(* Title: HOL/Probability/Caratheodory.thy
Author: Lawrence C Paulson
Author: Johannes Hölzl, TU München
*)
header {*Caratheodory Extension Theorem*}
theory Caratheodory
imports Measure_Space
begin
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 SUP_upper2
setsum_nonneg suminf_ereal_eq_SUPR SUPR_pair
SUPR_ereal_setsum[symmetric] incseq_setsumI setsum_nonneg)
qed
subsection {* Measure Spaces *}
definition subadditive where "subadditive M f \<longleftrightarrow>
(\<forall>x\<in>M. \<forall>y\<in>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> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>
(f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))))"
definition lambda_system where "lambda_system \<Omega> M f = {l \<in> M.
\<forall>x \<in> M. f (l \<inter> x) + f ((\<Omega> - 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> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}"
lemma subadditiveD:
"subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) \<le> f x + f y"
by (auto simp add: subadditive_def)
subsection {* Lambda Systems *}
lemma (in algebra) lambda_system_eq:
shows "lambda_system \<Omega> M f = {l \<in> M. \<forall>x \<in> M. f (x \<inter> l) + f (x - l) = f x}"
proof -
have [simp]: "!!l x. l \<in> M \<Longrightarrow> x \<in> M \<Longrightarrow> (\<Omega> - 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 \<Omega> M f"
by (auto simp add: positive_def lambda_system_eq)
lemma lambda_system_sets:
"x \<in> lambda_system \<Omega> M f \<Longrightarrow> x \<in> 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 \<Omega> M f"
shows "\<Omega> - x \<in> lambda_system \<Omega> M f"
proof -
have "x \<subseteq> \<Omega>"
by (metis sets_into_space lambda_system_sets x)
hence "\<Omega> - (\<Omega> - 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 \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
shows "x \<inter> y \<in> lambda_system \<Omega> M f"
proof -
from xl yl show ?thesis
proof (auto simp add: positive_def lambda_system_eq Int)
fix u
assume x: "x \<in> M" and y: "y \<in> M" and u: "u \<in> M"
and fx: "\<forall>z\<in>M. f (z \<inter> x) + f (z - x) = f z"
and fy: "\<forall>z\<in>M. f (z \<inter> y) + f (z - y) = f z"
have "u - x \<inter> y \<in> 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 \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
shows "x \<union> y \<in> lambda_system \<Omega> M f"
proof -
have "(\<Omega> - x) \<inter> (\<Omega> - y) \<in> M"
by (metis Diff_Un Un compl_sets lambda_system_sets xl yl)
moreover
have "x \<union> y = \<Omega> - ((\<Omega> - x) \<inter> (\<Omega> - 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 \<Omega> (lambda_system \<Omega> M f)"
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> M" and disj: "x \<inter> y = {}"
and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> 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> 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 (lambda_system \<Omega> M f) f"
proof (auto simp add: additive_def)
fix x and y
assume disj: "x \<inter> y = {}"
and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
hence "x \<in> M" "y \<in> 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) 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> 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) \<le> (\<Sum> n. f (binaryset x y n))"
using cs by (auto simp add: countably_subadditive_def)
hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> 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 lambda_system_increasing:
"increasing M f \<Longrightarrow> increasing (lambda_system \<Omega> M f) f"
by (simp add: increasing_def lambda_system_def)
lemma lambda_system_positive:
"positive M f \<Longrightarrow> positive (lambda_system \<Omega> 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> M"
and A: "range A \<subseteq> lambda_system \<Omega> 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 \<Omega> M f" using A
by blast
interpret l: algebra \<Omega> "lambda_system \<Omega> M f"
using f by (rule lambda_system_algebra)
have 4: "UNION {0..<n} A \<in> lambda_system \<Omega> 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 \<Omega> M f"
and disj: "disjoint_family A"
shows "(\<Union>i. A i) \<in> lambda_system \<Omega> 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': "\<And>S. A`S \<subseteq> (lambda_system \<Omega> M f)" using A
by auto
interpret ls: algebra \<Omega> "lambda_system \<Omega> M f"
using pos by (rule lambda_system_algebra)
have A'': "range A \<subseteq> M"
by (metis A image_subset_iff lambda_system_sets)
have U_in: "(\<Union>i. A i) \<in> 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] countable_UN)
qed
{
fix a
assume a [iff]: "a \<in> 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> 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> 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> 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 \<Omega> 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> 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> 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"
defines "L \<equiv> lambda_system \<Omega> M f"
shows "measure_space \<Omega> L f"
proof -
have pos: "positive M f"
by (metis oms outer_measure_space_def)
have alg: "algebra \<Omega> L"
using lambda_system_algebra [of f, OF pos]
by (simp add: algebra_iff_Un L_def)
then
have "sigma_algebra \<Omega> L"
using lambda_system_caratheodory [OF oms]
by (simp add: sigma_algebra_disjoint_iff L_def)
moreover
have "countably_additive L f" "positive L f"
using pos lambda_system_caratheodory [OF oms]
by (auto simp add: lambda_system_sets L_def countably_additive_def positive_def)
ultimately
show ?thesis
using pos by (simp add: measure_space_def)
qed
lemma inf_measure_nonempty:
assumes f: "positive M f" and b: "b \<in> M" and a: "a \<subseteq> b" "{} \<in> 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_all 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> M"
shows "Inf (measure_set M f s) = f s"
proof (intro Inf_eqI)
fix z
assume z: "z \<in> measure_set M f s"
from this obtain A where
A: "range A \<subseteq> 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> 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> 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> 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" .
