--- a/src/HOL/Complete_Lattice.thy Thu Dec 02 14:56:16 2010 +0100
+++ b/src/HOL/Complete_Lattice.thy Thu Dec 02 15:32:48 2010 +0100
@@ -172,6 +172,18 @@
"(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<le> g m) \<Longrightarrow> (INF n:A. f n) \<le> (INF n:B. g n)"
by (force intro!: Inf_mono simp: INFI_def)
+lemma SUP_subset: "A \<subseteq> B \<Longrightarrow> SUPR A f \<le> SUPR B f"
+ by (intro SUP_mono) auto
+
+lemma INF_subset: "A \<subseteq> B \<Longrightarrow> INFI B f \<le> INFI A f"
+ by (intro INF_mono) auto
+
+lemma SUP_commute: "(SUP i:A. SUP j:B. f i j) = (SUP j:B. SUP i:A. f i j)"
+ by (iprover intro: SUP_leI le_SUPI order_trans antisym)
+
+lemma INF_commute: "(INF i:A. INF j:B. f i j) = (INF j:B. INF i:A. f i j)"
+ by (iprover intro: INF_leI le_INFI order_trans antisym)
+
end
lemma less_Sup_iff:
@@ -184,6 +196,16 @@
shows "Inf S < a \<longleftrightarrow> (\<exists>x\<in>S. x < a)"
unfolding not_le[symmetric] le_Inf_iff by auto
+lemma less_SUP_iff:
+ fixes a :: "'a::{complete_lattice,linorder}"
+ shows "a < (SUP i:A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)"
+ unfolding SUPR_def less_Sup_iff by auto
+
+lemma INF_less_iff:
+ fixes a :: "'a::{complete_lattice,linorder}"
+ shows "(INF i:A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"
+ unfolding INFI_def Inf_less_iff by auto
+
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
instantiation bool :: complete_lattice
--- a/src/HOL/Probability/Borel_Space.thy Thu Dec 02 14:56:16 2010 +0100
+++ b/src/HOL/Probability/Borel_Space.thy Thu Dec 02 15:32:48 2010 +0100
@@ -6,12 +6,6 @@
imports Sigma_Algebra Positive_Infinite_Real Multivariate_Analysis
begin
-lemma (in sigma_algebra) sets_sigma_subset:
- assumes "space N = space M"
- assumes "sets N \<subseteq> sets M"
- shows "sets (sigma N) \<subseteq> sets M"
- by (unfold assms sets_sigma, rule sigma_sets_subset, rule assms)
-
lemma LIMSEQ_max:
"u ----> (x::real) \<Longrightarrow> (\<lambda>i. max (u i) 0) ----> max x 0"
by (fastsimp intro!: LIMSEQ_I dest!: LIMSEQ_D)
@@ -612,13 +606,10 @@
then show ?thesis by (intro sets_sigma_subset) auto
qed
-lemma algebra_eqI: assumes "sets A = sets (B::'a algebra)" "space A = space B"
- shows "A = B" using assms by auto
-
lemma borel_eq_atMost:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> a. {.. a::'a\<Colon>ordered_euclidean_space})\<rparr>)"
(is "_ = ?SIGMA")
-proof (rule algebra_eqI, rule antisym)
+proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using halfspace_le_span_atMost halfspace_span_halfspace_le open_span_halfspace
by auto
@@ -629,7 +620,7 @@
lemma borel_eq_atLeastAtMost:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space, b). {a .. b})\<rparr>)"
(is "_ = ?SIGMA")
-proof (rule algebra_eqI, rule antisym)
+proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using atMost_span_atLeastAtMost halfspace_le_span_atMost
halfspace_span_halfspace_le open_span_halfspace
@@ -641,7 +632,7 @@
lemma borel_eq_greaterThan:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {a <..})\<rparr>)"
(is "_ = ?SIGMA")
-proof (rule algebra_eqI, rule antisym)
+proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using halfspace_le_span_greaterThan
halfspace_span_halfspace_le open_span_halfspace
@@ -653,7 +644,7 @@
lemma borel_eq_lessThan:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {..< a})\<rparr>)"
(is "_ = ?SIGMA")
-proof (rule algebra_eqI, rule antisym)
+proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using halfspace_le_span_lessThan
halfspace_span_halfspace_ge open_span_halfspace
@@ -665,7 +656,7 @@
lemma borel_eq_greaterThanLessThan:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, b). {a <..< (b :: 'a \<Colon> ordered_euclidean_space)})\<rparr>)"
(is "_ = ?SIGMA")
-proof (rule algebra_eqI, rule antisym)
+proof (intro algebra.equality antisym)
show "sets ?SIGMA \<subseteq> sets borel"
by (rule borel.sets_sigma_subset) auto
show "sets borel \<subseteq> sets ?SIGMA"
@@ -686,7 +677,7 @@
lemma borel_eq_halfspace_le:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x$$i \<le> a})\<rparr>)"
(is "_ = ?SIGMA")
-proof (rule algebra_eqI, rule antisym)
+proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using open_span_halfspace halfspace_span_halfspace_le by auto
show "sets ?SIGMA \<subseteq> sets borel"
@@ -696,7 +687,7 @@
lemma borel_eq_halfspace_less:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x$$i < a})\<rparr>)"
(is "_ = ?SIGMA")
-proof (rule algebra_eqI, rule antisym)
+proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using open_span_halfspace .
show "sets ?SIGMA \<subseteq> sets borel"
@@ -706,7 +697,7 @@
lemma borel_eq_halfspace_gt:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a < x$$i})\<rparr>)"
(is "_ = ?SIGMA")
-proof (rule algebra_eqI, rule antisym)
+proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using halfspace_le_span_halfspace_gt open_span_halfspace halfspace_span_halfspace_le by auto
show "sets ?SIGMA \<subseteq> sets borel"
@@ -716,7 +707,7 @@
lemma borel_eq_halfspace_ge:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a \<le> x$$i})\<rparr>)"
(is "_ = ?SIGMA")
-proof (rule algebra_eqI, rule antisym)
+proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using halfspace_span_halfspace_ge open_span_halfspace by auto
show "sets ?SIGMA \<subseteq> sets borel"
@@ -1025,7 +1016,6 @@
then obtain T x where T: "open T" "Real ` (T \<inter> {0..}) = B - {\<omega>}" and
x: "\<omega> \<in> B \<Longrightarrow> 0 \<le> x" "\<omega> \<in> B \<Longrightarrow> {Real x <..} \<subseteq> B"
unfolding open_pinfreal_def by blast
-
have "Real -` B = Real -` (B - {\<omega>})" by auto
also have "\<dots> = Real -` (Real ` (T \<inter> {0..}))" using T by simp
also have "\<dots> = (if 0 \<in> T then T \<union> {.. 0} else T \<inter> {0..})"
@@ -1231,12 +1221,10 @@
hence **: "\<forall>a. {x\<in>space M. f x < a} \<in> sets M"
unfolding less_eq_le_pinfreal_measurable
unfolding greater_eq_le_measurable .
-
show "f \<in> borel_measurable M" unfolding borel_measurable_pinfreal_eq_real borel_measurable_iff_greater
proof safe
have "f -` {\<omega>} \<inter> space M = space M - {x\<in>space M. f x < \<omega>}" by auto
then show \<omega>: "f -` {\<omega>} \<inter> space M \<in> sets M" using ** by auto
-
fix a
have "{w \<in> space M. a < real (f w)} =
(if 0 \<le> a then {w\<in>space M. Real a < f w} - (f -` {\<omega>} \<inter> space M) else space M)"
@@ -1367,14 +1355,14 @@
by induct auto
qed (simp add: borel_measurable_const)
-lemma (in sigma_algebra) borel_measurable_pinfreal_min[intro, simp]:
+lemma (in sigma_algebra) borel_measurable_pinfreal_min[simp, intro]:
fixes f g :: "'a \<Rightarrow> pinfreal"
assumes "f \<in> borel_measurable M"
assumes "g \<in> borel_measurable M"
shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
using assms unfolding min_def by (auto intro!: measurable_If)
-lemma (in sigma_algebra) borel_measurable_pinfreal_max[intro]:
+lemma (in sigma_algebra) borel_measurable_pinfreal_max[simp, intro]:
fixes f g :: "'a \<Rightarrow> pinfreal"
assumes "f \<in> borel_measurable M"
and "g \<in> borel_measurable M"
@@ -1421,7 +1409,7 @@
using assms by auto
qed
-lemma (in sigma_algebra) borel_measurable_psuminf:
+lemma (in sigma_algebra) borel_measurable_psuminf[simp, intro]:
assumes "\<And>i. f i \<in> borel_measurable M"
shows "(\<lambda>x. (\<Sum>\<^isub>\<infinity> i. f i x)) \<in> borel_measurable M"
using assms unfolding psuminf_def
@@ -1437,7 +1425,6 @@
proof -
let "?pu x i" = "max (u i x) 0"
let "?nu x i" = "max (- u i x) 0"
-
{ fix x assume x: "x \<in> space M"
have "(?pu x) ----> max (u' x) 0"
"(?nu x) ----> max (- u' x) 0"
@@ -1447,10 +1434,8 @@
"(SUP n. INF m. Real (- u (n + m) x)) = Real (- u' x)"
by (simp_all add: Real_max'[symmetric]) }
note eq = this
-
have *: "\<And>x. real (Real (u' x)) - real (Real (- u' x)) = u' x"
by auto
-
have "(SUP n. INF m. (\<lambda>x. Real (u (n + m) x))) \<in> borel_measurable M"
"(SUP n. INF m. (\<lambda>x. Real (- u (n + m) x))) \<in> borel_measurable M"
using u by (auto intro: borel_measurable_SUP borel_measurable_INF borel_measurable_Real)
--- a/src/HOL/Probability/Complete_Measure.thy Thu Dec 02 14:56:16 2010 +0100
+++ b/src/HOL/Probability/Complete_Measure.thy Thu Dec 02 15:32:48 2010 +0100
@@ -189,56 +189,13 @@
qed
qed
-lemma (in sigma_algebra) simple_functionD':
- assumes "simple_function f"
- shows "f -` {x} \<inter> space M \<in> sets M"
-proof cases
- assume "x \<in> f`space M" from simple_functionD(2)[OF assms this] show ?thesis .
