--- a/src/HOL/Probability/Lebesgue_Measure.thy Wed Dec 01 06:50:54 2010 -0800
+++ b/src/HOL/Probability/Lebesgue_Measure.thy Wed Dec 01 19:20:30 2010 +0100
@@ -1,38 +1,115 @@
-
+(* Author: Robert Himmelmann, TU Muenchen *)
header {* Lebsegue measure *}
-(* Author: Robert Himmelmann, TU Muenchen *)
-
theory Lebesgue_Measure
- imports Gauge_Measure Measure Lebesgue_Integration
+ imports Product_Measure Gauge_Measure Complete_Measure
begin
-subsection {* Various *}
+lemma (in complete_lattice) SUP_pair:
+ "(SUP i:A. SUP j:B. f i j) = (SUP p:A\<times>B. (\<lambda> (i, j). f i j) p)" (is "?l = ?r")
+proof (intro antisym SUP_leI)
+ fix i j assume "i \<in> A" "j \<in> B"
+ then have "(case (i,j) of (i,j) \<Rightarrow> f i j) \<le> ?r"
+ by (intro SUPR_upper) auto
+ then show "f i j \<le> ?r" by auto
+next
+ fix p assume "p \<in> A \<times> B"
+ then obtain i j where "p = (i,j)" "i \<in> A" "j \<in> B" by auto
+ have "f i j \<le> (SUP j:B. f i j)" using `j \<in> B` by (intro SUPR_upper)
+ also have "(SUP j:B. f i j) \<le> ?l" using `i \<in> A` by (intro SUPR_upper)
+ finally show "(case p of (i, j) \<Rightarrow> f i j) \<le> ?l" using `p = (i,j)` by simp
+qed
-lemma seq_offset_iff:"f ----> l \<longleftrightarrow> (\<lambda>i. f (i + k)) ----> l"
- using seq_offset_rev seq_offset[of f l k] by auto
+lemma (in complete_lattice) SUP_surj_compose:
+ assumes *: "f`A = B" shows "SUPR A (g \<circ> f) = SUPR B g"
+ unfolding SUPR_def unfolding *[symmetric]
+ by (simp add: image_compose)
+
+lemma (in complete_lattice) SUP_swap:
+ "(SUP i:A. SUP j:B. f i j) = (SUP j:B. SUP i:A. f i j)"
+proof -
+ have *: "(\<lambda>(i,j). (j,i)) ` (B \<times> A) = A \<times> B" by auto
+ show ?thesis
+ unfolding SUP_pair SUP_surj_compose[symmetric, OF *]
+ by (auto intro!: arg_cong[where f=Sup] image_eqI simp: comp_def SUPR_def)
+qed
-lemma has_integral_disjoint_union: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
- assumes "(f has_integral i) s" "(f has_integral j) t" "s \<inter> t = {}"
- shows "(f has_integral (i + j)) (s \<union> t)"
- apply(rule has_integral_union[OF assms(1-2)]) unfolding assms by auto
+lemma SUP_\<omega>: "(SUP i:A. f i) = \<omega> \<longleftrightarrow> (\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)"
+proof
+ assume *: "(SUP i:A. f i) = \<omega>"
+ show "(\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)" unfolding *[symmetric]
+ proof (intro allI impI)
+ fix x assume "x < SUPR A f" then show "\<exists>i\<in>A. x < f i"
+ unfolding less_SUP_iff by auto
+ qed
+next
+ assume *: "\<forall>x<\<omega>. \<exists>i\<in>A. x < f i"
+ show "(SUP i:A. f i) = \<omega>"
+ proof (rule pinfreal_SUPI)
+ fix y assume **: "\<And>i. i \<in> A \<Longrightarrow> f i \<le> y"
+ show "\<omega> \<le> y"
+ proof cases
+ assume "y < \<omega>"
+ from *[THEN spec, THEN mp, OF this]
+ obtain i where "i \<in> A" "\<not> (f i \<le> y)" by auto
+ with ** show ?thesis by auto
+ qed auto
+ qed auto
+qed
-lemma lim_eq: assumes "\<forall>n>N. f n = g n" shows "(f ----> l) \<longleftrightarrow> (g ----> l)" using assms
-proof(induct N arbitrary: f g) case 0
- hence *:"\<And>n. f (Suc n) = g (Suc n)" by auto
- show ?case apply(subst LIMSEQ_Suc_iff[THEN sym]) apply(subst(2) LIMSEQ_Suc_iff[THEN sym])
- unfolding * ..
-next case (Suc n)
- show ?case apply(subst LIMSEQ_Suc_iff[THEN sym]) apply(subst(2) LIMSEQ_Suc_iff[THEN sym])
- apply(rule Suc(1)) using Suc(2) by auto
+lemma psuminf_commute:
+ shows "(\<Sum>\<^isub>\<infinity> i j. f i j) = (\<Sum>\<^isub>\<infinity> j i. f i j)"
+proof -
+ have "(SUP n. \<Sum> i < n. SUP m. \<Sum> j < m. f i j) = (SUP n. SUP m. \<Sum> i < n. \<Sum> j < m. f i j)"
+ apply (subst SUPR_pinfreal_setsum)
+ by auto
+ also have "\<dots> = (SUP m n. \<Sum> j < m. \<Sum> i < n. f i j)"
+ apply (subst SUP_swap)
+ apply (subst setsum_commute)
+ by auto
+ also have "\<dots> = (SUP m. \<Sum> j < m. SUP n. \<Sum> i < n. f i j)"
+ apply (subst SUPR_pinfreal_setsum)
+ by auto
+ finally show ?thesis
+ unfolding psuminf_def by auto
+qed
+
+lemma psuminf_SUP_eq:
+ assumes "\<And>n i. f n i \<le> f (Suc n) i"
+ shows "(\<Sum>\<^isub>\<infinity> i. SUP n::nat. f n i) = (SUP n::nat. \<Sum>\<^isub>\<infinity> i. f n i)"
+proof -
+ { fix n :: nat
+ have "(\<Sum>i<n. SUP k. f k i) = (SUP k. \<Sum>i<n. f k i)"
+ using assms by (auto intro!: SUPR_pinfreal_setsum[symmetric]) }
+ note * = this
+ show ?thesis
+ unfolding psuminf_def
+ unfolding *
+ apply (subst SUP_swap) ..
qed
subsection {* Standard Cubes *}
-definition cube where
- "cube (n::nat) \<equiv> {\<chi>\<chi> i. - real n .. (\<chi>\<chi> i. real n)::_::ordered_euclidean_space}"
+definition cube :: "nat \<Rightarrow> 'a::ordered_euclidean_space set" where
+ "cube n \<equiv> {\<chi>\<chi> i. - real n .. \<chi>\<chi> i. real n}"
+
+lemma cube_closed[intro]: "closed (cube n)"
+ unfolding cube_def by auto
+
+lemma cube_subset[intro]: "n \<le> N \<Longrightarrow> cube n \<subseteq> cube N"
+ by (fastsimp simp: eucl_le[where 'a='a] cube_def)
-lemma cube_subset[intro]:"n\<le>N \<Longrightarrow> cube n \<subseteq> (cube N::'a::ordered_euclidean_space set)"
- apply(auto simp: eucl_le[where 'a='a] cube_def) apply(erule_tac[!] x=i in allE)+ by auto
+lemma cube_subset_iff:
+ "cube n \<subseteq> cube N \<longleftrightarrow> n \<le> N"
+proof
+ assume subset: "cube n \<subseteq> (cube N::'a set)"
+ then have "((\<chi>\<chi> i. real n)::'a) \<in> cube N"
+ using DIM_positive[where 'a='a]
+ by (fastsimp simp: cube_def eucl_le[where 'a='a])
+ then show "n \<le> N"
+ by (fastsimp simp: cube_def eucl_le[where 'a='a])
+next
+ assume "n \<le> N" then show "cube n \<subseteq> (cube N::'a set)" by (rule cube_subset)
+qed
lemma ball_subset_cube:"ball (0::'a::ordered_euclidean_space) (real n) \<subseteq> cube n"
unfolding cube_def subset_eq mem_interval apply safe unfolding euclidean_lambda_beta'
@@ -63,202 +140,277 @@
apply(rule has_gmeasure_subset[of "s\<inter>cube n" _ "cube n"])
unfolding has_gmeasure_measure[THEN sym] using assms by auto
+lemma has_gmeasure_cube[intro]: "(cube n::('a::ordered_euclidean_space) set)
+ has_gmeasure ((2 * real n) ^ (DIM('a)))"
+proof-
+ have "content {\<chi>\<chi> i. - real n..(\<chi>\<chi> i. real n)::'a} = (2 * real n) ^ (DIM('a))"
+ apply(subst content_closed_interval) defer
+ by (auto simp add:setprod_constant)
+ thus ?thesis unfolding cube_def
+ using has_gmeasure_interval(1)[of "(\<chi>\<chi> i. - real n)::'a" "(\<chi>\<chi> i. real n)::'a"]
+ by auto
+qed
+
+lemma gmeasure_cube_eq[simp]:
+ "gmeasure (cube n::('a::ordered_euclidean_space) set) = (2 * real n) ^ DIM('a)"
+ by (intro measure_unique) auto
+
+lemma gmeasure_cube_ge_n: "gmeasure (cube n::('a::ordered_euclidean_space) set) \<ge> real n"
+proof cases
+ assume "n = 0" then show ?thesis by simp
+next
+ assume "n \<noteq> 0"
+ have "real n \<le> (2 * real n)^1" by simp
+ also have "\<dots> \<le> (2 * real n)^DIM('a)"
+ using DIM_positive[where 'a='a] `n \<noteq> 0`
+ by (intro power_increasing) auto
+ also have "\<dots> = gmeasure (cube n::'a set)" by simp
+ finally show ?thesis .
+qed
+
+lemma gmeasure_setsum:
+ assumes "finite A" and "\<And>s t. s \<in> A \<Longrightarrow> t \<in> A \<Longrightarrow> s \<noteq> t \<Longrightarrow> f s \<inter> f t = {}"
+ and "\<And>i. i \<in> A \<Longrightarrow> gmeasurable (f i)"
+ shows "gmeasure (\<Union>i\<in>A. f i) = (\<Sum>i\<in>A. gmeasure (f i))"
+proof -
+ have "gmeasure (\<Union>i\<in>A. f i) = gmeasure (\<Union>f ` A)" by auto
+ also have "\<dots> = setsum gmeasure (f ` A)" using assms
+ proof (intro measure_negligible_unions)
+ fix X Y assume "X \<in> f`A" "Y \<in> f`A" "X \<noteq> Y"
+ then have "X \<inter> Y = {}" using assms by auto
+ then show "negligible (X \<inter> Y)" by auto
+ qed auto
+ also have "\<dots> = setsum gmeasure (f ` A - {{}})"
+ using assms by (intro setsum_mono_zero_cong_right) auto
+ also have "\<dots> = (\<Sum>i\<in>A - {i. f i = {}}. gmeasure (f i))"
+ proof (intro setsum_reindex_cong inj_onI)
+ fix s t assume *: "s \<in> A - {i. f i = {}}" "t \<in> A - {i. f i = {}}" "f s = f t"
+ show "s = t"
+ proof (rule ccontr)
+ assume "s \<noteq> t" with assms(2)[of s t] * show False by auto
+ qed
+ qed auto
+ also have "\<dots> = (\<Sum>i\<in>A. gmeasure (f i))"
+ using assms by (intro setsum_mono_zero_cong_left) auto
+ finally show ?thesis .
