--- a/src/HOL/Probability/Lebesgue_Measure.thy Wed Feb 02 10:35:41 2011 +0100
+++ b/src/HOL/Probability/Lebesgue_Measure.thy Wed Feb 02 12:34:45 2011 +0100
@@ -1,7 +1,7 @@
(* Author: Robert Himmelmann, TU Muenchen *)
header {* Lebsegue measure *}
theory Lebesgue_Measure
- imports Product_Measure Complete_Measure
+ imports Product_Measure
begin
subsection {* Standard Cubes *}
@@ -42,10 +42,16 @@
by (auto simp add:dist_norm)
qed
-definition lebesgue :: "'a::ordered_euclidean_space algebra" where
- "lebesgue = \<lparr> space = UNIV, sets = {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n} \<rparr>"
+lemma cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)"
+ unfolding cube_def_raw subset_eq apply safe unfolding mem_interval by auto
-definition "lmeasure A = (SUP n. Real (integral (cube n) (indicator A)))"
+lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
+ unfolding Pi_def by auto
+
+definition lebesgue :: "'a::ordered_euclidean_space measure_space" where
+ "lebesgue = \<lparr> space = UNIV,
+ sets = {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n},
+ measure = \<lambda>A. SUP n. Real (integral (cube n) (indicator A)) \<rparr>"
lemma space_lebesgue[simp]: "space lebesgue = UNIV"
unfolding lebesgue_def by simp
@@ -106,12 +112,12 @@
qed (auto intro: LIMSEQ_indicator_UN simp: cube_def)
qed simp
-interpretation lebesgue: measure_space lebesgue lmeasure
+interpretation lebesgue: measure_space lebesgue
proof
have *: "indicator {} = (\<lambda>x. 0 :: real)" by (simp add: fun_eq_iff)
- show "lmeasure {} = 0" by (simp add: integral_0 * lmeasure_def)
+ show "measure lebesgue {} = 0" by (simp add: integral_0 * lebesgue_def)
next
- show "countably_additive lebesgue lmeasure"
+ show "countably_additive lebesgue (measure lebesgue)"
proof (intro countably_additive_def[THEN iffD2] allI impI)
fix A :: "nat \<Rightarrow> 'b set" assume rA: "range A \<subseteq> sets lebesgue" "disjoint_family A"
then have A[simp, intro]: "\<And>i n. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n"
@@ -122,8 +128,8 @@
assume "(\<Union>i. A i) \<in> sets lebesgue"
then have UN_A[simp, intro]: "\<And>i n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n"
by (auto dest: lebesgueD)
- show "(\<Sum>\<^isub>\<infinity>n. lmeasure (A n)) = lmeasure (\<Union>i. A i)" unfolding lmeasure_def
- proof (subst psuminf_SUP_eq)
+ show "(\<Sum>\<^isub>\<infinity>n. measure lebesgue (A n)) = measure lebesgue (\<Union>i. A i)"
+ proof (simp add: lebesgue_def, subst psuminf_SUP_eq)
fix n i show "Real (?m n i) \<le> Real (?m (Suc n) i)"
using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le)
next
@@ -213,20 +219,19 @@
using assms by (force simp: cube_def integrable_on_def negligible_def intro!: lebesgueI)
lemma lmeasure_eq_0:
- fixes S :: "'a::ordered_euclidean_space set" assumes "negligible S" shows "lmeasure S = 0"
+ fixes S :: "'a::ordered_euclidean_space set" assumes "negligible S" shows "lebesgue.\<mu> S = 0"
proof -
have "\<And>n. integral (cube n) (indicator S :: 'a\<Rightarrow>real) = 0"
- unfolding integral_def using assms
- by (intro some1_equality ex_ex1I has_integral_unique)
- (auto simp: cube_def negligible_def intro: )
- then show ?thesis unfolding lmeasure_def by auto
+ unfolding lebesgue_integral_def using assms
+ by (intro integral_unique some1_equality ex_ex1I)
+ (auto simp: cube_def negligible_def)
+ then show ?thesis by (auto simp: lebesgue_def)
qed
lemma lmeasure_iff_LIMSEQ:
assumes "A \<in> sets lebesgue" "0 \<le> m"
- shows "lmeasure A = Real m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) ----> m"
- unfolding lmeasure_def
-proof (intro SUP_eq_LIMSEQ)
+ shows "lebesgue.\<mu> A = Real m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) ----> m"
+proof (simp add: lebesgue_def, intro SUP_eq_LIMSEQ)
show "mono (\<lambda>n. integral (cube n) (indicator A::_=>real))"
using cube_subset assms by (intro monoI integral_subset_le) (auto dest!: lebesgueD)
fix n show "0 \<le> integral (cube n) (indicator A::_=>real)"
@@ -253,7 +258,7 @@
lemma lmeasure_finite_has_integral:
fixes s :: "'a::ordered_euclidean_space set"
- assumes "s \<in> sets lebesgue" "lmeasure s = Real m" "0 \<le> m"
+ assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s = Real m" "0 \<le> m"
shows "(indicator s has_integral m) UNIV"
proof -
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
@@ -295,9 +300,9 @@
unfolding m by (intro integrable_integral **)
qed
-lemma lmeasure_finite_integrable: assumes "s \<in> sets lebesgue" "lmeasure s \<noteq> \<omega>"
+lemma lmeasure_finite_integrable: assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s \<noteq> \<omega>"
shows "(indicator s :: _ \<Rightarrow> real) integrable_on UNIV"
-proof (cases "lmeasure s")
+proof (cases "lebesgue.\<mu> s")
case (preal m) from lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this]
show ?thesis unfolding integrable_on_def by auto
qed (insert assms, auto)
@@ -314,7 +319,7 @@
qed
lemma has_integral_lmeasure: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
- shows "lmeasure s = Real m"
+ shows "lebesgue.\<mu> s = Real m"
proof (intro lmeasure_iff_LIMSEQ[THEN iffD2])
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
show "s \<in> sets lebesgue" using has_integral_lebesgue[OF assms] .
