--- a/src/HOL/Probability/Radon_Nikodym.thy Wed Feb 02 10:35:41 2011 +0100
+++ b/src/HOL/Probability/Radon_Nikodym.thy Wed Feb 02 12:34:45 2011 +0100
@@ -11,7 +11,7 @@
qed auto
lemma (in sigma_finite_measure) Ex_finite_integrable_function:
- shows "\<exists>h\<in>borel_measurable M. positive_integral h \<noteq> \<omega> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<omega>)"
+ shows "\<exists>h\<in>borel_measurable M. integral\<^isup>P M h \<noteq> \<omega> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<omega>)"
proof -
obtain A :: "nat \<Rightarrow> 'a set" where
range: "range A \<subseteq> sets M" and
@@ -42,7 +42,7 @@
proof (safe intro!: bexI[of _ ?h] del: notI)
have "\<And>i. A i \<in> sets M"
using range by fastsimp+
- then have "positive_integral ?h = (\<Sum>\<^isub>\<infinity> i. n i * \<mu> (A i))"
+ then have "integral\<^isup>P M ?h = (\<Sum>\<^isub>\<infinity> i. n i * \<mu> (A i))"
by (simp add: positive_integral_psuminf positive_integral_cmult_indicator)
also have "\<dots> \<le> (\<Sum>\<^isub>\<infinity> i. Real ((1 / 2)^Suc i))"
proof (rule psuminf_le)
@@ -56,7 +56,7 @@
qed
also have "\<dots> = Real 1"
by (rule suminf_imp_psuminf, rule power_half_series, auto)
- finally show "positive_integral ?h \<noteq> \<omega>" by auto
+ finally show "integral\<^isup>P M ?h \<noteq> \<omega>" by auto
next
fix x assume "x \<in> space M"
then obtain i where "x \<in> A i" using space[symmetric] by auto
@@ -75,46 +75,47 @@
"absolutely_continuous \<nu> = (\<forall>N\<in>null_sets. \<nu> N = (0 :: pextreal))"
lemma (in sigma_finite_measure) absolutely_continuous_AE:
- assumes "measure_space M \<nu>" "absolutely_continuous \<nu>" "AE x. P x"
- shows "measure_space.almost_everywhere M \<nu> P"
+ assumes "measure_space M'" and [simp]: "sets M' = sets M" "space M' = space M"
+ and "absolutely_continuous (measure M')" "AE x. P x"
+ shows "measure_space.almost_everywhere M' P"
proof -
- interpret \<nu>: measure_space M \<nu> by fact
+ interpret \<nu>: measure_space M' by fact
from `AE x. P x` obtain N where N: "N \<in> null_sets" and "{x\<in>space M. \<not> P x} \<subseteq> N"
unfolding almost_everywhere_def by auto
show "\<nu>.almost_everywhere P"
proof (rule \<nu>.AE_I')
- show "{x\<in>space M. \<not> P x} \<subseteq> N" by fact
- from `absolutely_continuous \<nu>` show "N \<in> \<nu>.null_sets"
+ show "{x\<in>space M'. \<not> P x} \<subseteq> N" by simp fact
+ from `absolutely_continuous (measure M')` show "N \<in> \<nu>.null_sets"
using N unfolding absolutely_continuous_def by auto
qed
qed
lemma (in finite_measure_space) absolutely_continuousI:
- assumes "finite_measure_space M \<nu>"
+ assumes "finite_measure_space (M\<lparr> measure := \<nu>\<rparr>)" (is "finite_measure_space ?\<nu>")
assumes v: "\<And>x. \<lbrakk> x \<in> space M ; \<mu> {x} = 0 \<rbrakk> \<Longrightarrow> \<nu> {x} = 0"
shows "absolutely_continuous \<nu>"
proof (unfold absolutely_continuous_def sets_eq_Pow, safe)
fix N assume "\<mu> N = 0" "N \<subseteq> space M"
- interpret v: finite_measure_space M \<nu> by fact
- have "\<nu> N = \<nu> (\<Union>x\<in>N. {x})" by simp
- also have "\<dots> = (\<Sum>x\<in>N. \<nu> {x})"
+ interpret v: finite_measure_space ?\<nu> by fact
+ have "\<nu> N = measure ?\<nu> (\<Union>x\<in>N. {x})" by simp
+ also have "\<dots> = (\<Sum>x\<in>N. measure ?\<nu> {x})"
proof (rule v.measure_finitely_additive''[symmetric])
show "finite N" using `N \<subseteq> space M` finite_space by (auto intro: finite_subset)
show "disjoint_family_on (\<lambda>i. {i}) N" unfolding disjoint_family_on_def by auto
- fix x assume "x \<in> N" thus "{x} \<in> sets M" using `N \<subseteq> space M` sets_eq_Pow by auto
+ fix x assume "x \<in> N" thus "{x} \<in> sets ?\<nu>" using `N \<subseteq> space M` sets_eq_Pow by auto
qed
also have "\<dots> = 0"
proof (safe intro!: setsum_0')
fix x assume "x \<in> N"
hence "\<mu> {x} \<le> \<mu> N" using sets_eq_Pow `N \<subseteq> space M` by (auto intro!: measure_mono)
hence "\<mu> {x} = 0" using `\<mu> N = 0` by simp
- thus "\<nu> {x} = 0" using v[of x] `x \<in> N` `N \<subseteq> space M` by auto
+ thus "measure ?\<nu> {x} = 0" using v[of x] `x \<in> N` `N \<subseteq> space M` by auto
qed
- finally show "\<nu> N = 0" .
+ finally show "\<nu> N = 0" by simp
qed
lemma (in measure_space) density_is_absolutely_continuous:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
+ assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
shows "absolutely_continuous \<nu>"
using assms unfolding absolutely_continuous_def
by (simp add: positive_integral_null_set)
@@ -123,13 +124,13 @@
lemma (in finite_measure) Radon_Nikodym_aux_epsilon:
fixes e :: real assumes "0 < e"
- assumes "finite_measure M s"
- shows "\<exists>A\<in>sets M. real (\<mu> (space M)) - real (s (space M)) \<le>
- real (\<mu> A) - real (s A) \<and>
- (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < real (\<mu> B) - real (s B))"
+ assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)"
+ shows "\<exists>A\<in>sets M. real (\<mu> (space M)) - real (\<nu> (space M)) \<le>
+ real (\<mu> A) - real (\<nu> A) \<and>
+ (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < real (\<mu> B) - real (\<nu> B))"
proof -
- let "?d A" = "real (\<mu> A) - real (s A)"
- interpret M': finite_measure M s by fact
+ let "?d A" = "real (\<mu> A) - real (\<nu> A)"
+ interpret M': finite_measure "M\<lparr>measure := \<nu>\<rparr>" 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)"
@@ -216,21 +217,24 @@
qed
lemma (in finite_measure) Radon_Nikodym_aux:
- assumes "finite_measure M s"
- shows "\<exists>A\<in>sets M. real (\<mu> (space M)) - real (s (space M)) \<le>
- real (\<mu> A) - real (s A) \<and>
- (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> real (\<mu> B) - real (s B))"
+ assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
+ shows "\<exists>A\<in>sets M. real (\<mu> (space M)) - real (\<nu> (space M)) \<le>
+ real (\<mu> A) - real (\<nu> A) \<and>
+ (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> real (\<mu> B) - real (\<nu> B))"
proof -
- let "?d A" = "real (\<mu> A) - real (s A)"
+ let "?d A" = "real (\<mu> A) - real (\<nu> 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
+ interpret M': finite_measure ?M' where
+ "space ?M' = space M" and "sets ?M' = sets M" and "measure ?M' = \<nu>" by fact auto
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
- from M'.restricted_finite_measure[of S] restricted_finite_measure[of S] S
- have "finite_measure (?r S) \<mu>" "0 < 1 / real (Suc n)"
- "finite_measure (?r S) s" by auto
+ have [simp]: "(restricted_space S\<lparr>measure := \<nu>\<rparr>) = M'.restricted_space S"
+ by (cases M) simp
+ from M'.restricted_finite_measure[of S] restricted_finite_measure[OF S] S
+ have "finite_measure (?r S)" "0 < 1 / real (Suc n)"
+ "finite_measure (?r S\<lparr>measure := \<nu>\<rparr>)" by auto
from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X ..
