(* Title: HOL/Probability/Measure_Space.thy
Author: Lawrence C Paulson
Author: Johannes Hölzl, TU München
Author: Armin Heller, TU München
*)
header {* Measure spaces and their properties *}
theory Measure_Space
imports
Sigma_Algebra
"~~/src/HOL/Multivariate_Analysis/Extended_Real_Limits"
begin
lemma suminf_cmult_indicator:
fixes f :: "nat \<Rightarrow> ereal"
assumes "disjoint_family A" "x \<in> A i" "\<And>i. 0 \<le> f i"
shows "(\<Sum>n. f n * indicator (A n) x) = f i"
proof -
have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: ereal)"
using `x \<in> A i` assms unfolding disjoint_family_on_def indicator_def by auto
then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ereal)"
by (auto simp: setsum_cases)
moreover have "(SUP n. if i < n then f i else 0) = (f i :: ereal)"
proof (rule ereal_SUPI)
fix y :: ereal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y"
from this[of "Suc i"] show "f i \<le> y" by auto
qed (insert assms, simp)
ultimately show ?thesis using assms
by (subst suminf_ereal_eq_SUPR) (auto simp: indicator_def)
qed
lemma suminf_indicator:
assumes "disjoint_family A"
shows "(\<Sum>n. indicator (A n) x :: ereal) = indicator (\<Union>i. A i) x"
proof cases
assume *: "x \<in> (\<Union>i. A i)"
then obtain i where "x \<in> A i" by auto
from suminf_cmult_indicator[OF assms(1), OF `x \<in> A i`, of "\<lambda>k. 1"]
show ?thesis using * by simp
qed simp
text {*
The type for emeasure spaces is already defined in @{theory Sigma_Algebra}, as it is also used to
represent sigma algebras (with an arbitrary emeasure).
*}
section "Extend binary sets"
lemma LIMSEQ_binaryset:
assumes f: "f {} = 0"
shows "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) ----> f A + f B"
proof -
have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
proof
fix n
show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B"
by (induct n) (auto simp add: binaryset_def f)
qed
moreover
have "... ----> f A + f B" by (rule tendsto_const)
ultimately
have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
by metis
hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) ----> f A + f B"
by simp
thus ?thesis by (rule LIMSEQ_offset [where k=2])
qed
lemma binaryset_sums:
assumes f: "f {} = 0"
shows "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"
by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan)
lemma suminf_binaryset_eq:
fixes f :: "'a set \<Rightarrow> 'b::{comm_monoid_add, t2_space}"
shows "f {} = 0 \<Longrightarrow> (\<Sum>n. f (binaryset A B n)) = f A + f B"
by (metis binaryset_sums sums_unique)
section {* Properties of a premeasure @{term \<mu>} *}
text {*
The definitions for @{const positive} and @{const countably_additive} should be here, by they are
necessary to define @{typ "'a measure"} in @{theory Sigma_Algebra}.
*}
definition additive where
"additive M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> \<mu> (x \<union> y) = \<mu> x + \<mu> y)"
definition increasing where
"increasing M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<subseteq> y \<longrightarrow> \<mu> x \<le> \<mu> y)"
lemma positiveD_empty:
"positive M f \<Longrightarrow> f {} = 0"
by (auto simp add: positive_def)
lemma additiveD:
"additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) = f x + f y"
by (auto simp add: additive_def)
lemma increasingD:
"increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>M \<Longrightarrow> y\<in>M \<Longrightarrow> f x \<le> f y"
by (auto simp add: increasing_def)
lemma countably_additiveI:
"(\<And>A. range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> M \<Longrightarrow> (\<Sum>i. f (A i)) = f (\<Union>i. A i))
\<Longrightarrow> countably_additive M f"
by (simp add: countably_additive_def)
lemma (in ring_of_sets) disjointed_additive:
assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> M" "incseq A"
shows "(\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"
proof (induct n)
case (Suc n)
then have "(\<Sum>i\<le>Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))"
by simp
also have "\<dots> = f (A n \<union> disjointed A (Suc n))"
using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_incseq)
also have "A n \<union> disjointed A (Suc n) = A (Suc n)"
using `incseq A` by (auto dest: incseq_SucD simp: disjointed_incseq)
finally show ?case .
qed simp
lemma (in ring_of_sets) additive_sum:
fixes A:: "'i \<Rightarrow> 'a set"
assumes f: "positive M f" and ad: "additive M f" and "finite S"
and A: "A`S \<subseteq> M"
and disj: "disjoint_family_on A S"
shows "(\<Sum>i\<in>S. f (A i)) = f (\<Union>i\<in>S. A i)"
using `finite S` disj A proof induct
case empty show ?case using f by (simp add: positive_def)
next
case (insert s S)
then have "A s \<inter> (\<Union>i\<in>S. A i) = {}"
by (auto simp add: disjoint_family_on_def neq_iff)
moreover
have "A s \<in> M" using insert by blast
moreover have "(\<Union>i\<in>S. A i) \<in> M"
using insert `finite S` by auto
moreover
ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)"
using ad UNION_in_sets A by (auto simp add: additive_def)
with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
by (auto simp add: additive_def subset_insertI)
qed
lemma (in ring_of_sets) additive_increasing:
assumes posf: "positive M f" and addf: "additive M f"
shows "increasing M f"
proof (auto simp add: increasing_def)
fix x y
assume xy: "x \<in> M" "y \<in> M" "x \<subseteq> y"
then have "y - x \<in> M" by auto
then have "0 \<le> f (y-x)" using posf[unfolded positive_def] by auto
then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono) auto
also have "... = f (x \<union> (y-x))" using addf
by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
also have "... = f y"
by (metis Un_Diff_cancel Un_absorb1 xy(3))
finally show "f x \<le> f y" by simp
qed
lemma (in ring_of_sets) countably_additive_additive:
assumes posf: "positive M f" and ca: "countably_additive M f"
shows "additive M f"
proof (auto simp add: additive_def)
fix x y
assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"
hence "disjoint_family (binaryset x y)"
by (auto simp add: disjoint_family_on_def binaryset_def)
hence "range (binaryset x y) \<subseteq> M \<longrightarrow>
(\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))"
using ca
by (simp add: countably_additive_def)
hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>
f (x \<union> y) = (\<Sum>n. f (binaryset x y n))"
by (simp add: range_binaryset_eq UN_binaryset_eq)
thus "f (x \<union> y) = f x + f y" using posf x y
by (auto simp add: Un suminf_binaryset_eq positive_def)
qed
lemma (in algebra) increasing_additive_bound:
fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ereal"
assumes f: "positive M f" and ad: "additive M f"
and inc: "increasing M f"
and A: "range A \<subseteq> M"
and disj: "disjoint_family A"
shows "(\<Sum>i. f (A i)) \<le> f \<Omega>"
proof (safe intro!: suminf_bound)
fix N
note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"]
have "(\<Sum>i<N. f (A i)) = f (\<Union>i\<in>{..<N}. A i)"
using A by (intro additive_sum [OF f ad _ _]) (auto simp: disj_N)
also have "... \<le> f \<Omega>" using space_closed A
by (intro increasingD[OF inc] finite_UN) auto
finally show "(\<Sum>i<N. f (A i)) \<le> f \<Omega>" by simp
qed (insert f A, auto simp: positive_def)
lemma (in ring_of_sets) countably_additiveI_finite:
assumes "finite \<Omega>" "positive M \<mu>" "additive M \<mu>"
shows "countably_additive M \<mu>"
proof (rule countably_additiveI)
fix F :: "nat \<Rightarrow> 'a set" assume F: "range F \<subseteq> M" "(\<Union>i. F i) \<in> M" and disj: "disjoint_family F"
have "\<forall>i\<in>{i. F i \<noteq> {}}. \<exists>x. x \<in> F i" by auto
from bchoice[OF this] obtain f where f: "\<And>i. F i \<noteq> {} \<Longrightarrow> f i \<in> F i" by auto
have inj_f: "inj_on f {i. F i \<noteq> {}}"
proof (rule inj_onI, simp)
fix i j a b assume *: "f i = f j" "F i \<noteq> {}" "F j \<noteq> {}"
then have "f i \<in> F i" "f j \<in> F j" using f by force+
with disj * show "i = j" by (auto simp: disjoint_family_on_def)
qed
have "finite (\<Union>i. F i)"
by (metis F(2) assms(1) infinite_super sets_into_space)
have F_subset: "{i. \<mu> (F i) \<noteq> 0} \<subseteq> {i. F i \<noteq> {}}"
by (auto simp: positiveD_empty[OF `positive M \<mu>`])
moreover have fin_not_empty: "finite {i. F i \<noteq> {}}"
proof (rule finite_imageD)
from f have "f`{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto
then show "finite (f`{i. F i \<noteq> {}})"
by (rule finite_subset) fact
qed fact
ultimately have fin_not_0: "finite {i. \<mu> (F i) \<noteq> 0}"
by (rule finite_subset)
have disj_not_empty: "disjoint_family_on F {i. F i \<noteq> {}}"
using disj by (auto simp: disjoint_family_on_def)
from fin_not_0 have "(\<Sum>i. \<mu> (F i)) = (\<Sum>i | \<mu> (F i) \<noteq> 0. \<mu> (F i))"
by (rule suminf_finite) auto
also have "\<dots> = (\<Sum>i | F i \<noteq> {}. \<mu> (F i))"
using fin_not_empty F_subset by (rule setsum_mono_zero_left) auto
also have "\<dots> = \<mu> (\<Union>i\<in>{i. F i \<noteq> {}}. F i)"
using `positive M \<mu>` `additive M \<mu>` fin_not_empty disj_not_empty F by (intro additive_sum) auto
also have "\<dots> = \<mu> (\<Union>i. F i)"
by (rule arg_cong[where f=\<mu>]) auto
finally show "(\<Sum>i. \<mu> (F i)) = \<mu> (\<Union>i. F i)" .
