header {*Caratheodory Extension Theorem*}
theory Caratheodory
imports Sigma_Algebra SeriesPlus
begin
text{*From the Hurd/Coble measure theory development, translated by Lawrence Paulson.*}
subsection {* Measure Spaces *}
text {*A measure assigns a nonnegative real to every measurable set.
It is countably additive for disjoint sets.*}
record 'a measure_space = "'a algebra" +
measure:: "'a set \<Rightarrow> real"
definition
disjoint_family_on where
"disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
abbreviation
"disjoint_family A \<equiv> disjoint_family_on A UNIV"
definition
positive where
"positive M f \<longleftrightarrow> f {} = (0::real) & (\<forall>x \<in> sets M. 0 \<le> f x)"
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 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>
(\<lambda>n. f (A n)) sums 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>
summable (f o A) \<longrightarrow>
f (\<Union>i. A i) \<le> suminf (\<lambda>n. f (A n)))"
definition
lambda_system where
"lambda_system M f =
{l. 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 & increasing M f & countably_subadditive M f"
definition
measure_set where
"measure_set M f X =
{r . \<exists>A. range A \<subseteq> sets M & disjoint_family A & X \<subseteq> (\<Union>i. A i) & (f \<circ> A) sums r}"
locale measure_space = sigma_algebra +
assumes positive: "!!a. a \<in> sets M \<Longrightarrow> 0 \<le> measure M a"
and empty_measure [simp]: "measure M {} = (0::real)"
and ca: "countably_additive M (measure M)"
subsection {* Basic Lemmas *}
lemma positive_imp_0: "positive M f \<Longrightarrow> f {} = 0"
by (simp add: positive_def)
lemma positive_imp_pos: "positive M f \<Longrightarrow> x \<in> sets M \<Longrightarrow> 0 \<le> f x"
by (simp add: positive_def)
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_additiveD:
"countably_additive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A
\<Longrightarrow> (\<Union>i. A i) \<in> sets M \<Longrightarrow> (\<lambda>n. f (A n)) sums f (\<Union>i. A i)"
by (simp add: countably_additive_def)
lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"
by blast
lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"
by blast
lemma disjoint_family_subset:
"disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B"
by (force simp add: disjoint_family_on_def)
subsection {* A Two-Element Series *}
definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
where "binaryset A B = (\<lambda>\<^isup>x. {})(0 := A, Suc 0 := B)"
lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"
apply (simp add: binaryset_def)
apply (rule set_ext)
apply (auto simp add: image_iff)
done
lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"
by (simp add: UNION_eq_Union_image range_binaryset_eq)
lemma LIMSEQ_binaryset:
assumes f: "f {} = 0"
shows "(\<lambda>n. \<Sum>i = 0..<n. f (binaryset A B i)) ----> f A + f B"
proof -
have "(\<lambda>n. \<Sum>i = 0..< Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
proof
fix n
show "(\<Sum>i\<Colon>nat = 0\<Colon>nat..<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 = 0..< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
by metis
hence "(\<lambda>n. \<Sum>i = 0..< 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])
lemma suminf_binaryset_eq:
"f {} = 0 \<Longrightarrow> suminf (\<lambda>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:
"lambda_system M f =
{l. 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> real"
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)
qed
lemma (in algebra) lambda_system_Int:
fixes f:: "'a set \<Rightarrow> real"
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)
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> real"
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 (|sets := lambda_system M f|))"
apply (auto simp add: algebra_def)
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) Int_space_eq1 [simp]: "x \<in> sets M \<Longrightarrow> space M \<inter> x = x"
by (metis Int_absorb1 sets_into_space)
lemma (in algebra) Int_space_eq2 [simp]: "x \<in> sets M \<Longrightarrow> x \<inter> space M = x"
by (metis Int_absorb2 sets_into_space)
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 algebra) 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>
summable (f o (binaryset x y)) \<longrightarrow>
