(* Title: HOL/Probability/Caratheodory.thy
Author: Lawrence C Paulson
Author: Johannes Hölzl, TU München
*)
header {*Caratheodory Extension Theorem*}
theory Caratheodory
imports Sigma_Algebra "~~/src/HOL/Multivariate_Analysis/Extended_Real_Limits"
begin
lemma sums_def2:
"f sums x \<longleftrightarrow> (\<lambda>n. (\<Sum>i\<le>n. f i)) ----> x"
unfolding sums_def
apply (subst LIMSEQ_Suc_iff[symmetric])
unfolding atLeastLessThanSuc_atLeastAtMost atLeast0AtMost ..
text {*
Originally from the Hurd/Coble measure theory development, translated by Lawrence Paulson.
*}
lemma suminf_ereal_2dimen:
fixes f:: "nat \<times> nat \<Rightarrow> ereal"
assumes pos: "\<And>p. 0 \<le> f p"
assumes "\<And>m. g m = (\<Sum>n. f (m,n))"
shows "(\<Sum>i. f (prod_decode i)) = suminf g"
proof -
have g_def: "g = (\<lambda>m. (\<Sum>n. f (m,n)))"
using assms by (simp add: fun_eq_iff)
have reindex: "\<And>B. (\<Sum>x\<in>B. f (prod_decode x)) = setsum f (prod_decode ` B)"
by (simp add: setsum_reindex[OF inj_prod_decode] comp_def)
{ fix n
let ?M = "\<lambda>f. Suc (Max (f ` prod_decode ` {..<n}))"
{ fix a b x assume "x < n" and [symmetric]: "(a, b) = prod_decode x"
then have "a < ?M fst" "b < ?M snd"
by (auto intro!: Max_ge le_imp_less_Suc image_eqI) }
then have "setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<?M fst} \<times> {..<?M snd})"
by (auto intro!: setsum_mono3 simp: pos)
then have "\<exists>a b. setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<a} \<times> {..<b})" by auto }
moreover
{ fix a b
let ?M = "prod_decode ` {..<Suc (Max (prod_encode ` ({..<a} \<times> {..<b})))}"
{ fix a' b' assume "a' < a" "b' < b" then have "(a', b') \<in> ?M"
by (auto intro!: Max_ge le_imp_less_Suc image_eqI[where x="prod_encode (a', b')"]) }
then have "setsum f ({..<a} \<times> {..<b}) \<le> setsum f ?M"
by (auto intro!: setsum_mono3 simp: pos) }
ultimately
show ?thesis unfolding g_def using pos
by (auto intro!: SUPR_eq simp: setsum_cartesian_product reindex le_SUPI2
setsum_nonneg suminf_ereal_eq_SUPR SUPR_pair
SUPR_ereal_setsum[symmetric] incseq_setsumI setsum_nonneg)
qed
subsection {* Measure Spaces *}
record 'a measure_space = "'a algebra" +
measure :: "'a set \<Rightarrow> ereal"
definition positive where "positive M f \<longleftrightarrow> f {} = (0::ereal) \<and> (\<forall>A\<in>sets M. 0 \<le> f A)"
definition additive where "additive M f \<longleftrightarrow>
(\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) = f x + f y)"
definition countably_additive :: "('a, 'b) algebra_scheme \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
"countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow>
(\<Sum>i. f (A i)) = f (\<Union>i. A i))"
definition increasing where "increasing M f \<longleftrightarrow>
(\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<subseteq> y \<longrightarrow> f x \<le> f y)"
definition subadditive where "subadditive M f \<longleftrightarrow>
(\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
definition countably_subadditive where "countably_subadditive M f \<longleftrightarrow>
(\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow>
(f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))))"
definition lambda_system where "lambda_system M f = {l \<in> sets M.
\<forall>x \<in> sets M. f (l \<inter> x) + f ((space M - l) \<inter> x) = f x}"
definition outer_measure_space where "outer_measure_space M f \<longleftrightarrow>
positive M f \<and> increasing M f \<and> countably_subadditive M f"
definition measure_set where "measure_set M f X = {r.
\<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}"
locale measure_space = sigma_algebra M for M :: "('a, 'b) measure_space_scheme" +
assumes measure_positive: "positive M (measure M)"
and ca: "countably_additive M (measure M)"
abbreviation (in measure_space) "\<mu> \<equiv> measure M"
lemma (in measure_space)
shows empty_measure[simp, intro]: "\<mu> {} = 0"
and positive_measure[simp, intro!]: "\<And>A. A \<in> sets M \<Longrightarrow> 0 \<le> \<mu> A"
using measure_positive unfolding positive_def by auto
lemma increasingD:
"increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M \<Longrightarrow> f x \<le> f y"
by (auto simp add: increasing_def)
lemma subadditiveD:
"subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> sets M \<Longrightarrow> y \<in> sets M
\<Longrightarrow> f (x \<union> y) \<le> f x + f y"
by (auto simp add: subadditive_def)
lemma additiveD:
"additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> sets M \<Longrightarrow> y \<in> sets M
\<Longrightarrow> f (x \<union> y) = f x + f y"
by (auto simp add: additive_def)
lemma countably_additiveI:
assumes "\<And>A x. range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> sets M
\<Longrightarrow> (\<Sum>i. f (A i)) = f (\<Union>i. A i)"
shows "countably_additive M f"
using assms by (simp add: countably_additive_def)
section "Extend binary sets"
lemma LIMSEQ_binaryset:
assumes f: "f {} = 0"
shows "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) ----> f A + f B"
proof -
have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
proof
fix n
show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B"
by (induct n) (auto simp add: binaryset_def f)
qed
moreover
have "... ----> f A + f B" by (rule LIMSEQ_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 {* Lambda Systems *}
lemma (in algebra) lambda_system_eq:
shows "lambda_system M f = {l \<in> sets M.
