nu> {x})"
6.45 + using real_measure[OF x] v.real_measure[of "{x}"] x by auto
6.46 + qed
6.47 + thus "0 \<le> KL_divergence b M \<nu>" using finite_sum_over_space_eq_1 by (simp add: \<mu>'_def)
6.48 qed
6.49
6.50 subsection {* Mutual Information *}
6.51
6.52 definition (in prob_space)
6.53 "mutual_information b S T X Y =
6.54 - KL_divergence b (S\<lparr>measure := distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := distribution Y\<rparr>)
6.55 - (joint_distribution X Y)"
6.56 + KL_divergence b (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>)
6.57 + (extreal\<circ>joint_distribution X Y)"
6.58
6.59 definition (in prob_space)
6.60 "entropy b s X = mutual_information b s s X X"
6.61 @@ -280,38 +286,33 @@
6.62 abbreviation (in information_space)
6.63 mutual_information_Pow ("\<I>'(_ ; _')") where
6.64 "\<I>(X ; Y) \<equiv> mutual_information b
6.65 - \<lparr> space = X`space M, sets = Pow (X`space M), measure = distribution X \<rparr>
6.66 - \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = distribution Y \<rparr> X Y"
6.67 + \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
6.68 + \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr> X Y"
6.69
6.70 lemma (in prob_space) finite_variables_absolutely_continuous:
6.71 assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
6.72 shows "measure_space.absolutely_continuous
6.73 - (S\<lparr>measure := distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := distribution Y\<rparr>)
6.74 - (joint_distribution X Y)"
6.75 + (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>)
6.76 + (extreal\<circ>joint_distribution X Y)"
6.77 proof -
6.78 - interpret X: finite_prob_space "S\<lparr>measure := distribution X\<rparr>"
6.79 + interpret X: finite_prob_space "S\<lparr>measure := extreal\<circ>distribution X\<rparr>"
6.80 using X by (rule distribution_finite_prob_space)
6.81 - interpret Y: finite_prob_space "T\<lparr>measure := distribution Y\<rparr>"
6.82 + interpret Y: finite_prob_space "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
6.83 using Y by (rule distribution_finite_prob_space)
6.84 interpret XY: pair_finite_prob_space
6.85 - "S\<lparr>measure := distribution X\<rparr>" "T\<lparr> measure := distribution Y\<rparr>" by default
6.86 - interpret P: finite_prob_space "XY.P\<lparr> measure := joint_distribution X Y\<rparr>"
6.87 + "S\<lparr>measure := extreal\<circ>distribution X\<rparr>" "T\<lparr> measure := extreal\<circ>distribution Y\<rparr>" by default
6.88 + interpret P: finite_prob_space "XY.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>"
6.89 using assms by (auto intro!: joint_distribution_finite_prob_space)
6.90 note rv = assms[THEN finite_random_variableD]
6.91 - show "XY.absolutely_continuous (joint_distribution X Y)"
6.92 + show "XY.absolutely_continuous (extreal\<circ>joint_distribution X Y)"
6.93 proof (rule XY.absolutely_continuousI)
6.94 - show "finite_measure_space (XY.P\<lparr> measure := joint_distribution X Y\<rparr>)" by default
6.95 + show "finite_measure_space (XY.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
6.96 fix x assume "x \<in> space XY.P" and "XY.\<mu> {x} = 0"
6.97 - then obtain a b where "(a, b) = x" and "a \<in> space S" "b \<in> space T"
6.98 - and distr: "distribution X {a} * distribution Y {b} = 0"
6.99 + then obtain a b where "x = (a, b)"
6.100 + and "distribution X {a} = 0 \<or> distribution Y {b} = 0"
6.101 by (cases x) (auto simp: space_pair_measure)
6.102 - with X.sets_eq_Pow Y.sets_eq_Pow
6.103 - joint_distribution_Times_le_fst[OF rv, of "{a}" "{b}"]
6.104 - joint_distribution_Times_le_snd[OF rv, of "{a}" "{b}"]
6.105 - have "joint_distribution X Y {x} \<le> distribution Y {b}"
6.106 - "joint_distribution X Y {x} \<le> distribution X {a}"
6.107 - by (auto simp del: X.sets_eq_Pow Y.sets_eq_Pow)
6.108 - with distr show "joint_distribution X Y {x} = 0" by auto
6.109 + with finite_distribution_order(5,6)[OF X Y]
6.110 + show "(extreal \<circ> joint_distribution X Y) {x} = 0" by auto
6.111 qed
6.112 qed
6.113
6.114 @@ -320,28 +321,28 @@
6.115 assumes MY: "finite_random_variable MY Y"
6.116 shows mutual_information_generic_eq:
6.117 "mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY.
6.118 - real (joint_distribution X Y {(x,y)}) *
6.119 - log b (real (joint_distribution X Y {(x,y)}) /
6.120 - (real (distribution X {x}) * real (distribution Y {y}))))"
6.121 + joint_distribution X Y {(x,y)} *
6.122 + log b (joint_distribution X Y {(x,y)} /
6.123 + (distribution X {x} * distribution Y {y})))"
6.124 (is ?sum)
6.125 and mutual_information_positive_generic:
6.126 "0 \<le> mutual_information b MX MY X Y" (is ?positive)
6.127 proof -
6.128 - interpret X: finite_prob_space "MX\<lparr>measure := distribution X\<rparr>"
6.129 + interpret X: finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>"
6.130 using MX by (rule distribution_finite_prob_space)
6.131 - interpret Y: finite_prob_space "MY\<lparr>measure := distribution Y\<rparr>"
6.132 + interpret Y: finite_prob_space "MY\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
6.133 using MY by (rule distribution_finite_prob_space)
6.134 - interpret XY: pair_finite_prob_space "MX\<lparr>measure := distribution X\<rparr>" "MY\<lparr>measure := distribution Y\<rparr>" by default
6.135 - interpret P: finite_prob_space "XY.P\<lparr>measure := joint_distribution X Y\<rparr>"
6.136 + interpret XY: pair_finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>" "MY\<lparr>measure := extreal\<circ>distribution Y\<rparr>" by default
6.137 + interpret P: finite_prob_space "XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>"
6.138 using assms by (auto intro!: joint_distribution_finite_prob_space)
6.139
6.140 - have P_ms: "finite_measure_space (XY.P\<lparr>measure :=joint_distribution X Y\<rparr>)" by default
6.141 - have P_ps: "finite_prob_space (XY.P\<lparr>measure := joint_distribution X Y\<rparr>)" by default
6.142 + have P_ms: "finite_measure_space (XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
6.143 + have P_ps: "finite_prob_space (XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
6.144
6.145 show ?sum
6.146 unfolding Let_def mutual_information_def
6.147 by (subst XY.KL_divergence_eq_finite[OF P_ms finite_variables_absolutely_continuous[OF MX MY]])
6.148 - (auto simp add: space_pair_measure setsum_cartesian_product' real_of_pextreal_mult[symmetric])
6.149 + (auto simp add: space_pair_measure setsum_cartesian_product')
6.150
6.151 show ?positive
6.152 using XY.KL_divergence_positive_finite[OF P_ps finite_variables_absolutely_continuous[OF MX MY] b_gt_1]
6.153 @@ -351,10 +352,10 @@
6.154 lemma (in information_space) mutual_information_commute_generic:
6.155 assumes X: "random_variable S X" and Y: "random_variable T Y"
6.156 assumes ac: "measure_space.absolutely_continuous
6.157 - (S\<lparr>measure := distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := distribution Y\<rparr>) (joint_distribution X Y)"
6.158 + (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>) (extreal\<circ>joint_distribution X Y)"
6.159 shows "mutual_information b S T X Y = mutual_information b T S Y X"
6.160 proof -
6.161 - let ?S = "S\<lparr>measure := distribution X\<rparr>" and ?T = "T\<lparr>measure := distribution Y\<rparr>"
6.162 + let ?S = "S\<lparr>measure := extreal\<circ>distribution X\<rparr>" and ?T = "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
6.163 interpret S: prob_space ?S using X by (rule distribution_prob_space)
6.164 interpret T: prob_space ?T using Y by (rule distribution_prob_space)
6.165 interpret P: pair_prob_space ?S ?T ..
6.166 @@ -363,13 +364,13 @@
6.167 unfolding mutual_information_def
6.168 proof (intro Q.KL_divergence_vimage[OF Q.measure_preserving_swap _ _ _ _ ac b_gt_1])
6.169 show "(\<lambda>(x,y). (y,x)) \<in> measure_preserving
6.170 - (P.P \<lparr> measure := joint_distribution X Y\<rparr>) (Q.P \<lparr> measure := joint_distribution Y X\<rparr>)"
6.171 + (P.P \<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>) (Q.P \<lparr> measure := extreal\<circ>joint_distribution Y X\<rparr>)"
6.172 using X Y unfolding measurable_def
6.173 unfolding measure_preserving_def using P.pair_sigma_algebra_swap_measurable
6.174 - by (auto simp add: space_pair_measure distribution_def intro!: arg_cong[where f=\<mu>])
6.175 - have "prob_space (P.P\<lparr> measure := joint_distribution X Y\<rparr>)"
6.176 + by (auto simp add: space_pair_measure distribution_def intro!: arg_cong[where f=\<mu>'])
6.177 + have "prob_space (P.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)"
6.178 using X Y by (auto intro!: distribution_prob_space random_variable_pairI)
6.179 - then show "measure_space (P.P\<lparr> measure := joint_distribution X Y\<rparr>)"
6.180 + then show "measure_space (P.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)"
6.181 unfolding prob_space_def by simp
6.182 qed auto
6.183 qed
6.184 @@ -389,8 +390,8 @@
6.185 lemma (in information_space) mutual_information_eq:
6.186 assumes "simple_function M X" "simple_function M Y"
6.187 shows "\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
6.188 - real (distribution (\<lambda>x. (X x, Y x)) {(x,y)}) * log b (real (distribution (\<lambda>x. (X x, Y x)) {(x,y)}) /
6.189 - (real (distribution X {x}) * real (distribution Y {y}))))"
6.190 + distribution (\<lambda>x. (X x, Y x)) {(x,y)} * log b (distribution (\<lambda>x. (X x, Y x)) {(x,y)} /
6.191 + (distribution X {x} * distribution Y {y})))"
6.192 using assms by (simp add: mutual_information_generic_eq)
6.193
6.194 lemma (in information_space) mutual_information_generic_cong:
6.195 @@ -416,22 +417,27 @@
6.196
6.197 abbreviation (in information_space)
6.198 entropy_Pow ("\<H>'(_')") where
6.199 - "\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M), measure = distribution X \<rparr> X"
6.200 + "\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr> X"
6.201
6.202 lemma (in information_space) entropy_generic_eq:
6.203 + fixes X :: "'a \<Rightarrow> 'c"
6.204 assumes MX: "finite_random_variable MX X"
6.205 - shows "entropy b MX X = -(\<Sum> x \<in> space MX. real (distribution X {x}) * log b (real (distribution X {x})))"
6.206 + shows "entropy b MX X = -(\<Sum> x \<in> space MX. distribution X {x} * log b (distribution X {x}))"
6.207 proof -
6.208 - interpret MX: finite_prob_space "MX\<lparr>measure := distribution X\<rparr>"
6.209 + interpret MX: finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>"
6.210 using MX by (rule distribution_finite_prob_space)
6.211 - let "?X x" = "real (distribution X {x})"
6.212 - let "?XX x y" = "real (joint_distribution X X {(x, y)})"
6.213 - { fix x y
6.214 - have "(\<lambda>x. (X x, X x)) -` {(x, y)} = (if x = y then X -` {x} else {})" by auto
6.215 + let "?X x" = "distribution X {x}"
6.216 + let "?XX x y" = "joint_distribution X X {(x, y)}"
6.217 +
6.218 + { fix x y :: 'c
6.219 + { assume "x \<noteq> y"
6.220 + then have "(\<lambda>x. (X x, X x)) -` {(x,y)} \<inter> space M = {}" by auto
6.221 + then have "joint_distribution X X {(x, y)} = 0" by (simp add: distribution_def) }
6.222 then have "?XX x y * log b (?XX x y / (?X x * ?X y)) =
6.223 (if x = y then - ?X y * log b (?X y) else 0)"
6.224 - unfolding distribution_def by (auto simp: log_simps zero_less_mult_iff) }
6.225 + by (auto simp: log_simps zero_less_mult_iff) }
6.226 note remove_XX = this
6.227 +
6.228 show ?thesis
6.229 unfolding entropy_def mutual_information_generic_eq[OF MX MX]
6.230 unfolding setsum_cartesian_product[symmetric] setsum_negf[symmetric] remove_XX
6.231 @@ -440,7 +446,7 @@
6.232
6.233 lemma (in information_space) entropy_eq:
6.234 assumes "simple_function M X"
6.235 - shows "\<H>(X) = -(\<Sum> x \<in> X ` space M. real (distribution X {x}) * log b (real (distribution X {x})))"
6.236 + shows "\<H>(X) = -(\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
6.237 using assms by (simp add: entropy_generic_eq)
6.238
6.239 lemma (in information_space) entropy_positive:
6.240 @@ -448,63 +454,77 @@
6.241 unfolding entropy_def by (simp add: mutual_information_positive)
6.242
6.243 lemma (in information_space) entropy_certainty_eq_0:
6.244 - assumes "simple_function M X" and "x \<in> X ` space M" and "distribution X {x} = 1"
6.245 + assumes X: "simple_function M X" and "x \<in> X ` space M" and "distribution X {x} = 1"
6.246 shows "\<H>(X) = 0"
6.247 proof -
6.248 - let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = distribution X\<rparr>"
6.249 + let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = extreal\<circ>distribution X\<rparr>"
6.250 note simple_function_imp_finite_random_variable[OF `simple_function M X`]
6.251 - from distribution_finite_prob_space[OF this, of "\<lparr> measure = distribution X \<rparr>"]
6.252 + from distribution_finite_prob_space[OF this, of "\<lparr> measure = extreal\<circ>distribution X \<rparr>"]
6.253 interpret X: finite_prob_space ?X by simp
6.254 have "distribution X (X ` space M - {x}) = distribution X (X ` space M) - distribution X {x}"
6.255 using X.measure_compl[of "{x}"] assms by auto
6.256 also have "\<dots> = 0" using X.prob_space assms by auto
6.257 finally have X0: "distribution X (X ` space M - {x}) = 0" by auto
6.258 - { fix y assume asm: "y \<noteq> x" "y \<in> X ` space M"
6.259 - hence "{y} \<subseteq> X ` space M - {x}" by auto
6.260 - from X.measure_mono[OF this] X0 asm
6.261 - have "distribution X {y} = 0" by auto }
6.262 - hence fi: "\<And> y. y \<in> X ` space M \<Longrightarrow> real (distribution X {y}) = (if x = y then 1 else 0)"
6.263 - using assms by auto
6.264 + { fix y assume *: "y \<in> X ` space M"
6.265 + { assume asm: "y \<noteq> x"
6.266 + with * have "{y} \<subseteq> X ` space M - {x}" by auto
6.267 + from X.measure_mono[OF this] X0 asm *
6.268 + have "distribution X {y} = 0" by (auto intro: antisym) }
6.269 + then have "distribution X {y} = (if x = y then 1 else 0)"
6.270 + using assms by auto }
6.271 + note fi = this
6.272 have y: "\<And>y. (if x = y then 1 else 0) * log b (if x = y then 1 else 0) = 0" by simp
6.273 show ?thesis unfolding entropy_eq[OF `simple_function M X`] by (auto simp: y fi)
6.274 qed
6.275
6.276 lemma (in information_space) entropy_le_card_not_0:
6.277 - assumes "simple_function M X"
6.278 - shows "\<H>(X) \<le> log b (real (card (X ` space M \<inter> {x . distribution X {x} \<noteq> 0})))"
6.279 + assumes X: "simple_function M X"
6.280 + shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}))"
6.281 proof -
6.282 - let "?d x" = "distribution X {x}"
6.283 - let "?p x" = "real (?d x)"
6.284 + let "?p x" = "distribution X {x}"
6.285 have "\<H>(X) = (\<Sum>x\<in>X`space M. ?p x * log b (1 / ?p x))"
6.286 - by (auto intro!: setsum_cong simp: entropy_eq[OF `simple_function M X`] setsum_negf[symmetric] log_simps not_less)
6.287 + unfolding entropy_eq[OF X] setsum_negf[symmetric]
6.288 + by (auto intro!: setsum_cong simp: log_simps)
6.289 also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. ?p x * (1 / ?p x))"
6.290 - apply (rule log_setsum')
6.291 - using not_empty b_gt_1 `simple_function M X` sum_over_space_real_distribution
6.292 - by (auto simp: simple_function_def)
6.293 - also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?d x \<noteq> 0 then 1 else 0)"
6.294 - using distribution_finite[OF `simple_function M X`[THEN simple_function_imp_random_variable], simplified]
6.295 - by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) (auto simp: real_of_pextreal_eq_0)
6.296 + using not_empty b_gt_1 `simple_function M X` sum_over_space_real_distribution[OF X]
6.297 + by (intro log_setsum') (auto simp: simple_function_def)
6.298 + also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?p x \<noteq> 0 then 1 else 0)"
6.299 + by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) auto
6.300 finally show ?thesis
6.301 using `simple_function M X` by (auto simp: setsum_cases real_eq_of_nat simple_function_def)
6.302 qed
6.303
6.304 +lemma (in prob_space) measure'_translate:
6.305 + assumes X: "random_variable S X" and A: "A \<in> sets S"
6.306 + shows "finite_measure.\<mu>' (S\<lparr> measure := extreal\<circ>distribution X \<rparr>) A = distribution X A"
6.307 +proof -
6.308 + interpret S: prob_space "S\<lparr> measure := extreal\<circ>distribution X \<rparr>"
6.309 + using distribution_prob_space[OF X] .
6.310 + from A show "S.\<mu>' A = distribution X A"
6.311 + unfolding S.\<mu>'_def by (simp add: distribution_def_raw \<mu>'_def)
6.312 +qed
6.313 +
6.314 lemma (in information_space) entropy_uniform_max:
6.315 - assumes "simple_function M X"
6.316 + assumes X: "simple_function M X"
6.317 assumes "\<And>x y. \<lbrakk> x \<in> X ` space M ; y \<in> X ` space M \<rbrakk> \<Longrightarrow> distribution X {x} = distribution X {y}"
6.318 shows "\<H>(X) = log b (real (card (X ` space M)))"
6.319 proof -
6.320 - let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = distribution X\<rparr>"
6.321 - note simple_function_imp_finite_random_variable[OF `simple_function M X`]
6.322 - from distribution_finite_prob_space[OF this, of "\<lparr> measure = distribution X \<rparr>"]
6.323 + let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = undefined\<rparr>\<lparr> measure := extreal\<circ>distribution X\<rparr>"
6.324 + note frv = simple_function_imp_finite_random_variable[OF X]
6.325 + from distribution_finite_prob_space[OF this, of "\<lparr> measure = extreal\<circ>distribution X \<rparr>"]
6.326 interpret X: finite_prob_space ?X by simp
6.327 + note rv = finite_random_variableD[OF frv]
6.328 have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
6.329 using `simple_function M X` not_empty by (auto simp: simple_function_def)
6.330 - { fix x assume "x \<in> X ` space M"
6.331 - hence "real (distribution X {x}) = 1 / real (card (X ` space M))"
6.332 - proof (rule X.uniform_prob[simplified])
6.333 - fix x y assume "x \<in> X`space M" "y \<in> X`space M"
6.334 - from assms(2)[OF this] show "real (distribution X {x}) = real (distribution X {y})" by simp
6.335 - qed }
6.336 + { fix x assume "x \<in> space ?X"
6.337 + moreover then have "X.\<mu>' {x} = 1 / card (space ?X)"
6.338 + proof (rule X.uniform_prob)
6.339 + fix x y assume "x \<in> space ?X" "y \<in> space ?X"
6.340 + with assms(2)[of x y] show "X.\<mu>' {x} = X.\<mu>' {y}"
6.341 + by (subst (1 2) measure'_translate[OF rv]) auto
6.342 + qed
6.343 + ultimately have "distribution X {x} = 1 / card (space ?X)"
6.344 + by (subst (asm) measure'_translate[OF rv]) auto }
6.345 thus ?thesis
6.346 using not_empty X.finite_space b_gt_1 card_gt0
6.347 by (simp add: entropy_eq[OF `simple_function M X`] real_eq_of_nat[symmetric] log_simps)
6.348 @@ -552,8 +572,7 @@
6.349 lemma (in information_space) entropy_eq_cartesian_product:
6.350 assumes "simple_function M X" "simple_function M Y"
6.351 shows "\<H>(\<lambda>x. (X x, Y x)) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
6.352 - real (joint_distribution X Y {(x,y)}) *
6.353 - log b (real (joint_distribution X Y {(x,y)})))"
6.354 + joint_distribution X Y {(x,y)} * log b (joint_distribution X Y {(x,y)}))"
6.355 proof -
6.356 have sf: "simple_function M (\<lambda>x. (X x, Y x))"
6.357 using assms by (auto intro: simple_function_Pair)
6.358 @@ -576,9 +595,9 @@
6.359 abbreviation (in information_space)
6.360 conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where
6.361 "\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
6.362 - \<lparr> space = X`space M, sets = Pow (X`space M), measure = distribution X \<rparr>
6.363 - \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = distribution Y \<rparr>
6.364 - \<lparr> space = Z`space M, sets = Pow (Z`space M), measure = distribution Z \<rparr>
6.365 + \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
6.366 + \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr>
6.367 + \<lparr> space = Z`space M, sets = Pow (Z`space M), measure = extreal\<circ>distribution Z \<rparr>
6.368 X Y Z"
6.369
6.370 lemma (in information_space) conditional_mutual_information_generic_eq:
6.371 @@ -586,58 +605,44 @@
6.372 and MY: "finite_random_variable MY Y"
6.373 and MZ: "finite_random_variable MZ Z"
6.374 shows "conditional_mutual_information b MX MY MZ X Y Z = (\<Sum>(x, y, z) \<in> space MX \<times> space MY \<times> space MZ.
6.375 - real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) *
6.376 - log b (real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) /
6.377 - (real (joint_distribution X Z {(x, z)}) * real (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
6.378 - (is "_ = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * ?YZdZ y z)))")
6.379 + distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
6.380 + log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
6.381 + (joint_distribution X Z {(x, z)} * (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
6.382 + (is "_ = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * (?YZ y z / ?Z z))))")
6.383 proof -
6.384 - let ?YZ = "\<lambda>y z. real (joint_distribution Y Z {(y, z)})"
6.385 - let ?X = "\<lambda>x. real (distribution X {x})"
6.386 - let ?Z = "\<lambda>z. real (distribution Z {z})"
6.387 -
6.388 - txt {* This proof is actually quiet easy, however we need to show that the
6.389 - distributions are finite and the joint distributions are zero when one of
6.390 - the variables distribution is also zero. *}
6.391 -
6.392 + let ?X = "\<lambda>x. distribution X {x}"
6.393 note finite_var = MX MY MZ
6.394 - note random_var = finite_var[THEN finite_random_variableD]
6.395 -
6.396 - note space_simps = space_pair_measure space_sigma algebra.simps
6.397 -
6.398 note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
6.399 + note XYZ = finite_random_variable_pairI[OF MX YZ]
6.400 note XZ = finite_random_variable_pairI[OF finite_var(1,3)]
6.401 note ZX = finite_random_variable_pairI[OF finite_var(3,1)]
6.402 note YZX = finite_random_variable_pairI[OF finite_var(2) ZX]
6.403 note order1 =
6.404 - finite_distribution_order(5,6)[OF finite_var(1) YZ, simplified space_simps]
6.405 - finite_distribution_order(5,6)[OF finite_var(1,3), simplified space_simps]
6.406 + finite_distribution_order(5,6)[OF finite_var(1) YZ]
6.407 + finite_distribution_order(5,6)[OF finite_var(1,3)]
6.408
6.409 + note random_var = finite_var[THEN finite_random_variableD]
6.410 note finite = finite_var(1) YZ finite_var(3) XZ YZX
6.411 - note finite[THEN finite_distribution_finite, simplified space_simps, simp]
6.412
6.413 have order2: "\<And>x y z. \<lbrakk>x \<in> space MX; y \<in> space MY; z \<in> space MZ; joint_distribution X Z {(x, z)} = 0\<rbrakk>
6.414 \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
6.415 unfolding joint_distribution_commute_singleton[of X]
6.416 unfolding joint_distribution_assoc_singleton[symmetric]
6.417 using finite_distribution_order(6)[OF finite_var(2) ZX]
6.418 - by (auto simp: space_simps)
6.419 + by auto
6.420
6.421 - have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * ?YZdZ y z))) =
6.422 + have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * (?YZ y z / ?Z z)))) =
6.423 (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * (log b (?XYZ x y z / (?X x * ?YZ y z)) - log b (?XZ x z / (?X x * ?Z z))))"
6.424 (is "(\<Sum>(x, y, z)\<in>?S. ?L x y z) = (\<Sum>(x, y, z)\<in>?S. ?R x y z)")
6.425 proof (safe intro!: setsum_cong)
6.426 fix x y z assume space: "x \<in> space MX" "y \<in> space MY" "z \<in> space MZ"
6.427 - then have *: "?XYZ x y z / (?XZ x z * ?YZdZ y z) =
6.428 - (?XYZ x y z / (?X x * ?YZ y z)) / (?XZ x z / (?X x * ?Z z))"
6.429 - using order1(3)
6.430 - by (auto simp: real_of_pextreal_mult[symmetric] real_of_pextreal_eq_0)
6.431 show "?L x y z = ?R x y z"
6.432 proof cases
6.433 assume "?XYZ x y z \<noteq> 0"
6.434 - with space b_gt_1 order1 order2 show ?thesis unfolding *
6.435 - by (subst log_divide)
6.436 - (auto simp: zero_less_divide_iff zero_less_real_of_pextreal
6.437 - real_of_pextreal_eq_0 zero_less_mult_iff)
6.438 + with space have "0 < ?X x" "0 < ?Z z" "0 < ?XZ x z" "0 < ?YZ y z" "0 < ?XYZ x y z"
6.439 + using order1 order2 by (auto simp: less_le)
6.440 + with b_gt_1 show ?thesis
6.441 + by (simp add: log_mult log_divide zero_less_mult_iff zero_less_divide_iff)
6.442 qed simp
6.443 qed
6.444 also have "\<dots> = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
6.445 @@ -649,8 +654,8 @@
6.446 setsum_left_distrib[symmetric]
6.447 unfolding joint_distribution_commute_singleton[of X]
6.448 unfolding joint_distribution_assoc_singleton[symmetric]
6.449 - using setsum_real_joint_distribution_singleton[OF finite_var(2) ZX, unfolded space_simps]
6.450 - by (intro setsum_cong refl) simp
6.451 + using setsum_joint_distribution_singleton[OF finite_var(2) ZX]
6.452 + by (intro setsum_cong refl) (simp add: space_pair_measure)
6.453 also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
6.454 (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z))) =
6.455 conditional_mutual_information b MX MY MZ X Y Z"
6.456 @@ -664,11 +669,11 @@
6.457 lemma (in information_space) conditional_mutual_information_eq:
6.458 assumes "simple_function M X" "simple_function M Y" "simple_function M Z"
6.459 shows "\<I>(X;Y|Z) = (\<Sum>(x, y, z) \<in> X`space M \<times> Y`space M \<times> Z`space M.
6.460 - real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) *
6.461 - log b (real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) /
6.462 - (real (joint_distribution X Z {(x, z)}) * real (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
6.463 - using conditional_mutual_information_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]]
6.464 - by simp
6.465 + distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
6.466 + log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
6.467 + (joint_distribution X Z {(x, z)} * joint_distribution Y Z {(y,z)} / distribution Z {z})))"
6.468 + by (subst conditional_mutual_information_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
6.469 + simp
6.470
6.471 lemma (in information_space) conditional_mutual_information_eq_mutual_information:
6.472 assumes X: "simple_function M X" and Y: "simple_function M Y"
6.473 @@ -683,10 +688,10 @@
6.474 qed
6.475
6.476 lemma (in prob_space) distribution_unit[simp]: "distribution (\<lambda>x. ()) {()} = 1"
6.477 - unfolding distribution_def using measure_space_1 by auto
6.478 + unfolding distribution_def using prob_space by auto
6.479
6.480 lemma (in prob_space) joint_distribution_unit[simp]: "distribution (\<lambda>x. (X x, ())) {(a, ())} = distribution X {a}"
6.481 - unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>])
6.482 + unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
6.483
6.484 lemma (in prob_space) setsum_distribution:
6.485 assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
6.486 @@ -695,12 +700,13 @@
6.487
6.488 lemma (in prob_space) setsum_real_distribution:
6.489 fixes MX :: "('c, 'd) measure_space_scheme"
6.490 - assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. real (distribution X {a})) = 1"
6.491 - using setsum_real_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV, measure = undefined \<rparr>" "\<lambda>x. ()" "{()}"]
6.492 - using sigma_algebra_Pow[of "UNIV::unit set" "\<lparr> measure = undefined \<rparr>"] by simp
6.493 + assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
6.494 + using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV, measure = undefined \<rparr>" "\<lambda>x. ()" "{()}"]
6.495 + using sigma_algebra_Pow[of "UNIV::unit set" "\<lparr> measure = undefined \<rparr>"]
6.496 + by auto
6.497
6.498 lemma (in information_space) conditional_mutual_information_generic_positive:
6.499 - assumes "finite_random_variable MX X" and "finite_random_variable MY Y" and "finite_random_variable MZ Z"
6.500 + assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" and Z: "finite_random_variable MZ Z"
6.501 shows "0 \<le> conditional_mutual_information b MX MY MZ X Y Z"
6.502 proof (cases "space MX \<times> space MY \<times> space MZ = {}")
6.503 case True show ?thesis
6.504 @@ -708,43 +714,35 @@
6.505 by simp
6.506 next
6.507 case False
6.508 - let "?dXYZ A" = "real (distribution (\<lambda>x. (X x, Y x, Z x)) A)"
6.509 - let "?dXZ A" = "real (joint_distribution X Z A)"
6.510 - let "?dYZ A" = "real (joint_distribution Y Z A)"
6.511 - let "?dX A" = "real (distribution X A)"
6.512 - let "?dZ A" = "real (distribution Z A)"
6.513 + let ?dXYZ = "distribution (\<lambda>x. (X x, Y x, Z x))"
6.514 + let ?dXZ = "joint_distribution X Z"
6.515 + let ?dYZ = "joint_distribution Y Z"
6.516 + let ?dX = "distribution X"
6.517 + let ?dZ = "distribution Z"
6.518 let ?M = "space MX \<times> space MY \<times> space MZ"
6.519
6.520 - have split_beta: "\<And>f. split f = (\<lambda>x. f (fst x) (snd x))" by (simp add: fun_eq_iff)
6.521 -
6.522 - note space_simps = space_pair_measure space_sigma algebra.simps
6.523 -
6.524 - note finite_var = assms
6.525 - note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
6.526 - note XZ = finite_random_variable_pairI[OF finite_var(1,3)]
6.527 - note ZX = finite_random_variable_pairI[OF finite_var(3,1)]
6.528 - note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
6.529 - note XYZ = finite_random_variable_pairI[OF finite_var(1) YZ]
6.530 - note finite = finite_var(3) YZ XZ XYZ
6.531 - note finite = finite[THEN finite_distribution_finite, simplified space_simps]
6.532 -
6.533 + note YZ = finite_random_variable_pairI[OF Y Z]
6.534 + note XZ = finite_random_variable_pairI[OF X Z]
6.535 + note ZX = finite_random_variable_pairI[OF Z X]
6.536 + note YZ = finite_random_variable_pairI[OF Y Z]
6.537 + note XYZ = finite_random_variable_pairI[OF X YZ]
6.538 + note finite = Z YZ XZ XYZ
6.539 have order: "\<And>x y z. \<lbrakk>x \<in> space MX; y \<in> space MY; z \<in> space MZ; joint_distribution X Z {(x, z)} = 0\<rbrakk>
6.540 \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
6.541 unfolding joint_distribution_commute_singleton[of X]
6.542 unfolding joint_distribution_assoc_singleton[symmetric]
6.543 - using finite_distribution_order(6)[OF finite_var(2) ZX]
6.544 - by (auto simp: space_simps)
6.545 + using finite_distribution_order(6)[OF Y ZX]
6.546 + by auto
6.547
6.548 note order = order
6.549 - finite_distribution_order(5,6)[OF finite_var(1) YZ, simplified space_simps]
6.550 - finite_distribution_order(5,6)[OF finite_var(2,3), simplified space_simps]
6.551 + finite_distribution_order(5,6)[OF X YZ]
6.552 + finite_distribution_order(5,6)[OF Y Z]
6.553
6.554 have "- conditional_mutual_information b MX MY MZ X Y Z = - (\<Sum>(x, y, z) \<in> ?M. ?dXYZ {(x, y, z)} *
6.555 log b (?dXYZ {(x, y, z)} / (?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})))"
6.556 - unfolding conditional_mutual_information_generic_eq[OF assms] neg_equal_iff_equal
6.557 - by (intro setsum_cong) (auto intro!: arg_cong[where f="log b"] simp: real_of_pextreal_mult[symmetric])
6.558 + unfolding conditional_mutual_information_generic_eq[OF assms] neg_equal_iff_equal by auto
6.559 also have "\<dots> \<le> log b (\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})"
6.560 - unfolding split_beta
6.561 + unfolding split_beta'
6.562 proof (rule log_setsum_divide)
6.563 show "?M \<noteq> {}" using False by simp
6.564 show "1 < b" using b_gt_1 .
6.565 @@ -757,33 +755,31 @@
6.566 unfolding setsum_commute[of _ "space MY"]
6.567 unfolding setsum_commute[of _ "space MZ"]
6.568 by (simp_all add: space_pair_measure
6.569 - setsum_real_joint_distribution_singleton[OF `finite_random_variable MX X` YZ]
6.570 - setsum_real_joint_distribution_singleton[OF `finite_random_variable MY Y` finite_var(3)]
6.571 - setsum_real_distribution[OF `finite_random_variable MZ Z`])
6.572 + setsum_joint_distribution_singleton[OF X YZ]
6.573 + setsum_joint_distribution_singleton[OF Y Z]
6.574 + setsum_distribution[OF Z])
6.575
6.576 fix x assume "x \<in> ?M"
6.577 let ?x = "(fst x, fst (snd x), snd (snd x))"
6.578
6.579 - show "0 \<le> ?dXYZ {?x}" using real_pextreal_nonneg .
6.580 - show "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
6.581 - by (simp add: real_pextreal_nonneg mult_nonneg_nonneg divide_nonneg_nonneg)
6.582 + show "0 \<le> ?dXYZ {?x}"
6.583 + "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
6.584 + by (simp_all add: mult_nonneg_nonneg divide_nonneg_nonneg)
6.585
6.586 assume *: "0 < ?dXYZ {?x}"
6.587 - with `x \<in> ?M` show "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
6.588 - using finite order
6.589 - by (cases x)
6.590 - (auto simp add: zero_less_real_of_pextreal zero_less_mult_iff zero_less_divide_iff)
6.591 + with `x \<in> ?M` finite order show "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
6.592 + by (cases x) (auto simp add: zero_le_mult_iff zero_le_divide_iff less_le)
6.593 qed
6.594 also have "(\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}) = (\<Sum>z\<in>space MZ. ?dZ {z})"
6.595 apply (simp add: setsum_cartesian_product')
6.596 apply (subst setsum_commute)
6.597 apply (subst (2) setsum_commute)
6.598 by (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric]
6.599 - setsum_real_joint_distribution_singleton[OF finite_var(1,3)]
6.600 - setsum_real_joint_distribution_singleton[OF finite_var(2,3)]
6.601 + setsum_joint_distribution_singleton[OF X Z]
6.602 + setsum_joint_distribution_singleton[OF Y Z]
6.603 intro!: setsum_cong)
6.604 also have "log b (\<Sum>z\<in>space MZ. ?dZ {z}) = 0"
6.605 - unfolding setsum_real_distribution[OF finite_var(3)] by simp
6.606 + unfolding setsum_real_distribution[OF Z] by simp
6.607 finally show ?thesis by simp
6.608 qed
6.609
6.610 @@ -800,57 +796,52 @@
6.611 abbreviation (in information_space)
6.612 conditional_entropy_Pow ("\<H>'(_ | _')") where
6.613 "\<H>(X | Y) \<equiv> conditional_entropy b
6.614 - \<lparr> space = X`space M, sets = Pow (X`space M), measure = distribution X \<rparr>
6.615 - \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = distribution Y \<rparr> X Y"
6.616 + \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
6.617 + \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr> X Y"
6.618
6.619 lemma (in information_space) conditional_entropy_positive:
6.620 "simple_function M X \<Longrightarrow> simple_function M Y \<Longrightarrow> 0 \<le> \<H>(X | Y)"
6.621 unfolding conditional_entropy_def by (auto intro!: conditional_mutual_information_positive)
6.622
6.623 -lemma (in measure_space) empty_measureI: "A = {} \<Longrightarrow> \<mu> A = 0" by simp
6.624 -
6.625 lemma (in information_space) conditional_entropy_generic_eq:
6.626 fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
6.627 assumes MX: "finite_random_variable MX X"
6.628 assumes MZ: "finite_random_variable MZ Z"
6.629 shows "conditional_entropy b MX MZ X Z =
6.630 - (\<Sum>(x, z)\<in>space MX \<times> space MZ.