-next
- assume "x \<notin> f`space M" then have "f -` {x} \<inter> space M = {}" by auto
- then show ?thesis by auto
-qed
-
-lemma (in sigma_algebra) simple_function_If:
- assumes sf: "simple_function f" "simple_function g" and A: "A \<in> sets M"
- shows "simple_function (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function ?IF")
-proof -
- def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
- show ?thesis unfolding simple_function_def
- proof safe
- have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
- from finite_subset[OF this] assms
- show "finite (?IF ` space M)" unfolding simple_function_def by auto
- next
- fix x assume "x \<in> space M"
- then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
- then ((F (f x) \<inter> A) \<union> (G (f x) - (G (f x) \<inter> A)))
- else ((F (g x) \<inter> A) \<union> (G (g x) - (G (g x) \<inter> A))))"
- using sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
- have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
- unfolding F_def G_def using sf[THEN simple_functionD'] by auto
- show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
- qed
-qed
-
-lemma (in measure_space) null_sets_finite_UN:
- assumes "finite S" "\<And>i. i \<in> S \<Longrightarrow> A i \<in> null_sets"
- shows "(\<Union>i\<in>S. A i) \<in> null_sets"
-proof (intro CollectI conjI)
- show "(\<Union>i\<in>S. A i) \<in> sets M" using assms by (intro finite_UN) auto
- have "\<mu> (\<Union>i\<in>S. A i) \<le> (\<Sum>i\<in>S. \<mu> (A i))"
- using assms by (intro measure_finitely_subadditive) auto
- then show "\<mu> (\<Union>i\<in>S. A i) = 0"
- using assms by auto
-qed
-
lemma (in completeable_measure_space) completion_ex_simple_function:
assumes f: "completion.simple_function f"
shows "\<exists>f'. simple_function f' \<and> (AE x. f x = f' x)"
proof -
let "?F x" = "f -` {x} \<inter> space M"
have F: "\<And>x. ?F x \<in> sets completion" and fin: "finite (f`space M)"
- using completion.simple_functionD'[OF f]
+ using completion.simple_functionD[OF f]
completion.simple_functionD[OF f] by simp_all
have "\<forall>x. \<exists>N. N \<in> null_sets \<and> null_part (?F x) \<subseteq> N"
using F null_part by auto
--- a/src/HOL/Probability/Lebesgue_Integration.thy Thu Dec 02 14:56:16 2010 +0100
+++ b/src/HOL/Probability/Lebesgue_Integration.thy Thu Dec 02 15:32:48 2010 +0100
@@ -6,20 +6,6 @@
imports Measure Borel_Space
begin
-lemma image_set_cong:
- assumes A: "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. f x = g y"
- assumes B: "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. g y = f x"
- shows "f ` A = g ` B"
-proof safe
- fix x assume "x \<in> A"
- with A obtain y where "f x = g y" "y \<in> B" by auto
- then show "f x \<in> g ` B" by auto
-next
- fix y assume "y \<in> B"
- with B obtain x where "g y = f x" "x \<in> A" by auto
- then show "g y \<in> f ` A" by auto
-qed
-
lemma sums_If_finite:
assumes finite: "finite {r. P r}"
shows "(\<lambda>r. if P r then f r else 0) sums (\<Sum>r\<in>{r. P r}. f r)" (is "?F sums _")
@@ -57,9 +43,15 @@
lemma (in sigma_algebra) simple_functionD:
assumes "simple_function g"
- shows "finite (g ` space M)"
- "x \<in> g ` space M \<Longrightarrow> g -` {x} \<inter> space M \<in> sets M"
- using assms unfolding simple_function_def by auto
+ shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
+proof -
+ show "finite (g ` space M)"
+ using assms unfolding simple_function_def by auto
+ have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto
+ also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto
+ finally show "g -` X \<inter> space M \<in> sets M" using assms
+ by (auto intro!: finite_UN simp del: UN_simps simp: simple_function_def)
+qed
lemma (in sigma_algebra) simple_function_indicator_representation:
fixes f ::"'a \<Rightarrow> pinfreal"
@@ -516,9 +508,7 @@
proof -
interpret v: measure_space M \<nu>
by (rule measure_space_cong) fact
- have "\<And>x. x \<in> space M \<Longrightarrow> f -` {f x} \<inter> space M \<in> sets M"
- using `simple_function f`[THEN simple_functionD(2)] by auto
- with assms show ?thesis
+ from simple_functionD[OF `simple_function f`] assms show ?thesis
unfolding simple_integral_def v.simple_integral_def
by (auto intro!: setsum_cong)
qed
@@ -629,6 +619,28 @@
by (auto simp: setsum_right_distrib field_simps intro!: setsum_cong)
qed
+lemma (in sigma_algebra) simple_function_If:
+ assumes sf: "simple_function f" "simple_function g" and A: "A \<in> sets M"
+ shows "simple_function (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function ?IF")
+proof -
+ def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
+ show ?thesis unfolding simple_function_def
+ proof safe
+ have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
+ from finite_subset[OF this] assms
+ show "finite (?IF ` space M)" unfolding simple_function_def by auto
+ next
+ fix x assume "x \<in> space M"
+ then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
+ then ((F (f x) \<inter> A) \<union> (G (f x) - (G (f x) \<inter> A)))
+ else ((F (g x) \<inter> A) \<union> (G (g x) - (G (g x) \<inter> A))))"
+ using sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
+ have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
+ unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
+ show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
+ qed
+qed
+
lemma (in measure_space) simple_integral_mono_AE:
assumes "simple_function f" and "simple_function g"
and mono: "AE x. f x \<le> g x"
@@ -652,8 +664,8 @@
obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
using mono by (auto elim!: AE_E)
have "?S x \<subseteq> N" using N `x \<in> space M` False by auto
- moreover have "?S x \<in> sets M" using assms `x \<in> space M`
- by (rule_tac Int) (auto intro!: simple_functionD(2))
+ moreover have "?S x \<in> sets M" using assms
+ by (rule_tac Int) (auto intro!: simple_functionD)
ultimately have "\<mu> (?S x) \<le> \<mu> N"
using `N \<in> sets M` by (auto intro!: measure_mono)
then show ?thesis using `\<mu> N = 0` by auto
@@ -831,8 +843,67 @@
section "Continuous posititve integration"
definition (in measure_space)
+ "positive_integral f = SUPR {g. simple_function g \<and> g \<le> f} simple_integral"
+
+lemma (in measure_space) positive_integral_alt:
"positive_integral f =
- (SUP g : {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}. simple_integral g)"
+ (SUPR {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M} simple_integral)" (is "_ = ?alt")
+proof (rule antisym SUP_leI)
+ show "positive_integral f \<le> ?alt" unfolding positive_integral_def
+ proof (safe intro!: SUP_leI)
+ fix g assume g: "simple_function g" "g \<le> f"
+ let ?G = "g -` {\<omega>} \<inter> space M"
+ show "simple_integral g \<le>
+ SUPR {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g ` space M} simple_integral"
+ (is "simple_integral g \<le> SUPR ?A simple_integral")
+ proof cases
+ let ?g = "\<lambda>x. indicator (space M - ?G) x * g x"
+ have g': "simple_function ?g"
+ using g by (auto intro: simple_functionD)
+ moreover
+ assume "\<mu> ?G = 0"
+ then have "AE x. g x = ?g x" using g
+ by (intro AE_I[where N="?G"])
+ (auto intro: simple_functionD simp: indicator_def)
+ with g(1) g' have "simple_integral g = simple_integral ?g"
+ by (rule simple_integral_cong_AE)
+ moreover have "?g \<le> g" by (auto simp: le_fun_def indicator_def)
+ from this `g \<le> f` have "?g \<le> f" by (rule order_trans)
+ moreover have "\<omega> \<notin> ?g ` space M"
+ by (auto simp: indicator_def split: split_if_asm)
+ ultimately show ?thesis by (auto intro!: le_SUPI)
+ next
+ assume "\<mu> ?G \<noteq> 0"
+ then have "?G \<noteq> {}" by auto
+ then have "\<omega> \<in> g`space M" by force
+ then have "space M \<noteq> {}" by auto
+ have "SUPR ?A simple_integral = \<omega>"
+ proof (intro SUP_\<omega>[THEN iffD2] allI impI)
+ fix x assume "x < \<omega>"
+ then guess n unfolding less_\<omega>_Ex_of_nat .. note n = this
+ then have "0 < n" by (intro neq0_conv[THEN iffD1] notI) simp
+ let ?g = "\<lambda>x. (of_nat n / (if \<mu> ?G = \<omega> then 1 else \<mu> ?G)) * indicator ?G x"
+ show "\<exists>i\<in>?A. x < simple_integral i"
+ proof (intro bexI impI CollectI conjI)
+ show "simple_function ?g" using g
+ by (auto intro!: simple_functionD simple_function_add)
+ have "?g \<le> g" by (auto simp: le_fun_def indicator_def)
+ from this g(2) show "?g \<le> f" by (rule order_trans)
+ show "\<omega> \<notin> ?g ` space M"
+ using `\<mu> ?G \<noteq> 0` by (auto simp: indicator_def split: split_if_asm)
+ have "x < (of_nat n / (if \<mu> ?G = \<omega> then 1 else \<mu> ?G)) * \<mu> ?G"
+ using n `\<mu> ?G \<noteq> 0` `0 < n`
+ by (auto simp: pinfreal_noteq_omega_Ex field_simps)
+ also have "\<dots> = simple_integral ?g" using g `space M \<noteq> {}`
+ by (subst simple_integral_indicator)
+ (auto simp: image_constant ac_simps dest: simple_functionD)
+ finally show "x < simple_integral ?g" .
+ qed
+ qed
+ then show ?thesis by simp
+ qed
+ qed
+qed (auto intro!: SUP_subset simp: positive_integral_def)
lemma (in measure_space) positive_integral_cong_measure:
assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<mu> A"
@@ -849,7 +920,7 @@
lemma (in measure_space) positive_integral_alt1:
"positive_integral f =
(SUP g : {g. simple_function g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}. simple_integral g)"
- unfolding positive_integral_def SUPR_def
+ unfolding positive_integral_alt SUPR_def
proof (safe intro!: arg_cong[where f=Sup])
fix g let ?g = "\<lambda>x. if x \<in> space M then g x else f x"
assume "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
@@ -866,75 +937,6 @@
by auto
qed
-lemma (in measure_space) positive_integral_alt:
- "positive_integral f =
- (SUP g : {g. simple_function g \<and> g \<le> f}. simple_integral g)"
- apply(rule order_class.antisym) unfolding positive_integral_def
- apply(rule SUPR_subset) apply blast apply(rule SUPR_mono_lim)
-proof safe fix u assume u:"simple_function u" and uf:"u \<le> f"
- let ?u = "\<lambda>n x. if u x = \<omega> then Real (real (n::nat)) else u x"
- have su:"\<And>n. simple_function (?u n)" using simple_function_compose1[OF u] .
- show "\<exists>b. \<forall>n. b n \<in> {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g ` space M} \<and>
- (\<lambda>n. simple_integral (b n)) ----> simple_integral u"
- apply(rule_tac x="?u" in exI, safe) apply(rule su)
- proof- fix n::nat have "?u n \<le> u" unfolding le_fun_def by auto
- also note uf finally show "?u n \<le> f" .