qed (blast intro: inf_measure_nonempty [of _ f, OF posf s subset_refl])
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> 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> 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> 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> 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> M"
qed (rule inf_measure_pos[OF p])
lemma (in ring_of_sets) inf_measure_increasing:
assumes posf: "positive M f"
shows "increasing (Pow \<Omega>) (\<lambda>x. Inf (measure_set M f x))"
apply (clarsimp 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> 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> 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> 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> (\<Omega>)" and "Inf (measure_set M f s) \<noteq> \<infinity>"
shows "\<exists>A. range A \<subseteq> 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 (Pow \<Omega>) (\<lambda>x. Inf (measure_set M f x))"
proof (simp add: countably_subadditive_def, safe)
fix A :: "nat \<Rightarrow> 'a set"
let ?outer = "\<lambda>B. Inf (measure_set M f B)"
assume A: "range A \<subseteq> Pow (\<Omega>)"
and disj: "disjoint_family A"
and sb: "(\<Union>i. A i) \<subseteq> \<Omega>"
{ fix e :: ereal assume e: "0 < e" and "\<forall>i. ?outer (A i) \<noteq> \<infinity>"
hence "\<exists>BB. \<forall>n. range (BB n) \<subseteq> 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> 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> 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 (Pow \<Omega>) (\<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 "M \<subseteq> lambda_system \<Omega> (Pow \<Omega>) (\<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> M"
and s: "s \<subseteq> \<Omega>"
have [simp]: "!!x. x \<in> M \<Longrightarrow> s \<inter> (\<Omega> - 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> 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 \<Omega> N \<mu> \<Longrightarrow> sigma_algebra \<Omega> M \<Longrightarrow> M \<subseteq> N \<Longrightarrow> measure_space \<Omega> M \<mu>"
by (simp add: measure_space_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> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
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 \<Omega> (Pow \<Omega>) ?infm"
have mls: "measure_space \<Omega> ls ?infm"
using sigma_algebra.caratheodory_lemma
[OF sigma_algebra_Pow inf_measure_outer [OF posf inc]]
by (simp add: ls_def)
hence sls: "sigma_algebra \<Omega> ls"
by (simp add: measure_space_def)
have "M \<subseteq> ls"
by (simp add: ls_def)
(metis ca posf inc countably_additive_additive algebra_subset_lambda_system)
hence sgs_sb: "sigma_sets (\<Omega>) (M) \<subseteq> ls"
using sigma_algebra.sigma_sets_subset [OF sls, of "M"]
by simp
have "measure_space \<Omega> (sigma_sets \<Omega> M) ?infm"
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) caratheodory_empty_continuous:
assumes f: "positive M f" "additive M f" and fin: "\<And>A. A \<in> M \<Longrightarrow> 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 "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
proof (intro caratheodory' empty_continuous_imp_countably_additive f)
show "\<forall>A\<in>M. f A \<noteq> \<infinity>" using fin by auto
qed (rule cont)
section {* Volumes *}
definition volume :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
"volume M f \<longleftrightarrow>
(f {} = 0) \<and> (\<forall>a\<in>M. 0 \<le> f a) \<and>
(\<forall>C\<subseteq>M. disjoint C \<longrightarrow> finite C \<longrightarrow> \<Union>C \<in> M \<longrightarrow> f (\<Union>C) = (\<Sum>c\<in>C. f c))"
lemma volumeI:
assumes "f {} = 0"
assumes "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> f a"