6.631 - real (joint_distribution X Z {(x, z)}) *
6.632 - log b (real (joint_distribution X Z {(x, z)}) / real (distribution Z {z})))"
6.633 + joint_distribution X Z {(x, z)} * log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
6.634 proof -
6.635 interpret MX: finite_sigma_algebra MX using MX by simp
6.636 interpret MZ: finite_sigma_algebra MZ using MZ by simp
6.637 let "?XXZ x y z" = "joint_distribution X (\<lambda>x. (X x, Z x)) {(x, y, z)}"
6.638 let "?XZ x z" = "joint_distribution X Z {(x, z)}"
6.639 let "?Z z" = "distribution Z {z}"
6.640 - let "?f x y z" = "log b (real (?XXZ x y z) / (real (?XZ x z) * real (?XZ y z / ?Z z)))"
6.641 + let "?f x y z" = "log b (?XXZ x y z * ?Z z / (?XZ x z * ?XZ y z))"
6.642 { fix x z have "?XXZ x x z = ?XZ x z"
6.643 - unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>]) }
6.644 + unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) }
6.645 note this[simp]
6.646 { fix x x' :: 'c and z assume "x' \<noteq> x"
6.647 then have "?XXZ x x' z = 0"
6.648 - by (auto simp: distribution_def intro!: arg_cong[where f=\<mu>] empty_measureI) }
6.649 + by (auto simp: distribution_def empty_measure'[symmetric]
6.650 + simp del: empty_measure' intro!: arg_cong[where f=\<mu>']) }
6.651 note this[simp]
6.652 { fix x x' z assume *: "x \<in> space MX" "z \<in> space MZ"
6.653 - then have "(\<Sum>x'\<in>space MX. real (?XXZ x x' z) * ?f x x' z)
6.654 - = (\<Sum>x'\<in>space MX. if x = x' then real (?XZ x z) * ?f x x z else 0)"
6.655 + then have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z)
6.656 + = (\<Sum>x'\<in>space MX. if x = x' then ?XZ x z * ?f x x z else 0)"
6.657 by (auto intro!: setsum_cong)
6.658 - also have "\<dots> = real (?XZ x z) * ?f x x z"
6.659 + also have "\<dots> = ?XZ x z * ?f x x z"
6.660 using `x \<in> space MX` by (simp add: setsum_cases[OF MX.finite_space])
6.661 - also have "\<dots> = real (?XZ x z) * log b (real (?Z z) / real (?XZ x z))"
6.662 - by (auto simp: real_of_pextreal_mult[symmetric])
6.663 - also have "\<dots> = - real (?XZ x z) * log b (real (?XZ x z) / real (?Z z))"
6.664 - using assms[THEN finite_distribution_finite]
6.665 + also have "\<dots> = ?XZ x z * log b (?Z z / ?XZ x z)" by auto
6.666 + also have "\<dots> = - ?XZ x z * log b (?XZ x z / ?Z z)"
6.667 using finite_distribution_order(6)[OF MX MZ]
6.668 - by (auto simp: log_simps field_simps zero_less_mult_iff zero_less_real_of_pextreal real_of_pextreal_eq_0)
6.669 - finally have "(\<Sum>x'\<in>space MX. real (?XXZ x x' z) * ?f x x' z) =
6.670 - - real (?XZ x z) * log b (real (?XZ x z) / real (?Z z))" . }
6.671 + by (auto simp: log_simps field_simps zero_less_mult_iff)
6.672 + finally have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z) = - ?XZ x z * log b (?XZ x z / ?Z z)" . }
6.673 note * = this
6.674 show ?thesis
6.675 unfolding conditional_entropy_def
6.676 unfolding conditional_mutual_information_generic_eq[OF MX MX MZ]
6.677 by (auto simp: setsum_cartesian_product' setsum_negf[symmetric]
6.678 - setsum_commute[of _ "space MZ"] * simp del: divide_pextreal_def
6.679 + setsum_commute[of _ "space MZ"] *
6.680 intro!: setsum_cong)
6.681 qed
6.682
6.683 @@ -858,29 +849,27 @@
6.684 assumes "simple_function M X" "simple_function M Z"
6.685 shows "\<H>(X | Z) =
6.686 - (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
6.687 - real (joint_distribution X Z {(x, z)}) *
6.688 - log b (real (joint_distribution X Z {(x, z)}) / real (distribution Z {z})))"
6.689 - using conditional_entropy_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]]
6.690 - by simp
6.691 + joint_distribution X Z {(x, z)} *
6.692 + log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
6.693 + by (subst conditional_entropy_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
6.694 + simp
6.695
6.696 lemma (in information_space) conditional_entropy_eq_ce_with_hypothesis:
6.697 assumes X: "simple_function M X" and Y: "simple_function M Y"
6.698 shows "\<H>(X | Y) =
6.699 - -(\<Sum>y\<in>Y`space M. real (distribution Y {y}) *
6.700 - (\<Sum>x\<in>X`space M. real (joint_distribution X Y {(x,y)}) / real (distribution Y {(y)}) *
6.701 - log b (real (joint_distribution X Y {(x,y)}) / real (distribution Y {(y)}))))"
6.702 + -(\<Sum>y\<in>Y`space M. distribution Y {y} *
6.703 + (\<Sum>x\<in>X`space M. joint_distribution X Y {(x,y)} / distribution Y {(y)} *
6.704 + log b (joint_distribution X Y {(x,y)} / distribution Y {(y)})))"
6.705 unfolding conditional_entropy_eq[OF assms]
6.706 - using finite_distribution_finite[OF finite_random_variable_pairI[OF assms[THEN simple_function_imp_finite_random_variable]]]
6.707 using finite_distribution_order(5,6)[OF assms[THEN simple_function_imp_finite_random_variable]]
6.708 - using finite_distribution_finite[OF Y[THEN simple_function_imp_finite_random_variable]]
6.709 - by (auto simp: setsum_cartesian_product' setsum_commute[of _ "Y`space M"] setsum_right_distrib real_of_pextreal_eq_0
6.710 + by (auto simp: setsum_cartesian_product' setsum_commute[of _ "Y`space M"] setsum_right_distrib
6.711 intro!: setsum_cong)
6.712
6.713 lemma (in information_space) conditional_entropy_eq_cartesian_product:
6.714 assumes "simple_function M X" "simple_function M Y"
6.715 shows "\<H>(X | Y) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
6.716 - real (joint_distribution X Y {(x,y)}) *
6.717 - log b (real (joint_distribution X Y {(x,y)}) / real (distribution Y {y})))"
6.718 + joint_distribution X Y {(x,y)} *
6.719 + log b (joint_distribution X Y {(x,y)} / distribution Y {y}))"
6.720 unfolding conditional_entropy_eq[OF assms]
6.721 by (auto intro!: setsum_cong simp: setsum_cartesian_product')
6.722
6.723 @@ -890,24 +879,22 @@
6.724 assumes X: "simple_function M X" and Z: "simple_function M Z"
6.725 shows "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
6.726 proof -
6.727 - let "?XZ x z" = "real (joint_distribution X Z {(x, z)})"
6.728 - let "?Z z" = "real (distribution Z {z})"
6.729 - let "?X x" = "real (distribution X {x})"
6.730 + let "?XZ x z" = "joint_distribution X Z {(x, z)}"
6.731 + let "?Z z" = "distribution Z {z}"
6.732 + let "?X x" = "distribution X {x}"
6.733 note fX = X[THEN simple_function_imp_finite_random_variable]
6.734 note fZ = Z[THEN simple_function_imp_finite_random_variable]
6.735 - note fX[THEN finite_distribution_finite, simp] and fZ[THEN finite_distribution_finite, simp]
6.736 note finite_distribution_order[OF fX fZ, simp]
6.737 { fix x z assume "x \<in> X`space M" "z \<in> Z`space M"
6.738 have "?XZ x z * log b (?XZ x z / (?X x * ?Z z)) =
6.739 ?XZ x z * log b (?XZ x z / ?Z z) - ?XZ x z * log b (?X x)"
6.740 - by (auto simp: log_simps real_of_pextreal_mult[symmetric] zero_less_mult_iff
6.741 - zero_less_real_of_pextreal field_simps real_of_pextreal_eq_0 abs_mult) }
6.742 + by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
6.743 note * = this
6.744 show ?thesis
6.745 unfolding entropy_eq[OF X] conditional_entropy_eq[OF X Z] mutual_information_eq[OF X Z]
6.746 - using setsum_real_joint_distribution_singleton[OF fZ fX, unfolded joint_distribution_commute_singleton[of Z X]]
6.747 + using setsum_joint_distribution_singleton[OF fZ fX, unfolded joint_distribution_commute_singleton[of Z X]]
6.748 by (simp add: * setsum_cartesian_product' setsum_subtractf setsum_left_distrib[symmetric]
6.749 - setsum_real_distribution)
6.750 + setsum_distribution)
6.751 qed
6.752
6.753 lemma (in information_space) conditional_entropy_less_eq_entropy:
6.754 @@ -923,21 +910,19 @@
6.755 assumes X: "simple_function M X" and Y: "simple_function M Y"
6.756 shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
6.757 proof -
6.758 - let "?XY x y" = "real (joint_distribution X Y {(x, y)})"
6.759 - let "?Y y" = "real (distribution Y {y})"
6.760 - let "?X x" = "real (distribution X {x})"
6.761 + let "?XY x y" = "joint_distribution X Y {(x, y)}"
6.762 + let "?Y y" = "distribution Y {y}"
6.763 + let "?X x" = "distribution X {x}"
6.764 note fX = X[THEN simple_function_imp_finite_random_variable]
6.765 note fY = Y[THEN simple_function_imp_finite_random_variable]
6.766 - note fX[THEN finite_distribution_finite, simp] and fY[THEN finite_distribution_finite, simp]
6.767 note finite_distribution_order[OF fX fY, simp]
6.768 { fix x y assume "x \<in> X`space M" "y \<in> Y`space M"
6.769 have "?XY x y * log b (?XY x y / ?X x) =
6.770 ?XY x y * log b (?XY x y) - ?XY x y * log b (?X x)"
6.771 - by (auto simp: log_simps real_of_pextreal_mult[symmetric] zero_less_mult_iff
6.772 - zero_less_real_of_pextreal field_simps real_of_pextreal_eq_0 abs_mult) }
6.773 + by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
6.774 note * = this
6.775 show ?thesis
6.776 - using setsum_real_joint_distribution_singleton[OF fY fX]
6.777 + using setsum_joint_distribution_singleton[OF fY fX]
6.778 unfolding entropy_eq[OF X] conditional_entropy_eq_cartesian_product[OF Y X] entropy_eq_cartesian_product[OF X Y]
6.779 unfolding joint_distribution_commute_singleton[of Y X] setsum_commute[of _ "X`space M"]
6.780 by (simp add: * setsum_subtractf setsum_left_distrib[symmetric])
6.781 @@ -1063,23 +1048,21 @@
6.782 assumes svi: "subvimage (space M) X P"
6.783 shows "\<H>(X) = \<H>(P) + \<H>(X|P)"
6.784 proof -
6.785 - let "?XP x p" = "real (joint_distribution X P {(x, p)})"
6.786 - let "?X x" = "real (distribution X {x})"
6.787 - let "?P p" = "real (distribution P {p})"
6.788 + let "?XP x p" = "joint_distribution X P {(x, p)}"
6.789 + let "?X x" = "distribution X {x}"
6.790 + let "?P p" = "distribution P {p}"
6.791 note fX = sf(1)[THEN simple_function_imp_finite_random_variable]
6.792 note fP = sf(2)[THEN simple_function_imp_finite_random_variable]
6.793 - note fX[THEN finite_distribution_finite, simp] and fP[THEN finite_distribution_finite, simp]
6.794 note finite_distribution_order[OF fX fP, simp]
6.795 - have "(\<Sum>x\<in>X ` space M. real (distribution X {x}) * log b (real (distribution X {x}))) =
6.796 - (\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M.
6.797 - real (joint_distribution X P {(x, y)}) * log b (real (joint_distribution X P {(x, y)})))"
6.798 + have "(\<Sum>x\<in>X ` space M. ?X x * log b (?X x)) =
6.799 + (\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M. ?XP x y * log b (?XP x y))"
6.800 proof (subst setsum_image_split[OF svi],
6.801 safe intro!: setsum_mono_zero_cong_left imageI)
6.802 show "finite (X ` space M)" "finite (X ` space M)" "finite (P ` space M)"
6.803 using sf unfolding simple_function_def by auto
6.804 next
6.805 fix p x assume in_space: "p \<in> space M" "x \<in> space M"
6.806 - assume "real (joint_distribution X P {(X x, P p)}) * log b (real (joint_distribution X P {(X x, P p)})) \<noteq> 0"
6.807 + assume "?XP (X x) (P p) * log b (?XP (X x) (P p)) \<noteq> 0"
6.808 hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M \<noteq> {}" by (auto simp: distribution_def)
6.809 with svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
6.810 show "x \<in> P -` {P p}" by auto
6.811 @@ -1091,20 +1074,16 @@
6.812 by auto
6.813 hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M = X -` {X x} \<inter> space M"
6.814 by auto
6.815 - thus "real (distribution X {X x}) * log b (real (distribution X {X x})) =
6.816 - real (joint_distribution X P {(X x, P p)}) *
6.817 - log b (real (joint_distribution X P {(X x, P p)}))"
6.818 + thus "?X (X x) * log b (?X (X x)) = ?XP (X x) (P p) * log b (?XP (X x) (P p))"
6.819 by (auto simp: distribution_def)
6.820 qed
6.821 - moreover have "\<And>x y. real (joint_distribution X P {(x, y)}) *
6.822 - log b (real (joint_distribution X P {(x, y)}) / real (distribution P {y})) =
6.823 - real (joint_distribution X P {(x, y)}) * log b (real (joint_distribution X P {(x, y)})) -
6.824 - real (joint_distribution X P {(x, y)}) * log b (real (distribution P {y}))"
6.825 + moreover have "\<And>x y. ?XP x y * log b (?XP x y / ?P y) =
6.826 + ?XP x y * log b (?XP x y) - ?XP x y * log b (?P y)"
6.827 by (auto simp add: log_simps zero_less_mult_iff field_simps)
6.828 ultimately show ?thesis
6.829 unfolding sf[THEN entropy_eq] conditional_entropy_eq[OF sf]
6.830 - using setsum_real_joint_distribution_singleton[OF fX fP]
6.831 - by (simp add: setsum_cartesian_product' setsum_subtractf setsum_real_distribution
6.832 + using setsum_joint_distribution_singleton[OF fX fP]
6.833 + by (simp add: setsum_cartesian_product' setsum_subtractf setsum_distribution
6.834 setsum_left_distrib[symmetric] setsum_commute[where B="P`space M"])
6.835 qed
6.836
7.1 --- a/src/HOL/Probability/Lebesgue_Integration.thy Mon Mar 14 14:37:47 2011 +0100
7.2 +++ b/src/HOL/Probability/Lebesgue_Integration.thy Mon Mar 14 14:37:49 2011 +0100
7.3 @@ -6,6 +6,88 @@
7.4 imports Measure Borel_Space
7.5 begin
7.6
7.7 +lemma extreal_indicator_pos[simp,intro]: "0 \<le> (indicator A x ::extreal)"
7.8 + unfolding indicator_def by auto
7.9 +
7.10 +lemma tendsto_real_max:
7.11 + fixes x y :: real
7.12 + assumes "(X ---> x) net"
7.13 + assumes "(Y ---> y) net"
7.14 + shows "((\<lambda>x. max (X x) (Y x)) ---> max x y) net"
7.15 +proof -
7.16 + have *: "\<And>x y :: real. max x y = y + ((x - y) + norm (x - y)) / 2"
7.17 + by (auto split: split_max simp: field_simps)
7.18 + show ?thesis
7.19 + unfolding *
7.20 + by (intro tendsto_add assms tendsto_divide tendsto_norm tendsto_diff) auto
7.21 +qed
7.22 +
7.23 +lemma (in measure_space) measure_Union:
7.24 + assumes "finite S" "S \<subseteq> sets M" "\<And>A B. A \<in> S \<Longrightarrow> B \<in> S \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<inter> B = {}"
7.25 + shows "setsum \<mu> S = \<mu> (\<Union>S)"
7.26 +proof -
7.27 + have "setsum \<mu> S = \<mu> (\<Union>i\<in>S. i)"
7.28 + using assms by (intro measure_setsum[OF `finite S`]) (auto simp: disjoint_family_on_def)
7.29 + also have "\<dots> = \<mu> (\<Union>S)" by (auto intro!: arg_cong[where f=\<mu>])
7.30 + finally show ?thesis .
7.31 +qed
7.32 +
7.33 +lemma (in sigma_algebra) measurable_sets2[intro]:
7.34 + assumes "f \<in> measurable M M'" "g \<in> measurable M M''"
7.35 + and "A \<in> sets M'" "B \<in> sets M''"
7.36 + shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
7.37 +proof -
7.38 + have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
7.39 + by auto
7.40 + then show ?thesis using assms by (auto intro: measurable_sets)
7.41 +qed
7.42 +
7.43 +lemma incseq_extreal: "incseq f \<Longrightarrow> incseq (\<lambda>x. extreal (f x))"
7.44 + unfolding incseq_def by auto
7.45 +
7.46 +lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
7.47 +proof
7.48 + assume "\<forall>n. f n \<le> f (Suc n)" then show "incseq f" by (auto intro!: incseq_SucI)
7.49 +qed (auto simp: incseq_def)
7.50 +
7.51 +lemma borel_measurable_real_floor:
7.52 + "(\<lambda>x::real. real \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
7.53 + unfolding borel.borel_measurable_iff_ge
7.54 +proof (intro allI)
7.55 + fix a :: real
7.56 + { fix x have "a \<le> real \<lfloor>x\<rfloor> \<longleftrightarrow> real \<lceil>a\<rceil> \<le> x"
7.57 + using le_floor_eq[of "\<lceil>a\<rceil>" x] ceiling_le_iff[of a "\<lfloor>x\<rfloor>"]
7.58 + unfolding real_eq_of_int by simp }
7.59 + then have "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} = {real \<lceil>a\<rceil>..}" by auto
7.60 + then show "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} \<in> sets borel" by auto
7.61 +qed
7.62 +
7.63 +lemma measure_preservingD2:
7.64 + "f \<in> measure_preserving A B \<Longrightarrow> f \<in> measurable A B"
7.65 + unfolding measure_preserving_def by auto
7.66 +
7.67 +lemma measure_preservingD3:
7.68 + "f \<in> measure_preserving A B \<Longrightarrow> f \<in> space A \<rightarrow> space B"
7.69 + unfolding measure_preserving_def measurable_def by auto
7.70 +
7.71 +lemma measure_preservingD:
7.72 + "T \<in> measure_preserving A B \<Longrightarrow> X \<in> sets B \<Longrightarrow> measure A (T -` X \<inter> space A) = measure B X"
7.73 + unfolding measure_preserving_def by auto
7.74 +
7.75 +lemma (in sigma_algebra) borel_measurable_real_natfloor[intro, simp]:
7.76 + assumes "f \<in> borel_measurable M"
7.77 + shows "(\<lambda>x. real (natfloor (f x))) \<in> borel_measurable M"
7.78 +proof -
7.79 + have "\<And>x. real (natfloor (f x)) = max 0 (real \<lfloor>f x\<rfloor>)"
7.80 + by (auto simp: max_def natfloor_def)
7.81 + with borel_measurable_max[OF measurable_comp[OF assms borel_measurable_real_floor] borel_measurable_const]
7.82 + show ?thesis by (simp add: comp_def)
7.83 +qed
7.84 +
7.85 +lemma (in measure_space) AE_not_in:
7.86 + assumes N: "N \<in> null_sets" shows "AE x. x \<notin> N"
7.87 + using N by (rule AE_I') auto
7.88 +
7.89 lemma sums_If_finite:
7.90 fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
7.91 assumes finite: "finite {r. P r}"
7.92 @@ -55,8 +137,17 @@
7.93 by (auto intro!: finite_UN simp del: UN_simps simp: simple_function_def)
7.94 qed
7.95
7.96 +lemma (in sigma_algebra) simple_function_measurable2[intro]:
7.97 + assumes "simple_function M f" "simple_function M g"
7.98 + shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
7.99 +proof -
7.100 + have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
7.101 + by auto
7.102 + then show ?thesis using assms[THEN simple_functionD(2)] by auto
7.103 +qed
7.104 +
7.105 lemma (in sigma_algebra) simple_function_indicator_representation:
7.106 - fixes f ::"'a \<Rightarrow> pextreal"
7.107 + fixes f ::"'a \<Rightarrow> extreal"
7.108 assumes f: "simple_function M f" and x: "x \<in> space M"
7.109 shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
7.110 (is "?l = ?r")
7.111 @@ -71,7 +162,7 @@
7.112 qed
7.113
7.114 lemma (in measure_space) simple_function_notspace:
7.115 - "simple_function M (\<lambda>x. h x * indicator (- space M) x::pextreal)" (is "simple_function M ?h")
7.116 + "simple_function M (\<lambda>x. h x * indicator (- space M) x::extreal)" (is "simple_function M ?h")
7.117 proof -
7.118 have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
7.119 hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
7.120 @@ -111,16 +202,22 @@
7.121 qed
7.122
7.123 lemma (in sigma_algebra) simple_function_borel_measurable:
7.124 - fixes f :: "'a \<Rightarrow> 'x::t2_space"
7.125 + fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
7.126 assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
7.127 shows "simple_function M f"
7.128 using assms unfolding simple_function_def
7.129 by (auto intro: borel_measurable_vimage)
7.130
7.131 +lemma (in sigma_algebra) simple_function_eq_borel_measurable:
7.132 + fixes f :: "'a \<Rightarrow> extreal"
7.133 + shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> borel_measurable M"
7.134 + using simple_function_borel_measurable[of f]
7.135 + borel_measurable_simple_function[of f]
7.136 + by (fastsimp simp: simple_function_def)
7.137 +
7.138 lemma (in sigma_algebra) simple_function_const[intro, simp]:
7.139 "simple_function M (\<lambda>x. c)"
7.140 by (auto intro: finite_subset simp: simple_function_def)
7.141 -
7.142 lemma (in sigma_algebra) simple_function_compose[intro, simp]:
7.143 assumes "simple_function M f"
7.144 shows "simple_function M (g \<circ> f)"
7.145 @@ -189,6 +286,7 @@
7.146 and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
7.147 and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
7.148 and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
7.149 + and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
7.150
7.151 lemma (in sigma_algebra) simple_function_setsum[intro, simp]:
7.152 assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
7.153 @@ -197,247 +295,168 @@
7.154 assume "finite P" from this assms show ?thesis by induct auto
7.155 qed auto
7.156
7.157 -lemma (in sigma_algebra) simple_function_le_measurable:
7.158 - assumes "simple_function M f" "simple_function M g"
7.159 - shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
7.160 -proof -
7.161 - have *: "{x \<in> space M. f x \<le> g x} =
7.162 - (\<Union>(F, G)\<in>f`space M \<times> g`space M.
7.163 - if F \<le> G then (f -` {F} \<inter> space M) \<inter> (g -` {G} \<inter> space M) else {})"
7.164 - apply (auto split: split_if_asm)
7.165 - apply (rule_tac x=x in bexI)
7.166 - apply (rule_tac x=x in bexI)
7.167 - by simp_all
7.168 - have **: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow>
7.169 - (f -` {f x} \<inter> space M) \<inter> (g -` {g y} \<inter> space M) \<in> sets M"
7.170 - using assms unfolding simple_function_def by auto
7.171 - have "finite (f`space M \<times> g`space M)"
7.172 - using assms unfolding simple_function_def by auto
7.173 - thus ?thesis unfolding *
7.174 - apply (rule finite_UN)
7.175 - using assms unfolding simple_function_def
7.176 - by (auto intro!: **)
7.177 -qed
7.178 +lemma (in sigma_algebra)
7.179 + fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
7.180 + shows simple_function_extreal[intro, simp]: "simple_function M (\<lambda>x. extreal (f x))"
7.181 + by (auto intro!: simple_function_compose1[OF sf])
7.182 +
7.183 +lemma (in sigma_algebra)
7.184 + fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
7.185 + shows simple_function_real_of_nat[intro, simp]: "simple_function M (\<lambda>x. real (f x))"
7.186 + by (auto intro!: simple_function_compose1[OF sf])
7.187
7.188 lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence:
7.189 - fixes u :: "'a \<Rightarrow> pextreal"
7.190 + fixes u :: "'a \<Rightarrow> extreal"
7.191 assumes u: "u \<in> borel_measurable M"
7.192 - shows "\<exists>f. (\<forall>i. simple_function M (f i) \<and> (\<forall>x\<in>space M. f i x \<noteq> \<omega>)) \<and> f \<up> u"
7.193 + shows "\<exists>f. incseq f \<and> (\<forall>i. \<infinity> \<notin> range (f i) \<and> simple_function M (f i)) \<and>
7.194 + (\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)"
7.195 proof -
7.196 - have "\<exists>f. \<forall>x j. (of_nat j \<le> u x \<longrightarrow> f x j = j*2^j) \<and>
7.197 - (u x < of_nat j \<longrightarrow> of_nat (f x j) \<le> u x * 2^j \<and> u x * 2^j < of_nat (Suc (f x j)))"
7.198 - (is "\<exists>f. \<forall>x j. ?P x j (f x j)")
7.199 - proof(rule choice, rule, rule choice, rule)
7.200 - fix x j show "\<exists>n. ?P x j n"
7.201 - proof cases
7.202 - assume *: "u x < of_nat j"
7.203 - then obtain r where r: "u x = Real r" "0 \<le> r" by (cases "u x") auto
7.204 - from reals_Archimedean6a[of "r * 2^j"]
7.205 - obtain n :: nat where "real n \<le> r * 2 ^ j" "r * 2 ^ j < real (Suc n)"
7.206 - using `0 \<le> r` by (auto simp: zero_le_power zero_le_mult_iff)
7.207 - thus ?thesis using r * by (auto intro!: exI[of _ n])
7.208 - qed auto
7.209 - qed
7.210 - then obtain f where top: "\<And>j x. of_nat j \<le> u x \<Longrightarrow> f x j = j*2^j" and
7.211 - upper: "\<And>j x. u x < of_nat j \<Longrightarrow> of_nat (f x j) \<le> u x * 2^j" and
7.212 - lower: "\<And>j x. u x < of_nat j \<Longrightarrow> u x * 2^j < of_nat (Suc (f x j))" by blast
7.213 -
7.214 - { fix j x P
7.215 - assume 1: "of_nat j \<le> u x \<Longrightarrow> P (j * 2^j)"
7.216 - assume 2: "\<And>k. \<lbrakk> u x < of_nat j ; of_nat k \<le> u x * 2^j ; u x * 2^j < of_nat (Suc k) \<rbrakk> \<Longrightarrow> P k"
7.217 - have "P (f x j)"
7.218 - proof cases
7.219 - assume "of_nat j \<le> u x" thus "P (f x j)"
7.220 - using top[of j x] 1 by auto
7.221 - next
7.222 - assume "\<not> of_nat j \<le> u x"
7.223 - hence "u x < of_nat j" "of_nat (f x j) \<le> u x * 2^j" "u x * 2^j < of_nat (Suc (f x j))"
7.224 - using upper lower by auto
7.225 - from 2[OF this] show "P (f x j)" .
7.226 - qed }
7.227 - note fI = this
7.228 -
7.229 - { fix j x
7.230 - have "f x j = j * 2 ^ j \<longleftrightarrow> of_nat j \<le> u x"
7.231 - by (rule fI, simp, cases "u x") (auto split: split_if_asm) }
7.232 - note f_eq = this
7.233 -
7.234 - { fix j x
7.235 - have "f x j \<le> j * 2 ^ j"
7.236 - proof (rule fI)
7.237 - fix k assume *: "u x < of_nat j"
7.238 - assume "of_nat k \<le> u x * 2 ^ j"
7.239 - also have "\<dots> \<le> of_nat (j * 2^j)"
7.240 - using * by (cases "u x") (auto simp: zero_le_mult_iff)
7.241 - finally show "k \<le> j*2^j" by (auto simp del: real_of_nat_mult)
7.242 - qed simp }
7.243 + def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else natfloor (real (u x) * 2 ^ i)"
7.244 + { fix x j have "f x j \<le> j * 2 ^ j" unfolding f_def
7.245 + proof (split split_if, intro conjI impI)
7.246 + assume "\<not> real j \<le> u x"
7.247 + then have "natfloor (real (u x) * 2 ^ j) \<le> natfloor (j * 2 ^ j)"
7.248 + by (cases "u x") (auto intro!: natfloor_mono simp: mult_nonneg_nonneg)
7.249 + moreover have "real (natfloor (j * 2 ^ j)) \<le> j * 2^j"
7.250 + by (intro real_natfloor_le) (auto simp: mult_nonneg_nonneg)
7.251 + ultimately show "natfloor (real (u x) * 2 ^ j) \<le> j * 2 ^ j"
7.252 + unfolding real_of_nat_le_iff by auto
7.253 + qed auto }
7.254 note f_upper = this
7.255
7.256 - let "?g j x" = "of_nat (f x j) / 2^j :: pextreal"
7.257 - show ?thesis unfolding simple_function_def isoton_fun_expand unfolding isoton_def
7.258 - proof (safe intro!: exI[of _ ?g])
7.259 - fix j
7.260 - have *: "?g j ` space M \<subseteq> (\<lambda>x. of_nat x / 2^j) ` {..j * 2^j}"
7.261 - using f_upper by auto
7.262 - thus "finite (?g j ` space M)" by (rule finite_subset) auto
7.263 - next
7.264 - fix j t assume "t \<in> space M"
7.265 - have **: "?g j -` {?g j t} \<inter> space M = {x \<in> space M. f t j = f x j}"
7.266 - by (auto simp: power_le_zero_eq Real_eq_Real mult_le_0_iff)
7.267 + have real_f:
7.268 + "\<And>i x. real (f x i) = (if real i \<le> u x then i * 2 ^ i else real (natfloor (real (u x) * 2 ^ i)))"
7.269 + unfolding f_def by auto
7.270
7.271 - show "?g j -` {?g j t} \<inter> space M \<in> sets M"
7.272 - proof cases
7.273 - assume "of_nat j \<le> u t"
7.274 - hence "?g j -` {?g j t} \<inter> space M = {x\<in>space M. of_nat j \<le> u x}"
7.275 - unfolding ** f_eq[symmetric] by auto
7.276 - thus "?g j -` {?g j t} \<inter> space M \<in> sets M"
7.277 - using u by auto
7.278 - next
7.279 - assume not_t: "\<not> of_nat j \<le> u t"
7.280 - hence less: "f t j < j*2^j" using f_eq[symmetric] `f t j \<le> j*2^j` by auto
7.281 - have split_vimage: "?g j -` {?g j t} \<inter> space M =
7.282 - {x\<in>space M. of_nat (f t j)/2^j \<le> u x} \<inter> {x\<in>space M. u x < of_nat (Suc (f t j))/2^j}"
7.283 - unfolding **
7.284 - proof safe
7.285 - fix x assume [simp]: "f t j = f x j"
7.286 - have *: "\<not> of_nat j \<le> u x" using not_t f_eq[symmetric] by simp
7.287 - hence "of_nat (f t j) \<le> u x * 2^j \<and> u x * 2^j < of_nat (Suc (f t j))"
7.288 - using upper lower by auto
7.289 - hence "of_nat (f t j) / 2^j \<le> u x \<and> u x < of_nat (Suc (f t j))/2^j" using *
7.290 - by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps)
7.291 - thus "of_nat (f t j) / 2^j \<le> u x" "u x < of_nat (Suc (f t j))/2^j" by auto
7.292 + let "?g j x" = "real (f x j) / 2^j :: extreal"
7.293 + show ?thesis
7.294 + proof (intro exI[of _ ?g] conjI allI ballI)
7.295 + fix i
7.296 + have "simple_function M (\<lambda>x. real (f x i))"
7.297 + proof (intro simple_function_borel_measurable)
7.298 + show "(\<lambda>x. real (f x i)) \<in> borel_measurable M"
7.299 + using u by (auto intro!: measurable_If simp: real_f)
7.300 + have "(\<lambda>x. real (f x i))`space M \<subseteq> real`{..i*2^i}"
7.301 + using f_upper[of _ i] by auto
7.302 + then show "finite ((\<lambda>x. real (f x i))`space M)"
7.303 + by (rule finite_subset) auto
7.304 + qed
7.305 + then show "simple_function M (?g i)"
7.306 + by (auto intro: simple_function_extreal simple_function_div)
7.307 + next
7.308 + show "incseq ?g"
7.309 + proof (intro incseq_extreal incseq_SucI le_funI)
7.310 + fix x and i :: nat
7.311 + have "f x i * 2 \<le> f x (Suc i)" unfolding f_def
7.312 + proof ((split split_if)+, intro conjI impI)
7.313 + assume "extreal (real i) \<le> u x" "\<not> extreal (real (Suc i)) \<le> u x"
7.314 + then show "i * 2 ^ i * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)"
7.315 + by (cases "u x") (auto intro!: le_natfloor)
7.316 next
7.317 - fix x
7.318 - assume "of_nat (f t j) / 2^j \<le> u x" "u x < of_nat (Suc (f t j))/2^j"
7.319 - hence "of_nat (f t j) \<le> u x * 2 ^ j \<and> u x * 2 ^ j < of_nat (Suc (f t j))"
7.320 - by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps)
7.321 - hence 1: "of_nat (f t j) \<le> u x * 2 ^ j" and 2: "u x * 2 ^ j < of_nat (Suc (f t j))" by auto
7.322 - note 2
7.323 - also have "\<dots> \<le> of_nat (j*2^j)"
7.324 - using less by (auto simp: zero_le_mult_iff simp del: real_of_nat_mult)
7.325 - finally have bound_ux: "u x < of_nat j"
7.326 - by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq)
7.327 - show "f t j = f x j"
7.328 - proof (rule antisym)
7.329 - from 1 lower[OF bound_ux]
7.330 - show "f t j \<le> f x j" by (cases "u x") (auto split: split_if_asm)
7.331 - from upper[OF bound_ux] 2
7.332 - show "f x j \<le> f t j" by (cases "u x") (auto split: split_if_asm)
7.333 + assume "\<not> extreal (real i) \<le> u x" "extreal (real (Suc i)) \<le> u x"
7.334 + then show "natfloor (real (u x) * 2 ^ i) * 2 \<le> Suc i * 2 ^ Suc i"
7.335 + by (cases "u x") auto
7.336 + next
7.337 + assume "\<not> extreal (real i) \<le> u x" "\<not> extreal (real (Suc i)) \<le> u x"
7.338 + have "natfloor (real (u x) * 2 ^ i) * 2 = natfloor (real (u x) * 2 ^ i) * natfloor 2"
7.339 + by simp
7.340 + also have "\<dots> \<le> natfloor (real (u x) * 2 ^ i * 2)"
7.341 + proof cases
7.342 + assume "0 \<le> u x" then show ?thesis
7.343 + by (intro le_mult_natfloor) (cases "u x", auto intro!: mult_nonneg_nonneg)
7.344 + next
7.345 + assume "\<not> 0 \<le> u x" then show ?thesis
7.346 + by (cases "u x") (auto simp: natfloor_neg mult_nonpos_nonneg)
7.347 qed
7.348 - qed
7.349 - show ?thesis unfolding split_vimage using u by auto
7.350 + also have "\<dots> = natfloor (real (u x) * 2 ^ Suc i)"
7.351 + by (simp add: ac_simps)
7.352 + finally show "natfloor (real (u x) * 2 ^ i) * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)" .
7.353 + qed simp
7.354 + then show "?g i x \<le> ?g (Suc i) x"
7.355 + by (auto simp: field_simps)
7.356 qed
7.357 next
7.358 - fix j t assume "?g j t = \<omega>" thus False by (auto simp: power_le_zero_eq)
7.359 - next
7.360 - fix t
7.361 - { fix i
7.362 - have "f t i * 2 \<le> f t (Suc i)"
7.363 - proof (rule fI)
7.364 - assume "of_nat (Suc i) \<le> u t"
7.365 - hence "of_nat i \<le> u t" by (cases "u t") auto
7.366 - thus "f t i * 2 \<le> Suc i * 2 ^ Suc i" unfolding f_eq[symmetric] by simp
7.367 - next
7.368 - fix k
7.369 - assume *: "u t * 2 ^ Suc i < of_nat (Suc k)"
7.370 - show "f t i * 2 \<le> k"
7.371 - proof (rule fI)
7.372 - assume "of_nat i \<le> u t"
7.373 - hence "of_nat (i * 2^Suc i) \<le> u t * 2^Suc i"
7.374 - by (cases "u t") (auto simp: zero_le_mult_iff power_le_zero_eq)
7.375 - also have "\<dots> < of_nat (Suc k)" using * by auto
7.376 - finally show "i * 2 ^ i * 2 \<le> k"
7.377 - by (auto simp del: real_of_nat_mult)
7.378 - next
7.379 - fix j assume "of_nat j \<le> u t * 2 ^ i"
7.380 - with * show "j * 2 \<le> k" by (cases "u t") (auto simp: zero_le_mult_iff power_le_zero_eq)
7.381 + fix x show "(SUP i. ?g i x) = max 0 (u x)"
7.382 + proof (rule extreal_SUPI)
7.383 + fix i show "?g i x \<le> max 0 (u x)" unfolding max_def f_def
7.384 + by (cases "u x") (auto simp: field_simps real_natfloor_le natfloor_neg
7.385 + mult_nonpos_nonneg mult_nonneg_nonneg)
7.386 + next
7.387 + fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> ?g i x \<le> y"
7.388 + have "\<And>i. 0 \<le> ?g i x" by (auto simp: divide_nonneg_pos)
7.389 + from order_trans[OF this *] have "0 \<le> y" by simp
7.390 + show "max 0 (u x) \<le> y"
7.391 + proof (cases y)
7.392 + case (real r)
7.393 + with * have *: "\<And>i. f x i \<le> r * 2^i" by (auto simp: divide_le_eq)
7.394 + from real_arch_lt[of r] * have "u x \<noteq> \<infinity>" by (auto simp: f_def) (metis less_le_not_le)
7.395 + then have "\<exists>p. max 0 (u x) = extreal p \<and> 0 \<le> p" by (cases "u x") (auto simp: max_def)
7.396 + then guess p .. note ux = this
7.397 + obtain m :: nat where m: "p < real m" using real_arch_lt ..