@@ -339,37 +344,37 @@
qed
lemma has_integral_iff_lmeasure:
- "(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m)"
+ "(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m)"
proof
assume "(indicator A has_integral m) UNIV"
with has_integral_lmeasure[OF this] has_integral_lebesgue[OF this]
- show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m"
+ show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m"
by (auto intro: has_integral_nonneg)
next
- assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m"
+ assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m"
then show "(indicator A has_integral m) UNIV" by (intro lmeasure_finite_has_integral) auto
qed
lemma lmeasure_eq_integral: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV"
- shows "lmeasure s = Real (integral UNIV (indicator s))"
+ shows "lebesgue.\<mu> s = Real (integral UNIV (indicator s))"
using assms unfolding integrable_on_def
proof safe
fix y :: real assume "(indicator s has_integral y) UNIV"
from this[unfolded has_integral_iff_lmeasure] integral_unique[OF this]
- show "lmeasure s = Real (integral UNIV (indicator s))" by simp
+ show "lebesgue.\<mu> s = Real (integral UNIV (indicator s))" by simp
qed
lemma lebesgue_simple_function_indicator:
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
- assumes f:"lebesgue.simple_function f"
+ assumes f:"simple_function lebesgue f"
shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f -` {y}) x))"
- apply(rule,subst lebesgue.simple_function_indicator_representation[OF f]) by auto
+ by (rule, subst lebesgue.simple_function_indicator_representation[OF f]) auto
lemma integral_eq_lmeasure:
- "(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (lmeasure s)"
+ "(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (lebesgue.\<mu> s)"
by (subst lmeasure_eq_integral) (auto intro!: integral_nonneg)
-lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "lmeasure s \<noteq> \<omega>"
+lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "lebesgue.\<mu> s \<noteq> \<omega>"
using lmeasure_eq_integral[OF assms] by auto
lemma negligible_iff_lebesgue_null_sets:
@@ -402,14 +407,13 @@
shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c"
by (rule integral_unique) (rule has_integral_const)
-lemma lmeasure_UNIV[intro]: "lmeasure (UNIV::'a::ordered_euclidean_space set) = \<omega>"
- unfolding lmeasure_def SUP_\<omega>
-proof (intro allI impI)
+lemma lmeasure_UNIV[intro]: "lebesgue.\<mu> (UNIV::'a::ordered_euclidean_space set) = \<omega>"
+proof (simp add: lebesgue_def SUP_\<omega>, intro allI impI)
fix x assume "x < \<omega>"
then obtain r where r: "x = Real r" "0 \<le> r" by (cases x) auto
then obtain n where n: "r < of_nat n" using ex_less_of_nat by auto
- show "\<exists>i\<in>UNIV. x < Real (integral (cube i) (indicator UNIV::'a\<Rightarrow>real))"
- proof (intro bexI[of _ n])
+ show "\<exists>i. x < Real (integral (cube i) (indicator UNIV::'a\<Rightarrow>real))"
+ proof (intro exI[of _ n])
have [simp]: "indicator UNIV = (\<lambda>x. 1)" by (auto simp: fun_eq_iff)
{ fix m :: nat assume "0 < m" then have "real n \<le> (\<Prod>x<m. 2 * real n)"
proof (induct m)
@@ -428,12 +432,12 @@
also have "Real (of_nat n) \<le> Real (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))"
by (auto simp: real_eq_of_nat[symmetric] cube_def content_closed_interval_cases)
finally show "x < Real (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))" .
- qed auto
+ qed
qed
lemma
fixes a b ::"'a::ordered_euclidean_space"
- shows lmeasure_atLeastAtMost[simp]: "lmeasure {a..b} = Real (content {a..b})"
+ shows lmeasure_atLeastAtMost[simp]: "lebesgue.\<mu> {a..b} = Real (content {a..b})"
proof -
have "(indicator (UNIV \<inter> {a..b})::_\<Rightarrow>real) integrable_on UNIV"
unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def_raw)
@@ -453,7 +457,7 @@
qed
lemma lmeasure_singleton[simp]:
- fixes a :: "'a::ordered_euclidean_space" shows "lmeasure {a} = 0"
+ fixes a :: "'a::ordered_euclidean_space" shows "lebesgue.\<mu> {a} = 0"