\<forall>x \<in> sets M. f (x \<inter> l) + f (x - l) = f x}"
proof -
have [simp]: "!!l x. l \<in> sets M \<Longrightarrow> x \<in> sets M \<Longrightarrow> (space M - l) \<inter> x = x - l"
by (metis Int_Diff Int_absorb1 Int_commute sets_into_space)
show ?thesis
by (auto simp add: lambda_system_def) (metis Int_commute)+
qed
lemma (in algebra) lambda_system_empty:
"positive M f \<Longrightarrow> {} \<in> lambda_system M f"
by (auto simp add: positive_def lambda_system_eq)
lemma lambda_system_sets:
"x \<in> lambda_system M f \<Longrightarrow> x \<in> sets M"
by (simp add: lambda_system_def)
lemma (in algebra) lambda_system_Compl:
fixes f:: "'a set \<Rightarrow> ereal"
assumes x: "x \<in> lambda_system M f"
shows "space M - x \<in> lambda_system M f"
proof -
have "x \<subseteq> space M"
by (metis sets_into_space lambda_system_sets x)
hence "space M - (space M - x) = x"
by (metis double_diff equalityE)
with x show ?thesis
by (force simp add: lambda_system_def ac_simps)
qed
lemma (in algebra) lambda_system_Int:
fixes f:: "'a set \<Rightarrow> ereal"
assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
shows "x \<inter> y \<in> lambda_system M f"
proof -
from xl yl show ?thesis
proof (auto simp add: positive_def lambda_system_eq Int)
fix u
assume x: "x \<in> sets M" and y: "y \<in> sets M" and u: "u \<in> sets M"
and fx: "\<forall>z\<in>sets M. f (z \<inter> x) + f (z - x) = f z"
and fy: "\<forall>z\<in>sets M. f (z \<inter> y) + f (z - y) = f z"
have "u - x \<inter> y \<in> sets M"
by (metis Diff Diff_Int Un u x y)
moreover
have "(u - (x \<inter> y)) \<inter> y = u \<inter> y - x" by blast
moreover
have "u - x \<inter> y - y = u - y" by blast
ultimately
have ey: "f (u - x \<inter> y) = f (u \<inter> y - x) + f (u - y)" using fy
by force
have "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y)
= (f (u \<inter> (x \<inter> y)) + f (u \<inter> y - x)) + f (u - y)"
by (simp add: ey ac_simps)
also have "... = (f ((u \<inter> y) \<inter> x) + f (u \<inter> y - x)) + f (u - y)"
by (simp add: Int_ac)
also have "... = f (u \<inter> y) + f (u - y)"
using fx [THEN bspec, of "u \<inter> y"] Int y u
by force
also have "... = f u"
by (metis fy u)
finally show "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) = f u" .
qed
qed
lemma (in algebra) lambda_system_Un:
fixes f:: "'a set \<Rightarrow> ereal"
assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
shows "x \<union> y \<in> lambda_system M f"
proof -
have "(space M - x) \<inter> (space M - y) \<in> sets M"
by (metis Diff_Un Un compl_sets lambda_system_sets xl yl)
moreover
have "x \<union> y = space M - ((space M - x) \<inter> (space M - y))"
by auto (metis subsetD lambda_system_sets sets_into_space xl yl)+
ultimately show ?thesis
by (metis lambda_system_Compl lambda_system_Int xl yl)
qed
lemma (in algebra) lambda_system_algebra:
"positive M f \<Longrightarrow> algebra (M\<lparr>sets := lambda_system M f\<rparr>)"
apply (auto simp add: algebra_iff_Un)
apply (metis lambda_system_sets set_mp sets_into_space)
apply (metis lambda_system_empty)
apply (metis lambda_system_Compl)
apply (metis lambda_system_Un)
done
lemma (in algebra) lambda_system_strong_additive:
assumes z: "z \<in> sets M" and disj: "x \<inter> y = {}"
and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)"
proof -
have "z \<inter> x = (z \<inter> (x \<union> y)) \<inter> x" using disj by blast
moreover
have "z \<inter> y = (z \<inter> (x \<union> y)) - x" using disj by blast
moreover
have "(z \<inter> (x \<union> y)) \<in> sets M"
by (metis Int Un lambda_system_sets xl yl z)
ultimately show ?thesis using xl yl
by (simp add: lambda_system_eq)
qed
lemma (in algebra) lambda_system_additive:
"additive (M (|sets := lambda_system M f|)) f"
proof (auto simp add: additive_def)
fix x and y
assume disj: "x \<inter> y = {}"
and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
hence "x \<in> sets M" "y \<in> sets M" by (blast intro: lambda_system_sets)+
thus "f (x \<union> y) = f x + f y"
using lambda_system_strong_additive [OF top disj xl yl]
by (simp add: Un)
qed
lemma (in ring_of_sets) disjointed_additive:
assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> sets M" "incseq A"
shows "(\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"
proof (induct n)
case (Suc n)
then have "(\<Sum>i\<le>Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))"
by simp
also have "\<dots> = f (A n \<union> disjointed A (Suc n))"
using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_incseq)
also have "A n \<union> disjointed A (Suc n) = A (Suc n)"
using `incseq A` by (auto dest: incseq_SucD simp: disjointed_incseq)
finally show ?case .
