(* Title: HOL/Probability/Measure_Space.thy
Author: Lawrence C Paulson
Author: Johannes Hölzl, TU München
Author: Armin Heller, TU München
*)
section {* Measure spaces and their properties *}
theory Measure_Space
imports
Measurable "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
begin
subsection "Relate extended reals and the indicator function"
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.If_cases)
moreover have "(SUP n. if i < n then f i else 0) = (f i :: ereal)"
proof (rule SUP_eqI)
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_SUP) (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
lemma setsum_indicator_disjoint_family:
fixes f :: "'d \<Rightarrow> 'e::semiring_1"
assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"
shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
proof -
have "P \<inter> {i. x \<in> A i} = {j}"
using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def
by auto
thus ?thesis
unfolding indicator_def
by (simp add: if_distrib setsum.If_cases[OF `finite P`])
qed
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).
*}
subsection "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)
subsection {* 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 positiveD1: "positive M f \<Longrightarrow> f {} = 0" by (auto simp: positive_def)
lemma positiveD2: "positive M f \<Longrightarrow> A \<in> M \<Longrightarrow> 0 \<le> f A" by (auto simp: positive_def)
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[case_names countably]:
"(\<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_mono)
also have "A n \<union> disjointed A (Suc n) = A (Suc n)"
using `incseq A` by (auto dest: incseq_SucD simp: disjointed_mono)
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
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) subadditive:
assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> M" and S: "finite S"
shows "f (\<Union>i\<in>S. A i) \<le> (\<Sum>i\<in>S. f (A i))"
using S
proof (induct S)
case empty thus ?case using f by (auto simp: positive_def)
next
case (insert x F)
hence in_M: "A x \<in> M" "(\<Union>i\<in>F. A i) \<in> M" "(\<Union>i\<in>F. A i) - A x \<in> M" using A by force+
have subs: "(\<Union>i\<in>F. A i) - A x \<subseteq> (\<Union>i\<in>F. A i)" by auto
have "(\<Union>i\<in>(insert x F). A i) = A x \<union> ((\<Union>i\<in>F. A i) - A x)" by auto
hence "f (\<Union>i\<in>(insert x F). A i) = f (A x \<union> ((\<Union>i\<in>F. A i) - A x))"
by simp
also have "\<dots> = f (A x) + f ((\<Union>i\<in>F. A i) - A x)"
using f(2) by (rule additiveD) (insert in_M, auto)
also have "\<dots> \<le> f (A x) + f (\<Union>i\<in>F. A i)"
using additive_increasing[OF f] in_M subs by (auto simp: increasing_def intro: add_left_mono)
also have "\<dots> \<le> f (A x) + (\<Sum>i\<in>F. f (A i))" using insert by (auto intro: add_left_mono)
finally show "f (\<Union>i\<in>(insert x F). A i) \<le> (\<Sum>i\<in>(insert x F). f (A i))" using insert 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_neutral_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
lemma (in ring_of_sets) countably_additive_iff_continuous_from_below:
assumes f: "positive M f" "additive M f"
shows "countably_additive M f \<longleftrightarrow>
(\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i))"
unfolding countably_additive_def
proof safe
assume count_sum: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> UNION UNIV A \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (UNION UNIV A)"
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M"
then have dA: "range (disjointed A) \<subseteq> M" by (auto simp: range_disjointed_sets)
with count_sum[THEN spec, of "disjointed A"] A(3)
have f_UN: "(\<Sum>i. f (disjointed A i)) = f (\<Union>i. A i)"
by (auto simp: UN_disjointed_eq disjoint_family_disjointed)
moreover have "(\<lambda>n. (\<Sum>i<n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))"
using f(1)[unfolded positive_def] dA
by (auto intro!: summable_LIMSEQ summable_ereal_pos)
from LIMSEQ_Suc[OF this]
have "(\<lambda>n. (\<Sum>i\<le>n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))"
unfolding lessThan_Suc_atMost .
moreover have "\<And>n. (\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"
using disjointed_additive[OF f A(1,2)] .
ultimately show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" by simp
next
assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i)"
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "disjoint_family A" "(\<Union>i. A i) \<in> M"
have *: "(\<Union>n. (\<Union>i<n. A i)) = (\<Union>i. A i)" by auto
have "(\<lambda>n. f (\<Union>i<n. A i)) ----> f (\<Union>i. A i)"
proof (unfold *[symmetric], intro cont[rule_format])
show "range (\<lambda>i. \<Union>i<i. A i) \<subseteq> M" "(\<Union>i. \<Union>i<i. A i) \<in> M"
using A * by auto
qed (force intro!: incseq_SucI)
moreover have "\<And>n. f (\<Union>i<n. A i) = (\<Sum>i<n. f (A i))"
using A
by (intro additive_sum[OF f, of _ A, symmetric])
(auto intro: disjoint_family_on_mono[where B=UNIV])
ultimately
have "(\<lambda>i. f (A i)) sums f (\<Union>i. A i)"
unfolding sums_def by simp
from sums_unique[OF this]
show "(\<Sum>i. f (A i)) = f (\<Union>i. A i)" by simp
qed
lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous:
assumes f: "positive M f" "additive M f"
shows "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))
\<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0)"
proof safe
assume cont: "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))"
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) = {}" "\<forall>i. f (A i) \<noteq> \<infinity>"
with cont[THEN spec, of A] show "(\<lambda>i. f (A i)) ----> 0"
using `positive M f`[unfolded positive_def] by auto
next
assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0"
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) \<in> M" "\<forall>i. f (A i) \<noteq> \<infinity>"
have f_mono: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<subseteq> b \<Longrightarrow> f a \<le> f b"
using additive_increasing[OF f] unfolding increasing_def by simp
have decseq_fA: "decseq (\<lambda>i. f (A i))"
using A by (auto simp: decseq_def intro!: f_mono)
have decseq: "decseq (\<lambda>i. A i - (\<Inter>i. A i))"
using A by (auto simp: decseq_def)
then have decseq_f: "decseq (\<lambda>i. f (A i - (\<Inter>i. A i)))"
using A unfolding decseq_def by (auto intro!: f_mono Diff)
have "f (\<Inter>x. A x) \<le> f (A 0)"
using A by (auto intro!: f_mono)
then have f_Int_fin: "f (\<Inter>x. A x) \<noteq> \<infinity>"
using A by auto
{ fix i
have "f (A i - (\<Inter>i. A i)) \<le> f (A i)" using A by (auto intro!: f_mono)
then have "f (A i - (\<Inter>i. A i)) \<noteq> \<infinity>"
using A by auto }
note f_fin = this
have "(\<lambda>i. f (A i - (\<Inter>i. A i))) ----> 0"
proof (intro cont[rule_format, OF _ decseq _ f_fin])
show "range (\<lambda>i. A i - (\<Inter>i. A i)) \<subseteq> M" "(\<Inter>i. A i - (\<Inter>i. A i)) = {}"
using A by auto
qed
from INF_Lim_ereal[OF decseq_f this]
have "(INF n. f (A n - (\<Inter>i. A i))) = 0" .