- let ?s = "{x \<in> space M. u x = \<omega>}"
- show "(\<lambda>n. simple_integral (?u n)) ----> simple_integral u"
- proof(cases "\<mu> ?s = 0")
- case True have *:"\<And>n. {x\<in>space M. ?u n x \<noteq> u x} = {x\<in>space M. u x = \<omega>}" by auto
- have *:"\<And>n. simple_integral (?u n) = simple_integral u"
- apply(rule simple_integral_cong'[OF su u]) unfolding * True ..
- show ?thesis unfolding * by auto
- next case False note m0=this
- have s:"{x \<in> space M. u x = \<omega>} \<in> sets M" using u by (auto simp: borel_measurable_simple_function)
- have "\<omega> = simple_integral (\<lambda>x. \<omega> * indicator {x\<in>space M. u x = \<omega>} x)"
- apply(subst simple_integral_mult) using s
- unfolding simple_integral_indicator_only[OF s] using False by auto
- also have "... \<le> simple_integral u"
- apply (rule simple_integral_mono)
- apply (rule simple_function_mult)
- apply (rule simple_function_const)
- apply(rule ) prefer 3 apply(subst indicator_def)
- using s u by auto
- finally have *:"simple_integral u = \<omega>" by auto
- show ?thesis unfolding * Lim_omega_pos
- proof safe case goal1
- from real_arch_simple[of "B / real (\<mu> ?s)"] guess N0 .. note N=this
- def N \<equiv> "Suc N0" have N:"real N \<ge> B / real (\<mu> ?s)" "N > 0"
- unfolding N_def using N by auto
- show ?case apply-apply(rule_tac x=N in exI,safe)
- proof- case goal1
- have "Real B \<le> Real (real N) * \<mu> ?s"
- proof(cases "\<mu> ?s = \<omega>")
- case True thus ?thesis using `B>0` N by auto
- next case False
- have *:"B \<le> real N * real (\<mu> ?s)"
- using N(1) apply-apply(subst (asm) pos_divide_le_eq)
- apply rule using m0 False by auto
- show ?thesis apply(subst Real_real'[THEN sym,OF False])
- apply(subst pinfreal_times,subst if_P) defer
- apply(subst pinfreal_less_eq,subst if_P) defer
- using * N `B>0` by(auto intro: mult_nonneg_nonneg)
- qed
- also have "... \<le> Real (real n) * \<mu> ?s"
- apply(rule mult_right_mono) using goal1 by auto
- also have "... = simple_integral (\<lambda>x. Real (real n) * indicator ?s x)"
- apply(subst simple_integral_mult) apply(rule simple_function_indicator[OF s])
- unfolding simple_integral_indicator_only[OF s] ..
- also have "... \<le> simple_integral (\<lambda>x. if u x = \<omega> then Real (real n) else u x)"
- apply(rule simple_integral_mono) apply(rule simple_function_mult)
- apply(rule simple_function_const)
- apply(rule simple_function_indicator) apply(rule s su)+ by auto
- finally show ?case .
- qed qed qed
- fix x assume x:"\<omega> = (if u x = \<omega> then Real (real n) else u x)" "x \<in> space M"
- hence "u x = \<omega>" apply-apply(rule ccontr) by auto
- hence "\<omega> = Real (real n)" using x by auto
- thus False by auto
- qed
-qed
-
lemma (in measure_space) positive_integral_cong:
assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
shows "positive_integral f = positive_integral g"
@@ -947,7 +949,7 @@
lemma (in measure_space) positive_integral_eq_simple_integral:
assumes "simple_function f"
shows "positive_integral f = simple_integral f"
- unfolding positive_integral_alt
+ unfolding positive_integral_def
proof (safe intro!: pinfreal_SUPI)
fix g assume "simple_function g" "g \<le> f"
with assms show "simple_integral g \<le> simple_integral f"
@@ -1008,6 +1010,12 @@
shows "positive_integral u \<le> positive_integral v"
using mono by (auto intro!: AE_cong positive_integral_mono_AE)
+lemma image_set_cong:
+ assumes A: "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. f x = g y"
+ assumes B: "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. g y = f x"
+ shows "f ` A = g ` B"
+ using assms by blast
+
lemma (in measure_space) positive_integral_vimage:
fixes g :: "'a \<Rightarrow> pinfreal" and f :: "'d \<Rightarrow> 'a"
assumes f: "bij_betw f S (space M)"
@@ -1020,14 +1028,12 @@
from assms have inv: "bij_betw (the_inv_into S f) (space M) S"
by (rule bij_betw_the_inv_into)
then have inv_fun: "the_inv_into S f \<in> space M \<rightarrow> S" unfolding bij_betw_def by auto
-
have surj: "f`S = space M"
using f unfolding bij_betw_def by simp
have inj: "inj_on f S"
using f unfolding bij_betw_def by simp
have inv_f: "\<And>x. x \<in> space M \<Longrightarrow> f (the_inv_into S f x) = x"
using f_the_inv_into_f[of f S] f unfolding bij_betw_def by auto
-
from simple_integral_vimage[OF assms, symmetric]
have *: "simple_integral = T.simple_integral \<circ> (\<lambda>g. g \<circ> f)" by (simp add: comp_def)
show ?thesis
@@ -1181,7 +1187,7 @@
by (auto intro!: SUP_leI positive_integral_mono)
next
show "positive_integral u \<le> (SUP i. positive_integral (f i))"
- unfolding positive_integral_def[of u]
+ unfolding positive_integral_alt[of u]
by (auto intro!: SUP_leI positive_integral_SUP_approx[OF assms])
qed
qed
@@ -1194,7 +1200,6 @@
proof -
have "?u \<in> borel_measurable M"
using borel_measurable_SUP[of _ f] assms by (simp add: SUPR_fun_expand)
-
show ?thesis
proof (rule antisym)
show "(SUP j. positive_integral (f j)) \<le> positive_integral ?u"
@@ -1205,9 +1210,10 @@
using assms by (simp cong: measurable_cong)
moreover have iso: "rf \<up> ru" using assms unfolding rf_def ru_def
unfolding isoton_def SUPR_fun_expand le_fun_def fun_eq_iff
+ using SUP_const[OF UNIV_not_empty]
by (auto simp: restrict_def le_fun_def SUPR_fun_expand fun_eq_iff)
ultimately have "positive_integral ru \<le> (SUP i. positive_integral (rf i))"
- unfolding positive_integral_def[of ru]
+ unfolding positive_integral_alt[of ru]
by (auto simp: le_fun_def intro!: SUP_leI positive_integral_SUP_approx)
then show "positive_integral ?u \<le> (SUP i. positive_integral (f i))"
unfolding ru_def rf_def by (simp cong: positive_integral_cong)
@@ -1523,19 +1529,18 @@
apply (rule arg_cong[where f=Sup])
proof (auto simp add: image_iff simple_integral_restricted[OF `A \<in> sets M`])
fix g assume "simple_function (\<lambda>x. g x * indicator A x)"
- "g \<le> f" "\<forall>x\<in>A. \<omega> \<noteq> g x"
- then show "\<exists>x. simple_function x \<and> x \<le> (\<lambda>x. f x * indicator A x) \<and> (\<forall>y\<in>space M. \<omega> \<noteq> x y) \<and>
+ "g \<le> f"
+ then show "\<exists>x. simple_function x \<and> x \<le> (\<lambda>x. f x * indicator A x) \<and>
simple_integral (\<lambda>x. g x * indicator A x) = simple_integral x"
apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
by (auto simp: indicator_def le_fun_def)
next
fix g assume g: "simple_function g" "g \<le> (\<lambda>x. f x * indicator A x)"
- "\<forall>x\<in>space M. \<omega> \<noteq> g x"
then have *: "(\<lambda>x. g x * indicator A x) = g"
"\<And>x. g x * indicator A x = g x"
"\<And>x. g x \<le> f x"
by (auto simp: le_fun_def fun_eq_iff indicator_def split: split_if_asm)
- from g show "\<exists>x. simple_function (\<lambda>xa. x xa * indicator A xa) \<and> x \<le> f \<and> (\<forall>xa\<in>A. \<omega> \<noteq> x xa) \<and>
+ from g show "\<exists>x. simple_function (\<lambda>xa. x xa * indicator A xa) \<and> x \<le> f \<and>
simple_integral g = simple_integral (\<lambda>xa. x xa * indicator A xa)"
using `A \<in> sets M`[THEN sets_into_space]
apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
@@ -2299,7 +2304,7 @@
qed
lemma (in finite_measure_space) simple_function_finite[simp, intro]: "simple_function f"
- unfolding simple_function_def sets_eq_Pow using finite_space by auto
+ unfolding simple_function_def using finite_space by auto
lemma (in finite_measure_space) borel_measurable_finite[intro, simp]: "f \<in> borel_measurable M"
by (auto intro: borel_measurable_simple_function)
@@ -2310,7 +2315,7 @@
have *: "positive_integral f = positive_integral (\<lambda>x. \<Sum>y\<in>space M. f y * indicator {y} x)"
by (auto intro!: positive_integral_cong simp add: indicator_def if_distrib setsum_cases[OF finite_space])
show ?thesis unfolding * using borel_measurable_finite[of f]
- by (simp add: positive_integral_setsum positive_integral_cmult_indicator sets_eq_Pow)
+ by (simp add: positive_integral_setsum positive_integral_cmult_indicator)
qed
lemma (in finite_measure_space) integral_finite_singleton:
@@ -2322,9 +2327,9 @@
"positive_integral (\<lambda>x. Real (- f x)) = (\<Sum>x \<in> space M. Real (- f x) * \<mu> {x})"
unfolding positive_integral_finite_eq_setsum by auto
show "integrable f" using finite_space finite_measure
- by (simp add: setsum_\<omega> integrable_def sets_eq_Pow)
+ by (simp add: setsum_\<omega> integrable_def)
show ?I using finite_measure
- apply (simp add: integral_def sets_eq_Pow real_of_pinfreal_setsum[symmetric]
+ apply (simp add: integral_def real_of_pinfreal_setsum[symmetric]
real_of_pinfreal_mult[symmetric] setsum_subtractf[symmetric])
by (rule setsum_cong) (simp_all split: split_if)
qed
--- a/src/HOL/Probability/Lebesgue_Measure.thy Thu Dec 02 14:56:16 2010 +0100
+++ b/src/HOL/Probability/Lebesgue_Measure.thy Thu Dec 02 15:32:48 2010 +0100
@@ -4,89 +4,6 @@
imports Product_Measure Gauge_Measure Complete_Measure
begin
-lemma (in complete_lattice) SUP_pair:
- "(SUP i:A. SUP j:B. f i j) = (SUP p:A\<times>B. (\<lambda> (i, j). f i j) p)" (is "?l = ?r")
-proof (intro antisym SUP_leI)
- fix i j assume "i \<in> A" "j \<in> B"
- then have "(case (i,j) of (i,j) \<Rightarrow> f i j) \<le> ?r"
- by (intro SUPR_upper) auto
- then show "f i j \<le> ?r" by auto
-next
- fix p assume "p \<in> A \<times> B"
- then obtain i j where "p = (i,j)" "i \<in> A" "j \<in> B" by auto
- have "f i j \<le> (SUP j:B. f i j)" using `j \<in> B` by (intro SUPR_upper)
- also have "(SUP j:B. f i j) \<le> ?l" using `i \<in> A` by (intro SUPR_upper)
- finally show "(case p of (i, j) \<Rightarrow> f i j) \<le> ?l" using `p = (i,j)` by simp
-qed
-
-lemma (in complete_lattice) SUP_surj_compose:
- assumes *: "f`A = B" shows "SUPR A (g \<circ> f) = SUPR B g"
- unfolding SUPR_def unfolding *[symmetric]
- by (simp add: image_compose)
-
-lemma (in complete_lattice) SUP_swap:
- "(SUP i:A. SUP j:B. f i j) = (SUP j:B. SUP i:A. f i j)"
-proof -
- have *: "(\<lambda>(i,j). (j,i)) ` (B \<times> A) = A \<times> B" by auto
- show ?thesis
- unfolding SUP_pair SUP_surj_compose[symmetric, OF *]
- by (auto intro!: arg_cong[where f=Sup] image_eqI simp: comp_def SUPR_def)
-qed
-
-lemma SUP_\<omega>: "(SUP i:A. f i) = \<omega> \<longleftrightarrow> (\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)"
-proof
- assume *: "(SUP i:A. f i) = \<omega>"
- show "(\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)" unfolding *[symmetric]
- proof (intro allI impI)
- fix x assume "x < SUPR A f" then show "\<exists>i\<in>A. x < f i"
- unfolding less_SUP_iff by auto
- qed
-next
- assume *: "\<forall>x<\<omega>. \<exists>i\<in>A. x < f i"
- show "(SUP i:A. f i) = \<omega>"
- proof (rule pinfreal_SUPI)
- fix y assume **: "\<And>i. i \<in> A \<Longrightarrow> f i \<le> y"
- show "\<omega> \<le> y"
- proof cases
- assume "y < \<omega>"
- from *[THEN spec, THEN mp, OF this]
- obtain i where "i \<in> A" "\<not> (f i \<le> y)" by auto
- with ** show ?thesis by auto
- qed auto
- qed auto
-qed
-
-lemma psuminf_commute:
- shows "(\<Sum>\<^isub>\<infinity> i j. f i j) = (\<Sum>\<^isub>\<infinity> j i. f i j)"
-proof -
- have "(SUP n. \<Sum> i < n. SUP m. \<Sum> j < m. f i j) = (SUP n. SUP m. \<Sum> i < n. \<Sum> j < m. f i j)"
- apply (subst SUPR_pinfreal_setsum)
- by auto
- also have "\<dots> = (SUP m n. \<Sum> j < m. \<Sum> i < n. f i j)"
- apply (subst SUP_swap)
- apply (subst setsum_commute)
- by auto
- also have "\<dots> = (SUP m. \<Sum> j < m. SUP n. \<Sum> i < n. f i j)"
- apply (subst SUPR_pinfreal_setsum)
- by auto
- finally show ?thesis
- unfolding psuminf_def by auto
-qed
-
-lemma psuminf_SUP_eq:
- assumes "\<And>n i. f n i \<le> f (Suc n) i"
- shows "(\<Sum>\<^isub>\<infinity> i. SUP n::nat. f n i) = (SUP n::nat. \<Sum>\<^isub>\<infinity> i. f n i)"
-proof -
- { fix n :: nat
- have "(\<Sum>i<n. SUP k. f k i) = (SUP k. \<Sum>i<n. f k i)"
- using assms by (auto intro!: SUPR_pinfreal_setsum[symmetric]) }
- note * = this
- show ?thesis
- unfolding psuminf_def
- unfolding *
- apply (subst SUP_swap) ..
-qed
-
subsection {* Standard Cubes *}
definition cube :: "nat \<Rightarrow> 'a::ordered_euclidean_space set" where
@@ -838,20 +755,6 @@
qed
qed
-lemma Real_mult_nonneg: assumes "x \<ge> 0" "y \<ge> 0"
- shows "Real (x * y) = Real x * Real y" using assms by auto
-
-lemma Real_setprod: assumes "\<forall>x\<in>A. f x \<ge> 0" shows "Real (setprod f A) = setprod (\<lambda>x. Real (f x)) A"
-proof(cases "finite A")
- case True thus ?thesis using assms
- proof(induct A) case (insert x A)
- have "0 \<le> setprod f A" apply(rule setprod_nonneg) using insert by auto
- thus ?case unfolding setprod_insert[OF insert(1-2)] apply-
- apply(subst Real_mult_nonneg) prefer 3 apply(subst insert(3)[THEN sym])
- using insert by auto
- qed auto
-qed auto
-
lemma e2p_Int:"e2p ` A \<inter> e2p ` B = e2p ` (A \<inter> B)" (is "?L = ?R")
apply(rule image_Int[THEN sym]) using bij_euclidean_component
unfolding bij_betw_def by auto
--- a/src/HOL/Probability/Measure.thy Thu Dec 02 14:56:16 2010 +0100
+++ b/src/HOL/Probability/Measure.thy Thu Dec 02 15:32:48 2010 +0100
@@ -651,27 +651,6 @@
abbreviation (in measure_space) "null_sets \<equiv> {N\<in>sets M. \<mu> N = 0}"
-definition (in measure_space)
- almost_everywhere :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "AE " 10) where
- "almost_everywhere P \<longleftrightarrow> (\<exists>N\<in>null_sets. { x \<in> space M. \<not> P x } \<subseteq> N)"
-
-lemma (in measure_space) AE_I':
- "N \<in> null_sets \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x. P x)"
- unfolding almost_everywhere_def by auto
-
-lemma (in measure_space) AE_iff_null_set:
- assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M")
- shows "(AE x. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets"
-proof
- assume "AE x. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "\<mu> N = 0"
- unfolding almost_everywhere_def by auto
- moreover have "\<mu> ?P \<le> \<mu> N"
- using assms N(1,2) by (auto intro: measure_mono)
- ultimately show "?P \<in> null_sets" using assms by auto
-next
- assume "?P \<in> null_sets" with assms show "AE x. P x" by (auto intro: AE_I')
-qed
-
lemma (in measure_space) null_sets_Un[intro]:
assumes "N \<in> null_sets" "N' \<in> null_sets"
shows "N \<union> N' \<in> null_sets"
@@ -703,6 +682,17 @@
using assms by auto
qed
+lemma (in measure_space) null_sets_finite_UN:
+ assumes "finite S" "\<And>i. i \<in> S \<Longrightarrow> A i \<in> null_sets"
+ shows "(\<Union>i\<in>S. A i) \<in> null_sets"
+proof (intro CollectI conjI)
+ show "(\<Union>i\<in>S. A i) \<in> sets M" using assms by (intro finite_UN) auto
+ have "\<mu> (\<Union>i\<in>S. A i) \<le> (\<Sum>i\<in>S. \<mu> (A i))"
+ using assms by (intro measure_finitely_subadditive) auto
+ then show "\<mu> (\<Union>i\<in>S. A i) = 0"
+ using assms by auto
+qed
+
lemma (in measure_space) null_set_Int1:
assumes "B \<in> null_sets" "A \<in> sets M" shows "A \<inter> B \<in> null_sets"
using assms proof (intro CollectI conjI)
@@ -741,6 +731,29 @@
by (subst measure_additive[symmetric]) auto
qed
+section "Formalise almost everywhere"
+
+definition (in measure_space)
+ almost_everywhere :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "AE " 10) where
+ "almost_everywhere P \<longleftrightarrow> (\<exists>N\<in>null_sets. { x \<in> space M. \<not> P x } \<subseteq> N)"
+
+lemma (in measure_space) AE_I':
+ "N \<in> null_sets \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x. P x)"
+ unfolding almost_everywhere_def by auto
+
+lemma (in measure_space) AE_iff_null_set:
+ assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M")
+ shows "(AE x. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets"
+proof
+ assume "AE x. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "\<mu> N = 0"
+ unfolding almost_everywhere_def by auto
+ moreover have "\<mu> ?P \<le> \<mu> N"
+ using assms N(1,2) by (auto intro: measure_mono)
+ ultimately show "?P \<in> null_sets" using assms by auto
+next
+ assume "?P \<in> null_sets" with assms show "AE x. P x" by (auto intro: AE_I')
+qed
+
lemma (in measure_space) AE_True[intro, simp]: "AE x. True"
unfolding almost_everywhere_def by auto
@@ -1409,7 +1422,7 @@
show "\<mu> {x} \<noteq> \<omega>" by (auto simp: insert_absorb[OF *] Diff_subset) }
qed
-sublocale finite_measure_space < finite_measure
+sublocale finite_measure_space \<subseteq> finite_measure
proof
show "\<mu> (space M) \<noteq> \<omega>"
unfolding sum_over_space[symmetric] setsum_\<omega>
--- a/src/HOL/Probability/Positive_Infinite_Real.thy Thu Dec 02 14:56:16 2010 +0100
+++ b/src/HOL/Probability/Positive_Infinite_Real.thy Thu Dec 02 15:32:48 2010 +0100
@@ -6,14 +6,6 @@
imports Complex_Main Nat_Bijection Multivariate_Analysis
begin
-lemma range_const[simp]: "range (\<lambda>x. c) = {c}" by auto
-
-lemma (in complete_lattice) SUPR_const[simp]: "(SUP i. c) = c"
- unfolding SUPR_def by simp
-
-lemma (in complete_lattice) INFI_const[simp]: "(INF i. c) = c"
- unfolding INFI_def by simp
-
lemma (in complete_lattice) Sup_start:
assumes *: "\<And>x. f x \<le> f 0"
shows "(SUP n. f n) = f 0"
@@ -94,6 +86,26 @@
ultimately show ?thesis by simp
qed
+lemma (in complete_lattice) lim_INF_le_lim_SUP:
+ fixes f :: "nat \<Rightarrow> 'a"
+ shows "(SUP n. INF m. f (n + m)) \<le> (INF n. SUP m. f (n + m))"
+proof (rule SUP_leI, rule le_INFI)
+ fix i j show "(INF m. f (i + m)) \<le> (SUP m. f (j + m))"
+ proof (cases rule: le_cases)
+ assume "i \<le> j"
+ have "(INF m. f (i + m)) \<le> f (i + (j - i))" by (rule INF_leI) simp
+ also have "\<dots> = f (j + 0)" using `i \<le> j` by auto
+ also have "\<dots> \<le> (SUP m. f (j + m))" by (rule le_SUPI) simp
+ finally show ?thesis .
+ next
+ assume "j \<le> i"
+ have "(INF m. f (i + m)) \<le> f (i + 0)" by (rule INF_leI) simp
+ also have "\<dots> = f (j + (i - j))" using `j \<le> i` by auto
+ also have "\<dots> \<le> (SUP m. f (j + m))" by (rule le_SUPI) simp
+ finally show ?thesis .