hence "?P X S n"
proof (simp add: **, safe)
@@ -287,12 +291,14 @@
qed
lemma (in finite_measure) Radon_Nikodym_finite_measure:
- assumes "finite_measure M \<nu>"
+ assumes "finite_measure (M\<lparr> measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
assumes "absolutely_continuous \<nu>"
- shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
+ shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
proof -
- interpret M': finite_measure M \<nu> using assms(1) .
- def G \<equiv> "{g \<in> borel_measurable M. \<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x) \<le> \<nu> A}"
+ interpret M': finite_measure ?M'
+ where "space ?M' = space M" and "sets ?M' = sets M" and "measure ?M' = \<nu>"
+ using assms(1) by auto
+ def G \<equiv> "{g \<in> borel_measurable M. \<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<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"
@@ -308,16 +314,16 @@
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)
- hence "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x) =
- (\<integral>\<^isup>+x. g x * indicator (?A \<inter> A) x) +
- (\<integral>\<^isup>+x. f x * indicator ((space M - ?A) \<inter> A) x)"
+ hence "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) =
+ (\<integral>\<^isup>+x. g x * indicator (?A \<inter> A) x \<partial>M) +
+ (\<integral>\<^isup>+x. f x * indicator ((space M - ?A) \<inter> A) x \<partial>M)"
using f g sets unfolding G_def
by (auto cong: positive_integral_cong intro!: positive_integral_add borel_measurable_indicator)
also have "\<dots> \<le> \<nu> (?A \<inter> A) + \<nu> ((space M - ?A) \<inter> A)"
using f g sets unfolding G_def by (auto intro!: add_mono)
also have "\<dots> = \<nu> A"
using M'.measure_additive[OF sets] union by auto
- finally show "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x) \<le> \<nu> A" .
+ finally show "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> \<nu> A" .
qed }
note max_in_G = this
{ fix f g assume "f \<up> g" and f: "\<And>i. f i \<in> G"
@@ -331,30 +337,30 @@
hence "\<And>i. (\<lambda>x. f i x * indicator A x) \<in> borel_measurable M"
using f_borel by (auto intro!: borel_measurable_indicator)
from positive_integral_isoton[OF isoton_indicator[OF `f \<up> g`] this]
- have SUP: "(\<integral>\<^isup>+x. g x * indicator A x) =
- (SUP i. (\<integral>\<^isup>+x. f i x * indicator A x))"
+ have SUP: "(\<integral>\<^isup>+x. g x * indicator A x \<partial>M) =
+ (SUP i. (\<integral>\<^isup>+x. f i x * indicator A x \<partial>M))"
unfolding isoton_def by simp
- show "(\<integral>\<^isup>+x. g x * indicator A x) \<le> \<nu> A" unfolding SUP
+ show "(\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A" unfolding SUP
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"
+ let ?y = "SUP g : G. integral\<^isup>P M g"
have "?y \<le> \<nu> (space M)" unfolding G_def
proof (safe intro!: SUP_leI)
- fix g assume "\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x) \<le> \<nu> A"
- from this[THEN bspec, OF top] show "positive_integral g \<le> \<nu> (space M)"
+ fix g assume "\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A"
+ from this[THEN bspec, OF top] show "integral\<^isup>P M g \<le> \<nu> (space M)"
by (simp cong: positive_integral_cong)
qed
hence "?y \<noteq> \<omega>" using M'.finite_measure_of_space by auto
from SUPR_countable_SUPR[OF this `G \<noteq> {}`] guess ys .. note ys = this
- hence "\<forall>n. \<exists>g. g\<in>G \<and> positive_integral g = ys n"
+ hence "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n"
proof safe
- fix n assume "range ys \<subseteq> positive_integral ` G"
- hence "ys n \<in> positive_integral ` G" by auto
- thus "\<exists>g. g\<in>G \<and> positive_integral g = ys n" by auto
+ fix n assume "range ys \<subseteq> integral\<^isup>P M ` G"
+ hence "ys n \<in> integral\<^isup>P M ` G" by auto
+ thus "\<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n" by auto
qed
- from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. positive_integral (gs n) = ys n" by auto
- hence y_eq: "?y = (SUP i. positive_integral (gs i))" using ys by auto
+ from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. integral\<^isup>P M (gs n) = ys n" by auto
+ hence y_eq: "?y = (SUP i. integral\<^isup>P M (gs i))" using ys by auto
let "?g i x" = "Max ((\<lambda>n. gs n x) ` {..i})"
def f \<equiv> "SUP i. ?g i"
have gs_not_empty: "\<And>i. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto
@@ -372,53 +378,53 @@
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"
+ have "(\<lambda>i. integral\<^isup>P M (?g i)) \<up> integral\<^isup>P M 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))"
+ hence "integral\<^isup>P M f = (SUP i. integral\<^isup>P M (?g i))"
unfolding isoton_def by simp
also have "\<dots> = ?y"
proof (rule antisym)
- show "(SUP i. positive_integral (?g i)) \<le> ?y"
+ show "(SUP i. integral\<^isup>P M (?g i)) \<le> ?y"
using g_in_G by (auto intro!: exI Sup_mono simp: SUPR_def)
- show "?y \<le> (SUP i. positive_integral (?g i))" unfolding y_eq
+ show "?y \<le> (SUP i. integral\<^isup>P M (?g i))" unfolding y_eq
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 - (\<integral>\<^isup>+x. f x * indicator A x)"
- have "finite_measure M ?t"
- proof
- show "?t {} = 0" by simp
- show "?t (space M) \<noteq> \<omega>" using M'.finite_measure by simp
- show "countably_additive M ?t" unfolding countably_additive_def
- proof safe
- fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "disjoint_family A"
- have "(\<Sum>\<^isub>\<infinity> n. (\<integral>\<^isup>+x. f x * indicator (A n) x))
- = (\<integral>\<^isup>+x. (\<Sum>\<^isub>\<infinity>n. f x * indicator (A n) x))"
- using `range A \<subseteq> sets M`
- by (rule_tac positive_integral_psuminf[symmetric]) (auto intro!: borel_measurable_indicator)
- also have "\<dots> = (\<integral>\<^isup>+x. f x * indicator (\<Union>n. A n) x)"
- apply (rule positive_integral_cong)
- apply (subst psuminf_cmult_right)
- unfolding psuminf_indicator[OF `disjoint_family A`] ..