moreover have "(INF n. f (\<Inter>i. A i)) = f (\<Inter>i. A i)"
by auto
ultimately have "(INF n. f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i)) = 0 + f (\<Inter>i. A i)"
using A(4) f_fin f_Int_fin
by (subst INF_ereal_add) (auto simp: decseq_f)
moreover {
fix n
have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f ((A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i))"
using A by (subst f(2)[THEN additiveD]) auto
also have "(A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i) = A n"
by auto
finally have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f (A n)" . }
ultimately have "(INF n. f (A n)) = f (\<Inter>i. A i)"
by simp
with LIMSEQ_INF[OF decseq_fA]
show "(\<lambda>i. f (A i)) ----> f (\<Inter>i. A i)" by simp
qed
lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below:
assumes f: "positive M f" "additive M f" "\<forall>A\<in>M. f A \<noteq> \<infinity>"
assumes cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<lambda>i. f (A i)) ----> 0"
assumes A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M"
shows "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)"
proof -
have "\<forall>A\<in>M. \<exists>x. f A = ereal x"
proof
fix A assume "A \<in> M" with f show "\<exists>x. f A = ereal x"
unfolding positive_def by (cases "f A") auto
qed
from bchoice[OF this] guess f' .. note f' = this[rule_format]
from A have "(\<lambda>i. f ((\<Union>i. A i) - A i)) ----> 0"
by (intro cont[rule_format]) (auto simp: decseq_def incseq_def)
moreover
{ fix i
have "f ((\<Union>i. A i) - A i) + f (A i) = f ((\<Union>i. A i) - A i \<union> A i)"
using A by (intro f(2)[THEN additiveD, symmetric]) auto
also have "(\<Union>i. A i) - A i \<union> A i = (\<Union>i. A i)"
by auto
finally have "f' (\<Union>i. A i) - f' (A i) = f' ((\<Union>i. A i) - A i)"
using A by (subst (asm) (1 2 3) f') auto
then have "f ((\<Union>i. A i) - A i) = ereal (f' (\<Union>i. A i) - f' (A i))"
using A f' by auto }
ultimately have "(\<lambda>i. f' (\<Union>i. A i) - f' (A i)) ----> 0"
by (simp add: zero_ereal_def)
then have "(\<lambda>i. f' (A i)) ----> f' (\<Union>i. A i)"
by (rule Lim_transform2[OF tendsto_const])
then show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)"
using A by (subst (1 2) f') auto
qed
lemma (in ring_of_sets) empty_continuous_imp_countably_additive:
assumes f: "positive M f" "additive M f" and fin: "\<forall>A\<in>M. f A \<noteq> \<infinity>"
assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0"
shows "countably_additive M f"
using countably_additive_iff_continuous_from_below[OF f]
using empty_continuous_imp_continuous_from_below[OF f fin] cont
by blast
subsection {* 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_le_0_iff: "emeasure M A \<le> 0 \<longleftrightarrow> emeasure M A = 0"
using emeasure_nonneg[of M A] by auto
lemma emeasure_less_0_iff: "emeasure M A < 0 \<longleftrightarrow> False"
using emeasure_nonneg[of M A] by auto
lemma emeasure_single_in_space: "emeasure M {x} \<noteq> 0 \<Longrightarrow> x \<in> space M"
using emeasure_notin_sets[of "{x}" M] by (auto dest: sets.sets_into_space)
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 sets.countable_UN[of A UNIV M] emeasure_countably_additive[of M]
by (simp add: countably_additive_def)
lemma sums_emeasure:
"disjoint_family F \<Longrightarrow> (\<And>i. F i \<in> sets M) \<Longrightarrow> (\<lambda>i. emeasure M (F i)) sums emeasure M (\<Union>i. F i)"
unfolding sums_iff by (intro conjI summable_ereal_pos emeasure_nonneg suminf_emeasure) auto
lemma emeasure_additive: "additive (sets M) (emeasure M)"
by (metis sets.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 sets.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 sets.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.sets_into_space sets.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.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" and "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"
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 Lim_emeasure_incseq:
"range A \<subseteq> sets M \<Longrightarrow> incseq A \<Longrightarrow> (\<lambda>i. (emeasure M (A i))) ----> emeasure M (\<Union>i. A i)"
using emeasure_countably_additive
by (auto simp add: sets.countably_additive_iff_continuous_from_below emeasure_positive
emeasure_additive)
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 SUP_emeasure_incseq:
assumes A: "range A \<subseteq> sets M" "incseq A"
shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)"
using LIMSEQ_SUP[OF incseq_emeasure, OF A] Lim_emeasure_incseq[OF A]
by (simp add: LIMSEQ_unique)
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_SUP_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 SUP_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 emeasure_INT_decseq_subset:
fixes F :: "nat \<Rightarrow> 'a set"
assumes I: "I \<noteq> {}" and F: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> i \<le> j \<Longrightarrow> F j \<subseteq> F i"
assumes F_sets[measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M"
and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (F i) \<noteq> \<infinity>"
shows "emeasure M (\<Inter>i\<in>I. F i) = (INF i:I. emeasure M (F i))"
proof cases
assume "finite I"
have "(\<Inter>i\<in>I. F i) = F (Max I)"
using I \<open>finite I\<close> by (intro antisym INF_lower INF_greatest F) auto
moreover have "(INF i:I. emeasure M (F i)) = emeasure M (F (Max I))"
using I \<open>finite I\<close> by (intro antisym INF_lower INF_greatest F emeasure_mono) auto
ultimately show ?thesis
by simp
next
assume "infinite I"
def L \<equiv> "\<lambda>n. LEAST i. i \<in> I \<and> i \<ge> n"
have L: "L n \<in> I \<and> n \<le> L n" for n
unfolding L_def
proof (rule LeastI_ex)
show "\<exists>x. x \<in> I \<and> n \<le> x"
using \<open>infinite I\<close> finite_subset[of I "{..< n}"]
by (rule_tac ccontr) (auto simp: not_le)
qed
have L_eq[simp]: "i \<in> I \<Longrightarrow> L i = i" for i
unfolding L_def by (intro Least_equality) auto
have L_mono: "i \<le> j \<Longrightarrow> L i \<le> L j" for i j
using L[of j] unfolding L_def by (intro Least_le) (auto simp: L_def)
have "emeasure M (\<Inter>i. F (L i)) = (INF i. emeasure M (F (L i)))"
proof (intro INF_emeasure_decseq[symmetric])
show "decseq (\<lambda>i. F (L i))"
using L by (intro antimonoI F L_mono) auto
qed (insert L fin, auto)
also have "\<dots> = (INF i:I. emeasure M (F i))"
proof (intro antisym INF_greatest)
show "i \<in> I \<Longrightarrow> (INF i. emeasure M (F (L i))) \<le> emeasure M (F i)" for i
by (intro INF_lower2[of i]) auto
qed (insert L, auto intro: INF_lower)
also have "(\<Inter>i. F (L i)) = (\<Inter>i\<in>I. F i)"
proof (intro antisym INF_greatest)
show "i \<in> I \<Longrightarrow> (\<Inter>i. F (L i)) \<subseteq> F i" for i
by (intro INF_lower2[of i]) auto
qed (insert L, auto)
finally show ?thesis .
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_INF[OF decseq_emeasure, OF A]
using INF_emeasure_decseq[OF A fin] by simp
lemma emeasure_lfp'[consumes 1, case_names cont measurable]:
assumes "P M"
assumes cont: "sup_continuous F"
assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
shows "emeasure M {x\<in>space M. lfp F x} = (SUP i. emeasure M {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"
proof -
have "emeasure M {x\<in>space M. lfp F x} = emeasure M (\<Union>i. {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"
using sup_continuous_lfp[OF cont] by (auto simp add: bot_fun_def intro!: arg_cong2[where f=emeasure])
moreover { fix i from `P M` have "{x\<in>space M. (F ^^ i) (\<lambda>x. False) x} \<in> sets M"
by (induct i arbitrary: M) (auto simp add: pred_def[symmetric] intro: *) }
moreover have "incseq (\<lambda>i. {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"
proof (rule incseq_SucI)
fix i
have "(F ^^ i) (\<lambda>x. False) \<le> (F ^^ (Suc i)) (\<lambda>x. False)"
proof (induct i)
case 0 show ?case by (simp add: le_fun_def)
next
case Suc thus ?case using monoD[OF sup_continuous_mono[OF cont] Suc] by auto
qed
then show "{x \<in> space M. (F ^^ i) (\<lambda>x. False) x} \<subseteq> {x \<in> space M. (F ^^ Suc i) (\<lambda>x. False) x}"
by auto
qed
ultimately show ?thesis
by (subst SUP_emeasure_incseq) auto
qed
lemma emeasure_lfp:
assumes [simp]: "\<And>s. sets (M s) = sets N"
