(* Title: HOL/Probability/Lebesgue_Measure.thy
Author: Johannes Hölzl, TU München
Author: Robert Himmelmann, TU München
*)
header {* Lebsegue measure *}
theory Lebesgue_Measure
imports Finite_Product_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 (fastforce 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 (fastforce simp: cube_def eucl_le[where 'a='a])
then show "n \<le> N"
by (fastforce 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 reals_Archimedean2[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 cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)"
unfolding cube_def_raw subset_eq apply safe unfolding mem_interval by auto
subsection {* Lebesgue measure *}
definition lebesgue :: "'a::ordered_euclidean_space measure_space" where
"lebesgue = \<lparr> space = UNIV,
sets = {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n},
measure = \<lambda>A. SUP n. ereal (integral (cube n) (indicator A)) \<rparr>"
lemma space_lebesgue[simp]: "space lebesgue = UNIV"
unfolding lebesgue_def by simp
lemma lebesgueD: "A \<in> sets lebesgue \<Longrightarrow> (indicator A :: _ \<Rightarrow> real) integrable_on cube n"
unfolding lebesgue_def by simp
lemma lebesgueI: "(\<And>n. (indicator A :: _ \<Rightarrow> real) integrable_on cube n) \<Longrightarrow> A \<in> sets lebesgue"
unfolding lebesgue_def by simp
lemma absolutely_integrable_on_indicator[simp]:
fixes A :: "'a::ordered_euclidean_space set"
shows "((indicator A :: _ \<Rightarrow> real) absolutely_integrable_on X) \<longleftrightarrow>
(indicator A :: _ \<Rightarrow> real) integrable_on X"
unfolding absolutely_integrable_on_def by simp
lemma LIMSEQ_indicator_UN:
"(\<lambda>k. indicator (\<Union> i<k. A i) x) ----> (indicator (\<Union>i. A i) x :: real)"
proof cases
assume "\<exists>i. x \<in> A i" then guess i .. note i = this
then have *: "\<And>n. (indicator (\<Union> i<n + Suc i. A i) x :: real) = 1"
"(indicator (\<Union> i. A i) x :: real) = 1" by (auto simp: indicator_def)
show ?thesis
apply (rule LIMSEQ_offset[of _ "Suc i"]) unfolding * by auto
qed (auto simp: indicator_def)
lemma indicator_add:
"A \<inter> B = {} \<Longrightarrow> (indicator A x::_::monoid_add) + indicator B x = indicator (A \<union> B) x"
unfolding indicator_def by auto
interpretation lebesgue: sigma_algebra lebesgue
proof (intro sigma_algebra_iff2[THEN iffD2] conjI allI ballI impI lebesgueI)
fix A n assume A: "A \<in> sets lebesgue"
have "indicator (space lebesgue - A) = (\<lambda>x. 1 - indicator A x :: real)"
by (auto simp: fun_eq_iff indicator_def)
then show "(indicator (space lebesgue - A) :: _ \<Rightarrow> real) integrable_on cube n"
using A by (auto intro!: integrable_sub dest: lebesgueD simp: cube_def)
next
fix n show "(indicator {} :: _\<Rightarrow>real) integrable_on cube n"
by (auto simp: cube_def indicator_def_raw)
next
fix A :: "nat \<Rightarrow> 'a set" and n ::nat assume "range A \<subseteq> sets lebesgue"
then have A: "\<And>i. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n"
by (auto dest: lebesgueD)
show "(indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n" (is "?g integrable_on _")
proof (intro dominated_convergence[where g="?g"] ballI)
fix k show "(indicator (\<Union>i<k. A i) :: _ \<Rightarrow> real) integrable_on cube n"
proof (induct k)
case (Suc k)
have *: "(\<Union> i<Suc k. A i) = (\<Union> i<k. A i) \<union> A k"
unfolding lessThan_Suc UN_insert by auto
have *: "(\<lambda>x. max (indicator (\<Union> i<k. A i) x) (indicator (A k) x) :: real) =
indicator (\<Union> i<Suc k. A i)" (is "(\<lambda>x. max (?f x) (?g x)) = _")
by (auto simp: fun_eq_iff * indicator_def)
show ?case
using absolutely_integrable_max[of ?f "cube n" ?g] A Suc by (simp add: *)
qed auto
qed (auto intro: LIMSEQ_indicator_UN simp: cube_def)
qed simp
interpretation lebesgue: measure_space lebesgue
proof
have *: "indicator {} = (\<lambda>x. 0 :: real)" by (simp add: fun_eq_iff)
show "positive lebesgue (measure lebesgue)"
proof (unfold positive_def, safe)
show "measure lebesgue {} = 0" by (simp add: integral_0 * lebesgue_def)
fix A assume "A \<in> sets lebesgue"
then show "0 \<le> measure lebesgue A"
unfolding lebesgue_def
by (auto intro!: le_SUPI2 integral_nonneg)
qed
next
show "countably_additive lebesgue (measure lebesgue)"
proof (intro countably_additive_def[THEN iffD2] allI impI)
fix A :: "nat \<Rightarrow> 'b set" assume rA: "range A \<subseteq> sets lebesgue" "disjoint_family A"
then have A[simp, intro]: "\<And>i n. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n"
by (auto dest: lebesgueD)
let "?m n i" = "integral (cube n) (indicator (A i) :: _\<Rightarrow>real)"
let "?M n I" = "integral (cube n) (indicator (\<Union>i\<in>I. A i) :: _\<Rightarrow>real)"
have nn[simp, intro]: "\<And>i n. 0 \<le> ?m n i" by (auto intro!: integral_nonneg)
assume "(\<Union>i. A i) \<in> sets lebesgue"
then have UN_A[simp, intro]: "\<And>i n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n"
by (auto dest: lebesgueD)
show "(\<Sum>n. measure lebesgue (A n)) = measure lebesgue (\<Union>i. A i)"
proof (simp add: lebesgue_def, subst suminf_SUP_eq, safe intro!: incseq_SucI)
fix i n show "ereal (?m n i) \<le> ereal (?m (Suc n) i)"