- finally have "(\<Sum>\<^isub>\<infinity> n. (\<integral>\<^isup>+x. f x * indicator (A n) x))
- = (\<integral>\<^isup>+x. f x * indicator (\<Union>n. A n) x)" .
- moreover have "(\<Sum>\<^isub>\<infinity>n. \<nu> (A n)) = \<nu> (\<Union>n. A n)"
- using M'.measure_countably_additive A by (simp add: comp_def)
- moreover have "\<And>i. (\<integral>\<^isup>+x. f x * indicator (A i) x) \<le> \<nu> (A i)"
- using A `f \<in> G` unfolding G_def by auto
- moreover have v_fin: "\<nu> (\<Union>i. A i) \<noteq> \<omega>" using M'.finite_measure A by (simp add: countable_UN)
- moreover {
- have "(\<integral>\<^isup>+x. f x * indicator (\<Union>i. A i) x) \<le> \<nu> (\<Union>i. A i)"
- using A `f \<in> G` unfolding G_def by (auto simp: countable_UN)
- also have "\<nu> (\<Union>i. A i) < \<omega>" using v_fin by (simp add: pextreal_less_\<omega>)
- finally have "(\<integral>\<^isup>+x. f x * indicator (\<Union>i. A i) x) \<noteq> \<omega>"
- by (simp add: pextreal_less_\<omega>) }
- ultimately
- show "(\<Sum>\<^isub>\<infinity> n. ?t (A n)) = ?t (\<Union>i. A i)"
- apply (subst psuminf_minus) by simp_all
- qed
+ finally have int_f_eq_y: "integral\<^isup>P M f = ?y" .
+ let "?t A" = "\<nu> A - (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
+ let ?M = "M\<lparr> measure := ?t\<rparr>"
+ interpret M: sigma_algebra ?M
+ by (intro sigma_algebra_cong) auto
+ have fmM: "finite_measure ?M"
+ proof (default, simp_all add: countably_additive_def, safe)
+ fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "disjoint_family A"
+ have "(\<Sum>\<^isub>\<infinity> n. (\<integral>\<^isup>+x. f x * indicator (A n) x \<partial>M))
+ = (\<integral>\<^isup>+x. (\<Sum>\<^isub>\<infinity>n. f x * indicator (A n) x) \<partial>M)"
+ using `range A \<subseteq> sets M`
+ by (rule_tac positive_integral_psuminf[symmetric]) (auto intro!: borel_measurable_indicator)
+ also have "\<dots> = (\<integral>\<^isup>+x. f x * indicator (\<Union>n. A n) x \<partial>M)"
+ apply (rule positive_integral_cong)
+ apply (subst psuminf_cmult_right)
+ unfolding psuminf_indicator[OF `disjoint_family A`] ..
+ finally have "(\<Sum>\<^isub>\<infinity> n. (\<integral>\<^isup>+x. f x * indicator (A n) x \<partial>M))
+ = (\<integral>\<^isup>+x. f x * indicator (\<Union>n. A n) x \<partial>M)" .
+ moreover have "(\<Sum>\<^isub>\<infinity>n. \<nu> (A n)) = \<nu> (\<Union>n. A n)"
+ using M'.measure_countably_additive A by (simp add: comp_def)
+ moreover have "\<And>i. (\<integral>\<^isup>+x. f x * indicator (A i) x \<partial>M) \<le> \<nu> (A i)"
+ using A `f \<in> G` unfolding G_def by auto
+ moreover have v_fin: "\<nu> (\<Union>i. A i) \<noteq> \<omega>" using M'.finite_measure A by (simp add: countable_UN)
+ moreover {
+ have "(\<integral>\<^isup>+x. f x * indicator (\<Union>i. A i) x \<partial>M) \<le> \<nu> (\<Union>i. A i)"
+ using A `f \<in> G` unfolding G_def by (auto simp: countable_UN)
+ also have "\<nu> (\<Union>i. A i) < \<omega>" using v_fin by (simp add: pextreal_less_\<omega>)
+ finally have "(\<integral>\<^isup>+x. f x * indicator (\<Union>i. A i) x \<partial>M) \<noteq> \<omega>"
+ by (simp add: pextreal_less_\<omega>) }
+ ultimately
+ show "(\<Sum>\<^isub>\<infinity> n. ?t (A n)) = ?t (\<Union>i. A i)"
+ apply (subst psuminf_minus) by simp_all
qed
- then interpret M: finite_measure M ?t .