using lmeasure_atLeastAtMost[of a a] by simp
declare content_real[simp]
@@ -461,74 +465,97 @@
lemma
fixes a b :: real
shows lmeasure_real_greaterThanAtMost[simp]:
- "lmeasure {a <.. b} = Real (if a \<le> b then b - a else 0)"
+ "lebesgue.\<mu> {a <.. b} = Real (if a \<le> b then b - a else 0)"
proof cases
assume "a < b"
- then have "lmeasure {a <.. b} = lmeasure {a .. b} - lmeasure {a}"
+ then have "lebesgue.\<mu> {a <.. b} = lebesgue.\<mu> {a .. b} - lebesgue.\<mu> {a}"
by (subst lebesgue.measure_Diff[symmetric])
- (auto intro!: arg_cong[where f=lmeasure])
+ (auto intro!: arg_cong[where f=lebesgue.\<mu>])
then show ?thesis by auto
qed auto
lemma
fixes a b :: real
shows lmeasure_real_atLeastLessThan[simp]:
- "lmeasure {a ..< b} = Real (if a \<le> b then b - a else 0)"
+ "lebesgue.\<mu> {a ..< b} = Real (if a \<le> b then b - a else 0)"
proof cases
assume "a < b"
- then have "lmeasure {a ..< b} = lmeasure {a .. b} - lmeasure {b}"
+ then have "lebesgue.\<mu> {a ..< b} = lebesgue.\<mu> {a .. b} - lebesgue.\<mu> {b}"
by (subst lebesgue.measure_Diff[symmetric])
- (auto intro!: arg_cong[where f=lmeasure])
+ (auto intro!: arg_cong[where f=lebesgue.\<mu>])
then show ?thesis by auto
qed auto
lemma
fixes a b :: real
shows lmeasure_real_greaterThanLessThan[simp]:
- "lmeasure {a <..< b} = Real (if a \<le> b then b - a else 0)"
+ "lebesgue.\<mu> {a <..< b} = Real (if a \<le> b then b - a else 0)"
proof cases
assume "a < b"
- then have "lmeasure {a <..< b} = lmeasure {a <.. b} - lmeasure {b}"
+ then have "lebesgue.\<mu> {a <..< b} = lebesgue.\<mu> {a <.. b} - lebesgue.\<mu> {b}"
by (subst lebesgue.measure_Diff[symmetric])
- (auto intro!: arg_cong[where f=lmeasure])
+ (auto intro!: arg_cong[where f=lebesgue.\<mu>])
then show ?thesis by auto
qed auto
-interpretation borel: measure_space borel lmeasure
-proof
- show "countably_additive borel lmeasure"
- using lebesgue.ca unfolding countably_additive_def
- apply safe apply (erule_tac x=A in allE) by auto
-qed auto
+definition "lborel = lebesgue \<lparr> sets := sets borel \<rparr>"
+
+lemma
+ shows space_lborel[simp]: "space lborel = UNIV"
+ and sets_lborel[simp]: "sets lborel = sets borel"
+ and measure_lborel[simp]: "measure lborel = lebesgue.\<mu>"
+ and measurable_lborel[simp]: "measurable lborel = measurable borel"
+ by (simp_all add: measurable_def_raw lborel_def)
-interpretation borel: sigma_finite_measure borel lmeasure
-proof (default, intro conjI exI[of _ "\<lambda>n. cube n"])
- show "range cube \<subseteq> sets borel" by (auto intro: borel_closed)
- { fix x have "\<exists>n. x\<in>cube n" using mem_big_cube by auto }
- thus "(\<Union>i. cube i) = space borel" by auto
- show "\<forall>i. lmeasure (cube i) \<noteq> \<omega>" unfolding cube_def by auto
-qed
+interpretation lborel: measure_space lborel
+ where "space lborel = UNIV"
+ and "sets lborel = sets borel"
+ and "measure lborel = lebesgue.\<mu>"
+ and "measurable lborel = measurable borel"
+proof -
+ show "measure_space lborel"
+ proof
+ show "countably_additive lborel (measure lborel)"
+ using lebesgue.ca unfolding countably_additive_def lborel_def
+ apply safe apply (erule_tac x=A in allE) by auto
+ qed (auto simp: lborel_def)
+qed simp_all
-interpretation lebesgue: sigma_finite_measure lebesgue lmeasure
+interpretation lborel: sigma_finite_measure lborel
+ where "space lborel = UNIV"
+ and "sets lborel = sets borel"
+ and "measure lborel = lebesgue.\<mu>"
+ and "measurable lborel = measurable borel"
+proof -
+ show "sigma_finite_measure lborel"
+ proof (default, intro conjI exI[of _ "\<lambda>n. cube n"])
+ show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed)
+ { fix x have "\<exists>n. x\<in>cube n" using mem_big_cube by auto }
+ thus "(\<Union>i. cube i) = space lborel" by auto
+ show "\<forall>i. measure lborel (cube i) \<noteq> \<omega>" by (simp add: cube_def)
+ qed
+qed simp_all
+
+interpretation lebesgue: sigma_finite_measure lebesgue
proof
- from borel.sigma_finite guess A ..
+ from lborel.sigma_finite guess A ..