+qed
+
+lemma gmeasurable_finite_UNION[intro]:
+ assumes "\<And>i. i \<in> S \<Longrightarrow> gmeasurable (A i)" "finite S"
+ shows "gmeasurable (\<Union>i\<in>S. A i)"
+ unfolding UNION_eq_Union_image using assms
+ by (intro gmeasurable_finite_unions) auto
+
+lemma gmeasurable_countable_UNION[intro]:
+ fixes A :: "nat \<Rightarrow> ('a::ordered_euclidean_space) set"
+ assumes measurable: "\<And>i. gmeasurable (A i)"
+ and finite: "\<And>n. gmeasure (UNION {.. n} A) \<le> B"
+ shows "gmeasurable (\<Union>i. A i)"
+proof -
+ have *: "\<And>n. \<Union>{A k |k. k \<le> n} = (\<Union>i\<le>n. A i)"
+ "(\<Union>{A n |n. n \<in> UNIV}) = (\<Union>i. A i)" by auto
+ show ?thesis
+ by (rule gmeasurable_countable_unions_strong[of A B, unfolded *, OF assms])
+qed
subsection {* Measurability *}
-definition lmeasurable :: "('a::ordered_euclidean_space) set => bool" where
- "lmeasurable s \<equiv> (\<forall>n::nat. gmeasurable (s \<inter> cube n))"
+definition lebesgue :: "'a::ordered_euclidean_space algebra" where
+ "lebesgue = \<lparr> space = UNIV, sets = {A. \<forall>n. gmeasurable (A \<inter> cube n)} \<rparr>"
+
+lemma space_lebesgue[simp]:"space lebesgue = UNIV"
+ unfolding lebesgue_def by auto
-lemma lmeasurableD[dest]:assumes "lmeasurable s"
- shows "\<And>n. gmeasurable (s \<inter> cube n)"
- using assms unfolding lmeasurable_def by auto
+lemma lebesgueD[dest]: assumes "S \<in> sets lebesgue"
+ shows "\<And>n. gmeasurable (S \<inter> cube n)"
+ using assms unfolding lebesgue_def by auto
-lemma measurable_imp_lmeasurable: assumes"gmeasurable s"
- shows "lmeasurable s" unfolding lmeasurable_def cube_def
+lemma lebesgueI[intro]: assumes "gmeasurable S"
+ shows "S \<in> sets lebesgue" unfolding lebesgue_def cube_def
using assms gmeasurable_interval by auto
-lemma lmeasurable_empty[intro]: "lmeasurable {}"
- using gmeasurable_empty apply- apply(drule_tac measurable_imp_lmeasurable) .
-
-lemma lmeasurable_union[intro]: assumes "lmeasurable s" "lmeasurable t"
- shows "lmeasurable (s \<union> t)"
- using assms unfolding lmeasurable_def Int_Un_distrib2
- by(auto intro:gmeasurable_union)
+lemma lebesgueI2: "(\<And>n. gmeasurable (S \<inter> cube n)) \<Longrightarrow> S \<in> sets lebesgue"
+ using assms unfolding lebesgue_def by auto
-lemma lmeasurable_countable_unions_strong:
- fixes s::"nat => 'a::ordered_euclidean_space set"
- assumes "\<And>n::nat. lmeasurable(s n)"
- shows "lmeasurable(\<Union>{ s(n) |n. n \<in> UNIV })"
- unfolding lmeasurable_def
-proof fix n::nat
- have *:"\<Union>{s n |n. n \<in> UNIV} \<inter> cube n = \<Union>{s k \<inter> cube n |k. k \<in> UNIV}" by auto
- show "gmeasurable (\<Union>{s n |n. n \<in> UNIV} \<inter> cube n)" unfolding *
- apply(rule gmeasurable_countable_unions_strong)
- apply(rule assms[unfolded lmeasurable_def,rule_format])
- proof- fix k::nat
- show "gmeasure (\<Union>{s ka \<inter> cube n |ka. ka \<le> k}) \<le> gmeasure (cube n::'a set)"
- apply(rule measure_subset) apply(rule gmeasurable_finite_unions)
- using assms(1)[unfolded lmeasurable_def] by auto
- qed
+interpretation lebesgue: sigma_algebra lebesgue
+proof
+ show "sets lebesgue \<subseteq> Pow (space lebesgue)"
+ unfolding lebesgue_def by auto
+ show "{} \<in> sets lebesgue"
+ using gmeasurable_empty by auto
+ { fix A B :: "'a set" assume "A \<in> sets lebesgue" "B \<in> sets lebesgue"
+ then show "A \<union> B \<in> sets lebesgue"
+ by (auto intro: gmeasurable_union simp: lebesgue_def Int_Un_distrib2) }
+ { fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets lebesgue"
+ show "(\<Union>i. A i) \<in> sets lebesgue"
+ proof (rule lebesgueI2)
+ fix n show "gmeasurable ((\<Union>i. A i) \<inter> cube n)" unfolding UN_extend_simps
+ using A
+ by (intro gmeasurable_countable_UNION[where B="gmeasure (cube n::'a set)"])
+ (auto intro!: measure_subset gmeasure_setsum simp: UN_extend_simps simp del: gmeasure_cube_eq UN_simps)
+ qed }
+ { fix A assume A: "A \<in> sets lebesgue" show "space lebesgue - A \<in> sets lebesgue"
+ proof (rule lebesgueI2)
+ fix n
+ have *: "(space lebesgue - A) \<inter> cube n = cube n - (A \<inter> cube n)"
+ unfolding lebesgue_def by auto
+ show "gmeasurable ((space lebesgue - A) \<inter> cube n)" unfolding *
+ using A by (auto intro!: gmeasurable_diff)
+ qed }
qed
-lemma lmeasurable_inter[intro]: fixes s::"'a :: ordered_euclidean_space set"
- assumes "lmeasurable s" "lmeasurable t" shows "lmeasurable (s \<inter> t)"
- unfolding lmeasurable_def
-proof fix n::nat
- have *:"s \<inter> t \<inter> cube n = (s \<inter> cube n) \<inter> (t \<inter> cube n)" by auto
- show "gmeasurable (s \<inter> t \<inter> cube n)"
- using assms unfolding lmeasurable_def *
- using gmeasurable_inter[of "s \<inter> cube n" "t \<inter> cube n"] by auto
+lemma lebesgueI_borel[intro, simp]: fixes s::"'a::ordered_euclidean_space set"
+ assumes "s \<in> sets borel" shows "s \<in> sets lebesgue"
+proof- let ?S = "range (\<lambda>(a, b). {a .. (b :: 'a\<Colon>ordered_euclidean_space)})"
+ have *:"?S \<subseteq> sets lebesgue" by auto
+ have "s \<in> sigma_sets UNIV ?S" using assms
+ unfolding borel_eq_atLeastAtMost by (simp add: sigma_def)
+ thus ?thesis
+ using lebesgue.sigma_subset[of "\<lparr> space = UNIV, sets = ?S\<rparr>", simplified, OF *]
+ by (auto simp: sigma_def)
qed
-lemma lmeasurable_complement[intro]: assumes "lmeasurable s"
- shows "lmeasurable (UNIV - s)"
- unfolding lmeasurable_def
-proof fix n::nat
- have *:"(UNIV - s) \<inter> cube n = cube n - (s \<inter> cube n)" by auto
- show "gmeasurable ((UNIV - s) \<inter> cube n)" unfolding *
- apply(rule gmeasurable_diff) using assms unfolding lmeasurable_def by auto
-qed
-
-lemma lmeasurable_finite_unions:
- assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> lmeasurable s"
- shows "lmeasurable (\<Union> f)" unfolding lmeasurable_def
-proof fix n::nat have *:"(\<Union>f \<inter> cube n) = \<Union>{x \<inter> cube n | x . x\<in>f}" by auto
- show "gmeasurable (\<Union>f \<inter> cube n)" unfolding *
- apply(rule gmeasurable_finite_unions) unfolding simple_image
- using assms unfolding lmeasurable_def by auto
-qed
-
-lemma negligible_imp_lmeasurable[dest]: fixes s::"'a::ordered_euclidean_space set"
- assumes "negligible s" shows "lmeasurable s"
- unfolding lmeasurable_def
-proof case goal1
+lemma lebesgueI_negligible[dest]: fixes s::"'a::ordered_euclidean_space set"
+ assumes "negligible s" shows "s \<in> sets lebesgue"
+proof (rule lebesgueI2)
+ fix n
have *:"\<And>x. (if x \<in> cube n then indicator s x else 0) = (if x \<in> s \<inter> cube n then 1 else 0)"
unfolding indicator_def_raw by auto
note assms[unfolded negligible_def,rule_format,of "(\<chi>\<chi> i. - real n)::'a" "\<chi>\<chi> i. real n"]
- thus ?case apply-apply(rule gmeasurableI[of _ 0]) unfolding has_gmeasure_def
+ thus "gmeasurable (s \<inter> cube n)" apply-apply(rule gmeasurableI[of _ 0]) unfolding has_gmeasure_def
apply(subst(asm) has_integral_restrict_univ[THEN sym]) unfolding cube_def[symmetric]
apply(subst has_integral_restrict_univ[THEN sym]) unfolding * .
