--- a/src/HOL/Probability/Lebesgue_Measure.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Lebesgue_Measure.thy Mon Apr 23 12:14:35 2012 +0200
@@ -9,6 +9,15 @@
imports Finite_Product_Measure
begin
+lemma 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 measurable_identity[intro,simp]:
+ "(\<lambda>x. x) \<in> measurable M M"
+ unfolding measurable_def by auto
+
subsection {* Standard Cubes *}
definition cube :: "nat \<Rightarrow> 'a::ordered_euclidean_space set" where
@@ -52,17 +61,13 @@
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>"
+definition lebesgue :: "'a::ordered_euclidean_space measure" where
+ "lebesgue = measure_of UNIV {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n}
+ (\<lambda>A. SUP n. ereal (integral (cube n) (indicator A)))"
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
@@ -86,23 +91,23 @@
"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)"
+lemma sigma_algebra_lebesgue:
+ defines "leb \<equiv> {A. \<forall>n. (indicator A :: 'a::ordered_euclidean_space \<Rightarrow> real) integrable_on cube n}"
+ shows "sigma_algebra UNIV leb"
+proof (safe intro!: sigma_algebra_iff2[THEN iffD2])
+ fix A assume A: "A \<in> leb"
+ moreover have "indicator (UNIV - 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)
+ ultimately show "UNIV - A \<in> leb"
+ using A by (auto intro!: integrable_sub simp: cube_def leb_def)
next
- fix n show "(indicator {} :: _\<Rightarrow>real) integrable_on cube n"
- by (auto simp: cube_def indicator_def [abs_def])
+ fix n show "{} \<in> leb"
+ by (auto simp: cube_def indicator_def[abs_def] leb_def)
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"
+ fix A :: "nat \<Rightarrow> _" assume A: "range A \<subseteq> leb"
+ have "\<forall>n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n" (is "\<forall>n. ?g integrable_on _")
+ proof (intro dominated_convergence[where g="?g"] ballI allI)
+ fix k n 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"
@@ -111,36 +116,45 @@
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: *)
+ using absolutely_integrable_max[of ?f "cube n" ?g] A Suc
+ by (simp add: * leb_def subset_eq)
qed auto
qed (auto intro: LIMSEQ_indicator_UN simp: cube_def)
+ then show "(\<Union>i. A i) \<in> leb" by (auto simp: leb_def)
qed simp
-interpretation lebesgue: measure_space lebesgue
-proof
+lemma sets_lebesgue: "sets lebesgue = {A. \<forall>n. (indicator A :: _ \<Rightarrow> real) integrable_on cube n}"
+ unfolding lebesgue_def sigma_algebra.sets_measure_of_eq[OF sigma_algebra_lebesgue] ..
+
+lemma lebesgueD: "A \<in> sets lebesgue \<Longrightarrow> (indicator A :: _ \<Rightarrow> real) integrable_on cube n"
+ unfolding sets_lebesgue by simp
+
+lemma emeasure_lebesgue:
+ assumes "A \<in> sets lebesgue"
+ shows "emeasure lebesgue A = (SUP n. ereal (integral (cube n) (indicator A)))"
+ (is "_ = ?\<mu> A")
+proof (rule emeasure_measure_of[OF lebesgue_def])
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!: SUP_upper2 integral_nonneg)
+ show "positive (sets lebesgue) ?\<mu>"
+ proof (unfold positive_def, intro conjI ballI)
+ show "?\<mu> {} = 0" by (simp add: integral_0 *)
+ fix A :: "'a set" assume "A \<in> sets lebesgue" then show "0 \<le> ?\<mu> A"
+ by (auto intro!: SUP_upper2 Integration.integral_nonneg simp: sets_lebesgue)
qed
next
- show "countably_additive lebesgue (measure lebesgue)"
+ show "countably_additive (sets lebesgue) ?\<mu>"
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"
+ fix A :: "nat \<Rightarrow> 'a 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 = "\<lambda>n i. integral (cube n) (indicator (A i) :: _\<Rightarrow>real)"
let ?M = "\<lambda>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)
+ have nn[simp, intro]: "\<And>i n. 0 \<le> ?m n i" by (auto intro!: Integration.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)
+ by (auto simp: sets_lebesgue)
+ show "(\<Sum>n. ?\<mu> (A n)) = ?\<mu> (\<Union>i. A i)"
+ proof (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
@@ -172,14 +186,15 @@
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])
+ using Suc A by (simp add: Integration.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
+next
+qed (auto, fact)
lemma has_integral_interval_cube:
fixes a b :: "'a::ordered_euclidean_space"
@@ -202,9 +217,10 @@
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)
+ have "s \<in> sigma_sets (space lebesgue) (range (\<lambda>(a, b). {a .. (b :: 'a\<Colon>ordered_euclidean_space)}))"
+ using assms by (simp add: borel_eq_atLeastAtMost)
+ also have "\<dots> \<subseteq> sets lebesgue"
+ proof (safe intro!: sigma_sets_subset 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)"
@@ -212,11 +228,7 @@
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)
+ finally show ?thesis .