moreover then have "range A \<subseteq> sets lebesgue" using lebesgueI_borel by blast
- ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lmeasure (A i) \<noteq> \<omega>)"
+ ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lebesgue.\<mu> (A i) \<noteq> \<omega>)"
by auto
qed
lemma simple_function_has_integral:
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
- assumes f:"lebesgue.simple_function f"
+ assumes f:"simple_function lebesgue f"
and f':"\<forall>x. f x \<noteq> \<omega>"
- and om:"\<forall>x\<in>range f. lmeasure (f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
- shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.simple_integral f))) UNIV"
- unfolding lebesgue.simple_integral_def
+ and om:"\<forall>x\<in>range f. lebesgue.\<mu> (f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
+ shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV"
+ unfolding simple_integral_def
apply(subst lebesgue_simple_function_indicator[OF f])
proof -
case goal1
have *:"\<And>x. \<forall>y\<in>range f. y * indicator (f -` {y}) x \<noteq> \<omega>"
- "\<forall>x\<in>range f. x * lmeasure (f -` {x} \<inter> UNIV) \<noteq> \<omega>"
+ "\<forall>x\<in>range f. x * lebesgue.\<mu> (f -` {x} \<inter> UNIV) \<noteq> \<omega>"
using f' om unfolding indicator_def by auto
show ?case unfolding space_lebesgue real_of_pextreal_setsum'[OF *(1),THEN sym]
unfolding real_of_pextreal_setsum'[OF *(2),THEN sym]
@@ -536,11 +563,11 @@
apply(rule has_integral_setsum)
proof safe show "finite (range f)" using f by (auto dest: lebesgue.simple_functionD)
fix y::'a show "((\<lambda>x. real (f y * indicator (f -` {f y}) x)) has_integral
- real (f y * lmeasure (f -` {f y} \<inter> UNIV))) UNIV"
+ real (f y * lebesgue.\<mu> (f -` {f y} \<inter> UNIV))) UNIV"
proof(cases "f y = 0") case False
have mea:"(indicator (f -` {f y}) ::_\<Rightarrow>real) integrable_on UNIV"
apply(rule lmeasure_finite_integrable)
- using assms unfolding lebesgue.simple_function_def using False by auto
+ using assms unfolding simple_function_def using False by auto
have *:"\<And>x. real (indicator (f -` {f y}) x::pextreal) = (indicator (f -` {f y}) x)"
by (auto simp: indicator_def)
show ?thesis unfolding real_of_pextreal_mult[THEN sym]
@@ -558,31 +585,31 @@
lemma simple_function_has_integral':
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
- assumes f:"lebesgue.simple_function f"
- and i: "lebesgue.simple_integral f \<noteq> \<omega>"
- shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.simple_integral f))) UNIV"
+ assumes f:"simple_function lebesgue f"
+ and i: "integral\<^isup>S lebesgue f \<noteq> \<omega>"
+ shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV"
proof- let ?f = "\<lambda>x. if f x = \<omega> then 0 else f x"
{ fix x have "real (f x) = real (?f x)" by (cases "f x") auto } note * = this
have **:"{x. f x \<noteq> ?f x} = f -` {\<omega>}" by auto
- have **:"lmeasure {x\<in>space lebesgue. f x \<noteq> ?f x} = 0"
+ have **:"lebesgue.\<mu> {x\<in>space lebesgue. f x \<noteq> ?f x} = 0"
using lebesgue.simple_integral_omega[OF assms] by(auto simp add:**)
show ?thesis apply(subst lebesgue.simple_integral_cong'[OF f _ **])
apply(rule lebesgue.simple_function_compose1[OF f])
unfolding * defer apply(rule simple_function_has_integral)
proof-
- show "lebesgue.simple_function ?f"
+ show "simple_function lebesgue ?f"
using lebesgue.simple_function_compose1[OF f] .
show "\<forall>x. ?f x \<noteq> \<omega>" by auto
- show "\<forall>x\<in>range ?f. lmeasure (?f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
+ show "\<forall>x\<in>range ?f. lebesgue.\<mu> (?f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
proof (safe, simp, safe, rule ccontr)
fix y assume "f y \<noteq> \<omega>" "f y \<noteq> 0"
hence "(\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y} = f -` {f y}"
by (auto split: split_if_asm)
- moreover assume "lmeasure ((\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y}) = \<omega>"
- ultimately have "lmeasure (f -` {f y}) = \<omega>" by simp
+ moreover assume "lebesgue.\<mu> ((\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y}) = \<omega>"
+ ultimately have "lebesgue.\<mu> (f -` {f y}) = \<omega>" by simp
moreover
- have "f y * lmeasure (f -` {f y}) \<noteq> \<omega>" using i f
- unfolding lebesgue.simple_integral_def setsum_\<omega> lebesgue.simple_function_def
+ have "f y * lebesgue.\<mu> (f -` {f y}) \<noteq> \<omega>" using i f
+ unfolding simple_integral_def setsum_\<omega> simple_function_def
by auto
ultimately have "f y = 0" by (auto split: split_if_asm)
then show False using `f y \<noteq> 0` by simp
@@ -595,7 +622,7 @@
assumes i: "\<And>i. f i \<in> borel_measurable M" and mono: "\<And>x. mono (\<lambda>n. f n x)"
and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
shows "u \<in> borel_measurable M"
- and "(\<lambda>i. positive_integral (f i)) ----> positive_integral u" (is ?ilim)
+ and "(\<lambda>i. integral\<^isup>P M (f i)) ----> integral\<^isup>P M u" (is ?ilim)
proof -
from positive_integral_isoton[unfolded isoton_fun_expand isoton_iff_Lim_mono, of f u]
show ?ilim using mono lim i by auto
@@ -609,19 +636,19 @@
lemma positive_integral_has_integral:
fixes f::"'a::ordered_euclidean_space => pextreal"
assumes f:"f \<in> borel_measurable lebesgue"
- and int_om:"lebesgue.positive_integral f \<noteq> \<omega>"
+ and int_om:"integral\<^isup>P lebesgue f \<noteq> \<omega>"
and f_om:"\<forall>x. f x \<noteq> \<omega>" (* TODO: remove this *)
- shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.positive_integral f))) UNIV"
-proof- let ?i = "lebesgue.positive_integral f"
+ shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>P lebesgue f))) UNIV"
+proof- let ?i = "integral\<^isup>P lebesgue f"
from lebesgue.borel_measurable_implies_simple_function_sequence[OF f]
guess u .. note conjunctD2[OF this,rule_format] note u = conjunctD2[OF this(1)] this(2)
let ?u = "\<lambda>i x. real (u i x)" and ?f = "\<lambda>x. real (f x)"
- have u_simple:"\<And>k. lebesgue.simple_integral (u k) = lebesgue.positive_integral (u k)"
+ have u_simple:"\<And>k. integral\<^isup>S lebesgue (u k) = integral\<^isup>P lebesgue (u k)"
apply(subst lebesgue.positive_integral_eq_simple_integral[THEN sym,OF u(1)]) ..