+ qed
+qed
+
text {*
We introduce the the positive real numbers as needed for measure theory.
@@ -348,6 +360,20 @@
lemma real_of_pinfreal_mult: "real X * real Y = real (X * Y :: pinfreal)"
by (cases X, cases Y) (auto simp: zero_le_mult_iff)
+lemma Real_mult_nonneg: assumes "x \<ge> 0" "y \<ge> 0"
+ shows "Real (x * y) = Real x * Real y" using assms by auto
+
+lemma Real_setprod: assumes "\<forall>x\<in>A. f x \<ge> 0" shows "Real (setprod f A) = setprod (\<lambda>x. Real (f x)) A"
+proof(cases "finite A")
+ case True thus ?thesis using assms
+ proof(induct A) case (insert x A)
+ have "0 \<le> setprod f A" apply(rule setprod_nonneg) using insert by auto
+ thus ?case unfolding setprod_insert[OF insert(1-2)] apply-
+ apply(subst Real_mult_nonneg) prefer 3 apply(subst insert(3)[THEN sym])
+ using insert by auto
+ qed auto
+qed auto
+
subsection "@{typ pinfreal} is a linear order"
instantiation pinfreal :: linorder
@@ -549,6 +575,14 @@
lemma pinfreal_of_nat[simp]: "of_nat m = Real (real m)"
by (induct m) (auto simp: real_of_nat_Suc one_pinfreal_def simp del: Real_1)
+lemma less_\<omega>_Ex_of_nat: "x < \<omega> \<longleftrightarrow> (\<exists>n. x < of_nat n)"
+proof safe
+ assume "x < \<omega>"
+ then obtain r where "0 \<le> r" "x = Real r" by (cases x) auto
+ moreover obtain n where "r < of_nat n" using ex_less_of_nat by auto
+ ultimately show "\<exists>n. x < of_nat n" by (auto simp: real_eq_of_nat)
+qed auto
+
lemma real_of_pinfreal_mono:
fixes a b :: pinfreal
assumes "b \<noteq> \<omega>" "a \<le> b"
@@ -831,6 +865,29 @@
qed simp
qed simp
+lemma SUP_\<omega>: "(SUP i:A. f i) = \<omega> \<longleftrightarrow> (\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)"
+proof
+ assume *: "(SUP i:A. f i) = \<omega>"
+ show "(\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)" unfolding *[symmetric]
+ proof (intro allI impI)
+ fix x assume "x < SUPR A f" then show "\<exists>i\<in>A. x < f i"
+ unfolding less_SUP_iff by auto
+ qed
+next
+ assume *: "\<forall>x<\<omega>. \<exists>i\<in>A. x < f i"
+ show "(SUP i:A. f i) = \<omega>"
+ proof (rule pinfreal_SUPI)
+ fix y assume **: "\<And>i. i \<in> A \<Longrightarrow> f i \<le> y"
+ show "\<omega> \<le> y"
+ proof cases
+ assume "y < \<omega>"
+ from *[THEN spec, THEN mp, OF this]
+ obtain i where "i \<in> A" "\<not> (f i \<le> y)" by auto
+ with ** show ?thesis by auto
+ qed auto
+ qed auto
+qed
+
subsubsection {* Equivalence between @{text "f ----> x"} and @{text SUP} on @{typ pinfreal} *}
lemma monoseq_monoI: "mono f \<Longrightarrow> monoseq f"
@@ -1241,7 +1298,6 @@
have [intro, simp]: "\<And>A. inj_on f A" using `bij f` unfolding bij_def by (auto intro: subset_inj_on)
have f[intro, simp]: "\<And>x. f (inv f x) = x"
using `bij f` unfolding bij_def by (auto intro: surj_f_inv_f)
-
show ?thesis
proof (rule psuminf_equality)
fix n
@@ -1266,49 +1322,6 @@
qed
qed
-lemma psuminf_2dimen:
- fixes f:: "nat * nat \<Rightarrow> pinfreal"
- assumes fsums: "\<And>m. g m = (\<Sum>\<^isub>\<infinity> n. f (m,n))"
- shows "psuminf (f \<circ> prod_decode) = psuminf g"
-proof (rule psuminf_equality)
- fix n :: nat
- let ?P = "prod_decode ` {..<n}"
- have "setsum (f \<circ> prod_decode) {..<n} = setsum f ?P"
- by (auto simp: setsum_reindex inj_prod_decode)
- also have "\<dots> \<le> setsum f ({..Max (fst ` ?P)} \<times> {..Max (snd ` ?P)})"
- proof (safe intro!: setsum_mono3 Max_ge image_eqI)
- fix a b x assume "(a, b) = prod_decode x"
- from this[symmetric] show "a = fst (prod_decode x)" "b = snd (prod_decode x)"
- by simp_all
- qed simp_all
- also have "\<dots> = (\<Sum>m\<le>Max (fst ` ?P). (\<Sum>n\<le>Max (snd ` ?P). f (m,n)))"
- unfolding setsum_cartesian_product by simp
- also have "\<dots> \<le> (\<Sum>m\<le>Max (fst ` ?P). g m)"
- by (auto intro!: setsum_mono psuminf_upper simp del: setsum_lessThan_Suc
- simp: fsums lessThan_Suc_atMost[symmetric])
- also have "\<dots> \<le> psuminf g"
- by (auto intro!: psuminf_upper simp del: setsum_lessThan_Suc
- simp: lessThan_Suc_atMost[symmetric])
- finally show "setsum (f \<circ> prod_decode) {..<n} \<le> psuminf g" .
-next
- fix y assume *: "\<And>n. setsum (f \<circ> prod_decode) {..<n} \<le> y"
- have g: "g = (\<lambda>m. \<Sum>\<^isub>\<infinity> n. f (m,n))" unfolding fsums[symmetric] ..
- show "psuminf g \<le> y" unfolding g
- proof (rule psuminf_bound, unfold setsum_pinfsum[symmetric], safe intro!: psuminf_bound)
- fix N M :: nat
- let ?P = "{..<N} \<times> {..<M}"
- let ?M = "Max (prod_encode ` ?P)"
- have "(\<Sum>n<M. \<Sum>m<N. f (m, n)) \<le> (\<Sum>(m, n)\<in>?P. f (m, n))"
- unfolding setsum_commute[of _ _ "{..<M}"] unfolding setsum_cartesian_product ..
- also have "\<dots> \<le> (\<Sum>(m,n)\<in>(prod_decode ` {..?M}). f (m, n))"
- by (auto intro!: setsum_mono3 image_eqI[where f=prod_decode, OF prod_encode_inverse[symmetric]])
- also have "\<dots> \<le> y" using *[of "Suc ?M"]
- by (simp add: lessThan_Suc_atMost[symmetric] setsum_reindex
- inj_prod_decode del: setsum_lessThan_Suc)
- finally show "(\<Sum>n<M. \<Sum>m<N. f (m, n)) \<le> y" .
- qed
-qed
-
lemma pinfreal_mult_less_right:
assumes "b * a < c * a" "0 < a" "a < \<omega>"
shows "b < c"
@@ -1384,6 +1397,80 @@
qed simp
qed simp
+lemma psuminf_SUP_eq:
+ assumes "\<And>n i. f n i \<le> f (Suc n) i"
+ shows "(\<Sum>\<^isub>\<infinity> i. SUP n::nat. f n i) = (SUP n::nat. \<Sum>\<^isub>\<infinity> i. f n i)"
+proof -
+ { fix n :: nat
+ have "(\<Sum>i<n. SUP k. f k i) = (SUP k. \<Sum>i<n. f k i)"
+ using assms by (auto intro!: SUPR_pinfreal_setsum[symmetric]) }
+ note * = this
+ show ?thesis
+ unfolding psuminf_def
+ unfolding *
+ apply (subst SUP_commute) ..
+qed
+
+lemma psuminf_commute:
+ shows "(\<Sum>\<^isub>\<infinity> i j. f i j) = (\<Sum>\<^isub>\<infinity> j i. f i j)"
+proof -
+ have "(SUP n. \<Sum> i < n. SUP m. \<Sum> j < m. f i j) = (SUP n. SUP m. \<Sum> i < n. \<Sum> j < m. f i j)"
+ apply (subst SUPR_pinfreal_setsum)
+ by auto
+ also have "\<dots> = (SUP m n. \<Sum> j < m. \<Sum> i < n. f i j)"
+ apply (subst SUP_commute)
+ apply (subst setsum_commute)
+ by auto
+ also have "\<dots> = (SUP m. \<Sum> j < m. SUP n. \<Sum> i < n. f i j)"
+ apply (subst SUPR_pinfreal_setsum)
+ by auto
+ finally show ?thesis
+ unfolding psuminf_def by auto
+qed
+
+lemma psuminf_2dimen:
+ fixes f:: "nat * nat \<Rightarrow> pinfreal"
+ assumes fsums: "\<And>m. g m = (\<Sum>\<^isub>\<infinity> n. f (m,n))"
+ shows "psuminf (f \<circ> prod_decode) = psuminf g"
+proof (rule psuminf_equality)
+ fix n :: nat
+ let ?P = "prod_decode ` {..<n}"
+ have "setsum (f \<circ> prod_decode) {..<n} = setsum f ?P"
+ by (auto simp: setsum_reindex inj_prod_decode)
+ also have "\<dots> \<le> setsum f ({..Max (fst ` ?P)} \<times> {..Max (snd ` ?P)})"
+ proof (safe intro!: setsum_mono3 Max_ge image_eqI)
+ fix a b x assume "(a, b) = prod_decode x"
+ from this[symmetric] show "a = fst (prod_decode x)" "b = snd (prod_decode x)"
+ by simp_all
+ qed simp_all
+ also have "\<dots> = (\<Sum>m\<le>Max (fst ` ?P). (\<Sum>n\<le>Max (snd ` ?P). f (m,n)))"
+ unfolding setsum_cartesian_product by simp
+ also have "\<dots> \<le> (\<Sum>m\<le>Max (fst ` ?P). g m)"
+ by (auto intro!: setsum_mono psuminf_upper simp del: setsum_lessThan_Suc
+ simp: fsums lessThan_Suc_atMost[symmetric])
+ also have "\<dots> \<le> psuminf g"
+ by (auto intro!: psuminf_upper simp del: setsum_lessThan_Suc
+ simp: lessThan_Suc_atMost[symmetric])
+ finally show "setsum (f \<circ> prod_decode) {..<n} \<le> psuminf g" .
+next
+ fix y assume *: "\<And>n. setsum (f \<circ> prod_decode) {..<n} \<le> y"
+ have g: "g = (\<lambda>m. \<Sum>\<^isub>\<infinity> n. f (m,n))" unfolding fsums[symmetric] ..