qed
-
section {* The Lebesgue Measure *}
-definition has_lmeasure (infixr "has'_lmeasure" 80) where
- "s has_lmeasure m \<equiv> lmeasurable s \<and> ((\<lambda>n. Real (gmeasure (s \<inter> cube n))) ---> m) sequentially"
+definition "lmeasure A = (SUP n. Real (gmeasure (A \<inter> cube n)))"
-lemma has_lmeasureD: assumes "s has_lmeasure m"
- shows "lmeasurable s" "gmeasurable (s \<inter> cube n)"
- "((\<lambda>n. Real (gmeasure (s \<inter> cube n))) ---> m) sequentially"
- using assms unfolding has_lmeasure_def lmeasurable_def by auto
+lemma lmeasure_eq_0: assumes "negligible S" shows "lmeasure S = 0"
+proof -
+ from lebesgueI_negligible[OF assms]
+ have "\<And>n. gmeasurable (S \<inter> cube n)" by auto
+ from gmeasurable_measure_eq_0[OF this]
+ have "\<And>n. gmeasure (S \<inter> cube n) = 0" using assms by auto
+ then show ?thesis unfolding lmeasure_def by simp
+qed
+
+lemma lmeasure_iff_LIMSEQ:
+ assumes "A \<in> sets lebesgue" "0 \<le> m"
+ shows "lmeasure A = Real m \<longleftrightarrow> (\<lambda>n. (gmeasure (A \<inter> cube n))) ----> m"
+ unfolding lmeasure_def using assms cube_subset[where 'a='a]
+ by (intro SUP_eq_LIMSEQ monoI measure_subset) force+
-lemma has_lmeasureI: assumes "lmeasurable s" "((\<lambda>n. Real (gmeasure (s \<inter> cube n))) ---> m) sequentially"
- shows "s has_lmeasure m" using assms unfolding has_lmeasure_def by auto
-
-definition lmeasure where
- "lmeasure s \<equiv> SOME m. s has_lmeasure m"
+interpretation lebesgue: measure_space lebesgue lmeasure
+proof
+ show "lmeasure {} = 0"
+ by (auto intro!: lmeasure_eq_0)
+ show "countably_additive lebesgue lmeasure"
+ proof (unfold countably_additive_def, intro allI impI conjI)
+ fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets lebesgue" "disjoint_family A"
+ then have A: "\<And>i. A i \<in> sets lebesgue" by auto
+ show "(\<Sum>\<^isub>\<infinity>n. lmeasure (A n)) = lmeasure (\<Union>i. A i)" unfolding lmeasure_def
+ proof (subst psuminf_SUP_eq)
+ { fix i n
+ have "gmeasure (A i \<inter> cube n) \<le> gmeasure (A i \<inter> cube (Suc n))"
+ using A cube_subset[of n "Suc n"] by (auto intro!: measure_subset)
+ then show "Real (gmeasure (A i \<inter> cube n)) \<le> Real (gmeasure (A i \<inter> cube (Suc n)))"
+ by auto }
+ show "(SUP n. \<Sum>\<^isub>\<infinity>i. Real (gmeasure (A i \<inter> cube n))) = (SUP n. Real (gmeasure ((\<Union>i. A i) \<inter> cube n)))"
+ proof (intro arg_cong[where f="SUPR UNIV"] ext)
+ fix n
+ have sums: "(\<lambda>i. gmeasure (A i \<inter> cube n)) sums gmeasure (\<Union>{A i \<inter> cube n |i. i \<in> UNIV})"
+ proof (rule has_gmeasure_countable_negligible_unions(2))
+ fix i show "gmeasurable (A i \<inter> cube n)" using A by auto
+ next
+ fix i m :: nat assume "m \<noteq> i"
+ then have "A m \<inter> cube n \<inter> (A i \<inter> cube n) = {}"
+ using `disjoint_family A` unfolding disjoint_family_on_def by auto
+ then show "negligible (A m \<inter> cube n \<inter> (A i \<inter> cube n))" by auto
+ next
+ fix i
+ have "(\<Sum>k = 0..i. gmeasure (A k \<inter> cube n)) = gmeasure (\<Union>k\<le>i . A k \<inter> cube n)"
+ unfolding atLeast0AtMost using A
+ proof (intro gmeasure_setsum[symmetric])
+ fix s t :: nat assume "s \<noteq> t" then have "A t \<inter> A s = {}"
+ using `disjoint_family A` unfolding disjoint_family_on_def by auto
+ then show "A s \<inter> cube n \<inter> (A t \<inter> cube n) = {}" by auto
+ qed auto
+ also have "\<dots> \<le> gmeasure (cube n :: 'b set)" using A
+ by (intro measure_subset gmeasurable_finite_UNION) auto
+ finally show "(\<Sum>k = 0..i. gmeasure (A k \<inter> cube n)) \<le> gmeasure (cube n :: 'b set)" .
+ qed
+ show "(\<Sum>\<^isub>\<infinity>i. Real (gmeasure (A i \<inter> cube n))) = Real (gmeasure ((\<Union>i. A i) \<inter> cube n))"
+ unfolding psuminf_def
+ apply (subst setsum_Real)
+ apply (simp add: measure_pos_le)
+ proof (rule SUP_eq_LIMSEQ[THEN iffD2])
+ have "(\<Union>{A i \<inter> cube n |i. i \<in> UNIV}) = (\<Union>i. A i) \<inter> cube n" by auto
+ with sums show "(\<lambda>i. \<Sum>k<i. gmeasure (A k \<inter> cube n)) ----> gmeasure ((\<Union>i. A i) \<inter> cube n)"
+ unfolding sums_def atLeast0LessThan by simp
+ qed (auto intro!: monoI setsum_nonneg setsum_mono2)
+ qed
+ qed
+ qed
+qed
-lemma has_lmeasure_has_gmeasure: assumes "s has_lmeasure (Real m)" "m\<ge>0"
+lemma lmeasure_finite_has_gmeasure: assumes "s \<in> sets lebesgue" "lmeasure s = Real m" "0 \<le> m"
shows "s has_gmeasure m"
-proof- note s = has_lmeasureD[OF assms(1)]
+proof-
have *:"(\<lambda>n. (gmeasure (s \<inter> cube n))) ----> m"
- using s(3) apply(subst (asm) lim_Real) using s(2) assms(2) by auto
-
+ using `lmeasure s = Real m` unfolding lmeasure_iff_LIMSEQ[OF `s \<in> sets lebesgue` `0 \<le> m`] .
+ have s: "\<And>n. gmeasurable (s \<inter> cube n)" using assms by auto
have "(\<lambda>x. if x \<in> s then 1 else (0::real)) integrable_on UNIV \<and>
(\<lambda>k. integral UNIV (\<lambda>x. if x \<in> s \<inter> cube k then 1 else (0::real)))
----> integral UNIV (\<lambda>x. if x \<in> s then 1 else 0)"
proof(rule monotone_convergence_increasing)
- have "\<forall>n. gmeasure (s \<inter> cube n) \<le> m" apply(rule ccontr) unfolding not_all not_le
- proof(erule exE) fix k assume k:"m < gmeasure (s \<inter> cube k)"
- hence "gmeasure (s \<inter> cube k) - m > 0" by auto
- from *[unfolded Lim_sequentially,rule_format,OF this] guess N ..
- note this[unfolded dist_real_def,rule_format,of "N + k"]
- moreover have "gmeasure (s \<inter> cube (N + k)) \<ge> gmeasure (s \<inter> cube k)" apply-
- apply(rule measure_subset) prefer 3 using s(2)
- using cube_subset[of k "N + k"] by auto
- ultimately show False by auto
- qed
- thus *:"bounded {integral UNIV (\<lambda>x. if x \<in> s \<inter> cube k then 1 else (0::real)) |k. True}"
- unfolding integral_measure_univ[OF s(2)] bounded_def apply-
+ have "lmeasure s \<le> Real m" using `lmeasure s = Real m` by simp
+ then have "\<forall>n. gmeasure (s \<inter> cube n) \<le> m"
+ unfolding lmeasure_def complete_lattice_class.SUP_le_iff
+ using `0 \<le> m` by (auto simp: measure_pos_le)
+ thus *:"bounded {integral UNIV (\<lambda>x. if x \<in> s \<inter> cube k then 1 else (0::real)) |k. True}"
+ unfolding integral_measure_univ[OF s] bounded_def apply-
apply(rule_tac x=0 in exI,rule_tac x=m in exI) unfolding dist_real_def
by (auto simp: measure_pos_le)
-
show "\<forall>k. (\<lambda>x. if x \<in> s \<inter> cube k then (1::real) else 0) integrable_on UNIV"
unfolding integrable_restrict_univ
- using s(2) unfolding gmeasurable_def has_gmeasure_def by auto
+ using s unfolding gmeasurable_def has_gmeasure_def by auto
have *:"\<And>n. n \<le> Suc n" by auto
show "\<forall>k. \<forall>x\<in>UNIV. (if x \<in> s \<inter> cube k then 1 else 0) \<le> (if x \<in> s \<inter> cube (Suc k) then 1 else (0::real))"
using cube_subset[OF *] by fastsimp
show "\<forall>x\<in>UNIV. (\<lambda>k. if x \<in> s \<inter> cube k then 1 else 0) ----> (if x \<in> s then 1 else (0::real))"
- unfolding Lim_sequentially
+ unfolding Lim_sequentially
proof safe case goal1 from real_arch_lt[of "norm x"] guess N .. note N = this
show ?case apply(rule_tac x=N in exI)
proof safe case goal1
have "x \<in> cube n" using cube_subset[OF goal1] N
- using ball_subset_cube[of N] by(auto simp: dist_norm)
+ using ball_subset_cube[of N] by(auto simp: dist_norm)
thus ?case using `e>0` by auto
qed
qed
qed note ** = conjunctD2[OF this]
hence *:"m = integral UNIV (\<lambda>x. if x \<in> s then 1 else 0)" apply-
- apply(rule LIMSEQ_unique[OF _ **(2)]) unfolding measure_integral_univ[THEN sym,OF s(2)] using * .