f (\<Union>i. binaryset x y i) \<le> suminf (\<lambda>n. f (binaryset x y n))"
using cs by (simp add: countably_subadditive_def)
hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
summable (f o (binaryset x y)) \<longrightarrow>
f (x \<union> y) \<le> suminf (\<lambda>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 binaryset_sums
by (auto simp add: Un sums_iff positive_def o_def)
qed
definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set "
where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"
lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)"
proof (induct n)
case 0 show ?case by simp
next
case (Suc n)
thus ?case by (simp add: atLeastLessThanSuc disjointed_def)
qed
lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)"
apply (rule UN_finite2_eq [where k=0])
apply (simp add: finite_UN_disjointed_eq)
done
lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}"
by (auto simp add: disjointed_def)
lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
by (simp add: disjoint_family_on_def)
(metis neq_iff Int_commute less_disjoint_disjointed)
lemma disjointed_subset: "disjointed A n \<subseteq> A n"
by (auto simp add: disjointed_def)
lemma (in algebra) UNION_in_sets:
fixes A:: "nat \<Rightarrow> 'a set"
assumes A: "range A \<subseteq> sets M "
shows "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
proof (induct n)
case 0 show ?case by simp
next
case (Suc n)
thus ?case
by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff)
qed
lemma (in algebra) range_disjointed_sets:
assumes A: "range A \<subseteq> sets M "
shows "range (disjointed A) \<subseteq> sets M"
proof (auto simp add: disjointed_def)
fix n
show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> sets M" using UNION_in_sets
by (metis A Diff UNIV_I image_subset_iff)
qed
lemma sigma_algebra_disjoint_iff:
"sigma_algebra M \<longleftrightarrow>
algebra M &
(\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow>
(\<Union>i::nat. A i) \<in> sets M)"
proof (auto simp add: sigma_algebra_iff)
fix A :: "nat \<Rightarrow> 'a set"
assume M: "algebra M"
and A: "range A \<subseteq> sets M"
and UnA: "\<forall>A. range A \<subseteq> sets M \<longrightarrow>
disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
hence "range (disjointed A) \<subseteq> sets M \<longrightarrow>
disjoint_family (disjointed A) \<longrightarrow>
(\<Union>i. disjointed A i) \<in> sets M" by blast
hence "(\<Union>i. disjointed A i) \<in> sets M"
by (simp add: algebra.range_disjointed_sets M A disjoint_family_disjointed)
thus "(\<Union>i::nat. A i) \<in> sets M" by (simp add: UN_disjointed_eq)
qed
lemma (in algebra) additive_sum:
fixes A:: "nat \<Rightarrow> 'a set"
assumes f: "positive M f" and ad: "additive M f"
and A: "range A \<subseteq> sets M"
and disj: "disjoint_family A"
shows "setsum (f o A) {0..<n} = f (\<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 "A n \<inter> (\<Union>i\<in>{0..<n}. A i) = {}" using disj
by (auto simp add: disjoint_family_on_def neq_iff) blast
moreover
have "A n \<in> sets M" using A by blast
moreover have "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
by (metis A UNION_in_sets atLeast0LessThan)
moreover
ultimately have "f (A n \<union> (\<Union>i\<in>{0..<n}. A i)) = f (A n) + f(\<Union>i\<in>{0..<n}. A i)"
using ad UNION_in_sets A by (auto simp add: additive_def)
with Suc.hyps show ?case using ad
by (auto simp add: atLeastLessThanSuc additive_def)
qed
lemma countably_subadditiveD:
"countably_subadditive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow>
(\<Union>i. A i) \<in> sets M \<Longrightarrow> summable (f o A) \<Longrightarrow> f (\<Union>i. A i) \<le> suminf (f o A)"
by (auto simp add: countably_subadditive_def o_def)
lemma (in algebra) increasing_additive_summable:
fixes A:: "nat \<Rightarrow> 'a set"
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 "summable (f o A)"
proof (rule pos_summable)
fix n
show "0 \<le> (f \<circ> A) n" using f A
by (force simp add: positive_def)
next
fix n
have "setsum (f \<circ> A) {0..<n} = f (\<Union>i\<in>{0..<n}. A i)"
by (rule additive_sum [OF f ad A disj])
also have "... \<le> f (space M)" using space_closed A
by (blast intro: increasingD [OF inc] UNION_in_sets top)
finally show "setsum (f \<circ> A) {0..<n} \<le> f (space M)" .