7.398 + have "p \<le> r"
7.399 + proof (rule ccontr)
7.400 + assume "\<not> p \<le> r"
7.401 + with LIMSEQ_inverse_realpow_zero[unfolded LIMSEQ_iff, rule_format, of 2 "p - r"]
7.402 + obtain N where "\<forall>n\<ge>N. r * 2^n < p * 2^n - 1" by (auto simp: inverse_eq_divide field_simps)
7.403 + then have "r * 2^max N m < p * 2^max N m - 1" by simp
7.404 + moreover
7.405 + have "real (natfloor (p * 2 ^ max N m)) \<le> r * 2 ^ max N m"
7.406 + using *[of "max N m"] m unfolding real_f using ux
7.407 + by (cases "0 \<le> u x") (simp_all add: max_def mult_nonneg_nonneg split: split_if_asm)
7.408 + then have "p * 2 ^ max N m - 1 < r * 2 ^ max N m"
7.409 + by (metis real_natfloor_gt_diff_one less_le_trans)
7.410 + ultimately show False by auto
7.411 qed
7.412 - qed
7.413 - thus "?g i t \<le> ?g (Suc i) t"
7.414 - by (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps) }
7.415 - hence mono: "mono (\<lambda>i. ?g i t)" unfolding mono_iff_le_Suc by auto
7.416 + then show "max 0 (u x) \<le> y" using real ux by simp
7.417 + qed (insert `0 \<le> y`, auto)
7.418 + qed
7.419 + qed (auto simp: divide_nonneg_pos)
7.420 +qed
7.421
7.422 - show "(SUP j. of_nat (f t j) / 2 ^ j) = u t"
7.423 - proof (rule pextreal_SUPI)
7.424 - fix j show "of_nat (f t j) / 2 ^ j \<le> u t"
7.425 - proof (rule fI)
7.426 - assume "of_nat j \<le> u t" thus "of_nat (j * 2 ^ j) / 2 ^ j \<le> u t"
7.427 - by (cases "u t") (auto simp: power_le_zero_eq divide_real_def[symmetric] field_simps)
7.428 - next
7.429 - fix k assume "of_nat k \<le> u t * 2 ^ j"
7.430 - thus "of_nat k / 2 ^ j \<le> u t"
7.431 - by (cases "u t")
7.432 - (auto simp: power_le_zero_eq divide_real_def[symmetric] field_simps zero_le_mult_iff)
7.433 - qed
7.434 - next
7.435 - fix y :: pextreal assume *: "\<And>j. j \<in> UNIV \<Longrightarrow> of_nat (f t j) / 2 ^ j \<le> y"
7.436 - show "u t \<le> y"
7.437 - proof (cases "u t")
7.438 - case (preal r)
7.439 - show ?thesis
7.440 - proof (rule ccontr)
7.441 - assume "\<not> u t \<le> y"
7.442 - then obtain p where p: "y = Real p" "0 \<le> p" "p < r" using preal by (cases y) auto
7.443 - with LIMSEQ_inverse_realpow_zero[of 2, unfolded LIMSEQ_iff, rule_format, of "r - p"]
7.444 - obtain n where n: "\<And>N. n \<le> N \<Longrightarrow> inverse (2^N) < r - p" by auto
7.445 - let ?N = "max n (natfloor r + 1)"
7.446 - have "u t < of_nat ?N" "n \<le> ?N"
7.447 - using ge_natfloor_plus_one_imp_gt[of r n] preal
7.448 - using real_natfloor_add_one_gt
7.449 - by (auto simp: max_def real_of_nat_Suc)
7.450 - from lower[OF this(1)]
7.451 - have "r < (real (f t ?N) + 1) / 2 ^ ?N" unfolding less_divide_eq
7.452 - using preal by (auto simp add: power_le_zero_eq zero_le_mult_iff real_of_nat_Suc split: split_if_asm)
7.453 - hence "u t < of_nat (f t ?N) / 2 ^ ?N + 1 / 2 ^ ?N"
7.454 - using preal by (auto simp: field_simps divide_real_def[symmetric])
7.455 - with n[OF `n \<le> ?N`] p preal *[of ?N]
7.456 - show False
7.457 - by (cases "f t ?N") (auto simp: power_le_zero_eq split: split_if_asm)
7.458 - qed
7.459 - next
7.460 - case infinite
7.461 - { fix j have "f t j = j*2^j" using top[of j t] infinite by simp
7.462 - hence "of_nat j \<le> y" using *[of j]
7.463 - by (cases y) (auto simp: power_le_zero_eq zero_le_mult_iff field_simps) }
7.464 - note all_less_y = this
7.465 - show ?thesis unfolding infinite
7.466 - proof (rule ccontr)
7.467 - assume "\<not> \<omega> \<le> y" then obtain r where r: "y = Real r" "0 \<le> r" by (cases y) auto
7.468 - moreover obtain n ::nat where "r < real n" using ex_less_of_nat by (auto simp: real_eq_of_nat)
7.469 - with all_less_y[of n] r show False by auto
7.470 - qed
7.471 - qed
7.472 - qed
7.473 +lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
7.474 + fixes u :: "'a \<Rightarrow> extreal"
7.475 + assumes u: "u \<in> borel_measurable M"
7.476 + obtains f where "\<And>i. simple_function M (f i)" "incseq f" "\<And>i. \<infinity> \<notin> range (f i)"
7.477 + "\<And>x. (SUP i. f i x) = max 0 (u x)" "\<And>i x. 0 \<le> f i x"
7.478 + using borel_measurable_implies_simple_function_sequence[OF u] by auto
7.479 +
7.480 +lemma (in sigma_algebra) simple_function_If_set:
7.481 + assumes sf: "simple_function M f" "simple_function M g" and A: "A \<inter> space M \<in> sets M"
7.482 + shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
7.483 +proof -
7.484 + def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
7.485 + show ?thesis unfolding simple_function_def
7.486 + proof safe
7.487 + have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
7.488 + from finite_subset[OF this] assms
7.489 + show "finite (?IF ` space M)" unfolding simple_function_def by auto
7.490 + next
7.491 + fix x assume "x \<in> space M"
7.492 + then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
7.493 + then ((F (f x) \<inter> (A \<inter> space M)) \<union> (G (f x) - (G (f x) \<inter> (A \<inter> space M))))
7.494 + else ((F (g x) \<inter> (A \<inter> space M)) \<union> (G (g x) - (G (g x) \<inter> (A \<inter> space M)))))"
7.495 + using sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
7.496 + have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
7.497 + unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
7.498 + show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
7.499 qed
7.500 qed
7.501
7.502 -lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
7.503 - fixes u :: "'a \<Rightarrow> pextreal"
7.504 - assumes "u \<in> borel_measurable M"
7.505 - obtains (x) f where "f \<up> u" "\<And>i. simple_function M (f i)" "\<And>i. \<omega>\<notin>f i`space M"
7.506 +lemma (in sigma_algebra) simple_function_If:
7.507 + assumes sf: "simple_function M f" "simple_function M g" and P: "{x\<in>space M. P x} \<in> sets M"
7.508 + shows "simple_function M (\<lambda>x. if P x then f x else g x)"
7.509 proof -
7.510 - from borel_measurable_implies_simple_function_sequence[OF assms]
7.511 - obtain f where x: "\<And>i. simple_function M (f i)" "f \<up> u"
7.512 - and fin: "\<And>i. \<And>x. x\<in>space M \<Longrightarrow> f i x \<noteq> \<omega>" by auto
7.513 - { fix i from fin[of _ i] have "\<omega> \<notin> f i`space M" by fastsimp }
7.514 - with x show thesis by (auto intro!: that[of f])
7.515 + have "{x\<in>space M. P x} = {x. P x} \<inter> space M" by auto
7.516 + with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp
7.517 qed
7.518
7.519 -lemma (in sigma_algebra) simple_function_eq_borel_measurable:
7.520 - fixes f :: "'a \<Rightarrow> pextreal"
7.521 - shows "simple_function M f \<longleftrightarrow>
7.522 - finite (f`space M) \<and> f \<in> borel_measurable M"
7.523 - using simple_function_borel_measurable[of f]
7.524 - borel_measurable_simple_function[of f]
7.525 - by (fastsimp simp: simple_function_def)
7.526 -
7.527 lemma (in measure_space) simple_function_restricted:
7.528 - fixes f :: "'a \<Rightarrow> pextreal" assumes "A \<in> sets M"
7.529 + fixes f :: "'a \<Rightarrow> extreal" assumes "A \<in> sets M"
7.530 shows "simple_function (restricted_space A) f \<longleftrightarrow> simple_function M (\<lambda>x. f x * indicator A x)"
7.531 (is "simple_function ?R f \<longleftrightarrow> simple_function M ?f")
7.532 proof -
7.533 interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
7.534 - have "finite (f`A) \<longleftrightarrow> finite (?f`space M)"
7.535 + have f: "finite (f`A) \<longleftrightarrow> finite (?f`space M)"
7.536 proof cases
7.537 assume "A = space M"
7.538 then have "f`A = ?f`space M" by (fastsimp simp: image_iff)
7.539 @@ -456,7 +475,7 @@
7.540 using `A \<in> sets M` sets_into_space by (auto intro!: bexI[of _ x])
7.541 next
7.542 fix x
7.543 - assume "indicator A x \<noteq> (0::pextreal)"
7.544 + assume "indicator A x \<noteq> (0::extreal)"
7.545 then have "x \<in> A" by (auto simp: indicator_def split: split_if_asm)
7.546 moreover assume "x \<in> space M" "\<forall>y\<in>A. ?f x \<noteq> f y"
7.547 ultimately show "f x = 0" by auto
7.548 @@ -467,7 +486,8 @@
7.549 unfolding simple_function_eq_borel_measurable
7.550 R.simple_function_eq_borel_measurable
7.551 unfolding borel_measurable_restricted[OF `A \<in> sets M`]
7.552 - by auto
7.553 + using assms(1)[THEN sets_into_space]
7.554 + by (auto simp: indicator_def)
7.555 qed
7.556
7.557 lemma (in sigma_algebra) simple_function_subalgebra:
7.558 @@ -504,7 +524,7 @@
7.559 "integral\<^isup>S M f = (\<Sum>x \<in> f ` space M. x * measure M (f -` {x} \<inter> space M))"
7.560
7.561 syntax
7.562 - "_simple_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> pextreal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> pextreal" ("\<integral>\<^isup>S _. _ \<partial>_" [60,61] 110)
7.563 + "_simple_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> extreal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> extreal" ("\<integral>\<^isup>S _. _ \<partial>_" [60,61] 110)
7.564
7.565 translations
7.566 "\<integral>\<^isup>S x. f \<partial>M" == "CONST integral\<^isup>S M (%x. f)"
7.567 @@ -540,7 +560,7 @@
7.568 qed
7.569
7.570 lemma (in measure_space) simple_function_partition:
7.571 - assumes "simple_function M f" and "simple_function M g"
7.572 + assumes f: "simple_function M f" and g: "simple_function M g"
7.573 shows "integral\<^isup>S M f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * \<mu> A)"
7.574 (is "_ = setsum _ (?p ` space M)")
7.575 proof-
7.576 @@ -559,23 +579,16 @@
7.577 hence "finite (?p ` (A \<inter> space M))"
7.578 by (rule finite_subset) auto }
7.579 note this[intro, simp]
7.580 + note sets = simple_function_measurable2[OF f g]
7.581
7.582 { fix x assume "x \<in> space M"
7.583 have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
7.584 - moreover {
7.585 - fix x y
7.586 - have *: "f -` {f x} \<inter> g -` {g x} \<inter> space M
7.587 - = (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)" by auto
7.588 - assume "x \<in> space M" "y \<in> space M"
7.589 - hence "f -` {f x} \<inter> g -` {g x} \<inter> space M \<in> sets M"
7.590 - using assms unfolding simple_function_def * by auto }
7.591 - ultimately
7.592 - have "\<mu> (f -` {f x} \<inter> space M) = setsum (\<mu>) (?sub (f x))"
7.593 - by (subst measure_finitely_additive) auto }
7.594 + with sets have "\<mu> (f -` {f x} \<inter> space M) = setsum \<mu> (?sub (f x))"
7.595 + by (subst measure_Union) auto }
7.596 hence "integral\<^isup>S M f = (\<Sum>(x,A)\<in>?SIGMA. x * \<mu> A)"
7.597 - unfolding simple_integral_def
7.598 - by (subst setsum_Sigma[symmetric],
7.599 - auto intro!: setsum_cong simp: setsum_right_distrib[symmetric])
7.600 + unfolding simple_integral_def using f sets
7.601 + by (subst setsum_Sigma[symmetric])
7.602 + (auto intro!: setsum_cong setsum_extreal_right_distrib)
7.603 also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * \<mu> A)"
7.604 proof -
7.605 have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
7.606 @@ -595,7 +608,7 @@
7.607 qed
7.608
7.609 lemma (in measure_space) simple_integral_add[simp]:
7.610 - assumes "simple_function M f" and "simple_function M g"
7.611 + assumes f: "simple_function M f" and "\<And>x. 0 \<le> f x" and g: "simple_function M g" and "\<And>x. 0 \<le> g x"
7.612 shows "(\<integral>\<^isup>Sx. f x + g x \<partial>M) = integral\<^isup>S M f + integral\<^isup>S M g"
7.613 proof -
7.614 { fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
7.615 @@ -603,63 +616,43 @@
7.616 hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}"
7.617 "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
7.618 by auto }
7.619 - thus ?thesis
7.620 + with assms show ?thesis
7.621 unfolding
7.622 - simple_function_partition[OF simple_function_add[OF assms] simple_function_Pair[OF assms]]
7.623 - simple_function_partition[OF `simple_function M f` `simple_function M g`]
7.624 - simple_function_partition[OF `simple_function M g` `simple_function M f`]
7.625 - apply (subst (3) Int_commute)
7.626 - by (auto simp add: field_simps setsum_addf[symmetric] intro!: setsum_cong)
7.627 + simple_function_partition[OF simple_function_add[OF f g] simple_function_Pair[OF f g]]
7.628 + simple_function_partition[OF f g]
7.629 + simple_function_partition[OF g f]
7.630 + by (subst (3) Int_commute)
7.631 + (auto simp add: extreal_left_distrib setsum_addf[symmetric] intro!: setsum_cong)
7.632 qed
7.633
7.634 lemma (in measure_space) simple_integral_setsum[simp]:
7.635 + assumes "\<And>i x. i \<in> P \<Longrightarrow> 0 \<le> f i x"
7.636 assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
7.637 shows "(\<integral>\<^isup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>S M (f i))"
7.638 proof cases
7.639 assume "finite P"
7.640 from this assms show ?thesis
7.641 - by induct (auto simp: simple_function_setsum simple_integral_add)
7.642 + by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg)
7.643 qed auto
7.644
7.645 lemma (in measure_space) simple_integral_mult[simp]:
7.646 - assumes "simple_function M f"
7.647 + assumes f: "simple_function M f" "\<And>x. 0 \<le> f x"
7.648 shows "(\<integral>\<^isup>Sx. c * f x \<partial>M) = c * integral\<^isup>S M f"
7.649 proof -
7.650 - note mult = simple_function_mult[OF simple_function_const[of c] assms]
7.651 + note mult = simple_function_mult[OF simple_function_const[of c] f(1)]
7.652 { fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
7.653 assume "x \<in> space M"
7.654 hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
7.655 by auto }
7.656 - thus ?thesis
7.657 - unfolding simple_function_partition[OF mult assms]
7.658 - simple_function_partition[OF assms mult]
7.659 - by (auto simp: setsum_right_distrib field_simps intro!: setsum_cong)
7.660 -qed
7.661 -
7.662 -lemma (in sigma_algebra) simple_function_If:
7.663 - assumes sf: "simple_function M f" "simple_function M g" and A: "A \<in> sets M"
7.664 - shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
7.665 -proof -
7.666 - def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
7.667 - show ?thesis unfolding simple_function_def
7.668 - proof safe
7.669 - have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
7.670 - from finite_subset[OF this] assms
7.671 - show "finite (?IF ` space M)" unfolding simple_function_def by auto
7.672 - next
7.673 - fix x assume "x \<in> space M"
7.674 - then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
7.675 - then ((F (f x) \<inter> A) \<union> (G (f x) - (G (f x) \<inter> A)))
7.676 - else ((F (g x) \<inter> A) \<union> (G (g x) - (G (g x) \<inter> A))))"
7.677 - using sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
7.678 - have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
7.679 - unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
7.680 - show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
7.681 - qed
7.682 + with assms show ?thesis
7.683 + unfolding simple_function_partition[OF mult f(1)]
7.684 + simple_function_partition[OF f(1) mult]
7.685 + by (subst setsum_extreal_right_distrib)
7.686 + (auto intro!: extreal_0_le_mult setsum_cong simp: mult_assoc)
7.687 qed
7.688
7.689 lemma (in measure_space) simple_integral_mono_AE:
7.690 - assumes "simple_function M f" and "simple_function M g"
7.691 + assumes f: "simple_function M f" and g: "simple_function M g"
7.692 and mono: "AE x. f x \<le> g x"
7.693 shows "integral\<^isup>S M f \<le> integral\<^isup>S M g"
7.694 proof -
7.695 @@ -668,14 +661,16 @@
7.696 "\<And>x. f -` {f x} \<inter> g -` {g x} \<inter> space M = ?S x" by auto
7.697 show ?thesis
7.698 unfolding *
7.699 - simple_function_partition[OF `simple_function M f` `simple_function M g`]
7.700 - simple_function_partition[OF `simple_function M g` `simple_function M f`]
7.701 + simple_function_partition[OF f g]
7.702 + simple_function_partition[OF g f]
7.703 proof (safe intro!: setsum_mono)
7.704 fix x assume "x \<in> space M"
7.705 then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto
7.706 show "the_elem (f`?S x) * \<mu> (?S x) \<le> the_elem (g`?S x) * \<mu> (?S x)"
7.707 proof (cases "f x \<le> g x")
7.708 - case True then show ?thesis using * by (auto intro!: mult_right_mono)
7.709 + case True then show ?thesis
7.710 + using * assms(1,2)[THEN simple_functionD(2)]
7.711 + by (auto intro!: extreal_mult_right_mono)
7.712 next
7.713 case False
7.714 obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
7.715 @@ -685,7 +680,10 @@
7.716 by (rule_tac Int) (auto intro!: simple_functionD)
7.717 ultimately have "\<mu> (?S x) \<le> \<mu> N"
7.718 using `N \<in> sets M` by (auto intro!: measure_mono)
7.719 - then show ?thesis using `\<mu> N = 0` by auto
7.720 + moreover have "0 \<le> \<mu> (?S x)"
7.721 + using assms(1,2)[THEN simple_functionD(2)] by auto
7.722 + ultimately have "\<mu> (?S x) = 0" using `\<mu> N = 0` by auto
7.723 + then show ?thesis by simp
7.724 qed
7.725 qed
7.726 qed
7.727 @@ -697,7 +695,8 @@
7.728 using assms by (intro simple_integral_mono_AE) auto
7.729
7.730 lemma (in measure_space) simple_integral_cong_AE:
7.731 - assumes "simple_function M f" "simple_function M g" and "AE x. f x = g x"
7.732 + assumes "simple_function M f" and "simple_function M g"
7.733 + and "AE x. f x = g x"
7.734 shows "integral\<^isup>S M f = integral\<^isup>S M g"
7.735 using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
7.736
7.737 @@ -765,7 +764,7 @@
7.738 assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto
7.739 thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
7.740 next
7.741 - assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::pextreal}" by auto
7.742 + assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::extreal}" by auto
7.743 thus ?thesis
7.744 using simple_integral_indicator[OF assms simple_function_const[of 1]]
7.745 using sets_into_space[OF assms]
7.746 @@ -773,13 +772,13 @@
7.747 qed
7.748
7.749 lemma (in measure_space) simple_integral_null_set:
7.750 - assumes "simple_function M u" "N \<in> null_sets"
7.751 + assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_sets"
7.752 shows "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = 0"
7.753 proof -
7.754 - have "AE x. indicator N x = (0 :: pextreal)"
7.755 + have "AE x. indicator N x = (0 :: extreal)"
7.756 using `N \<in> null_sets` by (auto simp: indicator_def intro!: AE_I[of _ N])
7.757 then have "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^isup>Sx. 0 \<partial>M)"
7.758 - using assms by (intro simple_integral_cong_AE) auto
7.759 + using assms apply (intro simple_integral_cong_AE) by auto
7.760 then show ?thesis by simp
7.761 qed
7.762
7.763 @@ -813,7 +812,7 @@
7.764 by (auto simp: indicator_def split: split_if_asm)
7.765 then show "f x * \<mu> (f -` {f x} \<inter> A) =
7.766 f x * \<mu> (?f -` {f x} \<inter> space M)"
7.767 - unfolding pextreal_mult_cancel_left by auto
7.768 + unfolding extreal_mult_cancel_left by auto
7.769 qed
7.770
7.771 lemma (in measure_space) simple_integral_subalgebra:
7.772 @@ -821,10 +820,6 @@
7.773 shows "integral\<^isup>S N = integral\<^isup>S M"
7.774 unfolding simple_integral_def_raw by simp
7.775
7.776 -lemma measure_preservingD:
7.777 - "T \<in> measure_preserving A B \<Longrightarrow> X \<in> sets B \<Longrightarrow> measure A (T -` X \<inter> space A) = measure B X"
7.778 - unfolding measure_preserving_def by auto
7.779 -
7.780 lemma (in measure_space) simple_integral_vimage:
7.781 assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
7.782 and f: "simple_function M' f"
7.783 @@ -853,196 +848,164 @@
7.784 qed
7.785 qed
7.786
7.787 +lemma (in measure_space) simple_integral_cmult_indicator:
7.788 + assumes A: "A \<in> sets M"
7.789 + shows "(\<integral>\<^isup>Sx. c * indicator A x \<partial>M) = c * \<mu> A"
7.790 + using simple_integral_mult[OF simple_function_indicator[OF A]]
7.791 + unfolding simple_integral_indicator_only[OF A] by simp
7.792 +
7.793 +lemma (in measure_space) simple_integral_positive:
7.794 + assumes f: "simple_function M f" and ae: "AE x. 0 \<le> f x"
7.795 + shows "0 \<le> integral\<^isup>S M f"
7.796 +proof -
7.797 + have "integral\<^isup>S M (\<lambda>x. 0) \<le> integral\<^isup>S M f"
7.798 + using simple_integral_mono_AE[OF _ f ae] by auto
7.799 + then show ?thesis by simp
7.800 +qed
7.801 +
7.802 section "Continuous positive integration"
7.803
7.804 definition positive_integral_def:
7.805 - "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> f}. integral\<^isup>S M g)"
7.806 + "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^isup>S M g)"
7.807
7.808 syntax
7.809 - "_positive_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> pextreal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> pextreal" ("\<integral>\<^isup>+ _. _ \<partial>_" [60,61] 110)
7.810 + "_positive_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> extreal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> extreal" ("\<integral>\<^isup>+ _. _ \<partial>_" [60,61] 110)
7.811
7.812 translations
7.813 "\<integral>\<^isup>+ x. f \<partial>M" == "CONST integral\<^isup>P M (%x. f)"
7.814
7.815 -lemma (in measure_space) positive_integral_alt: "integral\<^isup>P M f =
7.816 - (SUP g : {g. simple_function M g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}. integral\<^isup>S M g)"
7.817 - (is "_ = ?alt")
7.818 -proof (rule antisym SUP_leI)
7.819 - show "integral\<^isup>P M f \<le> ?alt" unfolding positive_integral_def
7.820 - proof (safe intro!: SUP_leI)
7.821 - fix g assume g: "simple_function M g" "g \<le> f"
7.822 - let ?G = "g -` {\<omega>} \<inter> space M"
7.823 - show "integral\<^isup>S M g \<le>
7.824 - (SUP h : {i. simple_function M i \<and> i \<le> f \<and> \<omega> \<notin> i ` space M}. integral\<^isup>S M h)"
7.825 - (is "integral\<^isup>S M g \<le> SUPR ?A _")
7.826 - proof cases
7.827 - let ?g = "\<lambda>x. indicator (space M - ?G) x * g x"
7.828 - have g': "simple_function M ?g"
7.829 - using g by (auto intro: simple_functionD)
7.830 - moreover
7.831 - assume "\<mu> ?G = 0"
7.832 - then have "AE x. g x = ?g x" using g
7.833 - by (intro AE_I[where N="?G"])
7.834 - (auto intro: simple_functionD simp: indicator_def)
7.835 - with g(1) g' have "integral\<^isup>S M g = integral\<^isup>S M ?g"
7.836 - by (rule simple_integral_cong_AE)
7.837 - moreover have "?g \<le> g" by (auto simp: le_fun_def indicator_def)
7.838 - from this `g \<le> f` have "?g \<le> f" by (rule order_trans)
7.839 - moreover have "\<omega> \<notin> ?g ` space M"
7.840 - by (auto simp: indicator_def split: split_if_asm)
7.841 - ultimately show ?thesis by (auto intro!: le_SUPI)
7.842 - next
7.843 - assume "\<mu> ?G \<noteq> 0"
7.844 - then have "?G \<noteq> {}" by auto
7.845 - then have "\<omega> \<in> g`space M" by force
7.846 - then have "space M \<noteq> {}" by auto
7.847 - have "SUPR ?A (integral\<^isup>S M) = \<omega>"
7.848 - proof (intro SUP_\<omega>[THEN iffD2] allI impI)
7.849 - fix x assume "x < \<omega>"
7.850 - then guess n unfolding less_\<omega>_Ex_of_nat .. note n = this
7.851 - then have "0 < n" by (intro neq0_conv[THEN iffD1] notI) simp
7.852 - let ?g = "\<lambda>x. (of_nat n / (if \<mu> ?G = \<omega> then 1 else \<mu> ?G)) * indicator ?G x"
7.853 - show "\<exists>i\<in>?A. x < integral\<^isup>S M i"
7.854 - proof (intro bexI impI CollectI conjI)
7.855 - show "simple_function M ?g" using g
7.856 - by (auto intro!: simple_functionD simple_function_add)
7.857 - have "?g \<le> g" by (auto simp: le_fun_def indicator_def)
7.858 - from this g(2) show "?g \<le> f" by (rule order_trans)
7.859 - show "\<omega> \<notin> ?g ` space M"
7.860 - using `\<mu> ?G \<noteq> 0` by (auto simp: indicator_def split: split_if_asm)
7.861 - have "x < (of_nat n / (if \<mu> ?G = \<omega> then 1 else \<mu> ?G)) * \<mu> ?G"
7.862 - using n `\<mu> ?G \<noteq> 0` `0 < n`
7.863 - by (auto simp: pextreal_noteq_omega_Ex field_simps)
7.864 - also have "\<dots> = integral\<^isup>S M ?g" using g `space M \<noteq> {}`
7.865 - by (subst simple_integral_indicator)
7.866 - (auto simp: image_constant ac_simps dest: simple_functionD)
7.867 - finally show "x < integral\<^isup>S M ?g" .
7.868 - qed
7.869 - qed
7.870 - then show ?thesis by simp
7.871 - qed
7.872 - qed
7.873 -qed (auto intro!: SUP_subset simp: positive_integral_def)
7.874 -
7.875 lemma (in measure_space) positive_integral_cong_measure:
7.876 assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
7.877 shows "integral\<^isup>P N f = integral\<^isup>P M f"
7.878 -proof -
7.879 - interpret v: measure_space N
7.880 - by (rule measure_space_cong) fact+
7.881 - with assms show ?thesis
7.882 - unfolding positive_integral_def SUPR_def
7.883 - by (auto intro!: arg_cong[where f=Sup] image_cong
7.884 - simp: simple_integral_cong_measure[OF assms]
7.885 - simple_function_cong_algebra[OF assms(2,3)])
7.886 -qed
7.887 + unfolding positive_integral_def
7.888 + unfolding simple_function_cong_algebra[OF assms(2,3), symmetric]
7.889 + using AE_cong_measure[OF assms]
7.890 + using simple_integral_cong_measure[OF assms]
7.891 + by (auto intro!: SUP_cong)
7.892 +
7.893 +lemma (in measure_space) positive_integral_positive:
7.894 + "0 \<le> integral\<^isup>P M f"
7.895 + by (auto intro!: le_SUPI2[of "\<lambda>x. 0"] simp: positive_integral_def le_fun_def)
7.896
7.897 -lemma (in measure_space) positive_integral_alt1:
7.898 - "integral\<^isup>P M f =
7.899 - (SUP g : {g. simple_function M g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}. integral\<^isup>S M g)"
7.900 - unfolding positive_integral_alt SUPR_def
7.901 -proof (safe intro!: arg_cong[where f=Sup])
7.902 - fix g let ?g = "\<lambda>x. if x \<in> space M then g x else f x"
7.903 - assume "simple_function M g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
7.904 - hence "?g \<le> f" "simple_function M ?g" "integral\<^isup>S M g = integral\<^isup>S M ?g"
7.905 - "\<omega> \<notin> g`space M"
7.906 - unfolding le_fun_def by (auto cong: simple_function_cong simple_integral_cong)
7.907 - thus "integral\<^isup>S M g \<in> integral\<^isup>S M ` {g. simple_function M g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}"
7.908 - by auto
7.909 -next
7.910 - fix g assume "simple_function M g" "g \<le> f" "\<omega> \<notin> g`space M"
7.911 - hence "simple_function M g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
7.912 - by (auto simp add: le_fun_def image_iff)
7.913 - thus "integral\<^isup>S M g \<in> integral\<^isup>S M ` {g. simple_function M g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}"
7.914 - by auto
7.915 -qed
7.916 +lemma (in measure_space) positive_integral_def_finite:
7.917 + "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f \<and> range g \<subseteq> {0 ..< \<infinity>}}. integral\<^isup>S M g)"
7.918 + (is "_ = SUPR ?A ?f")
7.919 + unfolding positive_integral_def
7.920 +proof (safe intro!: antisym SUP_leI)
7.921 + fix g assume g: "simple_function M g" "g \<le> max 0 \<circ> f"
7.922 + let ?G = "{x \<in> space M. \<not> g x \<noteq> \<infinity>}"
7.923 + note gM = g(1)[THEN borel_measurable_simple_function]
7.924 + have \<mu>G_pos: "0 \<le> \<mu> ?G" using gM by auto
7.925 + let "?g y x" = "if g x = \<infinity> then y else max 0 (g x)"
7.926 + from g gM have g_in_A: "\<And>y. 0 \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> ?g y \<in> ?A"
7.927 + apply (safe intro!: simple_function_max simple_function_If)
7.928 + apply (force simp: max_def le_fun_def split: split_if_asm)+
7.929 + done
7.930 + show "integral\<^isup>S M g \<le> SUPR ?A ?f"
7.931 + proof cases
7.932 + have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto
7.933 + assume "\<mu> ?G = 0"
7.934 + with gM have "AE x. x \<notin> ?G" by (simp add: AE_iff_null_set)
7.935 + with gM g show ?thesis
7.936 + by (intro le_SUPI2[OF g0] simple_integral_mono_AE)
7.937 + (auto simp: max_def intro!: simple_function_If)
7.938 + next
7.939 + assume \<mu>G: "\<mu> ?G \<noteq> 0"
7.940 + have "SUPR ?A (integral\<^isup>S M) = \<infinity>"
7.941 + proof (intro SUP_PInfty)
7.942 + fix n :: nat
7.943 + let ?y = "extreal (real n) / (if \<mu> ?G = \<infinity> then 1 else \<mu> ?G)"
7.944 + have "0 \<le> ?y" "?y \<noteq> \<infinity>" using \<mu>G \<mu>G_pos by (auto simp: extreal_divide_eq)
7.945 + then have "?g ?y \<in> ?A" by (rule g_in_A)
7.946 + have "real n \<le> ?y * \<mu> ?G"
7.947 + using \<mu>G \<mu>G_pos by (cases "\<mu> ?G") (auto simp: field_simps)
7.948 + also have "\<dots> = (\<integral>\<^isup>Sx. ?y * indicator ?G x \<partial>M)"
7.949 + using `0 \<le> ?y` `?g ?y \<in> ?A` gM
7.950 + by (subst simple_integral_cmult_indicator) auto
7.951 + also have "\<dots> \<le> integral\<^isup>S M (?g ?y)" using `?g ?y \<in> ?A` gM
7.952 + by (intro simple_integral_mono) auto
7.953 + finally show "\<exists>i\<in>?A. real n \<le> integral\<^isup>S M i"
7.954 + using `?g ?y \<in> ?A` by blast
7.955 + qed
7.956 + then show ?thesis by simp
7.957 + qed
7.958 +qed (auto intro: le_SUPI)
7.959
7.960 -lemma (in measure_space) positive_integral_cong:
7.961 - assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
7.962 - shows "integral\<^isup>P M f = integral\<^isup>P M g"
7.963 -proof -
7.964 - have "\<And>h. (\<forall>x\<in>space M. h x \<le> f x \<and> h x \<noteq> \<omega>) = (\<forall>x\<in>space M. h x \<le> g x \<and> h x \<noteq> \<omega>)"
7.965 - using assms by auto
7.966 - thus ?thesis unfolding positive_integral_alt1 by auto
7.967 +lemma (in measure_space) positive_integral_mono_AE:
7.968 + assumes ae: "AE x. u x \<le> v x" shows "integral\<^isup>P M u \<le> integral\<^isup>P M v"
7.969 + unfolding positive_integral_def
7.970 +proof (safe intro!: SUP_mono)
7.971 + fix n assume n: "simple_function M n" "n \<le> max 0 \<circ> u"
7.972 + from ae[THEN AE_E] guess N . note N = this
7.973 + then have ae_N: "AE x. x \<notin> N" by (auto intro: AE_not_in)
7.974 + let "?n x" = "n x * indicator (space M - N) x"
7.975 + have "AE x. n x \<le> ?n x" "simple_function M ?n"
7.976 + using n N ae_N by auto
7.977 + moreover
7.978 + { fix x have "?n x \<le> max 0 (v x)"
7.979 + proof cases
7.980 + assume x: "x \<in> space M - N"
7.981 + with N have "u x \<le> v x" by auto
7.982 + with n(2)[THEN le_funD, of x] x show ?thesis
7.983 + by (auto simp: max_def split: split_if_asm)
7.984 + qed simp }
7.985 + then have "?n \<le> max 0 \<circ> v" by (auto simp: le_funI)
7.986 + moreover have "integral\<^isup>S M n \<le> integral\<^isup>S M ?n"
7.987 + using ae_N N n by (auto intro!: simple_integral_mono_AE)
7.988 + ultimately show "\<exists>m\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> v}. integral\<^isup>S M n \<le> integral\<^isup>S M m"
7.989 + by force
7.990 qed
7.991
7.992 -lemma (in measure_space) positive_integral_eq_simple_integral:
7.993 - assumes "simple_function M f"
7.994 - shows "integral\<^isup>P M f = integral\<^isup>S M f"
7.995 - unfolding positive_integral_def
7.996 -proof (safe intro!: pextreal_SUPI)
7.997 - fix g assume "simple_function M g" "g \<le> f"
7.998 - with assms show "integral\<^isup>S M g \<le> integral\<^isup>S M f"
7.999 - by (auto intro!: simple_integral_mono simp: le_fun_def)
7.1000 -next
7.1001 - fix y assume "\<forall>x. x\<in>{g. simple_function M g \<and> g \<le> f} \<longrightarrow> integral\<^isup>S M x \<le> y"
7.1002 - with assms show "integral\<^isup>S M f \<le> y" by auto
7.1003 -qed
7.1004 -
7.1005 -lemma (in measure_space) positive_integral_mono_AE:
7.1006 - assumes ae: "AE x. u x \<le> v x"
7.1007 - shows "integral\<^isup>P M u \<le> integral\<^isup>P M v"
7.1008 - unfolding positive_integral_alt1
7.1009 -proof (safe intro!: SUPR_mono)
7.1010 - fix a assume a: "simple_function M a" and mono: "\<forall>x\<in>space M. a x \<le> u x \<and> a x \<noteq> \<omega>"
7.1011 - from ae obtain N where N: "{x\<in>space M. \<not> u x \<le> v x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
7.1012 - by (auto elim!: AE_E)
7.1013 - have "simple_function M (\<lambda>x. a x * indicator (space M - N) x)"
7.1014 - using `N \<in> sets M` a by auto
7.1015 - with a show "\<exists>b\<in>{g. simple_function M g \<and> (\<forall>x\<in>space M. g x \<le> v x \<and> g x \<noteq> \<omega>)}.
7.1016 - integral\<^isup>S M a \<le> integral\<^isup>S M b"
7.1017 - proof (safe intro!: bexI[of _ "\<lambda>x. a x * indicator (space M - N) x"]
7.1018 - simple_integral_mono_AE)
7.1019 - show "AE x. a x \<le> a x * indicator (space M - N) x"
7.1020 - proof (rule AE_I, rule subset_refl)
7.1021 - have *: "{x \<in> space M. \<not> a x \<le> a x * indicator (space M - N) x} =
7.1022 - N \<inter> {x \<in> space M. a x \<noteq> 0}" (is "?N = _")
7.1023 - using `N \<in> sets M`[THEN sets_into_space] by (auto simp: indicator_def)
7.1024 - then show "?N \<in> sets M"
7.1025 - using `N \<in> sets M` `simple_function M a`[THEN borel_measurable_simple_function]
7.1026 - by (auto intro!: measure_mono Int)
7.1027 - then have "\<mu> ?N \<le> \<mu> N"
7.1028 - unfolding * using `N \<in> sets M` by (auto intro!: measure_mono)
7.1029 - then show "\<mu> ?N = 0" using `\<mu> N = 0` by auto
7.1030 - qed
7.1031 - next
7.1032 - fix x assume "x \<in> space M"
7.1033 - show "a x * indicator (space M - N) x \<le> v x"
7.1034 - proof (cases "x \<in> N")
7.1035 - case True then show ?thesis by simp
7.1036 - next
7.1037 - case False
7.1038 - with N mono have "a x \<le> u x" "u x \<le> v x" using `x \<in> space M` by auto
7.1039 - with False `x \<in> space M` show "a x * indicator (space M - N) x \<le> v x" by auto
7.1040 - qed
7.1041 - assume "a x * indicator (space M - N) x = \<omega>"
7.1042 - with mono `x \<in> space M` show False
7.1043 - by (simp split: split_if_asm add: indicator_def)
7.1044 - qed
7.1045 -qed
7.1046 +lemma (in measure_space) positive_integral_mono:
7.1047 + "(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^isup>P M u \<le> integral\<^isup>P M v"
7.1048 + by (auto intro: positive_integral_mono_AE)
7.1049
7.1050 lemma (in measure_space) positive_integral_cong_AE:
7.1051 "AE x. u x = v x \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
7.1052 by (auto simp: eq_iff intro!: positive_integral_mono_AE)
7.1053
7.1054 -lemma (in measure_space) positive_integral_mono:
7.1055 - "(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^isup>P M u \<le> integral\<^isup>P M v"
7.1056 - by (auto intro: positive_integral_mono_AE)
7.1057 +lemma (in measure_space) positive_integral_cong:
7.1058 + "(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
7.1059 + by (auto intro: positive_integral_cong_AE)
7.1060
7.1061 -lemma image_set_cong:
7.1062 - assumes A: "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. f x = g y"
7.1063 - assumes B: "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. g y = f x"
7.1064 - shows "f ` A = g ` B"
7.1065 - using assms by blast
7.1066 +lemma (in measure_space) positive_integral_eq_simple_integral:
7.1067 + assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
7.1068 +proof -
7.1069 + let "?f x" = "f x * indicator (space M) x"
7.1070 + have f': "simple_function M ?f" using f by auto
7.1071 + with f(2) have [simp]: "max 0 \<circ> ?f = ?f"
7.1072 + by (auto simp: fun_eq_iff max_def split: split_indicator)
7.1073 + have "integral\<^isup>P M ?f \<le> integral\<^isup>S M ?f" using f'
7.1074 + by (force intro!: SUP_leI simple_integral_mono simp: le_fun_def positive_integral_def)
7.1075 + moreover have "integral\<^isup>S M ?f \<le> integral\<^isup>P M ?f"
7.1076 + unfolding positive_integral_def
7.1077 + using f' by (auto intro!: le_SUPI)
7.1078 + ultimately show ?thesis
7.1079 + by (simp cong: positive_integral_cong simple_integral_cong)
7.1080 +qed
7.1081 +
7.1082 +lemma (in measure_space) positive_integral_eq_simple_integral_AE:
7.1083 + assumes f: "simple_function M f" "AE x. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
7.1084 +proof -
7.1085 + have "AE x. f x = max 0 (f x)" using f by (auto split: split_max)
7.1086 + with f have "integral\<^isup>P M f = integral\<^isup>S M (\<lambda>x. max 0 (f x))"
7.1087 + by (simp cong: positive_integral_cong_AE simple_integral_cong_AE
7.1088 + add: positive_integral_eq_simple_integral)
7.1089 + with assms show ?thesis
7.1090 + by (auto intro!: simple_integral_cong_AE split: split_max)
7.1091 +qed
7.1092
7.1093 lemma (in measure_space) positive_integral_SUP_approx:
7.1094 - assumes "f \<up> s"
7.1095 - and f: "\<And>i. f i \<in> borel_measurable M"
7.1096 - and "simple_function M u"
7.1097 - and le: "u \<le> s" and real: "\<omega> \<notin> u`space M"
7.1098 + assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
7.1099 + and u: "simple_function M u" "u \<le> (SUP i. f i)" "u`space M \<subseteq> {0..<\<infinity>}"
7.1100 shows "integral\<^isup>S M u \<le> (SUP i. integral\<^isup>P M (f i))" (is "_ \<le> ?S")
7.1101 -proof (rule pextreal_le_mult_one_interval)
7.1102 - fix a :: pextreal assume "0 < a" "a < 1"
7.1103 +proof (rule extreal_le_mult_one_interval)
7.1104 + have "0 \<le> (SUP i. integral\<^isup>P M (f i))"
7.1105 + using f(3) by (auto intro!: le_SUPI2 positive_integral_positive)
7.1106 + then show "(SUP i. integral\<^isup>P M (f i)) \<noteq> -\<infinity>" by auto
7.1107 + have u_range: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> u x \<and> u x \<noteq> \<infinity>"
7.1108 + using u(3) by auto
7.1109 + fix a :: extreal assume "0 < a" "a < 1"
7.1110 hence "a \<noteq> 0" by auto
7.1111 let "?B i" = "{x \<in> space M. a * u x \<le> f i x}"
7.1112 have B: "\<And>i. ?B i \<in> sets M"
7.1113 @@ -1054,203 +1017,269 @@
7.1114 proof safe
7.1115 fix i x assume "a * u x \<le> f i x"
7.1116 also have "\<dots> \<le> f (Suc i) x"
7.1117 - using `f \<up> s` unfolding isoton_def le_fun_def by auto
7.1118 + using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto
7.1119 finally show "a * u x \<le> f (Suc i) x" .
7.1120 qed }
7.1121 note B_mono = this
7.1122
7.1123 - have u: "\<And>x. x \<in> space M \<Longrightarrow> u -` {u x} \<inter> space M \<in> sets M"
7.1124 - using `simple_function M u` by (auto simp add: simple_function_def)
7.1125 + note B_u = Int[OF u(1)[THEN simple_functionD(2)] B]
7.1126
7.1127 - have "\<And>i. (\<lambda>n. (u -` {i} \<inter> space M) \<inter> ?B n) \<up> (u -` {i} \<inter> space M)" using B_mono unfolding isoton_def
7.1128 - proof safe
7.1129 - fix x i assume "x \<in> space M"
7.1130 - show "x \<in> (\<Union>i. (u -` {u x} \<inter> space M) \<inter> ?B i)"
7.1131 - proof cases
7.1132 - assume "u x = 0" thus ?thesis using `x \<in> space M` by simp
7.1133 - next
7.1134 - assume "u x \<noteq> 0"
7.1135 - with `a < 1` real `x \<in> space M`
7.1136 - have "a * u x < 1 * u x" by (rule_tac pextreal_mult_strict_right_mono) (auto simp: image_iff)
7.1137 - also have "\<dots> \<le> (SUP i. f i x)" using le `f \<up> s`
7.1138 - unfolding isoton_fun_expand by (auto simp: isoton_def le_fun_def)
7.1139 - finally obtain i where "a * u x < f i x" unfolding SUPR_def
7.1140 - by (auto simp add: less_Sup_iff)
7.1141 - hence "a * u x \<le> f i x" by auto
7.1142 - thus ?thesis using `x \<in> space M` by auto
7.1143 + let "?B' i n" = "(u -` {i} \<inter> space M) \<inter> ?B n"
7.1144 + have measure_conv: "\<And>i. \<mu> (u -` {i} \<inter> space M) = (SUP n. \<mu> (?B' i n))"
7.1145 + proof -
7.1146 + fix i
7.1147 + have 1: "range (?B' i) \<subseteq> sets M" using B_u by auto
7.1148 + have 2: "incseq (?B' i)" using B_mono by (auto intro!: incseq_SucI)
7.1149 + have "(\<Union>n. ?B' i n) = u -` {i} \<inter> space M"
7.1150 + proof safe
7.1151 + fix x i assume x: "x \<in> space M"
7.1152 + show "x \<in> (\<Union>i. ?B' (u x) i)"
7.1153 + proof cases
7.1154 + assume "u x = 0" thus ?thesis using `x \<in> space M` f(3) by simp
7.1155 + next
7.1156 + assume "u x \<noteq> 0"
7.1157 + with `a < 1` u_range[OF `x \<in> space M`]
7.1158 + have "a * u x < 1 * u x"
7.1159 + by (intro extreal_mult_strict_right_mono) (auto simp: image_iff)
7.1160 + also have "\<dots> \<le> (SUP i. f i x)" using u(2) by (auto simp: le_fun_def SUPR_apply)
7.1161 + finally obtain i where "a * u x < f i x" unfolding SUPR_def
7.1162 + by (auto simp add: less_Sup_iff)
7.1163 + hence "a * u x \<le> f i x" by auto
7.1164 + thus ?thesis using `x \<in> space M` by auto
7.1165 + qed
7.1166 qed
7.1167 - qed auto
7.1168 - note measure_conv = measure_up[OF Int[OF u B] this]
7.1169 + then show "?thesis i" using continuity_from_below[OF 1 2] by simp
7.1170 + qed
7.1171
7.1172 have "integral\<^isup>S M u = (SUP i. integral\<^isup>S M (?uB i))"
7.1173 unfolding simple_integral_indicator[OF B `simple_function M u`]
7.1174 - proof (subst SUPR_pextreal_setsum, safe)
7.1175 + proof (subst SUPR_extreal_setsum, safe)
7.1176 fix x n assume "x \<in> space M"
7.1177 - have "\<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f n x})
7.1178 - \<le> \<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f (Suc n) x})"
7.1179 - using B_mono Int[OF u[OF `x \<in> space M`] B] by (auto intro!: measure_mono)
7.1180 - thus "u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B n)
7.1181 - \<le> u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B (Suc n))"
7.1182 - by (auto intro: mult_left_mono)
7.1183 + with u_range show "incseq (\<lambda>i. u x * \<mu> (?B' (u x) i))" "\<And>i. 0 \<le> u x * \<mu> (?B' (u x) i)"
7.1184 + using B_mono B_u by (auto intro!: measure_mono extreal_mult_left_mono incseq_SucI simp: extreal_zero_le_0_iff)
7.1185 next
7.1186 - show "integral\<^isup>S M u =
7.1187 - (\<Sum>i\<in>u ` space M. SUP n. i * \<mu> (u -` {i} \<inter> space M \<inter> ?B n))"
7.1188 - using measure_conv unfolding simple_integral_def isoton_def
7.1189 - by (auto intro!: setsum_cong simp: pextreal_SUP_cmult)
7.1190 + show "integral\<^isup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * \<mu> (?B' i n))"
7.1191 + using measure_conv u_range B_u unfolding simple_integral_def
7.1192 + by (auto intro!: setsum_cong SUPR_extreal_cmult[symmetric])
7.1193 qed
7.1194 moreover
7.1195 have "a * (SUP i. integral\<^isup>S M (?uB i)) \<le> ?S"
7.1196 - unfolding pextreal_SUP_cmult[symmetric]
7.1197 + apply (subst SUPR_extreal_cmult[symmetric])
7.1198 proof (safe intro!: SUP_mono bexI)
7.1199 fix i
7.1200 have "a * integral\<^isup>S M (?uB i) = (\<integral>\<^isup>Sx. a * ?uB i x \<partial>M)"
7.1201 - using B `simple_function M u`
7.1202 - by (subst simple_integral_mult[symmetric]) (auto simp: field_simps)
7.1203 + using B `simple_function M u` u_range
7.1204 + by (subst simple_integral_mult) (auto split: split_indicator)
7.1205 also have "\<dots> \<le> integral\<^isup>P M (f i)"
7.1206 proof -
7.1207 - have "\<And>x. a * ?uB i x \<le> f i x" unfolding indicator_def by auto
7.1208 - hence *: "simple_function M (\<lambda>x. a * ?uB i x)" using B assms(3)
7.1209 - by (auto intro!: simple_integral_mono)
7.1210 - show ?thesis unfolding positive_integral_eq_simple_integral[OF *, symmetric]
7.1211 - by (auto intro!: positive_integral_mono simp: indicator_def)
7.1212 + have *: "simple_function M (\<lambda>x. a * ?uB i x)" using B `0 < a` u(1) by auto
7.1213 + show ?thesis using f(3) * u_range `0 < a`
7.1214 + by (subst positive_integral_eq_simple_integral[symmetric])
7.1215 + (auto intro!: positive_integral_mono split: split_indicator)
7.1216 qed
7.1217 finally show "a * integral\<^isup>S M (?uB i) \<le> integral\<^isup>P M (f i)"
7.1218 by auto
7.1219 - qed simp
7.1220 + next
7.1221 + fix i show "0 \<le> \<integral>\<^isup>S x. ?uB i x \<partial>M" using B `0 < a` u(1) u_range
7.1222 + by (intro simple_integral_positive) (auto split: split_indicator)
7.1223 + qed (insert `0 < a`, auto)
7.1224 ultimately show "a * integral\<^isup>S M u \<le> ?S" by simp
7.1225 qed
7.1226
7.1227 +lemma (in measure_space) incseq_positive_integral:
7.1228 + assumes "incseq f" shows "incseq (\<lambda>i. integral\<^isup>P M (f i))"
7.1229 +proof -
7.1230 + have "\<And>i x. f i x \<le> f (Suc i) x"
7.1231 + using assms by (auto dest!: incseq_SucD simp: le_fun_def)
7.1232 + then show ?thesis
7.1233 + by (auto intro!: incseq_SucI positive_integral_mono)
7.1234 +qed
7.1235 +
7.1236 text {* Beppo-Levi monotone convergence theorem *}
7.1237 -lemma (in measure_space) positive_integral_isoton:
7.1238 - assumes "f \<up> u" "\<And>i. f i \<in> borel_measurable M"
7.1239 - shows "(\<lambda>i. integral\<^isup>P M (f i)) \<up> integral\<^isup>P M u"
7.1240 - unfolding isoton_def
7.1241 -proof safe
7.1242 - fix i show "integral\<^isup>P M (f i) \<le> integral\<^isup>P M (f (Suc i))"
7.1243 - apply (rule positive_integral_mono)
7.1244 - using `f \<up> u` unfolding isoton_def le_fun_def by auto
7.1245 +lemma (in measure_space) positive_integral_monotone_convergence_SUP:
7.1246 + assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
7.1247 + shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
7.1248 +proof (rule antisym)
7.1249 + show "(SUP j. integral\<^isup>P M (f j)) \<le> (\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M)"
7.1250 + by (auto intro!: SUP_leI le_SUPI positive_integral_mono)
7.1251 next
7.1252 - have u: "u = (SUP i. f i)" using `f \<up> u` unfolding isoton_def by auto
7.1253 - show "(SUP i. integral\<^isup>P M (f i)) = integral\<^isup>P M u"
7.1254 - proof (rule antisym)
7.1255 - from `f \<up> u`[THEN isoton_Sup, unfolded le_fun_def]
7.1256 - show "(SUP j. integral\<^isup>P M (f j)) \<le> integral\<^isup>P M u"
7.1257 - by (auto intro!: SUP_leI positive_integral_mono)
7.1258 - next
7.1259 - show "integral\<^isup>P M u \<le> (SUP i. integral\<^isup>P M (f i))"
7.1260 - unfolding positive_integral_alt[of u]
7.1261 - by (auto intro!: SUP_leI positive_integral_SUP_approx[OF assms])
7.1262 + show "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^isup>P M (f j))"
7.1263 + unfolding positive_integral_def_finite[of "\<lambda>x. SUP i. f i x"]
7.1264 + proof (safe intro!: SUP_leI)
7.1265 + fix g assume g: "simple_function M g"
7.1266 + and "g \<le> max 0 \<circ> (\<lambda>x. SUP i. f i x)" "range g \<subseteq> {0..<\<infinity>}"
7.1267 + moreover then have "\<And>x. 0 \<le> (SUP i. f i x)" and g': "g`space M \<subseteq> {0..<\<infinity>}"
7.1268 + using f by (auto intro!: le_SUPI2)
7.1269 + ultimately show "integral\<^isup>S M g \<le> (SUP j. integral\<^isup>P M (f j))"
7.1270 + by (intro positive_integral_SUP_approx[OF f g _ g'])
7.1271 + (auto simp: le_fun_def max_def SUPR_apply)
7.1272 qed
7.1273 qed
7.1274
7.1275 -lemma (in measure_space) positive_integral_monotone_convergence_SUP:
7.1276 - assumes "\<And>i x. x \<in> space M \<Longrightarrow> f i x \<le> f (Suc i) x"
7.1277 - assumes "\<And>i. f i \<in> borel_measurable M"
7.1278 - shows "(SUP i. integral\<^isup>P M (f i)) = (\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M)"
7.1279 - (is "_ = integral\<^isup>P M ?u")
7.1280 +lemma (in measure_space) positive_integral_monotone_convergence_SUP_AE:
7.1281 + assumes f: "\<And>i. AE x. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x" "\<And>i. f i \<in> borel_measurable M"
7.1282 + shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
7.1283 proof -
7.1284 - show ?thesis
7.1285 - proof (rule antisym)
7.1286 - show "(SUP j. integral\<^isup>P M (f j)) \<le> integral\<^isup>P M ?u"
7.1287 - by (auto intro!: SUP_leI positive_integral_mono le_SUPI)
7.1288 - next
7.1289 - def rf \<equiv> "\<lambda>i. \<lambda>x\<in>space M. f i x" and ru \<equiv> "\<lambda>x\<in>space M. ?u x"
7.1290 - have "\<And>i. rf i \<in> borel_measurable M" unfolding rf_def
7.1291 - using assms by (simp cong: measurable_cong)
7.1292 - moreover have iso: "rf \<up> ru" using assms unfolding rf_def ru_def
7.1293 - unfolding isoton_def le_fun_def fun_eq_iff SUPR_apply
7.1294 - using SUP_const[OF UNIV_not_empty]
7.1295 - by (auto simp: restrict_def le_fun_def fun_eq_iff)
7.1296 - ultimately have "integral\<^isup>P M ru \<le> (SUP i. integral\<^isup>P M (rf i))"
7.1297 - unfolding positive_integral_alt[of ru]
7.1298 - by (auto simp: le_fun_def intro!: SUP_leI positive_integral_SUP_approx)
7.1299 - then show "integral\<^isup>P M ?u \<le> (SUP i. integral\<^isup>P M (f i))"
7.1300 - unfolding ru_def rf_def by (simp cong: positive_integral_cong)
7.1301 + from f have "AE x. \<forall>i. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x"
7.1302 + by (simp add: AE_all_countable)
7.1303 + from this[THEN AE_E] guess N . note N = this
7.1304 + let "?f i x" = "if x \<in> space M - N then f i x else 0"
7.1305 + have f_eq: "AE x. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ N])
7.1306 + then have "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (\<integral>\<^isup>+ x. (SUP i. ?f i x) \<partial>M)"
7.1307 + by (auto intro!: positive_integral_cong_AE)
7.1308 + also have "\<dots> = (SUP i. (\<integral>\<^isup>+ x. ?f i x \<partial>M))"
7.1309 + proof (rule positive_integral_monotone_convergence_SUP)
7.1310 + show "incseq ?f" using N(1) by (force intro!: incseq_SucI le_funI)
7.1311 + { fix i show "(\<lambda>x. if x \<in> space M - N then f i x else 0) \<in> borel_measurable M"
7.1312 + using f N(3) by (intro measurable_If_set) auto
7.1313 + fix x show "0 \<le> ?f i x"
7.1314 + using N(1) by auto }
7.1315 qed
7.1316 + also have "\<dots> = (SUP i. (\<integral>\<^isup>+ x. f i x \<partial>M))"
7.1317 + using f_eq by (force intro!: arg_cong[where f="SUPR UNIV"] positive_integral_cong_AE ext)
7.1318 + finally show ?thesis .