qed
lemma lebesgueI_negligible[dest]: fixes s::"'a::ordered_euclidean_space set"
@@ -224,19 +236,21 @@
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"
+ fixes S :: "'a::ordered_euclidean_space set"
+ assumes "negligible S" shows "emeasure lebesgue 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)
+ then show ?thesis
+ using assms by (simp add: emeasure_lebesgue lebesgueI_negligible)
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)
+ assumes A: "A \<in> sets lebesgue" and "0 \<le> m"
+ shows "emeasure lebesgue A = ereal m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) ----> m"
+proof (subst emeasure_lebesgue[OF A], 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
@@ -261,7 +275,7 @@
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"
+ assumes "s \<in> sets lebesgue" "emeasure lebesgue s = ereal m" "0 \<le> m"
shows "(indicator s has_integral m) UNIV"
proof -
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
@@ -275,7 +289,7 @@
(auto dest!: lebesgueD) }
moreover
{ fix n have "0 \<le> integral (cube n) (?I s)"
- using assms by (auto dest!: lebesgueD intro!: integral_nonneg) }
+ using assms by (auto dest!: lebesgueD intro!: Integration.integral_nonneg) }
ultimately
show "bounded {integral UNIV (?I (s \<inter> cube k)) |k. True}"
unfolding bounded_def
@@ -303,14 +317,13 @@
unfolding m by (intro integrable_integral **)
qed
-lemma lmeasure_finite_integrable: assumes s: "s \<in> sets lebesgue" and "lebesgue.\<mu> s \<noteq> \<infinity>"
+lemma lmeasure_finite_integrable: assumes s: "s \<in> sets lebesgue" and "emeasure lebesgue s \<noteq> \<infinity>"
shows "(indicator s :: _ \<Rightarrow> real) integrable_on UNIV"
-proof (cases "lebesgue.\<mu> s")
+proof (cases "emeasure lebesgue s")
case (real m)
- with lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this]
- lebesgue.positive_measure[OF s]
+ with lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this] emeasure_nonneg[of lebesgue s]
show ?thesis unfolding integrable_on_def by auto
-qed (insert assms lebesgue.positive_measure[OF s], auto)
+qed (insert assms emeasure_nonneg[of lebesgue s], auto)
lemma has_integral_lebesgue: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
shows "s \<in> sets lebesgue"
@@ -324,7 +337,7 @@
qed
lemma has_integral_lmeasure: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
- shows "lebesgue.\<mu> s = ereal m"
+ shows "emeasure lebesgue 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] .