qed
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 lambda_system_increasing:
"increasing M f \<Longrightarrow> increasing (M (|sets := lambda_system M f|)) f"
by (simp add: increasing_def lambda_system_def)
lemma (in algebra) lambda_system_strong_sum:
fixes A:: "nat \<Rightarrow> 'a set"
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
have 4: "UNION {0..<n} A \<in> lambda_system M f"
using A algebra.UNION_in_sets [OF local.lambda_system_algebra [OF f]]
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 & (f \<circ> A) sums 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
have alg_ls: "algebra (M(|sets := lambda_system M f|))"
by (rule lambda_system_algebra [OF pos])
have A'': "range A \<subseteq> sets M"
by (metis A image_subset_iff lambda_system_sets)
have sumfA: "summable (f \<circ> A)"
by (metis algebra.increasing_additive_summable [OF alg_ls]
lambda_system_positive lambda_system_additive lambda_system_increasing
A' oms outer_measure_space_def disj)
have U_in: "(\<Union>i. A i) \<in> sets M"
by (metis A'' countable_UN)
have U_eq: "f (\<Union>i. A i) = suminf (f o A)"
proof (rule antisym)
show "f (\<Union>i. A i) \<le> suminf (f \<circ> A)"
by (rule countably_subadditiveD [OF csa A'' disj U_in sumfA])
show "suminf (f \<circ> A) \<le> f (\<Union>i. A i)"
by (rule suminf_le [OF sumfA])
(metis algebra.additive_sum [OF alg_ls] pos disj UN_Un Un_UNIV_right
lambda_system_positive lambda_system_additive
subset_Un_eq increasingD [OF inc] A' A'' UNION_in_sets U_in)
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 -
have summ: "summable (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)" using pos A''
apply -
apply (rule summable_comparison_test [OF _ sumfA])
apply (rule_tac x="0" in exI)
apply (simp add: positive_def)
apply (auto simp add: )
apply (subst abs_of_nonneg)
apply (metis A'' Int UNIV_I a image_subset_iff)
apply (blast intro: increasingD [OF inc])
done
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> suminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)"
using countably_subadditiveD [OF csa, of "(\<lambda>i. a \<inter> A i)"] summ
by (simp add: o_def)
moreover
have "suminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) \<le> f a - f (a - (\<Union>i. A i))"
proof (rule suminf_le [OF summ])
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 algebra.UNION_in_sets [OF lambda_system_algebra [OF pos]]
by (simp add: A)
hence eq_fa: "f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i)) = f a"
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_eq_Union_image
UNION_in U_in)
thus "setsum (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) {0..<n} \<le> f a - f (a - (\<Union>i. A i))"
using eq_fa
by (simp add: suminf_le [OF summ] lambda_system_strong_sum pos
a A disj)
qed
ultimately show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a"
by arith
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 sumfA)
qed
lemma (in sigma_algebra) caratheodory_lemma:
assumes oms: "outer_measure_space M f"
shows "measure_space (|space = space M, sets = lambda_system M f, measure = f|)"
proof -
have pos: "positive M f"
by (metis oms outer_measure_space_def)
have alg: "algebra (|space = space M, sets = lambda_system M f, measure = f|)"
using lambda_system_algebra [OF pos]
by (simp add: algebra_def)
then moreover
have "sigma_algebra (|space = space M, sets = lambda_system M f, measure = f|)"
using lambda_system_caratheodory [OF oms]
by (simp add: sigma_algebra_disjoint_iff)
moreover
have "measure_space_axioms (|space = space M, sets = lambda_system M f, measure = f|)"
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 intro_locales (auto simp add: sigma_algebra_def)
qed
lemma (in algebra) inf_measure_nonempty:
assumes f: "positive M f" and b: "b \<in> sets M" and a: "a \<subseteq> b"
shows "f b \<in> measure_set M f a"
proof -
have "(f \<circ> (\<lambda>i. {})(0 := b)) sums setsum (f \<circ> (\<lambda>i. {})(0 := b)) {0..<1::nat}"
by (rule series_zero) (simp add: positive_imp_0 [OF f])
also have "... = f b"
by simp
finally have "(f \<circ> (\<lambda>i. {})(0 := b)) sums f b" .