+ apply(rule LIMSEQ_unique[OF _ **(2)]) unfolding measure_integral_univ[THEN sym,OF s] using * .
show ?thesis unfolding has_gmeasure * apply(rule integrable_integral) using ** by auto
qed
-lemma has_lmeasure_unique: "s has_lmeasure m1 \<Longrightarrow> s has_lmeasure m2 \<Longrightarrow> m1 = m2"
- unfolding has_lmeasure_def apply(rule Lim_unique) using trivial_limit_sequentially by auto
-
-lemma lmeasure_unique[intro]: assumes "A has_lmeasure m" shows "lmeasure A = m"
- using assms unfolding lmeasure_def lmeasurable_def apply-
- apply(rule some_equality) defer apply(rule has_lmeasure_unique) by auto
-
-lemma glmeasurable_finite: assumes "lmeasurable s" "lmeasure s \<noteq> \<omega>"
+lemma lmeasure_finite_gmeasurable: assumes "s \<in> sets lebesgue" "lmeasure s \<noteq> \<omega>"
shows "gmeasurable s"
-proof- have "\<exists>B. \<forall>n. gmeasure (s \<inter> cube n) \<le> B"
- proof(rule ccontr) case goal1
- note as = this[unfolded not_ex not_all not_le]
- have "s has_lmeasure \<omega>" apply- apply(rule has_lmeasureI[OF assms(1)])
- unfolding Lim_omega
- proof fix B::real
- from as[rule_format,of B] guess N .. note N = this
- have "\<And>n. N \<le> n \<Longrightarrow> B \<le> gmeasure (s \<inter> cube n)"
- apply(rule order_trans[where y="gmeasure (s \<inter> cube N)"]) defer
- apply(rule measure_subset) prefer 3
- using cube_subset N assms(1)[unfolded lmeasurable_def] by auto
- thus "\<exists>N. \<forall>n\<ge>N. Real B \<le> Real (gmeasure (s \<inter> cube n))" apply-
- apply(subst Real_max') apply(rule_tac x=N in exI,safe)
- unfolding pinfreal_less_eq apply(subst if_P) by auto
- qed note lmeasure_unique[OF this]
- thus False using assms(2) by auto
- qed then guess B .. note B=this
+proof (cases "lmeasure s")
+ case (preal m) from lmeasure_finite_has_gmeasure[OF `s \<in> sets lebesgue` this]
+ show ?thesis unfolding gmeasurable_def by auto
+qed (insert assms, auto)
- show ?thesis apply(rule gmeasurable_nested_unions[of "\<lambda>n. s \<inter> cube n",
- unfolded Union_inter_cube,THEN conjunct1, where B1=B])
- proof- fix n::nat
- show " gmeasurable (s \<inter> cube n)" using assms by auto
- show "gmeasure (s \<inter> cube n) \<le> B" using B by auto
- show "s \<inter> cube n \<subseteq> s \<inter> cube (Suc n)"
- by (rule Int_mono) (simp_all add: cube_subset)
- qed
-qed
-
-lemma lmeasure_empty[intro]:"lmeasure {} = 0"
- apply(rule lmeasure_unique)
- unfolding has_lmeasure_def by auto
-
-lemma lmeasurableI[dest]:"s has_lmeasure m \<Longrightarrow> lmeasurable s"
- unfolding has_lmeasure_def by auto
-
-lemma has_gmeasure_has_lmeasure: assumes "s has_gmeasure m"
- shows "s has_lmeasure (Real m)"
-proof- have gmea:"gmeasurable s" using assms by auto
+lemma has_gmeasure_lmeasure: assumes "s has_gmeasure m"
+ shows "lmeasure s = Real m"
+proof-
+ have gmea:"gmeasurable s" using assms by auto
+ then have s: "s \<in> sets lebesgue" by auto
have m:"m \<ge> 0" using assms by auto
have *:"m = gmeasure (\<Union>{s \<inter> cube n |n. n \<in> UNIV})" unfolding Union_inter_cube
using assms by(rule measure_unique[THEN sym])
- show ?thesis unfolding has_lmeasure_def
- apply(rule,rule measurable_imp_lmeasurable[OF gmea])
- apply(subst lim_Real) apply(rule,rule,rule m) unfolding *
+ show ?thesis
+ unfolding lmeasure_iff_LIMSEQ[OF s `0 \<le> m`] unfolding *
apply(rule gmeasurable_nested_unions[THEN conjunct2, where B1="gmeasure s"])
proof- fix n::nat show *:"gmeasurable (s \<inter> cube n)"
using gmeasurable_inter[OF gmea gmeasurable_cube] .
@@ -266,287 +418,26 @@
apply(rule * gmea)+ by auto
show "s \<inter> cube n \<subseteq> s \<inter> cube (Suc n)" using cube_subset[of n "Suc n"] by auto
qed
-qed
-
-lemma gmeasure_lmeasure: assumes "gmeasurable s" shows "lmeasure s = Real (gmeasure s)"
-proof- note has_gmeasure_measureI[OF assms]
- note has_gmeasure_has_lmeasure[OF this]
- thus ?thesis by(rule lmeasure_unique)
-qed
-
-lemma has_lmeasure_lmeasure: "lmeasurable s \<longleftrightarrow> s has_lmeasure (lmeasure s)" (is "?l = ?r")
-proof assume ?l let ?f = "\<lambda>n. Real (gmeasure (s \<inter> cube n))"
- have "\<forall>n m. n\<ge>m \<longrightarrow> ?f n \<ge> ?f m" unfolding pinfreal_less_eq apply safe
- apply(subst if_P) defer apply(rule measure_subset) prefer 3
- apply(drule cube_subset) using `?l` by auto
- from lim_pinfreal_increasing[OF this] guess l . note l=this
- hence "s has_lmeasure l" using `?l` apply-apply(rule has_lmeasureI) by auto
- thus ?r using lmeasure_unique by auto
-next assume ?r thus ?l unfolding has_lmeasure_def by auto
-qed
-
-lemma lmeasure_subset[dest]: assumes "lmeasurable s" "lmeasurable t" "s \<subseteq> t"
- shows "lmeasure s \<le> lmeasure t"
-proof(cases "lmeasure t = \<omega>")
- case False have som:"lmeasure s \<noteq> \<omega>"
- proof(rule ccontr,unfold not_not) assume as:"lmeasure s = \<omega>"
- have "t has_lmeasure \<omega>" using assms(2) apply(rule has_lmeasureI)
- unfolding Lim_omega
- proof case goal1
- note assms(1)[unfolded has_lmeasure_lmeasure]
- note has_lmeasureD(3)[OF this,unfolded as Lim_omega,rule_format,of B]
- then guess N .. note N = this
- show ?case apply(rule_tac x=N in exI) apply safe
- apply(rule order_trans) apply(rule N[rule_format],assumption)
- unfolding pinfreal_less_eq apply(subst if_P)defer
- apply(rule measure_subset) using assms by auto
- qed
- thus False using lmeasure_unique False by auto
- qed
-
- note assms(1)[unfolded has_lmeasure_lmeasure] note has_lmeasureD(3)[OF this]
- hence "(\<lambda>n. Real (gmeasure (s \<inter> cube n))) ----> Real (real (lmeasure s))"
- unfolding Real_real'[OF som] .
- hence l1:"(\<lambda>n. gmeasure (s \<inter> cube n)) ----> real (lmeasure s)"
- apply-apply(subst(asm) lim_Real) by auto
-
- note assms(2)[unfolded has_lmeasure_lmeasure] note has_lmeasureD(3)[OF this]
- hence "(\<lambda>n. Real (gmeasure (t \<inter> cube n))) ----> Real (real (lmeasure t))"
- unfolding Real_real'[OF False] .
- hence l2:"(\<lambda>n. gmeasure (t \<inter> cube n)) ----> real (lmeasure t)"
- apply-apply(subst(asm) lim_Real) by auto
-
- have "real (lmeasure s) \<le> real (lmeasure t)" apply(rule LIMSEQ_le[OF l1 l2])
- apply(rule_tac x=0 in exI,safe) apply(rule measure_subset) using assms by auto
- hence "Real (real (lmeasure s)) \<le> Real (real (lmeasure t))"
- unfolding pinfreal_less_eq by auto
- thus ?thesis unfolding Real_real'[OF som] Real_real'[OF False] .
-qed auto
-
-lemma has_lmeasure_negligible_unions_image:
- assumes "finite s" "\<And>x. x \<in> s ==> lmeasurable(f x)"
- "\<And>x y. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x \<noteq> y \<Longrightarrow> negligible((f x) \<inter> (f y))"
- shows "(\<Union> (f ` s)) has_lmeasure (setsum (\<lambda>x. lmeasure(f x)) s)"
- unfolding has_lmeasure_def
-proof show lmeaf:"lmeasurable (\<Union>f ` s)" apply(rule lmeasurable_finite_unions)
- using assms(1-2) by auto
- show "(\<lambda>n. Real (gmeasure (\<Union>f ` s \<inter> cube n))) ----> (\<Sum>x\<in>s. lmeasure (f x))" (is ?l)
- proof(cases "\<exists>x\<in>s. lmeasure (f x) = \<omega>")
- case False hence *:"(\<Sum>x\<in>s. lmeasure (f x)) \<noteq> \<omega>" apply-
- apply(rule setsum_neq_omega) using assms(1) by auto
- have gmea:"\<And>x. x\<in>s \<Longrightarrow> gmeasurable (f x)" apply(rule glmeasurable_finite) using False assms(2) by auto
- have "(\<Sum>x\<in>s. lmeasure (f x)) = (\<Sum>x\<in>s. Real (gmeasure (f x)))" apply(rule setsum_cong2)
- apply(rule gmeasure_lmeasure) using False assms(2) gmea by auto
- also have "... = Real (\<Sum>x\<in>s. (gmeasure (f x)))" apply(rule setsum_Real) by auto
- finally have sum:"(\<Sum>x\<in>s. lmeasure (f x)) = Real (\<Sum>x\<in>s. gmeasure (f x))" .