+ then interpret M: finite_measure ?M
+ where "space ?M = space M" and "sets ?M = sets M" and "measure ?M = ?t"
+ by (simp_all add: fmM)
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)
@@ -433,23 +439,21 @@
hence pos_M: "0 < \<mu> (space M)"
using ac top unfolding absolutely_continuous_def by auto
moreover
- have "(\<integral>\<^isup>+x. f x * indicator (space M) x) \<le> \<nu> (space M)"
+ have "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<le> \<nu> (space M)"
using `f \<in> G` unfolding G_def by auto
- hence "(\<integral>\<^isup>+x. f x * indicator (space M) x) \<noteq> \<omega>"
+ hence "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<noteq> \<omega>"
using M'.finite_measure_of_space by auto
moreover
def b \<equiv> "?t (space M) / \<mu> (space M) / 2"
ultimately have b: "b \<noteq> 0" "b \<noteq> \<omega>"
using M'.finite_measure_of_space
by (auto simp: pextreal_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
- show "?b (space M) \<noteq> \<omega>" using finite_measure_of_space b by auto
- show "countably_additive M ?b"
- unfolding countably_additive_def psuminf_cmult_right
- using measure_countably_additive by auto
- qed
+ let ?Mb = "?M\<lparr>measure := \<lambda>A. b * \<mu> A\<rparr>"
+ interpret b: sigma_algebra ?Mb by (intro sigma_algebra_cong) auto
+ have "finite_measure ?Mb"
+ by default
+ (insert finite_measure_of_space b measure_countably_additive,
+ auto simp: psuminf_cmult_right countably_additive_def)
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
@@ -462,30 +466,30 @@
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 "(\<integral>\<^isup>+x. ?f0 x * indicator A x) =
- (\<integral>\<^isup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x)"
+ have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) =
+ (\<integral>\<^isup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x \<partial>M)"
by (auto intro!: positive_integral_cong simp: field_simps indicator_inter_arith)
- hence "(\<integral>\<^isup>+x. ?f0 x * indicator A x) =
- (\<integral>\<^isup>+x. f x * indicator A x) + b * \<mu> (A \<inter> A0)"
+ hence "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) =
+ (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * \<mu> (A \<inter> A0)"
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: "(\<integral>\<^isup>+x. f x * indicator A x) \<le> \<nu> A"
+ have f_le_v: "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) \<le> \<nu> A"
using `f \<in> G` A unfolding G_def by auto
note f0_eq[OF A]
- also have "(\<integral>\<^isup>+x. f x * indicator A x) + b * \<mu> (A \<inter> A0) \<le>
- (\<integral>\<^isup>+x. f x * indicator A x) + ?t (A \<inter> A0)"
+ also have "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * \<mu> (A \<inter> A0) \<le>
+ (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?t (A \<inter> A0)"
using bM_le_t[OF `A \<inter> A0 \<in> sets M`] `A \<in> sets M` `A0 \<in> sets M`
by (auto intro!: add_left_mono)
- also have "\<dots> \<le> (\<integral>\<^isup>+x. f x * indicator A x) + ?t A"
+ also have "\<dots> \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?t A"
using M.measure_mono[simplified, OF _ `A \<inter> A0 \<in> sets M` `A \<in> sets M`]
by (auto intro!: add_left_mono)
also have "\<dots> \<le> \<nu> A"
using f_le_v M'.finite_measure[simplified, OF `A \<in> sets M`]
- by (cases "(\<integral>\<^isup>+x. f x * indicator A x)", cases "\<nu> A", auto)
- finally have "(\<integral>\<^isup>+x. ?f0 x * indicator A x) \<le> \<nu> A" . }
+ by (cases "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M)", cases "\<nu> A", auto)
+ finally have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) \<le> \<nu> A" . }
hence "?f0 \<in> G" using `A0 \<in> sets M` unfolding G_def
by (auto intro!: borel_measurable_indicator borel_measurable_pextreal_add borel_measurable_pextreal_times)
have real: "?t (space M) \<noteq> \<omega>" "?t A0 \<noteq> \<omega>"
@@ -494,7 +498,7 @@
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>"
+ have int_f_finite: "integral\<^isup>P M f \<noteq> \<omega>"
using M'.finite_measure_of_space pos_t unfolding pextreal_zero_less_diff_iff
by (auto cong: positive_integral_cong)
have "?t (space M) > b * \<mu> (space M)" unfolding b_def
@@ -514,22 +518,22 @@
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
+ have "?y + 0 < integral\<^isup>P M f + b * \<mu> A0" unfolding int_f_eq_y
by (rule pextreal_less_add)
- also have "\<dots> = positive_integral ?f0" using f0_eq[OF top] `A0 \<in> sets M` sets_into_space
+ also have "\<dots> = integral\<^isup>P M ?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)
+ finally have "?y < integral\<^isup>P M ?f0" by simp
+ moreover from `?f0 \<in> G` have "integral\<^isup>P M ?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"
- show "\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
+ show "\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
proof (rule antisym)
- show "(\<integral>\<^isup>+x. f x * indicator A x) \<le> \<nu> A"
+ show "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) \<le> \<nu> A"
using `f \<in> G` `A \<in> sets M` unfolding G_def by auto
- show "\<nu> A \<le> (\<integral>\<^isup>+x. f x * indicator A x)"
+ show "\<nu> A \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
using upper_bound[THEN bspec, OF `A \<in> sets M`]
by (simp add: pextreal_zero_le_diff)
qed
@@ -537,13 +541,15 @@
qed
lemma (in finite_measure) split_space_into_finite_sets_and_rest:
- assumes "measure_space M \<nu>"
+ assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
assumes ac: "absolutely_continuous \<nu>"
shows "\<exists>\<Omega>0\<in>sets M. \<exists>\<Omega>::nat\<Rightarrow>'a set. disjoint_family \<Omega> \<and> range \<Omega> \<subseteq> sets M \<and> \<Omega>0 = space M - (\<Union>i. \<Omega> i) \<and>
(\<forall>A\<in>sets M. A \<subseteq> \<Omega>0 \<longrightarrow> (\<mu> A = 0 \<and> \<nu> A = 0) \<or> (\<mu> A > 0 \<and> \<nu> A = \<omega>)) \<and>
(\<forall>i. \<nu> (\<Omega> i) \<noteq> \<omega>)"
proof -
- interpret v: measure_space M \<nu> by fact
+ interpret v: measure_space ?N
+ where "space ?N = space M" and "sets ?N = sets M" and "measure ?N = \<nu>"
+ by fact auto
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
@@ -667,11 +673,13 @@
qed
lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite:
- assumes "measure_space M \<nu>"
+ assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
assumes "absolutely_continuous \<nu>"
- shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
+ shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
proof -
- interpret v: measure_space M \<nu> by fact
+ interpret v: measure_space ?N
+ where "space ?N = space M" and "sets ?N = sets M" and "measure ?N = \<nu>"
+ by fact auto
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"
@@ -680,39 +688,38 @@
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) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x))"
+ \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x \<partial>M))"
proof
fix i
have indicator_eq: "\<And>f x A. (f x :: pextreal) * 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]])
+ have fm: "finite_measure (restricted_space (Q i))"
+ (is "finite_measure ?R") by (rule restricted_finite_measure[OF Q_sets[of i]])
then interpret R: finite_measure ?R .
- have fmv: "finite_measure ?R \<nu>"
+ have fmv: "finite_measure (restricted_space (Q i) \<lparr> measure := \<nu>\<rparr>)" (is "finite_measure ?Q")
unfolding finite_measure_def finite_measure_axioms_def
proof
- show "measure_space ?R \<nu>"
+ show "measure_space ?Q"
using v.restricted_measure_space Q_sets[of i] by auto
- show "\<nu> (space ?R) \<noteq> \<omega>"
- using Q_fin by simp
+ show "measure ?Q (space ?Q) \<noteq> \<omega>" using Q_fin by simp
qed
have "R.absolutely_continuous \<nu>"
using `absolutely_continuous \<nu>` `Q i \<in> sets M`
by (auto simp: R.absolutely_continuous_def absolutely_continuous_def)
- from finite_measure.Radon_Nikodym_finite_measure[OF fm fmv this]
+ from R.Radon_Nikodym_finite_measure[OF fmv this]
obtain f where f: "(\<lambda>x. f x * indicator (Q i) x) \<in> borel_measurable M"
- and f_int: "\<And>A. A\<in>sets M \<Longrightarrow> \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. (f x * indicator (Q i) x) * indicator A x)"
+ and f_int: "\<And>A. A\<in>sets M \<Longrightarrow> \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. (f x * indicator (Q i) x) * indicator A x \<partial>M)"
unfolding Bex_def borel_measurable_restricted[OF `Q i \<in> sets M`]
positive_integral_restricted[OF `Q i \<in> sets M`] by (auto simp: indicator_eq)
then show "\<exists>f. f\<in>borel_measurable M \<and> (\<forall>A\<in>sets M.