qed
section {* Properties of @{const emeasure} *}
lemma emeasure_positive: "positive (sets M) (emeasure M)"
by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
lemma emeasure_empty[simp, intro]: "emeasure M {} = 0"
using emeasure_positive[of M] by (simp add: positive_def)
lemma emeasure_nonneg[intro!]: "0 \<le> emeasure M A"
using emeasure_notin_sets[of A M] emeasure_positive[of M]
by (cases "A \<in> sets M") (auto simp: positive_def)
lemma emeasure_not_MInf[simp]: "emeasure M A \<noteq> - \<infinity>"
using emeasure_nonneg[of M A] by auto
lemma emeasure_countably_additive: "countably_additive (sets M) (emeasure M)"
by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
lemma suminf_emeasure:
"range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
using countable_UN[of A UNIV M] emeasure_countably_additive[of M]
by (simp add: countably_additive_def)
lemma emeasure_additive: "additive (sets M) (emeasure M)"
by (metis countably_additive_additive emeasure_positive emeasure_countably_additive)
lemma plus_emeasure:
"a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> emeasure M a + emeasure M b = emeasure M (a \<union> b)"
using additiveD[OF emeasure_additive] ..
lemma setsum_emeasure:
"F`I \<subseteq> sets M \<Longrightarrow> disjoint_family_on F I \<Longrightarrow> finite I \<Longrightarrow>
(\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union>i\<in>I. F i)"
by (metis additive_sum emeasure_positive emeasure_additive)
lemma emeasure_mono:
"a \<subseteq> b \<Longrightarrow> b \<in> sets M \<Longrightarrow> emeasure M a \<le> emeasure M b"
by (metis additive_increasing emeasure_additive emeasure_nonneg emeasure_notin_sets
emeasure_positive increasingD)
lemma emeasure_space:
"emeasure M A \<le> emeasure M (space M)"
by (metis emeasure_mono emeasure_nonneg emeasure_notin_sets sets_into_space top)
lemma emeasure_compl:
assumes s: "s \<in> sets M" and fin: "emeasure M s \<noteq> \<infinity>"
shows "emeasure M (space M - s) = emeasure M (space M) - emeasure M s"
proof -
from s have "0 \<le> emeasure M s" by auto
have "emeasure M (space M) = emeasure M (s \<union> (space M - s))" using s
by (metis Un_Diff_cancel Un_absorb1 s sets_into_space)
also have "... = emeasure M s + emeasure M (space M - s)"
by (rule plus_emeasure[symmetric]) (auto simp add: s)
finally have "emeasure M (space M) = emeasure M s + emeasure M (space M - s)" .
then show ?thesis
using fin `0 \<le> emeasure M s`
unfolding ereal_eq_minus_iff by (auto simp: ac_simps)
qed
lemma emeasure_Diff:
assumes finite: "emeasure M B \<noteq> \<infinity>"
and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
shows "emeasure M (A - B) = emeasure M A - emeasure M B"
proof -
have "0 \<le> emeasure M B" using assms by auto
have "(A - B) \<union> B = A" using `B \<subseteq> A` by auto
then have "emeasure M A = emeasure M ((A - B) \<union> B)" by simp
also have "\<dots> = emeasure M (A - B) + emeasure M B"
using measurable by (subst plus_emeasure[symmetric]) auto
finally show "emeasure M (A - B) = emeasure M A - emeasure M B"
unfolding ereal_eq_minus_iff
using finite `0 \<le> emeasure M B` by auto
qed
lemma emeasure_countable_increasing:
assumes A: "range A \<subseteq> sets M"
and A0: "A 0 = {}"
and ASuc: "\<And>n. A n \<subseteq> A (Suc n)"
shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)"
proof -
{ fix n
have "emeasure M (A n) = (\<Sum>i<n. emeasure M (A (Suc i) - A i))"
proof (induct n)
case 0 thus ?case by (auto simp add: A0)
next
case (Suc m)
have "A (Suc m) = A m \<union> (A (Suc m) - A m)"
by (metis ASuc Un_Diff_cancel Un_absorb1)
hence "emeasure M (A (Suc m)) =
emeasure M (A m) + emeasure M (A (Suc m) - A m)"
by (subst plus_emeasure)
(auto simp add: emeasure_additive range_subsetD [OF A])
with Suc show ?case
by simp
qed }
note Meq = this
have Aeq: "(\<Union>i. A (Suc i) - A i) = (\<Union>i. A i)"
proof (rule UN_finite2_eq [where k=1], simp)
fix i
show "(\<Union>i\<in>{0..<i}. A (Suc i) - A i) = (\<Union>i\<in>{0..<Suc i}. A i)"
proof (induct i)
case 0 thus ?case by (simp add: A0)
next
case (Suc i)
thus ?case
by (auto simp add: atLeastLessThanSuc intro: subsetD [OF ASuc])
qed
qed
have A1: "\<And>i. A (Suc i) - A i \<in> sets M"
by (metis A Diff range_subsetD)
have A2: "(\<Union>i. A (Suc i) - A i) \<in> sets M"
by (blast intro: range_subsetD [OF A])
have "(SUP n. \<Sum>i<n. emeasure M (A (Suc i) - A i)) = (\<Sum>i. emeasure M (A (Suc i) - A i))"
using A by (auto intro!: suminf_ereal_eq_SUPR[symmetric])
also have "\<dots> = emeasure M (\<Union>i. A (Suc i) - A i)"
by (rule suminf_emeasure)
(auto simp add: disjoint_family_Suc ASuc A1 A2)
also have "... = emeasure M (\<Union>i. A i)"
by (simp add: Aeq)
finally have "(SUP n. \<Sum>i<n. emeasure M (A (Suc i) - A i)) = emeasure M (\<Union>i. A i)" .