- have sum_0:"(\<Sum>x\<in>s. gmeasure (f x)) \<ge> 0" apply(rule setsum_nonneg) by auto
- have int_un:"\<Union>f ` s has_gmeasure (\<Sum>x\<in>s. gmeasure (f x))"
- apply(rule has_gmeasure_negligible_unions_image) using assms gmea by auto
-
- have unun:"\<Union>{\<Union>f ` s \<inter> cube n |n. n \<in> UNIV} = \<Union>f ` s" unfolding simple_image
- proof safe fix x y assume as:"x \<in> f y" "y \<in> s"
- from mem_big_cube[of x] guess n . note n=this
- thus "x \<in> \<Union>range (\<lambda>n. \<Union>f ` s \<inter> cube n)" unfolding Union_iff
- apply-apply(rule_tac x="\<Union>f ` s \<inter> cube n" in bexI) using as by auto
- qed
- show ?l apply(subst Real_real'[OF *,THEN sym])apply(subst lim_Real)
- apply rule apply rule unfolding sum real_Real if_P[OF sum_0] apply(rule sum_0)
- unfolding measure_unique[OF int_un,THEN sym] apply(subst(2) unun[THEN sym])
- apply(rule has_gmeasure_nested_unions[THEN conjunct2])
- proof- fix n::nat
- show *:"gmeasurable (\<Union>f ` s \<inter> cube n)" using lmeaf unfolding lmeasurable_def by auto
- thus "gmeasure (\<Union>f ` s \<inter> cube n) \<le> gmeasure (\<Union>f ` s)"
- apply(rule measure_subset) using int_un by auto
- show "\<Union>f ` s \<inter> cube n \<subseteq> \<Union>f ` s \<inter> cube (Suc n)"
- using cube_subset[of n "Suc n"] by auto
- qed
-
- next case True then guess X .. note X=this
- hence sum:"(\<Sum>x\<in>s. lmeasure (f x)) = \<omega>" using setsum_\<omega>[THEN iffD2, of s] assms by fastsimp
- show ?l unfolding sum Lim_omega
- proof fix B::real
- have Xm:"(f X) has_lmeasure \<omega>" using X by (metis assms(2) has_lmeasure_lmeasure)
- note this[unfolded has_lmeasure_def,THEN conjunct2, unfolded Lim_omega]
- from this[rule_format,of B] guess N .. note N=this[rule_format]
- show "\<exists>N. \<forall>n\<ge>N. Real B \<le> Real (gmeasure (\<Union>f ` s \<inter> cube n))"
- apply(rule_tac x=N in exI)
- proof safe case goal1 show ?case apply(rule order_trans[OF N[OF goal1]])
- unfolding pinfreal_less_eq apply(subst if_P) defer
- apply(rule measure_subset) using has_lmeasureD(2)[OF Xm]
- using lmeaf unfolding lmeasurable_def using X(1) by auto
- qed qed qed qed
-
-lemma has_lmeasure_negligible_unions:
- assumes"finite f" "\<And>s. s \<in> f ==> s has_lmeasure (m s)"
- "\<And>s t. s \<in> f \<Longrightarrow> t \<in> f \<Longrightarrow> s \<noteq> t ==> negligible (s\<inter>t)"
- shows "(\<Union> f) has_lmeasure (setsum m f)"
-proof- have *:"setsum m f = setsum lmeasure f" apply(rule setsum_cong2)
- apply(subst lmeasure_unique[OF assms(2)]) by auto
- show ?thesis unfolding *
- apply(rule has_lmeasure_negligible_unions_image[where s=f and f=id,unfolded image_id id_apply])
- using assms by auto
-qed
-
-lemma has_lmeasure_disjoint_unions:
- assumes"finite f" "\<And>s. s \<in> f ==> s has_lmeasure (m s)"
- "\<And>s t. s \<in> f \<Longrightarrow> t \<in> f \<Longrightarrow> s \<noteq> t ==> s \<inter> t = {}"
- shows "(\<Union> f) has_lmeasure (setsum m f)"
-proof- have *:"setsum m f = setsum lmeasure f" apply(rule setsum_cong2)
- apply(subst lmeasure_unique[OF assms(2)]) by auto
- show ?thesis unfolding *
- apply(rule has_lmeasure_negligible_unions_image[where s=f and f=id,unfolded image_id id_apply])
- using assms by auto
qed
-lemma has_lmeasure_nested_unions:
- assumes "\<And>n. lmeasurable(s n)" "\<And>n. s(n) \<subseteq> s(Suc n)"
- shows "lmeasurable(\<Union> { s n | n. n \<in> UNIV }) \<and>
- (\<lambda>n. lmeasure(s n)) ----> lmeasure(\<Union> { s(n) | n. n \<in> UNIV })" (is "?mea \<and> ?lim")
-proof- have cube:"\<And>m. \<Union> { s(n) | n. n \<in> UNIV } \<inter> cube m = \<Union> { s(n) \<inter> cube m | n. n \<in> UNIV }" by blast
- have 3:"\<And>n. \<forall>m\<ge>n. s n \<subseteq> s m" apply(rule transitive_stepwise_le) using assms(2) by auto
- have mea:"?mea" unfolding lmeasurable_def cube apply rule
- apply(rule_tac B1="gmeasure (cube n)" in has_gmeasure_nested_unions[THEN conjunct1])
- prefer 3 apply rule using assms(1) unfolding lmeasurable_def
- by(auto intro!:assms(2)[unfolded subset_eq,rule_format])
- show ?thesis apply(rule,rule mea)
- proof(cases "lmeasure(\<Union> { s(n) | n. n \<in> UNIV }) = \<omega>")
- case True show ?lim unfolding True Lim_omega
- proof(rule ccontr) case goal1 note this[unfolded not_all not_ex]
- hence "\<exists>B. \<forall>n. \<exists>m\<ge>n. Real B > lmeasure (s m)" by(auto simp add:not_le)
- from this guess B .. note B=this[rule_format]
-
- have "\<forall>n. gmeasurable (s n) \<and> gmeasure (s n) \<le> max B 0"
- proof safe fix n::nat from B[of n] guess m .. note m=this
- hence *:"lmeasure (s n) < Real B" apply-apply(rule le_less_trans)
- apply(rule lmeasure_subset[OF assms(1,1)]) apply(rule 3[rule_format]) by auto
- thus **:"gmeasurable (s n)" apply-apply(rule glmeasurable_finite[OF assms(1)]) by auto
- thus "gmeasure (s n) \<le> max B 0" using * unfolding gmeasure_lmeasure[OF **] Real_max'[of B]
- unfolding pinfreal_less apply- apply(subst(asm) if_P) by auto
- qed
- hence "\<And>n. gmeasurable (s n)" "\<And>n. gmeasure (s n) \<le> max B 0" by auto
- note g = conjunctD2[OF has_gmeasure_nested_unions[of s, OF this assms(2)]]
- show False using True unfolding gmeasure_lmeasure[OF g(1)] by auto
- qed
- next let ?B = "lmeasure (\<Union>{s n |n. n \<in> UNIV})"
- case False note gmea_lim = glmeasurable_finite[OF mea this]
- have ls:"\<And>n. lmeasure (s n) \<le> lmeasure (\<Union>{s n |n. n \<in> UNIV})"
- apply(rule lmeasure_subset) using assms(1) mea by auto
- have "\<And>n. lmeasure (s n) \<noteq> \<omega>"
- proof(rule ccontr,safe) case goal1
- show False using False ls[of n] unfolding goal1 by auto
- qed
- note gmea = glmeasurable_finite[OF assms(1) this]
-
- have "\<And>n. gmeasure (s n) \<le> real ?B" unfolding gmeasure_lmeasure[OF gmea_lim]
- unfolding real_Real apply(subst if_P,rule) apply(rule measure_subset)
- using gmea gmea_lim by auto
- note has_gmeasure_nested_unions[of s, OF gmea this assms(2)]
- thus ?lim unfolding gmeasure_lmeasure[OF gmea] gmeasure_lmeasure[OF gmea_lim]
- apply-apply(subst lim_Real) by auto
- qed
-qed
-
-lemma has_lmeasure_countable_negligible_unions:
- assumes "\<And>n. lmeasurable(s n)" "\<And>m n. m \<noteq> n \<Longrightarrow> negligible(s m \<inter> s n)"
- shows "(\<lambda>m. setsum (\<lambda>n. lmeasure(s n)) {..m}) ----> (lmeasure(\<Union> { s(n) |n. n \<in> UNIV }))"
-proof- have *:"\<And>n. (\<Union> (s ` {0..n})) has_lmeasure (setsum (\<lambda>k. lmeasure(s k)) {0..n})"
- apply(rule has_lmeasure_negligible_unions_image) using assms by auto
- have **:"(\<Union>{\<Union>s ` {0..n} |n. n \<in> UNIV}) = (\<Union>{s n |n. n \<in> UNIV})" unfolding simple_image by fastsimp
- have "lmeasurable (\<Union>{\<Union>s ` {0..n} |n. n \<in> UNIV}) \<and>
- (\<lambda>n. lmeasure (\<Union>(s ` {0..n}))) ----> lmeasure (\<Union>{\<Union>s ` {0..n} |n. n \<in> UNIV})"
- apply(rule has_lmeasure_nested_unions) apply(rule has_lmeasureD(1)[OF *])
- apply(rule Union_mono,rule image_mono) by auto
- note lem = conjunctD2[OF this,unfolded **]
- show ?thesis using lem(2) unfolding lmeasure_unique[OF *] unfolding atLeast0AtMost .
-qed
-
-lemma lmeasure_eq_0: assumes "negligible s" shows "lmeasure s = 0"
-proof- note mea=negligible_imp_lmeasurable[OF assms]
- have *:"\<And>n. (gmeasure (s \<inter> cube n)) = 0"
- unfolding gmeasurable_measure_eq_0[OF mea[unfolded lmeasurable_def,rule_format]]
- using assms by auto
- show ?thesis
- apply(rule lmeasure_unique) unfolding has_lmeasure_def
- apply(rule,rule mea) unfolding * by auto
+lemma has_gmeasure_iff_lmeasure:
+ "A has_gmeasure m \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m)"
+proof
+ assume "A has_gmeasure m"
+ with has_gmeasure_lmeasure[OF this]
+ have "gmeasurable A" "0 \<le> m" "lmeasure A = Real m" by auto
+ then show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m" by auto
+next
+ assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m"
+ then show "A has_gmeasure m" by (intro lmeasure_finite_has_gmeasure) auto
qed
-lemma negligible_img_gmeasurable: fixes s::"'a::ordered_euclidean_space set"
- assumes "negligible s" shows "gmeasurable s"
- apply(rule glmeasurable_finite)
- using lmeasure_eq_0[OF assms] negligible_imp_lmeasurable[OF assms] by auto
-
-
-
-
-section {* Instantiation of _::euclidean_space as measure_space *}
-
-definition lebesgue_space :: "'a::ordered_euclidean_space algebra" where
- "lebesgue_space = \<lparr> space = UNIV, sets = lmeasurable \<rparr>"
-
-lemma lebesgue_measurable[simp]:"A \<in> sets lebesgue_space \<longleftrightarrow> lmeasurable A"
- unfolding lebesgue_space_def by(auto simp: mem_def)
-
-lemma mem_gmeasurable[simp]: "A \<in> gmeasurable \<longleftrightarrow> gmeasurable A"
- unfolding mem_def ..
