theory Complete_Measure
imports Bochner_Integration Probability_Measure
begin
definition
"split_completion M A p = (if A ∈ sets M then p = (A, {}) else
∃N'. A = fst p ∪ snd p ∧ fst p ∩ snd p = {} ∧ fst p ∈ sets M ∧ snd p ⊆ N' ∧ N' ∈ 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 ⇒ 'a measure" where
"completion M = measure_of (space M) { S ∪ N |S N N'. S ∈ sets M ∧ N' ∈ null_sets M ∧ N ⊆ N' }
(emeasure M ∘ main_part M)"
lemma completion_into_space:
"{ S ∪ N |S N N'. S ∈ sets M ∧ N' ∈ null_sets M ∧ N ⊆ N' } ⊆ 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 ∪ N" "N ⊆ N'" "N' ∈ null_sets M" "S ∈ sets M"
shows "A ∈ { S ∪ N |S N N'. S ∈ sets M ∧ N' ∈ null_sets M ∧ N ⊆ N' }"
using assms by auto
lemma completionE:
assumes "A ∈ { S ∪ N |S N N'. S ∈ sets M ∧ N' ∈ null_sets M ∧ N ⊆ N' }"
obtains S N N' where "A = S ∪ N" "N ⊆ N'" "N' ∈ null_sets M" "S ∈ sets M"
using assms by auto
lemma sigma_algebra_completion:
"sigma_algebra (space M) { S ∪ N |S N N'. S ∈ sets M ∧ N' ∈ null_sets M ∧ N ⊆ N' }"
(is "sigma_algebra _ ?A")
unfolding sigma_algebra_iff2
proof (intro conjI ballI allI impI)
show "?A ⊆ Pow (space M)"
using sets.sets_into_space by auto
next
show "{} ∈ ?A" by auto
next
let ?C = "space M"
fix A assume "A ∈ ?A" from completionE[OF this] guess S N N' .
then show "space M - A ∈ ?A"
by (intro completionI[of _ "(?C - S) ∩ (?C - N')" "(?C - S) ∩ N' ∩ (?C - N)"]) auto
next
fix A :: "nat ⇒ 'a set" assume A: "range A ⊆ ?A"
then have "∀n. ∃S N N'. A n = S ∪ N ∧ S ∈ sets M ∧ N' ∈ null_sets M ∧ N ⊆ 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 ∈ ?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 ∪ N |S N N'. S ∈ sets M ∧ N' ∈ null_sets M ∧ N ⊆ N' }"
using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_completion] by (simp add: completion_def)
lemma sets_completionE:
assumes "A ∈ sets (completion M)"
obtains S N N' where "A = S ∪ N" "N ⊆ N'" "N' ∈ null_sets M" "S ∈ sets M"
using assms unfolding sets_completion by auto
lemma sets_completionI:
assumes "A = S ∪ N" "N ⊆ N'" "N' ∈ null_sets M" "S ∈ sets M"
shows "A ∈ sets (completion M)"
using assms unfolding sets_completion by auto
lemma sets_completionI_sets[intro, simp]:
"A ∈ sets M ⟹ A ∈ sets (completion M)"
unfolding sets_completion by force
lemma null_sets_completion:
assumes "N' ∈ null_sets M" "N ⊆ N'" shows "N ∈ sets (completion M)"
using assms by (intro sets_completionI[of N "{}" N N']) auto
lemma split_completion:
assumes "A ∈ sets (completion M)"
shows "split_completion M A (main_part M A, null_part M A)"
proof cases
assume "A ∈ sets M" then show ?thesis
by (simp add: split_completion_def[abs_def] main_part_def null_part_def)
next
assume nA: "A ∉ 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 "∃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 ∪ snd ?P" using A by auto
show "snd ?P ⊆ N'" using A by auto
qed auto
qed auto
qed
lemma
assumes "S ∈ sets (completion M)"
shows main_part_sets[intro, simp]: "main_part M S ∈ sets M"
and main_part_null_part_Un[simp]: "main_part M S ∪ null_part M S = S"
and main_part_null_part_Int[simp]: "main_part M S ∩ null_part M S = {}"
using split_completion[OF assms]
by (auto simp: split_completion_def split: if_split_asm)
lemma main_part[simp]: "S ∈ sets M ⟹ main_part M S = S"
using split_completion[of S M]