assumes cont: "sup_continuous F" "sup_continuous f"
assumes nonneg: "\<And>P s. 0 \<le> f P s"
assumes meas: "\<And>P. Measurable.pred N P \<Longrightarrow> Measurable.pred N (F P)"
assumes iter: "\<And>P s. Measurable.pred N P \<Longrightarrow> P \<le> lfp F \<Longrightarrow> emeasure (M s) {x\<in>space N. F P x} = f (\<lambda>s. emeasure (M s) {x\<in>space N. P x}) s"
shows "emeasure (M s) {x\<in>space N. lfp F x} = lfp f s"
proof (subst lfp_transfer_bounded[where \<alpha>="\<lambda>F s. emeasure (M s) {x\<in>space N. F x}" and g=f and f=F and P="Measurable.pred N", symmetric])
fix C assume "incseq C" "\<And>i. Measurable.pred N (C i)"
then show "(\<lambda>s. emeasure (M s) {x \<in> space N. (SUP i. C i) x}) = (SUP i. (\<lambda>s. emeasure (M s) {x \<in> space N. C i x}))"
unfolding SUP_apply[abs_def]
by (subst SUP_emeasure_incseq) (auto simp: mono_def fun_eq_iff intro!: arg_cong2[where f=emeasure])
qed (auto simp add: iter nonneg le_fun_def SUP_apply[abs_def] intro!: meas cont)
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)" by simp
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) 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 sets.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 sets.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_insert_ne:
"A \<noteq> {} \<Longrightarrow> {x} \<in> sets M \<Longrightarrow> A \<in> sets M \<Longrightarrow> x \<notin> A \<Longrightarrow> emeasure M (insert x A) = emeasure M {x} + emeasure M A"
by (rule emeasure_insert)
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" "(\<Union>i. A i) = \<Omega>" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
shows "M = N"
proof -
let ?\<mu> = "emeasure M" and ?\<nu> = "emeasure N"
interpret S: sigma_algebra \<Omega> "sigma_sets \<Omega> E" by (rule sigma_algebra_sigma_sets) fact
have "space M = \<Omega>"
using sets.top[of M] sets.space_closed[of M] S.top S.space_closed `sets M = sigma_sets \<Omega> E`
by blast
{ fix F D assume "F \<in> E" and "?\<mu> F \<noteq> \<infinity>"
then have [intro]: "F \<in> sigma_sets \<Omega> E" by auto
have "?\<nu> F \<noteq> \<infinity>" using `?\<mu> F \<noteq> \<infinity>` `F \<in> E` eq by simp
assume "D \<in> sets M"
with `Int_stable E` `E \<subseteq> Pow \<Omega>` have "emeasure M (F \<inter> D) = emeasure N (F \<inter> D)"
unfolding M
proof (induct rule: sigma_sets_induct_disjoint)
case (basic A)
then have "F \<inter> A \<in> E" using `Int_stable E` `F \<in> E` by (auto simp: Int_stable_def)
then show ?case using eq by auto
next
case empty then show ?case by simp
next
case (compl 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 simp: M)
have "?\<nu> (F \<inter> A) \<le> ?\<nu> F" by (auto intro!: emeasure_mono simp: M N)
then have "?\<nu> (F \<inter> A) \<noteq> \<infinity>" using `?\<nu> F \<noteq> \<infinity>` by auto
have "?\<mu> (F \<inter> A) \<le> ?\<mu> F" by (auto intro!: emeasure_mono simp: M N)
then have "?\<mu> (F \<inter> A) \<noteq> \<infinity>" using `?\<mu> F \<noteq> \<infinity>` by auto
then have "?\<mu> (F \<inter> (\<Omega> - A)) = ?\<mu> F - ?\<mu> (F \<inter> A)" unfolding **
using `F \<inter> A \<in> sigma_sets \<Omega> E` by (auto intro!: emeasure_Diff simp: M N)
also have "\<dots> = ?\<nu> F - ?\<nu> (F \<inter> A)" using eq `F \<in> E` compl by simp
also have "\<dots> = ?\<nu> (F \<inter> (\<Omega> - A))" unfolding **
using `F \<inter> A \<in> sigma_sets \<Omega> E` `?\<nu> (F \<inter> A) \<noteq> \<infinity>`
by (auto intro!: emeasure_Diff[symmetric] simp: M N)
finally show ?case
using `space M = \<Omega>` by auto
next
case (union A)
then have "?\<mu> (\<Union>x. F \<inter> A x) = ?\<nu> (\<Union>x. F \<inter> A x)"
by (subst (1 2) suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def subset_eq M N)
with A show ?case
by auto
qed }
note * = this
show "M = N"
proof (rule measure_eqI)
show "sets M = sets N"
using M N by simp
have [simp, intro]: "\<And>i. A i \<in> sets M"
using A(1) by (auto simp: subset_eq M)
fix F assume "F \<in> sets M"
let ?D = "disjointed (\<lambda>i. F \<inter> A i)"
from `space M = \<Omega>` have F_eq: "F = (\<Union>i. ?D i)"
using `F \<in> sets M`[THEN sets.sets_into_space] A(2)[symmetric] by (auto simp: UN_disjointed_eq)
have [simp, intro]: "\<And>i. ?D i \<in> sets M"
using sets.range_disjointed_sets[of "\<lambda>i. F \<inter> A i" M] `F \<in> sets M`
by (auto simp: subset_eq)
have "disjoint_family ?D"
by (auto simp: disjoint_family_disjointed)
moreover
have "(\<Sum>i. emeasure M (?D i)) = (\<Sum>i. emeasure N (?D i))"
proof (intro arg_cong[where f=suminf] ext)
fix i
have "A i \<inter> ?D i = ?D i"
by (auto simp: disjointed_def)
then show "emeasure M (?D i) = emeasure N (?D i)"
using *[of "A i" "?D i", OF _ A(3)] A(1) by auto
qed
ultimately show "emeasure M F = emeasure N F"
by (simp add: image_subset_iff `sets M = sets N`[symmetric] F_eq[symmetric] suminf_emeasure)
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
subsection {* @{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 (rule ring_of_setsI)
show "null_sets M \<subseteq> Pow (space M)"
using sets.sets_into_space by auto
show "{} \<in> null_sets M"
by auto
fix A B assume null_sets: "A \<in> null_sets M" "B \<in> null_sets M"
then have sets: "A \<in> sets M" "B \<in> sets M"
by auto
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)
then have "emeasure M B = 0" "emeasure M A = 0"
using null_sets by auto
with sets * show "A - B \<in> null_sets M" "A \<union> B \<in> null_sets M"
by (auto intro!: antisym)
qed
lemma UN_from_nat_into:
assumes I: "countable I" "I \<noteq> {}"
shows "(\<Union>i\<in>I. N i) = (\<Union>i. N (from_nat_into I i))"
proof -
have "(\<Union>i\<in>I. N i) = \<Union>(N ` range (from_nat_into I))"
using I by simp
also have "\<dots> = (\<Union>i. (N \<circ> from_nat_into I) i)"
by (simp only: SUP_def image_comp)
finally show ?thesis by simp
qed
lemma null_sets_UN':
assumes "countable I"
assumes "\<And>i. i \<in> I \<Longrightarrow> N i \<in> null_sets M"
shows "(\<Union>i\<in>I. N i) \<in> null_sets M"
proof cases
assume "I = {}" then show ?thesis by simp
next
assume "I \<noteq> {}"
show ?thesis
proof (intro conjI CollectI null_setsI)
show "(\<Union>i\<in>I. N i) \<in> sets M"
using assms by (intro sets.countable_UN') auto
have "emeasure M (\<Union>i\<in>I. N i) \<le> (\<Sum>n. emeasure M (N (from_nat_into I n)))"
unfolding UN_from_nat_into[OF `countable I` `I \<noteq> {}`]
using assms `I \<noteq> {}` by (intro emeasure_subadditive_countably) (auto intro: from_nat_into)
also have "(\<lambda>n. emeasure M (N (from_nat_into I n))) = (\<lambda>_. 0)"
using assms `I \<noteq> {}` by (auto intro: from_nat_into)
finally show "emeasure M (\<Union>i\<in>I. N i) = 0"
by (intro antisym emeasure_nonneg) simp
qed
qed
lemma null_sets_UN[intro]:
"(\<And>i::'i::countable. N i \<in> null_sets M) \<Longrightarrow> (\<Union>i. N i) \<in> null_sets M"
by (rule null_sets_UN') auto
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
subsection {* The almost everywhere filter (i.e.\ quantifier) *}
definition ae_filter :: "'a measure \<Rightarrow> 'a filter" where
"ae_filter M = (INF N:null_sets M. principal (space M - 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 (\<lambda>x. P)"
lemma eventually_ae_filter: "eventually P (ae_filter M) \<longleftrightarrow> (\<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N)"
unfolding ae_filter_def by (subst eventually_INF_base) (auto simp: eventually_principal subset_eq)
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.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_ball_countable:
assumes [intro]: "countable X"
shows "(AE x in M. \<forall>y\<in>X. P x y) \<longleftrightarrow> (\<forall>y\<in>X. AE x in M. P x y)"
proof
assume "\<forall>y\<in>X. AE x in M. P x y"
from this[unfolded eventually_ae_filter Bex_def, THEN bchoice]
obtain N where N: "\<And>y. y \<in> X \<Longrightarrow> N y \<in> null_sets M" "\<And>y. y \<in> X \<Longrightarrow> {x\<in>space M. \<not> P x y} \<subseteq> N y"
by auto
have "{x\<in>space M. \<not> (\<forall>y\<in>X. P x y)} \<subseteq> (\<Union>y\<in>X. {x\<in>space M. \<not> P x y})"
by auto
also have "\<dots> \<subseteq> (\<Union>y\<in>X. N y)"
using N by auto
finally have "{x\<in>space M. \<not> (\<forall>y\<in>X. P x y)} \<subseteq> (\<Union>y\<in>X. N y)" .