- \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x))"
+ \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x \<partial>M))"
by (fastsimp intro!: exI[of _ "\<lambda>x. f x * indicator (Q i) x"] positive_integral_cong
simp: indicator_def)
qed
from choice[OF this] obtain f where borel: "\<And>i. f i \<in> borel_measurable M"
and f: "\<And>A i. A \<in> sets M \<Longrightarrow>
- \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x)"
+ \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)"
by auto
let "?f x" =
"(\<Sum>\<^isub>\<infinity> i. f i x * indicator (Q i) x) + \<omega> * indicator Q0 x"
@@ -728,7 +735,7 @@
f i x * indicator (Q i \<inter> A) x"
"\<And>x i. (indicator A x * indicator Q0 x :: pextreal) =
indicator (Q0 \<inter> A) x" by (auto simp: indicator_def)
- have "(\<integral>\<^isup>+x. ?f x * indicator A x) =
+ have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) =
(\<Sum>\<^isub>\<infinity> i. \<nu> (Q i \<inter> A)) + \<omega> * \<mu> (Q0 \<inter> A)"
unfolding f[OF `A \<in> sets M`]
apply (simp del: pextreal_times(2) add: field_simps *)
@@ -755,27 +762,29 @@
using Q_sets `A \<in> sets M` Q0(1) by (auto intro!: countable_UN)
moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}"
using `A \<in> sets M` sets_into_space Q0 by auto
- ultimately show "\<nu> A = (\<integral>\<^isup>+x. ?f x * indicator A x)"
+ ultimately show "\<nu> A = (\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M)"
using v.measure_additive[simplified, of "(\<Union>i. Q i) \<inter> A" "Q0 \<inter> A"]
by simp
qed
qed
lemma (in sigma_finite_measure) Radon_Nikodym:
- assumes "measure_space M \<nu>"
+ assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
assumes "absolutely_continuous \<nu>"
- shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
+ shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
proof -
from Ex_finite_integrable_function
- obtain h where finite: "positive_integral h \<noteq> \<omega>" and
+ obtain h where finite: "integral\<^isup>P M h \<noteq> \<omega>" and
borel: "h \<in> borel_measurable M" and
pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and
"\<And>x. x \<in> space M \<Longrightarrow> h x < \<omega>" by auto
- let "?T A" = "(\<integral>\<^isup>+x. h x * indicator A x)"
+ let "?T A" = "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M)"
+ let ?MT = "M\<lparr> measure := ?T \<rparr>"
from measure_space_density[OF borel] finite
- interpret T: finite_measure M ?T
+ interpret T: finite_measure ?MT
+ where "space ?MT = space M" and "sets ?MT = sets M" and "measure ?MT = ?T"
unfolding finite_measure_def finite_measure_axioms_def
- by (simp cong: positive_integral_cong)
+ by (simp_all cong: positive_integral_cong)
have "\<And>N. N \<in> sets M \<Longrightarrow> {x \<in> space M. h x \<noteq> 0 \<and> indicator N x \<noteq> (0::pextreal)} = N"
using sets_into_space pos by (force simp: indicator_def)
then have "T.absolutely_continuous \<nu>" using assms(2) borel
@@ -783,7 +792,8 @@
by (fastsimp simp: borel_measurable_indicator positive_integral_0_iff)
from T.Radon_Nikodym_finite_measure_infinite[simplified, OF assms(1) this]
obtain f where f_borel: "f \<in> borel_measurable M" and
- fT: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = T.positive_integral (\<lambda>x. f x * indicator A x)" by auto
+ fT: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>?MT)"
+ by (auto simp: measurable_def)
show ?thesis
proof (safe intro!: bexI[of _ "\<lambda>x. h x * f x"])
show "(\<lambda>x. h x * f x) \<in> borel_measurable M"
@@ -792,7 +802,7 @@
then have "(\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
using f_borel by (auto intro: borel_measurable_pextreal_times borel_measurable_indicator)
from positive_integral_translated_density[OF borel this]
- show "\<nu> A = (\<integral>\<^isup>+x. h x * f x * indicator A x)"
+ show "\<nu> A = (\<integral>\<^isup>+x. h x * f x * indicator A x \<partial>M)"
unfolding fT[OF `A \<in> sets M`] by (simp add: field_simps)
qed
qed
@@ -801,8 +811,8 @@
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>"
- shows "(\<forall>A\<in>sets M. (\<integral>\<^isup>+x. f x * indicator A x) = (\<integral>\<^isup>+x. g x * indicator A x))
+ and fin: "integral\<^isup>P M f < \<omega>"
+ shows "(\<forall>A\<in>sets M. (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. g x * indicator A x \<partial>M))
\<longleftrightarrow> (AE x. f x = g x)"
(is "(\<forall>A\<in>sets M. ?P f A = ?P g A) \<longleftrightarrow> _")
proof (intro iffI ballI)
@@ -812,18 +822,18 @@
next
assume eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
from this[THEN bspec, OF top] fin
- have g_fin: "positive_integral g < \<omega>" by (simp cong: positive_integral_cong)
+ have g_fin: "integral\<^isup>P M g < \<omega>" by (simp cong: positive_integral_cong)
{ fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
- and g_fin: "positive_integral g < \<omega>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
+ and g_fin: "integral\<^isup>P M g < \<omega>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
let ?N = "{x\<in>space M. g x < f x}"
have N: "?N \<in> sets M" using borel by simp
- have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^isup>+x. f x * indicator ?N x - g x * indicator ?N x)"
+ have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^isup>+x. f x * indicator ?N x - g x * indicator ?N x \<partial>M)"
by (auto intro!: positive_integral_cong simp: indicator_def)
also have "\<dots> = ?P f ?N - ?P g ?N"
proof (rule positive_integral_diff)
show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M"
using borel N by auto
- have "?P g ?N \<le> positive_integral g"
+ have "?P g ?N \<le> integral\<^isup>P M g"
by (auto intro!: positive_integral_mono simp: indicator_def)
then show "?P g ?N \<noteq> \<omega>" using g_fin by auto
fix x assume "x \<in> space M"
@@ -848,17 +858,17 @@
lemma (in finite_measure) density_unique_finite_measure:
assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
- assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x) = (\<integral>\<^isup>+x. f' x * indicator A x)"
+ assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)"
(is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