then show ?thesis by (auto simp add: Meq)
qed
lemma SUP_emeasure_incseq:
assumes A: "range A \<subseteq> sets M" and "incseq A"
shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)"
proof -
have *: "(SUP n. emeasure M (nat_case {} A (Suc n))) = (SUP n. emeasure M (nat_case {} A n))"
using A by (auto intro!: SUPR_eq exI split: nat.split)
have ueq: "(\<Union>i. nat_case {} A i) = (\<Union>i. A i)"
by (auto simp add: split: nat.splits)
have meq: "\<And>n. emeasure M (A n) = (emeasure M \<circ> nat_case {} A) (Suc n)"
by simp
have "(SUP n. emeasure M (nat_case {} A n)) = emeasure M (\<Union>i. nat_case {} A i)"
using range_subsetD[OF A] incseq_SucD[OF `incseq A`]
by (force split: nat.splits intro!: emeasure_countable_increasing)
also have "emeasure M (\<Union>i. nat_case {} A i) = emeasure M (\<Union>i. A i)"
by (simp add: ueq)
finally have "(SUP n. emeasure M (nat_case {} A n)) = emeasure M (\<Union>i. A i)" .
thus ?thesis unfolding meq * comp_def .
qed
lemma incseq_emeasure:
assumes "range B \<subseteq> sets M" "incseq B"
shows "incseq (\<lambda>i. emeasure M (B i))"
using assms by (auto simp: incseq_def intro!: emeasure_mono)
lemma Lim_emeasure_incseq:
assumes A: "range A \<subseteq> sets M" "incseq A"
shows "(\<lambda>i. (emeasure M (A i))) ----> emeasure M (\<Union>i. A i)"
using LIMSEQ_ereal_SUPR[OF incseq_emeasure, OF A]
SUP_emeasure_incseq[OF A] by simp
lemma decseq_emeasure:
assumes "range B \<subseteq> sets M" "decseq B"
shows "decseq (\<lambda>i. emeasure M (B i))"
using assms by (auto simp: decseq_def intro!: emeasure_mono)
lemma INF_emeasure_decseq:
assumes A: "range A \<subseteq> sets M" and "decseq A"
and finite: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter>i. A i)"
proof -
have le_MI: "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"
using A by (auto intro!: emeasure_mono)
hence *: "emeasure M (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by auto
have A0: "0 \<le> emeasure M (A 0)" using A by auto
have "emeasure M (A 0) - (INF n. emeasure M (A n)) = emeasure M (A 0) + (SUP n. - emeasure M (A n))"
by (simp add: ereal_SUPR_uminus minus_ereal_def)
also have "\<dots> = (SUP n. emeasure M (A 0) - emeasure M (A n))"
unfolding minus_ereal_def using A0 assms
by (subst SUPR_ereal_add) (auto simp add: decseq_emeasure)
also have "\<dots> = (SUP n. emeasure M (A 0 - A n))"
using A finite `decseq A`[unfolded decseq_def] by (subst emeasure_Diff) auto
also have "\<dots> = emeasure M (\<Union>i. A 0 - A i)"
proof (rule SUP_emeasure_incseq)
show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"
using A by auto
show "incseq (\<lambda>n. A 0 - A n)"
using `decseq A` by (auto simp add: incseq_def decseq_def)
qed
also have "\<dots> = emeasure M (A 0) - emeasure M (\<Inter>i. A i)"
using A finite * by (simp, subst emeasure_Diff) auto
finally show ?thesis
unfolding ereal_minus_eq_minus_iff using finite A0 by auto
qed
lemma Lim_emeasure_decseq:
assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
shows "(\<lambda>i. emeasure M (A i)) ----> emeasure M (\<Inter>i. A i)"
using LIMSEQ_ereal_INFI[OF decseq_emeasure, OF A]
using INF_emeasure_decseq[OF A fin] by simp
lemma emeasure_subadditive:
assumes measurable: "A \<in> sets M" "B \<in> sets M"
shows "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
proof -
from plus_emeasure[of A M "B - A"]
have "emeasure M (A \<union> B) = emeasure M A + emeasure M (B - A)"
using assms by (simp add: Diff)
also have "\<dots> \<le> emeasure M A + emeasure M B"
using assms by (auto intro!: add_left_mono emeasure_mono)
finally show ?thesis .
qed
lemma emeasure_subadditive_finite:
assumes "finite I" "A ` I \<subseteq> sets M"
shows "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
using assms proof induct
case (insert i I)
then have "emeasure M (\<Union>i\<in>insert i I. A i) = emeasure M (A i \<union> (\<Union>i\<in>I. A i))"
by simp
also have "\<dots> \<le> emeasure M (A i) + emeasure M (\<Union>i\<in>I. A i)"
using insert by (intro emeasure_subadditive finite_UN) auto
also have "\<dots> \<le> emeasure M (A i) + (\<Sum>i\<in>I. emeasure M (A i))"
using insert by (intro add_mono) auto
also have "\<dots> = (\<Sum>i\<in>insert i I. emeasure M (A i))"
using insert by auto
finally show ?case .
qed simp
lemma emeasure_subadditive_countably:
assumes "range f \<subseteq> sets M"
shows "emeasure M (\<Union>i. f i) \<le> (\<Sum>i. emeasure M (f i))"
proof -
have "emeasure M (\<Union>i. f i) = emeasure M (\<Union>i. disjointed f i)"
unfolding UN_disjointed_eq ..
also have "\<dots> = (\<Sum>i. emeasure M (disjointed f i))"
using range_disjointed_sets[OF assms] suminf_emeasure[of "disjointed f"]
by (simp add: disjoint_family_disjointed comp_def)
also have "\<dots> \<le> (\<Sum>i. emeasure M (f i))"
using range_disjointed_sets[OF assms] assms
by (auto intro!: suminf_le_pos emeasure_mono disjointed_subset)
finally show ?thesis .
qed
lemma emeasure_insert:
assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A"
shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A"
proof -
have "{x} \<inter> A = {}" using `x \<notin> A` by auto
from plus_emeasure[OF sets this] show ?thesis by simp
qed
lemma emeasure_eq_setsum_singleton:
assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
shows "emeasure M S = (\<Sum>x\<in>S. emeasure M {x})"
using setsum_emeasure[of "\<lambda>x. {x}" S M] assms
by (auto simp: disjoint_family_on_def subset_eq)
lemma setsum_emeasure_cover:
assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M"
assumes A: "A \<subseteq> (\<Union>i\<in>S. B i)"
assumes disj: "disjoint_family_on B S"
shows "emeasure M A = (\<Sum>i\<in>S. emeasure M (A \<inter> (B i)))"
proof -
have "(\<Sum>i\<in>S. emeasure M (A \<inter> (B i))) = emeasure M (\<Union>i\<in>S. A \<inter> (B i))"
proof (rule setsum_emeasure)
show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
using `disjoint_family_on B S`
unfolding disjoint_family_on_def by auto
qed (insert assms, auto)
also have "(\<Union>i\<in>S. A \<inter> (B i)) = A"
using A by auto
finally show ?thesis by simp
qed
lemma emeasure_eq_0:
"N \<in> sets M \<Longrightarrow> emeasure M N = 0 \<Longrightarrow> K \<subseteq> N \<Longrightarrow> emeasure M K = 0"
by (metis emeasure_mono emeasure_nonneg order_eq_iff)
lemma emeasure_UN_eq_0:
assumes "\<And>i::nat. emeasure M (N i) = 0" and "range N \<subseteq> sets M"
shows "emeasure M (\<Union> i. N i) = 0"
proof -
have "0 \<le> emeasure M (\<Union> i. N i)" using assms by auto
moreover have "emeasure M (\<Union> i. N i) \<le> 0"
using emeasure_subadditive_countably[OF assms(2)] assms(1) by simp
ultimately show ?thesis by simp
qed
lemma measure_eqI_finite:
assumes [simp]: "sets M = Pow A" "sets N = Pow A" and "finite A"
assumes eq: "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}"
shows "M = N"
proof (rule measure_eqI)
fix X assume "X \<in> sets M"
then have X: "X \<subseteq> A" by auto
then have "emeasure M X = (\<Sum>a\<in>X. emeasure M {a})"
using `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
also have "\<dots> = (\<Sum>a\<in>X. emeasure N {a})"
using X eq by (auto intro!: setsum_cong)
also have "\<dots> = emeasure N X"
using X `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
finally show "emeasure M X = emeasure N X" .