moreover from N have "(\<Union>y\<in>X. N y) \<in> null_sets M"
by (intro null_sets_UN') auto
ultimately show "AE x in M. \<forall>y\<in>X. P x y"
unfolding eventually_ae_filter by auto
qed auto
lemma AE_discrete_difference:
assumes X: "countable X"
assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"
assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
shows "AE x in M. x \<notin> X"
proof -
have "(\<Union>x\<in>X. {x}) \<in> null_sets M"
using assms by (intro null_sets_UN') auto
from AE_not_in[OF this] show "AE x in M. x \<notin> X"
by auto
qed
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.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
lemma emeasure_Collect_eq_AE:
"AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> Measurable.pred M Q \<Longrightarrow> Measurable.pred M P \<Longrightarrow>
emeasure M {x\<in>space M. P x} = emeasure M {x\<in>space M. Q x}"
by (intro emeasure_eq_AE) auto
lemma emeasure_eq_0_AE: "AE x in M. \<not> P x \<Longrightarrow> emeasure M {x\<in>space M. P x} = 0"
using AE_iff_measurable[OF _ refl, of M "\<lambda>x. \<not> P x"]
by (cases "{x\<in>space M. P x} \<in> sets M") (simp_all add: emeasure_notin_sets)
lemma emeasure_add_AE:
assumes [measurable]: "A \<in> sets M" "B \<in> sets M" "C \<in> sets M"
assumes 1: "AE x in M. x \<in> C \<longleftrightarrow> x \<in> A \<or> x \<in> B"
assumes 2: "AE x in M. \<not> (x \<in> A \<and> x \<in> B)"
shows "emeasure M C = emeasure M A + emeasure M B"
proof -
have "emeasure M C = emeasure M (A \<union> B)"
by (rule emeasure_eq_AE) (insert 1, auto)
also have "\<dots> = emeasure M A + emeasure M (B - A)"
by (subst plus_emeasure) auto
also have "emeasure M (B - A) = emeasure M B"
by (rule emeasure_eq_AE) (insert 2, auto)
finally show ?thesis .
qed
subsection {* @{text \<sigma>}-finite Measures *}
locale sigma_finite_measure =
fixes M :: "'a measure"
assumes sigma_finite_countable:
"\<exists>A::'a set set. countable A \<and> A \<subseteq> sets M \<and> (\<Union>A) = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)"
lemma (in sigma_finite_measure) sigma_finite:
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>"
proof -
obtain A :: "'a set set" where
[simp]: "countable A" and
A: "A \<subseteq> sets M" "(\<Union>A) = space M" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
using sigma_finite_countable by metis
show thesis
proof cases
assume "A = {}" with `(\<Union>A) = space M` show thesis
by (intro that[of "\<lambda>_. {}"]) auto
next
assume "A \<noteq> {}"
show thesis
proof
show "range (from_nat_into A) \<subseteq> sets M"
using `A \<noteq> {}` A by auto
have "(\<Union>i. from_nat_into A i) = \<Union>A"
using range_from_nat_into[OF `A \<noteq> {}` `countable A`] by auto
with A show "(\<Union>i. from_nat_into A i) = space M"
by auto
qed (intro A from_nat_into `A \<noteq> {}`)
qed
qed
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 -
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
show thesis
proof (rule that[of "disjointed A"])
show "range (disjointed A) \<subseteq> sets M"
by (rule sets.range_disjointed_sets[OF range])
show "(\<Union>i. disjointed A i) = space M"
and "disjoint_family (disjointed A)"
using disjoint_family_disjointed UN_disjointed_eq[of A] space range
by auto
show "emeasure M (disjointed A i) \<noteq> \<infinity>" for i
proof -
have "emeasure M (disjointed A i) \<le> emeasure M (A i)"
using range disjointed_subset[of A i] by (auto intro!: emeasure_mono)
then show ?thesis using measure[of i] by auto
qed
qed
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 -
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
show thesis
proof (rule that[of "\<lambda>n. \<Union>i\<le>n. F i"])
show "range (\<lambda>n. \<Union>i\<le>n. F i) \<subseteq> sets M"
using F by (force simp: incseq_def)
show "(\<Union>n. \<Union>i\<le>n. F i) = space M"
proof -
from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto
with F show ?thesis by fastforce
qed
show "emeasure M (\<Union>i\<le>n. F i) \<noteq> \<infinity>" for n
proof -
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 ?thesis by simp
qed
show "incseq (\<lambda>n. \<Union>i\<le>n. F i)"
by (force simp: incseq_def)
qed
qed
subsection {* 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, measurable_cong]: "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 distr_cong:
"M = K \<Longrightarrow> sets N = sets L \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> distr M N f = distr K L g"
using sets_eq_imp_space_eq[of N L] by (simp add: distr_def Int_def cong: rev_conj_cong)
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 emeasure_Collect_distr:
assumes X[measurable]: "X \<in> measurable M N" "Measurable.pred N P"
shows "emeasure (distr M N X) {x\<in>space N. P x} = emeasure M {x\<in>space M. P (X x)}"
by (subst emeasure_distr)
(auto intro!: arg_cong2[where f=emeasure] X(1)[THEN measurable_space])
lemma emeasure_lfp2[consumes 1, case_names cont f measurable]:
assumes "P M"
assumes cont: "sup_continuous F"
assumes f: "\<And>M. P M \<Longrightarrow> f \<in> measurable M' M"
assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
shows "emeasure M' {x\<in>space M'. lfp F (f x)} = (SUP i. emeasure M' {x\<in>space M'. (F ^^ i) (\<lambda>x. False) (f x)})"
proof (subst (1 2) emeasure_Collect_distr[symmetric, where X=f])
show "f \<in> measurable M' M" "f \<in> measurable M' M"
using f[OF `P M`] by auto
{ fix i show "Measurable.pred M ((F ^^ i) (\<lambda>x. False))"
using `P M` by (induction i arbitrary: M) (auto intro!: *) }
show "Measurable.pred M (lfp F)"
using `P M` cont * by (rule measurable_lfp_coinduct[of P])
have "emeasure (distr M' M f) {x \<in> space (distr M' M f). lfp F x} =
(SUP i. emeasure (distr M' M f) {x \<in> space (distr M' M f). (F ^^ i) (\<lambda>x. False) x})"
using `P M`
proof (coinduction arbitrary: M rule: emeasure_lfp')
case (measurable A N) then have "\<And>N. P N \<Longrightarrow> Measurable.pred (distr M' N f) A"
by metis
then have "\<And>N. P N \<Longrightarrow> Measurable.pred N A"
by simp
with `P N`[THEN *] show ?case
by auto
qed fact
then show "emeasure (distr M' M f) {x \<in> space M. lfp F x} =
(SUP i. emeasure (distr M' M f) {x \<in> space M. (F ^^ i) (\<lambda>x. False) x})"
by simp
qed
lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N"
by (rule measure_eqI) (auto simp: emeasure_distr)
lemma measure_distr:
"f \<in> measurable M N \<Longrightarrow> S \<in> sets N \<Longrightarrow> measure (distr M N f) S = measure M (f -` S \<inter> space M)"
by (simp add: emeasure_distr measure_def)
lemma distr_cong_AE:
assumes 1: "M = K" "sets N = sets L" and
2: "(AE x in M. f x = g x)" and "f \<in> measurable M N" and "g \<in> measurable K L"
shows "distr M N f = distr K L g"
proof (rule measure_eqI)
fix A assume "A \<in> sets (distr M N f)"
with assms show "emeasure (distr M N f) A = emeasure (distr K L g) A"
by (auto simp add: emeasure_distr intro!: emeasure_eq_AE measurable_sets)
qed (insert 1, simp)
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 AE_distr_iff:
assumes f[measurable]: "f \<in> measurable M N" and P[measurable]: "{x \<in> space N. P x} \<in> sets N"
shows "(AE x in distr M N f. P x) \<longleftrightarrow> (AE x in M. P (f x))"
proof (subst (1 2) AE_iff_measurable[OF _ refl])
have "f -` {x\<in>space N. \<not> P x} \<inter> space M = {x \<in> space M. \<not> P (f x)}"
using f[THEN measurable_space] by auto
then show "(emeasure (distr M N f) {x \<in> space (distr M N f). \<not> P x} = 0) =
(emeasure M {x \<in> space M. \<not> P (f x)} = 0)"
by (simp add: emeasure_distr)
qed auto
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)
lemma distr_distr:
"g \<in> measurable N L \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> distr (distr M N f) L g = distr M L (g \<circ> f)"
by (auto simp add: emeasure_distr measurable_space
intro!: arg_cong[where f="emeasure M"] measure_eqI)