thus ?thesis using a
by (auto intro!: exI [of _ "(\<lambda>i. {})(0 := b)"]
simp add: measure_set_def disjoint_family_on_def b split_if_mem2)
qed
lemma (in algebra) inf_measure_pos0:
"positive M f \<Longrightarrow> x \<in> measure_set M f a \<Longrightarrow> 0 \<le> x"
apply (auto simp add: positive_def measure_set_def sums_iff intro!: suminf_ge_zero)
apply blast
done
lemma (in algebra) inf_measure_pos:
shows "positive M f \<Longrightarrow> x \<subseteq> space M \<Longrightarrow> 0 \<le> Inf (measure_set M f x)"
apply (rule Inf_greatest)
apply (metis emptyE inf_measure_nonempty top)
apply (metis inf_measure_pos0)
done
lemma (in algebra) 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"
have "f x \<le> f x + f (y-x)" using posf
by (simp add: positive_def) (metis Diff xy(1,2))
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" .
qed
lemma (in algebra) 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) = suminf (\<lambda>n. f (binaryset x y n))"
using ca
by (simp add: countably_additive_def) (metis UN_binaryset_eq sums_unique)
hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
f (x \<union> y) = suminf (\<lambda>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 (simp add: Un suminf_binaryset_eq positive_def)
qed
lemma (in algebra) 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"
proof (rule Inf_eq)
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 sm: "summable (f \<circ> A)"
and si: "suminf (f \<circ> A) = z"
by (auto simp add: measure_set_def sums_iff)
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: "(\<lambda>i. f (A i \<inter> s)) sums f (\<Union>i. A i \<inter> s)"
proof (rule countably_additiveD [OF ca])
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 = suminf (\<lambda>i. f (A i \<inter> s))"
using seq [symmetric] by (simp add: sums_iff)
also have "... \<le> suminf (f \<circ> A)"
proof (rule summable_le [OF _ _ sm])
show "\<forall>n. f (A n \<inter> s) \<le> (f \<circ> A) n" using A s
by (force intro: increasingD [OF inc])
show "summable (\<lambda>i. f (A i \<inter> s))" using sums
by (simp add: sums_iff)
qed
also have "... = z" by (rule si)
finally show "f s \<le> z" .
next
fix y
assume y: "!!u. u \<in> measure_set M f s \<Longrightarrow> y \<le> u"
thus "y \<le> f s"
by (blast intro: inf_measure_nonempty [OF posf s subset_refl])
qed
lemma (in algebra) inf_measure_empty:
assumes posf: "positive M f"
shows "Inf (measure_set M f {}) = 0"
proof (rule antisym)
show "0 \<le> Inf (measure_set M f {})"
by (metis empty_subsetI inf_measure_pos posf)
show "Inf (measure_set M f {}) \<le> 0"
by (metis Inf_lower empty_sets inf_measure_pos0 inf_measure_nonempty posf
positive_imp_0 subset_refl)
qed
lemma (in algebra) inf_measure_positive:
"positive M f \<Longrightarrow>
positive (| space = space M, sets = Pow (space M) |)
(\<lambda>x. Inf (measure_set M f x))"