using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le incseq_SucI)
next
fix i n show "0 \<le> ereal (?m n i)"
using rA unfolding lebesgue_def
by (auto intro!: le_SUPI2 integral_nonneg)
next
show "(SUP n. \<Sum>i. ereal (?m n i)) = (SUP n. ereal (?M n UNIV))"
proof (intro arg_cong[where f="SUPR UNIV"] ext sums_unique[symmetric] sums_ereal[THEN iffD2] sums_def[THEN iffD2])
fix n
have "\<And>m. (UNION {..<m} A) \<in> sets lebesgue" using rA by auto
from lebesgueD[OF this]
have "(\<lambda>m. ?M n {..< m}) ----> ?M n UNIV"
(is "(\<lambda>m. integral _ (?A m)) ----> ?I")
by (intro dominated_convergence(2)[where f="?A" and h="\<lambda>x. 1::real"])
(auto intro: LIMSEQ_indicator_UN simp: cube_def)
moreover
{ fix m have *: "(\<Sum>x<m. ?m n x) = ?M n {..< m}"
proof (induct m)
case (Suc m)
have "(\<Union>i<m. A i) \<in> sets lebesgue" using rA by auto
then have "(indicator (\<Union>i<m. A i) :: _\<Rightarrow>real) integrable_on (cube n)"
by (auto dest!: lebesgueD)
moreover
have "(\<Union>i<m. A i) \<inter> A m = {}"
using rA(2)[unfolded disjoint_family_on_def, THEN bspec, of m]
by auto
then have "\<And>x. indicator (\<Union>i<Suc m. A i) x =
indicator (\<Union>i<m. A i) x + (indicator (A m) x :: real)"
by (auto simp: indicator_add lessThan_Suc ac_simps)
ultimately show ?case
using Suc A by (simp add: integral_add[symmetric])
qed auto }
ultimately show "(\<lambda>m. \<Sum>x = 0..<m. ?m n x) ----> ?M n UNIV"
by (simp add: atLeast0LessThan)
qed
qed
qed
qed
lemma has_integral_interval_cube:
fixes a b :: "'a::ordered_euclidean_space"
shows "(indicator {a .. b} has_integral
content ({\<chi>\<chi> i. max (- real n) (a $$ i) .. \<chi>\<chi> i. min (real n) (b $$ i)} :: 'a set)) (cube n)"
(is "(?I has_integral content ?R) (cube n)")
proof -
let "{?N .. ?P}" = ?R
have [simp]: "(\<lambda>x. if x \<in> cube n then ?I x else 0) = indicator ?R"
by (auto simp: indicator_def cube_def fun_eq_iff eucl_le[where 'a='a])
have "(?I has_integral content ?R) (cube n) \<longleftrightarrow> (indicator ?R has_integral content ?R) UNIV"
unfolding has_integral_restrict_univ[where s="cube n", symmetric] by simp
also have "\<dots> \<longleftrightarrow> ((\<lambda>x. 1) has_integral content ?R) ?R"
unfolding indicator_def_raw has_integral_restrict_univ ..
finally show ?thesis
using has_integral_const[of "1::real" "?N" "?P"] by simp
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"
proof (safe intro!: lebesgueI)
fix n :: nat and a b :: 'a
let ?N = "\<chi>\<chi> i. max (- real n) (a $$ i)"
let ?P = "\<chi>\<chi> i. min (real n) (b $$ i)"
show "(indicator {a..b} :: 'a\<Rightarrow>real) integrable_on cube n"
unfolding integrable_on_def
using has_integral_interval_cube[of a b] by auto
qed
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"
using assms by (force simp: cube_def integrable_on_def negligible_def intro!: lebesgueI)
lemma lmeasure_eq_0:
fixes S :: "'a::ordered_euclidean_space set" assumes "negligible S" shows "lebesgue.\<mu> S = 0"
proof -
have "\<And>n. integral (cube n) (indicator S :: 'a\<Rightarrow>real) = 0"
unfolding lebesgue_integral_def using assms
by (intro integral_unique some1_equality ex_ex1I)
(auto simp: cube_def negligible_def)
then show ?thesis by (auto simp: lebesgue_def)
qed
lemma lmeasure_iff_LIMSEQ:
assumes "A \<in> sets lebesgue" "0 \<le> m"
shows "lebesgue.\<mu> A = ereal m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) ----> m"
proof (simp add: lebesgue_def, intro SUP_eq_LIMSEQ)
show "mono (\<lambda>n. integral (cube n) (indicator A::_=>real))"
using cube_subset assms by (intro monoI integral_subset_le) (auto dest!: lebesgueD)
qed
lemma has_integral_indicator_UNIV:
fixes s A :: "'a::ordered_euclidean_space set" and x :: real
shows "((indicator (s \<inter> A) :: 'a\<Rightarrow>real) has_integral x) UNIV = ((indicator s :: _\<Rightarrow>real) has_integral x) A"
proof -
have "(\<lambda>x. if x \<in> A then indicator s x else 0) = (indicator (s \<inter> A) :: _\<Rightarrow>real)"
by (auto simp: fun_eq_iff indicator_def)
then show ?thesis
unfolding has_integral_restrict_univ[where s=A, symmetric] by simp
qed
lemma
fixes s a :: "'a::ordered_euclidean_space set"
shows integral_indicator_UNIV:
"integral UNIV (indicator (s \<inter> A) :: 'a\<Rightarrow>real) = integral A (indicator s :: _\<Rightarrow>real)"
and integrable_indicator_UNIV:
"(indicator (s \<inter> A) :: 'a\<Rightarrow>real) integrable_on UNIV \<longleftrightarrow> (indicator s :: 'a\<Rightarrow>real) integrable_on A"
unfolding integral_def integrable_on_def has_integral_indicator_UNIV by auto
lemma lmeasure_finite_has_integral:
fixes s :: "'a::ordered_euclidean_space set"
assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s = ereal m" "0 \<le> m"
shows "(indicator s has_integral m) UNIV"
proof -
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
have **: "(?I s) integrable_on UNIV \<and> (\<lambda>k. integral UNIV (?I (s \<inter> cube k))) ----> integral UNIV (?I s)"
proof (intro monotone_convergence_increasing allI ballI)
have LIMSEQ: "(\<lambda>n. integral (cube n) (?I s)) ----> m"
using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1, 3)] .