subsection {* 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_le_0_iff: "measure M X \<le> 0 \<longleftrightarrow> measure M X = 0"
using measure_nonneg[of M X] by auto
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 max_0_ereal_measure [simp]: "max 0 (ereal (measure M X)) = ereal (measure M X)"
by(simp add: max_def measure_nonneg)
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
subsection {* 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!]:
"emeasure M (space M) \<noteq> \<infinity> \<Longrightarrow> finite_measure M"
proof qed (auto intro!: exI[of _ "{space M}"])
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 _ sets.top S sets.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)
lemma (in finite_measure) measure_increasing: "increasing M (measure M)"
by (auto intro!: finite_measure_mono simp: increasing_def)
lemma (in finite_measure) measure_zero_union:
assumes "s \<in> sets M" "t \<in> sets M" "measure M t = 0"
shows "measure M (s \<union> t) = measure M s"
using assms
proof -
have "measure M (s \<union> t) \<le> measure M s"
using finite_measure_subadditive[of s t] assms by auto
moreover have "measure M (s \<union> t) \<ge> measure M s"
using assms by (blast intro: finite_measure_mono)
ultimately show ?thesis by simp
qed
lemma (in finite_measure) measure_eq_compl:
assumes "s \<in> sets M" "t \<in> sets M"
assumes "measure M (space M - s) = measure M (space M - t)"
shows "measure M s = measure M t"
using assms finite_measure_compl by auto
lemma (in finite_measure) measure_eq_bigunion_image:
assumes "range f \<subseteq> sets M" "range g \<subseteq> sets M"
assumes "disjoint_family f" "disjoint_family g"
assumes "\<And> n :: nat. measure M (f n) = measure M (g n)"
shows "measure M (\<Union>i. f i) = measure M (\<Union>i. g i)"
using assms
proof -
have a: "(\<lambda> i. measure M (f i)) sums (measure M (\<Union>i. f i))"
by (rule finite_measure_UNION[OF assms(1,3)])
have b: "(\<lambda> i. measure M (g i)) sums (measure M (\<Union>i. g i))"
by (rule finite_measure_UNION[OF assms(2,4)])
show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp
qed
lemma (in finite_measure) measure_countably_zero:
assumes "range c \<subseteq> sets M"
assumes "\<And> i. measure M (c i) = 0"
shows "measure M (\<Union>i :: nat. c i) = 0"
proof (rule antisym)
show "measure M (\<Union>i :: nat. c i) \<le> 0"
using finite_measure_subadditive_countably[OF assms(1)] by (simp add: assms(2))
qed (simp add: measure_nonneg)
lemma (in finite_measure) measure_space_inter:
assumes events:"s \<in> sets M" "t \<in> sets M"
assumes "measure M t = measure M (space M)"
shows "measure M (s \<inter> t) = measure M s"
proof -
have "measure M ((space M - s) \<union> (space M - t)) = measure M (space M - s)"
using events assms finite_measure_compl[of "t"] by (auto intro!: measure_zero_union)
also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)"
by blast
finally show "measure M (s \<inter> t) = measure M s"
using events by (auto intro!: measure_eq_compl[of "s \<inter> t" s])
qed
lemma (in finite_measure) measure_equiprobable_finite_unions:
assumes s: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> sets M"
assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> measure M {x} = measure M {y}"
shows "measure M s = real (card s) * measure M {SOME x. x \<in> s}"
proof cases
assume "s \<noteq> {}"
then have "\<exists> x. x \<in> s" by blast
from someI_ex[OF this] assms
have prob_some: "\<And> x. x \<in> s \<Longrightarrow> measure M {x} = measure M {SOME y. y \<in> s}" by blast
have "measure M s = (\<Sum> x \<in> s. measure M {x})"
using finite_measure_eq_setsum_singleton[OF s] by simp
also have "\<dots> = (\<Sum> x \<in> s. measure M {SOME y. y \<in> s})" using prob_some by auto
also have "\<dots> = real (card s) * measure M {(SOME x. x \<in> s)}"
using setsum_constant assms by simp
finally show ?thesis by simp
qed simp
lemma (in finite_measure) measure_real_sum_image_fn:
assumes "e \<in> sets M"
assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> sets M"
assumes "finite s"
assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"
assumes upper: "space M \<subseteq> (\<Union>i \<in> s. f i)"
shows "measure M e = (\<Sum> x \<in> s. measure M (e \<inter> f x))"
proof -
have e: "e = (\<Union>i \<in> s. e \<inter> f i)"
using `e \<in> sets M` sets.sets_into_space upper by blast
hence "measure M e = measure M (\<Union>i \<in> s. e \<inter> f i)" by simp
also have "\<dots> = (\<Sum> x \<in> s. measure M (e \<inter> f x))"
proof (rule finite_measure_finite_Union)
show "finite s" by fact
show "(\<lambda>i. e \<inter> f i)`s \<subseteq> sets M" using assms(2) by auto
show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"
using disjoint by (auto simp: disjoint_family_on_def)
qed
finally show ?thesis .
qed
lemma (in finite_measure) measure_exclude:
assumes "A \<in> sets M" "B \<in> sets M"
assumes "measure M A = measure M (space M)" "A \<inter> B = {}"
shows "measure M B = 0"
using measure_space_inter[of B A] assms by (auto simp: ac_simps)
lemma (in finite_measure) finite_measure_distr:
assumes f: "f \<in> measurable M M'"
shows "finite_measure (distr M M' f)"
proof (rule finite_measureI)
have "f -` space M' \<inter> space M = space M" using f by (auto dest: measurable_space)
with f show "emeasure (distr M M' f) (space (distr M M' f)) \<noteq> \<infinity>" by (auto simp: emeasure_distr)
qed
lemma emeasure_gfp[consumes 1, case_names cont measurable]:
assumes sets[simp]: "\<And>s. sets (M s) = sets N"
assumes "\<And>s. finite_measure (M s)"
assumes cont: "inf_continuous F" "inf_continuous f"
assumes meas: "\<And>P. Measurable.pred N P \<Longrightarrow> Measurable.pred N (F P)"
assumes iter: "\<And>P s. Measurable.pred N P \<Longrightarrow> emeasure (M s) {x\<in>space N. F P x} = f (\<lambda>s. emeasure (M s) {x\<in>space N. P x}) s"
assumes bound: "\<And>P. f P \<le> f (\<lambda>s. emeasure (M s) (space (M s)))"
shows "emeasure (M s) {x\<in>space N. gfp F x} = gfp f s"
proof (subst gfp_transfer_bounded[where \<alpha>="\<lambda>F s. emeasure (M s) {x\<in>space N. F x}" and g=f and f=F and
P="Measurable.pred N", symmetric])
interpret finite_measure "M s" for s by fact
fix C assume "decseq C" "\<And>i. Measurable.pred N (C i)"
then show "(\<lambda>s. emeasure (M s) {x \<in> space N. (INF i. C i) x}) = (INF i. (\<lambda>s. emeasure (M s) {x \<in> space N. C i x}))"
unfolding INF_apply[abs_def]
by (subst INF_emeasure_decseq) (auto simp: antimono_def fun_eq_iff intro!: arg_cong2[where f=emeasure])
next
show "f x \<le> (\<lambda>s. emeasure (M s) {x \<in> space N. F top x})" for x
using bound[of x] sets_eq_imp_space_eq[OF sets] by (simp add: iter)
qed (auto simp add: iter le_fun_def INF_apply[abs_def] intro!: meas cont)
subsection {* Counting space *}
lemma strict_monoI_Suc:
assumes ord [simp]: "(\<And>n. f n < f (Suc n))" shows "strict_mono f"
unfolding strict_mono_def
proof safe
fix n m :: nat assume "n < m" then show "f n < f m"
by (induct m) (auto simp: less_Suc_eq intro: less_trans ord)
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 "X \<in> Pow A" using `X \<subseteq> A` by auto
show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow)
show positive: "positive (Pow A) ?M"
by (auto simp: positive_def)
have additive: "additive (Pow A) ?M"
by (auto simp: card_Un_disjoint additive_def)
interpret ring_of_sets A "Pow A"
by (rule ring_of_setsI) auto
show "countably_additive (Pow A) ?M"
unfolding countably_additive_iff_continuous_from_below[OF positive additive]
proof safe
fix F :: "nat \<Rightarrow> 'a set" assume "incseq F"
show "(\<lambda>i. ?M (F i)) ----> ?M (\<Union>i. F i)"
proof cases
assume "\<exists>i. \<forall>j\<ge>i. F i = F j"
then guess i .. note i = this
{ fix j from i `incseq F` have "F j \<subseteq> F i"
by (cases "i \<le> j") (auto simp: incseq_def) }
