(* Author: Robert Himmelmann, TU Muenchen *)
header {* Lebsegue measure *}
theory Lebesgue_Measure
imports Product_Measure Gauge_Measure Complete_Measure
begin
subsection {* Standard Cubes *}
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_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'
proof- fix x::'a and i assume as:"x\<in>ball 0 (real n)" "i<DIM('a)"
thus "- real n \<le> x $$ i" "real n \<ge> x $$ i"
using component_le_norm[of x i] by(auto simp: dist_norm)
qed
lemma mem_big_cube: obtains n where "x \<in> cube n"
proof- from real_arch_lt[of "norm x"] guess n ..
thus ?thesis apply-apply(rule that[where n=n])
apply(rule ball_subset_cube[unfolded subset_eq,rule_format])
by (auto simp add:dist_norm)
qed
lemma Union_inter_cube:"\<Union>{s \<inter> cube n |n. n \<in> UNIV} = s"
proof safe case goal1
from mem_big_cube[of x] guess n . note n=this
show ?case unfolding Union_iff apply(rule_tac x="s \<inter> cube n" in bexI)
using n goal1 by auto
qed
lemma gmeasurable_cube[intro]:"gmeasurable (cube n)"
unfolding cube_def by auto
lemma gmeasure_le_inter_cube[intro]: fixes s::"'a::ordered_euclidean_space set"
assumes "gmeasurable (s \<inter> cube n)" shows "gmeasure (s \<inter> cube n) \<le> gmeasure (cube n::'a set)"
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 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 lebesgueD[dest]: assumes "S \<in> sets lebesgue"
shows "\<And>n. gmeasurable (S \<inter> cube n)"
using assms unfolding lebesgue_def by auto
lemma lebesgueI[intro]: assumes "gmeasurable S"
shows "S \<in> sets lebesgue" unfolding lebesgue_def cube_def
using assms gmeasurable_interval by auto
lemma lebesgueI2: "(\<And>n. gmeasurable (S \<inter> cube n)) \<Longrightarrow> S \<in> sets lebesgue"
using assms unfolding lebesgue_def by auto
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 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 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 "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 "lmeasure A = (SUP n. Real (gmeasure (A \<inter> cube n)))"
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+
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 lmeasure_finite_has_gmeasure: assumes "s \<in> sets lebesgue" "lmeasure s = Real m" "0 \<le> m"
shows "s has_gmeasure m"
proof-
have *:"(\<lambda>n. (gmeasure (s \<inter> cube n))) ----> m"
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 "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 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
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)
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] using * .
show ?thesis unfolding has_gmeasure * apply(rule integrable_integral) using ** by auto
qed
lemma lmeasure_finite_gmeasurable: assumes "s \<in> sets lebesgue" "lmeasure s \<noteq> \<omega>"
shows "gmeasurable s"
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)
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 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] .