@@ -349,55 +362,56 @@
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)"
+ "(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> emeasure lebesgue 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"
+ show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> emeasure lebesgue 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"
+ assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> emeasure lebesgue 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))"
+ shows "emeasure lebesgue 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
+ show "emeasure lebesgue 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
+ by (rule, subst 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)"
+ "(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (emeasure lebesgue 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>"
+lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "emeasure lebesgue 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"
+ "negligible A \<longleftrightarrow> A \<in> null_sets lebesgue"
proof
assume "negligible A"
from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0]
- show "A \<in> lebesgue.null_sets" by auto
+ show "A \<in> null_sets lebesgue" 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
+ assume A: "A \<in> null_sets lebesgue"
+ then have *:"((indicator A) has_integral (0::real)) UNIV" using lmeasure_finite_has_integral[of A]
+ by (auto simp: null_sets_def)
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)
+ using * by (auto intro!: integral_subset_le)
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)"
@@ -412,8 +426,8 @@
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)
+lemma lmeasure_UNIV[intro]: "emeasure lebesgue (UNIV::'a::ordered_euclidean_space set) = \<infinity>"
+proof (simp add: emeasure_lebesgue, intro SUP_PInfty bexI)
fix n :: nat
have "indicator UNIV = (\<lambda>x::'a. 1 :: real)" by auto
moreover
@@ -434,7 +448,7 @@
lemma
fixes a b ::"'a::ordered_euclidean_space"
- shows lmeasure_atLeastAtMost[simp]: "lebesgue.\<mu> {a..b} = ereal (content {a..b})"
+ shows lmeasure_atLeastAtMost[simp]: "emeasure lebesgue {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 [abs_def])
@@ -454,7 +468,7 @@
qed
lemma lmeasure_singleton[simp]:
- fixes a :: "'a::ordered_euclidean_space" shows "lebesgue.\<mu> {a} = 0"
+ fixes a :: "'a::ordered_euclidean_space" shows "emeasure lebesgue {a} = 0"
using lmeasure_atLeastAtMost[of a a] by simp
declare content_real[simp]
@@ -462,82 +476,68 @@
lemma
fixes a b :: real
shows lmeasure_real_greaterThanAtMost[simp]:
- "lebesgue.\<mu> {a <.. b} = ereal (if a \<le> b then b - a else 0)"
+ "emeasure lebesgue {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 have "emeasure lebesgue {a <.. b} = emeasure lebesgue {a .. b} - emeasure lebesgue {a}"
+ by (subst emeasure_Diff[symmetric])
+ (auto intro!: arg_cong[where f="emeasure lebesgue"])
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)"
+ "emeasure lebesgue {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 have "emeasure lebesgue {a ..< b} = emeasure lebesgue {a .. b} - emeasure lebesgue {b}"
+ by (subst emeasure_Diff[symmetric])
+ (auto intro!: arg_cong[where f="emeasure lebesgue"])
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)"
+ "emeasure lebesgue {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 have "emeasure lebesgue {a <..< b} = emeasure lebesgue {a <.. b} - emeasure lebesgue {b}"
+ by (subst emeasure_Diff[symmetric])
+ (auto intro!: arg_cong[where f="emeasure lebesgue"])
then show ?thesis by auto
qed auto
subsection {* Lebesgue-Borel measure *}
-definition "lborel = lebesgue \<lparr> sets := sets borel \<rparr>"
+definition "lborel = measure_of UNIV (sets borel) (emeasure lebesgue)"
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 [abs_def] lborel_def)
+ and measurable_lborel1[simp]: "measurable lborel = measurable borel"
+ and measurable_lborel2[simp]: "measurable A lborel = measurable A borel"
+ using sigma_sets_eq[of borel]
+ by (auto simp add: lborel_def measurable_def[abs_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
+lemma emeasure_lborel[simp]: "A \<in> sets borel \<Longrightarrow> emeasure lborel A = emeasure lebesgue A"
+ by (rule emeasure_measure_of[OF lborel_def])
+ (auto simp: positive_def emeasure_nonneg countably_additive_def intro!: suminf_emeasure)
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
+proof (default, intro conjI exI[of _ "\<lambda>n. cube n"])
+ show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed)
+ { fix x :: 'a have "\<exists>n. x\<in>cube n" using mem_big_cube by auto }
+ then show "(\<Union>i. cube i) = (space lborel :: 'a set)" using mem_big_cube by auto
+ show "\<forall>i. emeasure lborel (cube i) \<noteq> \<infinity>" by (simp add: cube_def)
+qed
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
+ from lborel.sigma_finite guess A :: "nat \<Rightarrow> 'a set" ..