then have eq: "(\<Union>i. F i) = F i"
by auto
with i show ?thesis
by (auto intro!: Lim_eventually eventually_sequentiallyI[where c=i])
next
assume "\<not> (\<exists>i. \<forall>j\<ge>i. F i = F j)"
then obtain f where f: "\<And>i. i \<le> f i" "\<And>i. F i \<noteq> F (f i)" by metis
then have "\<And>i. F i \<subseteq> F (f i)" using `incseq F` by (auto simp: incseq_def)
with f have *: "\<And>i. F i \<subset> F (f i)" by auto
have "incseq (\<lambda>i. ?M (F i))"
using `incseq F` unfolding incseq_def by (auto simp: card_mono dest: finite_subset)
then have "(\<lambda>i. ?M (F i)) ----> (SUP n. ?M (F n))"
by (rule LIMSEQ_SUP)
moreover have "(SUP n. ?M (F n)) = \<infinity>"
proof (rule SUP_PInfty)
fix n :: nat show "\<exists>k::nat\<in>UNIV. ereal n \<le> ?M (F k)"
proof (induct n)
case (Suc n)
then guess k .. note k = this
moreover have "finite (F k) \<Longrightarrow> finite (F (f k)) \<Longrightarrow> card (F k) < card (F (f k))"
using `F k \<subset> F (f k)` by (simp add: psubset_card_mono)
moreover have "finite (F (f k)) \<Longrightarrow> finite (F k)"
using `k \<le> f k` `incseq F` by (auto simp: incseq_def dest: finite_subset)
ultimately show ?case
by (auto intro!: exI[of _ "f k"])
qed auto
qed
moreover
have "inj (\<lambda>n. F ((f ^^ n) 0))"
by (intro strict_mono_imp_inj_on strict_monoI_Suc) (simp add: *)
then have 1: "infinite (range (\<lambda>i. F ((f ^^ i) 0)))"
by (rule range_inj_infinite)
have "infinite (Pow (\<Union>i. F i))"
by (rule infinite_super[OF _ 1]) auto
then have "infinite (\<Union>i. F i)"
by auto
ultimately show ?thesis by auto
qed
qed
qed
lemma distr_bij_count_space:
assumes f: "bij_betw f A B"
shows "distr (count_space A) (count_space B) f = count_space B"
proof (rule measure_eqI)
have f': "f \<in> measurable (count_space A) (count_space B)"
using f unfolding Pi_def bij_betw_def by auto
fix X assume "X \<in> sets (distr (count_space A) (count_space B) f)"
then have X: "X \<in> sets (count_space B)" by auto
moreover then have "f -` X \<inter> A = the_inv_into A f ` X"
using f by (auto simp: bij_betw_def subset_image_iff image_iff the_inv_into_f_f intro: the_inv_into_f_f[symmetric])
moreover have "inj_on (the_inv_into A f) B"
using X f by (auto simp: bij_betw_def inj_on_the_inv_into)
with X have "inj_on (the_inv_into A f) X"
by (auto intro: subset_inj_on)
ultimately show "emeasure (distr (count_space A) (count_space B) f) X = emeasure (count_space B) X"
using f unfolding emeasure_distr[OF f' X]
by (subst (1 2) emeasure_count_space) (auto simp: card_image dest: finite_imageD)
qed simp
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 measure_count_space: "measure (count_space A) X = (if X \<subseteq> A then card X else 0)"
unfolding measure_def
by (cases "finite X") (simp_all add: emeasure_notin_sets)
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 space_empty: "space M = {} \<Longrightarrow> M = count_space {}"
by (rule measure_eqI) (simp_all add: space_empty_iff)
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_countable:
assumes A: "countable A"
shows "sigma_finite_measure (count_space A)"
proof qed (auto intro!: exI[of _ "(\<lambda>a. {a}) ` A"] simp: A)
lemma sigma_finite_measure_count_space:
fixes A :: "'a::countable set" shows "sigma_finite_measure (count_space A)"
by (rule sigma_finite_measure_count_space_countable) auto
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
subsection {* Measure restricted to space *}
lemma emeasure_restrict_space:
assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"
shows "emeasure (restrict_space M \<Omega>) A = emeasure M A"
proof cases
assume "A \<in> sets M"
show ?thesis
proof (rule emeasure_measure_of[OF restrict_space_def])
show "op \<inter> \<Omega> ` sets M \<subseteq> Pow (\<Omega> \<inter> space M)" "A \<in> sets (restrict_space M \<Omega>)"
using `A \<subseteq> \<Omega>` `A \<in> sets M` sets.space_closed by (auto simp: sets_restrict_space)
show "positive (sets (restrict_space M \<Omega>)) (emeasure M)"
by (auto simp: positive_def emeasure_nonneg)
show "countably_additive (sets (restrict_space M \<Omega>)) (emeasure M)"
proof (rule countably_additiveI)
fix A :: "nat \<Rightarrow> _" assume "range A \<subseteq> sets (restrict_space M \<Omega>)" "disjoint_family A"
with assms have "\<And>i. A i \<in> sets M" "\<And>i. A i \<subseteq> space M" "disjoint_family A"
by (fastforce simp: sets_restrict_space_iff[OF assms(1)] image_subset_iff
dest: sets.sets_into_space)+
then show "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
by (subst suminf_emeasure) (auto simp: disjoint_family_subset)
qed
qed
next
assume "A \<notin> sets M"
moreover with assms have "A \<notin> sets (restrict_space M \<Omega>)"
by (simp add: sets_restrict_space_iff)
ultimately show ?thesis
by (simp add: emeasure_notin_sets)
qed
lemma measure_restrict_space:
assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"
shows "measure (restrict_space M \<Omega>) A = measure M A"
using emeasure_restrict_space[OF assms] by (simp add: measure_def)
lemma AE_restrict_space_iff:
assumes "\<Omega> \<inter> space M \<in> sets M"
shows "(AE x in restrict_space M \<Omega>. P x) \<longleftrightarrow> (AE x in M. x \<in> \<Omega> \<longrightarrow> P x)"
proof -
have ex_cong: "\<And>P Q f. (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> (\<And>x. Q x \<Longrightarrow> P (f x)) \<Longrightarrow> (\<exists>x. P x) \<longleftrightarrow> (\<exists>x. Q x)"
by auto
{ fix X assume X: "X \<in> sets M" "emeasure M X = 0"
then have "emeasure M (\<Omega> \<inter> space M \<inter> X) \<le> emeasure M X"
by (intro emeasure_mono) auto
then have "emeasure M (\<Omega> \<inter> space M \<inter> X) = 0"
using X by (auto intro!: antisym) }
with assms show ?thesis
unfolding eventually_ae_filter
by (auto simp add: space_restrict_space null_sets_def sets_restrict_space_iff
emeasure_restrict_space cong: conj_cong
intro!: ex_cong[where f="\<lambda>X. (\<Omega> \<inter> space M) \<inter> X"])
qed
lemma restrict_restrict_space:
assumes "A \<inter> space M \<in> sets M" "B \<inter> space M \<in> sets M"
shows "restrict_space (restrict_space M A) B = restrict_space M (A \<inter> B)" (is "?l = ?r")
proof (rule measure_eqI[symmetric])
show "sets ?r = sets ?l"
unfolding sets_restrict_space image_comp by (intro image_cong) auto
next
fix X assume "X \<in> sets (restrict_space M (A \<inter> B))"
then obtain Y where "Y \<in> sets M" "X = Y \<inter> A \<inter> B"
by (auto simp: sets_restrict_space)
with assms sets.Int[OF assms] show "emeasure ?r X = emeasure ?l X"
by (subst (1 2) emeasure_restrict_space)
(auto simp: space_restrict_space sets_restrict_space_iff emeasure_restrict_space ac_simps)
qed
lemma restrict_count_space: "restrict_space (count_space B) A = count_space (A \<inter> B)"
proof (rule measure_eqI)
show "sets (restrict_space (count_space B) A) = sets (count_space (A \<inter> B))"
by (subst sets_restrict_space) auto
moreover fix X assume "X \<in> sets (restrict_space (count_space B) A)"
ultimately have "X \<subseteq> A \<inter> B" by auto
then show "emeasure (restrict_space (count_space B) A) X = emeasure (count_space (A \<inter> B)) X"
by (cases "finite X") (auto simp add: emeasure_restrict_space)
qed
lemma sigma_finite_measure_restrict_space:
assumes "sigma_finite_measure M"
and A: "A \<in> sets M"
shows "sigma_finite_measure (restrict_space M A)"
proof -
interpret sigma_finite_measure M by fact
from sigma_finite_countable obtain C
where C: "countable C" "C \<subseteq> sets M" "(\<Union>C) = space M" "\<forall>a\<in>C. emeasure M a \<noteq> \<infinity>"
by blast
let ?C = "op \<inter> A ` C"
from C have "countable ?C" "?C \<subseteq> sets (restrict_space M A)" "(\<Union>?C) = space (restrict_space M A)"
by(auto simp add: sets_restrict_space space_restrict_space)
moreover {
fix a
assume "a \<in> ?C"
then obtain a' where "a = A \<inter> a'" "a' \<in> C" ..