-
-interpretation lebesgue: measure_space lebesgue_space lmeasure
- apply(intro_locales) unfolding measure_space_axioms_def countably_additive_def
- unfolding sigma_algebra_axioms_def algebra_def
- unfolding lebesgue_measurable
-proof safe
- fix A::"nat => _" assume as:"range A \<subseteq> sets lebesgue_space" "disjoint_family A"
- "lmeasurable (UNION UNIV A)"
- have *:"UNION UNIV A = \<Union>range A" by auto
- show "(\<Sum>\<^isub>\<infinity>n. lmeasure (A n)) = lmeasure (UNION UNIV A)"
- unfolding psuminf_def apply(rule SUP_Lim_pinfreal)
- proof- fix n m::nat assume mn:"m\<le>n"
- have *:"\<And>m. (\<Sum>n<m. lmeasure (A n)) = lmeasure (\<Union>A ` {..<m})"
- apply(subst lmeasure_unique[OF has_lmeasure_negligible_unions[where m=lmeasure]])
- apply(rule finite_imageI) apply rule apply(subst has_lmeasure_lmeasure[THEN sym])
- proof- fix m::nat
- show "(\<Sum>n<m. lmeasure (A n)) = setsum lmeasure (A ` {..<m})"
- apply(subst setsum_reindex_nonzero) unfolding o_def apply rule
- apply(rule lmeasure_eq_0) using as(2) unfolding disjoint_family_on_def
- apply(erule_tac x=x in ballE,safe,erule_tac x=y in ballE) by auto
- next fix m s assume "s \<in> A ` {..<m}"
- hence "s \<in> range A" by auto thus "lmeasurable s" using as(1) by fastsimp
- next fix m s t assume st:"s \<in> A ` {..<m}" "t \<in> A ` {..<m}" "s \<noteq> t"
- from st(1-2) guess sa ta unfolding image_iff apply-by(erule bexE)+ note a=this
- from st(3) have "sa \<noteq> ta" unfolding a by auto
- thus "negligible (s \<inter> t)"
- using as(2) unfolding disjoint_family_on_def a
- apply(erule_tac x=sa in ballE,erule_tac x=ta in ballE) by auto
- qed
-
- have "\<And>m. lmeasurable (\<Union>A ` {..<m})" apply(rule lmeasurable_finite_unions)
- apply(rule finite_imageI,rule) using as(1) by fastsimp
- from this this show "(\<Sum>n<m. lmeasure (A n)) \<le> (\<Sum>n<n. lmeasure (A n))" unfolding *
- apply(rule lmeasure_subset) apply(rule Union_mono) apply(rule image_mono) using mn by auto
-
- next have *:"UNION UNIV A = \<Union>{A n |n. n \<in> UNIV}" by auto
- show "(\<lambda>n. \<Sum>n<n. lmeasure (A n)) ----> lmeasure (UNION UNIV A)"
- apply(rule LIMSEQ_imp_Suc) unfolding lessThan_Suc_atMost *
- apply(rule has_lmeasure_countable_negligible_unions)
- using as unfolding disjoint_family_on_def subset_eq by auto
- qed
-
-next show "lmeasure {} = 0" by auto
-next fix A::"nat => _" assume as:"range A \<subseteq> sets lebesgue_space"
- have *:"UNION UNIV A = (\<Union>{A n |n. n \<in> UNIV})" unfolding simple_image by auto
- show "lmeasurable (UNION UNIV A)" unfolding * using as unfolding subset_eq
- using lmeasurable_countable_unions_strong[of A] by auto
-qed(auto simp: lebesgue_space_def mem_def)
-
-
-
-lemma lmeasurbale_closed_interval[intro]:
- "lmeasurable {a..b::'a::ordered_euclidean_space}"
- unfolding lmeasurable_def cube_def inter_interval by auto
-
-lemma space_lebesgue_space[simp]:"space lebesgue_space = UNIV"
- unfolding lebesgue_space_def by auto
-
-abbreviation "gintegral \<equiv> Integration.integral"
+lemma gmeasure_lmeasure: assumes "gmeasurable s" shows "lmeasure s = Real (gmeasure s)"
+proof -
+ note has_gmeasure_measureI[OF assms]
+ note has_gmeasure_lmeasure[OF this]
+ thus ?thesis .
+qed
lemma lebesgue_simple_function_indicator:
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pinfreal"
@@ -556,21 +447,614 @@
lemma lmeasure_gmeasure:
"gmeasurable s \<Longrightarrow> gmeasure s = real (lmeasure s)"
- apply(subst gmeasure_lmeasure) by auto
+ by (subst gmeasure_lmeasure) auto
lemma lmeasure_finite: assumes "gmeasurable s" shows "lmeasure s \<noteq> \<omega>"
using gmeasure_lmeasure[OF assms] by auto
-lemma negligible_lmeasure: assumes "lmeasurable s"
- shows "lmeasure s = 0 \<longleftrightarrow> negligible s" (is "?l = ?r")
-proof assume ?l
- hence *:"gmeasurable s" using glmeasurable_finite[of s] assms by auto
- show ?r unfolding gmeasurable_measure_eq_0[THEN sym,OF *]
- unfolding lmeasure_gmeasure[OF *] using `?l` by auto
-next assume ?r
- note g=negligible_img_gmeasurable[OF this] and measure_eq_0[OF this]
- hence "real (lmeasure s) = 0" using lmeasure_gmeasure[of s] by auto
- thus ?l using lmeasure_finite[OF g] apply- apply(rule real_0_imp_eq_0) by auto
+lemma negligible_iff_lebesgue_null_sets:
+ "negligible A \<longleftrightarrow> A \<in> lebesgue.null_sets"
+proof
+ assume "negligible A"
+ from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0]
+ show "A \<in> lebesgue.null_sets" by auto
+next
+ assume A: "A \<in> lebesgue.null_sets"
+ then have *:"gmeasurable A" using lmeasure_finite_gmeasurable[of A] by auto
+ show "negligible A"
+ unfolding gmeasurable_measure_eq_0[OF *, symmetric]
+ unfolding lmeasure_gmeasure[OF *] using A by auto
+qed
+
+lemma
+ fixes a b ::"'a::ordered_euclidean_space"
+ shows lmeasure_atLeastAtMost[simp]: "lmeasure {a..b} = Real (content {a..b})"
+ and lmeasure_greaterThanLessThan[simp]: "lmeasure {a <..< b} = Real (content {a..b})"
+ using has_gmeasure_interval[of a b] by (auto intro!: has_gmeasure_lmeasure)
+
+lemma lmeasure_cube:
+ "lmeasure (cube n::('a::ordered_euclidean_space) set) = (Real ((2 * real n) ^ (DIM('a))))"
+ by (intro has_gmeasure_lmeasure) auto
+
+lemma lmeasure_UNIV[intro]: "lmeasure UNIV = \<omega>"
+ unfolding lmeasure_def SUP_\<omega>
+proof (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 (gmeasure (UNIV \<inter> cube i))"
+ proof (intro bexI[of _ n])
+ have "x < Real (of_nat n)" using n r by auto
+ also have "Real (of_nat n) \<le> Real (gmeasure (UNIV \<inter> cube n))"
+ using gmeasure_cube_ge_n[of n] by (auto simp: real_eq_of_nat[symmetric])
+ finally show "x < Real (gmeasure (UNIV \<inter> cube n))" .
+ qed auto
+qed
+
+lemma atLeastAtMost_singleton_euclidean[simp]:
+ fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}"
+ by (force simp: eucl_le[where 'a='a] euclidean_eq[where 'a='a])
+
+lemma content_singleton[simp]: "content {a} = 0"
+proof -
+ have "content {a .. a} = 0"
+ by (subst content_closed_interval) auto
+ then show ?thesis by simp
+qed
+
+lemma lmeasure_singleton[simp]:
+ fixes a :: "'a::ordered_euclidean_space" shows "lmeasure {a} = 0"
+ using has_gmeasure_interval[of a a] unfolding zero_pinfreal_def
+ by (intro has_gmeasure_lmeasure)
+ (simp add: content_closed_interval DIM_positive)
+
+declare content_real[simp]
+
+lemma
+ fixes a b :: real
+ shows lmeasure_real_greaterThanAtMost[simp]:
+ "lmeasure {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}"
+ by (subst lebesgue.measure_additive)
+ (auto intro!: lebesgueI_borel arg_cong[where f=lmeasure])
+ 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)" (is ?eqlt)
+proof cases
+ assume "a < b"
+ then have "lmeasure {a ..< b} = lmeasure {a} + lmeasure {a <..< b}"
+ by (subst lebesgue.measure_additive)
+ (auto intro!: lebesgueI_borel arg_cong[where f=lmeasure])
+ 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
+
+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 lmeasure_cube by auto
+qed
+
+interpretation lebesgue: sigma_finite_measure lebesgue lmeasure
+proof
+ from borel.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>)"
+ by auto
+qed
+
+lemma simple_function_has_integral:
+ fixes f::"'a::ordered_euclidean_space \<Rightarrow> pinfreal"
+ assumes f:"lebesgue.simple_function 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
+ 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>"
+ using f' om unfolding indicator_def by auto
+ show ?case unfolding space_lebesgue real_of_pinfreal_setsum'[OF *(1),THEN sym]
+ unfolding real_of_pinfreal_setsum'[OF *(2),THEN sym]
+ unfolding real_of_pinfreal_setsum space_lebesgue
+ 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"
+ proof(cases "f y = 0") case False
+ have mea:"gmeasurable (f -` {f y})" apply(rule lmeasure_finite_gmeasurable)
+ using assms unfolding lebesgue.simple_function_def using False by auto
+ have *:"\<And>x. real (indicator (f -` {f y}) x::pinfreal) = (if x \<in> f -` {f y} then 1 else 0)" by auto
+ show ?thesis unfolding real_of_pinfreal_mult[THEN sym]
+ apply(rule has_integral_cmul[where 'b=real, unfolded real_scaleR_def])
+ unfolding Int_UNIV_right lmeasure_gmeasure[OF mea,THEN sym]
+ unfolding measure_integral_univ[OF mea] * apply(rule integrable_integral)
+ unfolding gmeasurable_integrable[THEN sym] using mea .
+ qed auto
+ qed qed
+
+lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"
+ unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)
+ using assms by auto
+
+lemma simple_function_has_integral':
+ fixes f::"'a::ordered_euclidean_space \<Rightarrow> pinfreal"
+ 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"
+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"
+ 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"
+ 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"
+ 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
+ have "f y * lmeasure (f -` {f y}) \<noteq> \<omega>" using i f
+ unfolding lebesgue.simple_integral_def setsum_\<omega> lebesgue.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
+ qed
+ qed
+qed
+
+lemma (in measure_space) positive_integral_monotone_convergence:
+ fixes f::"nat \<Rightarrow> 'a \<Rightarrow> pinfreal"
+ 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)
+proof -
+ from positive_integral_isoton[unfolded isoton_fun_expand isoton_iff_Lim_mono, of f u]
+ show ?ilim using mono lim i by auto
+ have "(SUP i. f i) = u" using mono lim SUP_Lim_pinfreal
+ unfolding fun_eq_iff SUPR_fun_expand mono_def by auto
+ moreover have "(SUP i. f i) \<in> borel_measurable M"
+ using i by (rule borel_measurable_SUP)
+ ultimately show "u \<in> borel_measurable M" by simp
+qed
+
+lemma positive_integral_has_integral:
+ fixes f::"'a::ordered_euclidean_space => pinfreal"
+ assumes f:"f \<in> borel_measurable lebesgue"
+ and int_om:"lebesgue.positive_integral 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"
+ 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)"
+ 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"
+ 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>"
+ 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]
+ have "(\<lambda>x. real (f x)) integrable_on UNIV \<and>
+ (\<lambda>k. Integration.integral UNIV (\<lambda>x. real (u k x))) ----> Integration.integral UNIV (\<lambda>x. real (f x))"
+ apply(rule monotone_convergence_increasing) apply(rule,rule,rule u_int)
+ proof safe case goal1 show ?case apply(rule real_of_pinfreal_mono) using u(2,3) by auto
+ next case goal2 show ?case using u(3) apply(subst lim_Real[THEN sym])
+ 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)"])
+ 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_pinfreal_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
+ 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-
+ 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
+ note * = LIMSEQ_unique[OF this int(2)[unfolded integral_unique[OF u_int]]]
+ show ?thesis unfolding * by(rule integrable_integral[OF int(1)])
+qed
+
+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"
+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 *)
+ 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)]
+ from positive_integral_has_integral[OF this f(2) *]
+ show ?case unfolding real_Real_max .