{ fix n have "integral (cube n) (?I s) \<le> m"
using cube_subset assms
by (intro incseq_le[where L=m] LIMSEQ incseq_def[THEN iffD2] integral_subset_le allI impI)
(auto dest!: lebesgueD) }
moreover
{ fix n have "0 \<le> integral (cube n) (?I s)"
using assms by (auto dest!: lebesgueD intro!: integral_nonneg) }
ultimately
show "bounded {integral UNIV (?I (s \<inter> cube k)) |k. True}"
unfolding bounded_def
apply (rule_tac exI[of _ 0])
apply (rule_tac exI[of _ m])
by (auto simp: dist_real_def integral_indicator_UNIV)
fix k show "?I (s \<inter> cube k) integrable_on UNIV"
unfolding integrable_indicator_UNIV using assms by (auto dest!: lebesgueD)
fix x show "?I (s \<inter> cube k) x \<le> ?I (s \<inter> cube (Suc k)) x"
using cube_subset[of k "Suc k"] by (auto simp: indicator_def)
next
fix x :: 'a
from mem_big_cube obtain k where k: "x \<in> cube k" .
{ fix n have "?I (s \<inter> cube (n + k)) x = ?I s x"
using k cube_subset[of k "n + k"] by (auto simp: indicator_def) }
note * = this
show "(\<lambda>k. ?I (s \<inter> cube k) x) ----> ?I s x"
by (rule LIMSEQ_offset[where k=k]) (auto simp: *)
qed
note ** = conjunctD2[OF this]
have m: "m = integral UNIV (?I s)"
apply (intro LIMSEQ_unique[OF _ **(2)])
using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1,3)] integral_indicator_UNIV .
show ?thesis
unfolding m by (intro integrable_integral **)
qed
lemma lmeasure_finite_integrable: assumes s: "s \<in> sets lebesgue" and "lebesgue.\<mu> s \<noteq> \<infinity>"
shows "(indicator s :: _ \<Rightarrow> real) integrable_on UNIV"
proof (cases "lebesgue.\<mu> s")
case (real m)
with lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this]
lebesgue.positive_measure[OF s]
show ?thesis unfolding integrable_on_def by auto
qed (insert assms lebesgue.positive_measure[OF s], auto)
lemma has_integral_lebesgue: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
shows "s \<in> sets lebesgue"
proof (intro lebesgueI)
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
fix n show "(?I s) integrable_on cube n" unfolding cube_def
proof (intro integrable_on_subinterval)
show "(?I s) integrable_on UNIV"
unfolding integrable_on_def using assms by auto
qed auto
qed
lemma has_integral_lmeasure: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
shows "lebesgue.\<mu> s = ereal m"
proof (intro lmeasure_iff_LIMSEQ[THEN iffD2])
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
show "s \<in> sets lebesgue" using has_integral_lebesgue[OF assms] .
show "0 \<le> m" using assms by (rule has_integral_nonneg) auto
have "(\<lambda>n. integral UNIV (?I (s \<inter> cube n))) ----> integral UNIV (?I s)"
proof (intro dominated_convergence(2) ballI)
show "(?I s) integrable_on UNIV" unfolding integrable_on_def using assms by auto
fix n show "?I (s \<inter> cube n) integrable_on UNIV"
unfolding integrable_indicator_UNIV using `s \<in> sets lebesgue` by (auto dest: lebesgueD)
fix x show "norm (?I (s \<inter> cube n) x) \<le> ?I s x" by (auto simp: indicator_def)
next
fix x :: 'a
from mem_big_cube obtain k where k: "x \<in> cube k" .
{ fix n have "?I (s \<inter> cube (n + k)) x = ?I s x"
using k cube_subset[of k "n + k"] by (auto simp: indicator_def) }
note * = this
show "(\<lambda>k. ?I (s \<inter> cube k) x) ----> ?I s x"
by (rule LIMSEQ_offset[where k=k]) (auto simp: *)
qed
then show "(\<lambda>n. integral (cube n) (?I s)) ----> m"
unfolding integral_unique[OF assms] integral_indicator_UNIV by simp
qed
lemma has_integral_iff_lmeasure:
"(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = ereal m)"
proof
assume "(indicator A has_integral m) UNIV"
with has_integral_lmeasure[OF this] has_integral_lebesgue[OF this]
show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = ereal m"
by (auto intro: has_integral_nonneg)
next
assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = ereal m"
then show "(indicator A has_integral m) UNIV" by (intro lmeasure_finite_has_integral) auto
qed
lemma lmeasure_eq_integral: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV"
shows "lebesgue.\<mu> s = ereal (integral UNIV (indicator s))"
using assms unfolding integrable_on_def
proof safe
fix y :: real assume "(indicator s has_integral y) UNIV"
from this[unfolded has_integral_iff_lmeasure] integral_unique[OF this]
show "lebesgue.\<mu> s = ereal (integral UNIV (indicator s))" by simp
qed
lemma lebesgue_simple_function_indicator:
fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal"
assumes f:"simple_function lebesgue f"
shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f -` {y}) x))"
by (rule, subst lebesgue.simple_function_indicator_representation[OF f]) auto
lemma integral_eq_lmeasure:
"(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (lebesgue.\<mu> s)"
by (subst lmeasure_eq_integral) (auto intro!: integral_nonneg)
lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "lebesgue.\<mu> s \<noteq> \<infinity>"
using lmeasure_eq_integral[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 *:"((indicator A) has_integral (0::real)) UNIV" using lmeasure_finite_has_integral[of A] by auto
show "negligible A" unfolding negligible_def
proof (intro allI)
fix a b :: 'a
have integrable: "(indicator A :: _\<Rightarrow>real) integrable_on {a..b}"
by (intro integrable_on_subinterval has_integral_integrable) (auto intro: *)
then have "integral {a..b} (indicator A) \<le> (integral UNIV (indicator A) :: real)"
using * by (auto intro!: integral_subset_le has_integral_integrable)
moreover have "(0::real) \<le> integral {a..b} (indicator A)"
using integrable by (auto intro!: integral_nonneg)
ultimately have "integral {a..b} (indicator A) = (0::real)"
using integral_unique[OF *] by auto
then show "(indicator A has_integral (0::real)) {a..b}"
using integrable_integral[OF integrable] by simp
qed
qed
lemma integral_const[simp]:
fixes a b :: "'a::ordered_euclidean_space"
shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c"
by (rule integral_unique) (rule has_integral_const)
lemma lmeasure_UNIV[intro]: "lebesgue.\<mu> (UNIV::'a::ordered_euclidean_space set) = \<infinity>"
proof (simp add: lebesgue_def, intro SUP_PInfty bexI)
fix n :: nat
have "indicator UNIV = (\<lambda>x::'a. 1 :: real)" by auto
moreover
{ have "real n \<le> (2 * real n) ^ DIM('a)"
proof (cases n)
case 0 then show ?thesis by auto
next
case (Suc n')
have "real n \<le> (2 * real n)^1" by auto
also have "(2 * real n)^1 \<le> (2 * real n) ^ DIM('a)"
using Suc DIM_positive[where 'a='a] by (intro power_increasing) (auto simp: real_of_nat_Suc)
finally show ?thesis .