7.1319 +qed
7.1320 +
7.1321 +lemma (in measure_space) positive_integral_monotone_convergence_SUP_AE_incseq:
7.1322 + assumes f: "incseq f" "\<And>i. AE x. 0 \<le> f i x" and borel: "\<And>i. f i \<in> borel_measurable M"
7.1323 + shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
7.1324 + using f[unfolded incseq_Suc_iff le_fun_def]
7.1325 + by (intro positive_integral_monotone_convergence_SUP_AE[OF _ borel])
7.1326 + auto
7.1327 +
7.1328 +lemma (in measure_space) positive_integral_monotone_convergence_simple:
7.1329 + assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
7.1330 + shows "(SUP i. integral\<^isup>S M (f i)) = (\<integral>\<^isup>+x. (SUP i. f i x) \<partial>M)"
7.1331 + using assms unfolding positive_integral_monotone_convergence_SUP[OF f(1)
7.1332 + f(3)[THEN borel_measurable_simple_function] f(2)]
7.1333 + by (auto intro!: positive_integral_eq_simple_integral[symmetric] arg_cong[where f="SUPR UNIV"] ext)
7.1334 +
7.1335 +lemma positive_integral_max_0:
7.1336 + "(\<integral>\<^isup>+x. max 0 (f x) \<partial>M) = integral\<^isup>P M f"
7.1337 + by (simp add: le_fun_def positive_integral_def)
7.1338 +
7.1339 +lemma (in measure_space) positive_integral_cong_pos:
7.1340 + assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> 0 \<and> g x \<le> 0 \<or> f x = g x"
7.1341 + shows "integral\<^isup>P M f = integral\<^isup>P M g"
7.1342 +proof -
7.1343 + have "integral\<^isup>P M (\<lambda>x. max 0 (f x)) = integral\<^isup>P M (\<lambda>x. max 0 (g x))"
7.1344 + proof (intro positive_integral_cong)
7.1345 + fix x assume "x \<in> space M"
7.1346 + from assms[OF this] show "max 0 (f x) = max 0 (g x)"
7.1347 + by (auto split: split_max)
7.1348 + qed
7.1349 + then show ?thesis by (simp add: positive_integral_max_0)
7.1350 qed
7.1351
7.1352 lemma (in measure_space) SUP_simple_integral_sequences:
7.1353 - assumes f: "f \<up> u" "\<And>i. simple_function M (f i)"
7.1354 - and g: "g \<up> u" "\<And>i. simple_function M (g i)"
7.1355 + assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
7.1356 + and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)"
7.1357 + and eq: "AE x. (SUP i. f i x) = (SUP i. g i x)"
7.1358 shows "(SUP i. integral\<^isup>S M (f i)) = (SUP i. integral\<^isup>S M (g i))"
7.1359 (is "SUPR _ ?F = SUPR _ ?G")
7.1360 proof -
7.1361 - have "(SUP i. ?F i) = (SUP i. integral\<^isup>P M (f i))"
7.1362 - using assms by (simp add: positive_integral_eq_simple_integral)
7.1363 - also have "\<dots> = integral\<^isup>P M u"
7.1364 - using positive_integral_isoton[OF f(1) borel_measurable_simple_function[OF f(2)]]
7.1365 - unfolding isoton_def by simp
7.1366 - also have "\<dots> = (SUP i. integral\<^isup>P M (g i))"
7.1367 - using positive_integral_isoton[OF g(1) borel_measurable_simple_function[OF g(2)]]
7.1368 - unfolding isoton_def by simp
7.1369 + have "(SUP i. integral\<^isup>S M (f i)) = (\<integral>\<^isup>+x. (SUP i. f i x) \<partial>M)"
7.1370 + using f by (rule positive_integral_monotone_convergence_simple)
7.1371 + also have "\<dots> = (\<integral>\<^isup>+x. (SUP i. g i x) \<partial>M)"
7.1372 + unfolding eq[THEN positive_integral_cong_AE] ..
7.1373 also have "\<dots> = (SUP i. ?G i)"
7.1374 - using assms by (simp add: positive_integral_eq_simple_integral)
7.1375 - finally show ?thesis .
7.1376 + using g by (rule positive_integral_monotone_convergence_simple[symmetric])
7.1377 + finally show ?thesis by simp
7.1378 qed
7.1379
7.1380 lemma (in measure_space) positive_integral_const[simp]:
7.1381 - "(\<integral>\<^isup>+ x. c \<partial>M) = c * \<mu> (space M)"
7.1382 + "0 \<le> c \<Longrightarrow> (\<integral>\<^isup>+ x. c \<partial>M) = c * \<mu> (space M)"
7.1383 by (subst positive_integral_eq_simple_integral) auto
7.1384
7.1385 -lemma (in measure_space) positive_integral_isoton_simple:
7.1386 - assumes "f \<up> u" and e: "\<And>i. simple_function M (f i)"
7.1387 - shows "(\<lambda>i. integral\<^isup>S M (f i)) \<up> integral\<^isup>P M u"
7.1388 - using positive_integral_isoton[OF `f \<up> u` e(1)[THEN borel_measurable_simple_function]]
7.1389 - unfolding positive_integral_eq_simple_integral[OF e] .
7.1390 -
7.1391 -lemma measure_preservingD2:
7.1392 - "f \<in> measure_preserving A B \<Longrightarrow> f \<in> measurable A B"
7.1393 - unfolding measure_preserving_def by auto
7.1394 -
7.1395 lemma (in measure_space) positive_integral_vimage:
7.1396 - assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'" and f: "f \<in> borel_measurable M'"
7.1397 + assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
7.1398 + and f: "f \<in> borel_measurable M'"
7.1399 shows "integral\<^isup>P M' f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
7.1400 proof -
7.1401 interpret T: measure_space M' by (rule measure_space_vimage[OF T])
7.1402 - obtain f' where f': "f' \<up> f" "\<And>i. simple_function M' (f' i)"
7.1403 - using T.borel_measurable_implies_simple_function_sequence[OF f] by blast
7.1404 - then have f: "(\<lambda>i x. f' i (T x)) \<up> (\<lambda>x. f (T x))" "\<And>i. simple_function M (\<lambda>x. f' i (T x))"
7.1405 - using simple_function_vimage[OF T(1) measure_preservingD2[OF T(2)]] unfolding isoton_fun_expand by auto
7.1406 + from T.borel_measurable_implies_simple_function_sequence'[OF f]
7.1407 + guess f' . note f' = this
7.1408 + let "?f i x" = "f' i (T x)"
7.1409 + have inc: "incseq ?f" using f' by (force simp: le_fun_def incseq_def)
7.1410 + have sup: "\<And>x. (SUP i. ?f i x) = max 0 (f (T x))"
7.1411 + using f'(4) .
7.1412 + have sf: "\<And>i. simple_function M (\<lambda>x. f' i (T x))"
7.1413 + using simple_function_vimage[OF T(1) measure_preservingD2[OF T(2)] f'(1)] .
7.1414 show "integral\<^isup>P M' f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
7.1415 - using positive_integral_isoton_simple[OF f]
7.1416 - using T.positive_integral_isoton_simple[OF f']
7.1417 - by (simp add: simple_integral_vimage[OF T f'(2)] isoton_def)
7.1418 + using
7.1419 + T.positive_integral_monotone_convergence_simple[OF f'(2,5,1)]
7.1420 + positive_integral_monotone_convergence_simple[OF inc f'(5) sf]
7.1421 + by (simp add: positive_integral_max_0 simple_integral_vimage[OF T f'(1)] f')
7.1422 qed
7.1423
7.1424 lemma (in measure_space) positive_integral_linear:
7.1425 - assumes f: "f \<in> borel_measurable M"
7.1426 - and g: "g \<in> borel_measurable M"
7.1427 + assumes f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and "0 \<le> a"
7.1428 + and g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
7.1429 shows "(\<integral>\<^isup>+ x. a * f x + g x \<partial>M) = a * integral\<^isup>P M f + integral\<^isup>P M g"
7.1430 (is "integral\<^isup>P M ?L = _")
7.1431 proof -
7.1432 - from borel_measurable_implies_simple_function_sequence'[OF f] guess u .
7.1433 - note u = this positive_integral_isoton_simple[OF this(1-2)]
7.1434 - from borel_measurable_implies_simple_function_sequence'[OF g] guess v .
7.1435 - note v = this positive_integral_isoton_simple[OF this(1-2)]
7.1436 + from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u .
7.1437 + note u = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
7.1438 + from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v .
7.1439 + note v = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
7.1440 let "?L' i x" = "a * u i x + v i x"
7.1441
7.1442 - have "?L \<in> borel_measurable M"
7.1443 - using assms by simp
7.1444 + have "?L \<in> borel_measurable M" using assms by auto
7.1445 from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
7.1446 - note positive_integral_isoton_simple[OF this(1-2)] and l = this
7.1447 - moreover have "(SUP i. integral\<^isup>S M (l i)) = (SUP i. integral\<^isup>S M (?L' i))"
7.1448 - proof (rule SUP_simple_integral_sequences[OF l(1-2)])
7.1449 - show "?L' \<up> ?L" "\<And>i. simple_function M (?L' i)"
7.1450 - using u v by (auto simp: isoton_fun_expand isoton_add isoton_cmult_right)
7.1451 + note l = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
7.1452 +
7.1453 + have inc: "incseq (\<lambda>i. a * integral\<^isup>S M (u i))" "incseq (\<lambda>i. integral\<^isup>S M (v i))"
7.1454 + using u v `0 \<le> a`
7.1455 + by (auto simp: incseq_Suc_iff le_fun_def
7.1456 + intro!: add_mono extreal_mult_left_mono simple_integral_mono)
7.1457 + have pos: "\<And>i. 0 \<le> integral\<^isup>S M (u i)" "\<And>i. 0 \<le> integral\<^isup>S M (v i)" "\<And>i. 0 \<le> a * integral\<^isup>S M (u i)"
7.1458 + using u v `0 \<le> a` by (auto simp: simple_integral_positive)
7.1459 + { fix i from pos[of i] have "a * integral\<^isup>S M (u i) \<noteq> -\<infinity>" "integral\<^isup>S M (v i) \<noteq> -\<infinity>"
7.1460 + by (auto split: split_if_asm) }
7.1461 + note not_MInf = this
7.1462 +
7.1463 + have l': "(SUP i. integral\<^isup>S M (l i)) = (SUP i. integral\<^isup>S M (?L' i))"
7.1464 + proof (rule SUP_simple_integral_sequences[OF l(3,6,2)])
7.1465 + show "incseq ?L'" "\<And>i x. 0 \<le> ?L' i x" "\<And>i. simple_function M (?L' i)"
7.1466 + using u v `0 \<le> a` unfolding incseq_Suc_iff le_fun_def
7.1467 + by (auto intro!: add_mono extreal_mult_left_mono extreal_add_nonneg_nonneg)
7.1468 + { fix x
7.1469 + { fix i have "a * u i x \<noteq> -\<infinity>" "v i x \<noteq> -\<infinity>" "u i x \<noteq> -\<infinity>" using `0 \<le> a` u(6)[of i x] v(6)[of i x]
7.1470 + by auto }
7.1471 + then have "(SUP i. a * u i x + v i x) = a * (SUP i. u i x) + (SUP i. v i x)"
7.1472 + using `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x]
7.1473 + by (subst SUPR_extreal_cmult[symmetric, OF u(6) `0 \<le> a`])
7.1474 + (auto intro!: SUPR_extreal_add
7.1475 + simp: incseq_Suc_iff le_fun_def add_mono extreal_mult_left_mono extreal_add_nonneg_nonneg) }
7.1476 + then show "AE x. (SUP i. l i x) = (SUP i. ?L' i x)"
7.1477 + unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2)
7.1478 + by (intro AE_I2) (auto split: split_max simp add: extreal_add_nonneg_nonneg)
7.1479 qed
7.1480 - moreover from u v have L'_isoton:
7.1481 - "(\<lambda>i. integral\<^isup>S M (?L' i)) \<up> a * integral\<^isup>P M f + integral\<^isup>P M g"
7.1482 - by (simp add: isoton_add isoton_cmult_right)
7.1483 - ultimately show ?thesis by (simp add: isoton_def)
7.1484 + also have "\<dots> = (SUP i. a * integral\<^isup>S M (u i) + integral\<^isup>S M (v i))"
7.1485 + using u(2, 6) v(2, 6) `0 \<le> a` by (auto intro!: arg_cong[where f="SUPR UNIV"] ext)
7.1486 + finally have "(\<integral>\<^isup>+ x. max 0 (a * f x + g x) \<partial>M) = a * (\<integral>\<^isup>+x. max 0 (f x) \<partial>M) + (\<integral>\<^isup>+x. max 0 (g x) \<partial>M)"
7.1487 + unfolding l(5)[symmetric] u(5)[symmetric] v(5)[symmetric]
7.1488 + unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric]
7.1489 + apply (subst SUPR_extreal_cmult[symmetric, OF pos(1) `0 \<le> a`])
7.1490 + apply (subst SUPR_extreal_add[symmetric, OF inc not_MInf]) .
7.1491 + then show ?thesis by (simp add: positive_integral_max_0)
7.1492 qed
7.1493
7.1494 lemma (in measure_space) positive_integral_cmult:
7.1495 - assumes "f \<in> borel_measurable M"
7.1496 + assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x" "0 \<le> c"
7.1497 shows "(\<integral>\<^isup>+ x. c * f x \<partial>M) = c * integral\<^isup>P M f"
7.1498 - using positive_integral_linear[OF assms, of "\<lambda>x. 0" c] by auto
7.1499 +proof -
7.1500 + have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c`
7.1501 + by (auto split: split_max simp: extreal_zero_le_0_iff)
7.1502 + have "(\<integral>\<^isup>+ x. c * f x \<partial>M) = (\<integral>\<^isup>+ x. c * max 0 (f x) \<partial>M)"
7.1503 + by (simp add: positive_integral_max_0)
7.1504 + then show ?thesis
7.1505 + using positive_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" "\<lambda>x. 0"] f
7.1506 + by (auto simp: positive_integral_max_0)
7.1507 +qed
7.1508
7.1509 lemma (in measure_space) positive_integral_multc:
7.1510 - assumes "f \<in> borel_measurable M"
7.1511 + assumes "f \<in> borel_measurable M" "AE x. 0 \<le> f x" "0 \<le> c"
7.1512 shows "(\<integral>\<^isup>+ x. f x * c \<partial>M) = integral\<^isup>P M f * c"
7.1513 unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
7.1514
7.1515 @@ -1260,143 +1289,172 @@
7.1516 (auto simp: simple_function_indicator simple_integral_indicator)
7.1517
7.1518 lemma (in measure_space) positive_integral_cmult_indicator:
7.1519 - "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x \<partial>M) = c * \<mu> A"
7.1520 + "0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x \<partial>M) = c * \<mu> A"
7.1521 by (subst positive_integral_eq_simple_integral)
7.1522 (auto simp: simple_function_indicator simple_integral_indicator)
7.1523
7.1524 lemma (in measure_space) positive_integral_add:
7.1525 - assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
7.1526 + assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
7.1527 + and g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
7.1528 shows "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = integral\<^isup>P M f + integral\<^isup>P M g"
7.1529 - using positive_integral_linear[OF assms, of 1] by simp
7.1530 +proof -
7.1531 + have ae: "AE x. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)"
7.1532 + using assms by (auto split: split_max simp: extreal_add_nonneg_nonneg)
7.1533 + have "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = (\<integral>\<^isup>+ x. max 0 (f x + g x) \<partial>M)"
7.1534 + by (simp add: positive_integral_max_0)
7.1535 + also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)"
7.1536 + unfolding ae[THEN positive_integral_cong_AE] ..
7.1537 + also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^isup>+ x. max 0 (g x) \<partial>M)"
7.1538 + using positive_integral_linear[of "\<lambda>x. max 0 (f x)" 1 "\<lambda>x. max 0 (g x)"] f g
7.1539 + by auto
7.1540 + finally show ?thesis
7.1541 + by (simp add: positive_integral_max_0)
7.1542 +qed
7.1543
7.1544 lemma (in measure_space) positive_integral_setsum:
7.1545 - assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M"
7.1546 + assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M" "\<And>i. i\<in>P \<Longrightarrow> AE x. 0 \<le> f i x"
7.1547 shows "(\<integral>\<^isup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>P M (f i))"
7.1548 proof cases
7.1549 - assume "finite P"
7.1550 - from this assms show ?thesis
7.1551 + assume f: "finite P"
7.1552 + from assms have "AE x. \<forall>i\<in>P. 0 \<le> f i x" unfolding AE_finite_all[OF f] by auto
7.1553 + from f this assms(1) show ?thesis
7.1554 proof induct
7.1555 case (insert i P)
7.1556 - have "f i \<in> borel_measurable M"
7.1557 - "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M"
7.1558 - using insert by (auto intro!: borel_measurable_pextreal_setsum)
7.1559 + then have "f i \<in> borel_measurable M" "AE x. 0 \<le> f i x"
7.1560 + "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M" "AE x. 0 \<le> (\<Sum>i\<in>P. f i x)"
7.1561 + by (auto intro!: borel_measurable_extreal_setsum setsum_nonneg)
7.1562 from positive_integral_add[OF this]
7.1563 show ?case using insert by auto
7.1564 qed simp
7.1565 qed simp
7.1566
7.1567 +lemma (in measure_space) positive_integral_Markov_inequality:
7.1568 + assumes u: "u \<in> borel_measurable M" "AE x. 0 \<le> u x" and "A \<in> sets M" and c: "0 \<le> c" "c \<noteq> \<infinity>"
7.1569 + shows "\<mu> ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
7.1570 + (is "\<mu> ?A \<le> _ * ?PI")
7.1571 +proof -
7.1572 + have "?A \<in> sets M"
7.1573 + using `A \<in> sets M` u by auto
7.1574 + hence "\<mu> ?A = (\<integral>\<^isup>+ x. indicator ?A x \<partial>M)"
7.1575 + using positive_integral_indicator by simp
7.1576 + also have "\<dots> \<le> (\<integral>\<^isup>+ x. c * (u x * indicator A x) \<partial>M)" using u c
7.1577 + by (auto intro!: positive_integral_mono_AE
7.1578 + simp: indicator_def extreal_zero_le_0_iff)
7.1579 + also have "\<dots> = c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
7.1580 + using assms
7.1581 + by (auto intro!: positive_integral_cmult borel_measurable_indicator simp: extreal_zero_le_0_iff)
7.1582 + finally show ?thesis .
7.1583 +qed
7.1584 +
7.1585 +lemma (in measure_space) positive_integral_noteq_infinite:
7.1586 + assumes g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
7.1587 + and "integral\<^isup>P M g \<noteq> \<infinity>"
7.1588 + shows "AE x. g x \<noteq> \<infinity>"
7.1589 +proof (rule ccontr)
7.1590 + assume c: "\<not> (AE x. g x \<noteq> \<infinity>)"
7.1591 + have "\<mu> {x\<in>space M. g x = \<infinity>} \<noteq> 0"
7.1592 + using c g by (simp add: AE_iff_null_set)
7.1593 + moreover have "0 \<le> \<mu> {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets)
7.1594 + ultimately have "0 < \<mu> {x\<in>space M. g x = \<infinity>}" by auto
7.1595 + then have "\<infinity> = \<infinity> * \<mu> {x\<in>space M. g x = \<infinity>}" by auto
7.1596 + also have "\<dots> \<le> (\<integral>\<^isup>+x. \<infinity> * indicator {x\<in>space M. g x = \<infinity>} x \<partial>M)"
7.1597 + using g by (subst positive_integral_cmult_indicator) auto
7.1598 + also have "\<dots> \<le> integral\<^isup>P M g"
7.1599 + using assms by (auto intro!: positive_integral_mono_AE simp: indicator_def)
7.1600 + finally show False using `integral\<^isup>P M g \<noteq> \<infinity>` by auto
7.1601 +qed
7.1602 +
7.1603 lemma (in measure_space) positive_integral_diff:
7.1604 - assumes f: "f \<in> borel_measurable M" and g: "g \<in> borel_measurable M"
7.1605 - and fin: "integral\<^isup>P M g \<noteq> \<omega>"
7.1606 - and mono: "\<And>x. x \<in> space M \<Longrightarrow> g x \<le> f x"
7.1607 + assumes f: "f \<in> borel_measurable M"
7.1608 + and g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
7.1609 + and fin: "integral\<^isup>P M g \<noteq> \<infinity>"
7.1610 + and mono: "AE x. g x \<le> f x"
7.1611 shows "(\<integral>\<^isup>+ x. f x - g x \<partial>M) = integral\<^isup>P M f - integral\<^isup>P M g"
7.1612 proof -
7.1613 - have borel: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
7.1614 - using f g by (rule borel_measurable_pextreal_diff)
7.1615 - have "(\<integral>\<^isup>+x. f x - g x \<partial>M) + integral\<^isup>P M g = integral\<^isup>P M f"
7.1616 - unfolding positive_integral_add[OF borel g, symmetric]
7.1617 - proof (rule positive_integral_cong)
7.1618 - fix x assume "x \<in> space M"
7.1619 - from mono[OF this] show "f x - g x + g x = f x"
7.1620 - by (cases "f x", cases "g x", simp, simp, cases "g x", auto)
7.1621 - qed
7.1622 - with mono show ?thesis
7.1623 - by (subst minus_pextreal_eq2[OF _ fin]) (auto intro!: positive_integral_mono)
7.1624 + have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x. 0 \<le> f x - g x"
7.1625 + using assms by (auto intro: extreal_diff_positive)
7.1626 + have pos_f: "AE x. 0 \<le> f x" using mono g by auto
7.1627 + { fix a b :: extreal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b"
7.1628 + by (cases rule: extreal2_cases[of a b]) auto }
7.1629 + note * = this
7.1630 + then have "AE x. f x = f x - g x + g x"
7.1631 + using mono positive_integral_noteq_infinite[OF g fin] assms by auto
7.1632 + then have **: "integral\<^isup>P M f = (\<integral>\<^isup>+x. f x - g x \<partial>M) + integral\<^isup>P M g"
7.1633 + unfolding positive_integral_add[OF diff g, symmetric]
7.1634 + by (rule positive_integral_cong_AE)
7.1635 + show ?thesis unfolding **
7.1636 + using fin positive_integral_positive[of g]
7.1637 + by (cases rule: extreal2_cases[of "\<integral>\<^isup>+ x. f x - g x \<partial>M" "integral\<^isup>P M g"]) auto
7.1638 qed
7.1639
7.1640 -lemma (in measure_space) positive_integral_psuminf:
7.1641 - assumes "\<And>i. f i \<in> borel_measurable M"
7.1642 - shows "(\<integral>\<^isup>+ x. (\<Sum>\<^isub>\<infinity> i. f i x) \<partial>M) = (\<Sum>\<^isub>\<infinity> i. integral\<^isup>P M (f i))"
7.1643 +lemma (in measure_space) positive_integral_suminf:
7.1644 + assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x. 0 \<le> f i x"
7.1645 + shows "(\<integral>\<^isup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^isup>P M (f i))"
7.1646 proof -
7.1647 - have "(\<lambda>i. (\<integral>\<^isup>+x. (\<Sum>i<i. f i x) \<partial>M)) \<up> (\<integral>\<^isup>+x. (\<Sum>\<^isub>\<infinity>i. f i x) \<partial>M)"
7.1648 - by (rule positive_integral_isoton)
7.1649 - (auto intro!: borel_measurable_pextreal_setsum assms positive_integral_mono
7.1650 - arg_cong[where f=Sup]
7.1651 - simp: isoton_def le_fun_def psuminf_def fun_eq_iff SUPR_def Sup_fun_def)
7.1652 - thus ?thesis
7.1653 - by (auto simp: isoton_def psuminf_def positive_integral_setsum[OF assms])
7.1654 + have all_pos: "AE x. \<forall>i. 0 \<le> f i x"
7.1655 + using assms by (auto simp: AE_all_countable)
7.1656 + have "(\<Sum>i. integral\<^isup>P M (f i)) = (SUP n. \<Sum>i<n. integral\<^isup>P M (f i))"
7.1657 + using positive_integral_positive by (rule suminf_extreal_eq_SUPR)
7.1658 + also have "\<dots> = (SUP n. \<integral>\<^isup>+x. (\<Sum>i<n. f i x) \<partial>M)"
7.1659 + unfolding positive_integral_setsum[OF f] ..
7.1660 + also have "\<dots> = \<integral>\<^isup>+x. (SUP n. \<Sum>i<n. f i x) \<partial>M" using f all_pos
7.1661 + by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
7.1662 + (elim AE_mp, auto simp: setsum_nonneg simp del: setsum_lessThan_Suc intro!: AE_I2 setsum_mono3)
7.1663 + also have "\<dots> = \<integral>\<^isup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos
7.1664 + by (intro positive_integral_cong_AE) (auto simp: suminf_extreal_eq_SUPR)
7.1665 + finally show ?thesis by simp
7.1666 qed
7.1667
7.1668 text {* Fatou's lemma: convergence theorem on limes inferior *}
7.1669 lemma (in measure_space) positive_integral_lim_INF:
7.1670 - fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> pextreal"
7.1671 - assumes "\<And>i. u i \<in> borel_measurable M"
7.1672 - shows "(\<integral>\<^isup>+ x. (SUP n. INF m. u (m + n) x) \<partial>M) \<le>
7.1673 - (SUP n. INF m. integral\<^isup>P M (u (m + n)))"
7.1674 + fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
7.1675 + assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x. 0 \<le> u i x"
7.1676 + shows "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^isup>P M (u n))"
7.1677 proof -
7.1678 - have "(\<integral>\<^isup>+x. (SUP n. INF m. u (m + n) x) \<partial>M)
7.1679 - = (SUP n. (\<integral>\<^isup>+x. (INF m. u (m + n) x) \<partial>M))"
7.1680 - using assms
7.1681 - by (intro positive_integral_monotone_convergence_SUP[symmetric] INF_mono)
7.1682 - (auto simp del: add_Suc simp add: add_Suc[symmetric])
7.1683 - also have "\<dots> \<le> (SUP n. INF m. integral\<^isup>P M (u (m + n)))"
7.1684 - by (auto intro!: SUP_mono bexI le_INFI positive_integral_mono INF_leI)
7.1685 + have pos: "AE x. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable)
7.1686 + have "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) =
7.1687 + (SUP n. \<integral>\<^isup>+ x. (INF i:{n..}. u i x) \<partial>M)"
7.1688 + unfolding liminf_SUPR_INFI using pos u
7.1689 + by (intro positive_integral_monotone_convergence_SUP_AE)
7.1690 + (elim AE_mp, auto intro!: AE_I2 intro: le_INFI INF_subset)
7.1691 + also have "\<dots> \<le> liminf (\<lambda>n. integral\<^isup>P M (u n))"
7.1692 + unfolding liminf_SUPR_INFI
7.1693 + by (auto intro!: SUP_mono exI le_INFI positive_integral_mono INF_leI)
7.1694 finally show ?thesis .
7.1695 qed
7.1696
7.1697 lemma (in measure_space) measure_space_density:
7.1698 - assumes borel: "u \<in> borel_measurable M"
7.1699 + assumes u: "u \<in> borel_measurable M" "AE x. 0 \<le> u x"
7.1700 and M'[simp]: "M' = (M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)\<rparr>)"
7.1701 shows "measure_space M'"
7.1702 proof -
7.1703 interpret M': sigma_algebra M' by (intro sigma_algebra_cong) auto
7.1704 show ?thesis
7.1705 proof
7.1706 - show "measure M' {} = 0" unfolding M' by simp
7.1707 + have pos: "\<And>A. AE x. 0 \<le> u x * indicator A x"
7.1708 + using u by (auto simp: extreal_zero_le_0_iff)
7.1709 + then show "positive M' (measure M')" unfolding M'
7.1710 + using u(1) by (auto simp: positive_def intro!: positive_integral_positive)
7.1711 show "countably_additive M' (measure M')"
7.1712 proof (intro countably_additiveI)
7.1713 fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M'"
7.1714 - then have "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
7.1715 - using borel by (auto intro: borel_measurable_indicator)
7.1716 - moreover assume "disjoint_family A"
7.1717 - note psuminf_indicator[OF this]
7.1718 - ultimately show "(\<Sum>\<^isub>\<infinity>n. measure M' (A n)) = measure M' (\<Union>x. A x)"
7.1719 - by (simp add: positive_integral_psuminf[symmetric])
7.1720 + then have *: "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
7.1721 + using u by (auto intro: borel_measurable_indicator)
7.1722 + assume disj: "disjoint_family A"
7.1723 + have "(\<Sum>n. measure M' (A n)) = (\<integral>\<^isup>+ x. (\<Sum>n. u x * indicator (A n) x) \<partial>M)"
7.1724 + unfolding M' using u(1) *
7.1725 + by (simp add: positive_integral_suminf[OF _ pos, symmetric])
7.1726 + also have "\<dots> = (\<integral>\<^isup>+ x. u x * (\<Sum>n. indicator (A n) x) \<partial>M)" using u
7.1727 + by (intro positive_integral_cong_AE)
7.1728 + (elim AE_mp, auto intro!: AE_I2 suminf_cmult_extreal)
7.1729 + also have "\<dots> = (\<integral>\<^isup>+ x. u x * indicator (\<Union>n. A n) x \<partial>M)"
7.1730 + unfolding suminf_indicator[OF disj] ..
7.1731 + finally show "(\<Sum>n. measure M' (A n)) = measure M' (\<Union>x. A x)"
7.1732 + unfolding M' by simp
7.1733 qed
7.1734 qed
7.1735 qed
7.1736
7.1737 -lemma (in measure_space) positive_integral_translated_density:
7.1738 - assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
7.1739 - and M': "M' = (M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)\<rparr>)"
7.1740 - shows "integral\<^isup>P M' g = (\<integral>\<^isup>+ x. f x * g x \<partial>M)"
7.1741 -proof -
7.1742 - from measure_space_density[OF assms(1) M']
7.1743 - interpret T: measure_space M' .
7.1744 - have borel[simp]:
7.1745 - "borel_measurable M' = borel_measurable M"
7.1746 - "simple_function M' = simple_function M"
7.1747 - unfolding measurable_def simple_function_def_raw by (auto simp: M')
7.1748 - from borel_measurable_implies_simple_function_sequence[OF assms(2)]
7.1749 - obtain G where G: "\<And>i. simple_function M (G i)" "G \<up> g" by blast
7.1750 - note G_borel = borel_measurable_simple_function[OF this(1)]
7.1751 - from T.positive_integral_isoton[unfolded borel, OF `G \<up> g` G_borel]
7.1752 - have *: "(\<lambda>i. integral\<^isup>P M' (G i)) \<up> integral\<^isup>P M' g" .
7.1753 - { fix i
7.1754 - have [simp]: "finite (G i ` space M)"
7.1755 - using G(1) unfolding simple_function_def by auto
7.1756 - have "integral\<^isup>P M' (G i) = integral\<^isup>S M' (G i)"
7.1757 - using G T.positive_integral_eq_simple_integral by simp
7.1758 - also have "\<dots> = (\<integral>\<^isup>+x. f x * (\<Sum>y\<in>G i`space M. y * indicator (G i -` {y} \<inter> space M) x) \<partial>M)"
7.1759 - apply (simp add: simple_integral_def M')
7.1760 - apply (subst positive_integral_cmult[symmetric])
7.1761 - using G_borel assms(1) apply (fastsimp intro: borel_measurable_vimage)
7.1762 - apply (subst positive_integral_setsum[symmetric])
7.1763 - using G_borel assms(1) apply (fastsimp intro: borel_measurable_vimage)
7.1764 - by (simp add: setsum_right_distrib field_simps)
7.1765 - also have "\<dots> = (\<integral>\<^isup>+x. f x * G i x \<partial>M)"
7.1766 - by (auto intro!: positive_integral_cong
7.1767 - simp: indicator_def if_distrib setsum_cases)
7.1768 - finally have "integral\<^isup>P M' (G i) = (\<integral>\<^isup>+x. f x * G i x \<partial>M)" . }
7.1769 - with * have eq_Tg: "(\<lambda>i. (\<integral>\<^isup>+x. f x * G i x \<partial>M)) \<up> integral\<^isup>P M' g" by simp
7.1770 - from G(2) have "(\<lambda>i x. f x * G i x) \<up> (\<lambda>x. f x * g x)"
7.1771 - unfolding isoton_fun_expand by (auto intro!: isoton_cmult_right)
7.1772 - then have "(\<lambda>i. (\<integral>\<^isup>+x. f x * G i x \<partial>M)) \<up> (\<integral>\<^isup>+x. f x * g x \<partial>M)"
7.1773 - using assms(1) G_borel by (auto intro!: positive_integral_isoton borel_measurable_pextreal_times)
7.1774 - with eq_Tg show "integral\<^isup>P M' g = (\<integral>\<^isup>+x. f x * g x \<partial>M)"
7.1775 - unfolding isoton_def by simp
7.1776 -qed
7.1777 -
7.1778 lemma (in measure_space) positive_integral_null_set:
7.1779 assumes "N \<in> null_sets" shows "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = 0"
7.1780 proof -
7.1781 @@ -1410,144 +1468,199 @@
7.1782 then show ?thesis by simp
7.1783 qed
7.1784
7.1785 -lemma (in measure_space) positive_integral_Markov_inequality:
7.1786 - assumes borel: "u \<in> borel_measurable M" and "A \<in> sets M" and c: "c \<noteq> \<omega>"
7.1787 - shows "\<mu> ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
7.1788 - (is "\<mu> ?A \<le> _ * ?PI")
7.1789 +lemma (in measure_space) positive_integral_translated_density:
7.1790 + assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
7.1791 + assumes g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
7.1792 + and M': "M' = (M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)\<rparr>)"
7.1793 + shows "integral\<^isup>P M' g = (\<integral>\<^isup>+ x. f x * g x \<partial>M)"
7.1794 proof -
7.1795 - have "?A \<in> sets M"
7.1796 - using `A \<in> sets M` borel by auto
7.1797 - hence "\<mu> ?A = (\<integral>\<^isup>+ x. indicator ?A x \<partial>M)"
7.1798 - using positive_integral_indicator by simp
7.1799 - also have "\<dots> \<le> (\<integral>\<^isup>+ x. c * (u x * indicator A x) \<partial>M)"
7.1800 - proof (rule positive_integral_mono)
7.1801 - fix x assume "x \<in> space M"
7.1802 - show "indicator ?A x \<le> c * (u x * indicator A x)"
7.1803 - by (cases "x \<in> ?A") auto
7.1804 - qed
7.1805 - also have "\<dots> = c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
7.1806 - using assms
7.1807 - by (auto intro!: positive_integral_cmult borel_measurable_indicator)
7.1808 - finally show ?thesis .
7.1809 + from measure_space_density[OF f M']
7.1810 + interpret T: measure_space M' .
7.1811 + have borel[simp]:
7.1812 + "borel_measurable M' = borel_measurable M"
7.1813 + "simple_function M' = simple_function M"
7.1814 + unfolding measurable_def simple_function_def_raw by (auto simp: M')
7.1815 + from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess G . note G = this
7.1816 + note G' = borel_measurable_simple_function[OF this(1)] simple_functionD[OF G(1)]
7.1817 + note G'(2)[simp]
7.1818 + { fix P have "AE x. P x \<Longrightarrow> AE x in M'. P x"
7.1819 + using positive_integral_null_set[of _ f]
7.1820 + unfolding T.almost_everywhere_def almost_everywhere_def
7.1821 + by (auto simp: M') }
7.1822 + note ac = this
7.1823 + from G(4) g(2) have G_M': "AE x in M'. (SUP i. G i x) = g x"
7.1824 + by (auto intro!: ac split: split_max)
7.1825 + { fix i
7.1826 + let "?I y x" = "indicator (G i -` {y} \<inter> space M) x"
7.1827 + { fix x assume *: "x \<in> space M" "0 \<le> f x" "0 \<le> g x"
7.1828 + then have [simp]: "G i ` space M \<inter> {y. G i x = y \<and> x \<in> space M} = {G i x}" by auto
7.1829 + from * G' G have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * (\<Sum>y\<in>G i`space M. (y * ?I y x))"
7.1830 + by (subst setsum_extreal_right_distrib) (auto simp: ac_simps)
7.1831 + also have "\<dots> = f x * G i x"
7.1832 + by (simp add: indicator_def if_distrib setsum_cases)
7.1833 + finally have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * G i x" . }
7.1834 + note to_singleton = this
7.1835 + have "integral\<^isup>P M' (G i) = integral\<^isup>S M' (G i)"
7.1836 + using G T.positive_integral_eq_simple_integral by simp
7.1837 + also have "\<dots> = (\<Sum>y\<in>G i`space M. y * (\<integral>\<^isup>+x. f x * ?I y x \<partial>M))"
7.1838 + unfolding simple_integral_def M' by simp
7.1839 + also have "\<dots> = (\<Sum>y\<in>G i`space M. (\<integral>\<^isup>+x. y * (f x * ?I y x) \<partial>M))"
7.1840 + using f G' G by (auto intro!: setsum_cong positive_integral_cmult[symmetric])
7.1841 + also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) \<partial>M)"
7.1842 + using f G' G by (auto intro!: positive_integral_setsum[symmetric])
7.1843 + finally have "integral\<^isup>P M' (G i) = (\<integral>\<^isup>+x. f x * G i x \<partial>M)"
7.1844 + using f g G' to_singleton by (auto intro!: positive_integral_cong_AE) }
7.1845 + note [simp] = this
7.1846 + have "integral\<^isup>P M' g = (SUP i. integral\<^isup>P M' (G i))" using G'(1) G_M'(1) G
7.1847 + using T.positive_integral_monotone_convergence_SUP[symmetric, OF `incseq G`]
7.1848 + by (simp cong: T.positive_integral_cong_AE)
7.1849 + also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f x * G i x \<partial>M))" by simp
7.1850 + also have "\<dots> = (\<integral>\<^isup>+x. (SUP i. f x * G i x) \<partial>M)"
7.1851 + using f G' G(2)[THEN incseq_SucD] G
7.1852 + by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
7.1853 + (auto simp: extreal_mult_left_mono le_fun_def extreal_zero_le_0_iff)
7.1854 + also have "\<dots> = (\<integral>\<^isup>+x. f x * g x \<partial>M)" using f G' G g
7.1855 + by (intro positive_integral_cong_AE)
7.1856 + (auto simp add: SUPR_extreal_cmult split: split_max)
7.1857 + finally show "integral\<^isup>P M' g = (\<integral>\<^isup>+x. f x * g x \<partial>M)" .