+ then show "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. emeasure lebesgue (A i) \<noteq> \<infinity>)"
+ by (intro exI[of _ A]) (auto simp: subset_eq)
qed
subsection {* Lebesgue integrable implies Gauge integrable *}
@@ -556,11 +556,11 @@
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"
+ and om:"\<And>x. x \<in> range f \<Longrightarrow> emeasure lebesgue (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 = "\<lambda>x. lebesgue.\<mu> (f -` {x} \<inter> UNIV)"
+ let ?M = "\<lambda>x. emeasure lebesgue (f -` {x} \<inter> UNIV)"
let ?F = "\<lambda>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)"
@@ -571,7 +571,7 @@
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
+ with subsetD[OF f' x] f[THEN simple_functionD(2)] show ?thesis
by (cases rule: ereal2_cases[of x "?M x"]) auto
qed simp }
ultimately
@@ -580,11 +580,11 @@
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
+ real_of_ereal_pos emeasure_nonneg ballI)
+ show *: "finite (range f)" "\<And>y. f -` {y} \<in> sets lebesgue"
+ using 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))"
+ with * om[OF this(2)] show "emeasure lebesgue (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" .
@@ -601,28 +601,28 @@
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)]
+ note f(1)[THEN 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
+ using f by (intro 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
+ using simple_integral_PInf[OF f i]
+ by (intro 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)
+ by (rule 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)
+ fix x assume x: "x \<in> range ?f" and inf: "emeasure lebesgue (?f -` {x} \<inter> UNIV) = \<infinity>"
+ have "x * emeasure lebesgue (?f -` {x} \<inter> UNIV) = (\<integral>\<^isup>S y. x * indicator (?f -` {x}) y \<partial>lebesgue)"
+ using f'[THEN simple_functionD(2)]
+ by (simp add: 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)
+ using f'[THEN simple_functionD(2)] f
+ by (intro simple_integral_mono simple_function_mult 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
@@ -633,16 +633,16 @@
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)]
+ from 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 = "\<lambda>i x. real (u i x)"
- note u_eq = lebesgue.positive_integral_eq_simple_integral[OF u(1,5), symmetric]
+ note u_eq = 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: SUP_upper simp: u(4)[symmetric])
+ by (intro positive_integral_mono) (auto intro: SUP_upper 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
@@ -684,10 +684,10 @@
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
+ using 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 SUP_upper simp: SUP_eq[symmetric])
+ by (auto intro!: real_of_ereal_positive_mono positive_integral_positive
+ positive_integral_mono SUP_upper simp: SUP_eq[symmetric])
finally show "\<bar>integral UNIV (u' k)\<bar> \<le> real (integral\<^isup>P lebesgue f)" .
qed
qed
@@ -695,21 +695,21 @@
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]
+ unfolding u_eq by (intro LIMSEQ_ereal_SUPR incseq_positive_integral u)
+ also note positive_integral_monotone_convergence_SUP
+ [OF u(2) 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)
+ using u by (auto intro!: 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')
+ by (subst *) (simp add: u')
qed
then show ?thesis using convergent by (simp add: f' integrable_integral)
qed
@@ -721,8 +721,8 @@
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) }
+ { fix f :: "'a \<Rightarrow> real" have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue) = (\<integral>\<^isup>+ x. ereal (max 0 (f x)) \<partial>lebesgue)"
+ by (intro positive_integral_cong_pos) (auto split: split_max) }
note eq = this
show ?thesis
unfolding lebesgue_integral_def
@@ -732,7 +732,7 @@
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)
+ intro!: measurable_If)
qed
lemma lebesgue_positive_integral_eq_borel:
@@ -740,7 +740,7 @@
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
+ by (auto intro!: positive_integral_subalgebra[symmetric])
then show ?thesis unfolding positive_integral_max_0 .