qed }
ultimately show "ereal (real n) \<le> ereal (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))"
using integral_const DIM_positive[where 'a='a]
by (auto simp: cube_def content_closed_interval_cases setprod_constant)
qed simp
lemma
fixes a b ::"'a::ordered_euclidean_space"
shows lmeasure_atLeastAtMost[simp]: "lebesgue.\<mu> {a..b} = ereal (content {a..b})"
proof -
have "(indicator (UNIV \<inter> {a..b})::_\<Rightarrow>real) integrable_on UNIV"
unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def_raw)
from lmeasure_eq_integral[OF this] show ?thesis unfolding integral_indicator_UNIV
by (simp add: indicator_def_raw)
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 "lebesgue.\<mu> {a} = 0"
using lmeasure_atLeastAtMost[of a a] by simp
declare content_real[simp]
lemma
fixes a b :: real
shows lmeasure_real_greaterThanAtMost[simp]:
"lebesgue.\<mu> {a <.. b} = ereal (if a \<le> b then b - a else 0)"
proof cases
assume "a < b"
then have "lebesgue.\<mu> {a <.. b} = lebesgue.\<mu> {a .. b} - lebesgue.\<mu> {a}"
by (subst lebesgue.measure_Diff[symmetric])
(auto intro!: arg_cong[where f=lebesgue.\<mu>])
then show ?thesis by auto
qed auto
lemma
fixes a b :: real
shows lmeasure_real_atLeastLessThan[simp]:
"lebesgue.\<mu> {a ..< b} = ereal (if a \<le> b then b - a else 0)"
proof cases
assume "a < b"
then have "lebesgue.\<mu> {a ..< b} = lebesgue.\<mu> {a .. b} - lebesgue.\<mu> {b}"
by (subst lebesgue.measure_Diff[symmetric])
(auto intro!: arg_cong[where f=lebesgue.\<mu>])
then show ?thesis by auto
qed auto
lemma
fixes a b :: real
shows lmeasure_real_greaterThanLessThan[simp]:
"lebesgue.\<mu> {a <..< b} = ereal (if a \<le> b then b - a else 0)"
proof cases
assume "a < b"
then have "lebesgue.\<mu> {a <..< b} = lebesgue.\<mu> {a <.. b} - lebesgue.\<mu> {b}"
by (subst lebesgue.measure_Diff[symmetric])
(auto intro!: arg_cong[where f=lebesgue.\<mu>])
then show ?thesis by auto
qed auto
subsection {* Lebesgue-Borel measure *}
definition "lborel = lebesgue \<lparr> sets := sets borel \<rparr>"
lemma
shows space_lborel[simp]: "space lborel = UNIV"
and sets_lborel[simp]: "sets lborel = sets borel"
and measure_lborel[simp]: "measure lborel = lebesgue.\<mu>"
and measurable_lborel[simp]: "measurable lborel = measurable borel"
by (simp_all add: measurable_def_raw lborel_def)
interpretation lborel: measure_space "lborel :: ('a::ordered_euclidean_space) measure_space"
where "space lborel = UNIV"
and "sets lborel = sets borel"
and "measure lborel = lebesgue.\<mu>"
and "measurable lborel = measurable borel"
proof (rule lebesgue.measure_space_subalgebra)
have "sigma_algebra (lborel::'a measure_space) \<longleftrightarrow> sigma_algebra (borel::'a algebra)"
unfolding sigma_algebra_iff2 lborel_def by simp
then show "sigma_algebra (lborel::'a measure_space)" by simp default
qed auto
interpretation lborel: sigma_finite_measure lborel
where "space lborel = UNIV"
and "sets lborel = sets borel"
and "measure lborel = lebesgue.\<mu>"
and "measurable lborel = measurable borel"
proof -
show "sigma_finite_measure lborel"
proof (default, intro conjI exI[of _ "\<lambda>n. cube n"])
show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed)
{ fix x have "\<exists>n. x\<in>cube n" using mem_big_cube by auto }
thus "(\<Union>i. cube i) = space lborel" by auto
show "\<forall>i. measure lborel (cube i) \<noteq> \<infinity>" by (simp add: cube_def)
qed
qed simp_all
interpretation lebesgue: sigma_finite_measure lebesgue
proof
from lborel.sigma_finite guess A ..
moreover then have "range A \<subseteq> sets lebesgue" using lebesgueI_borel by blast
ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lebesgue.\<mu> (A i) \<noteq> \<infinity>)"
by auto
qed
subsection {* Lebesgue integrable implies Gauge integrable *}
lemma has_integral_cmult_real:
fixes c :: real
assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A"
shows "((\<lambda>x. c * f x) has_integral c * x) A"
proof cases
assume "c \<noteq> 0"
from has_integral_cmul[OF assms[OF this], of c] show ?thesis
unfolding real_scaleR_def .
qed simp
lemma simple_function_has_integral:
fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal"
assumes f:"simple_function lebesgue f"
and f':"range f \<subseteq> {0..<\<infinity>}"
and om:"\<And>x. x \<in> range f \<Longrightarrow> lebesgue.\<mu> (f -` {x} \<inter> UNIV) = \<infinity> \<Longrightarrow> x = 0"
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV"
unfolding simple_integral_def space_lebesgue
proof (subst lebesgue_simple_function_indicator)
let "?M x" = "lebesgue.\<mu> (f -` {x} \<inter> UNIV)"
let "?F x" = "indicator (f -` {x})"
{ fix x y assume "y \<in> range f"
from subsetD[OF f' this] have "y * ?F y x = ereal (real y * ?F y x)"
by (cases rule: ereal2_cases[of y "?F y x"])
(auto simp: indicator_def one_ereal_def split: split_if_asm) }
moreover
{ fix x assume x: "x\<in>range f"
have "x * ?M x = real x * real (?M x)"
proof cases
assume "x \<noteq> 0" with om[OF x] have "?M x \<noteq> \<infinity>" by auto
with subsetD[OF f' x] f[THEN lebesgue.simple_functionD(2)] show ?thesis
by (cases rule: ereal2_cases[of x "?M x"]) auto
qed simp }
ultimately
have "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV \<longleftrightarrow>
((\<lambda>x. \<Sum>y\<in>range f. real y * ?F y x) has_integral (\<Sum>x\<in>range f. real x * real (?M x))) UNIV"
by simp
also have \<dots>
proof (intro has_integral_setsum has_integral_cmult_real lmeasure_finite_has_integral
real_of_ereal_pos lebesgue.positive_measure ballI)
show *: "finite (range f)" "\<And>y. f -` {y} \<in> sets lebesgue" "\<And>y. f -` {y} \<inter> UNIV \<in> sets lebesgue"
using lebesgue.simple_functionD[OF f] by auto
fix y assume "real y \<noteq> 0" "y \<in> range f"
with * om[OF this(2)] show "lebesgue.\<mu> (f -` {y}) = ereal (real (?M y))"
by (auto simp: ereal_real)
qed
finally show "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV" .