+ next case goal2
+ have *:"\<forall>x. Real (- f x) \<noteq> \<omega>" by auto
+ note lebesgue.borel_measurable_uminus[OF f(1)]
+ note lebesgue.borel_measurable_Real[OF this]
+ from positive_integral_has_integral[OF this f(3) *]
+ show ?case unfolding real_Real_max minus_min_eq_max by auto
+ qed
+qed
+
+lemma continuous_on_imp_borel_measurable:
+ fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
+ assumes "continuous_on UNIV f"
+ shows "f \<in> borel_measurable lebesgue"
+ apply(rule lebesgue.borel_measurableI)
+ 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)"
+ 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"
+ using integral_monotone_convergence_pos[OF assms] by auto
+
+definition e2p :: "'a::ordered_euclidean_space \<Rightarrow> (nat \<Rightarrow> real)" where
+ "e2p x = (\<lambda>i\<in>{..<DIM('a)}. x$$i)"
+
+definition p2e :: "(nat \<Rightarrow> real) \<Rightarrow> 'a::ordered_euclidean_space" where
+ "p2e x = (\<chi>\<chi> i. x i)"
+
+lemma bij_euclidean_component:
+ "bij_betw (e2p::'a::ordered_euclidean_space \<Rightarrow> _) (UNIV :: 'a set)
+ ({..<DIM('a)} \<rightarrow>\<^isub>E (UNIV :: real set))"
+ unfolding bij_betw_def e2p_def_raw
+proof let ?e = "\<lambda>x.\<lambda>i\<in>{..<DIM('a::ordered_euclidean_space)}. (x::'a)$$i"
+ show "inj ?e" unfolding inj_on_def restrict_def apply(subst euclidean_eq) apply safe
+ apply(drule_tac x=i in fun_cong) by auto
+ { fix x::"nat \<Rightarrow> real" assume x:"\<forall>i. i \<notin> {..<DIM('a)} \<longrightarrow> x i = undefined"
+ hence "x = ?e (\<chi>\<chi> i. x i)" apply-apply(rule,case_tac "xa<DIM('a)") by auto
+ hence "x \<in> range ?e" by fastsimp
+ } thus "range ?e = ({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)}"
+ unfolding extensional_def using DIM_positive by auto
+qed
+
+lemma bij_p2e:
+ "bij_betw (p2e::_ \<Rightarrow> 'a::ordered_euclidean_space) ({..<DIM('a)} \<rightarrow>\<^isub>E (UNIV :: real set))
+ (UNIV :: 'a set)" (is "bij_betw ?p ?U _")
+ unfolding bij_betw_def
+proof show "inj_on ?p ?U" unfolding inj_on_def p2e_def
+ apply(subst euclidean_eq) apply(safe,rule) unfolding extensional_def
+ apply(case_tac "xa<DIM('a)") by auto
+ { fix x::'a have "x \<in> ?p ` extensional {..<DIM('a)}"
+ unfolding image_iff apply(rule_tac x="\<lambda>i. if i<DIM('a) then x$$i else undefined" in bexI)
+ apply(subst euclidean_eq,safe) unfolding p2e_def extensional_def by auto
+ } thus "?p ` ?U = UNIV" by auto
+qed
+
+lemma e2p_p2e[simp]: fixes z::"'a::ordered_euclidean_space"
+ assumes "x \<in> extensional {..<DIM('a)}"
+ shows "e2p (p2e x::'a) = x"
+proof fix i::nat
+ show "e2p (p2e x::'a) i = x i" unfolding e2p_def p2e_def restrict_def
+ using assms unfolding extensional_def by auto
+qed
+
+lemma p2e_e2p[simp]: fixes x::"'a::ordered_euclidean_space"
+ shows "p2e (e2p x) = x"
+ apply(subst euclidean_eq) unfolding e2p_def p2e_def restrict_def by auto
+
+interpretation borel_product: product_sigma_finite "\<lambda>x. borel::real algebra" "\<lambda>x. lmeasure"
+ 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 borel_vimage_algebra_eq:
+ "sigma_algebra.vimage_algebra
+ (borel :: ('a::ordered_euclidean_space) algebra) ({..<DIM('a)} \<rightarrow>\<^isub>E UNIV) p2e =
+ sigma (product_algebra (\<lambda>x. \<lparr> space = UNIV::real set, sets = range (\<lambda>a. {..<a}) \<rparr>) {..<DIM('a)} )"
+proof- note bor = borel_eq_lessThan
+ def F \<equiv> "product_algebra (\<lambda>x. \<lparr> space = UNIV::real set, sets = range (\<lambda>a. {..<a}) \<rparr>) {..<DIM('a)}"
+ def E \<equiv> "\<lparr>space = (UNIV::'a set), sets = range lessThan\<rparr>"
+ have *:"(({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)}) = space F" unfolding F_def by auto
+ show ?thesis unfolding F_def[symmetric] * bor
+ proof(rule vimage_algebra_sigma,unfold E_def[symmetric])
+ show "sets E \<subseteq> Pow (space E)" "p2e \<in> space F \<rightarrow> space E" unfolding E_def by auto
+ next fix A assume "A \<in> sets F"
+ hence A:"A \<in> (Pi\<^isub>E {..<DIM('a)}) ` ({..<DIM('a)} \<rightarrow> range lessThan)"
+ unfolding F_def product_algebra_def algebra.simps .
+ then guess B unfolding image_iff .. note B=this
+ hence "\<forall>x<DIM('a). B x \<in> range lessThan" by auto
+ hence "\<forall>x. \<exists>xa. x<DIM('a) \<longrightarrow> B x = {..<xa}" unfolding image_iff by auto
+ from choice[OF this] guess b .. note b=this
+ hence b':"\<forall>i<DIM('a). Sup (B i) = b i" using Sup_lessThan by auto
+
+ show "A \<in> (\<lambda>X. p2e -` X \<inter> space F) ` sets E" unfolding image_iff B
+ proof(rule_tac x="{..< \<chi>\<chi> i. Sup (B i)}" in bexI)
+ show "Pi\<^isub>E {..<DIM('a)} B = p2e -` {..<(\<chi>\<chi> i. Sup (B i))::'a} \<inter> space F"
+ unfolding F_def E_def product_algebra_def algebra.simps
+ proof(rule,unfold subset_eq,rule_tac[!] ballI)
+ fix x assume "x \<in> Pi\<^isub>E {..<DIM('a)} B"
+ hence *:"\<forall>i<DIM('a). x i < b i" "\<forall>i\<ge>DIM('a). x i = undefined"
+ unfolding Pi_def extensional_def using b by auto
+ have "(p2e x::'a) < (\<chi>\<chi> i. Sup (B i))" unfolding less_prod_def eucl_less[of "p2e x"]
+ apply safe unfolding euclidean_lambda_beta b'[rule_format] p2e_def using * by auto
+ moreover have "x \<in> extensional {..<DIM('a)}"
+ using *(2) unfolding extensional_def by auto
+ ultimately show "x \<in> p2e -` {..<(\<chi>\<chi> i. Sup (B i)) ::'a} \<inter>
+ (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})" by auto
+ next fix x assume as:"x \<in> p2e -` {..<(\<chi>\<chi> i. Sup (B i))::'a} \<inter>
+ (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
+ hence "p2e x < ((\<chi>\<chi> i. Sup (B i))::'a)" by auto
+ hence "\<forall>i<DIM('a). x i \<in> B i" apply-apply(subst(asm) eucl_less)
+ unfolding p2e_def using b b' by auto
+ thus "x \<in> Pi\<^isub>E {..<DIM('a)} B" using as unfolding Pi_def extensional_def by auto
+ qed
+ show "{..<(\<chi>\<chi> i. Sup (B i))::'a} \<in> sets E" unfolding E_def algebra.simps by auto
+ qed
+ next fix A assume "A \<in> sets E"
+ then guess a unfolding E_def algebra.simps image_iff .. note a = this(2)
+ def B \<equiv> "\<lambda>i. {..<a $$ i}"
+ show "p2e -` A \<inter> space F \<in> sets F" unfolding F_def
+ unfolding product_algebra_def algebra.simps image_iff
+ apply(rule_tac x=B in bexI) apply rule unfolding subset_eq apply(rule_tac[1-2] ballI)
+ proof- show "B \<in> {..<DIM('a)} \<rightarrow> range lessThan" unfolding B_def by auto
+ fix x assume as:"x \<in> p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
+ hence "p2e x \<in> A" by auto
+ hence "\<forall>i<DIM('a). x i \<in> B i" unfolding B_def a lessThan_iff
+ apply-apply(subst (asm) eucl_less) unfolding p2e_def by auto
+ thus "x \<in> Pi\<^isub>E {..<DIM('a)} B" using as unfolding Pi_def extensional_def by auto
+ next fix x assume x:"x \<in> Pi\<^isub>E {..<DIM('a)} B"
+ moreover have "p2e x \<in> A" unfolding a lessThan_iff p2e_def apply(subst eucl_less)
+ using x unfolding Pi_def extensional_def B_def by auto
+ ultimately show "x \<in> p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})" by auto
+ qed
+ qed
+qed
+
+lemma Real_mult_nonneg: assumes "x \<ge> 0" "y \<ge> 0"
+ shows "Real (x * y) = Real x * Real y" using assms by auto
+
+lemma Real_setprod: assumes "\<forall>x\<in>A. f x \<ge> 0" shows "Real (setprod f A) = setprod (\<lambda>x. Real (f x)) A"
+proof(cases "finite A")
+ case True thus ?thesis using assms
+ proof(induct A) case (insert x A)
+ have "0 \<le> setprod f A" apply(rule setprod_nonneg) using insert by auto
+ thus ?case unfolding setprod_insert[OF insert(1-2)] apply-
+ apply(subst Real_mult_nonneg) prefer 3 apply(subst insert(3)[THEN sym])
+ using insert by auto
+ qed auto
+qed auto
+
+lemma e2p_Int:"e2p ` A \<inter> e2p ` B = e2p ` (A \<inter> B)" (is "?L = ?R")
+ apply(rule image_Int[THEN sym]) using bij_euclidean_component
+ unfolding bij_betw_def by auto
+
+lemma Int_stable_cuboids: fixes x::"'a::ordered_euclidean_space"
+ shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). e2p ` {a..b})\<rparr>"
+ unfolding Int_stable_def algebra.select_convs
+proof safe fix a b x y::'a
+ have *:"e2p ` {a..b} \<inter> e2p ` {x..y} =
+ (\<lambda>(a, b). e2p ` {a..b}) (\<chi>\<chi> i. max (a $$ i) (x $$ i), \<chi>\<chi> i. min (b $$ i) (y $$ i)::'a)"
+ unfolding e2p_Int inter_interval by auto
+ show "e2p ` {a..b} \<inter> e2p ` {x..y} \<in> range (\<lambda>(a, b). e2p ` {a..b::'a})" unfolding *
+ apply(rule range_eqI) ..