assumes "\<And>C. C \<subseteq> M \<Longrightarrow> disjoint C \<Longrightarrow> finite C \<Longrightarrow> \<Union>C \<in> M \<Longrightarrow> f (\<Union>C) = (\<Sum>c\<in>C. f c)"
shows "volume M f"
using assms by (auto simp: volume_def)
lemma volume_positive:
"volume M f \<Longrightarrow> a \<in> M \<Longrightarrow> 0 \<le> f a"
by (auto simp: volume_def)
lemma volume_empty:
"volume M f \<Longrightarrow> f {} = 0"
by (auto simp: volume_def)
lemma volume_finite_additive:
assumes "volume M f"
assumes A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" "disjoint_family_on A I" "finite I" "UNION I A \<in> M"
shows "f (UNION I A) = (\<Sum>i\<in>I. f (A i))"
proof -
have "A`I \<subseteq> M" "disjoint (A`I)" "finite (A`I)" "\<Union>A`I \<in> M"
using A unfolding SUP_def by (auto simp: disjoint_family_on_disjoint_image)
with `volume M f` have "f (\<Union>A`I) = (\<Sum>a\<in>A`I. f a)"
unfolding volume_def by blast
also have "\<dots> = (\<Sum>i\<in>I. f (A i))"
proof (subst setsum_reindex_nonzero)
fix i j assume "i \<in> I" "j \<in> I" "i \<noteq> j" "A i = A j"
with `disjoint_family_on A I` have "A i = {}"
by (auto simp: disjoint_family_on_def)
then show "f (A i) = 0"
using volume_empty[OF `volume M f`] by simp
qed (auto intro: `finite I`)
finally show "f (UNION I A) = (\<Sum>i\<in>I. f (A i))"
by simp
qed
lemma (in ring_of_sets) volume_additiveI:
assumes pos: "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> \<mu> a"
assumes [simp]: "\<mu> {} = 0"
assumes add: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> \<mu> (a \<union> b) = \<mu> a + \<mu> b"
shows "volume M \<mu>"
proof (unfold volume_def, safe)
fix C assume "finite C" "C \<subseteq> M" "disjoint C"
then show "\<mu> (\<Union>C) = setsum \<mu> C"
proof (induct C)
case (insert c C)
from insert(1,2,4,5) have "\<mu> (\<Union>insert c C) = \<mu> c + \<mu> (\<Union>C)"
by (auto intro!: add simp: disjoint_def)
with insert show ?case
by (simp add: disjoint_def)
qed simp
qed fact+
lemma (in semiring_of_sets) extend_volume:
assumes "volume M \<mu>"
shows "\<exists>\<mu>'. volume generated_ring \<mu>' \<and> (\<forall>a\<in>M. \<mu>' a = \<mu> a)"
proof -
let ?R = generated_ring
have "\<forall>a\<in>?R. \<exists>m. \<exists>C\<subseteq>M. a = \<Union>C \<and> finite C \<and> disjoint C \<and> m = (\<Sum>c\<in>C. \<mu> c)"
by (auto simp: generated_ring_def)
from bchoice[OF this] guess \<mu>' .. note \<mu>'_spec = this
{ fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
fix D assume D: "D \<subseteq> M" "finite D" "disjoint D"
assume "\<Union>C = \<Union>D"
have "(\<Sum>d\<in>D. \<mu> d) = (\<Sum>d\<in>D. \<Sum>c\<in>C. \<mu> (c \<inter> d))"
proof (intro setsum_cong refl)
fix d assume "d \<in> D"
have Un_eq_d: "(\<Union>c\<in>C. c \<inter> d) = d"
using `d \<in> D` `\<Union>C = \<Union>D` by auto
moreover have "\<mu> (\<Union>c\<in>C. c \<inter> d) = (\<Sum>c\<in>C. \<mu> (c \<inter> d))"
proof (rule volume_finite_additive)
{ fix c assume "c \<in> C" then show "c \<inter> d \<in> M"
using C D `d \<in> D` by auto }
show "(\<Union>a\<in>C. a \<inter> d) \<in> M"
unfolding Un_eq_d using `d \<in> D` D by auto
show "disjoint_family_on (\<lambda>a. a \<inter> d) C"
using `disjoint C` by (auto simp: disjoint_family_on_def disjoint_def)
qed fact+
ultimately show "\<mu> d = (\<Sum>c\<in>C. \<mu> (c \<inter> d))" by simp
qed }
note split_sum = this
{ fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
fix D assume D: "D \<subseteq> M" "finite D" "disjoint D"
assume "\<Union>C = \<Union>D"
with split_sum[OF C D] split_sum[OF D C]
have "(\<Sum>d\<in>D. \<mu> d) = (\<Sum>c\<in>C. \<mu> c)"
by (simp, subst setsum_commute, simp add: ac_simps) }
note sum_eq = this
{ fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
then have "\<Union>C \<in> ?R" by (auto simp: generated_ring_def)