then have "emeasure (restrict_space M A) a \<le> emeasure M a'"
using A C by(auto simp add: emeasure_restrict_space intro: emeasure_mono)
also have "\<dots> < \<infinity>" using C(4)[rule_format, of a'] \<open>a' \<in> C\<close> by simp
finally have "emeasure (restrict_space M A) a \<noteq> \<infinity>" by simp }
ultimately show ?thesis
by unfold_locales (rule exI conjI|assumption|blast)+
qed
lemma finite_measure_restrict_space:
assumes "finite_measure M"
and A: "A \<in> sets M"
shows "finite_measure (restrict_space M A)"
proof -
interpret finite_measure M by fact
show ?thesis
by(rule finite_measureI)(simp add: emeasure_restrict_space A space_restrict_space)
qed
lemma restrict_distr:
assumes [measurable]: "f \<in> measurable M N"
assumes [simp]: "\<Omega> \<inter> space N \<in> sets N" and restrict: "f \<in> space M \<rightarrow> \<Omega>"
shows "restrict_space (distr M N f) \<Omega> = distr M (restrict_space N \<Omega>) f"
(is "?l = ?r")
proof (rule measure_eqI)
fix A assume "A \<in> sets (restrict_space (distr M N f) \<Omega>)"
with restrict show "emeasure ?l A = emeasure ?r A"
by (subst emeasure_distr)
(auto simp: sets_restrict_space_iff emeasure_restrict_space emeasure_distr
intro!: measurable_restrict_space2)
qed (simp add: sets_restrict_space)
lemma measure_eqI_restrict_generator:
assumes E: "Int_stable E" "E \<subseteq> Pow \<Omega>" "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"
assumes sets_eq: "sets M = sets N" and \<Omega>: "\<Omega> \<in> sets M"
assumes "sets (restrict_space M \<Omega>) = sigma_sets \<Omega> E"
assumes "sets (restrict_space N \<Omega>) = sigma_sets \<Omega> E"
assumes ae: "AE x in M. x \<in> \<Omega>" "AE x in N. x \<in> \<Omega>"
assumes A: "countable A" "A \<noteq> {}" "A \<subseteq> E" "\<Union>A = \<Omega>" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
shows "M = N"
proof (rule measure_eqI)
fix X assume X: "X \<in> sets M"
then have "emeasure M X = emeasure (restrict_space M \<Omega>) (X \<inter> \<Omega>)"
using ae \<Omega> by (auto simp add: emeasure_restrict_space intro!: emeasure_eq_AE)
also have "restrict_space M \<Omega> = restrict_space N \<Omega>"
proof (rule measure_eqI_generator_eq)
fix X assume "X \<in> E"
then show "emeasure (restrict_space M \<Omega>) X = emeasure (restrict_space N \<Omega>) X"
using E \<Omega> by (subst (1 2) emeasure_restrict_space) (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq])
next
show "range (from_nat_into A) \<subseteq> E" "(\<Union>i. from_nat_into A i) = \<Omega>"
unfolding Sup_image_eq[symmetric, where f="from_nat_into A"] using A by auto
next
fix i
have "emeasure (restrict_space M \<Omega>) (from_nat_into A i) = emeasure M (from_nat_into A i)"
using A \<Omega> by (subst emeasure_restrict_space)
(auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq] intro: from_nat_into)
with A show "emeasure (restrict_space M \<Omega>) (from_nat_into A i) \<noteq> \<infinity>"
by (auto intro: from_nat_into)
qed fact+
also have "emeasure (restrict_space N \<Omega>) (X \<inter> \<Omega>) = emeasure N X"
using X ae \<Omega> by (auto simp add: emeasure_restrict_space sets_eq intro!: emeasure_eq_AE)
finally show "emeasure M X = emeasure N X" .
qed fact
subsection {* Null measure *}
definition "null_measure M = sigma (space M) (sets M)"
lemma space_null_measure[simp]: "space (null_measure M) = space M"
by (simp add: null_measure_def)
lemma sets_null_measure[simp, measurable_cong]: "sets (null_measure M) = sets M"
by (simp add: null_measure_def)
lemma emeasure_null_measure[simp]: "emeasure (null_measure M) X = 0"
by (cases "X \<in> sets M", rule emeasure_measure_of)
(auto simp: positive_def countably_additive_def emeasure_notin_sets null_measure_def
dest: sets.sets_into_space)
lemma measure_null_measure[simp]: "measure (null_measure M) X = 0"
by (simp add: measure_def)
lemma null_measure_idem [simp]: "null_measure (null_measure M) = null_measure M"
by(rule measure_eqI) simp_all
subsection \<open>Scaling a measure\<close>
definition scale_measure :: "real \<Rightarrow> 'a measure \<Rightarrow> 'a measure"
where "scale_measure r M = measure_of (space M) (sets M) (\<lambda>A. (max 0 r) * emeasure M A)"
lemma space_scale_measure: "space (scale_measure r M) = space M"
by(simp add: scale_measure_def)
lemma sets_scale_measure [simp, measurable_cong]: "sets (scale_measure r M) = sets M"
by(simp add: scale_measure_def)
lemma emeasure_scale_measure [simp]:
"emeasure (scale_measure r M) A = max 0 r * emeasure M A"
(is "_ = ?\<mu> A")
proof(cases "A \<in> sets M")
case True
show ?thesis unfolding scale_measure_def
proof(rule emeasure_measure_of_sigma)
show "sigma_algebra (space M) (sets M)" ..
show "positive (sets M) ?\<mu>" by(simp add: positive_def emeasure_nonneg)
show "countably_additive (sets M) ?\<mu>"
proof (rule countably_additiveI)
fix A :: "nat \<Rightarrow> _" assume *: "range A \<subseteq> sets M" "disjoint_family A"
have "(\<Sum>i. ?\<mu> (A i)) = max 0 (ereal r) * (\<Sum>i. emeasure M (A i))"
by(rule suminf_cmult_ereal)(simp_all add: emeasure_nonneg)
also have "\<dots> = ?\<mu> (\<Union>i. A i)" using * by(simp add: suminf_emeasure)
finally show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)" .
qed
qed(fact True)
qed(simp add: emeasure_notin_sets)
lemma measure_scale_measure [simp]: "measure (scale_measure r M) A = max 0 r * measure M A"
by(simp add: measure_def max_def)
lemma scale_measure_1 [simp]: "scale_measure 1 M = M"
by(rule measure_eqI)(simp_all add: max_def)
lemma scale_measure_le_0: "r \<le> 0 \<Longrightarrow> scale_measure r M = null_measure M"
by(rule measure_eqI)(simp_all add: max_def)
lemma scale_measure_0 [simp]: "scale_measure 0 M = null_measure M"
by(simp add: scale_measure_le_0)
lemma scale_scale_measure [simp]:
"scale_measure r (scale_measure r' M) = scale_measure (max 0 r * max 0 r') M"
by(rule measure_eqI)(simp_all add: max_def mult.assoc times_ereal.simps(1)[symmetric] del: times_ereal.simps(1))
lemma scale_null_measure [simp]: "scale_measure r (null_measure M) = null_measure M"
by(rule measure_eqI) simp_all
subsection \<open>Measures form a chain-complete partial order\<close>
instantiation measure :: (type) order_bot
begin
definition bot_measure :: "'a measure" where
"bot_measure = sigma {} {{}}"
lemma space_bot[simp]: "space bot = {}"
unfolding bot_measure_def by (rule space_measure_of) auto
lemma sets_bot[simp]: "sets bot = {{}}"
unfolding bot_measure_def by (subst sets_measure_of) auto
lemma emeasure_bot[simp]: "emeasure bot = (\<lambda>x. 0)"
unfolding bot_measure_def by (rule emeasure_sigma)
inductive less_eq_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where
"sets N = sets M \<Longrightarrow> (\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A \<le> emeasure N A) \<Longrightarrow> less_eq_measure M N"
| "less_eq_measure bot N"
definition less_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where
"less_measure M N \<longleftrightarrow> (M \<le> N \<and> \<not> N \<le> M)"
instance
proof (standard, goal_cases)
case 1 then show ?case
unfolding less_measure_def ..
