(* Title: HOL/Probability/Complete_Measure.thy
Author: Robert Himmelmann, Johannes Hoelzl, TU Muenchen
*)
theory Complete_Measure
imports Product_Measure
begin
locale completeable_measure_space = measure_space
definition (in completeable_measure_space)
"split_completion A p = (\<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)"
definition (in completeable_measure_space)
"main_part A = fst (Eps (split_completion A))"
definition (in completeable_measure_space)
"null_part A = snd (Eps (split_completion A))"
abbreviation (in completeable_measure_space) "\<mu>' A \<equiv> \<mu> (main_part A)"
definition (in completeable_measure_space) completion :: "('a, 'b) measure_space_scheme" where
"completion = \<lparr> space = space M,
sets = { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets \<and> N \<subseteq> N' },
measure = \<mu>',
\<dots> = more M \<rparr>"
lemma (in completeable_measure_space) space_completion[simp]:
"space completion = space M" unfolding completion_def by simp
lemma (in completeable_measure_space) sets_completionE:
assumes "A \<in> sets completion"
obtains S N N' where "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets" "S \<in> sets M"
using assms unfolding completion_def by auto
lemma (in completeable_measure_space) sets_completionI:
assumes "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets" "S \<in> sets M"
shows "A \<in> sets completion"
using assms unfolding completion_def by auto
lemma (in completeable_measure_space) sets_completionI_sets[intro]:
"A \<in> sets M \<Longrightarrow> A \<in> sets completion"
unfolding completion_def by force
lemma (in completeable_measure_space) null_sets_completion:
assumes "N' \<in> null_sets" "N \<subseteq> N'" shows "N \<in> sets completion"
apply(rule sets_completionI[of N "{}" N N'])
using assms by auto
sublocale completeable_measure_space \<subseteq> completion!: sigma_algebra completion
proof (unfold sigma_algebra_iff2, safe)
fix A x assume "A \<in> sets completion" "x \<in> A"
with sets_into_space show "x \<in> space completion"
by (auto elim!: sets_completionE)
next
fix A assume "A \<in> sets completion"
from this[THEN sets_completionE] guess S N N' . note A = this
let ?C = "space completion"
show "?C - A \<in> sets completion" using A
by (intro sets_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> sets completion"
then have "\<forall>n. \<exists>S N N'. A n = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets \<and> N \<subseteq> N'"
unfolding completion_def 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> sets completion"
using null_sets_UN[of N']
by (intro sets_completionI[of _ "UNION UNIV S" "UNION UNIV N" "UNION UNIV N'"])
auto
qed auto
lemma (in completeable_measure_space) split_completion:
assumes "A \<in> sets completion"
shows "split_completion A (main_part A, null_part A)"
unfolding main_part_def null_part_def
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 A p"
unfolding split_completion_def 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
lemma (in completeable_measure_space)
assumes "S \<in> sets completion"
shows main_part_sets[intro, simp]: "main_part S \<in> sets M"
and main_part_null_part_Un[simp]: "main_part S \<union> null_part S = S"
and main_part_null_part_Int[simp]: "main_part S \<inter> null_part S = {}"
using split_completion[OF assms] by (auto simp: split_completion_def)
lemma (in completeable_measure_space) null_part:
assumes "S \<in> sets completion" shows "\<exists>N. N\<in>null_sets \<and> null_part S \<subseteq> N"
using split_completion[OF assms] by (auto simp: split_completion_def)
lemma (in completeable_measure_space) null_part_sets[intro, simp]:
assumes "S \<in> sets M" shows "null_part S \<in> sets M" "\<mu> (null_part S) = 0"
proof -
have S: "S \<in> sets completion" using assms by auto
have "S - main_part 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 S = null_part S" by auto
ultimately show sets: "null_part S \<in> sets M" by auto
from null_part[OF S] guess N ..
with measure_eq_0[of N "null_part S"] sets
show "\<mu> (null_part S) = 0" by auto
qed
lemma (in completeable_measure_space) \<mu>'_set[simp]:
assumes "S \<in> sets M" shows "\<mu>' S = \<mu> S"
proof -
have S: "S \<in> sets completion" using assms by auto
then have "\<mu> S = \<mu> (main_part S \<union> null_part S)" by simp
also have "\<dots> = \<mu>' S"
using S assms measure_additive[of "main_part S" "null_part S"]
by (auto simp: measure_additive)
finally show ?thesis by simp
qed
lemma (in completeable_measure_space) sets_completionI_sub:
assumes N: "N' \<in> null_sets" "N \<subseteq> N'"
shows "N \<in> sets completion"
using assms by (intro sets_completionI[of _ "{}" N N']) auto
lemma (in completeable_measure_space) \<mu>_main_part_UN:
fixes S :: "nat \<Rightarrow> 'a set"
assumes "range S \<subseteq> sets completion"
shows "\<mu>' (\<Union>i. (S i)) = \<mu> (\<Union>i. main_part (S i))"
proof -
have S: "\<And>i. S i \<in> sets completion" using assms by auto
then have UN: "(\<Union>i. S i) \<in> sets completion" by auto
have "\<forall>i. \<exists>N. N \<in> null_sets \<and> null_part (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" by (intro null_sets_UN) auto
have "(\<Union>i. S i) \<in> sets completion" 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" using N' UN_N by (intro null_sets_Un) auto
have "main_part (\<Union>i. S i) \<union> ?N = (main_part (\<Union>i. S i) \<union> null_part (\<Union>i. S i)) \<union> ?N"
using N' by auto
also have "\<dots> = (\<Union>i. main_part (S i) \<union> null_part (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 (S i)) \<union> ?N"
using N by auto
finally have *: "main_part (\<Union>i. S i) \<union> ?N = (\<Union>i. main_part (S i)) \<union> ?N" .
