(* Title: HOL/Probability/Complete_Measure.thy
Author: Robert Himmelmann, Johannes Hoelzl, TU Muenchen
*)
theory Complete_Measure
imports Lebesgue_Integration
begin
definition
"split_completion M A p = (if A \<in> sets M then p = (A, {}) else
\<exists>N'. A = fst p \<union> snd p \<and> fst p \<inter> snd p = {} \<and> fst p \<in> sets M \<and> snd p \<subseteq> N' \<and> N' \<in> null_sets M)"
definition
"main_part M A = fst (Eps (split_completion M A))"
definition
"null_part M A = snd (Eps (split_completion M A))"
definition completion :: "'a measure \<Rightarrow> 'a measure" where
"completion M = measure_of (space M) { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }
(emeasure M \<circ> main_part M)"
lemma completion_into_space:
"{ S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' } \<subseteq> Pow (space M)"
using sets.sets_into_space by auto
lemma space_completion[simp]: "space (completion M) = space M"
unfolding completion_def using space_measure_of[OF completion_into_space] by simp
lemma completionI:
assumes "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
shows "A \<in> { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
using assms by auto
lemma completionE:
assumes "A \<in> { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
obtains S N N' where "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
using assms by auto
lemma sigma_algebra_completion:
"sigma_algebra (space M) { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
(is "sigma_algebra _ ?A")
unfolding sigma_algebra_iff2
proof (intro conjI ballI allI impI)
show "?A \<subseteq> Pow (space M)"
using sets.sets_into_space by auto
next
show "{} \<in> ?A" by auto
next
let ?C = "space M"
fix A assume "A \<in> ?A" from completionE[OF this] guess S N N' .
then show "space M - A \<in> ?A"
by (intro completionI[of _ "(?C - S) \<inter> (?C - N')" "(?C - S) \<inter> N' \<inter> (?C - N)"]) auto
next
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> ?A"
then have "\<forall>n. \<exists>S N N'. A n = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N'"
by (auto simp: image_subset_iff)
from choice[OF this] guess S ..
from choice[OF this] guess N ..
from choice[OF this] guess N' ..
then show "UNION UNIV A \<in> ?A"
using null_sets_UN[of N']
by (intro completionI[of _ "UNION UNIV S" "UNION UNIV N" "UNION UNIV N'"]) auto
qed
lemma sets_completion:
"sets (completion M) = { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_completion] by (simp add: completion_def)
lemma sets_completionE:
assumes "A \<in> sets (completion M)"
obtains S N N' where "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
using assms unfolding sets_completion by auto
lemma sets_completionI:
assumes "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
shows "A \<in> sets (completion M)"
using assms unfolding sets_completion by auto
lemma sets_completionI_sets[intro, simp]:
"A \<in> sets M \<Longrightarrow> A \<in> sets (completion M)"
unfolding sets_completion by force
lemma null_sets_completion:
assumes "N' \<in> null_sets M" "N \<subseteq> N'" shows "N \<in> sets (completion M)"
using assms by (intro sets_completionI[of N "{}" N N']) auto
lemma split_completion:
assumes "A \<in> sets (completion M)"
shows "split_completion M A (main_part M A, null_part M A)"
proof cases
assume "A \<in> sets M" then show ?thesis
by (simp add: split_completion_def[abs_def] main_part_def null_part_def)
next
assume nA: "A \<notin> sets M"
show ?thesis