qed simp
lemma (in ring_of_sets) countably_subadditive_subadditive:
assumes f: "positive M f" and cs: "countably_subadditive M f"
shows "subadditive M f"
proof (auto simp add: subadditive_def)
fix x y
assume x: "x \<in> sets M" and y: "y \<in> sets 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> sets M \<longrightarrow>
(\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
f (\<Union>i. binaryset x y i) \<le> (\<Sum> n. f (binaryset x y n))"
using cs by (auto simp add: countably_subadditive_def)
hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
f (x \<union> y) \<le> (\<Sum> n. f (binaryset x y n))"
by (simp add: range_binaryset_eq UN_binaryset_eq)
thus "f (x \<union> y) \<le> f x + f y" using f x y
by (auto simp add: Un o_def suminf_binaryset_eq positive_def)
qed
lemma (in ring_of_sets) additive_sum:
fixes A:: "nat \<Rightarrow> 'a set"
assumes f: "positive M f" and ad: "additive M f" and "finite S"
and A: "range A \<subseteq> sets 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 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> sets M" using A by blast
moreover have "(\<Union>i\<in>S. A i) \<in> sets M"
using A `finite S` by auto
moreover
ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)"
using ad UNION_in_sets A by (auto simp add: additive_def)
with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
by (auto simp add: additive_def subset_insertI)
qed
lemma (in 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> sets M"
and disj: "disjoint_family A"
shows "(\<Sum>i. f (A i)) \<le> f (space M)"
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)"
by (rule additive_sum [OF f ad _ A]) (auto simp: disj_N)
also have "... \<le> f (space M)" using space_closed A
by (intro increasingD[OF inc] finite_UN) auto
finally show "(\<Sum>i<N. f (A i)) \<le> f (space M)" by simp
qed (insert f A, auto simp: positive_def)
lemma lambda_system_increasing:
"increasing M f \<Longrightarrow> increasing (M (|sets := lambda_system M f|)) f"
by (simp add: increasing_def lambda_system_def)
lemma lambda_system_positive:
"positive M f \<Longrightarrow> positive (M (|sets := lambda_system M f|)) f"
by (simp add: positive_def lambda_system_def)
lemma (in algebra) lambda_system_strong_sum:
fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ereal"
assumes f: "positive M f" and a: "a \<in> sets M"
and A: "range A \<subseteq> lambda_system M f"
and disj: "disjoint_family A"
shows "(\<Sum>i = 0..<n. f (a \<inter>A i)) = f (a \<inter> (\<Union>i\<in>{0..<n}. A i))"
proof (induct n)
case 0 show ?case using f by (simp add: positive_def)
next
case (Suc n)
have 2: "A n \<inter> UNION {0..<n} A = {}" using disj
by (force simp add: disjoint_family_on_def neq_iff)
have 3: "A n \<in> lambda_system M f" using A
by blast
interpret l: algebra "M\<lparr>sets := lambda_system M f\<rparr>"
using f by (rule lambda_system_algebra)
have 4: "UNION {0..<n} A \<in> lambda_system M f"
using A l.UNION_in_sets by simp
from Suc.hyps show ?case
by (simp add: atLeastLessThanSuc lambda_system_strong_additive [OF a 2 3 4])
qed
lemma (in sigma_algebra) lambda_system_caratheodory:
assumes oms: "outer_measure_space M f"
and A: "range A \<subseteq> lambda_system M f"
and disj: "disjoint_family A"
shows "(\<Union>i. A i) \<in> lambda_system M f \<and> (\<Sum>i. f (A i)) = f (\<Union>i. A i)"
proof -
have pos: "positive M f" and inc: "increasing M f"
and csa: "countably_subadditive M f"
by (metis oms outer_measure_space_def)+
have sa: "subadditive M f"
by (metis countably_subadditive_subadditive csa pos)
have A': "range A \<subseteq> sets (M(|sets := lambda_system M f|))" using A
by simp
interpret ls: algebra "M\<lparr>sets := lambda_system M f\<rparr>"
using pos by (rule lambda_system_algebra)
have A'': "range A \<subseteq> sets M"
by (metis A image_subset_iff lambda_system_sets)
have U_in: "(\<Union>i. A i) \<in> sets M"
by (metis A'' countable_UN)
have U_eq: "f (\<Union>i. A i) = (\<Sum>i. f (A i))"
proof (rule antisym)
show "f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))"