have "\<mu> (main_part (\<Union>i. S i)) = \<mu> (main_part (\<Union>i. S i) \<union> ?N)"
using null_set UN by (intro measure_Un_null_set[symmetric]) auto
also have "\<dots> = \<mu> ((\<Union>i. main_part (S i)) \<union> ?N)"
unfolding * ..
also have "\<dots> = \<mu> (\<Union>i. main_part (S i))"
using null_set S by (intro measure_Un_null_set) auto
finally show ?thesis .
qed
lemma (in completeable_measure_space) \<mu>_main_part_Un:
assumes S: "S \<in> sets completion" and T: "T \<in> sets completion"
shows "\<mu>' (S \<union> T) = \<mu> (main_part S \<union> main_part T)"
proof -
have UN: "(\<Union>i. binary (main_part S) (main_part T) i) = (\<Union>i. main_part (binary S T i))"
unfolding binary_def by (auto split: split_if_asm)
show ?thesis
using \<mu>_main_part_UN[of "binary S T"] assms
unfolding range_binary_eq Un_range_binary UN by auto
qed
sublocale completeable_measure_space \<subseteq> completion!: measure_space completion
where "measure completion = \<mu>'"
proof -
show "measure_space completion"
proof
show "measure completion {} = 0" by (auto simp: completion_def)
next
show "countably_additive completion (measure completion)"
proof (intro countably_additiveI)
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets completion" "disjoint_family A"
have "disjoint_family (\<lambda>i. main_part (A i))"
proof (intro disjoint_family_on_bisimulation[OF A(2)])
fix n m assume "A n \<inter> A m = {}"
then have "(main_part (A n) \<union> null_part (A n)) \<inter> (main_part (A m) \<union> null_part (A m)) = {}"
using A by (subst (1 2) main_part_null_part_Un) auto
then show "main_part (A n) \<inter> main_part (A m) = {}" by auto
qed
then have "(\<Sum>\<^isub>\<infinity>n. measure completion (A n)) = \<mu> (\<Union>i. main_part (A i))"
unfolding completion_def using A by (auto intro!: measure_countably_additive)
then show "(\<Sum>\<^isub>\<infinity>n. measure completion (A n)) = measure completion (UNION UNIV A)"
by (simp add: completion_def \<mu>_main_part_UN[OF A(1)])
qed
qed
show "measure completion = \<mu>'" unfolding completion_def by simp
qed
lemma (in completeable_measure_space) completion_ex_simple_function:
assumes f: "simple_function completion f"
shows "\<exists>f'. simple_function M f' \<and> (AE x. f x = f' x)"
proof -
let "?F x" = "f -` {x} \<inter> space M"
have F: "\<And>x. ?F x \<in> sets completion" and fin: "finite (f`space M)"
using completion.simple_functionD[OF f]
completion.simple_functionD[OF f] by simp_all
have "\<forall>x. \<exists>N. N \<in> null_sets \<and> null_part (?F x) \<subseteq> N"
using F null_part by auto
from choice[OF this] obtain N where
N: "\<And>x. null_part (?F x) \<subseteq> N x" "\<And>x. N x \<in> null_sets" by auto
let ?N = "\<Union>x\<in>f`space M. N x" let "?f' x" = "if x \<in> ?N then undefined else f x"
have sets: "?N \<in> null_sets" 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] completion.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_into_space by (auto split: split_if_asm)
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 (?F y) \<union> null_part (?F y)) \<union> ?N"
using main_part_null_part_Un[OF F] by auto
also have "\<dots> = main_part (?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 (?F (f x)) \<union> null_part (?F (f x)) - ?N"
using main_part_null_part_Un[OF F] by auto
also have "\<dots> = main_part (?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. f x = ?f' x"
by (rule AE_I', rule sets) auto
qed
qed
lemma (in completeable_measure_space) completion_ex_borel_measurable:
fixes g :: "'a \<Rightarrow> pextreal"
assumes g: "g \<in> borel_measurable completion"
shows "\<exists>g'\<in>borel_measurable M. (AE x. g x = g' x)"
proof -
from g[THEN completion.borel_measurable_implies_simple_function_sequence]
obtain f where "\<And>i. simple_function completion (f i)" "f \<up> g" by auto
then have "\<forall>i. \<exists>f'. simple_function M f' \<and> (AE x. f i x = f' x)"
using completion_ex_simple_function by auto
from this[THEN choice] obtain f' where
sf: "\<And>i. simple_function M (f' i)" and
AE: "\<forall>i. AE x. f i x = f' i x" by auto
show ?thesis
proof (intro bexI)
from AE[unfolded all_AE_countable]
show "AE x. g x = (SUP i. f' i x)" (is "AE x. 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 = SUPR UNIV f" using `f \<up> g` unfolding isoton_def by simp
ultimately show "g x = ?f x" by (simp add: SUPR_apply)
qed
show "?f \<in> borel_measurable M"
using sf by (auto intro: borel_measurable_simple_function)
qed
qed
end