+qed
+
+lemma Int_stable_cuboids': fixes x::"'a::ordered_euclidean_space"
+ shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). {a..b})\<rparr>"
+ unfolding Int_stable_def algebra.select_convs
+ apply safe unfolding inter_interval by auto
+
+lemma product_borel_eq_vimage:
+ "sigma (product_algebra (\<lambda>x. borel) {..<DIM('a::ordered_euclidean_space)}) =
+ sigma_algebra.vimage_algebra borel (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})
+ (p2e:: _ \<Rightarrow> 'a::ordered_euclidean_space)"
+ unfolding borel_vimage_algebra_eq unfolding borel_eq_lessThan
+ apply(subst sigma_product_algebra_sigma_eq[where S="\<lambda>i. \<lambda>n. lessThan (real n)"])
+ unfolding lessThan_iff
+proof- fix i assume i:"i<DIM('a)"
+ show "(\<lambda>n. {..<real n}) \<up> space \<lparr>space = UNIV, sets = range lessThan\<rparr>"
+ by(auto intro!:real_arch_lt isotoneI)
+qed auto
+
+lemma inj_on_disjoint_family_on: assumes "disjoint_family_on A S" "inj f"
+ shows "disjoint_family_on (\<lambda>x. f ` A x) S"
+ unfolding disjoint_family_on_def
+proof(rule,rule,rule)
+ fix x1 x2 assume x:"x1 \<in> S" "x2 \<in> S" "x1 \<noteq> x2"
+ show "f ` A x1 \<inter> f ` A x2 = {}"
+ proof(rule ccontr) case goal1
+ then obtain z where z:"z \<in> f ` A x1 \<inter> f ` A x2" by auto
+ then obtain z1 z2 where z12:"z1 \<in> A x1" "z2 \<in> A x2" "f z1 = z" "f z2 = z" by auto
+ hence "z1 = z2" using assms(2) unfolding inj_on_def by blast
+ hence "x1 = x2" using z12(1-2) using assms[unfolded disjoint_family_on_def] using x by auto
+ thus False using x(3) by auto
+ qed
+qed
+
+declare restrict_extensional[intro]
+
+lemma e2p_extensional[intro]:"e2p (y::'a::ordered_euclidean_space) \<in> extensional {..<DIM('a)}"
+ unfolding e2p_def by auto
+
+lemma e2p_image_vimage: fixes A::"'a::ordered_euclidean_space set"
+ shows "e2p ` A = p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
+proof(rule set_eqI,rule)
+ fix x assume "x \<in> e2p ` A" then guess y unfolding image_iff .. note y=this
+ show "x \<in> p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
+ apply safe apply(rule vimageI[OF _ y(1)]) unfolding y p2e_e2p by auto
+next fix x assume "x \<in> p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
+ thus "x \<in> e2p ` A" unfolding image_iff apply(rule_tac x="p2e x" in bexI) apply(subst e2p_p2e) by auto
+qed
+
+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)"
+proof (rule measure_unique_Int_stable[where X=A and A=cube])
+ interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<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 lmeasure_cube by auto
+ show "\<And>X. X \<in> sets ?E \<Longrightarrow>
+ lmeasure X = borel_product.product_measure {..<DIM('a)} (e2p ` X :: (nat \<Rightarrow> real) set)"
+ 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 }
+ def XX \<equiv> "\<lambda>i. {a $$ i .. b $$ i}" assume "X \<noteq> {}" note X' = this[unfolded X interval_ne_empty]
+ have *:"Pi\<^isub>E {..<DIM('a)} XX = e2p ` X" apply(rule set_eqI)
+ proof fix x assume "x \<in> Pi\<^isub>E {..<DIM('a)} XX"
+ thus "x \<in> e2p ` X" unfolding image_iff apply(rule_tac x="\<chi>\<chi> i. x i" in bexI)
+ unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by rule auto
+ next fix x assume "x \<in> e2p ` X" then guess y unfolding image_iff .. note y = this
+ 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
+ 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)
+ 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
+ finally show ?case .
+ qed
+
+ show "range cube \<subseteq> sets \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>"
+ unfolding cube_def_raw by auto
+ 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))"
+ apply default unfolding countably_additive_def
+ proof safe fix A::"nat \<Rightarrow> 'a set" assume A:"range A \<subseteq> sets borel" "disjoint_family A"
+ "(\<Union>i. A i) \<in> sets borel"
+ note fprod.ca[unfolded countably_additive_def,rule_format]
+ note ca = this[of "\<lambda> n. e2p ` (A n)"]
+ show "(\<Sum>\<^isub>\<infinity>n. finite_product_sigma_finite.measure
+ (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` A n)) =
+ finite_product_sigma_finite.measure (\<lambda>x. borel)
+ (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` (\<Union>i. A i))" unfolding image_UN
+ proof(rule ca) show "range (\<lambda>n. e2p ` A n) \<subseteq> sets
+ (sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)}))"
+ unfolding product_borel_eq_vimage
+ proof case goal1
+ then guess y unfolding image_iff .. note y=this(2)
+ show ?case unfolding borel.in_vimage_algebra y apply-
+ apply(rule_tac x="A y" in bexI,rule e2p_image_vimage)
+ using A(1) by auto
+ qed
+
+ show "disjoint_family (\<lambda>n. e2p ` A n)" apply(rule inj_on_disjoint_family_on)
+ using bij_euclidean_component using A(2) unfolding bij_betw_def by auto
+ show "(\<Union>n. e2p ` A n) \<in> sets (sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)}))"
+ unfolding product_borel_eq_vimage borel.in_vimage_algebra
+ proof(rule bexI[OF _ A(3)],rule set_eqI,rule)
+ fix x assume x:"x \<in> (\<Union>n. e2p ` A n)" hence "p2e x \<in> (\<Union>i. A i)" by auto
+ moreover have "x \<in> extensional {..<DIM('a)}"
+ using x unfolding extensional_def e2p_def_raw by auto
+ ultimately show "x \<in> p2e -` (\<Union>i. A i) \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter>
+ extensional {..<DIM('a)})" by auto
+ next fix x assume x:"x \<in> p2e -` (\<Union>i. A i) \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter>
+ extensional {..<DIM('a)})"
+ hence "p2e x \<in> (\<Union>i. A i)" by auto
+ hence "\<exists>n. x \<in> e2p ` A n" apply safe apply(rule_tac x=i in exI)
+ unfolding image_iff apply(rule_tac x="p2e x" in bexI)
+ apply(subst e2p_p2e) using x by auto
+ thus "x \<in> (\<Union>n. e2p ` A n)" by auto
+ qed
+ qed
+ qed auto
+qed
+
+lemma e2p_p2e'[simp]: fixes x::"'a::ordered_euclidean_space"
+ assumes "A \<subseteq> extensional {..<DIM('a)}"
+ shows "e2p ` (p2e ` A ::'a set) = A"
+ apply(rule set_eqI) unfolding image_iff Bex_def apply safe defer
+ apply(rule_tac x="p2e x" in exI,safe) using assms by auto
+
+lemma range_p2e:"range (p2e::_\<Rightarrow>'a::ordered_euclidean_space) = UNIV"
+ apply safe defer unfolding image_iff apply(rule_tac x="\<lambda>i. x $$ i" in bexI)
+ unfolding p2e_def by auto
+
+lemma p2e_inv_extensional:"(A::'a::ordered_euclidean_space set)
+ = p2e ` (p2e -` A \<inter> extensional {..<DIM('a)})"
+ unfolding p2e_def_raw apply safe unfolding image_iff
+proof- fix x assume "x\<in>A"
+ let ?y = "\<lambda>i. if i<DIM('a) then x$$i else undefined"
+ have *:"Chi ?y = x" apply(subst euclidean_eq) by auto
+ show "\<exists>xa\<in>Chi -` A \<inter> extensional {..<DIM('a)}. x = Chi xa" apply(rule_tac x="?y" in bexI)
+ apply(subst euclidean_eq) unfolding extensional_def using `x\<in>A` by(auto simp: *)
+qed
+
+lemma borel_fubini_positiv_integral:
+ fixes f :: "'a::ordered_euclidean_space \<Rightarrow> pinfreal"
+ 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> "(({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)}):: (nat \<Rightarrow> real) set"
+ interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
+ have "\<And>x. \<exists>i::nat. x < real i" by (metis real_arch_lt)
+ hence "(\<lambda>n::nat. {..<real n}) \<up> UNIV" apply-apply(rule isotoneI) by auto
+ hence *:"sigma_algebra.vimage_algebra borel U (p2e:: _ \<Rightarrow> 'a)
+ = sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)})"
+ unfolding U_def apply-apply(subst borel_vimage_algebra_eq)
+ apply(subst sigma_product_algebra_sigma_eq[where S="\<lambda>x. \<lambda>n. {..<(\<chi>\<chi> i. real n)}", THEN sym])
+ unfolding borel_eq_lessThan[THEN sym] by auto
+ show ?thesis unfolding borel.positive_integral_vimage[unfolded space_borel,OF bij_p2e]
+ apply(subst fprod.positive_integral_cong_measure[THEN sym, of "\<lambda>A. lmeasure (p2e ` A)"])
+ unfolding U_def[symmetric] *[THEN sym] o_def
+ proof- fix A assume A:"A \<in> sets (sigma_algebra.vimage_algebra borel U (p2e ::_ \<Rightarrow> 'a))"
+ hence *:"A \<subseteq> extensional {..<DIM('a)}" unfolding U_def by auto
+ from A guess B unfolding borel.in_vimage_algebra U_def .. note B=this
+ have "(p2e ` A::'a set) \<in> sets borel" unfolding B apply(subst Int_left_commute)
+ apply(subst Int_absorb1) unfolding p2e_inv_extensional[of B,THEN sym] using B(1) by auto
+ from lmeasure_measure_eq_borel_prod[OF this] show "lmeasure (p2e ` A::'a set) =
+ finite_product_sigma_finite.measure (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} A"
+ unfolding e2p_p2e'[OF *] .
+ qed auto
+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
+ 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
+ unfolding borel_fubini_positiv_integral[OF 1] borel_fubini_positiv_integral[OF 2]
+ unfolding o_def ..
qed
end