qed simp
lemma measure_eqI_generator_eq:
fixes M N :: "'a measure" and E :: "'a set set" and A :: "nat \<Rightarrow> 'a set"
assumes "Int_stable E" "E \<subseteq> Pow \<Omega>"
and eq: "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"
and M: "sets M = sigma_sets \<Omega> E"
and N: "sets N = sigma_sets \<Omega> E"
and A: "range A \<subseteq> E" "incseq A" "(\<Union>i. A i) = \<Omega>" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
shows "M = N"
proof -
let ?D = "\<lambda>F. {D. D \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> D) = emeasure N (F \<inter> D)}"
interpret S: sigma_algebra \<Omega> "sigma_sets \<Omega> E" by (rule sigma_algebra_sigma_sets) fact
{ fix F assume "F \<in> E" and "emeasure M F \<noteq> \<infinity>"
then have [intro]: "F \<in> sigma_sets \<Omega> E" by auto
have "emeasure N F \<noteq> \<infinity>" using `emeasure M F \<noteq> \<infinity>` `F \<in> E` eq by simp
interpret D: dynkin_system \<Omega> "?D F"
proof (rule dynkin_systemI, simp_all)
fix A assume "A \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> A) = emeasure N (F \<inter> A)"
then show "A \<subseteq> \<Omega>" using S.sets_into_space by auto
next
have "F \<inter> \<Omega> = F" using `F \<in> E` `E \<subseteq> Pow \<Omega>` by auto
then show "emeasure M (F \<inter> \<Omega>) = emeasure N (F \<inter> \<Omega>)"
using `F \<in> E` eq by (auto intro: sigma_sets_top)
next
fix A assume *: "A \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> A) = emeasure N (F \<inter> A)"
then have **: "F \<inter> (\<Omega> - A) = F - (F \<inter> A)"
and [intro]: "F \<inter> A \<in> sigma_sets \<Omega> E"
using `F \<in> E` S.sets_into_space by auto
have "emeasure N (F \<inter> A) \<le> emeasure N F" by (auto intro!: emeasure_mono simp: M N)
then have "emeasure N (F \<inter> A) \<noteq> \<infinity>" using `emeasure N F \<noteq> \<infinity>` by auto
have "emeasure M (F \<inter> A) \<le> emeasure M F" by (auto intro!: emeasure_mono simp: M N)
then have "emeasure M (F \<inter> A) \<noteq> \<infinity>" using `emeasure M F \<noteq> \<infinity>` by auto
then have "emeasure M (F \<inter> (\<Omega> - A)) = emeasure M F - emeasure M (F \<inter> A)" unfolding **
using `F \<inter> A \<in> sigma_sets \<Omega> E` by (auto intro!: emeasure_Diff simp: M N)
also have "\<dots> = emeasure N F - emeasure N (F \<inter> A)" using eq `F \<in> E` * by simp
also have "\<dots> = emeasure N (F \<inter> (\<Omega> - A))" unfolding **
using `F \<inter> A \<in> sigma_sets \<Omega> E` `emeasure N (F \<inter> A) \<noteq> \<infinity>`
by (auto intro!: emeasure_Diff[symmetric] simp: M N)
finally show "\<Omega> - A \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> (\<Omega> - A)) = emeasure N (F \<inter> (\<Omega> - A))"
using * by auto
next
fix A :: "nat \<Rightarrow> 'a set"
assume "disjoint_family A" "range A \<subseteq> {X \<in> sigma_sets \<Omega> E. emeasure M (F \<inter> X) = emeasure N (F \<inter> X)}"
then have A: "range (\<lambda>i. F \<inter> A i) \<subseteq> sigma_sets \<Omega> E" "F \<inter> (\<Union>x. A x) = (\<Union>x. F \<inter> A x)"
"disjoint_family (\<lambda>i. F \<inter> A i)" "\<And>i. emeasure M (F \<inter> A i) = emeasure N (F \<inter> A i)" "range A \<subseteq> sigma_sets \<Omega> E"
by (auto simp: disjoint_family_on_def subset_eq)
then show "(\<Union>x. A x) \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> (\<Union>x. A x)) = emeasure N (F \<inter> (\<Union>x. A x))"
by (auto simp: M N suminf_emeasure[symmetric] simp del: UN_simps)
qed
have *: "sigma_sets \<Omega> E = ?D F"
using `F \<in> E` `Int_stable E`
by (intro D.dynkin_lemma) (auto simp add: Int_stable_def eq)
have "\<And>D. D \<in> sigma_sets \<Omega> E \<Longrightarrow> emeasure M (F \<inter> D) = emeasure N (F \<inter> D)"
by (subst (asm) *) auto }
note * = this
show "M = N"
proof (rule measure_eqI)
show "sets M = sets N"
using M N by simp
fix X assume "X \<in> sets M"
then have "X \<in> sigma_sets \<Omega> E"
using M by simp
let ?A = "\<lambda>i. A i \<inter> X"
have "range ?A \<subseteq> sigma_sets \<Omega> E" "incseq ?A"
using A(1,2) `X \<in> sigma_sets \<Omega> E` by (auto simp: incseq_def)
moreover
{ fix i have "emeasure M (?A i) = emeasure N (?A i)"
using *[of "A i" X] `X \<in> sigma_sets \<Omega> E` A finite by auto }
ultimately show "emeasure M X = emeasure N X"
using SUP_emeasure_incseq[of ?A M] SUP_emeasure_incseq[of ?A N] A(3) `X \<in> sigma_sets \<Omega> E`
by (auto simp: M N SUP_emeasure_incseq)
qed
qed
lemma measure_of_of_measure: "measure_of (space M) (sets M) (emeasure M) = M"
proof (intro measure_eqI emeasure_measure_of_sigma)
show "sigma_algebra (space M) (sets M)" ..
show "positive (sets M) (emeasure M)"
by (simp add: positive_def emeasure_nonneg)
show "countably_additive (sets M) (emeasure M)"
by (simp add: emeasure_countably_additive)
qed simp_all
section "@{text \<mu>}-null sets"
definition null_sets :: "'a measure \<Rightarrow> 'a set set" where
"null_sets M = {N\<in>sets M. emeasure M N = 0}"
lemma null_setsD1[dest]: "A \<in> null_sets M \<Longrightarrow> emeasure M A = 0"
by (simp add: null_sets_def)
lemma null_setsD2[dest]: "A \<in> null_sets M \<Longrightarrow> A \<in> sets M"
unfolding null_sets_def by simp
lemma null_setsI[intro]: "emeasure M A = 0 \<Longrightarrow> A \<in> sets M \<Longrightarrow> A \<in> null_sets M"
unfolding null_sets_def by simp
interpretation null_sets: ring_of_sets "space M" "null_sets M" for M
proof
show "null_sets M \<subseteq> Pow (space M)"
using sets_into_space by auto
show "{} \<in> null_sets M"
by auto
fix A B assume sets: "A \<in> null_sets M" "B \<in> null_sets M"
then have "A \<in> sets M" "B \<in> sets M"
by auto
moreover then have "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
"emeasure M (A - B) \<le> emeasure M A"
by (auto intro!: emeasure_subadditive emeasure_mono)
moreover have "emeasure M B = 0" "emeasure M A = 0"
using sets by auto
ultimately show "A - B \<in> null_sets M" "A \<union> B \<in> null_sets M"
by (auto intro!: antisym)
qed
lemma UN_from_nat: "(\<Union>i. N i) = (\<Union>i. N (Countable.from_nat i))"
proof -
have "(\<Union>i. N i) = (\<Union>i. (N \<circ> Countable.from_nat) i)"
unfolding SUP_def image_compose
unfolding surj_from_nat ..