using csa[unfolded countably_subadditive_def] A'' disj U_in by auto
have *: "\<And>i. 0 \<le> f (A i)" using pos A'' unfolding positive_def by auto
have dis: "\<And>N. disjoint_family_on A {..<N}" by (intro disjoint_family_on_mono[OF _ disj]) auto
show "(\<Sum>i. f (A i)) \<le> f (\<Union>i. A i)"
using ls.additive_sum [OF lambda_system_positive[OF pos] lambda_system_additive _ A' dis]
using A''
by (intro suminf_bound[OF _ *]) (auto intro!: increasingD[OF inc] allI countable_UN)
qed
{
fix a
assume a [iff]: "a \<in> sets M"
have "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) = f a"
proof -
show ?thesis
proof (rule antisym)
have "range (\<lambda>i. a \<inter> A i) \<subseteq> sets M" using A''
by blast
moreover
have "disjoint_family (\<lambda>i. a \<inter> A i)" using disj
by (auto simp add: disjoint_family_on_def)
moreover
have "a \<inter> (\<Union>i. A i) \<in> sets M"
by (metis Int U_in a)
ultimately
have "f (a \<inter> (\<Union>i. A i)) \<le> (\<Sum>i. f (a \<inter> A i))"
using csa[unfolded countably_subadditive_def, rule_format, of "(\<lambda>i. a \<inter> A i)"]
by (simp add: o_def)
hence "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le>
(\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i))"
by (rule add_right_mono)
moreover
have "(\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
proof (intro suminf_bound_add allI)
fix n
have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
by (metis A'' UNION_in_sets)
have le_fa: "f (UNION {0..<n} A \<inter> a) \<le> f a" using A''
by (blast intro: increasingD [OF inc] A'' UNION_in_sets)
have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system M f"
using ls.UNION_in_sets by (simp add: A)
hence eq_fa: "f a = f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i))"
by (simp add: lambda_system_eq UNION_in)
have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))"
by (blast intro: increasingD [OF inc] UNION_in U_in)
thus "(\<Sum>i<n. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric])
next
have "\<And>i. a \<inter> A i \<in> sets M" using A'' by auto
then show "\<And>i. 0 \<le> f (a \<inter> A i)" using pos[unfolded positive_def] by auto
have "\<And>i. a - (\<Union>i. A i) \<in> sets M" using A'' by auto
then have "\<And>i. 0 \<le> f (a - (\<Union>i. A i))" using pos[unfolded positive_def] by auto
then show "f (a - (\<Union>i. A i)) \<noteq> -\<infinity>" by auto
qed
ultimately show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a"
by (rule order_trans)
next
have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))"
by (blast intro: increasingD [OF inc] U_in)
also have "... \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))"
by (blast intro: subadditiveD [OF sa] U_in)
finally show "f a \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" .
qed
qed
}
thus ?thesis
by (simp add: lambda_system_eq sums_iff U_eq U_in)
qed
lemma (in sigma_algebra) caratheodory_lemma:
assumes oms: "outer_measure_space M f"
shows "measure_space \<lparr> space = space M, sets = lambda_system M f, measure = f \<rparr>"
(is "measure_space ?M")
proof -
have pos: "positive M f"
by (metis oms outer_measure_space_def)
have alg: "algebra ?M"
using lambda_system_algebra [of f, OF pos]
by (simp add: algebra_iff_Un)
then
have "sigma_algebra ?M"
using lambda_system_caratheodory [OF oms]
by (simp add: sigma_algebra_disjoint_iff)
moreover
have "measure_space_axioms ?M"
using pos lambda_system_caratheodory [OF oms]
by (simp add: measure_space_axioms_def positive_def lambda_system_sets
countably_additive_def o_def)
ultimately
show ?thesis
by (simp add: measure_space_def)
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> sets M" "y \<in> sets M" "x \<subseteq> y"
then have "y - x \<in> sets M" by auto
then have "0 \<le> f (y-x)" using posf[unfolded positive_def] by auto
then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono) auto
also have "... = f (x \<union> (y-x))" using addf
by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
also have "... = f y"
by (metis Un_Diff_cancel Un_absorb1 xy(3))
finally show "f x \<le> f y" by simp
qed