7.1858 qed
7.1859
7.1860 lemma (in measure_space) positive_integral_0_iff:
7.1861 - assumes borel: "u \<in> borel_measurable M"
7.1862 + assumes u: "u \<in> borel_measurable M" and pos: "AE x. 0 \<le> u x"
7.1863 shows "integral\<^isup>P M u = 0 \<longleftrightarrow> \<mu> {x\<in>space M. u x \<noteq> 0} = 0"
7.1864 (is "_ \<longleftrightarrow> \<mu> ?A = 0")
7.1865 proof -
7.1866 - have A: "?A \<in> sets M" using borel by auto
7.1867 - have u: "(\<integral>\<^isup>+ x. u x * indicator ?A x \<partial>M) = integral\<^isup>P M u"
7.1868 + have u_eq: "(\<integral>\<^isup>+ x. u x * indicator ?A x \<partial>M) = integral\<^isup>P M u"
7.1869 by (auto intro!: positive_integral_cong simp: indicator_def)
7.1870 -
7.1871 show ?thesis
7.1872 proof
7.1873 assume "\<mu> ?A = 0"
7.1874 - hence "?A \<in> null_sets" using `?A \<in> sets M` by auto
7.1875 - from positive_integral_null_set[OF this]
7.1876 - have "0 = (\<integral>\<^isup>+ x. u x * indicator ?A x \<partial>M)" by simp
7.1877 - thus "integral\<^isup>P M u = 0" unfolding u by simp
7.1878 + with positive_integral_null_set[of ?A u] u
7.1879 + show "integral\<^isup>P M u = 0" by (simp add: u_eq)
7.1880 next
7.1881 + { fix r :: extreal and n :: nat assume gt_1: "1 \<le> real n * r"
7.1882 + then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_extreal_def)
7.1883 + then have "0 \<le> r" by (auto simp add: extreal_zero_less_0_iff) }
7.1884 + note gt_1 = this
7.1885 assume *: "integral\<^isup>P M u = 0"
7.1886 - let "?M n" = "{x \<in> space M. 1 \<le> of_nat n * u x}"
7.1887 + let "?M n" = "{x \<in> space M. 1 \<le> real (n::nat) * u x}"
7.1888 have "0 = (SUP n. \<mu> (?M n \<inter> ?A))"
7.1889 proof -
7.1890 - { fix n
7.1891 - from positive_integral_Markov_inequality[OF borel `?A \<in> sets M`, of "of_nat n"]
7.1892 - have "\<mu> (?M n \<inter> ?A) = 0" unfolding * u by simp }
7.1893 + { fix n :: nat
7.1894 + from positive_integral_Markov_inequality[OF u pos, of ?A "extreal (real n)"]
7.1895 + have "\<mu> (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp
7.1896 + moreover have "0 \<le> \<mu> (?M n \<inter> ?A)" using u by auto
7.1897 + ultimately have "\<mu> (?M n \<inter> ?A) = 0" by auto }
7.1898 thus ?thesis by simp
7.1899 qed
7.1900 also have "\<dots> = \<mu> (\<Union>n. ?M n \<inter> ?A)"
7.1901 proof (safe intro!: continuity_from_below)
7.1902 fix n show "?M n \<inter> ?A \<in> sets M"
7.1903 - using borel by (auto intro!: Int)
7.1904 + using u by (auto intro!: Int)
7.1905 next
7.1906 - fix n x assume "1 \<le> of_nat n * u x"
7.1907 - also have "\<dots> \<le> of_nat (Suc n) * u x"
7.1908 - by (subst (1 2) mult_commute) (auto intro!: pextreal_mult_cancel)
7.1909 - finally show "1 \<le> of_nat (Suc n) * u x" .
7.1910 + show "incseq (\<lambda>n. {x \<in> space M. 1 \<le> real n * u x} \<inter> {x \<in> space M. u x \<noteq> 0})"
7.1911 + proof (safe intro!: incseq_SucI)
7.1912 + fix n :: nat and x
7.1913 + assume *: "1 \<le> real n * u x"
7.1914 + also from gt_1[OF this] have "real n * u x \<le> real (Suc n) * u x"
7.1915 + using `0 \<le> u x` by (auto intro!: extreal_mult_right_mono)
7.1916 + finally show "1 \<le> real (Suc n) * u x" by auto
7.1917 + qed
7.1918 qed
7.1919 - also have "\<dots> = \<mu> ?A"
7.1920 - proof (safe intro!: arg_cong[where f="\<mu>"])
7.1921 - fix x assume "u x \<noteq> 0" and [simp, intro]: "x \<in> space M"
7.1922 + also have "\<dots> = \<mu> {x\<in>space M. 0 < u x}"
7.1923 + proof (safe intro!: arg_cong[where f="\<mu>"] dest!: gt_1)
7.1924 + fix x assume "0 < u x" and [simp, intro]: "x \<in> space M"
7.1925 show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
7.1926 proof (cases "u x")
7.1927 - case (preal r)
7.1928 - obtain j where "1 / r \<le> of_nat j" using ex_le_of_nat ..
7.1929 - hence "1 / r * r \<le> of_nat j * r" using preal unfolding mult_le_cancel_right by auto
7.1930 - hence "1 \<le> of_nat j * r" using preal `u x \<noteq> 0` by auto
7.1931 - thus ?thesis using `u x \<noteq> 0` preal by (auto simp: real_of_nat_def[symmetric])
7.1932 - qed auto
7.1933 - qed
7.1934 - finally show "\<mu> ?A = 0" by simp
7.1935 + case (real r) with `0 < u x` have "0 < r" by auto
7.1936 + obtain j :: nat where "1 / r \<le> real j" using real_arch_simple ..
7.1937 + hence "1 / r * r \<le> real j * r" unfolding mult_le_cancel_right using `0 < r` by auto
7.1938 + hence "1 \<le> real j * r" using real `0 < r` by auto
7.1939 + thus ?thesis using `0 < r` real by (auto simp: one_extreal_def)
7.1940 + qed (insert `0 < u x`, auto)
7.1941 + qed auto
7.1942 + finally have "\<mu> {x\<in>space M. 0 < u x} = 0" by simp
7.1943 + moreover
7.1944 + from pos have "AE x. \<not> (u x < 0)" by auto
7.1945 + then have "\<mu> {x\<in>space M. u x < 0} = 0"
7.1946 + using AE_iff_null_set u by auto
7.1947 + moreover have "\<mu> {x\<in>space M. u x \<noteq> 0} = \<mu> {x\<in>space M. u x < 0} + \<mu> {x\<in>space M. 0 < u x}"
7.1948 + using u by (subst measure_additive) (auto intro!: arg_cong[where f=\<mu>])
7.1949 + ultimately show "\<mu> ?A = 0" by simp
7.1950 qed
7.1951 qed
7.1952
7.1953 lemma (in measure_space) positive_integral_0_iff_AE:
7.1954 assumes u: "u \<in> borel_measurable M"
7.1955 - shows "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x. u x = 0)"
7.1956 + shows "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x. u x \<le> 0)"
7.1957 proof -
7.1958 - have sets: "{x\<in>space M. u x \<noteq> 0} \<in> sets M"
7.1959 + have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M"
7.1960 using u by auto
7.1961 - then show ?thesis
7.1962 - using positive_integral_0_iff[OF u] AE_iff_null_set[OF sets] by auto
7.1963 + from positive_integral_0_iff[of "\<lambda>x. max 0 (u x)"]
7.1964 + have "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x. max 0 (u x) = 0)"
7.1965 + unfolding positive_integral_max_0
7.1966 + using AE_iff_null_set[OF sets] u by auto
7.1967 + also have "\<dots> \<longleftrightarrow> (AE x. u x \<le> 0)" by (auto split: split_max)
7.1968 + finally show ?thesis .
7.1969 qed
7.1970
7.1971 lemma (in measure_space) positive_integral_restricted:
7.1972 - assumes "A \<in> sets M"
7.1973 + assumes A: "A \<in> sets M"
7.1974 shows "integral\<^isup>P (restricted_space A) f = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)"
7.1975 (is "integral\<^isup>P ?R f = integral\<^isup>P M ?f")
7.1976 proof -
7.1977 - have msR: "measure_space ?R" by (rule restricted_measure_space[OF `A \<in> sets M`])
7.1978 - then interpret R: measure_space ?R .
7.1979 - have saR: "sigma_algebra ?R" by fact
7.1980 - have *: "integral\<^isup>P ?R f = integral\<^isup>P ?R ?f"
7.1981 - by (intro R.positive_integral_cong) auto
7.1982 + interpret R: measure_space ?R
7.1983 + by (rule restricted_measure_space) fact
7.1984 + let "?I g x" = "g x * indicator A x :: extreal"
7.1985 show ?thesis
7.1986 - unfolding * positive_integral_def
7.1987 - unfolding simple_function_restricted[OF `A \<in> sets M`]
7.1988 - apply (simp add: SUPR_def)
7.1989 - apply (rule arg_cong[where f=Sup])
7.1990 - proof (auto simp add: image_iff simple_integral_restricted[OF `A \<in> sets M`])
7.1991 - fix g assume "simple_function M (\<lambda>x. g x * indicator A x)"
7.1992 - "g \<le> f"
7.1993 - then show "\<exists>x. simple_function M x \<and> x \<le> (\<lambda>x. f x * indicator A x) \<and>
7.1994 - (\<integral>\<^isup>Sx. g x * indicator A x \<partial>M) = integral\<^isup>S M x"
7.1995 - apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
7.1996 - by (auto simp: indicator_def le_fun_def)
7.1997 + unfolding positive_integral_def
7.1998 + unfolding simple_function_restricted[OF A]
7.1999 + unfolding AE_restricted[OF A]
7.2000 + proof (safe intro!: SUPR_eq)
7.2001 + fix g assume g: "simple_function M (?I g)" and le: "g \<le> max 0 \<circ> f"
7.2002 + show "\<exists>j\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> ?I f}.
7.2003 + integral\<^isup>S (restricted_space A) g \<le> integral\<^isup>S M j"
7.2004 + proof (safe intro!: bexI[of _ "?I g"])
7.2005 + show "integral\<^isup>S (restricted_space A) g \<le> integral\<^isup>S M (?I g)"
7.2006 + using g A by (simp add: simple_integral_restricted)
7.2007 + show "?I g \<le> max 0 \<circ> ?I f"
7.2008 + using le by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
7.2009 + qed fact
7.2010 next
7.2011 - fix g assume g: "simple_function M g" "g \<le> (\<lambda>x. f x * indicator A x)"
7.2012 - then have *: "(\<lambda>x. g x * indicator A x) = g"
7.2013 - "\<And>x. g x * indicator A x = g x"
7.2014 - "\<And>x. g x \<le> f x"
7.2015 - by (auto simp: le_fun_def fun_eq_iff indicator_def split: split_if_asm)
7.2016 - from g show "\<exists>x. simple_function M (\<lambda>xa. x xa * indicator A xa) \<and> x \<le> f \<and>
7.2017 - integral\<^isup>S M g = integral\<^isup>S M (\<lambda>xa. x xa * indicator A xa)"
7.2018 - using `A \<in> sets M`[THEN sets_into_space]
7.2019 - apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
7.2020 - by (fastsimp simp: le_fun_def *)
7.2021 + fix g assume g: "simple_function M g" and le: "g \<le> max 0 \<circ> ?I f"
7.2022 + show "\<exists>i\<in>{g. simple_function M (?I g) \<and> g \<le> max 0 \<circ> f}.
7.2023 + integral\<^isup>S M g \<le> integral\<^isup>S (restricted_space A) i"
7.2024 + proof (safe intro!: bexI[of _ "?I g"])
7.2025 + show "?I g \<le> max 0 \<circ> f"
7.2026 + using le by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
7.2027 + from le have "\<And>x. g x \<le> ?I (?I g) x"
7.2028 + by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
7.2029 + then show "integral\<^isup>S M g \<le> integral\<^isup>S (restricted_space A) (?I g)"
7.2030 + using A g by (auto intro!: simple_integral_mono simp: simple_integral_restricted)
7.2031 + show "simple_function M (?I (?I g))" using g A by auto
7.2032 + qed
7.2033 qed
7.2034 qed
7.2035
7.2036 lemma (in measure_space) positive_integral_subalgebra:
7.2037 - assumes borel: "f \<in> borel_measurable N"
7.2038 + assumes f: "f \<in> borel_measurable N" "AE x in N. 0 \<le> f x"
7.2039 and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
7.2040 and sa: "sigma_algebra N"
7.2041 shows "integral\<^isup>P N f = integral\<^isup>P M f"
7.2042 proof -
7.2043 interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
7.2044 - from N.borel_measurable_implies_simple_function_sequence[OF borel]
7.2045 - obtain fs where Nsf: "\<And>i. simple_function N (fs i)" and "fs \<up> f" by blast
7.2046 - then have sf: "\<And>i. simple_function M (fs i)"
7.2047 - using simple_function_subalgebra[OF _ N(1,2)] by blast
7.2048 - from N.positive_integral_isoton_simple[OF `fs \<up> f` Nsf]
7.2049 + from N.borel_measurable_implies_simple_function_sequence'[OF f(1)] guess fs . note fs = this
7.2050 + note sf = simple_function_subalgebra[OF fs(1) N(1,2)]
7.2051 + from N.positive_integral_monotone_convergence_simple[OF fs(2,5,1), symmetric]
7.2052 have "integral\<^isup>P N f = (SUP i. \<Sum>x\<in>fs i ` space M. x * N.\<mu> (fs i -` {x} \<inter> space M))"
7.2053 - unfolding isoton_def simple_integral_def `space N = space M` by simp
7.2054 + unfolding fs(4) positive_integral_max_0
7.2055 + unfolding simple_integral_def `space N = space M` by simp
7.2056 also have "\<dots> = (SUP i. \<Sum>x\<in>fs i ` space M. x * \<mu> (fs i -` {x} \<inter> space M))"
7.2057 - using N N.simple_functionD(2)[OF Nsf] unfolding `space N = space M` by auto
7.2058 + using N N.simple_functionD(2)[OF fs(1)] unfolding `space N = space M` by auto
7.2059 also have "\<dots> = integral\<^isup>P M f"
7.2060 - using positive_integral_isoton_simple[OF `fs \<up> f` sf]
7.2061 - unfolding isoton_def simple_integral_def `space N = space M` by simp
7.2062 + using positive_integral_monotone_convergence_simple[OF fs(2,5) sf, symmetric]
7.2063 + unfolding fs(4) positive_integral_max_0
7.2064 + unfolding simple_integral_def `space N = space M` by simp
7.2065 finally show ?thesis .
7.2066 qed
7.2067
7.2068 @@ -1555,16 +1668,15 @@
7.2069
7.2070 definition integrable where
7.2071 "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and>
7.2072 - (\<integral>\<^isup>+ x. Real (f x) \<partial>M) \<noteq> \<omega> \<and>
7.2073 - (\<integral>\<^isup>+ x. Real (- f x) \<partial>M) \<noteq> \<omega>"
7.2074 + (\<integral>\<^isup>+ x. extreal (f x) \<partial>M) \<noteq> \<infinity> \<and> (\<integral>\<^isup>+ x. extreal (- f x) \<partial>M) \<noteq> \<infinity>"
7.2075
7.2076 lemma integrableD[dest]:
7.2077 assumes "integrable M f"
7.2078 - shows "f \<in> borel_measurable M" "(\<integral>\<^isup>+ x. Real (f x) \<partial>M) \<noteq> \<omega>" "(\<integral>\<^isup>+ x. Real (- f x) \<partial>M) \<noteq> \<omega>"
7.2079 + shows "f \<in> borel_measurable M" "(\<integral>\<^isup>+ x. extreal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^isup>+ x. extreal (- f x) \<partial>M) \<noteq> \<infinity>"
7.2080 using assms unfolding integrable_def by auto
7.2081
7.2082 definition lebesgue_integral_def:
7.2083 - "integral\<^isup>L M f = real ((\<integral>\<^isup>+ x. Real (f x) \<partial>M)) - real ((\<integral>\<^isup>+ x. Real (- f x) \<partial>M))"
7.2084 + "integral\<^isup>L M f = real ((\<integral>\<^isup>+ x. extreal (f x) \<partial>M)) - real ((\<integral>\<^isup>+ x. extreal (- f x) \<partial>M))"
7.2085
7.2086 syntax
7.2087 "_lebesgue_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110)
7.2088 @@ -1572,6 +1684,17 @@
7.2089 translations
7.2090 "\<integral> x. f \<partial>M" == "CONST integral\<^isup>L M (%x. f)"
7.2091
7.2092 +lemma (in measure_space) integrableE:
7.2093 + assumes "integrable M f"
7.2094 + obtains r q where
7.2095 + "(\<integral>\<^isup>+x. extreal (f x)\<partial>M) = extreal r"
7.2096 + "(\<integral>\<^isup>+x. extreal (-f x)\<partial>M) = extreal q"
7.2097 + "f \<in> borel_measurable M" "integral\<^isup>L M f = r - q"
7.2098 + using assms unfolding integrable_def lebesgue_integral_def
7.2099 + using positive_integral_positive[of "\<lambda>x. extreal (f x)"]
7.2100 + using positive_integral_positive[of "\<lambda>x. extreal (-f x)"]
7.2101 + by (cases rule: extreal2_cases[of "(\<integral>\<^isup>+x. extreal (-f x)\<partial>M)" "(\<integral>\<^isup>+x. extreal (f x)\<partial>M)"]) auto
7.2102 +
7.2103 lemma (in measure_space) integral_cong:
7.2104 assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
7.2105 shows "integral\<^isup>L M f = integral\<^isup>L M g"
7.2106 @@ -1580,21 +1703,16 @@
7.2107 lemma (in measure_space) integral_cong_measure:
7.2108 assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
7.2109 shows "integral\<^isup>L N f = integral\<^isup>L M f"
7.2110 -proof -
7.2111 - interpret v: measure_space N
7.2112 - by (rule measure_space_cong) fact+
7.2113 - show ?thesis
7.2114 - by (simp add: positive_integral_cong_measure[OF assms] lebesgue_integral_def)
7.2115 -qed
7.2116 + by (simp add: positive_integral_cong_measure[OF assms] lebesgue_integral_def)
7.2117
7.2118 lemma (in measure_space) integral_cong_AE:
7.2119 assumes cong: "AE x. f x = g x"
7.2120 shows "integral\<^isup>L M f = integral\<^isup>L M g"
7.2121 proof -
7.2122 - have "AE x. Real (f x) = Real (g x)" using cong by auto
7.2123 - moreover have "AE x. Real (- f x) = Real (- g x)" using cong by auto
7.2124 - ultimately show ?thesis
7.2125 - by (simp cong: positive_integral_cong_AE add: lebesgue_integral_def)
7.2126 + have *: "AE x. extreal (f x) = extreal (g x)"
7.2127 + "AE x. extreal (- f x) = extreal (- g x)" using cong by auto
7.2128 + show ?thesis
7.2129 + unfolding *[THEN positive_integral_cong_AE] lebesgue_integral_def ..
7.2130 qed
7.2131
7.2132 lemma (in measure_space) integrable_cong:
7.2133 @@ -1602,11 +1720,14 @@
7.2134 by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
7.2135
7.2136 lemma (in measure_space) integral_eq_positive_integral:
7.2137 - assumes "\<And>x. 0 \<le> f x"
7.2138 - shows "integral\<^isup>L M f = real (\<integral>\<^isup>+ x. Real (f x) \<partial>M)"
7.2139 + assumes f: "\<And>x. 0 \<le> f x"
7.2140 + shows "integral\<^isup>L M f = real (\<integral>\<^isup>+ x. extreal (f x) \<partial>M)"
7.2141 proof -
7.2142 - have "\<And>x. Real (- f x) = 0" using assms by simp
7.2143 - thus ?thesis by (simp del: Real_eq_0 add: lebesgue_integral_def)
7.2144 + { fix x have "max 0 (extreal (- f x)) = 0" using f[of x] by (simp split: split_max) }
7.2145 + then have "0 = (\<integral>\<^isup>+ x. max 0 (extreal (- f x)) \<partial>M)" by simp
7.2146 + also have "\<dots> = (\<integral>\<^isup>+ x. extreal (- f x) \<partial>M)" unfolding positive_integral_max_0 ..
7.2147 + finally show ?thesis
7.2148 + unfolding lebesgue_integral_def by simp
7.2149 qed
7.2150
7.2151 lemma (in measure_space) integral_vimage:
7.2152 @@ -1616,7 +1737,7 @@
7.2153 proof -
7.2154 interpret T: measure_space M' by (rule measure_space_vimage[OF T])
7.2155 from measurable_comp[OF measure_preservingD2[OF T(2)], of f borel]
7.2156 - have borel: "(\<lambda>x. Real (f x)) \<in> borel_measurable M'" "(\<lambda>x. Real (- f x)) \<in> borel_measurable M'"
7.2157 + have borel: "(\<lambda>x. extreal (f x)) \<in> borel_measurable M'" "(\<lambda>x. extreal (- f x)) \<in> borel_measurable M'"
7.2158 and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
7.2159 using f by (auto simp: comp_def)
7.2160 then show ?thesis
7.2161 @@ -1631,7 +1752,7 @@
7.2162 proof -
7.2163 interpret T: measure_space M' by (rule measure_space_vimage[OF T])
7.2164 from measurable_comp[OF measure_preservingD2[OF T(2)], of f borel]
7.2165 - have borel: "(\<lambda>x. Real (f x)) \<in> borel_measurable M'" "(\<lambda>x. Real (- f x)) \<in> borel_measurable M'"
7.2166 + have borel: "(\<lambda>x. extreal (f x)) \<in> borel_measurable M'" "(\<lambda>x. extreal (- f x)) \<in> borel_measurable M'"
7.2167 and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
7.2168 using f by (auto simp: comp_def)
7.2169 then show ?thesis
7.2170 @@ -1649,10 +1770,10 @@
7.2171 and f_def: "\<And>x. f x = u x - v x" and pos: "\<And>x. 0 \<le> u x" "\<And>x. 0 \<le> v x"
7.2172 shows "integrable M f" and "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
7.2173 proof -
7.2174 - let "?f x" = "Real (f x)"
7.2175 - let "?mf x" = "Real (- f x)"
7.2176 - let "?u x" = "Real (u x)"
7.2177 - let "?v x" = "Real (v x)"
7.2178 + let "?f x" = "max 0 (extreal (f x))"
7.2179 + let "?mf x" = "max 0 (extreal (- f x))"
7.2180 + let "?u x" = "max 0 (extreal (u x))"
7.2181 + let "?v x" = "max 0 (extreal (v x))"
7.2182
7.2183 from borel_measurable_diff[of u v] integrable
7.2184 have f_borel: "?f \<in> borel_measurable M" and
7.2185 @@ -1662,73 +1783,62 @@
7.2186 "f \<in> borel_measurable M"
7.2187 by (auto simp: f_def[symmetric] integrable_def)
7.2188
7.2189 - have "(\<integral>\<^isup>+ x. Real (u x - v x) \<partial>M) \<le> integral\<^isup>P M ?u"
7.2190 - using pos by (auto intro!: positive_integral_mono)
7.2191 - moreover have "(\<integral>\<^isup>+ x. Real (v x - u x) \<partial>M) \<le> integral\<^isup>P M ?v"
7.2192 - using pos by (auto intro!: positive_integral_mono)
7.2193 + have "(\<integral>\<^isup>+ x. extreal (u x - v x) \<partial>M) \<le> integral\<^isup>P M ?u"
7.2194 + using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
7.2195 + moreover have "(\<integral>\<^isup>+ x. extreal (v x - u x) \<partial>M) \<le> integral\<^isup>P M ?v"
7.2196 + using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
7.2197 ultimately show f: "integrable M f"
7.2198 using `integrable M u` `integrable M v` `f \<in> borel_measurable M`
7.2199 - by (auto simp: integrable_def f_def)
7.2200 - hence mf: "integrable M (\<lambda>x. - f x)" ..
7.2201 -
7.2202 - have *: "\<And>x. Real (- v x) = 0" "\<And>x. Real (- u x) = 0"
7.2203 - using pos by auto
7.2204 + by (auto simp: integrable_def f_def positive_integral_max_0)
7.2205
7.2206 have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
7.2207 - unfolding f_def using pos by simp
7.2208 - hence "(\<integral>\<^isup>+ x. ?u x + ?mf x \<partial>M) = (\<integral>\<^isup>+ x. ?v x + ?f x \<partial>M)" by simp
7.2209 - hence "real (integral\<^isup>P M ?u + integral\<^isup>P M ?mf) =
7.2210 + unfolding f_def using pos by (simp split: split_max)
7.2211 + then have "(\<integral>\<^isup>+ x. ?u x + ?mf x \<partial>M) = (\<integral>\<^isup>+ x. ?v x + ?f x \<partial>M)" by simp
7.2212 + then have "real (integral\<^isup>P M ?u + integral\<^isup>P M ?mf) =
7.2213 real (integral\<^isup>P M ?v + integral\<^isup>P M ?f)"
7.2214 - using positive_integral_add[OF u_borel mf_borel]
7.2215 - using positive_integral_add[OF v_borel f_borel]
7.2216 + using positive_integral_add[OF u_borel _ mf_borel]
7.2217 + using positive_integral_add[OF v_borel _ f_borel]
7.2218 by auto
7.2219 then show "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
7.2220 - using f mf `integrable M u` `integrable M v`
7.2221 - unfolding lebesgue_integral_def integrable_def *
7.2222 - by (cases "integral\<^isup>P M ?f", cases "integral\<^isup>P M ?mf", cases "integral\<^isup>P M ?v", cases "integral\<^isup>P M ?u")
7.2223 - (auto simp add: field_simps)
7.2224 + unfolding positive_integral_max_0
7.2225 + unfolding pos[THEN integral_eq_positive_integral]
7.2226 + using integrable f by (auto elim!: integrableE)
7.2227 qed
7.2228
7.2229 lemma (in measure_space) integral_linear:
7.2230 assumes "integrable M f" "integrable M g" and "0 \<le> a"
7.2231 shows "integrable M (\<lambda>t. a * f t + g t)"
7.2232 - and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^isup>L M f + integral\<^isup>L M g"
7.2233 + and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^isup>L M f + integral\<^isup>L M g" (is ?EQ)
7.2234 proof -
7.2235 - let ?PI = "integral\<^isup>P M"
7.2236 - let "?f x" = "Real (f x)"
7.2237 - let "?g x" = "Real (g x)"
7.2238 - let "?mf x" = "Real (- f x)"
7.2239 - let "?mg x" = "Real (- g x)"
7.2240 + let "?f x" = "max 0 (extreal (f x))"
7.2241 + let "?g x" = "max 0 (extreal (g x))"
7.2242 + let "?mf x" = "max 0 (extreal (- f x))"
7.2243 + let "?mg x" = "max 0 (extreal (- g x))"
7.2244 let "?p t" = "max 0 (a * f t) + max 0 (g t)"
7.2245 let "?n t" = "max 0 (- (a * f t)) + max 0 (- g t)"
7.2246
7.2247 - have pos: "?f \<in> borel_measurable M" "?g \<in> borel_measurable M"
7.2248 - and neg: "?mf \<in> borel_measurable M" "?mg \<in> borel_measurable M"
7.2249 - and p: "?p \<in> borel_measurable M"
7.2250 - and n: "?n \<in> borel_measurable M"
7.2251 - using assms by (simp_all add: integrable_def)
7.2252 + from assms have linear:
7.2253 + "(\<integral>\<^isup>+ x. extreal a * ?f x + ?g x \<partial>M) = extreal a * integral\<^isup>P M ?f + integral\<^isup>P M ?g"
7.2254 + "(\<integral>\<^isup>+ x. extreal a * ?mf x + ?mg x \<partial>M) = extreal a * integral\<^isup>P M ?mf + integral\<^isup>P M ?mg"
7.2255 + by (auto intro!: positive_integral_linear simp: integrable_def)
7.2256
7.2257 - have *: "\<And>x. Real (?p x) = Real a * ?f x + ?g x"
7.2258 - "\<And>x. Real (?n x) = Real a * ?mf x + ?mg x"
7.2259 - "\<And>x. Real (- ?p x) = 0"
7.2260 - "\<And>x. Real (- ?n x) = 0"
7.2261 - using `0 \<le> a` by (auto simp: max_def min_def zero_le_mult_iff mult_le_0_iff add_nonpos_nonpos)
7.2262 -
7.2263 - note linear =
7.2264 - positive_integral_linear[OF pos]
7.2265 - positive_integral_linear[OF neg]
7.2266 + have *: "(\<integral>\<^isup>+x. extreal (- ?p x) \<partial>M) = 0" "(\<integral>\<^isup>+x. extreal (- ?n x) \<partial>M) = 0"
7.2267 + using `0 \<le> a` assms by (auto simp: positive_integral_0_iff_AE integrable_def)
7.2268 + have **: "\<And>x. extreal a * ?f x + ?g x = max 0 (extreal (?p x))"
7.2269 + "\<And>x. extreal a * ?mf x + ?mg x = max 0 (extreal (?n x))"
7.2270 + using `0 \<le> a` by (auto split: split_max simp: zero_le_mult_iff mult_le_0_iff)
7.2271
7.2272 have "integrable M ?p" "integrable M ?n"
7.2273 "\<And>t. a * f t + g t = ?p t - ?n t" "\<And>t. 0 \<le> ?p t" "\<And>t. 0 \<le> ?n t"
7.2274 - using assms p n unfolding integrable_def * linear by auto
7.2275 + using linear assms unfolding integrable_def ** *
7.2276 + by (auto simp: positive_integral_max_0)
7.2277 note diff = integral_of_positive_diff[OF this]
7.2278
7.2279 show "integrable M (\<lambda>t. a * f t + g t)" by (rule diff)
7.2280 -
7.2281 - from assms show "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^isup>L M f + integral\<^isup>L M g"
7.2282 - unfolding diff(2) unfolding lebesgue_integral_def * linear integrable_def
7.2283 - by (cases "?PI ?f", cases "?PI ?mf", cases "?PI ?g", cases "?PI ?mg")
7.2284 - (auto simp add: field_simps zero_le_mult_iff)
7.2285 + from assms linear show ?EQ
7.2286 + unfolding diff(2) ** positive_integral_max_0
7.2287 + unfolding lebesgue_integral_def *
7.2288 + by (auto elim!: integrableE simp: field_simps)
7.2289 qed
7.2290
7.2291 lemma (in measure_space) integral_add[simp, intro]:
7.2292 @@ -1772,13 +1882,13 @@
7.2293 and mono: "AE t. f t \<le> g t"
7.2294 shows "integral\<^isup>L M f \<le> integral\<^isup>L M g"
7.2295 proof -
7.2296 - have "AE x. Real (f x) \<le> Real (g x)"
7.2297 + have "AE x. extreal (f x) \<le> extreal (g x)"
7.2298 using mono by auto
7.2299 - moreover have "AE x. Real (- g x) \<le> Real (- f x)"
7.2300 + moreover have "AE x. extreal (- g x) \<le> extreal (- f x)"
7.2301 using mono by auto
7.2302 ultimately show ?thesis using fg
7.2303 - by (auto simp: lebesgue_integral_def integrable_def diff_minus
7.2304 - intro!: add_mono real_of_pextreal_mono positive_integral_mono_AE)
7.2305 + by (auto intro!: add_mono positive_integral_mono_AE real_of_extreal_positive_mono
7.2306 + simp: positive_integral_positive lebesgue_integral_def diff_minus)
7.2307 qed
7.2308
7.2309 lemma (in measure_space) integral_mono:
7.2310 @@ -1795,20 +1905,21 @@
7.2311 by auto
7.2312
7.2313 lemma (in measure_space) integral_indicator[simp, intro]:
7.2314 - assumes "a \<in> sets M" and "\<mu> a \<noteq> \<omega>"
7.2315 - shows "integral\<^isup>L M (indicator a) = real (\<mu> a)" (is ?int)
7.2316 - and "integrable M (indicator a)" (is ?able)
7.2317 + assumes "A \<in> sets M" and "\<mu> A \<noteq> \<infinity>"
7.2318 + shows "integral\<^isup>L M (indicator A) = real (\<mu> A)" (is ?int)
7.2319 + and "integrable M (indicator A)" (is ?able)
7.2320 proof -
7.2321 - have *:
7.2322 - "\<And>A x. Real (indicator A x) = indicator A x"
7.2323 - "\<And>A x. Real (- indicator A x) = 0" unfolding indicator_def by auto
7.2324 + from `A \<in> sets M` have *:
7.2325 + "\<And>x. extreal (indicator A x) = indicator A x"
7.2326 + "(\<integral>\<^isup>+x. extreal (- indicator A x) \<partial>M) = 0"
7.2327 + by (auto split: split_indicator simp: positive_integral_0_iff_AE one_extreal_def)
7.2328 show ?int ?able
7.2329 using assms unfolding lebesgue_integral_def integrable_def
7.2330 by (auto simp: * positive_integral_indicator borel_measurable_indicator)
7.2331 qed
7.2332
7.2333 lemma (in measure_space) integral_cmul_indicator:
7.2334 - assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> \<mu> A \<noteq> \<omega>"
7.2335 + assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> \<mu> A \<noteq> \<infinity>"
7.2336 shows "integrable M (\<lambda>x. c * indicator A x)" (is ?P)
7.2337 and "(\<integral>x. c * indicator A x \<partial>M) = c * real (\<mu> A)" (is ?I)
7.2338 proof -
7.2339 @@ -1840,15 +1951,11 @@
7.2340 assumes "integrable M f"
7.2341 shows "integrable M (\<lambda> x. \<bar>f x\<bar>)"
7.2342 proof -
7.2343 - have *:
7.2344 - "\<And>x. Real \<bar>f x\<bar> = Real (f x) + Real (- f x)"
7.2345 - "\<And>x. Real (- \<bar>f x\<bar>) = 0" by auto
7.2346 - have abs: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M" and
7.2347 - f: "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
7.2348 - "(\<lambda>x. Real (- f x)) \<in> borel_measurable M"
7.2349 - using assms unfolding integrable_def by auto
7.2350 - from abs assms show ?thesis unfolding integrable_def *
7.2351 - using positive_integral_linear[OF f, of 1] by simp
7.2352 + from assms have *: "(\<integral>\<^isup>+x. extreal (- \<bar>f x\<bar>)\<partial>M) = 0"
7.2353 + "\<And>x. extreal \<bar>f x\<bar> = max 0 (extreal (f x)) + max 0 (extreal (- f x))"
7.2354 + by (auto simp: integrable_def positive_integral_0_iff_AE split: split_max)
7.2355 + with assms show ?thesis
7.2356 + by (simp add: positive_integral_add positive_integral_max_0 integrable_def)
7.2357 qed
7.2358
7.2359 lemma (in measure_space) integral_subalgebra:
7.2360 @@ -1858,12 +1965,13 @@
7.2361 and "integral\<^isup>L N f = integral\<^isup>L M f" (is ?I)
7.2362 proof -
7.2363 interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
7.2364 - have "(\<lambda>x. Real (f x)) \<in> borel_measurable N" "(\<lambda>x. Real (- f x)) \<in> borel_measurable N"
7.2365 - using borel by auto
7.2366 - note * = this[THEN positive_integral_subalgebra[OF _ N sa]]
7.2367 - have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
7.2368 + have "(\<integral>\<^isup>+ x. max 0 (extreal (f x)) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (extreal (f x)) \<partial>M)"
7.2369 + "(\<integral>\<^isup>+ x. max 0 (extreal (- f x)) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (extreal (- f x)) \<partial>M)"
7.2370 + using borel by (auto intro!: positive_integral_subalgebra N sa)
7.2371 + moreover have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
7.2372 using assms unfolding measurable_def by auto
7.2373 - then show ?P ?I by (auto simp: * integrable_def lebesgue_integral_def)
7.2374 + ultimately show ?P ?I
7.2375 + by (auto simp: integrable_def lebesgue_integral_def positive_integral_max_0)
7.2376 qed
7.2377
7.2378 lemma (in measure_space) integrable_bound:
7.2379 @@ -1873,21 +1981,21 @@
7.2380 assumes borel: "g \<in> borel_measurable M"
7.2381 shows "integrable M g"
7.2382 proof -
7.2383 - have "(\<integral>\<^isup>+ x. Real (g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. Real \<bar>g x\<bar> \<partial>M)"
7.2384 + have "(\<integral>\<^isup>+ x. extreal (g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. extreal \<bar>g x\<bar> \<partial>M)"
7.2385 by (auto intro!: positive_integral_mono)
7.2386 - also have "\<dots> \<le> (\<integral>\<^isup>+ x. Real (f x) \<partial>M)"
7.2387 + also have "\<dots> \<le> (\<integral>\<^isup>+ x. extreal (f x) \<partial>M)"
7.2388 using f by (auto intro!: positive_integral_mono)
7.2389 - also have "\<dots> < \<omega>"
7.2390 - using `integrable M f` unfolding integrable_def by (auto simp: pextreal_less_\<omega>)
7.2391 - finally have pos: "(\<integral>\<^isup>+ x. Real (g x) \<partial>M) < \<omega>" .
7.2392 + also have "\<dots> < \<infinity>"
7.2393 + using `integrable M f` unfolding integrable_def by auto
7.2394 + finally have pos: "(\<integral>\<^isup>+ x. extreal (g x) \<partial>M) < \<infinity>" .
7.2395
7.2396 - have "(\<integral>\<^isup>+ x. Real (- g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. Real (\<bar>g x\<bar>) \<partial>M)"
7.2397 + have "(\<integral>\<^isup>+ x. extreal (- g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. extreal (\<bar>g x\<bar>) \<partial>M)"
7.2398 by (auto intro!: positive_integral_mono)
7.2399 - also have "\<dots> \<le> (\<integral>\<^isup>+ x. Real (f x) \<partial>M)"
7.2400 + also have "\<dots> \<le> (\<integral>\<^isup>+ x. extreal (f x) \<partial>M)"
7.2401 using f by (auto intro!: positive_integral_mono)
7.2402 - also have "\<dots> < \<omega>"
7.2403 - using `integrable M f` unfolding integrable_def by (auto simp: pextreal_less_\<omega>)
7.2404 - finally have neg: "(\<integral>\<^isup>+ x. Real (- g x) \<partial>M) < \<omega>" .
7.2405 + also have "\<dots> < \<infinity>"
7.2406 + using `integrable M f` unfolding integrable_def by auto
7.2407 + finally have neg: "(\<integral>\<^isup>+ x. extreal (- g x) \<partial>M) < \<infinity>" .
7.2408
7.2409 from neg pos borel show ?thesis
7.2410 unfolding integrable_def by auto
7.2411 @@ -1959,41 +2067,34 @@
7.2412 by (simp add: mono_def incseq_def) }
7.2413 note pos_u = this
7.2414
7.2415 - hence [simp]: "\<And>i x. Real (- f i x) = 0" "\<And>x. Real (- u x) = 0"
7.2416 - using pos by auto
7.2417 + have SUP_F: "\<And>x. (SUP n. extreal (f n x)) = extreal (u x)"
7.2418 + unfolding SUP_eq_LIMSEQ[OF mono] by (rule lim)
7.2419
7.2420 - have SUP_F: "\<And>x. (SUP n. Real (f n x)) = Real (u x)"
7.2421 - using mono pos pos_u lim by (rule SUP_eq_LIMSEQ[THEN iffD2])
7.2422 -
7.2423 - have borel_f: "\<And>i. (\<lambda>x. Real (f i x)) \<in> borel_measurable M"
7.2424 + have borel_f: "\<And>i. (\<lambda>x. extreal (f i x)) \<in> borel_measurable M"
7.2425 using i unfolding integrable_def by auto
7.2426 - hence "(\<lambda>x. SUP i. Real (f i x)) \<in> borel_measurable M"
7.2427 + hence "(\<lambda>x. SUP i. extreal (f i x)) \<in> borel_measurable M"
7.2428 by auto
7.2429 hence borel_u: "u \<in> borel_measurable M"
7.2430 - using pos_u by (auto simp: borel_measurable_Real_eq SUP_F)
7.2431 + by (auto simp: borel_measurable_extreal_iff SUP_F)
7.2432
7.2433 - have integral_eq: "\<And>n. (\<integral>\<^isup>+ x. Real (f n x) \<partial>M) = Real (integral\<^isup>L M (f n))"
7.2434 - using i unfolding lebesgue_integral_def integrable_def by (auto simp: Real_real)
7.2435 + hence [simp]: "\<And>i. (\<integral>\<^isup>+x. extreal (- f i x) \<partial>M) = 0" "(\<integral>\<^isup>+x. extreal (- u x) \<partial>M) = 0"
7.2436 + using i borel_u pos pos_u by (auto simp: positive_integral_0_iff_AE integrable_def)
7.2437 +
7.2438 + have integral_eq: "\<And>n. (\<integral>\<^isup>+ x. extreal (f n x) \<partial>M) = extreal (integral\<^isup>L M (f n))"
7.2439 + using i positive_integral_positive by (auto simp: extreal_real lebesgue_integral_def integrable_def)
7.2440
7.2441 have pos_integral: "\<And>n. 0 \<le> integral\<^isup>L M (f n)"
7.2442 using pos i by (auto simp: integral_positive)
7.2443 hence "0 \<le> x"
7.2444 using LIMSEQ_le_const[OF ilim, of 0] by auto
7.2445
7.2446 - have "(\<lambda>i. (\<integral>\<^isup>+ x. Real (f i x) \<partial>M)) \<up> (\<integral>\<^isup>+ x. Real (u x) \<partial>M)"
7.2447 - proof (rule positive_integral_isoton)
7.2448 - from SUP_F mono pos
7.2449 - show "(\<lambda>i x. Real (f i x)) \<up> (\<lambda>x. Real (u x))"
7.2450 - unfolding isoton_fun_expand by (auto simp: isoton_def mono_def)
7.2451 - qed (rule borel_f)
7.2452 - hence pI: "(\<integral>\<^isup>+ x. Real (u x) \<partial>M) = (SUP n. (\<integral>\<^isup>+ x. Real (f n x) \<partial>M))"
7.2453 - unfolding isoton_def by simp
7.2454 - also have "\<dots> = Real x" unfolding integral_eq
7.2455 + from mono pos i have pI: "(\<integral>\<^isup>+ x. extreal (u x) \<partial>M) = (SUP n. (\<integral>\<^isup>+ x. extreal (f n x) \<partial>M))"
7.2456 + by (auto intro!: positive_integral_monotone_convergence_SUP
7.2457 + simp: integrable_def incseq_mono incseq_Suc_iff le_fun_def SUP_F[symmetric])
7.2458 + also have "\<dots> = extreal x" unfolding integral_eq
7.2459 proof (rule SUP_eq_LIMSEQ[THEN iffD2])
7.2460 show "mono (\<lambda>n. integral\<^isup>L M (f n))"
7.2461 using mono i by (auto simp: mono_def intro!: integral_mono)
7.2462 - show "\<And>n. 0 \<le> integral\<^isup>L M (f n)" using pos_integral .