qed fact
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> ereal"
assumes f: "simple_function lebesgue f" "\<And>x. 0 \<le> f x"
and i: "integral\<^isup>S lebesgue f \<noteq> \<infinity>"
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV"
proof -
let ?f = "\<lambda>x. if x \<in> f -` {\<infinity>} then 0 else f x"
note f(1)[THEN lebesgue.simple_functionD(2)]
then have [simp, intro]: "\<And>X. f -` X \<in> sets lebesgue" by auto
have f': "simple_function lebesgue ?f"
using f by (intro lebesgue.simple_function_If_set) auto
have rng: "range ?f \<subseteq> {0..<\<infinity>}" using f by auto
have "AE x in lebesgue. f x = ?f x"
using lebesgue.simple_integral_PInf[OF f i]
by (intro lebesgue.AE_I[where N="f -` {\<infinity>} \<inter> space lebesgue"]) auto
from f(1) f' this have eq: "integral\<^isup>S lebesgue f = integral\<^isup>S lebesgue ?f"
by (rule lebesgue.simple_integral_cong_AE)
have real_eq: "\<And>x. real (f x) = real (?f x)" by auto
show ?thesis
unfolding eq real_eq
proof (rule simple_function_has_integral[OF f' rng])
fix x assume x: "x \<in> range ?f" and inf: "lebesgue.\<mu> (?f -` {x} \<inter> UNIV) = \<infinity>"
have "x * lebesgue.\<mu> (?f -` {x} \<inter> UNIV) = (\<integral>\<^isup>S y. x * indicator (?f -` {x}) y \<partial>lebesgue)"
using f'[THEN lebesgue.simple_functionD(2)]
by (simp add: lebesgue.simple_integral_cmult_indicator)
also have "\<dots> \<le> integral\<^isup>S lebesgue f"
using f'[THEN lebesgue.simple_functionD(2)] f
by (intro lebesgue.simple_integral_mono lebesgue.simple_function_mult lebesgue.simple_function_indicator)
(auto split: split_indicator)
finally show "x = 0" unfolding inf using i subsetD[OF rng x] by (auto split: split_if_asm)
qed
qed
lemma positive_integral_has_integral:
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> ereal"
assumes f: "f \<in> borel_measurable lebesgue" "range f \<subseteq> {0..<\<infinity>}" "integral\<^isup>P lebesgue f \<noteq> \<infinity>"
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>P lebesgue f))) UNIV"
proof -
from lebesgue.borel_measurable_implies_simple_function_sequence'[OF f(1)]
guess u . note u = this
have SUP_eq: "\<And>x. (SUP i. u i x) = f x"
using u(4) f(2)[THEN subsetD] by (auto split: split_max)
let "?u i x" = "real (u i x)"
note u_eq = lebesgue.positive_integral_eq_simple_integral[OF u(1,5), symmetric]
{ fix i
note u_eq
also have "integral\<^isup>P lebesgue (u i) \<le> (\<integral>\<^isup>+x. max 0 (f x) \<partial>lebesgue)"
by (intro lebesgue.positive_integral_mono) (auto intro: le_SUPI simp: u(4)[symmetric])
finally have "integral\<^isup>S lebesgue (u i) \<noteq> \<infinity>"
unfolding positive_integral_max_0 using f by auto }
note u_fin = this
then have u_int: "\<And>i. (?u i has_integral real (integral\<^isup>S lebesgue (u i))) UNIV"
by (rule simple_function_has_integral'[OF u(1,5)])
have "\<forall>x. \<exists>r\<ge>0. f x = ereal r"
proof
fix x from f(2) have "0 \<le> f x" "f x \<noteq> \<infinity>" by (auto simp: subset_eq)
then show "\<exists>r\<ge>0. f x = ereal r" by (cases "f x") auto
qed
from choice[OF this] obtain f' where f': "f = (\<lambda>x. ereal (f' x))" "\<And>x. 0 \<le> f' x" by auto
have "\<forall>i. \<exists>r. \<forall>x. 0 \<le> r x \<and> u i x = ereal (r x)"
proof
fix i show "\<exists>r. \<forall>x. 0 \<le> r x \<and> u i x = ereal (r x)"
proof (intro choice allI)
fix i x have "u i x \<noteq> \<infinity>" using u(3)[of i] by (auto simp: image_iff) metis
then show "\<exists>r\<ge>0. u i x = ereal r" using u(5)[of i x] by (cases "u i x") auto
qed
qed
from choice[OF this] obtain u' where
u': "u = (\<lambda>i x. ereal (u' i x))" "\<And>i x. 0 \<le> u' i x" by (auto simp: fun_eq_iff)
have convergent: "f' integrable_on UNIV \<and>
(\<lambda>k. integral UNIV (u' k)) ----> integral UNIV f'"
proof (intro monotone_convergence_increasing allI ballI)
show int: "\<And>k. (u' k) integrable_on UNIV"
using u_int unfolding integrable_on_def u' by auto
show "\<And>k x. u' k x \<le> u' (Suc k) x" using u(2,3,5)
by (auto simp: incseq_Suc_iff le_fun_def image_iff u' intro!: real_of_ereal_positive_mono)
show "\<And>x. (\<lambda>k. u' k x) ----> f' x"
using SUP_eq u(2)
by (intro SUP_eq_LIMSEQ[THEN iffD1]) (auto simp: u' f' incseq_mono incseq_Suc_iff le_fun_def)
show "bounded {integral UNIV (u' k)|k. True}"
proof (safe intro!: bounded_realI)
fix k
have "\<bar>integral UNIV (u' k)\<bar> = integral UNIV (u' k)"
by (intro abs_of_nonneg integral_nonneg int ballI u')
also have "\<dots> = real (integral\<^isup>S lebesgue (u k))"
using u_int[THEN integral_unique] by (simp add: u')
also have "\<dots> = real (integral\<^isup>P lebesgue (u k))"
using lebesgue.positive_integral_eq_simple_integral[OF u(1,5)] by simp
also have "\<dots> \<le> real (integral\<^isup>P lebesgue f)" using f
by (auto intro!: real_of_ereal_positive_mono lebesgue.positive_integral_positive
lebesgue.positive_integral_mono le_SUPI simp: SUP_eq[symmetric])
finally show "\<bar>integral UNIV (u' k)\<bar> \<le> real (integral\<^isup>P lebesgue f)" .