- have int_u_le:"\<And>k. lebesgue.simple_integral (u k) \<le> lebesgue.positive_integral f"
+ have int_u_le:"\<And>k. integral\<^isup>S lebesgue (u k) \<le> integral\<^isup>P lebesgue f"
unfolding u_simple apply(rule lebesgue.positive_integral_mono)
using isoton_Sup[OF u(3)] unfolding le_fun_def by auto
- have u_int_om:"\<And>i. lebesgue.simple_integral (u i) \<noteq> \<omega>"
+ have u_int_om:"\<And>i. integral\<^isup>S lebesgue (u i) \<noteq> \<omega>"
proof- case goal1 thus ?case using int_u_le[of i] int_om by auto qed
note u_int = simple_function_has_integral'[OF u(1) this]
@@ -633,17 +660,17 @@
prefer 3 apply(subst Real_real') defer apply(subst Real_real')
using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] using f_om u by auto
next case goal3
- show ?case apply(rule bounded_realI[where B="real (lebesgue.positive_integral f)"])
+ show ?case apply(rule bounded_realI[where B="real (integral\<^isup>P lebesgue f)"])
apply safe apply(subst abs_of_nonneg) apply(rule integral_nonneg,rule) apply(rule u_int)
unfolding integral_unique[OF u_int] defer apply(rule real_of_pextreal_mono[OF _ int_u_le])
using u int_om by auto
qed note int = conjunctD2[OF this]
- have "(\<lambda>i. lebesgue.simple_integral (u i)) ----> ?i" unfolding u_simple
+ have "(\<lambda>i. integral\<^isup>S lebesgue (u i)) ----> ?i" unfolding u_simple
apply(rule lebesgue.positive_integral_monotone_convergence(2))
apply(rule lebesgue.borel_measurable_simple_function[OF u(1)])
using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] by auto
- hence "(\<lambda>i. real (lebesgue.simple_integral (u i))) ----> real ?i" apply-
+ hence "(\<lambda>i. real (integral\<^isup>S lebesgue (u i))) ----> real ?i" apply-
apply(subst lim_Real[THEN sym]) prefer 3
apply(subst Real_real') defer apply(subst Real_real')
using u f_om int_om u_int_om by auto
@@ -653,12 +680,12 @@
lemma lebesgue_integral_has_integral:
fixes f::"'a::ordered_euclidean_space => real"
- assumes f:"lebesgue.integrable f"
- shows "(f has_integral (lebesgue.integral f)) UNIV"
+ assumes f:"integrable lebesgue f"
+ shows "(f has_integral (integral\<^isup>L lebesgue f)) UNIV"
proof- let ?n = "\<lambda>x. - min (f x) 0" and ?p = "\<lambda>x. max (f x) 0"
have *:"f = (\<lambda>x. ?p x - ?n x)" apply rule by auto
- note f = lebesgue.integrableD[OF f]
- show ?thesis unfolding lebesgue.integral_def apply(subst *)
+ note f = integrableD[OF f]
+ show ?thesis unfolding lebesgue_integral_def apply(subst *)
proof(rule has_integral_sub) case goal1
have *:"\<forall>x. Real (f x) \<noteq> \<omega>" by auto
note lebesgue.borel_measurable_Real[OF f(1)]
@@ -674,27 +701,27 @@
qed
lemma lebesgue_positive_integral_eq_borel:
- "f \<in> borel_measurable borel \<Longrightarrow> lebesgue.positive_integral f = borel.positive_integral f "
+ "f \<in> borel_measurable borel \<Longrightarrow> integral\<^isup>P lebesgue f = integral\<^isup>P lborel f"
by (auto intro!: lebesgue.positive_integral_subalgebra[symmetric]) default
lemma lebesgue_integral_eq_borel:
assumes "f \<in> borel_measurable borel"
- shows "lebesgue.integrable f = borel.integrable f" (is ?P)
- and "lebesgue.integral f = borel.integral f" (is ?I)
+ shows "integrable lebesgue f \<longleftrightarrow> integrable lborel f" (is ?P)
+ and "integral\<^isup>L lebesgue f = integral\<^isup>L lborel f" (is ?I)
proof -
- have *: "sigma_algebra borel" by default
- have "sets borel \<subseteq> sets lebesgue" by auto
- from lebesgue.integral_subalgebra[OF assms this _ *]
+ have *: "sigma_algebra lborel" by default
+ have "sets lborel \<subseteq> sets lebesgue" by auto
+ from lebesgue.integral_subalgebra[of f lborel, OF _ this _ _ *] assms
show ?P ?I by auto
qed
lemma borel_integral_has_integral:
fixes f::"'a::ordered_euclidean_space => real"
- assumes f:"borel.integrable f"
- shows "(f has_integral (borel.integral f)) UNIV"
+ assumes f:"integrable lborel f"
+ shows "(f has_integral (integral\<^isup>L lborel f)) UNIV"
proof -
have borel: "f \<in> borel_measurable borel"
- using f unfolding borel.integrable_def by auto
+ using f unfolding integrable_def by auto
from f show ?thesis
using lebesgue_integral_has_integral[of f]
unfolding lebesgue_integral_eq_borel[OF borel] by simp
@@ -708,11 +735,11 @@
using continuous_open_preimage[OF assms] unfolding vimage_def by auto
lemma (in measure_space) integral_monotone_convergence_pos':
- assumes i: "\<And>i. integrable (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