7.2463 - show "0 \<le> x" using `0 \<le> x` .
7.2464 show "(\<lambda>n. integral\<^isup>L M (f n)) ----> x" using ilim .
7.2465 qed
7.2466 finally show "integrable M u" "integral\<^isup>L M u = x" using borel_u `0 \<le> x`
7.2467 @@ -2028,61 +2129,72 @@
7.2468 assumes "integrable M f"
7.2469 shows "(\<integral>x. \<bar>f x\<bar> \<partial>M) = 0 \<longleftrightarrow> \<mu> {x\<in>space M. f x \<noteq> 0} = 0"
7.2470 proof -
7.2471 - have *: "\<And>x. Real (- \<bar>f x\<bar>) = 0" by auto
7.2472 + have *: "(\<integral>\<^isup>+x. extreal (- \<bar>f x\<bar>) \<partial>M) = 0"
7.2473 + using assms by (auto simp: positive_integral_0_iff_AE integrable_def)
7.2474 have "integrable M (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs)
7.2475 - hence "(\<lambda>x. Real (\<bar>f x\<bar>)) \<in> borel_measurable M"
7.2476 - "(\<integral>\<^isup>+ x. Real \<bar>f x\<bar> \<partial>M) \<noteq> \<omega>" unfolding integrable_def by auto
7.2477 + hence "(\<lambda>x. extreal (\<bar>f x\<bar>)) \<in> borel_measurable M"
7.2478 + "(\<integral>\<^isup>+ x. extreal \<bar>f x\<bar> \<partial>M) \<noteq> \<infinity>" unfolding integrable_def by auto
7.2479 from positive_integral_0_iff[OF this(1)] this(2)
7.2480 show ?thesis unfolding lebesgue_integral_def *
7.2481 - by (simp add: real_of_pextreal_eq_0)
7.2482 + using positive_integral_positive[of "\<lambda>x. extreal \<bar>f x\<bar>"]
7.2483 + by (auto simp add: real_of_extreal_eq_0)
7.2484 qed
7.2485
7.2486 -lemma (in measure_space) positive_integral_omega:
7.2487 - assumes "f \<in> borel_measurable M"
7.2488 - and "integral\<^isup>P M f \<noteq> \<omega>"
7.2489 - shows "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
7.2490 +lemma (in measure_space) positive_integral_PInf:
7.2491 + assumes f: "f \<in> borel_measurable M"
7.2492 + and not_Inf: "integral\<^isup>P M f \<noteq> \<infinity>"
7.2493 + shows "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
7.2494 proof -
7.2495 - have "\<omega> * \<mu> (f -` {\<omega>} \<inter> space M) = (\<integral>\<^isup>+ x. \<omega> * indicator (f -` {\<omega>} \<inter> space M) x \<partial>M)"
7.2496 - using positive_integral_cmult_indicator[OF borel_measurable_vimage, OF assms(1), of \<omega> \<omega>] by simp
7.2497 - also have "\<dots> \<le> integral\<^isup>P M f"
7.2498 - by (auto intro!: positive_integral_mono simp: indicator_def)
7.2499 - finally show ?thesis
7.2500 - using assms(2) by (cases ?thesis) auto
7.2501 + have "\<infinity> * \<mu> (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^isup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)"
7.2502 + using f by (subst positive_integral_cmult_indicator) (auto simp: measurable_sets)
7.2503 + also have "\<dots> \<le> integral\<^isup>P M (\<lambda>x. max 0 (f x))"
7.2504 + by (auto intro!: positive_integral_mono simp: indicator_def max_def)
7.2505 + finally have "\<infinity> * \<mu> (f -` {\<infinity>} \<inter> space M) \<le> integral\<^isup>P M f"
7.2506 + by (simp add: positive_integral_max_0)
7.2507 + moreover have "0 \<le> \<mu> (f -` {\<infinity>} \<inter> space M)"
7.2508 + using f by (simp add: measurable_sets)
7.2509 + ultimately show ?thesis
7.2510 + using assms by (auto split: split_if_asm)
7.2511 qed
7.2512
7.2513 -lemma (in measure_space) positive_integral_omega_AE:
7.2514 - assumes "f \<in> borel_measurable M" "integral\<^isup>P M f \<noteq> \<omega>" shows "AE x. f x \<noteq> \<omega>"
7.2515 +lemma (in measure_space) positive_integral_PInf_AE:
7.2516 + assumes "f \<in> borel_measurable M" "integral\<^isup>P M f \<noteq> \<infinity>" shows "AE x. f x \<noteq> \<infinity>"
7.2517 proof (rule AE_I)
7.2518 - show "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
7.2519 - by (rule positive_integral_omega[OF assms])
7.2520 - show "f -` {\<omega>} \<inter> space M \<in> sets M"
7.2521 + show "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
7.2522 + by (rule positive_integral_PInf[OF assms])
7.2523 + show "f -` {\<infinity>} \<inter> space M \<in> sets M"
7.2524 using assms by (auto intro: borel_measurable_vimage)
7.2525 qed auto
7.2526
7.2527 -lemma (in measure_space) simple_integral_omega:
7.2528 - assumes "simple_function M f"
7.2529 - and "integral\<^isup>S M f \<noteq> \<omega>"
7.2530 - shows "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
7.2531 -proof (rule positive_integral_omega)
7.2532 +lemma (in measure_space) simple_integral_PInf:
7.2533 + assumes "simple_function M f" "\<And>x. 0 \<le> f x"
7.2534 + and "integral\<^isup>S M f \<noteq> \<infinity>"
7.2535 + shows "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
7.2536 +proof (rule positive_integral_PInf)
7.2537 show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
7.2538 - show "integral\<^isup>P M f \<noteq> \<omega>"
7.2539 + show "integral\<^isup>P M f \<noteq> \<infinity>"
7.2540 using assms by (simp add: positive_integral_eq_simple_integral)
7.2541 qed
7.2542
7.2543 lemma (in measure_space) integral_real:
7.2544 - fixes f :: "'a \<Rightarrow> pextreal"
7.2545 - assumes [simp]: "AE x. f x \<noteq> \<omega>"
7.2546 - shows "(\<integral>x. real (f x) \<partial>M) = real (integral\<^isup>P M f)" (is ?plus)
7.2547 - and "(\<integral>x. - real (f x) \<partial>M) = - real (integral\<^isup>P M f)" (is ?minus)
7.2548 -proof -
7.2549 - have "(\<integral>\<^isup>+ x. Real (real (f x)) \<partial>M) = integral\<^isup>P M f"
7.2550 - by (auto intro!: positive_integral_cong_AE simp: Real_real)
7.2551 - moreover
7.2552 - have "(\<integral>\<^isup>+ x. Real (- real (f x)) \<partial>M) = (\<integral>\<^isup>+ x. 0 \<partial>M)"
7.2553 - by (intro positive_integral_cong) auto
7.2554 - ultimately show ?plus ?minus
7.2555 - by (auto simp: lebesgue_integral_def integrable_def)
7.2556 -qed
7.2557 + "AE x. \<bar>f x\<bar> \<noteq> \<infinity> \<Longrightarrow> (\<integral>x. real (f x) \<partial>M) = real (integral\<^isup>P M f) - real (integral\<^isup>P M (\<lambda>x. - f x))"
7.2558 + using assms unfolding lebesgue_integral_def
7.2559 + by (subst (1 2) positive_integral_cong_AE) (auto simp add: extreal_real)
7.2560 +
7.2561 +lemma liminf_extreal_cminus:
7.2562 + fixes f :: "nat \<Rightarrow> extreal" assumes "c \<noteq> -\<infinity>"
7.2563 + shows "liminf (\<lambda>x. c - f x) = c - limsup f"
7.2564 +proof (cases c)
7.2565 + case PInf then show ?thesis by (simp add: Liminf_const)
7.2566 +next
7.2567 + case (real r) then show ?thesis
7.2568 + unfolding liminf_SUPR_INFI limsup_INFI_SUPR
7.2569 + apply (subst INFI_extreal_cminus)
7.2570 + apply auto
7.2571 + apply (subst SUPR_extreal_cminus)
7.2572 + apply auto
7.2573 + done
7.2574 +qed (insert `c \<noteq> -\<infinity>`, simp)
7.2575
7.2576 lemma (in measure_space) integral_dominated_convergence:
7.2577 assumes u: "\<And>i. integrable M (u i)" and bound: "\<And>x j. x\<in>space M \<Longrightarrow> \<bar>u j x\<bar> \<le> w x"
7.2578 @@ -2129,61 +2241,76 @@
7.2579 finally have "\<bar>u j x - u' x\<bar> \<le> 2 * w x" by simp }
7.2580 note diff_less_2w = this
7.2581
7.2582 - have PI_diff: "\<And>m n. (\<integral>\<^isup>+ x. Real (?diff (m + n) x) \<partial>M) =
7.2583 - (\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) - (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M)"
7.2584 + have PI_diff: "\<And>n. (\<integral>\<^isup>+ x. extreal (?diff n x) \<partial>M) =
7.2585 + (\<integral>\<^isup>+ x. extreal (2 * w x) \<partial>M) - (\<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)"
7.2586 using diff w diff_less_2w w_pos
7.2587 by (subst positive_integral_diff[symmetric])
7.2588 (auto simp: integrable_def intro!: positive_integral_cong)
7.2589
7.2590 have "integrable M (\<lambda>x. 2 * w x)"
7.2591 using w by (auto intro: integral_cmult)
7.2592 - hence I2w_fin: "(\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) \<noteq> \<omega>" and
7.2593 - borel_2w: "(\<lambda>x. Real (2 * w x)) \<in> borel_measurable M"
7.2594 + hence I2w_fin: "(\<integral>\<^isup>+ x. extreal (2 * w x) \<partial>M) \<noteq> \<infinity>" and
7.2595 + borel_2w: "(\<lambda>x. extreal (2 * w x)) \<in> borel_measurable M"
7.2596 unfolding integrable_def by auto
7.2597
7.2598 - have "(INF n. SUP m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M)) = 0" (is "?lim_SUP = 0")
7.2599 + have "limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M) = 0" (is "limsup ?f = 0")
7.2600 proof cases
7.2601 - assume eq_0: "(\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) = 0"
7.2602 - have "\<And>i. (\<integral>\<^isup>+ x. Real \<bar>u i x - u' x\<bar> \<partial>M) \<le> (\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M)"
7.2603 - proof (rule positive_integral_mono)
7.2604 - fix i x assume "x \<in> space M" from diff_less_2w[OF this, of i]
7.2605 - show "Real \<bar>u i x - u' x\<bar> \<le> Real (2 * w x)" by auto
7.2606 - qed
7.2607 - hence "\<And>i. (\<integral>\<^isup>+ x. Real \<bar>u i x - u' x\<bar> \<partial>M) = 0" using eq_0 by auto
7.2608 - thus ?thesis by simp
7.2609 + assume eq_0: "(\<integral>\<^isup>+ x. max 0 (extreal (2 * w x)) \<partial>M) = 0" (is "?wx = 0")
7.2610 + { fix n
7.2611 + have "?f n \<le> ?wx" (is "integral\<^isup>P M ?f' \<le> _")
7.2612 + using diff_less_2w[of _ n] unfolding positive_integral_max_0
7.2613 + by (intro positive_integral_mono) auto
7.2614 + then have "?f n = 0"
7.2615 + using positive_integral_positive[of ?f'] eq_0 by auto }
7.2616 + then show ?thesis by (simp add: Limsup_const)
7.2617 next
7.2618 - assume neq_0: "(\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) \<noteq> 0"
7.2619 - have "(\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) = (\<integral>\<^isup>+ x. (SUP n. INF m. Real (?diff (m + n) x)) \<partial>M)"
7.2620 - proof (rule positive_integral_cong, subst add_commute)
7.2621 + assume neq_0: "(\<integral>\<^isup>+ x. max 0 (extreal (2 * w x)) \<partial>M) \<noteq> 0" (is "?wx \<noteq> 0")
7.2622 + have "0 = limsup (\<lambda>n. 0 :: extreal)" by (simp add: Limsup_const)
7.2623 + also have "\<dots> \<le> limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)"
7.2624 + by (intro limsup_mono positive_integral_positive)
7.2625 + finally have pos: "0 \<le> limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)" .
7.2626 + have "?wx = (\<integral>\<^isup>+ x. liminf (\<lambda>n. max 0 (extreal (?diff n x))) \<partial>M)"
7.2627 + proof (rule positive_integral_cong)
7.2628 fix x assume x: "x \<in> space M"
7.2629 - show "Real (2 * w x) = (SUP n. INF m. Real (?diff (n + m) x))"
7.2630 - proof (rule LIMSEQ_imp_lim_INF[symmetric])
7.2631 - fix j show "0 \<le> ?diff j x" using diff_less_2w[OF x, of j] by simp
7.2632 - next
7.2633 + show "max 0 (extreal (2 * w x)) = liminf (\<lambda>n. max 0 (extreal (?diff n x)))"
7.2634 + unfolding extreal_max_0
7.2635 + proof (rule lim_imp_Liminf[symmetric], unfold lim_extreal)
7.2636 have "(\<lambda>i. ?diff i x) ----> 2 * w x - \<bar>u' x - u' x\<bar>"
7.2637 using u'[OF x] by (safe intro!: LIMSEQ_diff LIMSEQ_const LIMSEQ_imp_rabs)
7.2638 - thus "(\<lambda>i. ?diff i x) ----> 2 * w x" by simp
7.2639 - qed
7.2640 + then show "(\<lambda>i. max 0 (?diff i x)) ----> max 0 (2 * w x)"
7.2641 + by (auto intro!: tendsto_real_max simp add: lim_extreal)
7.2642 + qed (rule trivial_limit_sequentially)
7.2643 qed
7.2644 - also have "\<dots> \<le> (SUP n. INF m. (\<integral>\<^isup>+ x. Real (?diff (m + n) x) \<partial>M))"
7.2645 + also have "\<dots> \<le> liminf (\<lambda>n. \<integral>\<^isup>+ x. max 0 (extreal (?diff n x)) \<partial>M)"
7.2646 using u'_borel w u unfolding integrable_def
7.2647 - by (auto intro!: positive_integral_lim_INF)
7.2648 - also have "\<dots> = (\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) -
7.2649 - (INF n. SUP m. \<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M)"
7.2650 - unfolding PI_diff pextreal_INF_minus[OF I2w_fin] pextreal_SUP_minus ..
7.2651 - finally show ?thesis using neq_0 I2w_fin by (rule pextreal_le_minus_imp_0)
7.2652 + by (intro positive_integral_lim_INF) (auto intro!: positive_integral_lim_INF)
7.2653 + also have "\<dots> = (\<integral>\<^isup>+ x. extreal (2 * w x) \<partial>M) -
7.2654 + limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)"
7.2655 + unfolding PI_diff positive_integral_max_0
7.2656 + using positive_integral_positive[of "\<lambda>x. extreal (2 * w x)"]
7.2657 + by (subst liminf_extreal_cminus) auto
7.2658 + finally show ?thesis
7.2659 + using neq_0 I2w_fin positive_integral_positive[of "\<lambda>x. extreal (2 * w x)"] pos
7.2660 + unfolding positive_integral_max_0
7.2661 + by (cases rule: extreal2_cases[of "\<integral>\<^isup>+ x. extreal (2 * w x) \<partial>M" "limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)"])
7.2662 + auto
7.2663 qed
7.2664
7.2665 - have [simp]: "\<And>n m x. Real (- \<bar>u (m + n) x - u' x\<bar>) = 0" by auto
7.2666 -
7.2667 - have [simp]: "\<And>n m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M) =
7.2668 - Real ((\<integral>x. \<bar>u (n + m) x - u' x\<bar> \<partial>M))"
7.2669 - using diff by (subst add_commute) (simp add: lebesgue_integral_def integrable_def Real_real)
7.2670 -
7.2671 - have "(SUP n. INF m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M)) \<le> ?lim_SUP"
7.2672 - (is "?lim_INF \<le> _") by (subst (1 2) add_commute) (rule lim_INF_le_lim_SUP)
7.2673 - hence "?lim_INF = Real 0" "?lim_SUP = Real 0" using `?lim_SUP = 0` by auto
7.2674 - thus ?lim_diff using diff by (auto intro!: integral_positive lim_INF_eq_lim_SUP)
7.2675 + have "liminf ?f \<le> limsup ?f"
7.2676 + by (intro extreal_Liminf_le_Limsup trivial_limit_sequentially)
7.2677 + moreover
7.2678 + { have "0 = liminf (\<lambda>n. 0 :: extreal)" by (simp add: Liminf_const)
7.2679 + also have "\<dots> \<le> liminf ?f"
7.2680 + by (intro liminf_mono positive_integral_positive)
7.2681 + finally have "0 \<le> liminf ?f" . }
7.2682 + ultimately have liminf_limsup_eq: "liminf ?f = extreal 0" "limsup ?f = extreal 0"
7.2683 + using `limsup ?f = 0` by auto
7.2684 + have "\<And>n. (\<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M) = extreal (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)"
7.2685 + using diff positive_integral_positive
7.2686 + by (subst integral_eq_positive_integral) (auto simp: extreal_real integrable_def)
7.2687 + then show ?lim_diff
7.2688 + using extreal_Liminf_eq_Limsup[OF trivial_limit_sequentially liminf_limsup_eq]
7.2689 + by (simp add: lim_extreal)
7.2690
7.2691 show ?lim
7.2692 proof (rule LIMSEQ_I)
7.2693 @@ -2266,7 +2393,7 @@
7.2694 assumes f: "f \<in> borel_measurable M"
7.2695 and bij: "bij_betw enum S (f ` space M)"
7.2696 and enum_zero: "enum ` (-S) \<subseteq> {0}"
7.2697 - and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<omega>"
7.2698 + and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<infinity>"
7.2699 and abs_summable: "summable (\<lambda>r. \<bar>enum r * real (\<mu> (f -` {enum r} \<inter> space M))\<bar>)"
7.2700 shows "integrable M f"
7.2701 and "(\<lambda>r. enum r * real (\<mu> (f -` {enum r} \<inter> space M))) sums integral\<^isup>L M f" (is ?sums)
7.2702 @@ -2303,7 +2430,7 @@
7.2703 also have "\<dots> = \<bar>enum r\<bar> * real (\<mu> (?A r))"
7.2704 using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
7.2705 finally have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = \<bar>enum r * real (\<mu> (?A r))\<bar>"
7.2706 - by (simp add: abs_mult_pos real_pextreal_pos) }
7.2707 + using f by (subst (2) abs_mult_pos[symmetric]) (auto intro!: real_of_extreal_pos measurable_sets) }
7.2708 note int_abs_F = this
7.2709
7.2710 have 1: "\<And>i. integrable M (\<lambda>x. ?F i x)"
7.2711 @@ -2323,7 +2450,7 @@
7.2712
7.2713 lemma (in measure_space) integral_on_finite:
7.2714 assumes f: "f \<in> borel_measurable M" and finite: "finite (f`space M)"
7.2715 - and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<omega>"
7.2716 + and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<infinity>"
7.2717 shows "integrable M f"
7.2718 and "(\<integral>x. f x \<partial>M) =
7.2719 (\<Sum> r \<in> f`space M. r * real (\<mu> (f -` {r} \<inter> space M)))" (is "?integral")
7.2720 @@ -2353,11 +2480,12 @@
7.2721 by (auto intro: borel_measurable_simple_function)
7.2722
7.2723 lemma (in finite_measure_space) positive_integral_finite_eq_setsum:
7.2724 - "integral\<^isup>P M f = (\<Sum>x \<in> space M. f x * \<mu> {x})"
7.2725 + assumes pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
7.2726 + shows "integral\<^isup>P M f = (\<Sum>x \<in> space M. f x * \<mu> {x})"
7.2727 proof -
7.2728 have *: "integral\<^isup>P M f = (\<integral>\<^isup>+ x. (\<Sum>y\<in>space M. f y * indicator {y} x) \<partial>M)"
7.2729 by (auto intro!: positive_integral_cong simp add: indicator_def if_distrib setsum_cases[OF finite_space])
7.2730 - show ?thesis unfolding * using borel_measurable_finite[of f]
7.2731 + show ?thesis unfolding * using borel_measurable_finite[of f] pos
7.2732 by (simp add: positive_integral_setsum positive_integral_cmult_indicator)
7.2733 qed
7.2734
7.2735 @@ -2365,16 +2493,20 @@
7.2736 shows "integrable M f"
7.2737 and "integral\<^isup>L M f = (\<Sum>x \<in> space M. f x * real (\<mu> {x}))" (is ?I)
7.2738 proof -
7.2739 - have [simp]:
7.2740 - "(\<integral>\<^isup>+ x. Real (f x) \<partial>M) = (\<Sum>x \<in> space M. Real (f x) * \<mu> {x})"
7.2741 - "(\<integral>\<^isup>+ x. Real (- f x) \<partial>M) = (\<Sum>x \<in> space M. Real (- f x) * \<mu> {x})"
7.2742 - unfolding positive_integral_finite_eq_setsum by auto
7.2743 - show "integrable M f" using finite_space finite_measure
7.2744 - by (simp add: setsum_\<omega> integrable_def)
7.2745 - show ?I using finite_measure
7.2746 - apply (simp add: lebesgue_integral_def real_of_pextreal_setsum[symmetric]
7.2747 - real_of_pextreal_mult[symmetric] setsum_subtractf[symmetric])
7.2748 - by (rule setsum_cong) (simp_all split: split_if)
7.2749 + have *:
7.2750 + "(\<integral>\<^isup>+ x. max 0 (extreal (f x)) \<partial>M) = (\<Sum>x \<in> space M. max 0 (extreal (f x)) * \<mu> {x})"
7.2751 + "(\<integral>\<^isup>+ x. max 0 (extreal (- f x)) \<partial>M) = (\<Sum>x \<in> space M. max 0 (extreal (- f x)) * \<mu> {x})"
7.2752 + by (simp_all add: positive_integral_finite_eq_setsum)
7.2753 + then show "integrable M f" using finite_space finite_measure
7.2754 + by (simp add: setsum_Pinfty integrable_def positive_integral_max_0
7.2755 + split: split_max)
7.2756 + show ?I using finite_measure *
7.2757 + apply (simp add: positive_integral_max_0 lebesgue_integral_def)
7.2758 + apply (subst (1 2) setsum_real_of_extreal[symmetric])
7.2759 + apply (simp_all split: split_max add: setsum_subtractf[symmetric])
7.2760 + apply (intro setsum_cong[OF refl])
7.2761 + apply (simp split: split_max)
7.2762 + done
7.2763 qed
7.2764
7.2765 end
8.1 --- a/src/HOL/Probability/Lebesgue_Measure.thy Mon Mar 14 14:37:47 2011 +0100
8.2 +++ b/src/HOL/Probability/Lebesgue_Measure.thy Mon Mar 14 14:37:49 2011 +0100
8.3 @@ -48,12 +48,12 @@
8.4 lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
8.5 unfolding Pi_def by auto
8.6
8.7 -subsection {* Lebesgue measure *}
8.8 +subsection {* Lebesgue measure *}
8.9
8.10 definition lebesgue :: "'a::ordered_euclidean_space measure_space" where
8.11 "lebesgue = \<lparr> space = UNIV,
8.12 sets = {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n},
8.13 - measure = \<lambda>A. SUP n. Real (integral (cube n) (indicator A)) \<rparr>"
8.14 + measure = \<lambda>A. SUP n. extreal (integral (cube n) (indicator A)) \<rparr>"
8.15
8.16 lemma space_lebesgue[simp]: "space lebesgue = UNIV"
8.17 unfolding lebesgue_def by simp
8.18 @@ -114,10 +114,33 @@
8.19 qed (auto intro: LIMSEQ_indicator_UN simp: cube_def)
8.20 qed simp
8.21
8.22 +lemma suminf_SUP_eq:
8.23 + fixes f :: "nat \<Rightarrow> nat \<Rightarrow> extreal"
8.24 + assumes "\<And>i. incseq (\<lambda>n. f n i)" "\<And>n i. 0 \<le> f n i"
8.25 + shows "(\<Sum>i. SUP n. f n i) = (SUP n. \<Sum>i. f n i)"
8.26 +proof -
8.27 + { fix n :: nat
8.28 + have "(\<Sum>i<n. SUP k. f k i) = (SUP k. \<Sum>i<n. f k i)"
8.29 + using assms by (auto intro!: SUPR_extreal_setsum[symmetric]) }
8.30 + note * = this
8.31 + show ?thesis using assms
8.32 + apply (subst (1 2) suminf_extreal_eq_SUPR)
8.33 + unfolding *
8.34 + apply (auto intro!: le_SUPI2)
8.35 + apply (subst SUP_commute) ..
8.36 +qed
8.37 +
8.38 interpretation lebesgue: measure_space lebesgue
8.39 proof
8.40 have *: "indicator {} = (\<lambda>x. 0 :: real)" by (simp add: fun_eq_iff)
8.41 - show "measure lebesgue {} = 0" by (simp add: integral_0 * lebesgue_def)
8.42 + show "positive lebesgue (measure lebesgue)"
8.43 + proof (unfold positive_def, safe)
8.44 + show "measure lebesgue {} = 0" by (simp add: integral_0 * lebesgue_def)
8.45 + fix A assume "A \<in> sets lebesgue"
8.46 + then show "0 \<le> measure lebesgue A"
8.47 + unfolding lebesgue_def
8.48 + by (auto intro!: le_SUPI2 integral_nonneg)
8.49 + qed
8.50 next
8.51 show "countably_additive lebesgue (measure lebesgue)"
8.52 proof (intro countably_additive_def[THEN iffD2] allI impI)
8.53 @@ -130,23 +153,17 @@
8.54 assume "(\<Union>i. A i) \<in> sets lebesgue"
8.55 then have UN_A[simp, intro]: "\<And>i n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n"
8.56 by (auto dest: lebesgueD)
8.57 - show "(\<Sum>\<^isub>\<infinity>n. measure lebesgue (A n)) = measure lebesgue (\<Union>i. A i)"
8.58 - proof (simp add: lebesgue_def, subst psuminf_SUP_eq)
8.59 - fix n i show "Real (?m n i) \<le> Real (?m (Suc n) i)"
8.60 - using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le)
8.61 + show "(\<Sum>n. measure lebesgue (A n)) = measure lebesgue (\<Union>i. A i)"
8.62 + proof (simp add: lebesgue_def, subst suminf_SUP_eq, safe intro!: incseq_SucI)
8.63 + fix i n show "extreal (?m n i) \<le> extreal (?m (Suc n) i)"
8.64 + using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le incseq_SucI)
8.65 next
8.66 - show "(SUP n. \<Sum>\<^isub>\<infinity>i. Real (?m n i)) = (SUP n. Real (?M n UNIV))"
8.67 - unfolding psuminf_def
8.68 - proof (subst setsum_Real, (intro arg_cong[where f="SUPR UNIV"] ext ballI nn SUP_eq_LIMSEQ[THEN iffD2])+)
8.69 - fix n :: nat show "mono (\<lambda>m. \<Sum>x<m. ?m n x)"
8.70 - proof (intro mono_iff_le_Suc[THEN iffD2] allI)
8.71 - fix m show "(\<Sum>x<m. ?m n x) \<le> (\<Sum>x<Suc m. ?m n x)"
8.72 - using nn[of n m] by auto
8.73 - qed
8.74 - show "0 \<le> ?M n UNIV"
8.75 - using UN_A by (auto intro!: integral_nonneg)
8.76 - fix m show "0 \<le> (\<Sum>x<m. ?m n x)" by (auto intro!: setsum_nonneg)
8.77 - next
8.78 + fix i n show "0 \<le> extreal (?m n i)"
8.79 + using rA unfolding lebesgue_def
8.80 + by (auto intro!: le_SUPI2 integral_nonneg)
8.81 + next
8.82 + show "(SUP n. \<Sum>i. extreal (?m n i)) = (SUP n. extreal (?M n UNIV))"
8.83 + proof (intro arg_cong[where f="SUPR UNIV"] ext sums_unique[symmetric] sums_extreal[THEN iffD2] sums_def[THEN iffD2])
8.84 fix n
8.85 have "\<And>m. (UNION {..<m} A) \<in> sets lebesgue" using rA by auto
8.86 from lebesgueD[OF this]
8.87 @@ -171,8 +188,8 @@
8.88 ultimately show ?case
8.89 using Suc A by (simp add: integral_add[symmetric])
8.90 qed auto }
8.91 - ultimately show "(\<lambda>m. \<Sum>x<m. ?m n x) ----> ?M n UNIV"
8.92 - by simp
8.93 + ultimately show "(\<lambda>m. \<Sum>x = 0..<m. ?m n x) ----> ?M n UNIV"
8.94 + by (simp add: atLeast0LessThan)
8.95 qed
8.96 qed
8.97 qed
8.98 @@ -232,13 +249,11 @@
8.99
8.100 lemma lmeasure_iff_LIMSEQ:
8.101 assumes "A \<in> sets lebesgue" "0 \<le> m"
8.102 - shows "lebesgue.\<mu> A = Real m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) ----> m"
8.103 + shows "lebesgue.\<mu> A = extreal m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) ----> m"
8.104 proof (simp add: lebesgue_def, intro SUP_eq_LIMSEQ)
8.105 show "mono (\<lambda>n. integral (cube n) (indicator A::_=>real))"
8.106 using cube_subset assms by (intro monoI integral_subset_le) (auto dest!: lebesgueD)
8.107 - fix n show "0 \<le> integral (cube n) (indicator A::_=>real)"
8.108 - using assms by (auto dest!: lebesgueD intro!: integral_nonneg)
8.109 -qed fact
8.110 +qed
8.111
8.112 lemma has_integral_indicator_UNIV:
8.113 fixes s A :: "'a::ordered_euclidean_space set" and x :: real
8.114 @@ -260,7 +275,7 @@
8.115
8.116 lemma lmeasure_finite_has_integral:
8.117 fixes s :: "'a::ordered_euclidean_space set"
8.118 - assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s = Real m" "0 \<le> m"
8.119 + assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s = extreal m" "0 \<le> m"
8.120 shows "(indicator s has_integral m) UNIV"
8.121 proof -
8.122 let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
8.123 @@ -302,12 +317,14 @@
8.124 unfolding m by (intro integrable_integral **)
8.125 qed
8.126
8.127 -lemma lmeasure_finite_integrable: assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s \<noteq> \<omega>"
8.128 +lemma lmeasure_finite_integrable: assumes s: "s \<in> sets lebesgue" and "lebesgue.\<mu> s \<noteq> \<infinity>"
8.129 shows "(indicator s :: _ \<Rightarrow> real) integrable_on UNIV"
8.130 proof (cases "lebesgue.\<mu> s")
8.131 - case (preal m) from lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this]
8.132 + case (real m)
8.133 + with lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this]
8.134 + lebesgue.positive_measure[OF s]
8.135 show ?thesis unfolding integrable_on_def by auto
8.136 -qed (insert assms, auto)
8.137 +qed (insert assms lebesgue.positive_measure[OF s], auto)
8.138
8.139 lemma has_integral_lebesgue: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
8.140 shows "s \<in> sets lebesgue"
8.141 @@ -321,7 +338,7 @@
8.142 qed
8.143
8.144 lemma has_integral_lmeasure: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
8.145 - shows "lebesgue.\<mu> s = Real m"
8.146 + shows "lebesgue.\<mu> s = extreal m"
8.147 proof (intro lmeasure_iff_LIMSEQ[THEN iffD2])
8.148 let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
8.149 show "s \<in> sets lebesgue" using has_integral_lebesgue[OF assms] .
8.150 @@ -346,28 +363,28 @@
8.151 qed
8.152
8.153 lemma has_integral_iff_lmeasure:
8.154 - "(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m)"
8.155 + "(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = extreal m)"
8.156 proof
8.157 assume "(indicator A has_integral m) UNIV"
8.158 with has_integral_lmeasure[OF this] has_integral_lebesgue[OF this]
8.159 - show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m"
8.160 + show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = extreal m"
8.161 by (auto intro: has_integral_nonneg)
8.162 next
8.163 - assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m"
8.164 + assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = extreal m"
8.165 then show "(indicator A has_integral m) UNIV" by (intro lmeasure_finite_has_integral) auto
8.166 qed
8.167
8.168 lemma lmeasure_eq_integral: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV"
8.169 - shows "lebesgue.\<mu> s = Real (integral UNIV (indicator s))"
8.170 + shows "lebesgue.\<mu> s = extreal (integral UNIV (indicator s))"
8.171 using assms unfolding integrable_on_def
8.172 proof safe
8.173 fix y :: real assume "(indicator s has_integral y) UNIV"
8.174 from this[unfolded has_integral_iff_lmeasure] integral_unique[OF this]
8.175 - show "lebesgue.\<mu> s = Real (integral UNIV (indicator s))" by simp
8.176 + show "lebesgue.\<mu> s = extreal (integral UNIV (indicator s))" by simp
8.177 qed
8.178
8.179 lemma lebesgue_simple_function_indicator:
8.180 - fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
8.181 + fixes f::"'a::ordered_euclidean_space \<Rightarrow> extreal"
8.182 assumes f:"simple_function lebesgue f"
8.183 shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f -` {y}) x))"
8.184 by (rule, subst lebesgue.simple_function_indicator_representation[OF f]) auto
8.185 @@ -376,7 +393,7 @@
8.186 "(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (lebesgue.\<mu> s)"
8.187 by (subst lmeasure_eq_integral) (auto intro!: integral_nonneg)
8.188
8.189 -lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "lebesgue.\<mu> s \<noteq> \<omega>"
8.190 +lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "lebesgue.\<mu> s \<noteq> \<infinity>"
8.191 using lmeasure_eq_integral[OF assms] by auto
8.192
8.193 lemma negligible_iff_lebesgue_null_sets:
8.194 @@ -409,37 +426,29 @@
8.195 shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c"
8.196 by (rule integral_unique) (rule has_integral_const)
8.197
8.198 -lemma lmeasure_UNIV[intro]: "lebesgue.\<mu> (UNIV::'a::ordered_euclidean_space set) = \<omega>"
8.199 -proof (simp add: lebesgue_def SUP_\<omega>, intro allI impI)
8.200 - fix x assume "x < \<omega>"
8.201 - then obtain r where r: "x = Real r" "0 \<le> r" by (cases x) auto
8.202 - then obtain n where n: "r < of_nat n" using ex_less_of_nat by auto
8.203 - show "\<exists>i. x < Real (integral (cube i) (indicator UNIV::'a\<Rightarrow>real))"
8.204 - proof (intro exI[of _ n])
8.205 - have [simp]: "indicator UNIV = (\<lambda>x. 1)" by (auto simp: fun_eq_iff)
8.206 - { fix m :: nat assume "0 < m" then have "real n \<le> (\<Prod>x<m. 2 * real n)"
8.207 - proof (induct m)
8.208 - case (Suc m)
8.209 - show ?case
8.210 - proof cases
8.211 - assume "m = 0" then show ?thesis by (simp add: lessThan_Suc)
8.212 - next
8.213 - assume "m \<noteq> 0" then have "real n \<le> (\<Prod>x<m. 2 * real n)" using Suc by auto
8.214 - then show ?thesis
8.215 - by (auto simp: lessThan_Suc field_simps mult_le_cancel_left1)
8.216 - qed
8.217 - qed auto } note this[OF DIM_positive[where 'a='a], simp]
8.218 - then have [simp]: "0 \<le> (\<Prod>x<DIM('a). 2 * real n)" using real_of_nat_ge_zero by arith
8.219 - have "x < Real (of_nat n)" using n r by auto
8.220 - also have "Real (of_nat n) \<le> Real (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))"
8.221 - by (auto simp: real_eq_of_nat[symmetric] cube_def content_closed_interval_cases)
8.222 - finally show "x < Real (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))" .
8.223 - qed
8.224 -qed
8.225 +lemma lmeasure_UNIV[intro]: "lebesgue.\<mu> (UNIV::'a::ordered_euclidean_space set) = \<infinity>"
8.226 +proof (simp add: lebesgue_def, intro SUP_PInfty bexI)
8.227 + fix n :: nat
8.228 + have "indicator UNIV = (\<lambda>x::'a. 1 :: real)" by auto
8.229 + moreover
8.230 + { have "real n \<le> (2 * real n) ^ DIM('a)"
8.231 + proof (cases n)
8.232 + case 0 then show ?thesis by auto
8.233 + next
8.234 + case (Suc n')
8.235 + have "real n \<le> (2 * real n)^1" by auto
8.236 + also have "(2 * real n)^1 \<le> (2 * real n) ^ DIM('a)"
8.237 + using Suc DIM_positive[where 'a='a] by (intro power_increasing) (auto simp: real_of_nat_Suc)
8.238 + finally show ?thesis .
8.239 + qed }
8.240 + ultimately show "extreal (real n) \<le> extreal (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))"
8.241 + using integral_const DIM_positive[where 'a='a]
8.242 + by (auto simp: cube_def content_closed_interval_cases setprod_constant)
8.243 +qed simp
8.244
8.245 lemma
8.246 fixes a b ::"'a::ordered_euclidean_space"
8.247 - shows lmeasure_atLeastAtMost[simp]: "lebesgue.\<mu> {a..b} = Real (content {a..b})"
8.248 + shows lmeasure_atLeastAtMost[simp]: "lebesgue.\<mu> {a..b} = extreal (content {a..b})"
8.249 proof -
8.250 have "(indicator (UNIV \<inter> {a..b})::_\<Rightarrow>real) integrable_on UNIV"
8.251 unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def_raw)
8.252 @@ -467,7 +476,7 @@
8.253 lemma
8.254 fixes a b :: real
8.255 shows lmeasure_real_greaterThanAtMost[simp]:
8.256 - "lebesgue.\<mu> {a <.. b} = Real (if a \<le> b then b - a else 0)"
8.257 + "lebesgue.\<mu> {a <.. b} = extreal (if a \<le> b then b - a else 0)"
8.258 proof cases
8.259 assume "a < b"
8.260 then have "lebesgue.\<mu> {a <.. b} = lebesgue.\<mu> {a .. b} - lebesgue.\<mu> {a}"
8.261 @@ -479,7 +488,7 @@
8.262 lemma
8.263 fixes a b :: real
8.264 shows lmeasure_real_atLeastLessThan[simp]:
8.265 - "lebesgue.\<mu> {a ..< b} = Real (if a \<le> b then b - a else 0)"
8.266 + "lebesgue.\<mu> {a ..< b} = extreal (if a \<le> b then b - a else 0)"
8.267 proof cases
8.268 assume "a < b"
8.269 then have "lebesgue.\<mu> {a ..< b} = lebesgue.\<mu> {a .. b} - lebesgue.\<mu> {b}"
8.270 @@ -491,7 +500,7 @@
8.271 lemma
8.272 fixes a b :: real
8.273 shows lmeasure_real_greaterThanLessThan[simp]:
8.274 - "lebesgue.\<mu> {a <..< b} = Real (if a \<le> b then b - a else 0)"
8.275 + "lebesgue.\<mu> {a <..< b} = extreal (if a \<le> b then b - a else 0)"
8.276 proof cases
8.277 assume "a < b"
8.278 then have "lebesgue.\<mu> {a <..< b} = lebesgue.\<mu> {a <.. b} - lebesgue.\<mu> {b}"
8.279 @@ -511,19 +520,16 @@
8.280 and measurable_lborel[simp]: "measurable lborel = measurable borel"
8.281 by (simp_all add: measurable_def_raw lborel_def)
8.282
8.283 -interpretation lborel: measure_space lborel
8.284 +interpretation lborel: measure_space "lborel :: ('a::ordered_euclidean_space) measure_space"
8.285 where "space lborel = UNIV"
8.286 and "sets lborel = sets borel"
8.287 and "measure lborel = lebesgue.\<mu>"
8.288 and "measurable lborel = measurable borel"
8.289 -proof -
8.290 - show "measure_space lborel"
8.291 - proof
8.292 - show "countably_additive lborel (measure lborel)"
8.293 - using lebesgue.ca unfolding countably_additive_def lborel_def
8.294 - apply safe apply (erule_tac x=A in allE) by auto
8.295 - qed (auto simp: lborel_def)
8.296 -qed simp_all
8.297 +proof (rule lebesgue.measure_space_subalgebra)
8.298 + have "sigma_algebra (lborel::'a measure_space) \<longleftrightarrow> sigma_algebra (borel::'a algebra)"
8.299 + unfolding sigma_algebra_iff2 lborel_def by simp
8.300 + then show "sigma_algebra (lborel::'a measure_space)" by simp default
8.301 +qed auto
8.302
8.303 interpretation lborel: sigma_finite_measure lborel
8.304 where "space lborel = UNIV"
8.305 @@ -536,7 +542,7 @@
8.306 show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed)
8.307 { fix x have "\<exists>n. x\<in>cube n" using mem_big_cube by auto }
8.308 thus "(\<Union>i. cube i) = space lborel" by auto
8.309 - show "\<forall>i. measure lborel (cube i) \<noteq> \<omega>" by (simp add: cube_def)
8.310 + show "\<forall>i. measure lborel (cube i) \<noteq> \<infinity>" by (simp add: cube_def)
8.311 qed
8.312 qed simp_all
8.313
8.314 @@ -544,171 +550,221 @@
8.315 proof
8.316 from lborel.sigma_finite guess A ..