with \<mu>'_spec[THEN bspec, of "\<Union>C"]
obtain D where
D: "D \<subseteq> M" "finite D" "disjoint D" "\<Union>C = \<Union>D" and "\<mu>' (\<Union>C) = (\<Sum>d\<in>D. \<mu> d)"
by blast
with sum_eq[OF C D] have "\<mu>' (\<Union>C) = (\<Sum>c\<in>C. \<mu> c)" by simp }
note \<mu>' = this
show ?thesis
proof (intro exI conjI ring_of_sets.volume_additiveI[OF generating_ring] ballI)
fix a assume "a \<in> M" with \<mu>'[of "{a}"] show "\<mu>' a = \<mu> a"
by (simp add: disjoint_def)
next
fix a assume "a \<in> ?R" then guess Ca .. note Ca = this
with \<mu>'[of Ca] `volume M \<mu>`[THEN volume_positive]
show "0 \<le> \<mu>' a"
by (auto intro!: setsum_nonneg)
next
show "\<mu>' {} = 0" using \<mu>'[of "{}"] by auto
next
fix a assume "a \<in> ?R" then guess Ca .. note Ca = this
fix b assume "b \<in> ?R" then guess Cb .. note Cb = this
assume "a \<inter> b = {}"
with Ca Cb have "Ca \<inter> Cb \<subseteq> {{}}" by auto
then have C_Int_cases: "Ca \<inter> Cb = {{}} \<or> Ca \<inter> Cb = {}" by auto
from `a \<inter> b = {}` have "\<mu>' (\<Union> (Ca \<union> Cb)) = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c)"
using Ca Cb by (intro \<mu>') (auto intro!: disjoint_union)
also have "\<dots> = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c) + (\<Sum>c\<in>Ca \<inter> Cb. \<mu> c)"
using C_Int_cases volume_empty[OF `volume M \<mu>`] by (elim disjE) simp_all
also have "\<dots> = (\<Sum>c\<in>Ca. \<mu> c) + (\<Sum>c\<in>Cb. \<mu> c)"
using Ca Cb by (simp add: setsum_Un_Int)
also have "\<dots> = \<mu>' a + \<mu>' b"
using Ca Cb by (simp add: \<mu>')
finally show "\<mu>' (a \<union> b) = \<mu>' a + \<mu>' b"
using Ca Cb by simp
qed
qed
section {* Caratheodory on semirings *}
theorem (in semiring_of_sets) caratheodory:
assumes pos: "positive M \<mu>" and ca: "countably_additive M \<mu>"
shows "\<exists>\<mu>' :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu>' s = \<mu> s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>'"
proof -
have "volume M \<mu>"
proof (rule volumeI)
{ fix a assume "a \<in> M" then show "0 \<le> \<mu> a"
using pos unfolding positive_def by auto }
note p = this
fix C assume sets_C: "C \<subseteq> M" "\<Union>C \<in> M" and "disjoint C" "finite C"
have "\<exists>F'. bij_betw F' {..<card C} C"
by (rule finite_same_card_bij[OF _ `finite C`]) auto
then guess F' .. note F' = this
then have F': "C = F' ` {..< card C}" "inj_on F' {..< card C}"
by (auto simp: bij_betw_def)
{ fix i j assume *: "i < card C" "j < card C" "i \<noteq> j"
with F' have "F' i \<in> C" "F' j \<in> C" "F' i \<noteq> F' j"
unfolding inj_on_def by auto
with `disjoint C`[THEN disjointD]
have "F' i \<inter> F' j = {}"
by auto }
note F'_disj = this
def F \<equiv> "\<lambda>i. if i < card C then F' i else {}"
then have "disjoint_family F"
using F'_disj by (auto simp: disjoint_family_on_def)
moreover from F' have "(\<Union>i. F i) = \<Union>C"
by (auto simp: F_def set_eq_iff split: split_if_asm)
moreover have sets_F: "\<And>i. F i \<in> M"
using F' sets_C by (auto simp: F_def)
moreover note sets_C
ultimately have "\<mu> (\<Union>C) = (\<Sum>i. \<mu> (F i))"
using ca[unfolded countably_additive_def, THEN spec, of F] by auto
also have "\<dots> = (\<Sum>i<card C. \<mu> (F' i))"
proof -
have "(\<lambda>i. if i \<in> {..< card C} then \<mu> (F' i) else 0) sums (\<Sum>i<card C. \<mu> (F' i))"
by (rule sums_If_finite_set) auto
also have "(\<lambda>i. if i \<in> {..< card C} then \<mu> (F' i) else 0) = (\<lambda>i. \<mu> (F i))"
using pos by (auto simp: positive_def F_def)
finally show "(\<Sum>i. \<mu> (F i)) = (\<Sum>i<card C. \<mu> (F' i))"
by (simp add: sums_iff)
qed
also have "\<dots> = (\<Sum>c\<in>C. \<mu> c)"
using F'(2) by (subst (2) F') (simp add: setsum_reindex)
finally show "\<mu> (\<Union>C) = (\<Sum>c\<in>C. \<mu> c)" .