lemma (in ring_of_sets) countably_additive_additive:
assumes posf: "positive M f" and ca: "countably_additive M f"
shows "additive M f"
proof (auto simp add: additive_def)
fix x y
assume x: "x \<in> sets M" and y: "y \<in> sets 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> sets M \<longrightarrow>
(\<Union>i. binaryset x y i) \<in> sets 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> sets M \<longrightarrow> x \<union> y \<in> sets 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 inf_measure_nonempty:
assumes f: "positive M f" and b: "b \<in> sets M" and a: "a \<subseteq> b" "{} \<in> sets M"
shows "f b \<in> measure_set M f a"
proof -
let ?A = "\<lambda>i::nat. (if i = 0 then b else {})"
have "(\<Sum>i. f (?A i)) = (\<Sum>i<1::nat. f (?A i))"
by (rule suminf_finite) (simp add: f[unfolded positive_def])
also have "... = f b"
by simp
finally show ?thesis using assms
by (auto intro!: exI [of _ ?A]
simp: measure_set_def disjoint_family_on_def split_if_mem2 comp_def)
qed
lemma (in ring_of_sets) inf_measure_agrees:
assumes posf: "positive M f" and ca: "countably_additive M f"
and s: "s \<in> sets M"
shows "Inf (measure_set M f s) = f s"
unfolding Inf_ereal_def
proof (safe intro!: Greatest_equality)
fix z
assume z: "z \<in> measure_set M f s"
from this obtain A where
A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
and "s \<subseteq> (\<Union>x. A x)" and si: "(\<Sum>i. f (A i)) = z"
by (auto simp add: measure_set_def comp_def)
hence seq: "s = (\<Union>i. A i \<inter> s)" by blast
have inc: "increasing M f"
by (metis additive_increasing ca countably_additive_additive posf)
have sums: "(\<Sum>i. f (A i \<inter> s)) = f (\<Union>i. A i \<inter> s)"
proof (rule ca[unfolded countably_additive_def, rule_format])
show "range (\<lambda>n. A n \<inter> s) \<subseteq> sets M" using A s
by blast
show "disjoint_family (\<lambda>n. A n \<inter> s)" using disj
by (auto simp add: disjoint_family_on_def)
show "(\<Union>i. A i \<inter> s) \<in> sets M" using A s
by (metis UN_extend_simps(4) s seq)
qed
hence "f s = (\<Sum>i. f (A i \<inter> s))"
using seq [symmetric] by (simp add: sums_iff)
also have "... \<le> (\<Sum>i. f (A i))"
proof (rule suminf_le_pos)
fix n show "f (A n \<inter> s) \<le> f (A n)" using A s
by (force intro: increasingD [OF inc])
fix N have "A N \<inter> s \<in> sets M" using A s by auto
then show "0 \<le> f (A N \<inter> s)" using posf unfolding positive_def by auto
qed
also have "... = z" by (rule si)
finally show "f s \<le> z" .
next
fix y
assume y: "\<forall>u \<in> measure_set M f s. y \<le> u"
thus "y \<le> f s"
by (blast intro: inf_measure_nonempty [of _ f, OF posf s subset_refl])
qed
lemma measure_set_pos:
assumes posf: "positive M f" "r \<in> measure_set M f X"
shows "0 \<le> r"
proof -
obtain A where "range A \<subseteq> sets M" and r: "r = (\<Sum>i. f (A i))"
using `r \<in> measure_set M f X` unfolding measure_set_def by auto
then show "0 \<le> r" using posf unfolding r positive_def
by (intro suminf_0_le) auto
qed
lemma inf_measure_pos:
assumes posf: "positive M f"
shows "0 \<le> Inf (measure_set M f X)"
proof (rule complete_lattice_class.Inf_greatest)
fix r assume "r \<in> measure_set M f X" with posf show "0 \<le> r"
by (rule measure_set_pos)
qed
lemma inf_measure_empty:
assumes posf: "positive M f" and "{} \<in> sets M"
shows "Inf (measure_set M f {}) = 0"
proof (rule antisym)
show "Inf (measure_set M f {}) \<le> 0"
by (metis complete_lattice_class.Inf_lower `{} \<in> sets M`
inf_measure_nonempty[OF posf] subset_refl posf[unfolded positive_def])
qed (rule inf_measure_pos[OF posf])
lemma (in ring_of_sets) inf_measure_positive:
assumes p: "positive M f" and "{} \<in> sets M"
shows "positive M (\<lambda>x. Inf (measure_set M f x))"
proof (unfold positive_def, intro conjI ballI)
show "Inf (measure_set M f {}) = 0" using inf_measure_empty[OF assms] by auto
fix A assume "A \<in> sets M"
qed (rule inf_measure_pos[OF p])
lemma (in ring_of_sets) inf_measure_increasing:
assumes posf: "positive M f"
shows "increasing \<lparr> space = space M, sets = Pow (space M) \<rparr>