by (auto simp: split_completion_def split: if_split_asm)
lemma null_part:
assumes "S ∈ sets (completion M)" shows "∃N. N∈null_sets M ∧ null_part M S ⊆ N"
using split_completion[OF assms] by (auto simp: split_completion_def split: if_split_asm)
lemma null_part_sets[intro, simp]:
assumes "S ∈ sets M" shows "null_part M S ∈ sets M" "emeasure M (null_part M S) = 0"
proof -
have S: "S ∈ sets (completion M)" using assms by auto
have "S - main_part M S ∈ 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 ∈ 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 ⇒ 'a set"
assumes "range S ⊆ sets (completion M)"
shows "emeasure M (main_part M (⋃i. (S i))) = emeasure M (⋃i. main_part M (S i))"
proof -
have S: "⋀i. S i ∈ sets (completion M)" using assms by auto
then have UN: "(⋃i. S i) ∈ sets (completion M)" by auto
have "∀i. ∃N. N ∈ null_sets M ∧ null_part M (S i) ⊆ N"
using null_part[OF S] by auto
from choice[OF this] guess N .. note N = this
then have UN_N: "(⋃i. N i) ∈ null_sets M" by (intro null_sets_UN) auto
have "(⋃i. S i) ∈ sets (completion M)" using S by auto
from null_part[OF this] guess N' .. note N' = this
let ?N = "(⋃i. N i) ∪ N'"
have null_set: "?N ∈ null_sets M" using N' UN_N by (intro null_sets.Un) auto
have "main_part M (⋃i. S i) ∪ ?N = (main_part M (⋃i. S i) ∪ null_part M (⋃i. S i)) ∪ ?N"
using N' by auto
also have "… = (⋃i. main_part M (S i) ∪ null_part M (S i)) ∪ ?N"
unfolding main_part_null_part_Un[OF S] main_part_null_part_Un[OF UN] by auto
also have "… = (⋃i. main_part M (S i)) ∪ ?N"
using N by auto
finally have *: "main_part M (⋃i. S i) ∪ ?N = (⋃i. main_part M (S i)) ∪ ?N" .
have "emeasure M (main_part M (⋃i. S i)) = emeasure M (main_part M (⋃i. S i) ∪ ?N)"
using null_set UN by (intro emeasure_Un_null_set[symmetric]) auto
also have "… = emeasure M ((⋃i. main_part M (S i)) ∪ ?N)"
unfolding * ..
also have "… = emeasure M (⋃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 ∈ 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 ?μ = "emeasure M ∘ main_part M"
show "S ∈ sets (completion M)" "?μ S = emeasure M (main_part M S) " using S by simp_all
show "positive (sets (completion M)) ?μ"
by (simp add: positive_def)
show "countably_additive (sets (completion M)) ?μ"
proof (intro countably_additiveI)
fix A :: "nat ⇒ 'a set" assume A: "range A ⊆ sets (completion M)" "disjoint_family A"
have "disjoint_family (λi. main_part M (A i))"
proof (intro disjoint_family_on_bisimulation[OF A(2)])
fix n m assume "A n ∩ A m = {}"
then have "(main_part M (A n) ∪ null_part M (A n)) ∩ (main_part M (A m) ∪ null_part M (A m)) = {}"
using A by (subst (1 2) main_part_null_part_Un) auto
then show "main_part M (A n) ∩ main_part M (A m) = {}" by auto
qed
then have "(∑n. emeasure M (main_part M (A n))) = emeasure M (⋃i. main_part M (A i))"
using A by (auto intro!: suminf_emeasure)
then show "(∑n. ?μ (A n)) = ?μ (UNION UNIV A)"
by (simp add: completion_def emeasure_main_part_UN[OF A(1)])
qed
qed
lemma emeasure_completion_UN:
"range S ⊆ sets (completion M) ⟹
emeasure (completion M) (⋃i::nat. (S i)) = emeasure M (⋃i. main_part M (S i))"
by (subst emeasure_completion) (auto simp add: emeasure_main_part_UN)
lemma emeasure_completion_Un:
assumes S: "S ∈ sets (completion M)" and T: "T ∈ sets (completion M)"
shows "emeasure (completion M) (S ∪ T) = emeasure M (main_part M S ∪ main_part M T)"