next
show "\<mu> {} = 0"
using `positive M \<mu>` by (rule positiveD1)
qed
from extend_volume[OF this] obtain \<mu>_r where
V: "volume generated_ring \<mu>_r" "\<And>a. a \<in> M \<Longrightarrow> \<mu> a = \<mu>_r a"
by auto
interpret G: ring_of_sets \<Omega> generated_ring
by (rule generating_ring)
have pos: "positive generated_ring \<mu>_r"
using V unfolding positive_def by (auto simp: positive_def intro!: volume_positive volume_empty)
have "countably_additive generated_ring \<mu>_r"
proof (rule countably_additiveI)
fix A' :: "nat \<Rightarrow> 'a set" assume A': "range A' \<subseteq> generated_ring" "disjoint_family A'"
and Un_A: "(\<Union>i. A' i) \<in> generated_ring"
from generated_ringE[OF Un_A] guess C' . note C' = this
{ fix c assume "c \<in> C'"
moreover def A \<equiv> "\<lambda>i. A' i \<inter> c"
ultimately have A: "range A \<subseteq> generated_ring" "disjoint_family A"
and Un_A: "(\<Union>i. A i) \<in> generated_ring"
using A' C'
by (auto intro!: G.Int G.finite_Union intro: generated_ringI_Basic simp: disjoint_family_on_def)
from A C' `c \<in> C'` have UN_eq: "(\<Union>i. A i) = c"
by (auto simp: A_def)
have "\<forall>i::nat. \<exists>f::nat \<Rightarrow> 'a set. \<mu>_r (A i) = (\<Sum>j. \<mu>_r (f j)) \<and> disjoint_family f \<and> \<Union>range f = A i \<and> (\<forall>j. f j \<in> M)"
(is "\<forall>i. ?P i")
proof
fix i
from A have Ai: "A i \<in> generated_ring" by auto
from generated_ringE[OF this] guess C . note C = this
have "\<exists>F'. bij_betw F' {..<card C} C"
by (rule finite_same_card_bij[OF _ `finite C`]) auto
then guess F .. note F = this
def f \<equiv> "\<lambda>i. if i < card C then F i else {}"
then have f: "bij_betw f {..< card C} C"
by (intro bij_betw_cong[THEN iffD1, OF _ F]) auto
with C have "\<forall>j. f j \<in> M"
by (auto simp: Pi_iff f_def dest!: bij_betw_imp_funcset)
moreover
from f C have d_f: "disjoint_family_on f {..<card C}"
by (intro disjoint_image_disjoint_family_on) (auto simp: bij_betw_def)
then have "disjoint_family f"
by (auto simp: disjoint_family_on_def f_def)
moreover
have Ai_eq: "A i = (\<Union> x<card C. f x)"
using f C Ai unfolding bij_betw_def by (simp add: Union_image_eq[symmetric])
then have "\<Union>range f = A i"
using f C Ai unfolding bij_betw_def by (auto simp: f_def)
moreover
{ have "(\<Sum>j. \<mu>_r (f j)) = (\<Sum>j. if j \<in> {..< card C} then \<mu>_r (f j) else 0)"
using volume_empty[OF V(1)] by (auto intro!: arg_cong[where f=suminf] simp: f_def)
also have "\<dots> = (\<Sum>j<card C. \<mu>_r (f j))"
by (rule sums_If_finite_set[THEN sums_unique, symmetric]) simp
also have "\<dots> = \<mu>_r (A i)"
using C f[THEN bij_betw_imp_funcset] unfolding Ai_eq
by (intro volume_finite_additive[OF V(1) _ d_f, symmetric])
(auto simp: Pi_iff Ai_eq intro: generated_ringI_Basic)
finally have "\<mu>_r (A i) = (\<Sum>j. \<mu>_r (f j))" .. }
ultimately show "?P i"
by blast
qed
from choice[OF this] guess f .. note f = this