unfolding main_part_def null_part_def if_not_P[OF nA]
proof (rule someI2_ex)
from assms[THEN sets_completionE] guess S N N' . note A = this
let ?P = "(S, N - S)"
show "\<exists>p. split_completion M A p"
unfolding split_completion_def if_not_P[OF nA] using A
proof (intro exI conjI)
show "A = fst ?P \<union> snd ?P" using A by auto
show "snd ?P \<subseteq> N'" using A by auto
qed auto
qed auto
qed
lemma
assumes "S \<in> sets (completion M)"
shows main_part_sets[intro, simp]: "main_part M S \<in> sets M"
and main_part_null_part_Un[simp]: "main_part M S \<union> null_part M S = S"
and main_part_null_part_Int[simp]: "main_part M S \<inter> null_part M S = {}"
using split_completion[OF assms]
by (auto simp: split_completion_def split: split_if_asm)
lemma main_part[simp]: "S \<in> sets M \<Longrightarrow> main_part M S = S"
using split_completion[of S M]
by (auto simp: split_completion_def split: split_if_asm)
lemma null_part:
assumes "S \<in> sets (completion M)" shows "\<exists>N. N\<in>null_sets M \<and> null_part M S \<subseteq> N"
using split_completion[OF assms] by (auto simp: split_completion_def split: split_if_asm)
lemma null_part_sets[intro, simp]:
assumes "S \<in> sets M" shows "null_part M S \<in> sets M" "emeasure M (null_part M S) = 0"
proof -
have S: "S \<in> sets (completion M)" using assms by auto
have "S - main_part M S \<in> sets M" using assms by auto
moreover
from main_part_null_part_Un[OF S] main_part_null_part_Int[OF S]
have "S - main_part M S = null_part M S" by auto
ultimately show sets: "null_part M S \<in> sets M" by auto
from null_part[OF S] guess N ..
with emeasure_eq_0[of N _ "null_part M S"] sets
show "emeasure M (null_part M S) = 0" by auto
qed
lemma emeasure_main_part_UN:
fixes S :: "nat \<Rightarrow> 'a set"
assumes "range S \<subseteq> sets (completion M)"
shows "emeasure M (main_part M (\<Union>i. (S i))) = emeasure M (\<Union>i. main_part M (S i))"
proof -
have S: "\<And>i. S i \<in> sets (completion M)" using assms by auto
then have UN: "(\<Union>i. S i) \<in> sets (completion M)" by auto
have "\<forall>i. \<exists>N. N \<in> null_sets M \<and> null_part M (S i) \<subseteq> N"
using null_part[OF S] by auto
from choice[OF this] guess N .. note N = this
then have UN_N: "(\<Union>i. N i) \<in> null_sets M" by (intro null_sets_UN) auto
have "(\<Union>i. S i) \<in> sets (completion M)" using S by auto
from null_part[OF this] guess N' .. note N' = this
let ?N = "(\<Union>i. N i) \<union> N'"
have null_set: "?N \<in> null_sets M" using N' UN_N by (intro null_sets.Un) auto
have "main_part M (\<Union>i. S i) \<union> ?N = (main_part M (\<Union>i. S i) \<union> null_part M (\<Union>i. S i)) \<union> ?N"
using N' by auto
also have "\<dots> = (\<Union>i. main_part M (S i) \<union> null_part M (S i)) \<union> ?N"
unfolding main_part_null_part_Un[OF S] main_part_null_part_Un[OF UN] by auto
also have "\<dots> = (\<Union>i. main_part M (S i)) \<union> ?N"
using N by auto
finally have *: "main_part M (\<Union>i. S i) \<union> ?N = (\<Union>i. main_part M (S i)) \<union> ?N" .
have "emeasure M (main_part M (\<Union>i. S i)) = emeasure M (main_part M (\<Union>i. S i) \<union> ?N)"
using null_set UN by (intro emeasure_Un_null_set[symmetric]) auto
also have "\<dots> = emeasure M ((\<Union>i. main_part M (S i)) \<union> ?N)"
unfolding * ..
also have "\<dots> = emeasure M (\<Union>i. main_part M (S i))"
using null_set S by (intro emeasure_Un_null_set) auto
finally show ?thesis .