qed
@@ -749,9 +749,8 @@
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
+ from integral_subalgebra[of f lborel, OF _ this _ _] assms
show ?P ?I by auto
qed
@@ -783,152 +782,109 @@
"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"
+interpretation lborel_product: product_sigma_finite "\<lambda>x. lborel::real measure"
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
+interpretation lborel_space: finite_product_sigma_finite "\<lambda>x. lborel::real measure" "{..<n}" for n :: nat
+ by default auto
+
+lemma bchoice_iff: "(\<forall>x\<in>A. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>A. P x (f x))"
+ by metis
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
+ assumes I: "finite I"
+ shows "sets (\<Pi>\<^isub>M i\<in>I. lborel) = sigma_sets (\<Pi>\<^isub>E i\<in>I. UNIV) { \<Pi>\<^isub>E i\<in>I. {..< x i :: real} | x. True}" (is "_ = ?G")
+proof (subst sigma_prod_algebra_sigma_eq[where S="\<lambda>_ i::nat. {..<real i}" and E="\<lambda>_. range lessThan", OF I])
+ show "sigma_sets (space (Pi\<^isub>M I (\<lambda>i. lborel))) {Pi\<^isub>E I F |F. \<forall>i\<in>I. F i \<in> range lessThan} = ?G"
+ by (intro arg_cong2[where f=sigma_sets]) (auto simp: space_PiM image_iff bchoice_iff)
+qed (auto simp: borel_eq_lessThan incseq_def reals_Archimedean2 image_iff intro: real_natceiling_ge)
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
+ "e2p \<in> measurable (borel::'a::ordered_euclidean_space measure) (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure))"
+proof (rule measurable_sigma_sets[OF sets_product_borel])
+ fix A :: "(nat \<Rightarrow> real) set" assume "A \<in> {\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..<x i} |x. True} "
+ then obtain x where "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..<x i})" by auto
+ then have "e2p -` A = {..< (\<chi>\<chi> i. x i) :: 'a}"
+ using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def
+ euclidean_eq[where 'a='a] eucl_less[where 'a='a])
+ then show "e2p -` A \<inter> space (borel::'a measure) \<in> sets borel" by simp
+qed (auto simp: e2p_def)
lemma measurable_p2e:
- "p2e \<in> measurable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))
- (borel :: 'a::ordered_euclidean_space algebra)"
+ "p2e \<in> measurable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure))
+ (borel :: 'a::ordered_euclidean_space measure)"
(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 $$ i})"
- using DIM_positive
- by (auto simp: Pi_iff set_eq_iff p2e_def
- euclidean_eq[where 'a='a] eucl_less[where 'a='a])
- then show "p2e -` A \<inter> space ?P \<in> sets ?P" by auto
-qed simp
+proof (safe intro!: borel_measurable_iff_halfspace_le[THEN iffD2])
+ fix x i
+ let ?A = "{w \<in> space ?P. (p2e w :: 'a) $$ i \<le> x}"
+ assume "i < DIM('a)"
+ then have "?A = (\<Pi>\<^isub>E j\<in>{..<DIM('a)}. if i = j then {.. x} else UNIV)"
+ using DIM_positive by (auto simp: space_PiM p2e_def split: split_if_asm)
+ then show "?A \<in> sets ?P"
+ by auto
+qed
+
+lemma Int_stable_atLeastAtMost:
+ fixes x::"'a::ordered_euclidean_space"
+ shows "Int_stable (range (\<lambda>(a, b::'a). {a..b}))"
+ by (auto simp: inter_interval Int_stable_def)
-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_eqI:
+ fixes M :: "'a::ordered_euclidean_space measure"
+ assumes emeasure_eq: "\<And>a b. emeasure M {a .. b} = content {a .. b}"
+ assumes sets_eq: "sets M = sets borel"
+ shows "lborel = M"
+proof (rule measure_eqI_generator_eq[OF Int_stable_atLeastAtMost])
+ let ?P = "\<Pi>\<^isub>M i\<in>{..<DIM('a::ordered_euclidean_space)}. lborel"
+ let ?E = "range (\<lambda>(a, b). {a..b} :: 'a set)"
+ show "?E \<subseteq> Pow UNIV" "sets lborel = sigma_sets UNIV ?E" "sets M = sigma_sets UNIV ?E"
+ by (simp_all add: borel_eq_atLeastAtMost sets_eq)
+
+ show "range cube \<subseteq> ?E" unfolding cube_def [abs_def] by auto