then show ?thesis by simp
qed
lemma null_sets_UN[intro]:
assumes "\<And>i::'i::countable. N i \<in> null_sets M"
shows "(\<Union>i. N i) \<in> null_sets M"
proof (intro conjI CollectI null_setsI)
show "(\<Union>i. N i) \<in> sets M" using assms by auto
have "0 \<le> emeasure M (\<Union>i. N i)" by (rule emeasure_nonneg)
moreover have "emeasure M (\<Union>i. N i) \<le> (\<Sum>n. emeasure M (N (Countable.from_nat n)))"
unfolding UN_from_nat[of N]
using assms by (intro emeasure_subadditive_countably) auto
ultimately show "emeasure M (\<Union>i. N i) = 0"
using assms by (auto simp: null_setsD1)
qed
lemma null_set_Int1:
assumes "B \<in> null_sets M" "A \<in> sets M" shows "A \<inter> B \<in> null_sets M"
proof (intro CollectI conjI null_setsI)
show "emeasure M (A \<inter> B) = 0" using assms
by (intro emeasure_eq_0[of B _ "A \<inter> B"]) auto
qed (insert assms, auto)
lemma null_set_Int2:
assumes "B \<in> null_sets M" "A \<in> sets M" shows "B \<inter> A \<in> null_sets M"
using assms by (subst Int_commute) (rule null_set_Int1)
lemma emeasure_Diff_null_set:
assumes "B \<in> null_sets M" "A \<in> sets M"
shows "emeasure M (A - B) = emeasure M A"
proof -
have *: "A - B = (A - (A \<inter> B))" by auto
have "A \<inter> B \<in> null_sets M" using assms by (rule null_set_Int1)
then show ?thesis
unfolding * using assms
by (subst emeasure_Diff) auto
qed
lemma null_set_Diff:
assumes "B \<in> null_sets M" "A \<in> sets M" shows "B - A \<in> null_sets M"
proof (intro CollectI conjI null_setsI)
show "emeasure M (B - A) = 0" using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto
qed (insert assms, auto)
lemma emeasure_Un_null_set:
assumes "A \<in> sets M" "B \<in> null_sets M"
shows "emeasure M (A \<union> B) = emeasure M A"
proof -
have *: "A \<union> B = A \<union> (B - A)" by auto
have "B - A \<in> null_sets M" using assms(2,1) by (rule null_set_Diff)
then show ?thesis
unfolding * using assms
by (subst plus_emeasure[symmetric]) auto
qed
section "Formalize almost everywhere"
definition ae_filter :: "'a measure \<Rightarrow> 'a filter" where
"ae_filter M = Abs_filter (\<lambda>P. \<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N)"
abbreviation
almost_everywhere :: "'a measure \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
"almost_everywhere M P \<equiv> eventually P (ae_filter M)"
syntax
"_almost_everywhere" :: "pttrn \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10)
translations
"AE x in M. P" == "CONST almost_everywhere M (%x. P)"
lemma eventually_ae_filter:
fixes M P
defines [simp]: "F \<equiv> \<lambda>P. \<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N"
shows "eventually P (ae_filter M) \<longleftrightarrow> F P"
unfolding ae_filter_def F_def[symmetric]
proof (rule eventually_Abs_filter)
show "is_filter F"
proof
fix P Q assume "F P" "F Q"
then obtain N L where N: "N \<in> null_sets M" "{x \<in> space M. \<not> P x} \<subseteq> N"
and L: "L \<in> null_sets M" "{x \<in> space M. \<not> Q x} \<subseteq> L"
by auto
then have "L \<union> N \<in> null_sets M" "{x \<in> space M. \<not> (P x \<and> Q x)} \<subseteq> L \<union> N" by auto
then show "F (\<lambda>x. P x \<and> Q x)" by auto
next
fix P Q assume "F P"
then obtain N where N: "N \<in> null_sets M" "{x \<in> space M. \<not> P x} \<subseteq> N" by auto
moreover assume "\<forall>x. P x \<longrightarrow> Q x"
ultimately have "N \<in> null_sets M" "{x \<in> space M. \<not> Q x} \<subseteq> N" by auto
then show "F Q" by auto
qed auto
qed
lemma AE_I':
"N \<in> null_sets M \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x in M. P x)"
unfolding eventually_ae_filter by auto
lemma AE_iff_null:
assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M")
shows "(AE x in M. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets M"
proof
assume "AE x in M. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "emeasure M N = 0"
unfolding eventually_ae_filter by auto
have "0 \<le> emeasure M ?P" by auto
moreover have "emeasure M ?P \<le> emeasure M N"
using assms N(1,2) by (auto intro: emeasure_mono)
ultimately have "emeasure M ?P = 0" unfolding `emeasure M N = 0` by auto
then show "?P \<in> null_sets M" using assms by auto
next
assume "?P \<in> null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I')
qed
lemma AE_iff_null_sets:
"N \<in> sets M \<Longrightarrow> N \<in> null_sets M \<longleftrightarrow> (AE x in M. x \<notin> N)"
using Int_absorb1[OF sets_into_space, of N M]
by (subst AE_iff_null) (auto simp: Int_def[symmetric])
lemma AE_not_in:
"N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N"
by (metis AE_iff_null_sets null_setsD2)
lemma AE_iff_measurable:
"N \<in> sets M \<Longrightarrow> {x\<in>space M. \<not> P x} = N \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> emeasure M N = 0"
using AE_iff_null[of _ P] by auto
lemma AE_E[consumes 1]:
assumes "AE x in M. P x"
obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
using assms unfolding eventually_ae_filter by auto
lemma AE_E2:
assumes "AE x in M. P x" "{x\<in>space M. P x} \<in> sets M"
shows "emeasure M {x\<in>space M. \<not> P x} = 0" (is "emeasure M ?P = 0")
proof -
have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}" by auto
with AE_iff_null[of M P] assms show ?thesis by auto
qed
lemma AE_I:
assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
shows "AE x in M. P x"
using assms unfolding eventually_ae_filter by auto
lemma AE_mp[elim!]:
assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x \<longrightarrow> Q x"
shows "AE x in M. Q x"
proof -
from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A"
and A: "A \<in> sets M" "emeasure M A = 0"
by (auto elim!: AE_E)
from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B"
and B: "B \<in> sets M" "emeasure M B = 0"
by (auto elim!: AE_E)
show ?thesis
proof (intro AE_I)
have "0 \<le> emeasure M (A \<union> B)" using A B by auto
moreover have "emeasure M (A \<union> B) \<le> 0"
using emeasure_subadditive[of A M B] A B by auto
ultimately show "A \<union> B \<in> sets M" "emeasure M (A \<union> B) = 0" using A B by auto
show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"
using P imp by auto
qed
qed
(* depricated replace by laws about eventually *)
lemma
shows AE_iffI: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> AE x in M. Q x"
and AE_disjI1: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<or> Q x"
and AE_disjI2: "AE x in M. Q x \<Longrightarrow> AE x in M. P x \<or> Q x"
and AE_conjI: "AE x in M. P x \<Longrightarrow> AE x in M. Q x \<Longrightarrow> AE x in M. P x \<and> Q x"
and AE_conj_iff[simp]: "(AE x in M. P x \<and> Q x) \<longleftrightarrow> (AE x in M. P x) \<and> (AE x in M. Q x)"
by auto
lemma AE_impI:
"(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x"
by (cases P) auto
lemma AE_measure:
assumes AE: "AE x in M. P x" and sets: "{x\<in>space M. P x} \<in> sets M" (is "?P \<in> sets M")
shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)"
proof -
from AE_E[OF AE] guess N . note N = this
with sets have "emeasure M (space M) \<le> emeasure M (?P \<union> N)"
by (intro emeasure_mono) auto
also have "\<dots> \<le> emeasure M ?P + emeasure M N"
using sets N by (intro emeasure_subadditive) auto
also have "\<dots> = emeasure M ?P" using N by simp
finally show "emeasure M ?P = emeasure M (space M)"
using emeasure_space[of M "?P"] by auto
qed
lemma AE_space: "AE x in M. x \<in> space M"
by (rule AE_I[where N="{}"]) auto
lemma AE_I2[simp, intro]:
"(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x in M. P x"
using AE_space by force
lemma AE_Ball_mp:
"\<forall>x\<in>space M. P x \<Longrightarrow> AE x in M. P x \<longrightarrow> Q x \<Longrightarrow> AE x in M. Q x"
by auto
lemma AE_cong[cong]:
"(\<And>x. x \<in> space M \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> (AE x in M. Q x)"
by auto
lemma AE_all_countable:
"(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)"
proof
assume "\<forall>i. AE x in M. P i x"
from this[unfolded eventually_ae_filter Bex_def, THEN choice]
obtain N where N: "\<And>i. N i \<in> null_sets M" "\<And>i. {x\<in>space M. \<not> P i x} \<subseteq> N i" by auto
have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto
also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto
finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .
moreover from N have "(\<Union>i. N i) \<in> null_sets M"
by (intro null_sets_UN) auto
ultimately show "AE x in M. \<forall>i. P i x"
unfolding eventually_ae_filter by auto
qed auto
lemma AE_finite_all:
assumes f: "finite S" shows "(AE x in M. \<forall>i\<in>S. P i x) \<longleftrightarrow> (\<forall>i\<in>S. AE x in M. P i x)"
using f by induct auto
lemma AE_finite_allI:
assumes "finite S"
shows "(\<And>s. s \<in> S \<Longrightarrow> AE x in M. Q s x) \<Longrightarrow> AE x in M. \<forall>s\<in>S. Q s x"
using AE_finite_all[OF `finite S`] by auto
lemma emeasure_mono_AE:
assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B"
and B: "B \<in> sets M"
shows "emeasure M A \<le> emeasure M B"
proof cases
assume A: "A \<in> sets M"
from imp obtain N where N: "{x\<in>space M. \<not> (x \<in> A \<longrightarrow> x \<in> B)} \<subseteq> N" "N \<in> null_sets M"
by (auto simp: eventually_ae_filter)
have "emeasure M A = emeasure M (A - N)"
using N A by (subst emeasure_Diff_null_set) auto
also have "emeasure M (A - N) \<le> emeasure M (B - N)"
using N A B sets_into_space by (auto intro!: emeasure_mono)
also have "emeasure M (B - N) = emeasure M B"
using N B by (subst emeasure_Diff_null_set) auto
finally show ?thesis .
qed (simp add: emeasure_nonneg emeasure_notin_sets)
lemma emeasure_eq_AE:
assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
assumes A: "A \<in> sets M" and B: "B \<in> sets M"
shows "emeasure M A = emeasure M B"
using assms by (safe intro!: antisym emeasure_mono_AE) auto
section {* @{text \<sigma>}-finite Measures *}
locale sigma_finite_measure =
fixes M :: "'a measure"
assumes sigma_finite: "\<exists>A::nat \<Rightarrow> 'a set.
range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. emeasure M (A i) \<noteq> \<infinity>)"
lemma (in sigma_finite_measure) sigma_finite_disjoint:
obtains A :: "nat \<Rightarrow> 'a set"
where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "disjoint_family A"
proof atomize_elim
case goal1
obtain A :: "nat \<Rightarrow> 'a set" where
range: "range A \<subseteq> sets M" and
space: "(\<Union>i. A i) = space M" and
measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
using sigma_finite by auto
note range' = range_disjointed_sets[OF range] range
{ fix i
have "emeasure M (disjointed A i) \<le> emeasure M (A i)"
using range' disjointed_subset[of A i] by (auto intro!: emeasure_mono)
then have "emeasure M (disjointed A i) \<noteq> \<infinity>"
using measure[of i] by auto }
with disjoint_family_disjointed UN_disjointed_eq[of A] space range'
show ?case by (auto intro!: exI[of _ "disjointed A"])
qed
lemma (in sigma_finite_measure) sigma_finite_incseq:
obtains A :: "nat \<Rightarrow> 'a set"
where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"
proof atomize_elim
case goal1
obtain F :: "nat \<Rightarrow> 'a set" where
F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. emeasure M (F i) \<noteq> \<infinity>"
using sigma_finite by auto
then show ?case
proof (intro exI[of _ "\<lambda>n. \<Union>i\<le>n. F i"] conjI allI)
from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto
then show "(\<Union>n. \<Union> i\<le>n. F i) = space M"
using F by fastforce
next
fix n
have "emeasure M (\<Union> i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))" using F
by (auto intro!: emeasure_subadditive_finite)
also have "\<dots> < \<infinity>"
using F by (auto simp: setsum_Pinfty)
finally show "emeasure M (\<Union> i\<le>n. F i) \<noteq> \<infinity>" by simp
qed (force simp: incseq_def)+
qed
section {* Measure space induced by distribution of @{const measurable}-functions *}
definition distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where
"distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f -` A \<inter> space M))"
lemma
shows sets_distr[simp]: "sets (distr M N f) = sets N"
and space_distr[simp]: "space (distr M N f) = space N"
by (auto simp: distr_def)
lemma
shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'"
and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng"
by (auto simp: measurable_def)
lemma emeasure_distr:
fixes f :: "'a \<Rightarrow> 'b"
assumes f: "f \<in> measurable M N" and A: "A \<in> sets N"
shows "emeasure (distr M N f) A = emeasure M (f -` A \<inter> space M)" (is "_ = ?\<mu> A")
unfolding distr_def
proof (rule emeasure_measure_of_sigma)
show "positive (sets N) ?\<mu>"
by (auto simp: positive_def)
show "countably_additive (sets N) ?\<mu>"
proof (intro countably_additiveI)
fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A"
then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto
then have *: "range (\<lambda>i. f -` (A i) \<inter> space M) \<subseteq> sets M"
using f by (auto simp: measurable_def)
moreover have "(\<Union>i. f -` A i \<inter> space M) \<in> sets M"
using * by blast
moreover have **: "disjoint_family (\<lambda>i. f -` A i \<inter> space M)"
using `disjoint_family A` by (auto simp: disjoint_family_on_def)
ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)"
using suminf_emeasure[OF _ **] A f
by (auto simp: comp_def vimage_UN)
qed
show "sigma_algebra (space N) (sets N)" ..
qed fact
lemma AE_distrD:
assumes f: "f \<in> measurable M M'"
and AE: "AE x in distr M M' f. P x"
shows "AE x in M. P (f x)"
proof -
from AE[THEN AE_E] guess N .