shows "AE x. f x = f' x"
proof -
let "?\<nu> A" = "?P f A" and "?\<nu>' A" = "?P f' A"
let "?f A x" = "f x * indicator A x" and "?f' A x" = "f' x * indicator A x"
- interpret M: measure_space M ?\<nu>
- using borel(1) by (rule measure_space_density)
+ interpret M: measure_space "M\<lparr> measure := ?\<nu>\<rparr>"
+ using borel(1) by (rule measure_space_density) simp
have ac: "absolutely_continuous ?\<nu>"
using f by (rule density_is_absolutely_continuous)
- from split_space_into_finite_sets_and_rest[OF `measure_space M ?\<nu>` ac]
+ from split_space_into_finite_sets_and_rest[OF `measure_space (M\<lparr> measure := ?\<nu>\<rparr>)` ac]
obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
@@ -876,13 +886,13 @@
have 2: "AE x. ?f Q0 x = ?f' Q0 x"
proof (rule AE_I')
{ fix f :: "'a \<Rightarrow> pextreal" assume borel: "f \<in> borel_measurable M"
- and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
+ and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
let "?A i" = "Q0 \<inter> {x \<in> space M. f x < of_nat i}"
have "(\<Union>i. ?A i) \<in> null_sets"
proof (rule null_sets_UN)
fix i have "?A i \<in> sets M"
using borel Q0(1) by auto
- have "?\<nu> (?A i) \<le> (\<integral>\<^isup>+x. of_nat i * indicator (?A i) x)"
+ have "?\<nu> (?A i) \<le> (\<integral>\<^isup>+x. of_nat i * indicator (?A i) x \<partial>M)"
unfolding eq[OF `?A i \<in> sets M`]
by (auto intro!: positive_integral_mono simp: indicator_def)
also have "\<dots> = of_nat i * \<mu> (?A i)"
@@ -912,63 +922,72 @@
lemma (in sigma_finite_measure) density_unique:
assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
- assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x) = (\<integral>\<^isup>+x. f' x * indicator A x)"
+ assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)"
(is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
shows "AE x. f x = f' x"
proof -
obtain h where h_borel: "h \<in> borel_measurable M"
- and fin: "positive_integral h \<noteq> \<omega>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<omega>"
+ and fin: "integral\<^isup>P M h \<noteq> \<omega>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<omega>"
using Ex_finite_integrable_function by auto
- interpret h: measure_space M "\<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x)"
- using h_borel by (rule measure_space_density)
- interpret h: finite_measure M "\<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x)"
+ interpret h: measure_space "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
+ using h_borel by (rule measure_space_density) simp
+ interpret h: finite_measure "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
by default (simp cong: positive_integral_cong add: fin)
- interpret f: measure_space M "\<lambda>A. (\<integral>\<^isup>+x. f x * indicator A x)"
- using borel(1) by (rule measure_space_density)
- interpret f': measure_space M "\<lambda>A. (\<integral>\<^isup>+x. f' x * indicator A x)"
- using borel(2) by (rule measure_space_density)
+ let ?fM = "M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)\<rparr>"
+ interpret f: measure_space ?fM
+ using borel(1) by (rule measure_space_density) simp
+ let ?f'M = "M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)\<rparr>"
+ interpret f': measure_space ?f'M
+ using borel(2) by (rule measure_space_density) simp
{ fix A assume "A \<in> sets M"
then have " {x \<in> space M. h x \<noteq> 0 \<and> indicator A x \<noteq> (0::pextreal)} = A"
using pos sets_into_space by (force simp: indicator_def)
- then have "(\<integral>\<^isup>+x. h x * indicator A x) = 0 \<longleftrightarrow> A \<in> null_sets"
+ then have "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M) = 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 "(\<integral>\<^isup>+x. h x * (f x * indicator A x)) =
- f.positive_integral (\<lambda>x. h x * indicator A x)"
+ have "(\<integral>\<^isup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?fM)"
+ using `A \<in> sets M` h_borel borel
+ by (simp add: positive_integral_translated_density ac_simps cong: positive_integral_cong)
+ also have "\<dots> = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?f'M)"
+ by (rule f'.positive_integral_cong_measure) (simp_all add: f)
+ also have "\<dots> = (\<integral>\<^isup>+x. h x * (f' x * indicator A x) \<partial>M)"
using `A \<in> sets M` h_borel borel
by (simp add: positive_integral_translated_density ac_simps cong: positive_integral_cong)
- also have "\<dots> = f'.positive_integral (\<lambda>x. h x * indicator A x)"
- by (rule f'.positive_integral_cong_measure) (rule f)
- also have "\<dots> = (\<integral>\<^isup>+x. h x * (f' x * indicator A x))"
- using `A \<in> sets M` h_borel borel
- by (simp add: positive_integral_translated_density ac_simps cong: positive_integral_cong)
- finally have "(\<integral>\<^isup>+x. h x * (f x * indicator A x)) = (\<integral>\<^isup>+x. h x * (f' x * indicator A x))" . }
+ finally have "(\<integral>\<^isup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * (f' x * indicator A x) \<partial>M)" . }
then have "h.almost_everywhere (\<lambda>x. f x = f' x)"
using h_borel borel
- by (intro h.density_unique_finite_measure[OF borel])
- (simp add: positive_integral_translated_density)
+ apply (intro h.density_unique_finite_measure)
+ apply (simp add: measurable_def)
+ apply (simp add: measurable_def)
+ by (simp add: positive_integral_translated_density)
then show "AE x. f x = f' x"
unfolding h.almost_everywhere_def almost_everywhere_def
by (auto simp add: h_null_sets)
qed
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
- assumes \<nu>: "measure_space M \<nu>" and f: "f \<in> borel_measurable M"
- and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
- shows "sigma_finite_measure M \<nu> \<longleftrightarrow> (AE x. f x \<noteq> \<omega>)"
+ assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
+ and f: "f \<in> borel_measurable M"
+ and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
+ shows "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>) \<longleftrightarrow> (AE x. f x \<noteq> \<omega>)"
proof
- assume "sigma_finite_measure M \<nu>"
- then interpret \<nu>: sigma_finite_measure M \<nu> .