8.317 moreover then have "range A \<subseteq> sets lebesgue" using lebesgueI_borel by blast
8.318 - ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lebesgue.\<mu> (A i) \<noteq> \<omega>)"
8.319 + ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lebesgue.\<mu> (A i) \<noteq> \<infinity>)"
8.320 by auto
8.321 qed
8.322
8.323 subsection {* Lebesgue integrable implies Gauge integrable *}
8.324
8.325 +lemma positive_not_Inf:
8.326 + "0 \<le> x \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> \<bar>x\<bar> \<noteq> \<infinity>"
8.327 + by (cases x) auto
8.328 +
8.329 +lemma has_integral_cmult_real:
8.330 + fixes c :: real
8.331 + assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A"
8.332 + shows "((\<lambda>x. c * f x) has_integral c * x) A"
8.333 +proof cases
8.334 + assume "c \<noteq> 0"
8.335 + from has_integral_cmul[OF assms[OF this], of c] show ?thesis
8.336 + unfolding real_scaleR_def .
8.337 +qed simp
8.338 +
8.339 lemma simple_function_has_integral:
8.340 - fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
8.341 + fixes f::"'a::ordered_euclidean_space \<Rightarrow> extreal"
8.342 assumes f:"simple_function lebesgue f"
8.343 - and f':"\<forall>x. f x \<noteq> \<omega>"
8.344 - and om:"\<forall>x\<in>range f. lebesgue.\<mu> (f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
8.345 + and f':"range f \<subseteq> {0..<\<infinity>}"
8.346 + and om:"\<And>x. x \<in> range f \<Longrightarrow> lebesgue.\<mu> (f -` {x} \<inter> UNIV) = \<infinity> \<Longrightarrow> x = 0"
8.347 shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV"
8.348 - unfolding simple_integral_def
8.349 - apply(subst lebesgue_simple_function_indicator[OF f])
8.350 -proof -
8.351 - case goal1
8.352 - have *:"\<And>x. \<forall>y\<in>range f. y * indicator (f -` {y}) x \<noteq> \<omega>"
8.353 - "\<forall>x\<in>range f. x * lebesgue.\<mu> (f -` {x} \<inter> UNIV) \<noteq> \<omega>"
8.354 - using f' om unfolding indicator_def by auto
8.355 - show ?case unfolding space_lebesgue real_of_pextreal_setsum'[OF *(1),THEN sym]
8.356 - unfolding real_of_pextreal_setsum'[OF *(2),THEN sym]
8.357 - unfolding real_of_pextreal_setsum space_lebesgue
8.358 - apply(rule has_integral_setsum)
8.359 - proof safe show "finite (range f)" using f by (auto dest: lebesgue.simple_functionD)
8.360 - fix y::'a show "((\<lambda>x. real (f y * indicator (f -` {f y}) x)) has_integral
8.361 - real (f y * lebesgue.\<mu> (f -` {f y} \<inter> UNIV))) UNIV"
8.362 - proof(cases "f y = 0") case False
8.363 - have mea:"(indicator (f -` {f y}) ::_\<Rightarrow>real) integrable_on UNIV"
8.364 - apply(rule lmeasure_finite_integrable)
8.365 - using assms unfolding simple_function_def using False by auto
8.366 - have *:"\<And>x. real (indicator (f -` {f y}) x::pextreal) = (indicator (f -` {f y}) x)"
8.367 - by (auto simp: indicator_def)
8.368 - show ?thesis unfolding real_of_pextreal_mult[THEN sym]
8.369 - apply(rule has_integral_cmul[where 'b=real, unfolded real_scaleR_def])
8.370 - unfolding Int_UNIV_right lmeasure_eq_integral[OF mea,THEN sym]
8.371 - unfolding integral_eq_lmeasure[OF mea, symmetric] *
8.372 - apply(rule integrable_integral) using mea .
8.373 - qed auto
8.374 + unfolding simple_integral_def space_lebesgue
8.375 +proof (subst lebesgue_simple_function_indicator)
8.376 + let "?M x" = "lebesgue.\<mu> (f -` {x} \<inter> UNIV)"
8.377 + let "?F x" = "indicator (f -` {x})"
8.378 + { fix x y assume "y \<in> range f"
8.379 + from subsetD[OF f' this] have "y * ?F y x = extreal (real y * ?F y x)"
8.380 + by (cases rule: extreal2_cases[of y "?F y x"])
8.381 + (auto simp: indicator_def one_extreal_def split: split_if_asm) }
8.382 + moreover
8.383 + { fix x assume x: "x\<in>range f"
8.384 + have "x * ?M x = real x * real (?M x)"
8.385 + proof cases
8.386 + assume "x \<noteq> 0" with om[OF x] have "?M x \<noteq> \<infinity>" by auto
8.387 + with subsetD[OF f' x] f[THEN lebesgue.simple_functionD(2)] show ?thesis
8.388 + by (cases rule: extreal2_cases[of x "?M x"]) auto
8.389 + qed simp }
8.390 + ultimately
8.391 + have "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV \<longleftrightarrow>
8.392 + ((\<lambda>x. \<Sum>y\<in>range f. real y * ?F y x) has_integral (\<Sum>x\<in>range f. real x * real (?M x))) UNIV"
8.393 + by simp
8.394 + also have \<dots>
8.395 + proof (intro has_integral_setsum has_integral_cmult_real lmeasure_finite_has_integral
8.396 + real_of_extreal_pos lebesgue.positive_measure ballI)
8.397 + show *: "finite (range f)" "\<And>y. f -` {y} \<in> sets lebesgue" "\<And>y. f -` {y} \<inter> UNIV \<in> sets lebesgue"
8.398 + using lebesgue.simple_functionD[OF f] by auto
8.399 + fix y assume "real y \<noteq> 0" "y \<in> range f"
8.400 + with * om[OF this(2)] show "lebesgue.\<mu> (f -` {y}) = extreal (real (?M y))"
8.401 + by (auto simp: extreal_real)
8.402 qed
8.403 -qed
8.404 + finally show "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV" .
8.405 +qed fact
8.406
8.407 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"
8.408 unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)
8.409 using assms by auto
8.410
8.411 lemma simple_function_has_integral':
8.412 - fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
8.413 - assumes f:"simple_function lebesgue f"
8.414 - and i: "integral\<^isup>S lebesgue f \<noteq> \<omega>"
8.415 + fixes f::"'a::ordered_euclidean_space \<Rightarrow> extreal"
8.416 + assumes f: "simple_function lebesgue f" "\<And>x. 0 \<le> f x"
8.417 + and i: "integral\<^isup>S lebesgue f \<noteq> \<infinity>"
8.418 shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV"
8.419 -proof- let ?f = "\<lambda>x. if f x = \<omega> then 0 else f x"
8.420 - { fix x have "real (f x) = real (?f x)" by (cases "f x") auto } note * = this
8.421 - have **:"{x. f x \<noteq> ?f x} = f -` {\<omega>}" by auto
8.422 - have **:"lebesgue.\<mu> {x\<in>space lebesgue. f x \<noteq> ?f x} = 0"
8.423 - using lebesgue.simple_integral_omega[OF assms] by(auto simp add:**)
8.424 - show ?thesis apply(subst lebesgue.simple_integral_cong'[OF f _ **])
8.425 - apply(rule lebesgue.simple_function_compose1[OF f])
8.426 - unfolding * defer apply(rule simple_function_has_integral)
8.427 - proof-
8.428 - show "simple_function lebesgue ?f"
8.429 - using lebesgue.simple_function_compose1[OF f] .
8.430 - show "\<forall>x. ?f x \<noteq> \<omega>" by auto
8.431 - show "\<forall>x\<in>range ?f. lebesgue.\<mu> (?f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
8.432 - proof (safe, simp, safe, rule ccontr)
8.433 - fix y assume "f y \<noteq> \<omega>" "f y \<noteq> 0"
8.434 - hence "(\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y} = f -` {f y}"
8.435 - by (auto split: split_if_asm)
8.436 - moreover assume "lebesgue.\<mu> ((\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y}) = \<omega>"
8.437 - ultimately have "lebesgue.\<mu> (f -` {f y}) = \<omega>" by simp
8.438 - moreover
8.439 - have "f y * lebesgue.\<mu> (f -` {f y}) \<noteq> \<omega>" using i f
8.440 - unfolding simple_integral_def setsum_\<omega> simple_function_def
8.441 - by auto
8.442 - ultimately have "f y = 0" by (auto split: split_if_asm)
8.443 - then show False using `f y \<noteq> 0` by simp
8.444 - qed
8.445 +proof -
8.446 + let ?f = "\<lambda>x. if x \<in> f -` {\<infinity>} then 0 else f x"
8.447 + note f(1)[THEN lebesgue.simple_functionD(2)]
8.448 + then have [simp, intro]: "\<And>X. f -` X \<in> sets lebesgue" by auto
8.449 + have f': "simple_function lebesgue ?f"
8.450 + using f by (intro lebesgue.simple_function_If_set) auto
8.451 + have rng: "range ?f \<subseteq> {0..<\<infinity>}" using f by auto
8.452 + have "AE x in lebesgue. f x = ?f x"
8.453 + using lebesgue.simple_integral_PInf[OF f i]
8.454 + by (intro lebesgue.AE_I[where N="f -` {\<infinity>} \<inter> space lebesgue"]) auto
8.455 + from f(1) f' this have eq: "integral\<^isup>S lebesgue f = integral\<^isup>S lebesgue ?f"
8.456 + by (rule lebesgue.simple_integral_cong_AE)
8.457 + have real_eq: "\<And>x. real (f x) = real (?f x)" by auto
8.458 +
8.459 + show ?thesis
8.460 + unfolding eq real_eq
8.461 + proof (rule simple_function_has_integral[OF f' rng])
8.462 + fix x assume x: "x \<in> range ?f" and inf: "lebesgue.\<mu> (?f -` {x} \<inter> UNIV) = \<infinity>"
8.463 + have "x * lebesgue.\<mu> (?f -` {x} \<inter> UNIV) = (\<integral>\<^isup>S y. x * indicator (?f -` {x}) y \<partial>lebesgue)"
8.464 + using f'[THEN lebesgue.simple_functionD(2)]
8.465 + by (simp add: lebesgue.simple_integral_cmult_indicator)
8.466 + also have "\<dots> \<le> integral\<^isup>S lebesgue f"
8.467 + using f'[THEN lebesgue.simple_functionD(2)] f
8.468 + by (intro lebesgue.simple_integral_mono lebesgue.simple_function_mult lebesgue.simple_function_indicator)
8.469 + (auto split: split_indicator)
8.470 + finally show "x = 0" unfolding inf using i subsetD[OF rng x] by (auto split: split_if_asm)
8.471 qed
8.472 qed
8.473
8.474 -lemma (in measure_space) positive_integral_monotone_convergence:
8.475 - fixes f::"nat \<Rightarrow> 'a \<Rightarrow> pextreal"
8.476 - assumes i: "\<And>i. f i \<in> borel_measurable M" and mono: "\<And>x. mono (\<lambda>n. f n x)"
8.477 - and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
8.478 - shows "u \<in> borel_measurable M"
8.479 - and "(\<lambda>i. integral\<^isup>P M (f i)) ----> integral\<^isup>P M u" (is ?ilim)
8.480 -proof -
8.481 - from positive_integral_isoton[unfolded isoton_fun_expand isoton_iff_Lim_mono, of f u]
8.482 - show ?ilim using mono lim i by auto
8.483 - have "\<And>x. (SUP i. f i x) = u x" using mono lim SUP_Lim_pextreal
8.484 - unfolding fun_eq_iff mono_def by auto
8.485 - moreover have "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M"
8.486 - using i by auto
8.487 - ultimately show "u \<in> borel_measurable M" by simp
8.488 -qed
8.489 +lemma real_of_extreal_positive_mono:
8.490 + "\<lbrakk>0 \<le> x; x \<le> y; y \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> real x \<le> real y"
8.491 + by (cases rule: extreal2_cases[of x y]) auto
8.492
8.493 lemma positive_integral_has_integral:
8.494 - fixes f::"'a::ordered_euclidean_space => pextreal"
8.495 - assumes f:"f \<in> borel_measurable lebesgue"
8.496 - and int_om:"integral\<^isup>P lebesgue f \<noteq> \<omega>"
8.497 - and f_om:"\<forall>x. f x \<noteq> \<omega>" (* TODO: remove this *)
8.498 + fixes f :: "'a::ordered_euclidean_space \<Rightarrow> extreal"
8.499 + assumes f: "f \<in> borel_measurable lebesgue" "range f \<subseteq> {0..<\<infinity>}" "integral\<^isup>P lebesgue f \<noteq> \<infinity>"
8.500 shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>P lebesgue f))) UNIV"
8.501 -proof- let ?i = "integral\<^isup>P lebesgue f"
8.502 - from lebesgue.borel_measurable_implies_simple_function_sequence[OF f]
8.503 - guess u .. note conjunctD2[OF this,rule_format] note u = conjunctD2[OF this(1)] this(2)
8.504 - let ?u = "\<lambda>i x. real (u i x)" and ?f = "\<lambda>x. real (f x)"
8.505 - have u_simple:"\<And>k. integral\<^isup>S lebesgue (u k) = integral\<^isup>P lebesgue (u k)"
8.506 - apply(subst lebesgue.positive_integral_eq_simple_integral[THEN sym,OF u(1)]) ..
8.507 - have int_u_le:"\<And>k. integral\<^isup>S lebesgue (u k) \<le> integral\<^isup>P lebesgue f"
8.508 - unfolding u_simple apply(rule lebesgue.positive_integral_mono)
8.509 - using isoton_Sup[OF u(3)] unfolding le_fun_def by auto
8.510 - have u_int_om:"\<And>i. integral\<^isup>S lebesgue (u i) \<noteq> \<omega>"
8.511 - proof- case goal1 thus ?case using int_u_le[of i] int_om by auto qed
8.512 +proof -
8.513 + from lebesgue.borel_measurable_implies_simple_function_sequence'[OF f(1)]
8.514 + guess u . note u = this
8.515 + have SUP_eq: "\<And>x. (SUP i. u i x) = f x"
8.516 + using u(4) f(2)[THEN subsetD] by (auto split: split_max)
8.517 + let "?u i x" = "real (u i x)"
8.518 + note u_eq = lebesgue.positive_integral_eq_simple_integral[OF u(1,5), symmetric]
8.519 + { fix i
8.520 + note u_eq
8.521 + also have "integral\<^isup>P lebesgue (u i) \<le> (\<integral>\<^isup>+x. max 0 (f x) \<partial>lebesgue)"
8.522 + by (intro lebesgue.positive_integral_mono) (auto intro: le_SUPI simp: u(4)[symmetric])
8.523 + finally have "integral\<^isup>S lebesgue (u i) \<noteq> \<infinity>"
8.524 + unfolding positive_integral_max_0 using f by auto }
8.525 + note u_fin = this
8.526 + then have u_int: "\<And>i. (?u i has_integral real (integral\<^isup>S lebesgue (u i))) UNIV"
8.527 + by (rule simple_function_has_integral'[OF u(1,5)])
8.528 + have "\<forall>x. \<exists>r\<ge>0. f x = extreal r"
8.529 + proof
8.530 + fix x from f(2) have "0 \<le> f x" "f x \<noteq> \<infinity>" by (auto simp: subset_eq)
8.531 + then show "\<exists>r\<ge>0. f x = extreal r" by (cases "f x") auto
8.532 + qed
8.533 + from choice[OF this] obtain f' where f': "f = (\<lambda>x. extreal (f' x))" "\<And>x. 0 \<le> f' x" by auto
8.534 +
8.535 + have "\<forall>i. \<exists>r. \<forall>x. 0 \<le> r x \<and> u i x = extreal (r x)"
8.536 + proof
8.537 + fix i show "\<exists>r. \<forall>x. 0 \<le> r x \<and> u i x = extreal (r x)"
8.538 + proof (intro choice allI)
8.539 + fix i x have "u i x \<noteq> \<infinity>" using u(3)[of i] by (auto simp: image_iff) metis
8.540 + then show "\<exists>r\<ge>0. u i x = extreal r" using u(5)[of i x] by (cases "u i x") auto
8.541 + qed
8.542 + qed
8.543 + from choice[OF this] obtain u' where
8.544 + u': "u = (\<lambda>i x. extreal (u' i x))" "\<And>i x. 0 \<le> u' i x" by (auto simp: fun_eq_iff)
8.545
8.546 - note u_int = simple_function_has_integral'[OF u(1) this]
8.547 - have "(\<lambda>x. real (f x)) integrable_on UNIV \<and>
8.548 - (\<lambda>k. Integration.integral UNIV (\<lambda>x. real (u k x))) ----> Integration.integral UNIV (\<lambda>x. real (f x))"
8.549 - apply(rule monotone_convergence_increasing) apply(rule,rule,rule u_int)
8.550 - proof safe case goal1 show ?case apply(rule real_of_pextreal_mono) using u(2,3) by auto
8.551 - next case goal2 show ?case using u(3) apply(subst lim_Real[THEN sym])
8.552 - prefer 3 apply(subst Real_real') defer apply(subst Real_real')
8.553 - using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] using f_om u by auto
8.554 - next case goal3
8.555 - show ?case apply(rule bounded_realI[where B="real (integral\<^isup>P lebesgue f)"])
8.556 - apply safe apply(subst abs_of_nonneg) apply(rule integral_nonneg,rule) apply(rule u_int)
8.557 - unfolding integral_unique[OF u_int] defer apply(rule real_of_pextreal_mono[OF _ int_u_le])
8.558 - using u int_om by auto
8.559 - qed note int = conjunctD2[OF this]
8.560 + have convergent: "f' integrable_on UNIV \<and>
8.561 + (\<lambda>k. integral UNIV (u' k)) ----> integral UNIV f'"
8.562 + proof (intro monotone_convergence_increasing allI ballI)
8.563 + show int: "\<And>k. (u' k) integrable_on UNIV"
8.564 + using u_int unfolding integrable_on_def u' by auto
8.565 + show "\<And>k x. u' k x \<le> u' (Suc k) x" using u(2,3,5)
8.566 + by (auto simp: incseq_Suc_iff le_fun_def image_iff u' intro!: real_of_extreal_positive_mono)
8.567 + show "\<And>x. (\<lambda>k. u' k x) ----> f' x"
8.568 + using SUP_eq u(2)
8.569 + by (intro SUP_eq_LIMSEQ[THEN iffD1]) (auto simp: u' f' incseq_mono incseq_Suc_iff le_fun_def)
8.570 + show "bounded {integral UNIV (u' k)|k. True}"
8.571 + proof (safe intro!: bounded_realI)
8.572 + fix k
8.573 + have "\<bar>integral UNIV (u' k)\<bar> = integral UNIV (u' k)"
8.574 + by (intro abs_of_nonneg integral_nonneg int ballI u')
8.575 + also have "\<dots> = real (integral\<^isup>S lebesgue (u k))"
8.576 + using u_int[THEN integral_unique] by (simp add: u')
8.577 + also have "\<dots> = real (integral\<^isup>P lebesgue (u k))"
8.578 + using lebesgue.positive_integral_eq_simple_integral[OF u(1,5)] by simp
8.579 + also have "\<dots> \<le> real (integral\<^isup>P lebesgue f)" using f
8.580 + by (auto intro!: real_of_extreal_positive_mono lebesgue.positive_integral_positive
8.581 + lebesgue.positive_integral_mono le_SUPI simp: SUP_eq[symmetric])
8.582 + finally show "\<bar>integral UNIV (u' k)\<bar> \<le> real (integral\<^isup>P lebesgue f)" .
8.583 + qed
8.584 + qed
8.585
8.586 - have "(\<lambda>i. integral\<^isup>S lebesgue (u i)) ----> ?i" unfolding u_simple
8.587 - apply(rule lebesgue.positive_integral_monotone_convergence(2))
8.588 - apply(rule lebesgue.borel_measurable_simple_function[OF u(1)])
8.589 - using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] by auto
8.590 - hence "(\<lambda>i. real (integral\<^isup>S lebesgue (u i))) ----> real ?i" apply-
8.591 - apply(subst lim_Real[THEN sym]) prefer 3
8.592 - apply(subst Real_real') defer apply(subst Real_real')
8.593 - using u f_om int_om u_int_om by auto
8.594 - note * = LIMSEQ_unique[OF this int(2)[unfolded integral_unique[OF u_int]]]
8.595 - show ?thesis unfolding * by(rule integrable_integral[OF int(1)])
8.596 + have "integral\<^isup>P lebesgue f = extreal (integral UNIV f')"
8.597 + proof (rule tendsto_unique[OF trivial_limit_sequentially])
8.598 + have "(\<lambda>i. integral\<^isup>S lebesgue (u i)) ----> (SUP i. integral\<^isup>P lebesgue (u i))"
8.599 + unfolding u_eq by (intro LIMSEQ_extreal_SUPR lebesgue.incseq_positive_integral u)
8.600 + also note lebesgue.positive_integral_monotone_convergence_SUP
8.601 + [OF u(2) lebesgue.borel_measurable_simple_function[OF u(1)] u(5), symmetric]
8.602 + finally show "(\<lambda>k. integral\<^isup>S lebesgue (u k)) ----> integral\<^isup>P lebesgue f"
8.603 + unfolding SUP_eq .
8.604 +
8.605 + { fix k
8.606 + have "0 \<le> integral\<^isup>S lebesgue (u k)"
8.607 + using u by (auto intro!: lebesgue.simple_integral_positive)
8.608 + then have "integral\<^isup>S lebesgue (u k) = extreal (real (integral\<^isup>S lebesgue (u k)))"
8.609 + using u_fin by (auto simp: extreal_real) }
8.610 + note * = this
8.611 + show "(\<lambda>k. integral\<^isup>S lebesgue (u k)) ----> extreal (integral UNIV f')"
8.612 + using convergent using u_int[THEN integral_unique, symmetric]
8.613 + by (subst *) (simp add: lim_extreal u')
8.614 + qed
8.615 + then show ?thesis using convergent by (simp add: f' integrable_integral)
8.616 qed
8.617
8.618 lemma lebesgue_integral_has_integral:
8.619 - fixes f::"'a::ordered_euclidean_space => real"
8.620 - assumes f:"integrable lebesgue f"
8.621 + fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
8.622 + assumes f: "integrable lebesgue f"
8.623 shows "(f has_integral (integral\<^isup>L lebesgue f)) UNIV"
8.624 -proof- let ?n = "\<lambda>x. - min (f x) 0" and ?p = "\<lambda>x. max (f x) 0"
8.625 - have *:"f = (\<lambda>x. ?p x - ?n x)" apply rule by auto
8.626 - note f = integrableD[OF f]
8.627 - show ?thesis unfolding lebesgue_integral_def apply(subst *)
8.628 - proof(rule has_integral_sub) case goal1
8.629 - have *:"\<forall>x. Real (f x) \<noteq> \<omega>" by auto
8.630 - note lebesgue.borel_measurable_Real[OF f(1)]
8.631 - from positive_integral_has_integral[OF this f(2) *]
8.632 - show ?case unfolding real_Real_max .
8.633 - next case goal2
8.634 - have *:"\<forall>x. Real (- f x) \<noteq> \<omega>" by auto
8.635 - note lebesgue.borel_measurable_uminus[OF f(1)]
8.636 - note lebesgue.borel_measurable_Real[OF this]
8.637 - from positive_integral_has_integral[OF this f(3) *]
8.638 - show ?case unfolding real_Real_max minus_min_eq_max by auto
8.639 - qed
8.640 +proof -
8.641 + let ?n = "\<lambda>x. real (extreal (max 0 (- f x)))" and ?p = "\<lambda>x. real (extreal (max 0 (f x)))"
8.642 + have *: "f = (\<lambda>x. ?p x - ?n x)" by (auto simp del: extreal_max)
8.643 + { fix f have "(\<integral>\<^isup>+ x. extreal (f x) \<partial>lebesgue) = (\<integral>\<^isup>+ x. extreal (max 0 (f x)) \<partial>lebesgue)"
8.644 + by (intro lebesgue.positive_integral_cong_pos) (auto split: split_max) }
8.645 + note eq = this
8.646 + show ?thesis
8.647 + unfolding lebesgue_integral_def
8.648 + apply (subst *)
8.649 + apply (rule has_integral_sub)
8.650 + unfolding eq[of f] eq[of "\<lambda>x. - f x"]
8.651 + apply (safe intro!: positive_integral_has_integral)
8.652 + using integrableD[OF f]
8.653 + by (auto simp: zero_extreal_def[symmetric] positive_integral_max_0 split: split_max
8.654 + intro!: lebesgue.measurable_If lebesgue.borel_measurable_extreal)
8.655 qed
8.656
8.657 lemma lebesgue_positive_integral_eq_borel:
8.658 - "f \<in> borel_measurable borel \<Longrightarrow> integral\<^isup>P lebesgue f = integral\<^isup>P lborel f"
8.659 - by (auto intro!: lebesgue.positive_integral_subalgebra[symmetric]) default
8.660 + assumes f: "f \<in> borel_measurable borel"
8.661 + shows "integral\<^isup>P lebesgue f = integral\<^isup>P lborel f"
8.662 +proof -
8.663 + from f have "integral\<^isup>P lebesgue (\<lambda>x. max 0 (f x)) = integral\<^isup>P lborel (\<lambda>x. max 0 (f x))"
8.664 + by (auto intro!: lebesgue.positive_integral_subalgebra[symmetric]) default
8.665 + then show ?thesis unfolding positive_integral_max_0 .
8.666 +qed
8.667
8.668 lemma lebesgue_integral_eq_borel:
8.669 assumes "f \<in> borel_measurable borel"
8.670 @@ -771,7 +827,7 @@
8.671 have "sets ?G = sets (\<Pi>\<^isub>M i\<in>I.
8.672 sigma \<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>)"
8.673 by (subst sigma_product_algebra_sigma_eq[of I "\<lambda>_ i. {..<real i}" ])
8.674 - (auto intro!: measurable_sigma_sigma isotoneI real_arch_lt
8.675 + (auto intro!: measurable_sigma_sigma incseq_SucI real_arch_lt
8.676 simp: product_algebra_def)
8.677 then show ?thesis
8.678 unfolding lborel_def borel_eq_lessThan lebesgue_def sigma_def by simp
8.679 @@ -838,9 +894,10 @@
8.680 let ?E = "\<lparr> space = UNIV :: 'a set, sets = range (\<lambda>(a,b). {a..b}) \<rparr>"
8.681 show "Int_stable ?E" using Int_stable_cuboids .
8.682 show "range cube \<subseteq> sets ?E" unfolding cube_def_raw by auto
8.683 + show "incseq cube" using cube_subset_Suc by (auto intro!: incseq_SucI)
8.684 { fix x have "\<exists>n. x \<in> cube n" using mem_big_cube[of x] by fastsimp }
8.685 - then show "cube \<up> space ?E" by (intro isotoneI cube_subset_Suc) auto
8.686 - { fix i show "lborel.\<mu> (cube i) \<noteq> \<omega>" unfolding cube_def by auto }
8.687 + then show "(\<Union>i. cube i) = space ?E" by auto
8.688 + { fix i show "lborel.\<mu> (cube i) \<noteq> \<infinity>" unfolding cube_def by auto }
8.689 show "A \<in> sets (sigma ?E)" "sets (sigma ?E) = sets lborel" "space ?E = space lborel"
8.690 using assms by (simp_all add: borel_eq_atLeastAtMost)
8.691
8.692 @@ -857,7 +914,7 @@
8.693 by (simp add: interval_ne_empty eucl_le[where 'a='a])
8.694 then have "lborel.\<mu> X = (\<Prod>x<DIM('a). lborel.\<mu> {a $$ x .. b $$ x})"
8.695 by (auto simp: content_closed_interval eucl_le[where 'a='a]
8.696 - intro!: Real_setprod )
8.697 + intro!: setprod_extreal[symmetric])
8.698 also have "\<dots> = measure ?P (?T X)"
8.699 unfolding * by (subst lborel_space.measure_times) auto
8.700 finally show ?thesis .
8.701 @@ -882,7 +939,7 @@
8.702 using lborel_eq_lborel_space[OF A] by simp
8.703
8.704 lemma borel_fubini_positiv_integral:
8.705 - fixes f :: "'a::ordered_euclidean_space \<Rightarrow> pextreal"
8.706 + fixes f :: "'a::ordered_euclidean_space \<Rightarrow> extreal"
8.707 assumes f: "f \<in> borel_measurable borel"
8.708 shows "integral\<^isup>P lborel f = \<integral>\<^isup>+x. f (p2e x) \<partial>(lborel_space.P DIM('a))"
8.709 proof (rule lborel_space.positive_integral_vimage[OF _ measure_preserving_p2e _])
9.1 --- a/src/HOL/Probability/Measure.thy Mon Mar 14 14:37:47 2011 +0100
9.2 +++ b/src/HOL/Probability/Measure.thy Mon Mar 14 14:37:49 2011 +0100
9.3 @@ -76,7 +76,7 @@
9.4
9.5 lemma (in measure_space) measure_countably_additive:
9.6 assumes "range A \<subseteq> sets M" "disjoint_family A"
9.7 - shows "psuminf (\<lambda>i. \<mu> (A i)) = \<mu> (\<Union>i. A i)"
9.8 + shows "(\<Sum>i. \<mu> (A i)) = \<mu> (\<Union>i. A i)"
9.9 proof -
9.10 have "(\<Union> i. A i) \<in> sets M" using assms(1) by (rule countable_UN)
9.11 with ca assms show ?thesis by (simp add: countably_additive_def)
9.12 @@ -94,13 +94,13 @@
9.13 interpret N: sigma_algebra N by (intro sigma_algebra_cong assms)
9.14 show ?thesis
9.15 proof
9.16 - show "measure N {} = 0" using assms by auto
9.17 + show "positive N (measure N)" using assms by (auto simp: positive_def)
9.18 show "countably_additive N (measure N)" unfolding countably_additive_def
9.19 proof safe
9.20 fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets N" "disjoint_family A"
9.21 then have "\<And>i. A i \<in> sets M" "(UNION UNIV A) \<in> sets M" unfolding assms by auto
9.22 from measure_countably_additive[of A] A this[THEN assms(1)]
9.23 - show "(\<Sum>\<^isub>\<infinity>n. measure N (A n)) = measure N (UNION UNIV A)"
9.24 + show "(\<Sum>n. measure N (A n)) = measure N (UNION UNIV A)"
9.25 unfolding assms by simp
9.26 qed
9.27 qed
9.28 @@ -124,51 +124,51 @@
9.29 have "b = a \<union> (b - a)" using assms by auto
9.30 moreover have "{} = a \<inter> (b - a)" by auto
9.31 ultimately have "\<mu> b = \<mu> a + \<mu> (b - a)"
9.32 - using measure_additive[of a "b - a"] local.Diff[of b a] assms by auto
9.33 - moreover have "\<mu> (b - a) \<ge> 0" using assms by auto
9.34 + using measure_additive[of a "b - a"] Diff[of b a] assms by auto
9.35 + moreover have "\<mu> a + 0 \<le> \<mu> a + \<mu> (b - a)" using assms by (intro add_mono) auto
9.36 ultimately show "\<mu> a \<le> \<mu> b" by auto
9.37 qed
9.38
9.39 lemma (in measure_space) measure_compl:
9.40 - assumes s: "s \<in> sets M" and fin: "\<mu> s \<noteq> \<omega>"
9.41 + assumes s: "s \<in> sets M" and fin: "\<mu> s \<noteq> \<infinity>"
9.42 shows "\<mu> (space M - s) = \<mu> (space M) - \<mu> s"
9.43 proof -
9.44 have s_less_space: "\<mu> s \<le> \<mu> (space M)"
9.45 using s by (auto intro!: measure_mono sets_into_space)
9.46 -
9.47 + from s have "0 \<le> \<mu> s" by auto
9.48 have "\<mu> (space M) = \<mu> (s \<union> (space M - s))" using s
9.49 by (metis Un_Diff_cancel Un_absorb1 s sets_into_space)
9.50 also have "... = \<mu> s + \<mu> (space M - s)"
9.51 by (rule additiveD [OF additive]) (auto simp add: s)
9.52 finally have "\<mu> (space M) = \<mu> s + \<mu> (space M - s)" .
9.53 - thus ?thesis
9.54 - unfolding minus_pextreal_eq2[OF s_less_space fin]
9.55 - by (simp add: ac_simps)
9.56 + then show ?thesis
9.57 + using fin `0 \<le> \<mu> s`
9.58 + unfolding extreal_eq_minus_iff by (auto simp: ac_simps)
9.59 qed
9.60
9.61 lemma (in measure_space) measure_Diff:
9.62 - assumes finite: "\<mu> B \<noteq> \<omega>"
9.63 + assumes finite: "\<mu> B \<noteq> \<infinity>"
9.64 and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
9.65 shows "\<mu> (A - B) = \<mu> A - \<mu> B"
9.66 proof -
9.67 - have *: "(A - B) \<union> B = A" using `B \<subseteq> A` by auto
9.68 -
9.69 - have "\<mu> ((A - B) \<union> B) = \<mu> (A - B) + \<mu> B"
9.70 - using measurable by (rule_tac measure_additive[symmetric]) auto
9.71 - thus ?thesis unfolding * using `\<mu> B \<noteq> \<omega>`
9.72 - by (simp add: pextreal_cancel_plus_minus)
9.73 + have "0 \<le> \<mu> B" using assms by auto
9.74 + have "(A - B) \<union> B = A" using `B \<subseteq> A` by auto
9.75 + then have "\<mu> A = \<mu> ((A - B) \<union> B)" by simp
9.76 + also have "\<dots> = \<mu> (A - B) + \<mu> B"
9.77 + using measurable by (subst measure_additive[symmetric]) auto
9.78 + finally show "\<mu> (A - B) = \<mu> A - \<mu> B"
9.79 + unfolding extreal_eq_minus_iff
9.80 + using finite `0 \<le> \<mu> B` by auto
9.81 qed
9.82
9.83 lemma (in measure_space) measure_countable_increasing:
9.84 assumes A: "range A \<subseteq> sets M"
9.85 and A0: "A 0 = {}"
9.86 - and ASuc: "\<And>n. A n \<subseteq> A (Suc n)"
9.87 + and ASuc: "\<And>n. A n \<subseteq> A (Suc n)"
9.88 shows "(SUP n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
9.89 proof -
9.90 - {
9.91 - fix n
9.92 - have "\<mu> (A n) =
9.93 - setsum (\<mu> \<circ> (\<lambda>i. A (Suc i) - A i)) {..<n}"
9.94 + { fix n
9.95 + have "\<mu> (A n) = (\<Sum>i<n. \<mu> (A (Suc i) - A i))"
9.96 proof (induct n)
9.97 case 0 thus ?case by (auto simp add: A0)
9.98 next
9.99 @@ -199,92 +199,83 @@
9.100 by (metis A Diff range_subsetD)
9.101 have A2: "(\<Union>i. A (Suc i) - A i) \<in> sets M"
9.102 by (blast intro: range_subsetD [OF A])
9.103 - have "psuminf ( (\<lambda>i. \<mu> (A (Suc i) - A i))) = \<mu> (\<Union>i. A (Suc i) - A i)"
9.104 + have "(SUP n. \<Sum>i<n. \<mu> (A (Suc i) - A i)) = (\<Sum>i. \<mu> (A (Suc i) - A i))"
9.105 + using A by (auto intro!: suminf_extreal_eq_SUPR[symmetric])
9.106 + also have "\<dots> = \<mu> (\<Union>i. A (Suc i) - A i)"
9.107 by (rule measure_countably_additive)
9.108 (auto simp add: disjoint_family_Suc ASuc A1 A2)
9.109 also have "... = \<mu> (\<Union>i. A i)"
9.110 by (simp add: Aeq)
9.111 - finally have "psuminf (\<lambda>i. \<mu> (A (Suc i) - A i)) = \<mu> (\<Union>i. A i)" .
9.112 - thus ?thesis
9.113 - by (auto simp add: Meq psuminf_def)
9.114 + finally have "(SUP n. \<Sum>i<n. \<mu> (A (Suc i) - A i)) = \<mu> (\<Union>i. A i)" .
9.115 + then show ?thesis by (auto simp add: Meq)
9.116 qed
9.117
9.118 lemma (in measure_space) continuity_from_below:
9.119 - assumes A: "range A \<subseteq> sets M"
9.120 - and ASuc: "!!n. A n \<subseteq> A (Suc n)"
9.121 + assumes A: "range A \<subseteq> sets M" and "incseq A"
9.122 shows "(SUP n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
9.123 proof -
9.124 have *: "(SUP n. \<mu> (nat_case {} A (Suc n))) = (SUP n. \<mu> (nat_case {} A n))"
9.125 - apply (rule Sup_mono_offset_Suc)
9.126 - apply (rule measure_mono)
9.127 - using assms by (auto split: nat.split)
9.128 -
9.129 + using A by (auto intro!: SUPR_eq exI split: nat.split)
9.130 have ueq: "(\<Union>i. nat_case {} A i) = (\<Union>i. A i)"
9.131 by (auto simp add: split: nat.splits)
9.132 have meq: "\<And>n. \<mu> (A n) = (\<mu> \<circ> nat_case {} A) (Suc n)"
9.133 by simp
9.134 have "(SUP n. \<mu> (nat_case {} A n)) = \<mu> (\<Union>i. nat_case {} A i)"
9.135 - by (rule measure_countable_increasing)
9.136 - (auto simp add: range_subsetD [OF A] subsetD [OF ASuc] split: nat.splits)
9.137 + using range_subsetD[OF A] incseq_SucD[OF `incseq A`]
9.138 + by (force split: nat.splits intro!: measure_countable_increasing)
9.139 also have "\<mu> (\<Union>i. nat_case {} A i) = \<mu> (\<Union>i. A i)"
9.140 by (simp add: ueq)
9.141 finally have "(SUP n. \<mu> (nat_case {} A n)) = \<mu> (\<Union>i. A i)" .
9.142 thus ?thesis unfolding meq * comp_def .