with f show ?thesis
unfolding eventually_ae_filter
by (intro bexI[of _ "f -` N \<inter> space M"])
(auto simp: emeasure_distr measurable_def)
qed
lemma null_sets_distr_iff:
"f \<in> measurable M N \<Longrightarrow> A \<in> null_sets (distr M N f) \<longleftrightarrow> f -` A \<inter> space M \<in> null_sets M \<and> A \<in> sets N"
by (auto simp add: null_sets_def emeasure_distr measurable_sets)
lemma distr_distr:
assumes f: "g \<in> measurable N L" and g: "f \<in> measurable M N"
shows "distr (distr M N f) L g = distr M L (g \<circ> f)" (is "?L = ?R")
using measurable_comp[OF g f] f g
by (auto simp add: emeasure_distr measurable_sets measurable_space
intro!: arg_cong[where f="emeasure M"] measure_eqI)
section {* Real measure values *}
lemma measure_nonneg: "0 \<le> measure M A"
using emeasure_nonneg[of M A] unfolding measure_def by (auto intro: real_of_ereal_pos)
lemma measure_empty[simp]: "measure M {} = 0"
unfolding measure_def by simp
lemma emeasure_eq_ereal_measure:
"emeasure M A \<noteq> \<infinity> \<Longrightarrow> emeasure M A = ereal (measure M A)"
using emeasure_nonneg[of M A]
by (cases "emeasure M A") (auto simp: measure_def)
lemma measure_Union:
assumes finite: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"
and measurable: "A \<in> sets M" "B \<in> sets M" "A \<inter> B = {}"
shows "measure M (A \<union> B) = measure M A + measure M B"
unfolding measure_def
using plus_emeasure[OF measurable, symmetric] finite
by (simp add: emeasure_eq_ereal_measure)
lemma measure_finite_Union:
assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S"
assumes finite: "\<And>i. i \<in> S \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
unfolding measure_def
using setsum_emeasure[OF measurable, symmetric] finite
by (simp add: emeasure_eq_ereal_measure)
lemma measure_Diff:
assumes finite: "emeasure M A \<noteq> \<infinity>"
and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
shows "measure M (A - B) = measure M A - measure M B"
proof -
have "emeasure M (A - B) \<le> emeasure M A" "emeasure M B \<le> emeasure M A"
using measurable by (auto intro!: emeasure_mono)
hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B"
using measurable finite by (rule_tac measure_Union) auto
thus ?thesis using `B \<subseteq> A` by (auto simp: Un_absorb2)
qed
lemma measure_UNION:
assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"
assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
proof -
from summable_sums[OF summable_ereal_pos, of "\<lambda>i. emeasure M (A i)"]
suminf_emeasure[OF measurable] emeasure_nonneg[of M]
have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))" by simp
moreover
{ fix i
have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)"
using measurable by (auto intro!: emeasure_mono)
then have "emeasure M (A i) = ereal ((measure M (A i)))"
using finite by (intro emeasure_eq_ereal_measure) auto }
ultimately show ?thesis using finite
unfolding sums_ereal[symmetric] by (simp add: emeasure_eq_ereal_measure)
qed
lemma measure_subadditive:
assumes measurable: "A \<in> sets M" "B \<in> sets M"
and fin: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"
shows "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
proof -
have "emeasure M (A \<union> B) \<noteq> \<infinity>"
using emeasure_subadditive[OF measurable] fin by auto
then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
using emeasure_subadditive[OF measurable] fin
by (auto simp: emeasure_eq_ereal_measure)
qed
lemma measure_subadditive_finite:
assumes A: "finite I" "A`I \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
proof -
{ have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
using emeasure_subadditive_finite[OF A] .
also have "\<dots> < \<infinity>"
using fin by (simp add: setsum_Pinfty)
finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> \<infinity>" by simp }
then show ?thesis
using emeasure_subadditive_finite[OF A] fin
unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg)
qed
lemma measure_subadditive_countably:
assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>"
shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
proof -
from emeasure_nonneg fin have "\<And>i. emeasure M (A i) \<noteq> \<infinity>" by (rule suminf_PInfty)
moreover
{ have "emeasure M (\<Union>i. A i) \<le> (\<Sum>i. emeasure M (A i))"
using emeasure_subadditive_countably[OF A] .
also have "\<dots> < \<infinity>"
using fin by simp
finally have "emeasure M (\<Union>i. A i) \<noteq> \<infinity>" by simp }
ultimately show ?thesis
using emeasure_subadditive_countably[OF A] fin
unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg)
qed
lemma measure_eq_setsum_singleton:
assumes S: "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
and fin: "\<And>x. x \<in> S \<Longrightarrow> emeasure M {x} \<noteq> \<infinity>"
shows "(measure M S) = (\<Sum>x\<in>S. (measure M {x}))"
unfolding measure_def
using emeasure_eq_setsum_singleton[OF S] fin
by simp (simp add: emeasure_eq_ereal_measure)
lemma Lim_measure_incseq:
assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
shows "(\<lambda>i. (measure M (A i))) ----> (measure M (\<Union>i. A i))"
proof -
have "ereal ((measure M (\<Union>i. A i))) = emeasure M (\<Union>i. A i)"
using fin by (auto simp: emeasure_eq_ereal_measure)
then show ?thesis
using Lim_emeasure_incseq[OF A]
unfolding measure_def
by (intro lim_real_of_ereal) simp
qed
lemma Lim_measure_decseq:
assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)"
proof -
have "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"
using A by (auto intro!: emeasure_mono)
also have "\<dots> < \<infinity>"
using fin[of 0] by auto
finally have "ereal ((measure M (\<Inter>i. A i))) = emeasure M (\<Inter>i. A i)"
by (auto simp: emeasure_eq_ereal_measure)
then show ?thesis
unfolding measure_def
using Lim_emeasure_decseq[OF A fin]
by (intro lim_real_of_ereal) simp
qed
section {* Measure spaces with @{term "emeasure M (space M) < \<infinity>"} *}
locale finite_measure = sigma_finite_measure M for M +
assumes finite_emeasure_space: "emeasure M (space M) \<noteq> \<infinity>"
lemma finite_measureI[Pure.intro!]:
assumes *: "emeasure M (space M) \<noteq> \<infinity>"
shows "finite_measure M"
proof
show "\<exists>A. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. emeasure M (A i) \<noteq> \<infinity>)"
using * by (auto intro!: exI[of _ "\<lambda>_. space M"])
qed fact
lemma (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A \<noteq> \<infinity>"
using finite_emeasure_space emeasure_space[of M A] by auto
lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ereal (measure M A)"
unfolding measure_def by (simp add: emeasure_eq_ereal_measure)
lemma (in finite_measure) emeasure_real: "\<exists>r. 0 \<le> r \<and> emeasure M A = ereal r"
using emeasure_finite[of A] emeasure_nonneg[of M A] by (cases "emeasure M A") auto
lemma (in finite_measure) bounded_measure: "measure M A \<le> measure M (space M)"
using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def)
lemma (in finite_measure) finite_measure_Diff:
assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
shows "measure M (A - B) = measure M A - measure M B"
using measure_Diff[OF _ assms] by simp
lemma (in finite_measure) finite_measure_Union:
assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"
shows "measure M (A \<union> B) = measure M A + measure M B"
using measure_Union[OF _ _ assms] by simp
lemma (in finite_measure) finite_measure_finite_Union:
assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S"
shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
using measure_finite_Union[OF assms] by simp
lemma (in finite_measure) finite_measure_UNION:
assumes A: "range A \<subseteq> sets M" "disjoint_family A"
shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
using measure_UNION[OF A] by simp
lemma (in finite_measure) finite_measure_mono:
assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B"
using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def)
lemma (in finite_measure) finite_measure_subadditive:
assumes m: "A \<in> sets M" "B \<in> sets M"
shows "measure M (A \<union> B) \<le> measure M A + measure M B"
using measure_subadditive[OF m] by simp
lemma (in finite_measure) finite_measure_subadditive_finite:
assumes "finite I" "A`I \<subseteq> sets M" shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
using measure_subadditive_finite[OF assms] by simp