show "gmeasure (s \<inter> cube n) \<le> gmeasure s" apply(rule measure_subset)
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 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 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> pextreal"
assumes f:"lebesgue.simple_function 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
lemma lmeasure_gmeasure:
"gmeasurable s \<Longrightarrow> gmeasure s = real (lmeasure s)"
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_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_pextreal_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> pextreal"
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_pextreal_setsum'[OF *(1),THEN sym]
unfolding real_of_pextreal_setsum'[OF *(2),THEN sym]
unfolding real_of_pextreal_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::pextreal) = (if x \<in> f -` {f y} then 1 else 0)" by auto
show ?thesis unfolding real_of_pextreal_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> 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"
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> pextreal"
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 "\<And>x. (SUP i. f i x) = u x" using mono lim SUP_Lim_pextreal
unfolding fun_eq_iff mono_def by auto
moreover have "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M"
using i by auto
ultimately show "u \<in> borel_measurable M" by simp
qed
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 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_pextreal_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_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
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 lebesgue_positive_integral_eq_borel:
"f \<in> borel_measurable borel \<Longrightarrow> lebesgue.positive_integral f = borel.positive_integral 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)
proof -
have *: "sigma_algebra borel" by default
have "sets borel \<subseteq> sets lebesgue" by auto
from lebesgue.integral_subalgebra[OF assms this _ *]
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"
proof -
have borel: "f \<in> borel_measurable borel"
using f unfolding borel.integrable_def by auto
from f show ?thesis
using lebesgue_integral_has_integral[of f]
unfolding lebesgue_integral_eq_borel[OF borel] by simp
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 borel"
apply(rule borel.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 e2p_p2e[simp]:
"x \<in> extensional {..<DIM('a)} \<Longrightarrow> e2p (p2e x::'a::ordered_euclidean_space) = x"
by (auto simp: fun_eq_iff extensional_def p2e_def e2p_def)
lemma p2e_e2p[simp]:
"p2e (e2p x) = (x::'a::ordered_euclidean_space)"
by (auto simp: euclidean_eq[where 'a='a] p2e_def e2p_def)
lemma bij_inv_p2e_e2p:
shows "bij_inv ({..<DIM('a)} \<rightarrow>\<^isub>E UNIV) (UNIV :: 'a::ordered_euclidean_space set)
p2e e2p" (is "bij_inv ?P ?U _ _")
proof (rule bij_invI)
show "p2e \<in> ?P \<rightarrow> ?U" "e2p \<in> ?U \<rightarrow> ?P" by (auto simp: e2p_def)
qed 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 Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
unfolding Pi_def by auto
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
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)
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
then have "p2e -` A \<inter> space ?P = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< x $$ i})"
using DIM_positive
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 borel_vimage_algebra_eq:
defines "F \<equiv> product_algebra (\<lambda>x. \<lparr> space = (UNIV::real set), sets = range lessThan \<rparr>) {..<DIM('a::ordered_euclidean_space)}"
shows "sigma_algebra.vimage_algebra (borel::'a::ordered_euclidean_space algebra) (space (sigma F)) p2e = sigma F"
unfolding borel_eq_lessThan
proof (intro vimage_algebra_sigma)
let ?E = "\<lparr>space = (UNIV::'a set), sets = range lessThan\<rparr>"
show "bij_inv (space (sigma F)) (space (sigma ?E)) p2e e2p"
using bij_inv_p2e_e2p unfolding F_def by simp
show "sets F \<subseteq> Pow (space F)" "sets ?E \<subseteq> Pow (space ?E)" unfolding F_def
by (intro product_algebra_sets_into_space) auto
show "p2e \<in> measurable F ?E"
"e2p \<in> measurable ?E F"
unfolding F_def using measurable_p2e_on_generator measurable_e2p_on_generator by auto
qed
lemma product_borel_eq_vimage:
"sigma (product_algebra (\<lambda>x. borel) {..<DIM('a::ordered_euclidean_space)}) =
sigma_algebra.vimage_algebra borel (extensional {..<DIM('a)})
(p2e:: _ \<Rightarrow> 'a::ordered_euclidean_space)"
unfolding borel_vimage_algebra_eq[simplified]
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 e2p_Int:"e2p ` A \<inter> e2p ` B = e2p ` (A \<inter> B)" (is "?L = ?R")
apply(rule image_Int[THEN sym])
using bij_inv_p2e_e2p[THEN bij_inv_bij_betw(2)]
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 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> 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> 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> 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_inv_p2e_e2p[THEN bij_inv_bij_betw(2)] 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> extensional {..<DIM('a)}" by auto
next fix x assume x:"x \<in> p2e -` (\<Union>i. A i) \<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> 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
have *:"sigma_algebra.vimage_algebra borel U (p2e:: _ \<Rightarrow> 'a)
= sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)})"
unfolding U_def product_borel_eq_vimage[symmetric] ..
show ?thesis
unfolding borel.positive_integral_vimage[unfolded space_borel, OF bij_inv_p2e_e2p[THEN bij_inv_bij_betw(1)]]
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 ..
then have "(p2e ` A::'a set) \<in> sets borel"
by (simp add: p2e_inv_extensional[of B, symmetric])
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