proof (subst emeasure_completion)
have UN: "(⋃i. binary (main_part M S) (main_part M T) i) = (⋃i. main_part M (binary S T i))"
unfolding binary_def by (auto split: if_split_asm)
show "emeasure M (main_part M (S ∪ T)) = emeasure M (main_part M S ∪ main_part M T)"
using emeasure_main_part_UN[of "binary S T" M] assms
by (simp add: range_binary_eq, simp add: Un_range_binary UN)
qed (auto intro: S T)
lemma sets_completionI_sub:
assumes N: "N' ∈ null_sets M" "N ⊆ N'"
shows "N ∈ 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 "∃f'. simple_function M f' ∧ (AE x in M. f x = f' x)"
proof -
let ?F = "λx. f -` {x} ∩ space M"
have F: "⋀x. ?F x ∈ sets (completion M)" and fin: "finite (f`space M)"
using simple_functionD[OF f] simple_functionD[OF f] by simp_all
have "∀x. ∃N. N ∈ null_sets M ∧ null_part M (?F x) ⊆ N"
using F null_part by auto
from choice[OF this] obtain N where
N: "⋀x. null_part M (?F x) ⊆ N x" "⋀x. N x ∈ null_sets M" by auto
let ?N = "⋃x∈f`space M. N x"
let ?f' = "λx. if x ∈ ?N then undefined else f x"
have sets: "?N ∈ 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 ⊆ f`space M ∪ {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 ∈ space M"
have "?f' -` {?f' x} ∩ space M =
(if x ∈ ?N then ?F undefined ∪ ?N
else if f x = undefined then ?F (f x) ∪ ?N
else ?F (f x) - ?N)"
using N(2) sets.sets_into_space by (auto split: if_split_asm simp: null_sets_def)
moreover { fix y have "?F y ∪ ?N ∈ sets M"
proof cases
assume y: "y ∈ f`space M"
have "?F y ∪ ?N = (main_part M (?F y) ∪ null_part M (?F y)) ∪ ?N"
using main_part_null_part_Un[OF F] by auto
also have "… = main_part M (?F y) ∪ ?N"
using y N by auto
finally show ?thesis
using F sets by auto
next
assume "y ∉ 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)) ∪ null_part M (?F (f x)) - ?N"
using main_part_null_part_Un[OF F] by auto
also have "… = main_part M (?F (f x)) - ?N"
using N ‹x ∈ space M› by auto
finally have "?F (f x) - ?N ∈ sets M"
using F sets by auto }
ultimately show "?f' -` {?f' x} ∩ space M ∈ 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:
fixes g :: "'a ⇒ ennreal"
assumes g: "g ∈ borel_measurable (completion M)"
shows "∃g'∈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 "∀i. ∃f'. simple_function M f' ∧ (AE x in M. f i x = f' x)" ..
from this[THEN choice] obtain f' where
sf: "⋀i. simple_function M (f' i)" and
AE: "∀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: "∀i. f i x = f' i x"
moreover have "g x = (SUP i. f i x)"
unfolding f by (auto split: split_max)
ultimately show "g x = ?f x" by auto
qed
show "?f ∈ borel_measurable M"
using sf[THEN borel_measurable_simple_function] by auto
qed
qed
lemma (in prob_space) prob_space_completion: "prob_space (completion M)"
by (rule prob_spaceI) (simp add: emeasure_space_1)
lemma null_sets_completionI: "N ∈ null_sets M ⟹ N ∈ null_sets (completion M)"
by (auto simp: null_sets_def)
lemma AE_completion: "(AE x in M. P x) ⟹ (AE x in completion M. P x)"
unfolding eventually_ae_filter by (auto intro: null_sets_completionI)
lemma null_sets_completion_iff: "N ∈ sets M ⟹ N ∈ null_sets (completion M) ⟷ N ∈ null_sets M"
by (auto simp: null_sets_def)
lemma AE_completion_iff: "{x∈space M. P x} ∈ sets M ⟹ (AE x in M. P x) ⟷ (AE x in completion M. P x)"
by (simp add: AE_iff_null null_sets_completion_iff)
end