lemma (in finite_measure) finite_measure_subadditive_countably:
assumes A: "range A \<subseteq> sets M" and sum: "summable (\<lambda>i. measure M (A i))"
shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
proof -
from `summable (\<lambda>i. measure M (A i))`
have "(\<lambda>i. ereal (measure M (A i))) sums ereal (\<Sum>i. measure M (A i))"
by (simp add: sums_ereal) (rule summable_sums)
from sums_unique[OF this, symmetric]
measure_subadditive_countably[OF A]
show ?thesis by (simp add: emeasure_eq_measure)
qed
lemma (in finite_measure) finite_measure_eq_setsum_singleton:
assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
shows "measure M S = (\<Sum>x\<in>S. measure M {x})"
using measure_eq_setsum_singleton[OF assms] by simp
lemma (in finite_measure) finite_Lim_measure_incseq:
assumes A: "range A \<subseteq> sets M" "incseq A"
shows "(\<lambda>i. measure M (A i)) ----> measure M (\<Union>i. A i)"
using Lim_measure_incseq[OF A] by simp
lemma (in finite_measure) finite_Lim_measure_decseq:
assumes A: "range A \<subseteq> sets M" "decseq A"
shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)"
using Lim_measure_decseq[OF A] by simp
lemma (in finite_measure) finite_measure_compl:
assumes S: "S \<in> sets M"
shows "measure M (space M - S) = measure M (space M) - measure M S"
using measure_Diff[OF _ top S sets_into_space] S by simp
lemma (in finite_measure) finite_measure_mono_AE:
assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M"
shows "measure M A \<le> measure M B"
using assms emeasure_mono_AE[OF imp B]
by (simp add: emeasure_eq_measure)
lemma (in finite_measure) finite_measure_eq_AE:
assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
assumes A: "A \<in> sets M" and B: "B \<in> sets M"
shows "measure M A = measure M B"
using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure)
section {* Counting space *}
definition count_space :: "'a set \<Rightarrow> 'a measure" where
"count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"
lemma
shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"
and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"
using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]
by (auto simp: count_space_def)
lemma measurable_count_space_eq1[simp]:
"f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"
unfolding measurable_def by simp
lemma measurable_count_space_eq2[simp]:
assumes "finite A"
shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
proof -
{ fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
with `finite A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "finite X"
by (auto dest: finite_subset)
moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
ultimately have "f -` X \<inter> space M \<in> sets M"
using `X \<subseteq> A` by (auto intro!: finite_UN simp del: UN_simps) }
then show ?thesis
unfolding measurable_def by auto
qed
lemma emeasure_count_space:
assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then ereal (card X) else \<infinity>)"
(is "_ = ?M X")
unfolding count_space_def
proof (rule emeasure_measure_of_sigma)
show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow)
show "positive (Pow A) ?M"
by (auto simp: positive_def)
show "countably_additive (Pow A) ?M"
proof (unfold countably_additive_def, safe)
fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F"
show "(\<Sum>i. ?M (F i)) = ?M (\<Union>i. F i)"
proof cases
assume "\<forall>i. finite (F i)"
then have finite_F: "\<And>i. finite (F i)" by auto
have "\<forall>i\<in>{i. F i \<noteq> {}}. \<exists>x. x \<in> F i" by auto
from bchoice[OF this] obtain f where f: "\<And>i. F i \<noteq> {} \<Longrightarrow> f i \<in> F i" by auto
have inj_f: "inj_on f {i. F i \<noteq> {}}"
proof (rule inj_onI, simp)
fix i j a b assume *: "f i = f j" "F i \<noteq> {}" "F j \<noteq> {}"
then have "f i \<in> F i" "f j \<in> F j" using f by force+
with disj * show "i = j" by (auto simp: disjoint_family_on_def)
qed
have fin_eq: "finite (\<Union>i. F i) \<longleftrightarrow> finite {i. F i \<noteq> {}}"
proof
assume "finite (\<Union>i. F i)"
show "finite {i. F i \<noteq> {}}"
proof (rule finite_imageD)
from f have "f`{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto
then show "finite (f`{i. F i \<noteq> {}})"
by (rule finite_subset) fact
qed fact
next
assume "finite {i. F i \<noteq> {}}"
with finite_F have "finite (\<Union>i\<in>{i. F i \<noteq> {}}. F i)"
by auto
also have "(\<Union>i\<in>{i. F i \<noteq> {}}. F i) = (\<Union>i. F i)"
by auto
finally show "finite (\<Union>i. F i)" .
qed
show ?thesis
proof cases
assume *: "finite (\<Union>i. F i)"
with finite_F have "finite {i. ?M (F i) \<noteq> 0} "
by (simp add: fin_eq)
then have "(\<Sum>i. ?M (F i)) = (\<Sum>i | ?M (F i) \<noteq> 0. ?M (F i))"
by (rule suminf_finite) auto
also have "\<dots> = ereal (\<Sum>i | F i \<noteq> {}. card (F i))"
using finite_F by simp
also have "\<dots> = ereal (card (\<Union>i \<in> {i. F i \<noteq> {}}. F i))"
using * finite_F disj
by (subst card_UN_disjoint) (auto simp: disjoint_family_on_def fin_eq)
also have "\<dots> = ?M (\<Union>i. F i)"
using * by (auto intro!: arg_cong[where f=card])
finally show ?thesis .
next
assume inf: "infinite (\<Union>i. F i)"
{ fix i
have "\<exists>N. i \<le> (\<Sum>i<N. card (F i))"
proof (induct i)
case (Suc j)
from Suc obtain N where N: "j \<le> (\<Sum>i<N. card (F i))" by auto
have "infinite ({i. F i \<noteq> {}} - {..< N})"
using inf by (auto simp: fin_eq)
then have "{i. F i \<noteq> {}} - {..< N} \<noteq> {}"
by (metis finite.emptyI)
then obtain i where i: "F i \<noteq> {}" "N \<le> i"
by (auto simp: not_less[symmetric])
note N
also have "(\<Sum>i<N. card (F i)) \<le> (\<Sum>i<i. card (F i))"
by (rule setsum_mono2) (auto simp: i)
also have "\<dots> < (\<Sum>i<i. card (F i)) + card (F i)"
using finite_F `F i \<noteq> {}` by (simp add: card_gt_0_iff)
finally have "j < (\<Sum>i<Suc i. card (F i))"
by simp
then show ?case unfolding Suc_le_eq by blast
qed simp }
with finite_F inf show ?thesis
by (auto simp del: real_of_nat_setsum intro!: SUP_PInfty
simp add: suminf_ereal_eq_SUPR real_of_nat_setsum[symmetric])
qed
next
assume "\<not> (\<forall>i. finite (F i))"
then obtain j where j: "infinite (F j)" by auto
then have "infinite (\<Union>i. F i)"
using finite_subset[of "F j" "\<Union>i. F i"] by auto
moreover have "\<And>i. 0 \<le> ?M (F i)" by auto
ultimately show ?thesis
using suminf_PInfty[of "\<lambda>i. ?M (F i)" j] j by auto
qed
qed
show "X \<in> Pow A" using `X \<subseteq> A` by simp
qed
lemma emeasure_count_space_finite[simp]:
"X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> emeasure (count_space A) X = ereal (card X)"
using emeasure_count_space[of X A] by simp
lemma emeasure_count_space_infinite[simp]:
"X \<subseteq> A \<Longrightarrow> infinite X \<Longrightarrow> emeasure (count_space A) X = \<infinity>"
using emeasure_count_space[of X A] by simp
lemma emeasure_count_space_eq_0:
"emeasure (count_space A) X = 0 \<longleftrightarrow> (X \<subseteq> A \<longrightarrow> X = {})"
proof cases
assume X: "X \<subseteq> A"
then show ?thesis
proof (intro iffI impI)
assume "emeasure (count_space A) X = 0"
with X show "X = {}"
by (subst (asm) emeasure_count_space) (auto split: split_if_asm)
qed simp
qed (simp add: emeasure_notin_sets)
lemma null_sets_count_space: "null_sets (count_space A) = { {} }"
unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0)
lemma AE_count_space: "(AE x in count_space A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
unfolding eventually_ae_filter by (auto simp add: null_sets_count_space)
lemma sigma_finite_measure_count_space:
fixes A :: "'a::countable set"
shows "sigma_finite_measure (count_space A)"
proof
show "\<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (count_space A) \<and> (\<Union>i. F i) = space (count_space A) \<and>
(\<forall>i. emeasure (count_space A) (F i) \<noteq> \<infinity>)"
using surj_from_nat by (intro exI[of _ "\<lambda>i. {from_nat i} \<inter> A"]) (auto simp del: surj_from_nat)
qed
lemma finite_measure_count_space:
assumes [simp]: "finite A"
shows "finite_measure (count_space A)"
by rule simp
lemma sigma_finite_measure_count_space_finite:
assumes A: "finite A" shows "sigma_finite_measure (count_space A)"
proof -
interpret finite_measure "count_space A" using A by (rule finite_measure_count_space)
show "sigma_finite_measure (count_space A)" ..
qed
end