(\<lambda>x. Inf (measure_set M f x))"
apply (auto simp add: increasing_def)
apply (rule complete_lattice_class.Inf_greatest)
apply (rule complete_lattice_class.Inf_lower)
apply (clarsimp simp add: measure_set_def, rule_tac x=A in exI, blast)
done
lemma (in ring_of_sets) inf_measure_le:
assumes posf: "positive M f" and inc: "increasing M f"
and x: "x \<in> {r . \<exists>A. range A \<subseteq> sets M \<and> s \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}"
shows "Inf (measure_set M f s) \<le> x"
proof -
obtain A where A: "range A \<subseteq> sets M" and ss: "s \<subseteq> (\<Union>i. A i)"
and xeq: "(\<Sum>i. f (A i)) = x"
using x by auto
have dA: "range (disjointed A) \<subseteq> sets M"
by (metis A range_disjointed_sets)
have "\<forall>n. f (disjointed A n) \<le> f (A n)"
by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A comp_def)
moreover have "\<forall>i. 0 \<le> f (disjointed A i)"
using posf dA unfolding positive_def by auto
ultimately have sda: "(\<Sum>i. f (disjointed A i)) \<le> (\<Sum>i. f (A i))"
by (blast intro!: suminf_le_pos)
hence ley: "(\<Sum>i. f (disjointed A i)) \<le> x"
by (metis xeq)
hence y: "(\<Sum>i. f (disjointed A i)) \<in> measure_set M f s"
apply (auto simp add: measure_set_def)
apply (rule_tac x="disjointed A" in exI)
apply (simp add: disjoint_family_disjointed UN_disjointed_eq ss dA comp_def)
done
show ?thesis
by (blast intro: y order_trans [OF _ ley] posf complete_lattice_class.Inf_lower)
qed
lemma (in ring_of_sets) inf_measure_close:
fixes e :: ereal
assumes posf: "positive M f" and e: "0 < e" and ss: "s \<subseteq> (space M)" and "Inf (measure_set M f s) \<noteq> \<infinity>"
shows "\<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> s \<subseteq> (\<Union>i. A i) \<and>
(\<Sum>i. f (A i)) \<le> Inf (measure_set M f s) + e"
proof -
from `Inf (measure_set M f s) \<noteq> \<infinity>` have fin: "\<bar>Inf (measure_set M f s)\<bar> \<noteq> \<infinity>"
using inf_measure_pos[OF posf, of s] by auto
obtain l where "l \<in> measure_set M f s" "l \<le> Inf (measure_set M f s) + e"
using Inf_ereal_close[OF fin e] by auto
thus ?thesis
by (auto intro!: exI[of _ l] simp: measure_set_def comp_def)
qed
lemma (in ring_of_sets) inf_measure_countably_subadditive:
assumes posf: "positive M f" and inc: "increasing M f"
shows "countably_subadditive (| space = space M, sets = Pow (space M) |)
(\<lambda>x. Inf (measure_set M f x))"
proof (simp add: countably_subadditive_def, safe)
fix A :: "nat \<Rightarrow> 'a set"
let "?outer B" = "Inf (measure_set M f B)"
assume A: "range A \<subseteq> Pow (space M)"
and disj: "disjoint_family A"
and sb: "(\<Union>i. A i) \<subseteq> space M"
{ fix e :: ereal assume e: "0 < e" and "\<forall>i. ?outer (A i) \<noteq> \<infinity>"
hence "\<exists>BB. \<forall>n. range (BB n) \<subseteq> sets M \<and> disjoint_family (BB n) \<and>
A n \<subseteq> (\<Union>i. BB n i) \<and> (\<Sum>i. f (BB n i)) \<le> ?outer (A n) + e * (1/2)^(Suc n)"
apply (safe intro!: choice inf_measure_close [of f, OF posf])
using e sb by (auto simp: ereal_zero_less_0_iff one_ereal_def)
then obtain BB
where BB: "\<And>n. (range (BB n) \<subseteq> sets M)"
and disjBB: "\<And>n. disjoint_family (BB n)"
and sbBB: "\<And>n. A n \<subseteq> (\<Union>i. BB n i)"
and BBle: "\<And>n. (\<Sum>i. f (BB n i)) \<le> ?outer (A n) + e * (1/2)^(Suc n)"
by auto blast
have sll: "(\<Sum>n. \<Sum>i. (f (BB n i))) \<le> (\<Sum>n. ?outer (A n)) + e"
proof -
have sum_eq_1: "(\<Sum>n. e*(1/2) ^ Suc n) = e"
using suminf_half_series_ereal e
by (simp add: ereal_zero_le_0_iff zero_le_divide_ereal suminf_cmult_ereal)
have "\<And>n i. 0 \<le> f (BB n i)" using posf[unfolded positive_def] BB by auto
then have "\<And>n. 0 \<le> (\<Sum>i. f (BB n i))" by (rule suminf_0_le)
then have "(\<Sum>n. \<Sum>i. (f (BB n i))) \<le> (\<Sum>n. ?outer (A n) + e*(1/2) ^ Suc n)"
by (rule suminf_le_pos[OF BBle])
also have "... = (\<Sum>n. ?outer (A n)) + e"
using sum_eq_1 inf_measure_pos[OF posf] e
by (subst suminf_add_ereal) (auto simp add: ereal_zero_le_0_iff)
finally show ?thesis .