+ assume "sigma_finite_measure ?N"
+ then interpret \<nu>: sigma_finite_measure ?N
+ where "borel_measurable ?N = borel_measurable M" and "measure ?N = \<nu>"
+ and "sets ?N = sets M" and "space ?N = space M" by (auto simp: measurable_def)
from \<nu>.Ex_finite_integrable_function obtain h where
- h: "h \<in> borel_measurable M" "\<nu>.positive_integral h \<noteq> \<omega>"
+ h: "h \<in> borel_measurable M" "integral\<^isup>P ?N h \<noteq> \<omega>"
and fin: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<omega>" by auto
have "AE x. f x * h x \<noteq> \<omega>"
proof (rule AE_I')
- have "\<nu>.positive_integral h = (\<integral>\<^isup>+x. f x * h x)"
- by (simp add: \<nu>.positive_integral_cong_measure[symmetric, OF eq[symmetric]])
- (intro positive_integral_translated_density f h)
- then have "(\<integral>\<^isup>+x. f x * h x) \<noteq> \<omega>"
+ have "integral\<^isup>P ?N h = (\<integral>\<^isup>+x. f x * h x \<partial>M)"
+ apply (subst \<nu>.positive_integral_cong_measure[symmetric,
+ of "M\<lparr> measure := \<lambda> A. \<integral>\<^isup>+x. f x * indicator A x \<partial>M \<rparr>"])
+ apply (simp_all add: eq)
+ apply (rule positive_integral_translated_density)
+ using f h by auto
+ then have "(\<integral>\<^isup>+x. f x * h x \<partial>M) \<noteq> \<omega>"
using h(2) by simp
then show "(\<lambda>x. f x * h x) -` {\<omega>} \<inter> space M \<in> null_sets"
using f h(1) by (auto intro!: positive_integral_omega borel_measurable_vimage)
@@ -981,7 +1000,9 @@
next
assume AE: "AE x. f x \<noteq> \<omega>"
from sigma_finite guess Q .. note Q = this
- interpret \<nu>: measure_space M \<nu> by fact
+ interpret \<nu>: measure_space ?N
+ where "borel_measurable ?N = borel_measurable M" and "measure ?N = \<nu>"
+ and "sets ?N = sets M" and "space ?N = space M" using \<nu> by (auto simp: measurable_def)
def A \<equiv> "\<lambda>i. f -` (case i of 0 \<Rightarrow> {\<omega>} | Suc n \<Rightarrow> {.. of_nat (Suc n)}) \<inter> space M"
{ fix i j have "A i \<inter> Q j \<in> sets M"
unfolding A_def using f Q
@@ -989,11 +1010,11 @@
by (cases i) (auto intro: measurable_sets[OF f]) }
note A_in_sets = this
let "?A n" = "case prod_decode n of (i,j) \<Rightarrow> A i \<inter> Q j"
- show "sigma_finite_measure M \<nu>"
+ show "sigma_finite_measure ?N"
proof (default, intro exI conjI subsetI allI)
fix x assume "x \<in> range ?A"
then obtain n where n: "x = ?A n" by auto
- then show "x \<in> sets M" using A_in_sets by (cases "prod_decode n") auto
+ then show "x \<in> sets ?N" using A_in_sets by (cases "prod_decode n") auto
next
have "(\<Union>i. ?A i) = (\<Union>i j. A i \<inter> Q j)"
proof safe
@@ -1014,16 +1035,16 @@
then show ?thesis using x preal unfolding A_def by (auto intro!: exI[of _ "Suc n"])
qed
qed (auto simp: A_def)
- finally show "(\<Union>i. ?A i) = space M" by simp
+ finally show "(\<Union>i. ?A i) = space ?N" by simp
next
fix n obtain i j where
[simp]: "prod_decode n = (i, j)" by (cases "prod_decode n") auto
- have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x) \<noteq> \<omega>"
+ have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<omega>"
proof (cases i)
case 0
have "AE x. f x * indicator (A i \<inter> Q j) x = 0"
using AE by (rule AE_mp) (auto intro!: AE_cong simp: A_def `i = 0`)
- then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x) = 0"
+ then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) = 0"
using A_in_sets f
apply (subst positive_integral_0_iff)
apply fast
@@ -1034,8 +1055,8 @@
then show ?thesis by simp
next
case (Suc n)
- then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x) \<le>
- (\<integral>\<^isup>+x. of_nat (Suc n) * indicator (Q j) x)"
+ then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le>
+ (\<integral>\<^isup>+x. of_nat (Suc n) * indicator (Q j) x \<partial>M)"
by (auto intro!: positive_integral_mono simp: indicator_def A_def)
also have "\<dots> = of_nat (Suc n) * \<mu> (Q j)"
using Q by (auto intro!: positive_integral_cmult_indicator)
@@ -1043,33 +1064,34 @@
using Q by auto
finally show ?thesis by simp
qed
- then show "\<nu> (?A n) \<noteq> \<omega>"
+ then show "measure ?N (?A n) \<noteq> \<omega>"
using A_in_sets Q eq by auto
qed
qed
section "Radon-Nikodym derivative"
-definition (in sigma_finite_measure)
- "RN_deriv \<nu> \<equiv> SOME f. f \<in> borel_measurable M \<and>
- (\<forall>A \<in> sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x))"
+definition
+ "RN_deriv M \<nu> \<equiv> SOME f. f \<in> borel_measurable M \<and>
+ (\<forall>A \<in> sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
lemma (in sigma_finite_measure) RN_deriv_cong:
- assumes cong: "\<And>A. A \<in> sets M \<Longrightarrow> \<mu>' A = \<mu> A" "\<And>A. A \<in> sets M \<Longrightarrow> \<nu>' A = \<nu> A"
- shows "sigma_finite_measure.RN_deriv M \<mu>' \<nu>' x = RN_deriv \<nu> x"
+ assumes cong: "\<And>A. A \<in> sets M \<Longrightarrow> measure M' A = \<mu> A" "sets M' = sets M" "space M' = space M"
+ and \<nu>: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu>' A = \<nu> A"
+ shows "RN_deriv M' \<nu>' x = RN_deriv M \<nu> x"
proof -
- interpret \<mu>': sigma_finite_measure M \<mu>'
- using cong(1) by (rule sigma_finite_measure_cong)
+ interpret \<mu>': sigma_finite_measure M'
+ using cong by (rule sigma_finite_measure_cong)
show ?thesis
- unfolding RN_deriv_def \<mu>'.RN_deriv_def
- by (simp add: cong positive_integral_cong_measure[OF cong(1)])
+ unfolding RN_deriv_def
+ by (simp add: cong \<nu> positive_integral_cong_measure[OF cong] measurable_def)
qed
lemma (in sigma_finite_measure) RN_deriv:
- assumes "measure_space M \<nu>"
+ assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
assumes "absolutely_continuous \<nu>"
- shows "RN_deriv \<nu> \<in> borel_measurable M" (is ?borel)
- and "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. RN_deriv \<nu> x * indicator A x)"
+ shows "RN_deriv M \<nu> \<in> borel_measurable M" (is ?borel)
+ and "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)"
(is "\<And>A. _ \<Longrightarrow> ?int A")
proof -
note Ex = Radon_Nikodym[OF assms, unfolded Bex_def]
@@ -1080,87 +1102,92 @@
qed
lemma (in sigma_finite_measure) RN_deriv_positive_integral:
- assumes \<nu>: "measure_space M \<nu>" "absolutely_continuous \<nu>"
+ assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
and f: "f \<in> borel_measurable M"
- shows "measure_space.positive_integral M \<nu> f = (\<integral>\<^isup>+x. RN_deriv \<nu> x * f x)"
+ shows "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * f x \<partial>M)"
proof -
- interpret \<nu>: measure_space M \<nu> by fact
- have "\<nu>.positive_integral f =
- measure_space.positive_integral M (\<lambda>A. (\<integral>\<^isup>+x. RN_deriv \<nu> x * indicator A x)) f"
- by (intro \<nu>.positive_integral_cong_measure[symmetric] RN_deriv(2)[OF \<nu>, symmetric])
- also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv \<nu> x * f x)"
- by (intro positive_integral_translated_density RN_deriv[OF \<nu>] f)
+ interpret \<nu>: measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
+ have "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f =
+ integral\<^isup>P (M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)\<rparr>) f"
+ by (intro \<nu>.positive_integral_cong_measure[symmetric])
+ (simp_all add: RN_deriv(2)[OF \<nu>, symmetric])
+ also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * f x \<partial>M)"
+ by (intro positive_integral_translated_density)
+ (simp_all add: RN_deriv[OF \<nu>] f)
finally show ?thesis .