+ assumes i: "\<And>i. integrable M (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
and pos: "\<And>x i. 0 \<le> f i x"
and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
- and ilim: "(\<lambda>i. integral (f i)) ----> x"
- shows "integrable u \<and> integral u = x"
+ and ilim: "(\<lambda>i. integral\<^isup>L M (f i)) ----> x"
+ shows "integrable M u \<and> integral\<^isup>L M u = x"
using integral_monotone_convergence_pos[OF assms] by auto
definition e2p :: "'a::ordered_euclidean_space \<Rightarrow> (nat \<Rightarrow> real)" where
@@ -751,53 +778,68 @@
thus "x \<in> e2p ` A" unfolding image_iff apply(rule_tac x="p2e x" in bexI) apply(subst e2p_p2e) by auto
qed
-interpretation borel_product: product_sigma_finite "\<lambda>x. borel::real algebra" "\<lambda>x. lmeasure"
+interpretation lborel_product: product_sigma_finite "\<lambda>x. lborel::real measure_space"
by default
-lemma cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)"
- unfolding cube_def_raw subset_eq apply safe unfolding mem_interval by auto
-
-lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
- unfolding Pi_def by auto
+interpretation lborel_space: finite_product_sigma_finite "\<lambda>x. lborel::real measure_space" "{..<DIM('a::ordered_euclidean_space)}"
+ where "space lborel = UNIV"
+ and "sets lborel = sets borel"
+ and "measure lborel = lebesgue.\<mu>"
+ and "measurable lborel = measurable borel"
+proof -
+ show "finite_product_sigma_finite (\<lambda>x. lborel::real measure_space) {..<DIM('a::ordered_euclidean_space)}"
+ by default simp
+qed simp_all
-lemma measurable_e2p_on_generator:
- "e2p \<in> measurable \<lparr> space = UNIV::'a set, sets = range lessThan \<rparr>
- (product_algebra
- (\<lambda>x. \<lparr> space = UNIV::real set, sets = range lessThan \<rparr>)
- {..<DIM('a::ordered_euclidean_space)})"
- (is "e2p \<in> measurable ?E ?P")
-proof (unfold measurable_def, intro CollectI conjI ballI)
- show "e2p \<in> space ?E \<rightarrow> space ?P" by (auto simp: e2p_def)
- fix A assume "A \<in> sets ?P"
- then obtain E where A: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. E i)"
- and "E \<in> {..<DIM('a)} \<rightarrow> (range lessThan)"
- by (auto elim!: product_algebraE)
- then have "\<forall>i\<in>{..<DIM('a)}. \<exists>xs. E i = {..< xs}" by auto
- from this[THEN bchoice] guess xs ..
- then have A_eq: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< xs i})"
- using A by auto
- have "e2p -` A = {..< (\<chi>\<chi> i. xs i) :: 'a}"
- using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def A_eq
- euclidean_eq[where 'a='a] eucl_less[where 'a='a])
- then show "e2p -` A \<inter> space ?E \<in> sets ?E" by simp
+lemma sets_product_borel:
+ assumes [intro]: "finite I"
+ shows "sets (\<Pi>\<^isub>M i\<in>I.
+ \<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>) =
+ sets (\<Pi>\<^isub>M i\<in>I. lborel)" (is "sets ?G = _")
+proof -
+ have "sets ?G = sets (\<Pi>\<^isub>M i\<in>I.
+ sigma \<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>)"
+ by (subst sigma_product_algebra_sigma_eq[of I "\<lambda>_ i. {..<real i}" ])
+ (auto intro!: measurable_sigma_sigma isotoneI real_arch_lt
+ simp: product_algebra_def)
+ then show ?thesis
+ unfolding lborel_def borel_eq_lessThan lebesgue_def sigma_def by simp
qed
lemma measurable_e2p:
- "e2p \<in> measurable (borel::'a algebra)
- (sigma (product_algebra (\<lambda>x. borel :: real algebra) {..<DIM('a::ordered_euclidean_space)}))"
- using measurable_e2p_on_generator[where 'a='a] unfolding borel_eq_lessThan
- by (subst sigma_product_algebra_sigma_eq[where S="\<lambda>_ i. {..<real i}"])
- (auto intro!: measurable_sigma_sigma isotoneI real_arch_lt
- simp: product_algebra_def)
+ "e2p \<in> measurable (borel::'a::ordered_euclidean_space algebra)
+ (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))"
+ (is "_ \<in> measurable ?E ?P")
+proof -
+ let ?B = "\<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>"
+ let ?G = "product_algebra_generator {..<DIM('a)} (\<lambda>_. ?B)"
+ have "e2p \<in> measurable ?E (sigma ?G)"
+ proof (rule borel.measurable_sigma)
+ show "e2p \<in> space ?E \<rightarrow> space ?G" by (auto simp: e2p_def)
+ fix A assume "A \<in> sets ?G"
+ then obtain E where A: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. E i)"
+ and "E \<in> {..<DIM('a)} \<rightarrow> (range lessThan)"
+ by (auto elim!: product_algebraE simp: )
+ then have "\<forall>i\<in>{..<DIM('a)}. \<exists>xs. E i = {..< xs}" by auto
+ from this[THEN bchoice] guess xs ..