+ show "incseq cube" using cube_subset_Suc by (auto intro!: incseq_SucI)
+ { fix x :: 'a have "\<exists>n. x \<in> cube n" using mem_big_cube[of x] by fastforce }
+ then show "(\<Union>i. cube i :: 'a set) = UNIV" by auto
+
+ { fix i show "emeasure lborel (cube i) \<noteq> \<infinity>" unfolding cube_def by auto }
+ { fix X assume "X \<in> ?E" then show "emeasure lborel X = emeasure M X"
+ by (auto simp: emeasure_eq) }
+qed
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 [abs_def] 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 }
+ "(lborel :: 'a measure) = distr (\<Pi>\<^isub>M i\<in>{..<DIM('a::ordered_euclidean_space)}. lborel) lborel p2e"
+ (is "?B = ?D")
+proof (rule lborel_eqI)
+ show "sets ?D = sets borel" by simp
+ let ?P = "(\<Pi>\<^isub>M i\<in>{..<DIM('a::ordered_euclidean_space)}. lborel)"
+ fix a b :: 'a
+ have *: "p2e -` {a .. b} \<inter> space ?P = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {a $$ i .. b $$ i})"
+ by (auto simp: Pi_iff eucl_le[where 'a='a] p2e_def space_PiM)
+ have "emeasure ?P (p2e -` {a..b} \<inter> space ?P) = content {a..b}"
+ proof cases
+ assume "{a..b} \<noteq> {}"
+ then have "a \<le> b"
+ by (simp add: interval_ne_empty eucl_le[where 'a='a])
+ then have "emeasure lborel {a..b} = (\<Prod>x<DIM('a). emeasure lborel {a $$ x .. b $$ x})"
+ by (auto simp: content_closed_interval eucl_le[where 'a='a]
+ intro!: setprod_ereal[symmetric])
+ also have "\<dots> = emeasure ?P (p2e -` {a..b} \<inter> space ?P)"
+ unfolding * by (subst lborel_space.measure_times) auto
+ finally show ?thesis by simp
+ qed simp
+ then show "emeasure ?D {a .. b} = content {a .. b}"
+ by (simp add: emeasure_distr measurable_p2e)
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
+ shows "integral\<^isup>P lborel f = \<integral>\<^isup>+x. f (p2e x) \<partial>(\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel)"
+ by (subst lborel_eq_lborel_space) (simp add: positive_integral_distr measurable_p2e f)
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))"
+ integrable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel) (\<lambda>x. f (p2e x))"
(is "_ \<longleftrightarrow> integrable ?B ?f")
proof
assume "integrable lborel f"
@@ -941,9 +897,9 @@
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)
+ moreover
+ then have "?f \<circ> e2p \<in> borel_measurable (borel::'a measure)"
+ by (auto intro!: measurable_e2p)
then have "f \<in> borel_measurable borel"
by (simp cong: measurable_cong)
ultimately show "integrable lborel f"
@@ -953,100 +909,35 @@
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))"
+ shows "integral\<^isup>L lborel f = \<integral>x. f (p2e x) \<partial>((\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel))"
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 borel_measurable_indicator':
+ "A \<in> sets borel \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. indicator A (f x)) \<in> borel_measurable M"
+ using measurable_comp[OF _ borel_measurable_indicator, of f M borel A] by (auto simp add: comp_def)
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 = "\<lambda>\<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 [abs_def] 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
+ fixes c :: real assumes "c \<noteq> 0"
+ shows "lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. \<bar>c\<bar>)" (is "_ = ?D")
+proof (rule lborel_eqI)
+ fix a b show "emeasure ?D {a..b} = content {a .. b}"
+ 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 emeasure_density positive_integral_distr positive_integral_cmult
+ borel_measurable_indicator' emeasure_distr)
+ 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 emeasure_density positive_integral_distr
+ positive_integral_cmult borel_measurable_indicator' emeasure_distr)
qed
-qed
+qed simp
end