qed
qed
have "integral\<^isup>P lebesgue f = ereal (integral UNIV f')"
proof (rule tendsto_unique[OF trivial_limit_sequentially])
have "(\<lambda>i. integral\<^isup>S lebesgue (u i)) ----> (SUP i. integral\<^isup>P lebesgue (u i))"
unfolding u_eq by (intro LIMSEQ_ereal_SUPR lebesgue.incseq_positive_integral u)
also note lebesgue.positive_integral_monotone_convergence_SUP
[OF u(2) lebesgue.borel_measurable_simple_function[OF u(1)] u(5), symmetric]
finally show "(\<lambda>k. integral\<^isup>S lebesgue (u k)) ----> integral\<^isup>P lebesgue f"
unfolding SUP_eq .
{ fix k
have "0 \<le> integral\<^isup>S lebesgue (u k)"
using u by (auto intro!: lebesgue.simple_integral_positive)
then have "integral\<^isup>S lebesgue (u k) = ereal (real (integral\<^isup>S lebesgue (u k)))"
using u_fin by (auto simp: ereal_real) }
note * = this
show "(\<lambda>k. integral\<^isup>S lebesgue (u k)) ----> ereal (integral UNIV f')"
using convergent using u_int[THEN integral_unique, symmetric]
by (subst *) (simp add: lim_ereal u')
qed
then show ?thesis using convergent by (simp add: f' integrable_integral)
qed
lemma lebesgue_integral_has_integral:
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
assumes f: "integrable lebesgue f"
shows "(f has_integral (integral\<^isup>L lebesgue f)) UNIV"
proof -
let ?n = "\<lambda>x. real (ereal (max 0 (- f x)))" and ?p = "\<lambda>x. real (ereal (max 0 (f x)))"
have *: "f = (\<lambda>x. ?p x - ?n x)" by (auto simp del: ereal_max)
{ fix f have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue) = (\<integral>\<^isup>+ x. ereal (max 0 (f x)) \<partial>lebesgue)"
by (intro lebesgue.positive_integral_cong_pos) (auto split: split_max) }
note eq = this
show ?thesis
unfolding lebesgue_integral_def
apply (subst *)
apply (rule has_integral_sub)
unfolding eq[of f] eq[of "\<lambda>x. - f x"]
apply (safe intro!: positive_integral_has_integral)
using integrableD[OF f]
by (auto simp: zero_ereal_def[symmetric] positive_integral_max_0 split: split_max
intro!: lebesgue.measurable_If lebesgue.borel_measurable_ereal)
qed
lemma lebesgue_positive_integral_eq_borel:
assumes f: "f \<in> borel_measurable borel"
shows "integral\<^isup>P lebesgue f = integral\<^isup>P lborel f"
proof -
from f have "integral\<^isup>P lebesgue (\<lambda>x. max 0 (f x)) = integral\<^isup>P lborel (\<lambda>x. max 0 (f x))"
by (auto intro!: lebesgue.positive_integral_subalgebra[symmetric]) default
then show ?thesis unfolding positive_integral_max_0 .
qed
lemma lebesgue_integral_eq_borel:
assumes "f \<in> borel_measurable borel"
shows "integrable lebesgue f \<longleftrightarrow> integrable lborel f" (is ?P)
and "integral\<^isup>L lebesgue f = integral\<^isup>L lborel f" (is ?I)
proof -
have *: "sigma_algebra lborel" by default
have "sets lborel \<subseteq> sets lebesgue" by auto
from lebesgue.integral_subalgebra[of f lborel, OF _ this _ _ *] assms
show ?P ?I by auto
qed
lemma borel_integral_has_integral:
fixes f::"'a::ordered_euclidean_space => real"
assumes f:"integrable lborel f"
shows "(f has_integral (integral\<^isup>L lborel f)) UNIV"
proof -
have borel: "f \<in> borel_measurable borel"
using f unfolding integrable_def by auto
from f show ?thesis
using lebesgue_integral_has_integral[of f]
unfolding lebesgue_integral_eq_borel[OF borel] by simp
qed
subsection {* Equivalence between product spaces and euclidean spaces *}
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)
interpretation lborel_product: product_sigma_finite "\<lambda>x. lborel::real measure_space"
by default
interpretation lborel_space: finite_product_sigma_finite "\<lambda>x. lborel::real measure_space" "{..<n}" for n :: nat
where "space lborel = UNIV"
and "sets lborel = sets borel"
and "measure lborel = lebesgue.\<mu>"
and "measurable lborel = measurable borel"
proof -
show "finite_product_sigma_finite (\<lambda>x. lborel::real measure_space) {..<n}"
by default simp
qed simp_all
lemma sets_product_borel:
assumes [intro]: "finite I"
shows "sets (\<Pi>\<^isub>M i\<in>I.