by (simp add: positive_def inf_measure_empty inf_measure_pos)
lemma (in algebra) inf_measure_increasing:
assumes posf: "positive M f"
shows "increasing (| space = space M, sets = Pow (space M) |)
(\<lambda>x. Inf (measure_set M f x))"
apply (auto simp add: increasing_def)
apply (rule Inf_greatest, metis emptyE inf_measure_nonempty top posf)
apply (rule Inf_lower)
apply (clarsimp simp add: measure_set_def, rule_tac x=A in exI, blast)
apply (blast intro: inf_measure_pos0 posf)
done
lemma (in algebra) 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 & s \<subseteq> (\<Union>i. A i) & (f \<circ> A) sums r}"
shows "Inf (measure_set M f s) \<le> x"
proof -
from x
obtain A where A: "range A \<subseteq> sets M" and ss: "s \<subseteq> (\<Union>i. A i)"
and sm: "summable (f \<circ> A)" and xeq: "suminf (f \<circ> A) = x"
by (auto simp add: sums_iff)
have dA: "range (disjointed A) \<subseteq> sets M"
by (metis A range_disjointed_sets)
have "\<forall>n. \<bar>(f o disjointed A) n\<bar> \<le> (f \<circ> A) n"
proof (auto)
fix n
have "\<bar>f (disjointed A n)\<bar> = f (disjointed A n)" using posf dA
by (auto simp add: positive_def image_subset_iff)
also have "... \<le> f (A n)"
by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A)
finally show "\<bar>f (disjointed A n)\<bar> \<le> f (A n)" .
qed
from Series.summable_le2 [OF this sm]
have sda: "summable (f o disjointed A)"
"suminf (f o disjointed A) \<le> suminf (f \<circ> A)"
by blast+
hence ley: "suminf (f o disjointed A) \<le> x"
by (metis xeq)
from sda have "(f \<circ> disjointed A) sums suminf (f \<circ> disjointed A)"
by (simp add: sums_iff)
hence y: "suminf (f o disjointed A) \<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)
done
show ?thesis
by (blast intro: y order_trans [OF _ ley] inf_measure_pos0 posf)
qed
lemma (in algebra) inf_measure_close:
assumes posf: "positive M f" and e: "0 < e" and ss: "s \<subseteq> (space M)"
shows "\<exists>A l. range A \<subseteq> sets M & disjoint_family A & s \<subseteq> (\<Union>i. A i) &
(f \<circ> A) sums l & l \<le> Inf (measure_set M f s) + e"
proof -
have " measure_set M f s \<noteq> {}"
by (metis emptyE ss inf_measure_nonempty [OF posf top])
hence "\<exists>l \<in> measure_set M f s. l < Inf (measure_set M f s) + e"
by (rule Inf_close [OF _ e])
thus ?thesis
by (auto simp add: measure_set_def, rule_tac x=" A" in exI, auto)
qed
lemma (in algebra) 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 (auto simp add: countably_subadditive_def o_def, rule field_le_epsilon)
fix A :: "nat \<Rightarrow> 'a set" and e :: real
assume A: "range A \<subseteq> Pow (space M)"
and disj: "disjoint_family A"
and sb: "(\<Union>i. A i) \<subseteq> space M"
and sum1: "summable (\<lambda>n. Inf (measure_set M f (A n)))"
and e: "0 < e"
have "!!n. \<exists>B l. range B \<subseteq> sets M \<and> disjoint_family B \<and> A n \<subseteq> (\<Union>i. B i) \<and>