qed
lemma emeasure_completion[simp]:
assumes S: "S \<in> sets (completion M)" shows "emeasure (completion M) S = emeasure M (main_part M S)"
proof (subst emeasure_measure_of[OF completion_def completion_into_space])
let ?\<mu> = "emeasure M \<circ> main_part M"
show "S \<in> sets (completion M)" "?\<mu> S = emeasure M (main_part M S) " using S by simp_all
show "positive (sets (completion M)) ?\<mu>"
by (simp add: positive_def emeasure_nonneg)
show "countably_additive (sets (completion M)) ?\<mu>"
proof (intro countably_additiveI)
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets (completion M)" "disjoint_family A"
have "disjoint_family (\<lambda>i. main_part M (A i))"
proof (intro disjoint_family_on_bisimulation[OF A(2)])
fix n m assume "A n \<inter> A m = {}"
then have "(main_part M (A n) \<union> null_part M (A n)) \<inter> (main_part M (A m) \<union> null_part M (A m)) = {}"
using A by (subst (1 2) main_part_null_part_Un) auto
then show "main_part M (A n) \<inter> main_part M (A m) = {}" by auto
qed
then have "(\<Sum>n. emeasure M (main_part M (A n))) = emeasure M (\<Union>i. main_part M (A i))"
using A by (auto intro!: suminf_emeasure)
then show "(\<Sum>n. ?\<mu> (A n)) = ?\<mu> (UNION UNIV A)"
by (simp add: completion_def emeasure_main_part_UN[OF A(1)])
qed
qed
lemma emeasure_completion_UN:
"range S \<subseteq> sets (completion M) \<Longrightarrow>
emeasure (completion M) (\<Union>i::nat. (S i)) = emeasure M (\<Union>i. main_part M (S i))"
by (subst emeasure_completion) (auto simp add: emeasure_main_part_UN)
lemma emeasure_completion_Un:
assumes S: "S \<in> sets (completion M)" and T: "T \<in> sets (completion M)"
shows "emeasure (completion M) (S \<union> T) = emeasure M (main_part M S \<union> main_part M T)"
proof (subst emeasure_completion)
have UN: "(\<Union>i. binary (main_part M S) (main_part M T) i) = (\<Union>i. main_part M (binary S T i))"
unfolding binary_def by (auto split: split_if_asm)
show "emeasure M (main_part M (S \<union> T)) = emeasure M (main_part M S \<union> main_part M T)"
using emeasure_main_part_UN[of "binary S T" M] assms
unfolding range_binary_eq Un_range_binary UN by auto
qed (auto intro: S T)
lemma sets_completionI_sub:
assumes N: "N' \<in> null_sets M" "N \<subseteq> N'"
shows "N \<in> sets (completion M)"
using assms by (intro sets_completionI[of _ "{}" N N']) auto
lemma completion_ex_simple_function:
assumes f: "simple_function (completion M) f"
shows "\<exists>f'. simple_function M f' \<and> (AE x in M. f x = f' x)"
proof -
let ?F = "\<lambda>x. f -` {x} \<inter> space M"
have F: "\<And>x. ?F x \<in> sets (completion M)" and fin: "finite (f`space M)"
using simple_functionD[OF f] simple_functionD[OF f] by simp_all
have "\<forall>x. \<exists>N. N \<in> null_sets M \<and> null_part M (?F x) \<subseteq> N"
using F null_part by auto
from choice[OF this] obtain N where
N: "\<And>x. null_part M (?F x) \<subseteq> N x" "\<And>x. N x \<in> null_sets M" by auto
let ?N = "\<Union>x\<in>f`space M. N x"
let ?f' = "\<lambda>x. if x \<in> ?N then undefined else f x"
have sets: "?N \<in> null_sets M" using N fin by (intro null_sets.finite_UN) auto
show ?thesis unfolding simple_function_def
proof (safe intro!: exI[of _ ?f'])
have "?f' ` space M \<subseteq> f`space M \<union> {undefined}" by auto