+ then have A_eq: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< xs i})"
+ using A by auto
+ have "e2p -` A = {..< (\<chi>\<chi> i. xs i) :: 'a}"
+ using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def A_eq
+ euclidean_eq[where 'a='a] eucl_less[where 'a='a])
+ then show "e2p -` A \<inter> space ?E \<in> sets ?E" by simp
+ qed (auto simp: product_algebra_generator_def)
+ with sets_product_borel[of "{..<DIM('a)}"] show ?thesis
+ unfolding measurable_def product_algebra_def by simp
+qed
-lemma measurable_p2e_on_generator:
- "p2e \<in> measurable
- (product_algebra
- (\<lambda>x. \<lparr> space = UNIV::real set, sets = range lessThan \<rparr>)
- {..<DIM('a::ordered_euclidean_space)})
- \<lparr> space = UNIV::'a set, sets = range lessThan \<rparr>"
- (is "p2e \<in> measurable ?P ?E")
-proof (unfold measurable_def, intro CollectI conjI ballI)
+lemma measurable_p2e:
+ "p2e \<in> measurable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))
+ (borel :: 'a::ordered_euclidean_space algebra)"
+ (is "p2e \<in> measurable ?P _")
+ unfolding borel_eq_lessThan
+proof (intro lborel_space.measurable_sigma)
+ let ?E = "\<lparr> space = UNIV :: 'a set, sets = range lessThan \<rparr>"
show "p2e \<in> space ?P \<rightarrow> space ?E" by simp
fix A assume "A \<in> sets ?E"
then obtain x where "A = {..<x}" by auto
@@ -806,15 +848,7 @@
by (auto simp: Pi_iff set_eq_iff p2e_def
euclidean_eq[where 'a='a] eucl_less[where 'a='a])
then show "p2e -` A \<inter> space ?P \<in> sets ?P" by auto
-qed
-
-lemma measurable_p2e:
- "p2e \<in> measurable (sigma (product_algebra (\<lambda>x. borel :: real algebra) {..<DIM('a::ordered_euclidean_space)}))
- (borel::'a algebra)"
- using measurable_p2e_on_generator[where 'a='a] unfolding borel_eq_lessThan
- by (subst sigma_product_algebra_sigma_eq[where S="\<lambda>_ i. {..<real i}"])
- (auto intro!: measurable_sigma_sigma isotoneI real_arch_lt
- simp: product_algebra_def)
+qed simp
lemma e2p_Int:"e2p ` A \<inter> e2p ` B = e2p ` (A \<inter> B)" (is "?L = ?R")
apply(rule image_Int[THEN sym])
@@ -840,15 +874,13 @@
lemma lmeasure_measure_eq_borel_prod:
fixes A :: "('a::ordered_euclidean_space) set"
assumes "A \<in> sets borel"
- shows "lmeasure A = borel_product.product_measure {..<DIM('a)} (e2p ` A :: (nat \<Rightarrow> real) set)"
+ shows "lebesgue.\<mu> A = lborel_space.\<mu> TYPE('a) (e2p ` A)" (is "_ = ?m A")
proof (rule measure_unique_Int_stable[where X=A and A=cube])
- interpret fprod: finite_product_sigma_finite "\<lambda>x. borel :: real algebra" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
show "Int_stable \<lparr> space = UNIV :: 'a set, sets = range (\<lambda>(a,b). {a..b}) \<rparr>"
(is "Int_stable ?E" ) using Int_stable_cuboids' .
- show "borel = sigma ?E" using borel_eq_atLeastAtMost .
- show "\<And>i. lmeasure (cube i) \<noteq> \<omega>" unfolding cube_def by auto
- show "\<And>X. X \<in> sets ?E \<Longrightarrow>
- lmeasure X = borel_product.product_measure {..<DIM('a)} (e2p ` X :: (nat \<Rightarrow> real) set)"
+ have [simp]: "sigma ?E = borel" using borel_eq_atLeastAtMost ..
+ show "\<And>i. lebesgue.\<mu> (cube i) \<noteq> \<omega>" unfolding cube_def by auto
+ show "\<And>X. X \<in> sets ?E \<Longrightarrow> lebesgue.\<mu> X = ?m X"
proof- case goal1 then obtain a b where X:"X = {a..b}" by auto
{ presume *:"X \<noteq> {} \<Longrightarrow> ?case"
show ?case apply(cases,rule *,assumption) by auto }
@@ -861,12 +893,12 @@
show "x \<in> Pi\<^isub>E {..<DIM('a)} XX" unfolding y using y(1)
unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by auto
qed
- have "lmeasure X = (\<Prod>x<DIM('a). Real (b $$ x - a $$ x))" using X' apply- unfolding X
+ have "lebesgue.\<mu> X = (\<Prod>x<DIM('a). Real (b $$ x - a $$ x))" using X' apply- unfolding X
unfolding lmeasure_atLeastAtMost content_closed_interval apply(subst Real_setprod) by auto
- also have "... = (\<Prod>i<DIM('a). lmeasure (XX i))" apply(rule setprod_cong2)
+ also have "... = (\<Prod>i<DIM('a). lebesgue.\<mu> (XX i))" apply(rule setprod_cong2)
unfolding XX_def lmeasure_atLeastAtMost apply(subst content_real) using X' by auto
- also have "... = borel_product.product_measure {..<DIM('a)} (e2p ` X)" unfolding *[THEN sym]
- apply(rule fprod.measure_times[THEN sym]) unfolding XX_def by auto
+ also have "... = ?m X" unfolding *[THEN sym]
+ apply(rule lborel_space.measure_times[symmetric]) unfolding XX_def by auto
finally show ?case .