+ show "psuminf g \<le> y" unfolding g
+ proof (rule psuminf_bound, unfold setsum_pinfsum[symmetric], safe intro!: psuminf_bound)
+ fix N M :: nat
+ let ?P = "{..<N} \<times> {..<M}"
+ let ?M = "Max (prod_encode ` ?P)"
+ have "(\<Sum>n<M. \<Sum>m<N. f (m, n)) \<le> (\<Sum>(m, n)\<in>?P. f (m, n))"
+ unfolding setsum_commute[of _ _ "{..<M}"] unfolding setsum_cartesian_product ..
+ also have "\<dots> \<le> (\<Sum>(m,n)\<in>(prod_decode ` {..?M}). f (m, n))"
+ by (auto intro!: setsum_mono3 image_eqI[where f=prod_decode, OF prod_encode_inverse[symmetric]])
+ also have "\<dots> \<le> y" using *[of "Suc ?M"]
+ by (simp add: lessThan_Suc_atMost[symmetric] setsum_reindex
+ inj_prod_decode del: setsum_lessThan_Suc)
+ finally show "(\<Sum>n<M. \<Sum>m<N. f (m, n)) \<le> y" .
+ qed
+qed
+
lemma Real_max:
assumes "x \<ge> 0" "y \<ge> 0"
shows "Real (max x y) = max (Real x) (Real y)"
@@ -2076,20 +2163,6 @@
lemma real_Real_max:"real (Real x) = max x 0"
unfolding real_Real by auto
-lemma (in complete_lattice) SUPR_upper:
- "x \<in> A \<Longrightarrow> f x \<le> SUPR A f"
- unfolding SUPR_def apply(rule Sup_upper) by auto
-
-lemma (in complete_lattice) SUPR_subset:
- assumes "A \<subseteq> B" shows "SUPR A f \<le> SUPR B f"
- apply(rule SUP_leI) apply(rule SUPR_upper) using assms by auto
-
-lemma (in complete_lattice) SUPR_mono:
- assumes "\<forall>a\<in>A. \<exists>b\<in>B. f b \<ge> f a"
- shows "SUPR A f \<le> SUPR B f"
- unfolding SUPR_def apply(rule Sup_mono)
- using assms by auto
-
lemma Sup_lim:
assumes "\<forall>n. b n \<in> s" "b ----> (a::pinfreal)"
shows "a \<le> Sup s"
@@ -2161,11 +2234,6 @@
unfolding Real_real using om by auto
qed qed
-lemma less_SUP_iff:
- fixes a :: pinfreal
- shows "(a < (SUP i:A. f i)) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)"
- unfolding SUPR_def less_Sup_iff by auto
-
lemma Sup_mono_lim:
assumes "\<forall>a\<in>A. \<exists>b. \<forall>n. b n \<in> B \<and> b ----> (a::pinfreal)"
shows "Sup A \<le> Sup B"
@@ -2371,26 +2439,6 @@
shows "a \<le> a - b \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> a \<noteq> \<omega> \<Longrightarrow> b = 0"
by (cases a, cases b, auto split: split_if_asm)
-lemma lim_INF_le_lim_SUP:
- fixes f :: "nat \<Rightarrow> pinfreal"
- shows "(SUP n. INF m. f (n + m)) \<le> (INF n. SUP m. f (n + m))"
-proof (rule complete_lattice_class.SUP_leI, rule complete_lattice_class.le_INFI)
- fix i j show "(INF m. f (i + m)) \<le> (SUP m. f (j + m))"
- proof (cases rule: le_cases)
- assume "i \<le> j"
- have "(INF m. f (i + m)) \<le> f (i + (j - i))" by (rule INF_leI) simp
- also have "\<dots> = f (j + 0)" using `i \<le> j` by auto
- also have "\<dots> \<le> (SUP m. f (j + m))" by (rule le_SUPI) simp
- finally show ?thesis .
- next
- assume "j \<le> i"
- have "(INF m. f (i + m)) \<le> f (i + 0)" by (rule INF_leI) simp
- also have "\<dots> = f (j + (i - j))" using `j \<le> i` by auto
- also have "\<dots> \<le> (SUP m. f (j + m))" by (rule le_SUPI) simp
- finally show ?thesis .
- qed
-qed
-
lemma lim_INF_eq_lim_SUP:
fixes X :: "nat \<Rightarrow> real"
assumes "\<And>i. 0 \<le> X i" and "0 \<le> x"
@@ -2707,4 +2755,21 @@
lemma lessThan_0_Empty: "{..< 0 :: pinfreal} = {}" by auto
+lemma real_of_pinfreal_inverse[simp]:
+ fixes X :: pinfreal
+ shows "real (inverse X) = 1 / real X"
+ by (cases X) (auto simp: inverse_eq_divide)
+
+lemma real_of_pinfreal_le_0[simp]: "real (X :: pinfreal) \<le> 0 \<longleftrightarrow> (X = 0 \<or> X = \<omega>)"
+ by (cases X) auto
+
+lemma real_of_pinfreal_less_0[simp]: "\<not> (real (X :: pinfreal) < 0)"
+ by (cases X) auto
+
+lemma abs_real_of_pinfreal[simp]: "\<bar>real (X :: pinfreal)\<bar> = real X"
+ by simp
+
+lemma zero_less_real_of_pinfreal: "0 < real (X :: pinfreal) \<longleftrightarrow> X \<noteq> 0 \<and> X \<noteq> \<omega>"
+ by (cases X) auto
+
end
--- a/src/HOL/Probability/Product_Measure.thy Thu Dec 02 14:56:16 2010 +0100
+++ b/src/HOL/Probability/Product_Measure.thy Thu Dec 02 15:32:48 2010 +0100
@@ -2,28 +2,6 @@
imports Lebesgue_Integration
begin
-lemma in_sigma[intro, simp]: "A \<in> sets M \<Longrightarrow> A \<in> sets (sigma M)"
- unfolding sigma_def by (auto intro!: sigma_sets.Basic)
-
-lemma (in sigma_algebra) sigma_eq[simp]: "sigma M = M"
- unfolding sigma_def sigma_sets_eq by simp
-
-lemma vimage_algebra_sigma:
- assumes E: "sets E \<subseteq> Pow (space E)"
- and f: "f \<in> space F \<rightarrow> space E"
- and "\<And>A. A \<in> sets F \<Longrightarrow> A \<in> (\<lambda>X. f -` X \<inter> space F) ` sets E"
- and "\<And>A. A \<in> sets E \<Longrightarrow> f -` A \<inter> space F \<in> sets F"
- shows "sigma_algebra.vimage_algebra (sigma E) (space F) f = sigma F"
-proof -
- interpret sigma_algebra "sigma E"
- using assms by (intro sigma_algebra_sigma) auto
- have eq: "sets F = (\<lambda>X. f -` X \<inter> space F) ` sets E"
- using assms by auto
- show "vimage_algebra (space F) f = sigma F"
- unfolding vimage_algebra_def using assms
- by (simp add: sigma_def eq sigma_sets_vimage)
-qed
-
lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
by auto
@@ -786,13 +764,10 @@
positive_integral f"
proof -
interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
-
have s: "\<And>x y. (case (x, y) of (x, y) \<Rightarrow> f (y, x)) = f (y, x)" by simp
have t: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> (y, x))) = (\<lambda>(x, y). f (y, x))" by (auto simp: fun_eq_iff)
-
have bij: "bij_betw (\<lambda>(x, y). (y, x)) (space M2 \<times> space M1) (space P)"
by (auto intro!: inj_onI simp: space_pair_algebra bij_betw_def)
-
note pair_sigma_algebra_measurable[OF f]
from Q.positive_integral_fst_measurable[OF this]
have "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =
@@ -890,7 +865,7 @@
lemma (in finite_product_sigma_algebra) P_empty:
"I = {} \<Longrightarrow> P = \<lparr> space = {\<lambda>k. undefined}, sets = { {}, {\<lambda>k. undefined} }\<rparr>"
- unfolding product_algebra_def by (simp add: sigma_def)
+ unfolding product_algebra_def by (simp add: sigma_def image_constant)
lemma (in finite_product_sigma_algebra) in_P[simp, intro]:
"\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P"
@@ -930,7 +905,6 @@
using E1 E2 by (auto simp add: pair_algebra_def)
interpret E: sigma_algebra ?E unfolding pair_algebra_def
using E1 E2 by (intro sigma_algebra_sigma) auto
-
{ fix A assume "A \<in> sets E1"
then have "fst -` A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"
using E1 2 unfolding isoton_def pair_algebra_def by auto
@@ -954,7 +928,6 @@
"fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"
using E1 E2 by (subst (1 2) E.measurable_iff_sigma)
(auto simp: pair_algebra_def sets_sigma)
-
{ fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)"
with proj have "fst -` A \<inter> space ?E \<in> sets ?E" "snd -` B \<inter> space ?E \<in> sets ?E"
unfolding measurable_def by simp_all
@@ -966,7 +939,6 @@
by (intro E.sigma_sets_subset) (auto simp add: pair_algebra_def sets_sigma)
then have subset: "sets ?S \<subseteq> sets ?E"
by (simp add: sets_sigma pair_algebra_def)
-
have "sets ?S = sets ?E"
proof (intro set_eqI iffI)
fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
@@ -1286,7 +1258,7 @@
by (auto intro!: exI[of _ "\<lambda>A. if A = {} then 0 else 1"] sigma_algebra_sigma
sigma_algebra.finite_additivity_sufficient
simp add: positive_def additive_def sets_sigma sigma_finite_measure_def
- sigma_finite_measure_axioms_def)
+ sigma_finite_measure_axioms_def image_constant)
next
case (insert i I)
interpret finite_product_sigma_finite M \<mu> I by default fact
@@ -1304,7 +1276,6 @@
unfolding product_singleton_vimage_algebra_eq[OF `i \<notin> I` `finite I`, symmetric]
using bij_betw_restrict_I_i[OF `i \<notin> I`, of M]
by (intro P.measure_space_isomorphic) auto
-
show ?case
proof (intro exI[of _ ?\<nu>] conjI ballI)
{ fix A assume A: "A \<in> (\<Pi> i\<in>insert i I. sets (M i))"
@@ -1322,7 +1293,6 @@
apply fastsimp
using `i \<notin> I` `finite I` prod[of A] by (auto simp: ac_simps) }
note product = this
-
show "sigma_finite_measure I'.P ?\<nu>"
proof
from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
@@ -1395,7 +1365,7 @@
have "\<And>A. measure (Pi\<^isub>E {} A) = 1"
using assms by (subst measure_times) auto
then show ?thesis
- unfolding positive_integral_alt simple_function_def simple_integral_def_raw
+ unfolding positive_integral_def simple_function_def simple_integral_def_raw
proof (simp add: P_empty, intro antisym)
show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> f}. f (\<lambda>k. undefined))"
by (intro le_SUPI) auto
@@ -1455,17 +1425,13 @@
have "finite (I \<union> J)" using fin by auto
interpret IJ: finite_product_sigma_finite M \<mu> "I \<union> J" by default fact
interpret P: pair_sigma_finite I.P I.measure J.P J.measure by default
-
let ?f = "\<lambda>x. ((\<lambda>i\<in>I. x i), (\<lambda>i\<in>J. x i))"
-
have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable P.P"
by (subst product_product_vimage_algebra_eq[OF IJ fin, symmetric])
(auto simp: space_pair_algebra intro!: IJ.measurable_vimage f)
-
have split_f_image[simp]: "\<And>F. ?f ` (Pi\<^isub>E (I \<union> J) F) = (Pi\<^isub>E I F) \<times> (Pi\<^isub>E J F)"
apply auto apply (rule_tac x="merge I a J b" in image_eqI)
by (auto dest: extensional_restrict)
-
have "IJ.positive_integral f = IJ.positive_integral (\<lambda>x. f (restrict x (I \<union> J)))"
by (auto intro!: IJ.positive_integral_cong arg_cong[where f=f] dest!: extensional_restrict)
also have "\<dots> = I.positive_integral (\<lambda>x. J.positive_integral (\<lambda>y. f (merge I x J y)))"
--- a/src/HOL/Probability/Radon_Nikodym.thy Thu Dec 02 14:56:16 2010 +0100
+++ b/src/HOL/Probability/Radon_Nikodym.thy Thu Dec 02 15:32:48 2010 +0100
@@ -69,6 +69,8 @@
qed
qed
+subsection "Absolutely continuous"
+
definition (in measure_space)
"absolutely_continuous \<nu> = (\<forall>N\<in>null_sets. \<nu> N = (0 :: pinfreal))"
@@ -111,6 +113,14 @@
finally show "\<nu> N = 0" .