qed
def C \<equiv> "(split BB) o prod_decode"
have C: "!!n. C n \<in> sets M"
apply (rule_tac p="prod_decode n" in PairE)
apply (simp add: C_def)
apply (metis BB subsetD rangeI)
done
have sbC: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)"
proof (auto simp add: C_def)
fix x i
assume x: "x \<in> A i"
with sbBB [of i] obtain j where "x \<in> BB i j"
by blast
thus "\<exists>i. x \<in> split BB (prod_decode i)"
by (metis prod_encode_inverse prod.cases)
qed
have "(f \<circ> C) = (f \<circ> (\<lambda>(x, y). BB x y)) \<circ> prod_decode"
by (rule ext) (auto simp add: C_def)
moreover have "suminf ... = (\<Sum>n. \<Sum>i. f (BB n i))" using BBle
using BB posf[unfolded positive_def]
by (force intro!: suminf_ereal_2dimen simp: o_def)
ultimately have Csums: "(\<Sum>i. f (C i)) = (\<Sum>n. \<Sum>i. f (BB n i))" by (simp add: o_def)
have "?outer (\<Union>i. A i) \<le> (\<Sum>n. \<Sum>i. f (BB n i))"
apply (rule inf_measure_le [OF posf(1) inc], auto)
apply (rule_tac x="C" in exI)
apply (auto simp add: C sbC Csums)
done
also have "... \<le> (\<Sum>n. ?outer (A n)) + e" using sll
by blast
finally have "?outer (\<Union>i. A i) \<le> (\<Sum>n. ?outer (A n)) + e" . }
note for_finite_Inf = this
show "?outer (\<Union>i. A i) \<le> (\<Sum>n. ?outer (A n))"
proof cases
assume "\<forall>i. ?outer (A i) \<noteq> \<infinity>"
with for_finite_Inf show ?thesis
by (intro ereal_le_epsilon) auto
next
assume "\<not> (\<forall>i. ?outer (A i) \<noteq> \<infinity>)"
then have "\<exists>i. ?outer (A i) = \<infinity>"
by auto
then have "(\<Sum>n. ?outer (A n)) = \<infinity>"
using suminf_PInfty[OF inf_measure_pos, OF posf]
by metis
then show ?thesis by simp
qed
qed
lemma (in ring_of_sets) inf_measure_outer:
"\<lbrakk> positive M f ; increasing M f \<rbrakk>
\<Longrightarrow> outer_measure_space \<lparr> space = space M, sets = Pow (space M) \<rparr>
(\<lambda>x. Inf (measure_set M f x))"
using inf_measure_pos[of M f]
by (simp add: outer_measure_space_def inf_measure_empty
inf_measure_increasing inf_measure_countably_subadditive positive_def)
lemma (in ring_of_sets) algebra_subset_lambda_system:
assumes posf: "positive M f" and inc: "increasing M f"
and add: "additive M f"
shows "sets M \<subseteq> lambda_system \<lparr> space = space M, sets = Pow (space M) \<rparr>
(\<lambda>x. Inf (measure_set M f x))"
proof (auto dest: sets_into_space
simp add: algebra.lambda_system_eq [OF algebra_Pow])
fix x s
assume x: "x \<in> sets M"
and s: "s \<subseteq> space M"
have [simp]: "!!x. x \<in> sets M \<Longrightarrow> s \<inter> (space M - x) = s-x" using s
by blast
have "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
\<le> Inf (measure_set M f s)"
proof cases
assume "Inf (measure_set M f s) = \<infinity>" then show ?thesis by simp
next
assume fin: "Inf (measure_set M f s) \<noteq> \<infinity>"
then have "measure_set M f s \<noteq> {}"
by (auto simp: top_ereal_def)
show ?thesis
proof (rule complete_lattice_class.Inf_greatest)
fix r assume "r \<in> measure_set M f s"
then obtain A where A: "disjoint_family A" "range A \<subseteq> sets M" "s \<subseteq> (\<Union>i. A i)"
and r: "r = (\<Sum>i. f (A i))" unfolding measure_set_def by auto
have "Inf (measure_set M f (s \<inter> x)) \<le> (\<Sum>i. f (A i \<inter> x))"
unfolding measure_set_def
proof (safe intro!: complete_lattice_class.Inf_lower exI[of _ "\<lambda>i. A i \<inter> x"])
from A(1) show "disjoint_family (\<lambda>i. A i \<inter> x)"
by (rule disjoint_family_on_bisimulation) auto
qed (insert x A, auto)
moreover
have "Inf (measure_set M f (s - x)) \<le> (\<Sum>i. f (A i - x))"
unfolding measure_set_def
proof (safe intro!: complete_lattice_class.Inf_lower exI[of _ "\<lambda>i. A i - x"])
from A(1) show "disjoint_family (\<lambda>i. A i - x)"
by (rule disjoint_family_on_bisimulation) auto
qed (insert x A, auto)
ultimately have "Inf (measure_set M f (s \<inter> x)) + Inf (measure_set M f (s - x)) \<le>
(\<Sum>i. f (A i \<inter> x)) + (\<Sum>i. f (A i - x))" by (rule add_mono)
also have "\<dots> = (\<Sum>i. f (A i \<inter> x) + f (A i - x))"
using A(2) x posf by (subst suminf_add_ereal) (auto simp: positive_def)
also have "\<dots> = (\<Sum>i. f (A i))"
using A x
by (subst add[THEN additiveD, symmetric])
(auto intro!: arg_cong[where f=suminf] arg_cong[where f=f])
finally show "Inf (measure_set M f (s \<inter> x)) + Inf (measure_set M f (s - x)) \<le> r"
using r by simp
qed
qed
moreover
have "Inf (measure_set M f s)
\<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
proof -
have "Inf (measure_set M f s) = Inf (measure_set M f ((s\<inter>x) \<union> (s-x)))"
by (metis Un_Diff_Int Un_commute)
also have "... \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
apply (rule subadditiveD)
apply (rule ring_of_sets.countably_subadditive_subadditive [OF ring_of_sets_Pow])
apply (simp add: positive_def inf_measure_empty[OF posf] inf_measure_pos[OF posf])
apply (rule inf_measure_countably_subadditive)
using s by (auto intro!: posf inc)
finally show ?thesis .