(f o B) sums l \<and>
l \<le> Inf (measure_set M f (A n)) + e * (1/2)^(Suc n)"
apply (rule inf_measure_close [OF posf])
apply (metis e half mult_pos_pos zero_less_power)
apply (metis UNIV_I UN_subset_iff sb)
done
hence "\<exists>BB ll. \<forall>n. range (BB n) \<subseteq> sets M \<and> disjoint_family (BB n) \<and>
A n \<subseteq> (\<Union>i. BB n i) \<and> (f o BB n) sums ll n \<and>
ll n \<le> Inf (measure_set M f (A n)) + e * (1/2)^(Suc n)"
by (rule choice2)
then obtain BB ll
where BB: "!!n. (range (BB n) \<subseteq> sets M)"
and disjBB: "!!n. disjoint_family (BB n)"
and sbBB: "!!n. A n \<subseteq> (\<Union>i. BB n i)"
and BBsums: "!!n. (f o BB n) sums ll n"
and ll: "!!n. ll n \<le> Inf (measure_set M f (A n)) + e * (1/2)^(Suc n)"
by auto blast
have llpos: "!!n. 0 \<le> ll n"
by (metis BBsums sums_iff o_apply posf positive_imp_pos suminf_ge_zero
range_subsetD BB)
have sll: "summable ll &
suminf ll \<le> suminf (\<lambda>n. Inf (measure_set M f (A n))) + e"
proof -
have "(\<lambda>n. e * (1/2)^(Suc n)) sums (e*1)"
by (rule sums_mult [OF power_half_series])
hence sum0: "summable (\<lambda>n. e * (1 / 2) ^ Suc n)"
and eqe: "(\<Sum>n. e * (1 / 2) ^ n / 2) = e"
by (auto simp add: sums_iff)
have 0: "suminf (\<lambda>n. Inf (measure_set M f (A n))) +
suminf (\<lambda>n. e * (1/2)^(Suc n)) =
suminf (\<lambda>n. Inf (measure_set M f (A n)) + e * (1/2)^(Suc n))"
by (rule suminf_add [OF sum1 sum0])
have 1: "\<forall>n. \<bar>ll n\<bar> \<le> Inf (measure_set M f (A n)) + e * (1/2) ^ Suc n"
by (metis ll llpos abs_of_nonneg)
have 2: "summable (\<lambda>n. Inf (measure_set M f (A n)) + e*(1/2)^(Suc n))"
by (rule summable_add [OF sum1 sum0])
have "suminf ll \<le> (\<Sum>n. Inf (measure_set M f (A n)) + e*(1/2) ^ Suc n)"
using Series.summable_le2 [OF 1 2] by auto
also have "... = (\<Sum>n. Inf (measure_set M f (A n))) +
(\<Sum>n. e * (1 / 2) ^ Suc n)"
by (metis 0)
also have "... = (\<Sum>n. Inf (measure_set M f (A n))) + e"
by (simp add: eqe)
finally show ?thesis using Series.summable_le2 [OF 1 2] by auto
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)
also have "... sums suminf ll"
proof (rule suminf_2dimen)
show "\<And>m n. 0 \<le> (f \<circ> (\<lambda>(x, y). BB x y)) (m, n)" using posf BB
by (force simp add: positive_def)
show "\<And>m. (\<lambda>n. (f \<circ> (\<lambda>(x, y). BB x y)) (m, n)) sums ll m"using BBsums BB
by (force simp add: o_def)
show "summable ll" using sll
by auto
qed
finally have Csums: "(f \<circ> C) sums suminf ll" .
have "Inf (measure_set M f (\<Union>i. A i)) \<le> suminf ll"
apply (rule inf_measure_le [OF posf inc], auto)
apply (rule_tac x="C" in exI)
apply (auto simp add: C sbC Csums)
done
also have "... \<le> (\<Sum>n. Inf (measure_set M f (A n))) + e" using sll
by blast
finally show "Inf (measure_set M f (\<Union>i. A i)) \<le>
(\<Sum>n. Inf (measure_set M f (A n))) + e" .