then have UN_f_eq: "(\<Union>i. split f (prod_decode i)) = (\<Union>i. A i)"
unfolding UN_extend_simps surj_prod_decode by (auto simp: set_eq_iff)
have d: "disjoint_family (\<lambda>i. split f (prod_decode i))"
unfolding disjoint_family_on_def
proof (intro ballI impI)
fix m n :: nat assume "m \<noteq> n"
then have neq: "prod_decode m \<noteq> prod_decode n"
using inj_prod_decode[of UNIV] by (auto simp: inj_on_def)
show "split f (prod_decode m) \<inter> split f (prod_decode n) = {}"
proof cases
assume "fst (prod_decode m) = fst (prod_decode n)"
then show ?thesis
using neq f by (fastforce simp: disjoint_family_on_def)
next
assume neq: "fst (prod_decode m) \<noteq> fst (prod_decode n)"
have "split f (prod_decode m) \<subseteq> A (fst (prod_decode m))"
"split f (prod_decode n) \<subseteq> A (fst (prod_decode n))"
using f[THEN spec, of "fst (prod_decode m)"]
using f[THEN spec, of "fst (prod_decode n)"]
by (auto simp: set_eq_iff)
with f A neq show ?thesis
by (fastforce simp: disjoint_family_on_def subset_eq set_eq_iff)
qed
qed
from f have "(\<Sum>n. \<mu>_r (A n)) = (\<Sum>n. \<mu>_r (split f (prod_decode n)))"
by (intro suminf_ereal_2dimen[symmetric] positiveD2[OF pos] generated_ringI_Basic)
(auto split: prod.split)
also have "\<dots> = (\<Sum>n. \<mu> (split f (prod_decode n)))"
using f V(2) by (auto intro!: arg_cong[where f=suminf] split: prod.split)
also have "\<dots> = \<mu> (\<Union>i. split f (prod_decode i))"
using f `c \<in> C'` C'
by (intro ca[unfolded countably_additive_def, rule_format])
(auto split: prod.split simp: UN_f_eq d UN_eq)
finally have "(\<Sum>n. \<mu>_r (A' n \<inter> c)) = \<mu> c"
using UN_f_eq UN_eq by (simp add: A_def) }
note eq = this
have "(\<Sum>n. \<mu>_r (A' n)) = (\<Sum>n. \<Sum>c\<in>C'. \<mu>_r (A' n \<inter> c))"
using C' A'
by (subst volume_finite_additive[symmetric, OF V(1)])
(auto simp: disjoint_def disjoint_family_on_def Union_image_eq[symmetric] simp del: Union_image_eq
intro!: G.Int G.finite_Union arg_cong[where f="\<lambda>X. suminf (\<lambda>i. \<mu>_r (X i))"] ext
intro: generated_ringI_Basic)
also have "\<dots> = (\<Sum>c\<in>C'. \<Sum>n. \<mu>_r (A' n \<inter> c))"
using C' A'
by (intro suminf_setsum_ereal positiveD2[OF pos] G.Int G.finite_Union)
(auto intro: generated_ringI_Basic)
also have "\<dots> = (\<Sum>c\<in>C'. \<mu>_r c)"
using eq V C' by (auto intro!: setsum_cong)
also have "\<dots> = \<mu>_r (\<Union>C')"
using C' Un_A
by (subst volume_finite_additive[symmetric, OF V(1)])
(auto simp: disjoint_family_on_def disjoint_def Union_image_eq[symmetric] simp del: Union_image_eq
intro: generated_ringI_Basic)
finally show "(\<Sum>n. \<mu>_r (A' n)) = \<mu>_r (\<Union>i. A' i)"
using C' by simp
qed
from G.caratheodory'[OF `positive generated_ring \<mu>_r` `countably_additive generated_ring \<mu>_r`]
guess \<mu>' ..
with V show ?thesis
unfolding sigma_sets_generated_ring_eq
by (intro exI[of _ \<mu>']) (auto intro: generated_ringI_Basic)
qed
end