qed
ultimately
show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
= Inf (measure_set M f s)"
by (rule order_antisym)
qed
lemma measure_down:
"measure_space N \<Longrightarrow> sigma_algebra M \<Longrightarrow> sets M \<subseteq> sets N \<Longrightarrow> measure N = measure M \<Longrightarrow> measure_space M"
by (simp add: measure_space_def measure_space_axioms_def positive_def
countably_additive_def)
blast
theorem (in ring_of_sets) caratheodory:
assumes posf: "positive M f" and ca: "countably_additive M f"
shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and>
measure_space \<lparr> space = space M, sets = sets (sigma M), measure = \<mu> \<rparr>"
proof -
have inc: "increasing M f"
by (metis additive_increasing ca countably_additive_additive posf)
let ?infm = "(\<lambda>x. Inf (measure_set M f x))"
def ls \<equiv> "lambda_system (|space = space M, sets = Pow (space M)|) ?infm"
have mls: "measure_space \<lparr>space = space M, sets = ls, measure = ?infm\<rparr>"
using sigma_algebra.caratheodory_lemma
[OF sigma_algebra_Pow inf_measure_outer [OF posf inc]]
by (simp add: ls_def)
hence sls: "sigma_algebra (|space = space M, sets = ls, measure = ?infm|)"
by (simp add: measure_space_def)
have "sets M \<subseteq> ls"
by (simp add: ls_def)
(metis ca posf inc countably_additive_additive algebra_subset_lambda_system)
hence sgs_sb: "sigma_sets (space M) (sets M) \<subseteq> ls"
using sigma_algebra.sigma_sets_subset [OF sls, of "sets M"]
by simp
have "measure_space \<lparr> space = space M, sets = sets (sigma M), measure = ?infm \<rparr>"
unfolding sigma_def
by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets)
(simp_all add: sgs_sb space_closed)
thus ?thesis using inf_measure_agrees [OF posf ca]
by (intro exI[of _ ?infm]) auto
qed
subsubsection {*Alternative instances of caratheodory*}
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> sets M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> sets 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> sets M \<longrightarrow> disjoint_family A \<longrightarrow> UNION UNIV A \<in> sets M \<longrightarrow> (\<Sum>i. f (A i)) = f (UNION UNIV A)"
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "incseq A" "(\<Union>i. A i) \<in> sets M"
then have dA: "range (disjointed A) \<subseteq> sets 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=0..<n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))"
using f(1)[unfolded positive_def] dA
by (auto intro!: summable_sumr_LIMSEQ_suminf 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 atLeastLessThanSuc_atLeastAtMost atLeast0AtMost .
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> sets M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i)"
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "disjoint_family A" "(\<Union>i. A i) \<in> sets M"
have *: "(\<Union>n. (\<Union>i\<le>n. A i)) = (\<Union>i. A i)" by auto
have "(\<lambda>n. f (\<Union>i\<le>n. A i)) ----> f (\<Union>i. A i)"
proof (unfold *[symmetric], intro cont[rule_format])
show "range (\<lambda>i. \<Union> i\<le>i. A i) \<subseteq> sets M" "(\<Union>i. \<Union> i\<le>i. A i) \<in> sets M"
using A * by auto
qed (force intro!: incseq_SucI)
moreover have "\<And>n. f (\<Union>i\<le>n. A i) = (\<Sum>i\<le>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_def2 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> sets M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> sets 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> sets 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> sets M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> sets 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> sets 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> sets 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> sets M" "decseq A" "(\<Inter>i. A i) \<in> sets M" "\<forall>i. f (A i) \<noteq> \<infinity>"
have f_mono: "\<And>a b. a \<in> sets M \<Longrightarrow> b \<in> sets 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> sets 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 INFI_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_ereal_INFI[OF decseq_fA]
show "(\<lambda>i. f (A i)) ----> f (\<Inter>i. A i)" by simp
qed
lemma positiveD1: "positive M f \<Longrightarrow> f {} = 0" by (auto simp: positive_def)
lemma positiveD2: "positive M f \<Longrightarrow> A \<in> sets M \<Longrightarrow> 0 \<le> f A" by (auto simp: positive_def)
lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below:
assumes f: "positive M f" "additive M f" "\<forall>A\<in>sets M. f A \<noteq> \<infinity>"
assumes cont: "\<forall>A. range A \<subseteq> sets M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<lambda>i. f (A i)) ----> 0"
assumes A: "range A \<subseteq> sets M" "incseq A" "(\<Union>i. A i) \<in> sets M"
shows "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)"
proof -
have "\<forall>A\<in>sets M. \<exists>x. f A = ereal x"
proof
fix A assume "A \<in> sets 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 LIMSEQ_diff_approach_zero2[OF LIMSEQ_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>sets M. f A \<noteq> \<infinity>"
assumes cont: "\<And>A. range A \<subseteq> sets 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
lemma (in ring_of_sets) caratheodory_empty_continuous:
assumes f: "positive M f" "additive M f" and fin: "\<And>A. A \<in> sets M \<Longrightarrow> f A \<noteq> \<infinity>"
assumes cont: "\<And>A. range A \<subseteq> sets M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0"
shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and>
measure_space \<lparr> space = space M, sets = sets (sigma M), measure = \<mu> \<rparr>"
proof (intro caratheodory empty_continuous_imp_countably_additive f)
show "\<forall>A\<in>sets M. f A \<noteq> \<infinity>" using fin by auto
qed (rule cont)
end