qed
lemma (in algebra) inf_measure_outer:
"positive M f \<Longrightarrow> increasing M f
\<Longrightarrow> outer_measure_space (| space = space M, sets = Pow (space M) |)
(\<lambda>x. Inf (measure_set M f x))"
by (simp add: outer_measure_space_def inf_measure_positive
inf_measure_increasing inf_measure_countably_subadditive)
(*MOVE UP*)
lemma (in algebra) 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 (| space = space M, sets = Pow (space M) |)
(\<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 (rule field_le_epsilon)
fix e :: real
assume e: "0 < e"
from inf_measure_close [OF posf e s]
obtain A l where A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
and sUN: "s \<subseteq> (\<Union>i. A i)" and fsums: "(f \<circ> A) sums l"
and l: "l \<le> Inf (measure_set M f s) + e"
by auto
have [simp]: "!!x. x \<in> sets M \<Longrightarrow>
(f o (\<lambda>z. z \<inter> (space M - x)) o A) = (f o (\<lambda>z. z - x) o A)"
by (rule ext, simp, metis A Int_Diff Int_space_eq2 range_subsetD)
have [simp]: "!!n. f (A n \<inter> x) + f (A n - x) = f (A n)"
by (subst additiveD [OF add, symmetric])
(auto simp add: x range_subsetD [OF A] Int_Diff_Un Int_Diff_disjoint)
have fsumb: "summable (f \<circ> A)"
by (metis fsums sums_iff)
{ fix u
assume u: "u \<in> sets M"
have [simp]: "\<And>n. \<bar>f (A n \<inter> u)\<bar> \<le> f (A n)"
by (simp add: positive_imp_pos [OF posf] increasingD [OF inc]
u Int range_subsetD [OF A])
have 1: "summable (f o (\<lambda>z. z\<inter>u) o A)"
by (rule summable_comparison_test [OF _ fsumb]) simp
have 2: "Inf (measure_set M f (s\<inter>u)) \<le> suminf (f o (\<lambda>z. z\<inter>u) o A)"
proof (rule Inf_lower)
show "suminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A) \<in> measure_set M f (s \<inter> u)"
apply (simp add: measure_set_def)
apply (rule_tac x="(\<lambda>z. z \<inter> u) o A" in exI)
apply (auto simp add: disjoint_family_subset [OF disj])
apply (blast intro: u range_subsetD [OF A])
apply (blast dest: subsetD [OF sUN])
apply (metis 1 o_assoc sums_iff)
done
next
show "\<And>x. x \<in> measure_set M f (s \<inter> u) \<Longrightarrow> 0 \<le> x"
by (blast intro: inf_measure_pos0 [OF posf])
qed
note 1 2
} note lesum = this
have sum1: "summable (f o (\<lambda>z. z\<inter>x) o A)"
and inf1: "Inf (measure_set M f (s\<inter>x)) \<le> suminf (f o (\<lambda>z. z\<inter>x) o A)"
and sum2: "summable (f o (\<lambda>z. z \<inter> (space M - x)) o A)"
and inf2: "Inf (measure_set M f (s \<inter> (space M - x)))
\<le> suminf (f o (\<lambda>z. z \<inter> (space M - x)) o A)"
by (metis Diff lesum top x)+
hence "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
\<le> suminf (f o (\<lambda>s. s\<inter>x) o A) + suminf (f o (\<lambda>s. s-x) o A)"
by (simp add: x)
also have "... \<le> suminf (f o A)" using suminf_add [OF sum1 sum2]
by (simp add: x) (simp add: o_def)
also have "... \<le> Inf (measure_set M f s) + e"
by (metis fsums l sums_unique)
finally show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
\<le> Inf (measure_set M f s) + e" .
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 (iprover intro: algebra.countably_subadditive_subadditive algebra_Pow
inf_measure_positive inf_measure_countably_subadditive posf inc)
apply (auto simp add: subsetD [OF s])
done
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 M = measure N) \<Longrightarrow> measure_space M"
by (simp add: measure_space_def measure_space_axioms_def positive_def
countably_additive_def)
blast
theorem (in algebra) caratheodory:
assumes posf: "positive M f" and ca: "countably_additive M f"
shows "\<exists>MS :: 'a measure_space.
(\<forall>s \<in> sets M. measure MS s = f s) \<and>
((|space = space MS, sets = sets MS|) = sigma (space M) (sets M)) \<and>
measure_space MS"
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 (|space = space M, sets = ls, measure = ?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 (|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 (|space = space M,
sets = sigma_sets (space M) (sets M),
measure = ?infm|)"
by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets)
(simp_all add: sgs_sb space_closed)
thus ?thesis
by (force simp add: sigma_def inf_measure_agrees [OF posf ca])
qed
end