next
case (2 M) then show ?case
by (intro less_eq_measure.intros) auto
next
case (3 M N L) then show ?case
apply (safe elim!: less_eq_measure.cases)
apply (simp_all add: less_eq_measure.intros)
apply (rule less_eq_measure.intros)
apply simp
apply (blast intro: order_trans) []
unfolding less_eq_measure.simps
apply (rule disjI2)
apply simp
apply (rule measure_eqI)
apply (auto intro!: antisym)
done
next
case (4 M N) then show ?case
apply (safe elim!: less_eq_measure.cases intro!: measure_eqI)
apply simp
apply simp
apply (blast intro: antisym)
apply (simp)
apply (blast intro: antisym)
apply simp
done
qed (rule less_eq_measure.intros)
end
lemma le_emeasureD: "M \<le> N \<Longrightarrow> emeasure M A \<le> emeasure N A"
by (cases "A \<in> sets M") (auto elim!: less_eq_measure.cases simp: emeasure_notin_sets)
lemma le_sets: "N \<le> M \<Longrightarrow> sets N \<le> sets M"
unfolding less_eq_measure.simps by auto
instantiation measure :: (type) ccpo
begin
definition Sup_measure :: "'a measure set \<Rightarrow> 'a measure" where
"Sup_measure A = measure_of (SUP a:A. space a) (SUP a:A. sets a) (SUP a:A. emeasure a)"
lemma
assumes A: "Complete_Partial_Order.chain op \<le> A" and a: "a \<noteq> bot" "a \<in> A"
shows space_Sup: "space (Sup A) = space a"
and sets_Sup: "sets (Sup A) = sets a"
proof -
have sets: "(SUP a:A. sets a) = sets a"
proof (intro antisym SUP_least)
fix a' show "a' \<in> A \<Longrightarrow> sets a' \<subseteq> sets a"
using a chainD[OF A, of a a'] by (auto elim!: less_eq_measure.cases)
qed (insert \<open>a\<in>A\<close>, auto)
have space: "(SUP a:A. space a) = space a"
proof (intro antisym SUP_least)
fix a' show "a' \<in> A \<Longrightarrow> space a' \<subseteq> space a"
using a chainD[OF A, of a a'] by (intro sets_le_imp_space_le) (auto elim!: less_eq_measure.cases)
qed (insert \<open>a\<in>A\<close>, auto)
show "space (Sup A) = space a"
unfolding Sup_measure_def sets space sets.space_measure_of_eq ..
show "sets (Sup A) = sets a"
unfolding Sup_measure_def sets space sets.sets_measure_of_eq ..
qed
lemma emeasure_Sup:
assumes A: "Complete_Partial_Order.chain op \<le> A" "A \<noteq> {}"
assumes "X \<in> sets (Sup A)"
shows "emeasure (Sup A) X = (SUP a:A. emeasure a) X"
proof (rule emeasure_measure_of[OF Sup_measure_def])
show "countably_additive (sets (Sup A)) (SUP a:A. emeasure a)"
unfolding countably_additive_def
proof safe
fix F :: "nat \<Rightarrow> 'a set" assume F: "range F \<subseteq> sets (Sup A)" "disjoint_family F"
show "(\<Sum>i. (SUP a:A. emeasure a) (F i)) = SUPREMUM A emeasure (UNION UNIV F)"
unfolding SUP_apply
proof (subst suminf_SUP_eq_directed)
fix N i j assume "i \<in> A" "j \<in> A"
with A(1)
show "\<exists>k\<in>A. \<forall>n\<in>N. emeasure i (F n) \<le> emeasure k (F n) \<and> emeasure j (F n) \<le> emeasure k (F n)"
by (blast elim: chainE dest: le_emeasureD)
next
show "(SUP n:A. \<Sum>i. emeasure n (F i)) = (SUP y:A. emeasure y (UNION UNIV F))"
proof (intro SUP_cong refl)
fix a assume "a \<in> A" then show "(\<Sum>i. emeasure a (F i)) = emeasure a (UNION UNIV F)"
using sets_Sup[OF A(1), of a] F by (cases "a = bot") (auto simp: suminf_emeasure)
qed
qed (insert F \<open>A \<noteq> {}\<close>, auto simp: suminf_emeasure intro!: SUP_cong)
qed
qed (insert \<open>A \<noteq> {}\<close> \<open>X \<in> sets (Sup A)\<close>, auto simp: positive_def dest: sets.sets_into_space intro: SUP_upper2)
instance
proof
fix A and x :: "'a measure" assume A: "Complete_Partial_Order.chain op \<le> A" and x: "x \<in> A"
show "x \<le> Sup A"
proof cases
assume "x \<noteq> bot"
show ?thesis
proof
show "sets (Sup A) = sets x"
using A \<open>x \<noteq> bot\<close> x by (rule sets_Sup)
with x show "\<And>a. a \<in> sets x \<Longrightarrow> emeasure x a \<le> emeasure (Sup A) a"
by (subst emeasure_Sup[OF A]) (auto intro: SUP_upper)
qed
qed simp
next
fix A and x :: "'a measure" assume A: "Complete_Partial_Order.chain op \<le> A" and x: "\<And>z. z \<in> A \<Longrightarrow> z \<le> x"
consider "A = {}" | "A = {bot}" | x where "x\<in>A" "x \<noteq> bot"
by blast
then show "Sup A \<le> x"
proof cases
assume "A = {}"
moreover have "Sup ({}::'a measure set) = bot"
by (auto simp add: Sup_measure_def sigma_sets_empty_eq intro!: measure_eqI)
ultimately show ?thesis
by simp
next
assume "A = {bot}"
moreover have "Sup ({bot}::'a measure set) = bot"
by (auto simp add: Sup_measure_def sigma_sets_empty_eq intro!: measure_eqI)
ultimately show ?thesis
by simp
next
fix a assume "a \<in> A" "a \<noteq> bot"
then have "a \<le> x" "x \<noteq> bot" "a \<noteq> bot"
using x[OF \<open>a \<in> A\<close>] by (auto simp: bot_unique)
then have "sets x = sets a"
by (auto elim: less_eq_measure.cases)
show "Sup A \<le> x"
proof (rule less_eq_measure.intros)
show "sets x = sets (Sup A)"
by (subst sets_Sup[OF A \<open>a \<noteq> bot\<close> \<open>a \<in> A\<close>]) fact
next
fix X assume "X \<in> sets (Sup A)"
then have "emeasure (Sup A) X \<le> (SUP a:A. emeasure a X)"
using \<open>a\<in>A\<close> by (subst emeasure_Sup[OF A _]) auto
also have "\<dots> \<le> emeasure x X"
by (intro SUP_least le_emeasureD x)
finally show "emeasure (Sup A) X \<le> emeasure x X" .
qed
qed
qed
end
lemma
assumes A: "Complete_Partial_Order.chain op \<le> (f ` A)" and a: "a \<in> A" "f a \<noteq> bot"
shows space_SUP: "space (SUP M:A. f M) = space (f a)"
and sets_SUP: "sets (SUP M:A. f M) = sets (f a)"
unfolding SUP_def by(rule space_Sup[OF A a(2) imageI[OF a(1)]] sets_Sup[OF A a(2) imageI[OF a(1)]])+
lemma emeasure_SUP:
assumes A: "Complete_Partial_Order.chain op \<le> (f ` A)" "A \<noteq> {}"
assumes "X \<in> sets (SUP M:A. f M)"
shows "emeasure (SUP M:A. f M) X = (SUP M:A. emeasure (f M)) X"
using \<open>X \<in> _\<close> unfolding SUP_def by(subst emeasure_Sup[OF A(1)]; simp add: A)
end