\<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>) =
sets (\<Pi>\<^isub>M i\<in>I. lborel)" (is "sets ?G = _")
proof -
have "sets ?G = sets (\<Pi>\<^isub>M i\<in>I.
sigma \<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>)"
by (subst sigma_product_algebra_sigma_eq[of I "\<lambda>_ i. {..<real i}" ])
(auto intro!: measurable_sigma_sigma incseq_SucI reals_Archimedean2
simp: product_algebra_def)
then show ?thesis
unfolding lborel_def borel_eq_lessThan lebesgue_def sigma_def by simp
qed
lemma measurable_e2p:
"e2p \<in> measurable (borel::'a::ordered_euclidean_space algebra)
(\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))"
(is "_ \<in> measurable ?E ?P")
proof -
let ?B = "\<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>"
let ?G = "product_algebra_generator {..<DIM('a)} (\<lambda>_. ?B)"
have "e2p \<in> measurable ?E (sigma ?G)"
proof (rule borel.measurable_sigma)
show "e2p \<in> space ?E \<rightarrow> space ?G" by (auto simp: e2p_def)
fix A assume "A \<in> sets ?G"
then obtain E where A: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. E i)"
and "E \<in> {..<DIM('a)} \<rightarrow> (range lessThan)"
by (auto elim!: product_algebraE simp: )
then have "\<forall>i\<in>{..<DIM('a)}. \<exists>xs. E i = {..< xs}" by auto
from this[THEN bchoice] guess xs ..
then have A_eq: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< xs i})"
using A by auto
have "e2p -` A = {..< (\<chi>\<chi> i. xs i) :: 'a}"
using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def A_eq
euclidean_eq[where 'a='a] eucl_less[where 'a='a])
then show "e2p -` A \<inter> space ?E \<in> sets ?E" by simp
qed (auto simp: product_algebra_generator_def)
with sets_product_borel[of "{..<DIM('a)}"] show ?thesis
unfolding measurable_def product_algebra_def by simp
qed
lemma measurable_p2e:
"p2e \<in> measurable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))
(borel :: 'a::ordered_euclidean_space algebra)"
(is "p2e \<in> measurable ?P _")
unfolding borel_eq_lessThan
proof (intro lborel_space.measurable_sigma)
let ?E = "\<lparr> space = UNIV :: 'a set, sets = range lessThan \<rparr>"
show "p2e \<in> space ?P \<rightarrow> space ?E" by simp
fix A assume "A \<in> sets ?E"
then obtain x where "A = {..<x}" by auto
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 simp
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>"
by (auto simp: inter_interval Int_stable_def)
lemma lborel_eq_lborel_space:
fixes A :: "('a::ordered_euclidean_space) set"
assumes "A \<in> sets borel"
shows "lborel.\<mu> A = lborel_space.\<mu> DIM('a) (p2e -` A \<inter> (space (lborel_space.P DIM('a))))"
(is "_ = measure ?P (?T A)")
proof (rule measure_unique_Int_stable_vimage)
show "measure_space ?P" by default
show "measure_space lborel" by default
let ?E = "\<lparr> space = UNIV :: 'a set, sets = range (\<lambda>(a,b). {a..b}) \<rparr>"
show "Int_stable ?E" using Int_stable_cuboids .
show "range cube \<subseteq> sets ?E" unfolding cube_def_raw by auto
show "incseq cube" using cube_subset_Suc by (auto intro!: incseq_SucI)
{ fix x have "\<exists>n. x \<in> cube n" using mem_big_cube[of x] by fastforce }
then show "(\<Union>i. cube i) = space ?E" by auto
{ fix i show "lborel.\<mu> (cube i) \<noteq> \<infinity>" unfolding cube_def by auto }
show "A \<in> sets (sigma ?E)" "sets (sigma ?E) = sets lborel" "space ?E = space lborel"
using assms by (simp_all add: borel_eq_atLeastAtMost)
show "p2e \<in> measurable ?P (lborel :: 'a measure_space)"
using measurable_p2e unfolding measurable_def by simp
{ fix X assume "X \<in> sets ?E"
then obtain a b where X[simp]: "X = {a .. b}" by auto
have *: "?T X = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {a $$ i .. b $$ i})"
by (auto simp: Pi_iff eucl_le[where 'a='a] p2e_def)
show "lborel.\<mu> X = measure ?P (?T X)"
proof cases
assume "X \<noteq> {}"
then have "a \<le> b"
by (simp add: interval_ne_empty eucl_le[where 'a='a])
then have "lborel.\<mu> X = (\<Prod>x<DIM('a). lborel.\<mu> {a $$ x .. b $$ x})"
by (auto simp: content_closed_interval eucl_le[where 'a='a]
intro!: setprod_ereal[symmetric])
also have "\<dots> = measure ?P (?T X)"
unfolding * by (subst lborel_space.measure_times) auto
finally show ?thesis .
qed simp }
qed
lemma measure_preserving_p2e:
"p2e \<in> measure_preserving (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))
(lborel::'a::ordered_euclidean_space measure_space)" (is "_ \<in> measure_preserving ?P ?E")
proof
show "p2e \<in> measurable ?P ?E"
using measurable_p2e by (simp add: measurable_def)
fix A :: "'a set" assume "A \<in> sets lborel"
then show "lborel_space.\<mu> DIM('a) (p2e -` A \<inter> (space (lborel_space.P DIM('a)))) = lborel.\<mu> A"
by (intro lborel_eq_lborel_space[symmetric]) simp
qed
lemma lebesgue_eq_lborel_space_in_borel:
fixes A :: "('a::ordered_euclidean_space) set"
assumes A: "A \<in> sets borel"
shows "lebesgue.\<mu> A = lborel_space.\<mu> DIM('a) (p2e -` A \<inter> (space (lborel_space.P DIM('a))))"
using lborel_eq_lborel_space[OF A] by simp
lemma borel_fubini_positiv_integral:
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> ereal"
assumes f: "f \<in> borel_measurable borel"
shows "integral\<^isup>P lborel f = \<integral>\<^isup>+x. f (p2e x) \<partial>(lborel_space.P DIM('a))"
proof (rule lborel_space.positive_integral_vimage[OF _ measure_preserving_p2e _])
show "f \<in> borel_measurable lborel"
using f by (simp_all add: measurable_def)
qed default
lemma borel_fubini_integrable:
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
shows "integrable lborel f \<longleftrightarrow>
integrable (lborel_space.P DIM('a)) (\<lambda>x. f (p2e x))"
(is "_ \<longleftrightarrow> integrable ?B ?f")
proof
assume "integrable lborel f"
moreover then have f: "f \<in> borel_measurable borel"
by auto
moreover with measurable_p2e
have "f \<circ> p2e \<in> borel_measurable ?B"
by (rule measurable_comp)
ultimately show "integrable ?B ?f"
by (simp add: comp_def borel_fubini_positiv_integral integrable_def)
next
assume "integrable ?B ?f"
moreover then
have "?f \<circ> e2p \<in> borel_measurable (borel::'a algebra)"
by (auto intro!: measurable_e2p measurable_comp)
then have "f \<in> borel_measurable borel"
by (simp cong: measurable_cong)
ultimately show "integrable lborel f"
by (simp add: borel_fubini_positiv_integral integrable_def)
qed
lemma borel_fubini:
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
assumes f: "f \<in> borel_measurable borel"
shows "integral\<^isup>L lborel f = \<integral>x. f (p2e x) \<partial>(lborel_space.P DIM('a))"
using f by (simp add: borel_fubini_positiv_integral lebesgue_integral_def)
lemma Int_stable_atLeastAtMost:
"Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a,b). {a::'a::ordered_euclidean_space .. b}) \<rparr>"
proof (simp add: Int_stable_def image_iff, intro allI)
fix a1 b1 a2 b2 :: 'a
have "\<forall>i<DIM('a). \<exists>a b. {a1$$i..b1$$i} \<inter> {a2$$i..b2$$i} = {a..b}" by auto
then have "\<exists>a b. \<forall>i<DIM('a). {a1$$i..b1$$i} \<inter> {a2$$i..b2$$i} = {a i..b i}"
unfolding choice_iff' .