qed
+lemma (in measure_space) density_is_absolutely_continuous:
+ assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
+ shows "absolutely_continuous \<nu>"
+ using assms unfolding absolutely_continuous_def
+ by (simp add: positive_integral_null_set)
+
+subsection "Existence of the Radon-Nikodym derivative"
+
lemma (in finite_measure) Radon_Nikodym_aux_epsilon:
fixes e :: real assumes "0 < e"
assumes "finite_measure M s"
@@ -120,21 +130,17 @@
proof -
let "?d A" = "real (\<mu> A) - real (s A)"
interpret M': finite_measure M s by fact
-
let "?A A" = "if (\<forall>B\<in>sets M. B \<subseteq> space M - A \<longrightarrow> -e < ?d B)
then {}
else (SOME B. B \<in> sets M \<and> B \<subseteq> space M - A \<and> ?d B \<le> -e)"
def A \<equiv> "\<lambda>n. ((\<lambda>B. B \<union> ?A B) ^^ n) {}"
-
have A_simps[simp]:
"A 0 = {}"
"\<And>n. A (Suc n) = (A n \<union> ?A (A n))" unfolding A_def by simp_all
-
{ fix A assume "A \<in> sets M"
have "?A A \<in> sets M"
by (auto intro!: someI2[of _ _ "\<lambda>A. A \<in> sets M"] simp: not_less) }
note A'_in_sets = this
-
{ fix n have "A n \<in> sets M"
proof (induct n)
case (Suc n) thus "A (Suc n) \<in> sets M"
@@ -142,7 +148,6 @@
qed (simp add: A_def) }
note A_in_sets = this
hence "range A \<subseteq> sets M" by auto
-
{ fix n B
assume Ex: "\<exists>B. B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> -e"
hence False: "\<not> (\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B)" by (auto simp: not_less)
@@ -156,7 +161,6 @@
finally show "?d (A n \<union> B) \<le> ?d (A n) - e" .
qed }
note dA_epsilon = this
-
{ fix n have "?d (A (Suc n)) \<le> ?d (A n)"
proof (cases "\<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e")
case True from dA_epsilon[OF this] show ?thesis using `0 < e` by simp
@@ -166,7 +170,6 @@
thus ?thesis by simp
qed }
note dA_mono = this
-
show ?thesis
proof (cases "\<exists>n. \<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B")
case True then obtain n where B: "\<And>B. \<lbrakk> B \<in> sets M; B \<subseteq> space M - A n\<rbrakk> \<Longrightarrow> -e < ?d B" by blast
@@ -220,11 +223,8 @@
proof -
let "?d A" = "real (\<mu> A) - real (s A)"
let "?P A B n" = "A \<in> sets M \<and> A \<subseteq> B \<and> ?d B \<le> ?d A \<and> (\<forall>C\<in>sets M. C \<subseteq> A \<longrightarrow> - 1 / real (Suc n) < ?d C)"
-
interpret M': finite_measure M s by fact
-
let "?r S" = "restricted_space S"
-
{ fix S n
assume S: "S \<in> sets M"
hence **: "\<And>X. X \<in> op \<inter> S ` sets M \<longleftrightarrow> X \<in> sets M \<and> X \<subseteq> S" by auto
@@ -242,11 +242,9 @@
qed
hence "\<exists>A. ?P A S n" by auto }
note Ex_P = this
-
def A \<equiv> "nat_rec (space M) (\<lambda>n A. SOME B. ?P B A n)"
have A_Suc: "\<And>n. A (Suc n) = (SOME B. ?P B (A n) n)" by (simp add: A_def)
have A_0[simp]: "A 0 = space M" unfolding A_def by simp
-
{ fix i have "A i \<in> sets M" unfolding A_def
proof (induct i)
case (Suc i)
@@ -254,19 +252,15 @@
by (rule someI2_ex) simp
qed simp }
note A_in_sets = this
-
{ fix n have "?P (A (Suc n)) (A n) n"
using Ex_P[OF A_in_sets] unfolding A_Suc
by (rule someI2_ex) simp }
note P_A = this
-
have "range A \<subseteq> sets M" using A_in_sets by auto
-
have A_mono: "\<And>i. A (Suc i) \<subseteq> A i" using P_A by simp
have mono_dA: "mono (\<lambda>i. ?d (A i))" using P_A by (simp add: mono_iff_le_Suc)
have epsilon: "\<And>C i. \<lbrakk> C \<in> sets M; C \<subseteq> A (Suc i) \<rbrakk> \<Longrightarrow> - 1 / real (Suc i) < ?d C"
using P_A by auto
-
show ?thesis
proof (safe intro!: bexI[of _ "\<Inter>i. A i"])
show "(\<Inter>i. A i) \<in> sets M" using A_in_sets by auto
@@ -298,24 +292,19 @@
shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
proof -
interpret M': finite_measure M \<nu> using assms(1) .
-
def G \<equiv> "{g \<in> borel_measurable M. \<forall>A\<in>sets M. positive_integral (\<lambda>x. g x * indicator A x) \<le> \<nu> A}"
have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto
hence "G \<noteq> {}" by auto
-
{ fix f g assume f: "f \<in> G" and g: "g \<in> G"
have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def
proof safe
show "?max \<in> borel_measurable M" using f g unfolding G_def by auto
-
let ?A = "{x \<in> space M. f x \<le> g x}"
have "?A \<in> sets M" using f g unfolding G_def by auto
-
fix A assume "A \<in> sets M"
hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using `?A \<in> sets M` by auto
have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A"
using sets_into_space[OF `A \<in> sets M`] by auto
-
have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x =
g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x"
by (auto simp: indicator_def max_def)
@@ -331,14 +320,12 @@
finally show "positive_integral (\<lambda>x. max (g x) (f x) * indicator A x) \<le> \<nu> A" .
qed }
note max_in_G = this
-
{ fix f g assume "f \<up> g" and f: "\<And>i. f i \<in> G"
have "g \<in> G" unfolding G_def
proof safe
from `f \<up> g` have [simp]: "g = (SUP i. f i)" unfolding isoton_def by simp
have f_borel: "\<And>i. f i \<in> borel_measurable M" using f unfolding G_def by simp
thus "g \<in> borel_measurable M" by (auto intro!: borel_measurable_SUP)
-
fix A assume "A \<in> sets M"
hence "\<And>i. (\<lambda>x. f i x * indicator A x) \<in> borel_measurable M"
using f_borel by (auto intro!: borel_measurable_indicator)
@@ -350,7 +337,6 @@
using f `A \<in> sets M` unfolding G_def by (auto intro!: SUP_leI)
qed }
note SUP_in_G = this
-
let ?y = "SUP g : G. positive_integral g"
have "?y \<le> \<nu> (space M)" unfolding G_def
proof (safe intro!: SUP_leI)
@@ -385,7 +371,6 @@
hence isoton_g: "?g \<up> f" by (simp add: isoton_def le_fun_def f_def)
from SUP_in_G[OF this g_in_G] have "f \<in> G" .
hence [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto
-
have "(\<lambda>i. positive_integral (?g i)) \<up> positive_integral f"
using isoton_g g_in_G by (auto intro!: positive_integral_isoton simp: G_def f_def)
hence "positive_integral f = (SUP i. positive_integral (?g i))"
@@ -398,9 +383,7 @@
by (auto intro!: SUP_mono positive_integral_mono Max_ge)
qed
finally have int_f_eq_y: "positive_integral f = ?y" .
-
let "?t A" = "\<nu> A - positive_integral (\<lambda>x. f x * indicator A x)"
-
have "finite_measure M ?t"
proof
show "?t {} = 0" by simp
@@ -435,9 +418,7 @@
qed
qed
then interpret M: finite_measure M ?t .