9.143 qed
9.144
9.145 -lemma (in measure_space) measure_up:
9.146 - assumes "\<And>i. B i \<in> sets M" "B \<up> P"
9.147 - shows "(\<lambda>i. \<mu> (B i)) \<up> \<mu> P"
9.148 - using assms unfolding isoton_def
9.149 - by (auto intro!: measure_mono continuity_from_below)
9.150 +lemma (in measure_space) measure_incseq:
9.151 + assumes "range B \<subseteq> sets M" "incseq B"
9.152 + shows "incseq (\<lambda>i. \<mu> (B i))"
9.153 + using assms by (auto simp: incseq_def intro!: measure_mono)
9.154
9.155 -lemma (in measure_space) continuity_from_below':
9.156 - assumes A: "range A \<subseteq> sets M" "\<And>i. A i \<subseteq> A (Suc i)"
9.157 - shows "(\<lambda>i. (\<mu> (A i))) ----> (\<mu> (\<Union>i. A i))"
9.158 -proof- let ?A = "\<Union>i. A i"
9.159 - have " (\<lambda>i. \<mu> (A i)) \<up> \<mu> ?A" apply(rule measure_up)
9.160 - using assms unfolding complete_lattice_class.isoton_def by auto
9.161 - thus ?thesis by(rule isotone_Lim(1))
9.162 -qed
9.163 +lemma (in measure_space) continuity_from_below_Lim:
9.164 + assumes A: "range A \<subseteq> sets M" "incseq A"
9.165 + shows "(\<lambda>i. (\<mu> (A i))) ----> \<mu> (\<Union>i. A i)"
9.166 + using LIMSEQ_extreal_SUPR[OF measure_incseq, OF A]
9.167 + continuity_from_below[OF A] by simp
9.168 +
9.169 +lemma (in measure_space) measure_decseq:
9.170 + assumes "range B \<subseteq> sets M" "decseq B"
9.171 + shows "decseq (\<lambda>i. \<mu> (B i))"
9.172 + using assms by (auto simp: decseq_def intro!: measure_mono)
9.173
9.174 lemma (in measure_space) continuity_from_above:
9.175 - assumes A: "range A \<subseteq> sets M"
9.176 - and mono_Suc: "\<And>n. A (Suc n) \<subseteq> A n"
9.177 - and finite: "\<And>i. \<mu> (A i) \<noteq> \<omega>"
9.178 + assumes A: "range A \<subseteq> sets M" and "decseq A"
9.179 + and finite: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
9.180 shows "(INF n. \<mu> (A n)) = \<mu> (\<Inter>i. A i)"
9.181 proof -
9.182 - { fix n have "A n \<subseteq> A 0" using mono_Suc by (induct n) auto }
9.183 - note mono = this
9.184 -
9.185 have le_MI: "\<mu> (\<Inter>i. A i) \<le> \<mu> (A 0)"
9.186 using A by (auto intro!: measure_mono)
9.187 - hence *: "\<mu> (\<Inter>i. A i) \<noteq> \<omega>" using finite[of 0] by auto
9.188 + hence *: "\<mu> (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by auto
9.189 +
9.190 + have A0: "0 \<le> \<mu> (A 0)" using A by auto
9.191
9.192 - have le_IM: "(INF n. \<mu> (A n)) \<le> \<mu> (A 0)"
9.193 - by (rule INF_leI) simp
9.194 -
9.195 - have "\<mu> (A 0) - (INF n. \<mu> (A n)) = (SUP n. \<mu> (A 0 - A n))"
9.196 - unfolding pextreal_SUP_minus[symmetric]
9.197 - using mono A finite by (subst measure_Diff) auto
9.198 + have "\<mu> (A 0) - (INF n. \<mu> (A n)) = \<mu> (A 0) + (SUP n. - \<mu> (A n))"
9.199 + by (simp add: extreal_SUPR_uminus minus_extreal_def)
9.200 + also have "\<dots> = (SUP n. \<mu> (A 0) - \<mu> (A n))"
9.201 + unfolding minus_extreal_def using A0 assms
9.202 + by (subst SUPR_extreal_add) (auto simp add: measure_decseq)
9.203 + also have "\<dots> = (SUP n. \<mu> (A 0 - A n))"
9.204 + using A finite `decseq A`[unfolded decseq_def] by (subst measure_Diff) auto
9.205 also have "\<dots> = \<mu> (\<Union>i. A 0 - A i)"
9.206 proof (rule continuity_from_below)
9.207 show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"
9.208 using A by auto
9.209 - show "\<And>n. A 0 - A n \<subseteq> A 0 - A (Suc n)"
9.210 - using mono_Suc by auto
9.211 + show "incseq (\<lambda>n. A 0 - A n)"
9.212 + using `decseq A` by (auto simp add: incseq_def decseq_def)
9.213 qed
9.214 also have "\<dots> = \<mu> (A 0) - \<mu> (\<Inter>i. A i)"
9.215 - using mono A finite * by (simp, subst measure_Diff) auto
9.216 + using A finite * by (simp, subst measure_Diff) auto
9.217 finally show ?thesis
9.218 - by (rule pextreal_diff_eq_diff_imp_eq[OF finite[of 0] le_IM le_MI])
9.219 + unfolding extreal_minus_eq_minus_iff using finite A0 by auto
9.220 qed
9.221
9.222 -lemma (in measure_space) measure_down:
9.223 - assumes "\<And>i. B i \<in> sets M" "B \<down> P"
9.224 - and finite: "\<And>i. \<mu> (B i) \<noteq> \<omega>"
9.225 - shows "(\<lambda>i. \<mu> (B i)) \<down> \<mu> P"
9.226 - using assms unfolding antiton_def
9.227 - by (auto intro!: measure_mono continuity_from_above)
9.228 lemma (in measure_space) measure_insert:
9.229 assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A"
9.230 shows "\<mu> (insert x A) = \<mu> {x} + \<mu> A"
9.231 @@ -293,109 +284,26 @@
9.232 from measure_additive[OF sets this] show ?thesis by simp
9.233 qed
9.234
9.235 -lemma (in measure_space) measure_finite_singleton:
9.236 - assumes fin: "finite S"
9.237 - and ssets: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
9.238 - shows "\<mu> S = (\<Sum>x\<in>S. \<mu> {x})"
9.239 +lemma (in measure_space) measure_setsum:
9.240 + assumes "finite S" and "\<And>i. i \<in> S \<Longrightarrow> A i \<in> sets M"
9.241 + assumes disj: "disjoint_family_on A S"
9.242 + shows "(\<Sum>i\<in>S. \<mu> (A i)) = \<mu> (\<Union>i\<in>S. A i)"
9.243 using assms proof induct
9.244 - case (insert x S)
9.245 - have *: "\<mu> S = (\<Sum>x\<in>S. \<mu> {x})" "{x} \<in> sets M"
9.246 - using insert.prems by (blast intro!: insert.hyps(3))+
9.247 -
9.248 - have "(\<Union>x\<in>S. {x}) \<in> sets M"
9.249 - using insert.prems `finite S` by (blast intro!: finite_UN)
9.250 - hence "S \<in> sets M" by auto
9.251 - from measure_insert[OF `{x} \<in> sets M` this `x \<notin> S`]
9.252 - show ?case using `x \<notin> S` `finite S` * by simp
9.253 + case (insert i S)
9.254 + then have "(\<Sum>i\<in>S. \<mu> (A i)) = \<mu> (\<Union>a\<in>S. A a)"
9.255 + by (auto intro: disjoint_family_on_mono)
9.256 + moreover have "A i \<inter> (\<Union>a\<in>S. A a) = {}"
9.257 + using `disjoint_family_on A (insert i S)` `i \<notin> S`
9.258 + by (auto simp: disjoint_family_on_def)
9.259 + ultimately show ?case using insert
9.260 + by (auto simp: measure_additive finite_UN)
9.261 qed simp
9.262
9.263 -lemma (in measure_space) measure_finitely_additive':
9.264 - assumes "f \<in> ({..< n :: nat} \<rightarrow> sets M)"
9.265 - assumes "\<And> a b. \<lbrakk>a < n ; b < n ; a \<noteq> b\<rbrakk> \<Longrightarrow> f a \<inter> f b = {}"
9.266 - assumes "s = \<Union> (f ` {..< n})"
9.267 - shows "(\<Sum>i<n. (\<mu> \<circ> f) i) = \<mu> s"
9.268 -proof -
9.269 - def f' == "\<lambda> i. (if i < n then f i else {})"
9.270 - have rf: "range f' \<subseteq> sets M" unfolding f'_def
9.271 - using assms empty_sets by auto
9.272 - have df: "disjoint_family f'" unfolding f'_def disjoint_family_on_def
9.273 - using assms by simp
9.274 - have "\<Union> range f' = (\<Union> i \<in> {..< n}. f i)"
9.275 - unfolding f'_def by auto
9.276 - then have "\<mu> s = \<mu> (\<Union> range f')"
9.277 - using assms by simp
9.278 - then have part1: "(\<Sum>\<^isub>\<infinity> i. \<mu> (f' i)) = \<mu> s"
9.279 - using df rf ca[unfolded countably_additive_def, rule_format, of f']
9.280 - by auto
9.281 -
9.282 - have "(\<Sum>\<^isub>\<infinity> i. \<mu> (f' i)) = (\<Sum> i< n. \<mu> (f' i))"
9.283 - by (rule psuminf_finite) (simp add: f'_def)
9.284 - also have "\<dots> = (\<Sum>i<n. \<mu> (f i))"
9.285 - unfolding f'_def by auto
9.286 - finally have part2: "(\<Sum>\<^isub>\<infinity> i. \<mu> (f' i)) = (\<Sum>i<n. \<mu> (f i))" by simp
9.287 - show ?thesis using part1 part2 by auto
9.288 -qed
9.289 -
9.290 -
9.291 -lemma (in measure_space) measure_finitely_additive:
9.292 - assumes "finite r"
9.293 - assumes "r \<subseteq> sets M"
9.294 - assumes d: "\<And> a b. \<lbrakk>a \<in> r ; b \<in> r ; a \<noteq> b\<rbrakk> \<Longrightarrow> a \<inter> b = {}"
9.295 - shows "(\<Sum> i \<in> r. \<mu> i) = \<mu> (\<Union> r)"
9.296 -using assms
9.297 -proof -
9.298 - (* counting the elements in r is enough *)
9.299 - let ?R = "{..<card r}"
9.300 - obtain f where f: "f ` ?R = r" "inj_on f ?R"
9.301 - using ex_bij_betw_nat_finite[unfolded bij_betw_def, OF `finite r`]
9.302 - unfolding atLeast0LessThan by auto
9.303 - hence f_into_sets: "f \<in> ?R \<rightarrow> sets M" using assms by auto
9.304 - have disj: "\<And> a b. \<lbrakk>a \<in> ?R ; b \<in> ?R ; a \<noteq> b\<rbrakk> \<Longrightarrow> f a \<inter> f b = {}"
9.305 - proof -
9.306 - fix a b assume asm: "a \<in> ?R" "b \<in> ?R" "a \<noteq> b"
9.307 - hence neq: "f a \<noteq> f b" using f[unfolded inj_on_def, rule_format] by blast
9.308 - from asm have "f a \<in> r" "f b \<in> r" using f by auto
9.309 - thus "f a \<inter> f b = {}" using d[of "f a" "f b"] f using neq by auto
9.310 - qed
9.311 - have "(\<Union> r) = (\<Union> i \<in> ?R. f i)"
9.312 - using f by auto
9.313 - hence "\<mu> (\<Union> r)= \<mu> (\<Union> i \<in> ?R. f i)" by simp
9.314 - also have "\<dots> = (\<Sum> i \<in> ?R. \<mu> (f i))"
9.315 - using measure_finitely_additive'[OF f_into_sets disj] by simp
9.316 - also have "\<dots> = (\<Sum> a \<in> r. \<mu> a)"
9.317 - using f[rule_format] setsum_reindex[of f ?R "\<lambda> a. \<mu> a"] by auto
9.318 - finally show ?thesis by simp
9.319 -qed
9.320 -
9.321 -lemma (in measure_space) measure_finitely_additive'':
9.322 - assumes "finite s"
9.323 - assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<in> sets M"
9.324 - assumes d: "disjoint_family_on a s"
9.325 - shows "(\<Sum> i \<in> s. \<mu> (a i)) = \<mu> (\<Union> i \<in> s. a i)"
9.326 -using assms
9.327 -proof -
9.328 - (* counting the elements in s is enough *)
9.329 - let ?R = "{..< card s}"
9.330 - obtain f where f: "f ` ?R = s" "inj_on f ?R"
9.331 - using ex_bij_betw_nat_finite[unfolded bij_betw_def, OF `finite s`]
9.332 - unfolding atLeast0LessThan by auto
9.333 - hence f_into_sets: "a \<circ> f \<in> ?R \<rightarrow> sets M" using assms unfolding o_def by auto
9.334 - have disj: "\<And> i j. \<lbrakk>i \<in> ?R ; j \<in> ?R ; i \<noteq> j\<rbrakk> \<Longrightarrow> (a \<circ> f) i \<inter> (a \<circ> f) j = {}"
9.335 - proof -
9.336 - fix i j assume asm: "i \<in> ?R" "j \<in> ?R" "i \<noteq> j"
9.337 - hence neq: "f i \<noteq> f j" using f[unfolded inj_on_def, rule_format] by blast
9.338 - from asm have "f i \<in> s" "f j \<in> s" using f by auto
9.339 - thus "(a \<circ> f) i \<inter> (a \<circ> f) j = {}"
9.340 - using d f neq unfolding disjoint_family_on_def by auto
9.341 - qed
9.342 - have "(\<Union> i \<in> s. a i) = (\<Union> i \<in> f ` ?R. a i)" using f by auto
9.343 - hence "(\<Union> i \<in> s. a i) = (\<Union> i \<in> ?R. a (f i))" by auto
9.344 - hence "\<mu> (\<Union> i \<in> s. a i) = (\<Sum> i \<in> ?R. \<mu> (a (f i)))"
9.345 - using measure_finitely_additive'[OF f_into_sets disj] by simp
9.346 - also have "\<dots> = (\<Sum> i \<in> s. \<mu> (a i))"
9.347 - using f[rule_format] setsum_reindex[of f ?R "\<lambda> i. \<mu> (a i)"] by auto
9.348 - finally show ?thesis by simp
9.349 -qed
9.350 +lemma (in measure_space) measure_finite_singleton:
9.351 + assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
9.352 + shows "\<mu> S = (\<Sum>x\<in>S. \<mu> {x})"
9.353 + using measure_setsum[of S "\<lambda>x. {x}", OF assms]
9.354 + by (auto simp: disjoint_family_on_def)
9.355
9.356 lemma finite_additivity_sufficient:
9.357 assumes "sigma_algebra M"
9.358 @@ -405,7 +313,7 @@
9.359 interpret sigma_algebra M by fact
9.360 show ?thesis
9.361 proof
9.362 - show [simp]: "measure M {} = 0" using pos by (simp add: positive_def)
9.363 + show [simp]: "positive M (measure M)" using pos by (simp add: positive_def)
9.364 show "countably_additive M (measure M)"
9.365 proof (auto simp add: countably_additive_def)
9.366 fix A :: "nat \<Rightarrow> 'a set"
9.367 @@ -434,12 +342,12 @@
9.368 by blast
9.369 qed
9.370 then obtain N where N: "\<forall>m\<ge>N. A m = {}" by blast
9.371 - then have "\<forall>m\<ge>N. measure M (A m) = 0" by simp
9.372 - then have "(\<Sum>\<^isub>\<infinity> n. measure M (A n)) = setsum (\<lambda>m. measure M (A m)) {..<N}"
9.373 - by (simp add: psuminf_finite)
9.374 + then have "\<forall>m\<ge>N. measure M (A m) = 0" using pos[unfolded positive_def] by simp
9.375 + then have "(\<Sum>n. measure M (A n)) = (\<Sum>m<N. measure M (A m))"
9.376 + by (simp add: suminf_finite)
9.377 also have "... = measure M (\<Union>i<N. A i)"
9.378 proof (induct N)
9.379 - case 0 thus ?case by simp
9.380 + case 0 thus ?case using pos[unfolded positive_def] by simp
9.381 next
9.382 case (Suc n)
9.383 have "measure M (A n \<union> (\<Union> x<n. A x)) = measure M (A n) + measure M (\<Union> i<n. A i)"
9.384 @@ -465,30 +373,25 @@
9.385 by auto (metis Int_absorb N disjoint_iff_not_equal lessThan_iff not_leE)
9.386 thus ?thesis by simp
9.387 qed
9.388 - finally show "(\<Sum>\<^isub>\<infinity> n. measure M (A n)) = measure M (\<Union>i. A i)" .
9.389 + finally show "(\<Sum>n. measure M (A n)) = measure M (\<Union>i. A i)" .
9.390 qed
9.391 qed
9.392 qed
9.393
9.394 lemma (in measure_space) measure_setsum_split:
9.395 - assumes "finite r" and "a \<in> sets M" and br_in_M: "b ` r \<subseteq> sets M"
9.396 - assumes "(\<Union>i \<in> r. b i) = space M"
9.397 - assumes "disjoint_family_on b r"
9.398 - shows "\<mu> a = (\<Sum> i \<in> r. \<mu> (a \<inter> (b i)))"
9.399 + assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M"
9.400 + assumes "(\<Union>i\<in>S. B i) = space M"
9.401 + assumes "disjoint_family_on B S"
9.402 + shows "\<mu> A = (\<Sum>i\<in>S. \<mu> (A \<inter> (B i)))"
9.403 proof -
9.404 - have *: "\<mu> a = \<mu> (\<Union>i \<in> r. a \<inter> b i)"
9.405 + have *: "\<mu> A = \<mu> (\<Union>i\<in>S. A \<inter> B i)"
9.406 using assms by auto
9.407 show ?thesis unfolding *
9.408 - proof (rule measure_finitely_additive''[symmetric])
9.409 - show "finite r" using `finite r` by auto
9.410 - { fix i assume "i \<in> r"
9.411 - hence "b i \<in> sets M" using br_in_M by auto
9.412 - thus "a \<inter> b i \<in> sets M" using `a \<in> sets M` by auto
9.413 - }
9.414 - show "disjoint_family_on (\<lambda>i. a \<inter> b i) r"
9.415 - using `disjoint_family_on b r`
9.416 + proof (rule measure_setsum[symmetric])
9.417 + show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
9.418 + using `disjoint_family_on B S`
9.419 unfolding disjoint_family_on_def by auto
9.420 - qed
9.421 + qed (insert assms, auto)
9.422 qed
9.423
9.424 lemma (in measure_space) measure_subadditive:
9.425 @@ -506,7 +409,7 @@
9.426 lemma (in measure_space) measure_eq_0:
9.427 assumes "N \<in> sets M" and "\<mu> N = 0" and "K \<subseteq> N" and "K \<in> sets M"
9.428 shows "\<mu> K = 0"
9.429 -using measure_mono[OF assms(3,4,1)] assms(2) by auto
9.430 + using measure_mono[OF assms(3,4,1)] assms(2) positive_measure[OF assms(4)] by auto
9.431
9.432 lemma (in measure_space) measure_finitely_subadditive:
9.433 assumes "finite I" "A ` I \<subseteq> sets M"
9.434 @@ -523,35 +426,38 @@
9.435
9.436 lemma (in measure_space) measure_countably_subadditive:
9.437 assumes "range f \<subseteq> sets M"
9.438 - shows "\<mu> (\<Union>i. f i) \<le> (\<Sum>\<^isub>\<infinity> i. \<mu> (f i))"
9.439 + shows "\<mu> (\<Union>i. f i) \<le> (\<Sum>i. \<mu> (f i))"
9.440 proof -
9.441 have "\<mu> (\<Union>i. f i) = \<mu> (\<Union>i. disjointed f i)"
9.442 unfolding UN_disjointed_eq ..
9.443 - also have "\<dots> = (\<Sum>\<^isub>\<infinity> i. \<mu> (disjointed f i))"
9.444 + also have "\<dots> = (\<Sum>i. \<mu> (disjointed f i))"
9.445 using range_disjointed_sets[OF assms] measure_countably_additive
9.446 by (simp add: disjoint_family_disjointed comp_def)
9.447 - also have "\<dots> \<le> (\<Sum>\<^isub>\<infinity> i. \<mu> (f i))"
9.448 - proof (rule psuminf_le, rule measure_mono)
9.449 - fix i show "disjointed f i \<subseteq> f i" by (rule disjointed_subset)
9.450 - show "f i \<in> sets M" "disjointed f i \<in> sets M"
9.451 - using assms range_disjointed_sets[OF assms] by auto
9.452 - qed
9.453 + also have "\<dots> \<le> (\<Sum>i. \<mu> (f i))"
9.454 + using range_disjointed_sets[OF assms] assms
9.455 + by (auto intro!: suminf_le_pos measure_mono positive_measure disjointed_subset)
9.456 finally show ?thesis .
9.457 qed
9.458
9.459 lemma (in measure_space) measure_UN_eq_0:
9.460 - assumes "\<And> i :: nat. \<mu> (N i) = 0" and "range N \<subseteq> sets M"
9.461 + assumes "\<And>i::nat. \<mu> (N i) = 0" and "range N \<subseteq> sets M"
9.462 shows "\<mu> (\<Union> i. N i) = 0"
9.463 -using measure_countably_subadditive[OF assms(2)] assms(1) by auto
9.464 +proof -
9.465 + have "0 \<le> \<mu> (\<Union> i. N i)" using assms by auto
9.466 + moreover have "\<mu> (\<Union> i. N i) \<le> 0"
9.467 + using measure_countably_subadditive[OF assms(2)] assms(1) by simp
9.468 + ultimately show ?thesis by simp
9.469 +qed
9.470
9.471 lemma (in measure_space) measure_inter_full_set:
9.472 - assumes "S \<in> sets M" "T \<in> sets M" and not_\<omega>: "\<mu> (T - S) \<noteq> \<omega>"
9.473 + assumes "S \<in> sets M" "T \<in> sets M" and fin: "\<mu> (T - S) \<noteq> \<infinity>"
9.474 assumes T: "\<mu> T = \<mu> (space M)"
9.475 shows "\<mu> (S \<inter> T) = \<mu> S"
9.476 proof (rule antisym)
9.477 show " \<mu> (S \<inter> T) \<le> \<mu> S"
9.478 using assms by (auto intro!: measure_mono)
9.479
9.480 + have pos: "0 \<le> \<mu> (T - S)" using assms by auto
9.481 show "\<mu> S \<le> \<mu> (S \<inter> T)"
9.482 proof (rule ccontr)
9.483 assume contr: "\<not> ?thesis"
9.484 @@ -560,7 +466,7 @@
9.485 also have "\<dots> \<le> \<mu> (T - S) + \<mu> (S \<inter> T)"
9.486 using assms by (auto intro!: measure_subadditive)
9.487 also have "\<dots> < \<mu> (T - S) + \<mu> S"
9.488 - by (rule pextreal_less_add[OF not_\<omega>]) (insert contr, auto)
9.489 + using fin contr pos by (intro extreal_less_add) auto
9.490 also have "\<dots> = \<mu> (T \<union> S)"
9.491 using assms by (subst measure_additive) auto
9.492 also have "\<dots> \<le> \<mu> (space M)"
9.493 @@ -572,11 +478,11 @@
9.494 lemma measure_unique_Int_stable:
9.495 fixes E :: "('a, 'b) algebra_scheme" and A :: "nat \<Rightarrow> 'a set"
9.496 assumes "Int_stable E"
9.497 - and A: "range A \<subseteq> sets E" "A \<up> space E"
9.498 + and A: "range A \<subseteq> sets E" "incseq A" "(\<Union>i. A i) = space E"
9.499 and M: "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = \<mu>\<rparr>" (is "measure_space ?M")
9.500 and N: "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = \<nu>\<rparr>" (is "measure_space ?N")
9.501 and eq: "\<And>X. X \<in> sets E \<Longrightarrow> \<mu> X = \<nu> X"
9.502 - and finite: "\<And>i. \<mu> (A i) \<noteq> \<omega>"
9.503 + and finite: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
9.504 assumes "X \<in> sets (sigma E)"
9.505 shows "\<mu> X = \<nu> X"
9.506 proof -
9.507 @@ -585,9 +491,9 @@
9.508 where "space ?M = space E" and "sets ?M = sets (sigma E)" and "measure ?M = \<mu>" by (simp_all add: M)
9.509 interpret N: measure_space ?N
9.510 where "space ?N = space E" and "sets ?N = sets (sigma E)" and "measure ?N = \<nu>" by (simp_all add: N)
9.511 - { fix F assume "F \<in> sets E" and "\<mu> F \<noteq> \<omega>"
9.512 + { fix F assume "F \<in> sets E" and "\<mu> F \<noteq> \<infinity>"
9.513 then have [intro]: "F \<in> sets (sigma E)" by auto
9.514 - have "\<nu> F \<noteq> \<omega>" using `\<mu> F \<noteq> \<omega>` `F \<in> sets E` eq by simp
9.515 + have "\<nu> F \<noteq> \<infinity>" using `\<mu> F \<noteq> \<infinity>` `F \<in> sets E` eq by simp
9.516 interpret D: dynkin_system "\<lparr>space=space E, sets=?D F\<rparr>"
9.517 proof (rule dynkin_systemI, simp_all)
9.518 fix A assume "A \<in> sets (sigma E) \<and> \<mu> (F \<inter> A) = \<nu> (F \<inter> A)"
9.519 @@ -602,14 +508,14 @@
9.520 and [intro]: "F \<inter> A \<in> sets (sigma E)"
9.521 using `F \<in> sets E` M.sets_into_space by auto
9.522 have "\<nu> (F \<inter> A) \<le> \<nu> F" by (auto intro!: N.measure_mono)
9.523 - then have "\<nu> (F \<inter> A) \<noteq> \<omega>" using `\<nu> F \<noteq> \<omega>` by auto
9.524 + then have "\<nu> (F \<inter> A) \<noteq> \<infinity>" using `\<nu> F \<noteq> \<infinity>` by auto
9.525 have "\<mu> (F \<inter> A) \<le> \<mu> F" by (auto intro!: M.measure_mono)
9.526 - then have "\<mu> (F \<inter> A) \<noteq> \<omega>" using `\<mu> F \<noteq> \<omega>` by auto
9.527 + then have "\<mu> (F \<inter> A) \<noteq> \<infinity>" using `\<mu> F \<noteq> \<infinity>` by auto
9.528 then have "\<mu> (F \<inter> (space (sigma E) - A)) = \<mu> F - \<mu> (F \<inter> A)" unfolding **
9.529 using `F \<inter> A \<in> sets (sigma E)` by (auto intro!: M.measure_Diff)
9.530 also have "\<dots> = \<nu> F - \<nu> (F \<inter> A)" using eq `F \<in> sets E` * by simp
9.531 also have "\<dots> = \<nu> (F \<inter> (space (sigma E) - A))" unfolding **
9.532 - using `F \<inter> A \<in> sets (sigma E)` `\<nu> (F \<inter> A) \<noteq> \<omega>` by (auto intro!: N.measure_Diff[symmetric])
9.533 + using `F \<inter> A \<in> sets (sigma E)` `\<nu> (F \<inter> A) \<noteq> \<infinity>` by (auto intro!: N.measure_Diff[symmetric])
9.534 finally show "space E - A \<in> sets (sigma E) \<and> \<mu> (F \<inter> (space E - A)) = \<nu> (F \<inter> (space E - A))"
9.535 using * by auto
9.536 next
9.537 @@ -630,15 +536,13 @@
9.538 have "\<And>D. D \<in> sets (sigma E) \<Longrightarrow> \<mu> (F \<inter> D) = \<nu> (F \<inter> D)"
9.539 by (subst (asm) *) auto }
9.540 note * = this
9.541 - { fix i have "\<mu> (A i \<inter> X) = \<nu> (A i \<inter> X)"
9.542 + let "?A i" = "A i \<inter> X"
9.543 + have A': "range ?A \<subseteq> sets (sigma E)" "incseq ?A"
9.544 + using A(1,2) `X \<in> sets (sigma E)` by (auto simp: incseq_def)
9.545 + { fix i have "\<mu> (?A i) = \<nu> (?A i)"
9.546 using *[of "A i" X] `X \<in> sets (sigma E)` A finite by auto }
9.547 - moreover
9.548 - have "(\<lambda>i. A i \<inter> X) \<up> X"
9.549 - using `X \<in> sets (sigma E)` M.sets_into_space A
9.550 - by (auto simp: isoton_def)
9.551 - then have "(\<lambda>i. \<mu> (A i \<inter> X)) \<up> \<mu> X" "(\<lambda>i. \<nu> (A i \<inter> X)) \<up> \<nu> X"
9.552 - using `X \<in> sets (sigma E)` A by (auto intro!: M.measure_up N.measure_up M.Int simp: subset_eq)
9.553 - ultimately show ?thesis by (simp add: isoton_def)
9.554 + with M.continuity_from_below[OF A'] N.continuity_from_below[OF A']
9.555 + show ?thesis using A(3) `X \<in> sets (sigma E)` by auto
9.556 qed
9.557
9.558 section "@{text \<mu>}-null sets"
9.559 @@ -650,10 +554,10 @@
9.560 shows "N \<union> N' \<in> null_sets"
9.561 proof (intro conjI CollectI)
9.562 show "N \<union> N' \<in> sets M" using assms by auto
9.563 - have "\<mu> (N \<union> N') \<le> \<mu> N + \<mu> N'"
9.564 + then have "0 \<le> \<mu> (N \<union> N')" by simp
9.565 + moreover have "\<mu> (N \<union> N') \<le> \<mu> N + \<mu> N'"
9.566 using assms by (intro measure_subadditive) auto
9.567 - then show "\<mu> (N \<union> N') = 0"
9.568 - using assms by auto
9.569 + ultimately show "\<mu> (N \<union> N') = 0" using assms by auto
9.570 qed
9.571
9.572 lemma UN_from_nat: "(\<Union>i. N i) = (\<Union>i. N (Countable.from_nat i))"
9.573 @@ -669,11 +573,11 @@
9.574 shows "(\<Union>i. N i) \<in> null_sets"
9.575 proof (intro conjI CollectI)
9.576 show "(\<Union>i. N i) \<in> sets M" using assms by auto
9.577 - have "\<mu> (\<Union>i. N i) \<le> (\<Sum>\<^isub>\<infinity> n. \<mu> (N (Countable.from_nat n)))"
9.578 + then have "0 \<le> \<mu> (\<Union>i. N i)" by simp
9.579 + moreover have "\<mu> (\<Union>i. N i) \<le> (\<Sum>n. \<mu> (N (Countable.from_nat n)))"
9.580 unfolding UN_from_nat[of N]
9.581 using assms by (intro measure_countably_subadditive) auto
9.582 - then show "\<mu> (\<Union>i. N i) = 0"
9.583 - using assms by auto
9.584 + ultimately show "\<mu> (\<Union>i. N i) = 0" using assms by auto
9.585 qed
9.586
9.587 lemma (in measure_space) null_sets_finite_UN:
9.588 @@ -681,10 +585,10 @@
9.589 shows "(\<Union>i\<in>S. A i) \<in> null_sets"
9.590 proof (intro CollectI conjI)
9.591 show "(\<Union>i\<in>S. A i) \<in> sets M" using assms by (intro finite_UN) auto
9.592 - have "\<mu> (\<Union>i\<in>S. A i) \<le> (\<Sum>i\<in>S. \<mu> (A i))"
9.593 + then have "0 \<le> \<mu> (\<Union>i\<in>S. A i)" by simp
9.594 + moreover have "\<mu> (\<Union>i\<in>S. A i) \<le> (\<Sum>i\<in>S. \<mu> (A i))"
9.595 using assms by (intro measure_finitely_subadditive) auto
9.596 - then show "\<mu> (\<Union>i\<in>S. A i) = 0"
9.597 - using assms by auto
9.598 + ultimately show "\<mu> (\<Union>i\<in>S. A i) = 0" using assms by auto
9.599 qed
9.600
9.601 lemma (in measure_space) null_set_Int1:
9.602 @@ -731,6 +635,23 @@
9.603 almost_everywhere :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "AE " 10) where
9.604 "almost_everywhere P \<longleftrightarrow> (\<exists>N\<in>null_sets. { x \<in> space M. \<not> P x } \<subseteq> N)"
9.605
9.606 +syntax
9.607 + "_almost_everywhere" :: "'a \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10)
9.608 +
9.609 +translations
9.610 + "AE x in M. P" == "CONST measure_space.almost_everywhere M (%x. P)"
9.611 +
9.612 +lemma (in measure_space) AE_cong_measure:
9.613 + assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
9.614 + shows "(AE x in N. P x) \<longleftrightarrow> (AE x. P x)"
9.615 +proof -
9.616 + interpret N: measure_space N
9.617 + by (rule measure_space_cong) fact+
9.618 + show ?thesis
9.619 + unfolding N.almost_everywhere_def almost_everywhere_def
9.620 + by (auto simp: assms)
9.621 +qed
9.622 +
9.623 lemma (in measure_space) AE_I':
9.624 "N \<in> null_sets \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x. P x)"
9.625 unfolding almost_everywhere_def by auto
9.626 @@ -741,13 +662,19 @@
9.627 proof
9.628 assume "AE x. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "\<mu> N = 0"
9.629 unfolding almost_everywhere_def by auto
9.630 + have "0 \<le> \<mu> ?P" using assms by simp
9.631 moreover have "\<mu> ?P \<le> \<mu> N"
9.632 using assms N(1,2) by (auto intro: measure_mono)
9.633 - ultimately show "?P \<in> null_sets" using assms by auto
9.634 + ultimately have "\<mu> ?P = 0" unfolding `\<mu> N = 0` by auto
9.635 + then show "?P \<in> null_sets" using assms by simp
9.636 next
9.637 assume "?P \<in> null_sets" with assms show "AE x. P x" by (auto intro: AE_I')
9.638 qed
9.639
9.640 +lemma (in measure_space) AE_iff_measurable:
9.641 + "N \<in> sets M \<Longrightarrow> {x\<in>space M. \<not> P x} = N \<Longrightarrow> (AE x. P x) \<longleftrightarrow> \<mu> N = 0"
9.642 + using AE_iff_null_set[of P] by simp
9.643 +
9.644 lemma (in measure_space) AE_True[intro, simp]: "AE x. True"
9.645 unfolding almost_everywhere_def by auto
9.646
9.647 @@ -760,13 +687,9 @@
9.648 assumes "AE x. P x" "{x\<in>space M. P x} \<in> sets M"
9.649 shows "\<mu> {x\<in>space M. \<not> P x} = 0" (is "\<mu> ?P = 0")
9.650 proof -
9.651 - obtain A where A: "?P \<subseteq> A" "A \<in> sets M" "\<mu> A = 0"
9.652 - using assms by (auto elim!: AE_E)
9.653 - have "?P = space M - {x\<in>space M. P x}" by auto
9.654 - then have "?P \<in> sets M" using assms by auto
9.655 - with assms `?P \<subseteq> A` `A \<in> sets M` have "\<mu> ?P \<le> \<mu> A"
9.656 - by (auto intro!: measure_mono)
9.657 - then show "\<mu> ?P = 0" using A by simp
9.658 + have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}"
9.659 + by auto
9.660 + with AE_iff_null_set[of P] assms show ?thesis by auto
9.661 qed
9.662
9.663 lemma (in measure_space) AE_I:
9.664 @@ -788,8 +711,10 @@
9.665
9.666 show ?thesis
9.667 proof (intro AE_I)
9.668 - show "A \<union> B \<in> sets M" "\<mu> (A \<union> B) = 0" using A B
9.669 - using measure_subadditive[of A B] by auto
9.670 + have "0 \<le> \<mu> (A \<union> B)" using A B by auto
9.671 + moreover have "\<mu> (A \<union> B) \<le> 0"
9.672 + using measure_subadditive[of A B] A B by auto
9.673 + ultimately show "A \<union> B \<in> sets M" "\<mu> (A \<union> B) = 0" using A B by auto
9.674 show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"
9.675 using P imp by auto
9.676 qed
9.677 @@ -818,8 +743,8 @@
9.678 "(\<And>x. x \<in> space M \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow> (AE x. P x) \<longleftrightarrow> (AE x. Q x)"
9.679 by auto
9.680
9.681 -lemma (in measure_space) all_AE_countable:
9.682 - "(\<forall>i::'i::countable. AE x. P i x) \<longleftrightarrow> (AE x. \<forall>i. P i x)"
9.683 +lemma (in measure_space) AE_all_countable:
9.684 + "(AE x. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x. P i x)"
9.685 proof
9.686 assume "\<forall>i. AE x. P i x"
9.687 from this[unfolded almost_everywhere_def Bex_def, THEN choice]
9.688 @@ -833,6 +758,10 @@
9.689 unfolding almost_everywhere_def by auto
9.690 qed auto
9.691
9.692 +lemma (in measure_space) AE_finite_all:
9.693 + assumes f: "finite S" shows "(AE x. \<forall>i\<in>S. P i x) \<longleftrightarrow> (\<forall>i\<in>S. AE x. P i x)"
9.694 + using f by induct auto
9.695 +
9.696 lemma (in measure_space) restricted_measure_space:
9.697 assumes "S \<in> sets M"
9.698 shows "measure_space (restricted_space S)"
9.699 @@ -840,7 +769,7 @@
9.700 unfolding measure_space_def measure_space_axioms_def
9.701 proof safe
9.702 show "sigma_algebra ?r" using restricted_sigma_algebra[OF assms] .
9.703 - show "measure ?r {} = 0" by simp
9.704 + show "positive ?r (measure ?r)" using `S \<in> sets M` by (auto simp: positive_def)
9.705
9.706 show "countably_additive ?r (measure ?r)"
9.707 unfolding countably_additive_def
9.708 @@ -848,13 +777,38 @@
9.709 fix A :: "nat \<Rightarrow> 'a set"
9.710 assume *: "range A \<subseteq> sets ?r" and **: "disjoint_family A"
9.711 from restriction_in_sets[OF assms *[simplified]] **
9.712 - show "(\<Sum>\<^isub>\<infinity> n. measure ?r (A n)) = measure ?r (\<Union>i. A i)"
9.713 + show "(\<Sum>n. measure ?r (A n)) = measure ?r (\<Union>i. A i)"
9.714 using measure_countably_additive by simp
9.715 qed
9.716 qed
9.717
9.718 +lemma (in measure_space) AE_restricted:
9.719 + assumes "A \<in> sets M"
9.720 + shows "(AE x in restricted_space A. P x) \<longleftrightarrow> (AE x. x \<in> A \<longrightarrow> P x)"
9.721 +proof -
9.722 + interpret R: measure_space "restricted_space A"
9.723 + by (rule restricted_measure_space[OF `A \<in> sets M`])
9.724 + show ?thesis
9.725 + proof
9.726 + assume "AE x in restricted_space A. P x"
9.727 + from this[THEN R.AE_E] guess N' .
9.728 + then obtain N where "{x \<in> A. \<not> P x} \<subseteq> A \<inter> N" "\<mu> (A \<inter> N) = 0" "N \<in> sets M"
9.729 + by auto
9.730 + moreover then have "{x \<in> space M. \<not> (x \<in> A \<longrightarrow> P x)} \<subseteq> A \<inter> N"
9.731 + using `A \<in> sets M` sets_into_space by auto
9.732 + ultimately show "AE x. x \<in> A \<longrightarrow> P x"
9.733 + using `A \<in> sets M` by (auto intro!: AE_I[where N="A \<inter> N"])
9.734 + next
9.735 + assume "AE x. x \<in> A \<longrightarrow> P x"
9.736 + from this[THEN AE_E] guess N .
9.737 + then show "AE x in restricted_space A. P x"
9.738 + using null_set_Int1[OF _ `A \<in> sets M`] `A \<in> sets M`[THEN sets_into_space]
9.739 + by (auto intro!: R.AE_I[where N="A \<inter> N"] simp: subset_eq)
9.740 + qed
9.741 +qed
9.742 +
9.743 lemma (in measure_space) measure_space_subalgebra:
9.744 - assumes "sigma_algebra N" and [simp]: "sets N \<subseteq> sets M" "space N = space M"
9.745 + assumes "sigma_algebra N" and "sets N \<subseteq> sets M" "space N = space M"
9.746 and measure[simp]: "\<And>X. X \<in> sets N \<Longrightarrow> measure N X = measure M X"
9.747 shows "measure_space N"
9.748 proof -
9.749 @@ -864,13 +818,26 @@
9.750 from `sets N \<subseteq> sets M` have "\<And>A. range A \<subseteq> sets N \<Longrightarrow> range A \<subseteq> sets M" by blast
9.751 then show "countably_additive N (measure N)"
9.752 by (auto intro!: measure_countably_additive simp: countably_additive_def subset_eq)
9.753 - qed simp
9.754 + show "positive N (measure_space.measure N)"
9.755 + using assms(2) by (auto simp add: positive_def)
9.756 + qed
9.757 +qed
9.758 +
9.759 +lemma (in measure_space) AE_subalgebra:
9.760 + assumes ae: "AE x in N. P x"
9.761 + and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
9.762 + and sa: "sigma_algebra N"
9.763 + shows "AE x. P x"
9.764 +proof -
9.765 + interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
9.766 + from ae[THEN N.AE_E] guess N .
9.767 + with N show ?thesis unfolding almost_everywhere_def by auto
9.768 qed
9.769
9.770 section "@{text \<sigma>}-finite Measures"
9.771
9.772 locale sigma_finite_measure = measure_space +
9.773 - assumes sigma_finite: "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. \<mu> (A i) \<noteq> \<omega>)"
9.774 + assumes sigma_finite: "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. \<mu> (A i) \<noteq> \<infinity>)"
9.775
9.776 lemma (in sigma_finite_measure) restricted_sigma_finite_measure:
9.777 assumes "S \<in> sets M"
9.778 @@ -881,9 +848,9 @@
9.779 show "measure_space ?r" using restricted_measure_space[OF assms] .
9.780 next
9.781 obtain A :: "nat \<Rightarrow> 'a set" where
9.782 - "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. \<mu> (A i) \<noteq> \<omega>"
9.783 + "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
9.784 using sigma_finite by auto
9.785 - show "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets ?r \<and> (\<Union>i. A i) = space ?r \<and> (\<forall>i. measure ?r (A i) \<noteq> \<omega>)"
9.786 + show "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets ?r \<and> (\<Union>i. A i) = space ?r \<and> (\<forall>i. measure ?r (A i) \<noteq> \<infinity>)"
9.787 proof (safe intro!: exI[of _ "\<lambda>i. A i \<inter> S"] del: notI)
9.788 fix i
9.789 show "A i \<inter> S \<in> sets ?r"
9.790 @@ -897,8 +864,7 @@
9.791 fix i
9.792 have "\<mu> (A i \<inter> S) \<le> \<mu> (A i)"
9.793 using `range A \<subseteq> sets M` `S \<in> sets M` by (auto intro!: measure_mono)
9.794 - also have "\<dots> < \<omega>" using `\<mu> (A i) \<noteq> \<omega>` by (auto simp: pextreal_less_\<omega>)
9.795 - finally show "measure ?r (A i \<inter> S) \<noteq> \<omega>" by (auto simp: pextreal_less_\<omega>)
9.796 + then show "measure ?r (A i \<inter> S) \<noteq> \<infinity>" using `\<mu> (A i) \<noteq> \<infinity>` by auto
9.797 qed
9.798 qed
9.799
9.800 @@ -909,7 +875,7 @@
9.801 interpret M': measure_space M' by (intro measure_space_cong cong)
9.802 from sigma_finite guess A .. note A = this
9.803 then have "\<And>i. A i \<in> sets M" by auto
9.804 - with A have fin: "(\<forall>i. measure M' (A i) \<noteq> \<omega>)" using cong by auto
9.805 + with A have fin: "\<forall>i. measure M' (A i) \<noteq> \<infinity>" using cong by auto
9.806 show ?thesis
9.807 apply default
9.808 using A fin cong by auto
9.809 @@ -917,30 +883,30 @@
9.810
9.811 lemma (in sigma_finite_measure) disjoint_sigma_finite:
9.812 "\<exists>A::nat\<Rightarrow>'a set. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and>
9.813 - (\<forall>i. \<mu> (A i) \<noteq> \<omega>) \<and> disjoint_family A"
9.814 + (\<forall>i. \<mu> (A i) \<noteq> \<infinity>) \<and> disjoint_family A"
9.815 proof -
9.816 obtain A :: "nat \<Rightarrow> 'a set" where
9.817 range: "range A \<subseteq> sets M" and
9.818 space: "(\<Union>i. A i) = space M" and
9.819 - measure: "\<And>i. \<mu> (A i) \<noteq> \<omega>"
9.820 + measure: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
9.821 using sigma_finite by auto
9.822 note range' = range_disjointed_sets[OF range] range
9.823 { fix i
9.824 have "\<mu> (disjointed A i) \<le> \<mu> (A i)"
9.825 using range' disjointed_subset[of A i] by (auto intro!: measure_mono)
9.826 - then have "\<mu> (disjointed A i) \<noteq> \<omega>"
9.827 + then have "\<mu> (disjointed A i) \<noteq> \<infinity>"
9.828 using measure[of i] by auto }
9.829 with disjoint_family_disjointed UN_disjointed_eq[of A] space range'
9.830 show ?thesis by (auto intro!: exI[of _ "disjointed A"])
9.831 qed
9.832
9.833 lemma (in sigma_finite_measure) sigma_finite_up:
9.834 - "\<exists>F. range F \<subseteq> sets M \<and> F \<up> space M \<and> (\<forall>i. \<mu> (F i) \<noteq> \<omega>)"
9.835 + "\<exists>F. range F \<subseteq> sets M \<and> incseq F \<and> (\<Union>i. F i) = space M \<and> (\<forall>i. \<mu> (F i) \<noteq> \<infinity>)"
9.836 proof -
9.837 obtain F :: "nat \<Rightarrow> 'a set" where
9.838 - F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. \<mu> (F i) \<noteq> \<omega>"
9.839 + F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. \<mu> (F i) \<noteq> \<infinity>"
9.840 using sigma_finite by auto
9.841 - then show ?thesis unfolding isoton_def
9.842 + then show ?thesis
9.843 proof (intro exI[of _ "\<lambda>n. \<Union>i\<le>n. F i"] conjI allI)
9.844 from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto
9.845 then show "(\<Union>n. \<Union> i\<le>n. F i) = space M"
9.846 @@ -949,16 +915,16 @@
9.847 fix n
9.848 have "\<mu> (\<Union> i\<le>n. F i) \<le> (\<Sum>i\<le>n. \<mu> (F i))" using F
9.849 by (auto intro!: measure_finitely_subadditive)
9.850 - also have "\<dots> < \<omega>"
9.851 - using F by (auto simp: setsum_\<omega>)
9.852 - finally show "\<mu> (\<Union> i\<le>n. F i) \<noteq> \<omega>" by simp
9.853 - qed force+
9.854 + also have "\<dots> < \<infinity>"
9.855 + using F by (auto simp: setsum_Pinfty)
9.856 + finally show "\<mu> (\<Union> i\<le>n. F i) \<noteq> \<infinity>" by simp
9.857 + qed (force simp: incseq_def)+
9.858 qed
9.859
9.860 section {* Measure preserving *}
9.861
9.862 definition "measure_preserving A B =