then guess a b by safe
then have "{a1..b1} \<inter> {a2..b2} = {(\<chi>\<chi> i. a i) .. (\<chi>\<chi> i. b i)}"
by (simp add: set_eq_iff eucl_le[where 'a='a] all_conj_distrib[symmetric]) blast
then show "\<exists>a' b'. {a1..b1} \<inter> {a2..b2} = {a'..b'}" by blast
qed
lemma (in sigma_algebra) borel_measurable_sets:
assumes "f \<in> measurable borel M" "A \<in> sets M"
shows "f -` A \<in> sets borel"
using measurable_sets[OF assms] by simp
lemma (in sigma_algebra) measurable_identity[intro,simp]:
"(\<lambda>x. x) \<in> measurable M M"
unfolding measurable_def by auto
lemma lebesgue_real_affine:
fixes X :: "real set"
assumes "X \<in> sets borel" and "c \<noteq> 0"
shows "measure lebesgue X = ereal \<bar>c\<bar> * measure lebesgue ((\<lambda>x. t + c * x) -` X)"
(is "_ = ?\<nu> X")
proof -
let ?E = "\<lparr>space = UNIV, sets = range (\<lambda>(a,b). {a::real .. b})\<rparr> :: real algebra"
let "?M \<nu>" = "\<lparr>space = space ?E, sets = sets (sigma ?E), measure = \<nu>\<rparr> :: real measure_space"
have *: "?M (measure lebesgue) = lborel"
unfolding borel_eq_atLeastAtMost[symmetric]
by (simp add: lborel_def lebesgue_def)
have **: "?M ?\<nu> = lborel \<lparr> measure := ?\<nu> \<rparr>"
unfolding borel_eq_atLeastAtMost[symmetric]
by (simp add: lborel_def lebesgue_def)
show ?thesis
proof (rule measure_unique_Int_stable[where X=X, OF Int_stable_atLeastAtMost], unfold * **)
show "X \<in> sets (sigma ?E)"
unfolding borel_eq_atLeastAtMost[symmetric] by fact
have "\<And>x. \<exists>xa. x \<in> cube xa" apply(rule_tac x=x in mem_big_cube) by fastforce
then show "(\<Union>i. cube i) = space ?E" by auto
show "incseq cube" by (intro incseq_SucI cube_subset_Suc)
show "range cube \<subseteq> sets ?E"
unfolding cube_def_raw by auto
show "\<And>i. measure lebesgue (cube i) \<noteq> \<infinity>"
by (simp add: cube_def)
show "measure_space lborel" by default
then interpret sigma_algebra "lborel\<lparr>measure := ?\<nu>\<rparr>"
by (auto simp add: measure_space_def)
show "measure_space (lborel\<lparr>measure := ?\<nu>\<rparr>)"
proof
show "positive (lborel\<lparr>measure := ?\<nu>\<rparr>) (measure (lborel\<lparr>measure := ?\<nu>\<rparr>))"
by (auto simp: positive_def intro!: ereal_0_le_mult borel.borel_measurable_sets)
show "countably_additive (lborel\<lparr>measure := ?\<nu>\<rparr>) (measure (lborel\<lparr>measure := ?\<nu>\<rparr>))"
proof (simp add: countably_additive_def, safe)
fix A :: "nat \<Rightarrow> real set" assume A: "range A \<subseteq> sets borel" "disjoint_family A"
then have Ai: "\<And>i. A i \<in> sets borel" by auto
have "(\<Sum>n. measure lebesgue ((\<lambda>x. t + c * x) -` A n)) = measure lebesgue (\<Union>n. (\<lambda>x. t + c * x) -` A n)"
proof (intro lborel.measure_countably_additive)
{ fix n have "(\<lambda>x. t + c * x) -` A n \<inter> space borel \<in> sets borel"
using A borel.measurable_ident unfolding id_def
by (intro measurable_sets[where A=borel] borel.borel_measurable_add[OF _ borel.borel_measurable_times]) auto }
then show "range (\<lambda>i. (\<lambda>x. t + c * x) -` A i) \<subseteq> sets borel" by auto
from `disjoint_family A`
show "disjoint_family (\<lambda>i. (\<lambda>x. t + c * x) -` A i)"
by (rule disjoint_family_on_bisimulation) auto
qed
with Ai show "(\<Sum>n. ?\<nu> (A n)) = ?\<nu> (UNION UNIV A)"
by (subst suminf_cmult_ereal)
(auto simp: vimage_UN borel.borel_measurable_sets)
qed
qed
fix X assume "X \<in> sets ?E"
then obtain a b where [simp]: "X = {a .. b}" by auto
show "measure lebesgue X = ?\<nu> X"
proof cases
assume "0 < c"
then have "(\<lambda>x. t + c * x) -` {a..b} = {(a - t) / c .. (b - t) / c}"
by (auto simp: field_simps)
with `0 < c` show ?thesis
by (cases "a \<le> b") (auto simp: field_simps)
next
assume "\<not> 0 < c" with `c \<noteq> 0` have "c < 0" by auto
then have *: "(\<lambda>x. t + c * x) -` {a..b} = {(b - t) / c .. (a - t) / c}"
by (auto simp: field_simps)
with `c < 0` show ?thesis
by (cases "a \<le> b") (auto simp: field_simps)
qed
qed
qed
end