-
have ac: "absolutely_continuous ?t" using `absolutely_continuous \<nu>` unfolding absolutely_continuous_def by auto
-
have upper_bound: "\<forall>A\<in>sets M. ?t A \<le> 0"
proof (rule ccontr)
assume "\<not> ?thesis"
@@ -460,7 +441,6 @@
ultimately have b: "b \<noteq> 0" "b \<noteq> \<omega>"
using M'.finite_measure_of_space
by (auto simp: pinfreal_inverse_eq_0 finite_measure_of_space)
-
have "finite_measure M (\<lambda>A. b * \<mu> A)" (is "finite_measure M ?b")
proof
show "?b {} = 0" by simp
@@ -469,7 +449,6 @@
unfolding countably_additive_def psuminf_cmult_right
using measure_countably_additive by auto
qed
-
from M.Radon_Nikodym_aux[OF this]
obtain A0 where "A0 \<in> sets M" and
space_less_A0: "real (?t (space M)) - real (b * \<mu> (space M)) \<le> real (?t A0) - real (b * \<mu> A0)" and
@@ -479,9 +458,7 @@
using M'.finite_measure b finite_measure
by (cases "b * \<mu> B", cases "?t B") (auto simp: field_simps) }
note bM_le_t = this
-
let "?f0 x" = "f x + b * indicator A0 x"
-
{ fix A assume A: "A \<in> sets M"
hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
have "positive_integral (\<lambda>x. ?f0 x * indicator A x) =
@@ -492,7 +469,6 @@
using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A
by (simp add: borel_measurable_indicator positive_integral_add positive_integral_cmult_indicator) }
note f0_eq = this
-
{ fix A assume A: "A \<in> sets M"
hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
have f_le_v: "positive_integral (\<lambda>x. f x * indicator A x) \<le> \<nu> A"
@@ -511,18 +487,15 @@
finally have "positive_integral (\<lambda>x. ?f0 x * indicator A x) \<le> \<nu> A" . }
hence "?f0 \<in> G" using `A0 \<in> sets M` unfolding G_def
by (auto intro!: borel_measurable_indicator borel_measurable_pinfreal_add borel_measurable_pinfreal_times)
-
have real: "?t (space M) \<noteq> \<omega>" "?t A0 \<noteq> \<omega>"
"b * \<mu> (space M) \<noteq> \<omega>" "b * \<mu> A0 \<noteq> \<omega>"
using `A0 \<in> sets M` b
finite_measure[of A0] M.finite_measure[of A0]
finite_measure_of_space M.finite_measure_of_space
by auto
-
have int_f_finite: "positive_integral f \<noteq> \<omega>"
using M'.finite_measure_of_space pos_t unfolding pinfreal_zero_less_diff_iff
by (auto cong: positive_integral_cong)
-
have "?t (space M) > b * \<mu> (space M)" unfolding b_def
apply (simp add: field_simps)
apply (subst mult_assoc[symmetric])
@@ -539,18 +512,15 @@
hence "0 < \<mu> A0" using ac unfolding absolutely_continuous_def
using `A0 \<in> sets M` by auto
hence "0 < b * \<mu> A0" using b by auto
-
from int_f_finite this
have "?y + 0 < positive_integral f + b * \<mu> A0" unfolding int_f_eq_y
by (rule pinfreal_less_add)
also have "\<dots> = positive_integral ?f0" using f0_eq[OF top] `A0 \<in> sets M` sets_into_space
by (simp cong: positive_integral_cong)
finally have "?y < positive_integral ?f0" by simp
-
moreover from `?f0 \<in> G` have "positive_integral ?f0 \<le> ?y" by (auto intro!: le_SUPI)
ultimately show False by auto
qed
-
show ?thesis
proof (safe intro!: bexI[of _ f])
fix A assume "A\<in>sets M"
@@ -575,10 +545,8 @@
interpret v: measure_space M \<nu> by fact
let ?Q = "{Q\<in>sets M. \<nu> Q \<noteq> \<omega>}"
let ?a = "SUP Q:?Q. \<mu> Q"
-
have "{} \<in> ?Q" using v.empty_measure by auto
then have Q_not_empty: "?Q \<noteq> {}" by blast
-
have "?a \<le> \<mu> (space M)" using sets_into_space
by (auto intro!: SUP_leI measure_mono top)
then have "?a \<noteq> \<omega>" using finite_measure_of_space
@@ -596,9 +564,7 @@
show "range ?O \<subseteq> sets M" using Q' by (auto intro!: finite_UN)
show "\<And>i. ?O i \<subseteq> ?O (Suc i)" by fastsimp
qed
-
have Q'_sets: "\<And>i. Q' i \<in> sets M" using Q' by auto
-
have O_sets: "\<And>i. ?O i \<in> sets M"
using Q' by (auto intro!: finite_UN Un)
then have O_in_G: "\<And>i. ?O i \<in> ?Q"
@@ -611,7 +577,6 @@
finally show "\<nu> (?O i) \<noteq> \<omega>" unfolding pinfreal_less_\<omega> by auto
qed auto
have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastsimp
-
have a_eq: "?a = \<mu> (\<Union>i. ?O i)" unfolding Union[symmetric]
proof (rule antisym)
show "?a \<le> (SUP i. \<mu> (?O i))" unfolding a_Lim
@@ -625,14 +590,11 @@
using O_in_G[of i] by (auto intro!: exI[of _ "?O i"])
qed
qed
-
let "?O_0" = "(\<Union>i. ?O i)"
have "?O_0 \<in> sets M" using Q' by auto
-
def Q \<equiv> "\<lambda>i. case i of 0 \<Rightarrow> Q' 0 | Suc n \<Rightarrow> ?O (Suc n) - ?O n"
{ fix i have "Q i \<in> sets M" unfolding Q_def using Q'[of 0] by (cases i) (auto intro: O_sets) }
note Q_sets = this
-
show ?thesis
proof (intro bexI exI conjI ballI impI allI)
show "disjoint_family Q"
@@ -640,7 +602,6 @@
split: nat.split_asm)
show "range Q \<subseteq> sets M"
using Q_sets by auto
-
{ fix A assume A: "A \<in> sets M" "A \<subseteq> space M - ?O_0"
show "\<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<omega>"
proof (rule disjCI, simp)
@@ -677,7 +638,6 @@
with `\<mu> A \<noteq> 0` show ?thesis by auto
qed
qed }
-
{ fix i show "\<nu> (Q i) \<noteq> \<omega>"
proof (cases i)
case 0 then show ?thesis
@@ -688,9 +648,7 @@
using `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q` O_mono
using v.measure_Diff[of "?O n" "?O (Suc n)"] by auto
qed }
-
show "space M - ?O_0 \<in> sets M" using Q'_sets by auto
-
{ fix j have "(\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q i)"
proof (induct j)
case 0 then show ?case by (simp add: Q_def)
@@ -713,7 +671,6 @@
shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
proof -
interpret v: measure_space M \<nu> by fact
-
from split_space_into_finite_sets_and_rest[OF assms]
obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
@@ -721,7 +678,6 @@
and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<omega>"
and Q_fin: "\<And>i. \<nu> (Q i) \<noteq> \<omega>" by force
from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
-
have "\<forall>i. \<exists>f. f\<in>borel_measurable M \<and> (\<forall>A\<in>sets M.
\<nu> (Q i \<inter> A) = positive_integral (\<lambda>x. f x * indicator (Q i \<inter> A) x))"
proof
@@ -729,7 +685,6 @@
have indicator_eq: "\<And>f x A. (f x :: pinfreal) * indicator (Q i \<inter> A) x * indicator (Q i) x
= (f x * indicator (Q i) x) * indicator A x"
unfolding indicator_def by auto
-
have fm: "finite_measure (restricted_space (Q i)) \<mu>"
(is "finite_measure ?R \<mu>") by (rule restricted_finite_measure[OF Q_sets[of i]])
then interpret R: finite_measure ?R .
@@ -843,12 +798,6 @@
section "Uniqueness of densities"
-lemma (in measure_space) density_is_absolutely_continuous:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
- shows "absolutely_continuous \<nu>"
- using assms unfolding absolutely_continuous_def
- by (simp add: positive_integral_null_set)
-
lemma (in measure_space) finite_density_unique:
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
and fin: "positive_integral f < \<omega>"
@@ -973,19 +922,16 @@
using h_borel by (rule measure_space_density)
interpret h: finite_measure M "\<lambda>A. positive_integral (\<lambda>x. h x * indicator A x)"
by default (simp cong: positive_integral_cong add: fin)
-
interpret f: measure_space M "\<lambda>A. positive_integral (\<lambda>x. f x * indicator A x)"
using borel(1) by (rule measure_space_density)
interpret f': measure_space M "\<lambda>A. positive_integral (\<lambda>x. f' x * indicator A x)"
using borel(2) by (rule measure_space_density)
-
{ fix A assume "A \<in> sets M"
then have " {x \<in> space M. h x \<noteq> 0 \<and> indicator A x \<noteq> (0::pinfreal)} = A"
using pos sets_into_space by (force simp: indicator_def)
then have "positive_integral (\<lambda>xa. h xa * indicator A xa) = 0 \<longleftrightarrow> A \<in> null_sets"
using h_borel `A \<in> sets M` by (simp add: positive_integral_0_iff) }
note h_null_sets = this
-
{ fix A assume "A \<in> sets M"
have "positive_integral (\<lambda>x. h x * (f x * indicator A x)) =
f.positive_integral (\<lambda>x. h x * indicator A x)"
@@ -1101,7 +1047,7 @@
qed
qed
-section "Radon Nikodym derivative"
+section "Radon-Nikodym derivative"
definition (in sigma_finite_measure)
"RN_deriv \<nu> \<equiv> SOME f. f \<in> borel_measurable M \<and>
--- a/src/HOL/Probability/Sigma_Algebra.thy Thu Dec 02 14:56:16 2010 +0100
+++ b/src/HOL/Probability/Sigma_Algebra.thy Thu Dec 02 15:32:48 2010 +0100
@@ -397,6 +397,18 @@
by (auto intro: sigma_sets.Empty sigma_sets_top)
qed
+lemma (in sigma_algebra) sets_sigma_subset:
+ assumes "space N = space M"
+ assumes "sets N \<subseteq> sets M"
+ shows "sets (sigma N) \<subseteq> sets M"
+ by (unfold assms sets_sigma, rule sigma_sets_subset, rule assms)
+
+lemma in_sigma[intro, simp]: "A \<in> sets M \<Longrightarrow> A \<in> sets (sigma M)"
+ unfolding sigma_def by (auto intro!: sigma_sets.Basic)
+
+lemma (in sigma_algebra) sigma_eq[simp]: "sigma M = M"
+ unfolding sigma_def sigma_sets_eq by simp
+
section {* Measurable functions *}
definition
@@ -859,6 +871,22 @@
qed
qed
+lemma vimage_algebra_sigma:
+ assumes E: "sets E \<subseteq> Pow (space E)"
+ and f: "f \<in> space F \<rightarrow> space E"
+ and "\<And>A. A \<in> sets F \<Longrightarrow> A \<in> (\<lambda>X. f -` X \<inter> space F) ` sets E"
+ and "\<And>A. A \<in> sets E \<Longrightarrow> f -` A \<inter> space F \<in> sets F"
+ shows "sigma_algebra.vimage_algebra (sigma E) (space F) f = sigma F"
+proof -
+ interpret sigma_algebra "sigma E"
+ using assms by (intro sigma_algebra_sigma) auto
+ have eq: "sets F = (\<lambda>X. f -` X \<inter> space F) ` sets E"
+ using assms by auto
+ show "vimage_algebra (space F) f = sigma F"
+ unfolding vimage_algebra_def using assms
+ by (simp add: sigma_def eq sigma_sets_vimage)
+qed
+
section {* Conditional space *}
definition (in algebra)
@@ -1149,7 +1177,6 @@
section {* Dynkin systems *}
-
locale dynkin_system =
fixes M :: "'a algebra"
assumes space_closed: "sets M \<subseteq> Pow (space M)"
--- a/src/HOL/Set.thy Thu Dec 02 14:56:16 2010 +0100
+++ b/src/HOL/Set.thy Thu Dec 02 15:32:48 2010 +0100
@@ -882,7 +882,6 @@
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
by blast
-
subsubsection {* Some rules with @{text "if"} *}
text{* Elimination of @{text"{x. \<dots> & x=t & \<dots>}"}. *}