qed
lemma (in sigma_finite_measure) RN_deriv_unique:
- assumes \<nu>: "measure_space M \<nu>" "absolutely_continuous \<nu>"
+ assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
and f: "f \<in> borel_measurable M"
- and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
- shows "AE x. f x = RN_deriv \<nu> x"
+ and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
+ shows "AE x. f x = RN_deriv M \<nu> x"
proof (rule density_unique[OF f RN_deriv(1)[OF \<nu>]])
fix A assume A: "A \<in> sets M"
- show "(\<integral>\<^isup>+x. f x * indicator A x) = (\<integral>\<^isup>+x. RN_deriv \<nu> x * indicator A x)"
+ show "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)"
unfolding eq[OF A, symmetric] RN_deriv(2)[OF \<nu> A, symmetric] ..
qed
-
lemma (in sigma_finite_measure) RN_deriv_finite:
- assumes sfm: "sigma_finite_measure M \<nu>" and ac: "absolutely_continuous \<nu>"
- shows "AE x. RN_deriv \<nu> x \<noteq> \<omega>"
+ assumes sfm: "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>)" and ac: "absolutely_continuous \<nu>"
+ shows "AE x. RN_deriv M \<nu> x \<noteq> \<omega>"
proof -
- interpret \<nu>: sigma_finite_measure M \<nu> by fact
- have \<nu>: "measure_space M \<nu>" by default
+ interpret \<nu>: sigma_finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
+ have \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
from sfm show ?thesis
using sigma_finite_iff_density_finite[OF \<nu> RN_deriv[OF \<nu> ac]] by simp
qed
lemma (in sigma_finite_measure)
- assumes \<nu>: "sigma_finite_measure M \<nu>" "absolutely_continuous \<nu>"
+ assumes \<nu>: "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
and f: "f \<in> borel_measurable M"
- shows RN_deriv_integral: "measure_space.integral M \<nu> f = (\<integral>x. real (RN_deriv \<nu> x) * f x)" (is ?integral)
- and RN_deriv_integrable: "measure_space.integrable M \<nu> f \<longleftrightarrow> integrable (\<lambda>x. real (RN_deriv \<nu> x) * f x)" (is ?integrable)
+ shows RN_deriv_integrable: "integrable (M\<lparr>measure := \<nu>\<rparr>) f \<longleftrightarrow>
+ integrable M (\<lambda>x. real (RN_deriv M \<nu> x) * f x)" (is ?integrable)
+ and RN_deriv_integral: "integral\<^isup>L (M\<lparr>measure := \<nu>\<rparr>) f =
+ (\<integral>x. real (RN_deriv M \<nu> x) * f x \<partial>M)" (is ?integral)
proof -
- interpret \<nu>: sigma_finite_measure M \<nu> by fact
- have ms: "measure_space M \<nu>" by default
+ interpret \<nu>: sigma_finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
+ have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
have minus_cong: "\<And>A B A' B'::pextreal. A = A' \<Longrightarrow> B = B' \<Longrightarrow> real A - real B = real A' - real B'" by simp
have f': "(\<lambda>x. - f x) \<in> borel_measurable M" using f by auto
- { fix f :: "'a \<Rightarrow> real" assume "f \<in> borel_measurable M"
- { fix x assume *: "RN_deriv \<nu> x \<noteq> \<omega>"
- have "Real (real (RN_deriv \<nu> x)) * Real (f x) = Real (real (RN_deriv \<nu> x) * f x)"
+ have Nf: "f \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)" using f unfolding measurable_def by auto
+ { fix f :: "'a \<Rightarrow> real"
+ { fix x assume *: "RN_deriv M \<nu> x \<noteq> \<omega>"
+ have "Real (real (RN_deriv M \<nu> x)) * Real (f x) = Real (real (RN_deriv M \<nu> x) * f x)"
by (simp add: mult_le_0_iff)
- then have "RN_deriv \<nu> x * Real (f x) = Real (real (RN_deriv \<nu> x) * f x)"
+ then have "RN_deriv M \<nu> x * Real (f x) = Real (real (RN_deriv M \<nu> x) * f x)"
using * by (simp add: Real_real) }
note * = this
- have "(\<integral>\<^isup>+x. RN_deriv \<nu> x * Real (f x)) = (\<integral>\<^isup>+x. Real (real (RN_deriv \<nu> x) * f x))"
+ have "(\<integral>\<^isup>+x. RN_deriv M \<nu> x * Real (f x) \<partial>M) = (\<integral>\<^isup>+x. Real (real (RN_deriv M \<nu> x) * f x) \<partial>M)"
apply (rule positive_integral_cong_AE)
apply (rule AE_mp[OF RN_deriv_finite[OF \<nu>]])
by (auto intro!: AE_cong simp: *) }
- with this[OF f] this[OF f'] f f'
+ with this this f f' Nf
show ?integral ?integrable
- unfolding \<nu>.integral_def integral_def \<nu>.integrable_def integrable_def
- by (auto intro!: RN_deriv(1)[OF ms \<nu>(2)] minus_cong simp: RN_deriv_positive_integral[OF ms \<nu>(2)])
+ unfolding lebesgue_integral_def integrable_def
+ by (auto intro!: RN_deriv(1)[OF ms \<nu>(2)] minus_cong
+ simp: RN_deriv_positive_integral[OF ms \<nu>(2)])
qed
lemma (in sigma_finite_measure) RN_deriv_singleton:
- assumes "measure_space M \<nu>"
+ assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
and ac: "absolutely_continuous \<nu>"
and "{x} \<in> sets M"
- shows "\<nu> {x} = RN_deriv \<nu> x * \<mu> {x}"
+ shows "\<nu> {x} = RN_deriv M \<nu> x * \<mu> {x}"
proof -
note deriv = RN_deriv[OF assms(1, 2)]
from deriv(2)[OF `{x} \<in> sets M`]
- have "\<nu> {x} = (\<integral>\<^isup>+w. RN_deriv \<nu> x * indicator {x} w)"
+ have "\<nu> {x} = (\<integral>\<^isup>+w. RN_deriv M \<nu> x * indicator {x} w \<partial>M)"
by (auto simp: indicator_def intro!: positive_integral_cong)
thus ?thesis using positive_integral_cmult_indicator[OF `{x} \<in> sets M`]
by auto
qed
theorem (in finite_measure_space) RN_deriv_finite_measure:
- assumes "measure_space M \<nu>"
+ assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
and ac: "absolutely_continuous \<nu>"
and "x \<in> space M"
- shows "\<nu> {x} = RN_deriv \<nu> x * \<mu> {x}"
+ shows "\<nu> {x} = RN_deriv M \<nu> x * \<mu> {x}"
proof -
have "{x} \<in> sets M" using sets_eq_Pow `x \<in> space M` by auto
from RN_deriv_singleton[OF assms(1,2) this] show ?thesis .