from finite_subset[OF this] simple_functionD(1)[OF f]
show "finite (?f' ` space M)" by auto
next
fix x assume "x \<in> space M"
have "?f' -` {?f' x} \<inter> space M =
(if x \<in> ?N then ?F undefined \<union> ?N
else if f x = undefined then ?F (f x) \<union> ?N
else ?F (f x) - ?N)"
using N(2) sets.sets_into_space by (auto split: split_if_asm simp: null_sets_def)
moreover { fix y have "?F y \<union> ?N \<in> sets M"
proof cases
assume y: "y \<in> f`space M"
have "?F y \<union> ?N = (main_part M (?F y) \<union> null_part M (?F y)) \<union> ?N"
using main_part_null_part_Un[OF F] by auto
also have "\<dots> = main_part M (?F y) \<union> ?N"
using y N by auto
finally show ?thesis
using F sets by auto
next
assume "y \<notin> f`space M" then have "?F y = {}" by auto
then show ?thesis using sets by auto
qed }
moreover {
have "?F (f x) - ?N = main_part M (?F (f x)) \<union> null_part M (?F (f x)) - ?N"
using main_part_null_part_Un[OF F] by auto
also have "\<dots> = main_part M (?F (f x)) - ?N"
using N `x \<in> space M` by auto
finally have "?F (f x) - ?N \<in> sets M"
using F sets by auto }
ultimately show "?f' -` {?f' x} \<inter> space M \<in> sets M" by auto
next
show "AE x in M. f x = ?f' x"
by (rule AE_I', rule sets) auto
qed
qed
lemma completion_ex_borel_measurable_pos:
fixes g :: "'a \<Rightarrow> ereal"
assumes g: "g \<in> borel_measurable (completion M)" and "\<And>x. 0 \<le> g x"
shows "\<exists>g'\<in>borel_measurable M. (AE x in M. g x = g' x)"
proof -
from g[THEN borel_measurable_implies_simple_function_sequence'] guess f . note f = this
from this(1)[THEN completion_ex_simple_function]
have "\<forall>i. \<exists>f'. simple_function M f' \<and> (AE x in M. f i x = f' x)" ..
from this[THEN choice] obtain f' where
sf: "\<And>i. simple_function M (f' i)" and
AE: "\<forall>i. AE x in M. f i x = f' i x" by auto
show ?thesis
proof (intro bexI)
from AE[unfolded AE_all_countable[symmetric]]
show "AE x in M. g x = (SUP i. f' i x)" (is "AE x in M. g x = ?f x")
proof (elim AE_mp, safe intro!: AE_I2)
fix x assume eq: "\<forall>i. f i x = f' i x"
moreover have "g x = (SUP i. f i x)"
unfolding f using `0 \<le> g x` by (auto split: split_max)
ultimately show "g x = ?f x" by auto
qed
show "?f \<in> borel_measurable M"
using sf by (auto intro: borel_measurable_simple_function)
qed
qed
lemma completion_ex_borel_measurable:
fixes g :: "'a \<Rightarrow> ereal"
assumes g: "g \<in> borel_measurable (completion M)"
shows "\<exists>g'\<in>borel_measurable M. (AE x in M. g x = g' x)"
proof -
have "(\<lambda>x. max 0 (g x)) \<in> borel_measurable (completion M)" "\<And>x. 0 \<le> max 0 (g x)" using g by auto
from completion_ex_borel_measurable_pos[OF this] guess g_pos ..
moreover
have "(\<lambda>x. max 0 (- g x)) \<in> borel_measurable (completion M)" "\<And>x. 0 \<le> max 0 (- g x)" using g by auto
from completion_ex_borel_measurable_pos[OF this] guess g_neg ..
ultimately
show ?thesis
proof (safe intro!: bexI[of _ "\<lambda>x. g_pos x - g_neg x"])
show "AE x in M. max 0 (- g x) = g_neg x \<longrightarrow> max 0 (g x) = g_pos x \<longrightarrow> g x = g_pos x - g_neg x"
proof (intro AE_I2 impI)
fix x assume g: "max 0 (- g x) = g_neg x" "max 0 (g x) = g_pos x"
show "g x = g_pos x - g_neg x" unfolding g[symmetric]
by (cases "g x") (auto split: split_max)
qed
qed auto
qed
end