qed
@@ -875,18 +907,21 @@
have "\<And>x. \<exists>xa. x \<in> cube xa" apply(rule_tac x=x in mem_big_cube) by fastsimp
thus "cube \<up> space \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>"
apply-apply(rule isotoneI) apply(rule cube_subset_Suc) by auto
- show "A \<in> sets borel " by fact
- show "measure_space borel lmeasure" by default
- show "measure_space borel
- (\<lambda>a::'a set. finite_product_sigma_finite.measure (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` a))"
- proof (rule fprod.measure_space_vimage)
- show "sigma_algebra borel" by default
- show "(p2e :: (nat \<Rightarrow> real) \<Rightarrow> 'a) \<in> measurable fprod.P borel" by (rule measurable_p2e)
- fix A :: "'a set" assume "A \<in> sets borel"
- show "fprod.measure (e2p ` A) = fprod.measure (p2e -` A \<inter> space fprod.P)"
+ show "A \<in> sets (sigma ?E)" using assms by simp
+ have "measure_space lborel" by default
+ then show "measure_space \<lparr> space = space ?E, sets = sets (sigma ?E), measure = measure lebesgue\<rparr>"
+ unfolding lebesgue_def lborel_def by simp
+ let ?M = "\<lparr> space = space ?E, sets = sets (sigma ?E), measure = ?m \<rparr>"
+ show "measure_space ?M"
+ proof (rule lborel_space.measure_space_vimage)
+ show "sigma_algebra ?M" by (rule lborel.sigma_algebra_cong) auto
+ show "p2e \<in> measurable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel) ?M"
+ using measurable_p2e unfolding measurable_def by auto
+ fix A :: "'a set" assume "A \<in> sets ?M"
+ show "measure ?M A = lborel_space.\<mu> TYPE('a) (p2e -` A \<inter> space (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel))"
by (simp add: e2p_image_vimage)
qed
-qed
+qed simp
lemma range_e2p:"range (e2p::'a::ordered_euclidean_space \<Rightarrow> _) = extensional {..<DIM('a)}"
unfolding e2p_def_raw
@@ -896,41 +931,30 @@
lemma borel_fubini_positiv_integral:
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> pextreal"
assumes f: "f \<in> borel_measurable borel"
- shows "borel.positive_integral f =
- borel_product.product_positive_integral {..<DIM('a)} (f \<circ> p2e)"
-proof- def U \<equiv> "extensional {..<DIM('a)} :: (nat \<Rightarrow> real) set"
- interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
- show ?thesis
- proof (subst borel.positive_integral_vimage[symmetric, of _ "e2p :: 'a \<Rightarrow> _" "(\<lambda>x. f (p2e x))", unfolded p2e_e2p])
- show "(e2p :: 'a \<Rightarrow> _) \<in> measurable borel fprod.P" by (rule measurable_e2p)
- show "sigma_algebra fprod.P" by default
- from measurable_comp[OF measurable_p2e f]
- show "(\<lambda>x. f (p2e x)) \<in> borel_measurable fprod.P" by (simp add: comp_def)
- let "?L A" = "lmeasure ((e2p::'a \<Rightarrow> (nat \<Rightarrow> real)) -` A \<inter> space borel)"
- show "measure_space.positive_integral fprod.P ?L (\<lambda>x. f (p2e x)) =
- fprod.positive_integral (f \<circ> p2e)"
- unfolding comp_def
- proof (rule fprod.positive_integral_cong_measure)
- fix A :: "(nat \<Rightarrow> real) set" assume "A \<in> sets fprod.P"
- then have A: "(e2p::'a \<Rightarrow> (nat \<Rightarrow> real)) -` A \<inter> space borel \<in> sets borel"
- by (rule measurable_sets[OF measurable_e2p])
- have [simp]: "A \<inter> extensional {..<DIM('a)} = A"
- using `A \<in> sets fprod.P`[THEN fprod.sets_into_space] by auto
- show "?L A = fprod.measure A"
- unfolding lmeasure_measure_eq_borel_prod[OF A]
- by (simp add: range_e2p)
- qed
- qed
+ shows "integral\<^isup>P lborel f = \<integral>\<^isup>+x. f (p2e x) \<partial>(lborel_space.P TYPE('a))"
+proof (rule lborel.positive_integral_vimage[symmetric, of _ "e2p :: 'a \<Rightarrow> _" "(\<lambda>x. f (p2e x))", unfolded p2e_e2p])
+ show "(e2p :: 'a \<Rightarrow> _) \<in> measurable borel (lborel_space.P TYPE('a))" by (rule measurable_e2p)
+ show "sigma_algebra (lborel_space.P TYPE('a))" by default
+ from measurable_comp[OF measurable_p2e f]
+ show "(\<lambda>x. f (p2e x)) \<in> borel_measurable (lborel_space.P TYPE('a))" by (simp add: comp_def)
+ let "?L A" = "lebesgue.\<mu> ((e2p::'a \<Rightarrow> (nat \<Rightarrow> real)) -` A \<inter> UNIV)"
+ fix A :: "(nat \<Rightarrow> real) set" assume "A \<in> sets (lborel_space.P TYPE('a))"
+ then have A: "(e2p::'a \<Rightarrow> (nat \<Rightarrow> real)) -` A \<inter> space borel \<in> sets borel"
+ by (rule measurable_sets[OF measurable_e2p])
+ have [simp]: "A \<inter> extensional {..<DIM('a)} = A"
+ using `A \<in> sets (lborel_space.P TYPE('a))`[THEN lborel_space.sets_into_space] by auto
+ show "lborel_space.\<mu> TYPE('a) A = ?L A"
+ using lmeasure_measure_eq_borel_prod[OF A] by (simp add: range_e2p)
qed
lemma borel_fubini:
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
assumes f: "f \<in> borel_measurable borel"
- shows "borel.integral f = borel_product.product_integral {..<DIM('a)} (f \<circ> p2e)"
-proof- interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
+ shows "integral\<^isup>L lborel f = \<integral>x. f (p2e x) \<partial>(lborel_space.P TYPE('a))"
+proof -
have 1:"(\<lambda>x. Real (f x)) \<in> borel_measurable borel" using f by auto
have 2:"(\<lambda>x. Real (- f x)) \<in> borel_measurable borel" using f by auto
- show ?thesis unfolding fprod.integral_def borel.integral_def
+ show ?thesis unfolding lebesgue_integral_def
unfolding borel_fubini_positiv_integral[OF 1] borel_fubini_positiv_integral[OF 2]
unfolding o_def ..
qed