
wenzelm@41983: (*  Title:      HOL/Probability/Complete_Measure.thy
hoelzl@40859:     Author:     Robert Himmelmann, Johannes Hoelzl, TU Muenchen
hoelzl@40859: *)
wenzelm@41983: 
hoelzl@40859: theory Complete_Measure
hoelzl@56993: imports Bochner_Integration
hoelzl@40859: begin
hoelzl@40859: 
hoelzl@47694: definition
hoelzl@47694:   "split_completion M A p = (if A \<in> sets M then p = (A, {}) else
hoelzl@47694:    \<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)"
hoelzl@41689: 
hoelzl@47694: definition
hoelzl@47694:   "main_part M A = fst (Eps (split_completion M A))"
hoelzl@41689: 
hoelzl@47694: definition
hoelzl@47694:   "null_part M A = snd (Eps (split_completion M A))"
hoelzl@40859: 
hoelzl@47694: definition completion :: "'a measure \<Rightarrow> 'a measure" where
hoelzl@47694:   "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' }
hoelzl@47694:     (emeasure M \<circ> main_part M)"
hoelzl@40859: 
hoelzl@47694: lemma completion_into_space:
hoelzl@47694:   "{ S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' } \<subseteq> Pow (space M)"
immler@50244:   using sets.sets_into_space by auto
hoelzl@40859: 
hoelzl@47694: lemma space_completion[simp]: "space (completion M) = space M"
hoelzl@47694:   unfolding completion_def using space_measure_of[OF completion_into_space] by simp
hoelzl@40859: 
hoelzl@47694: lemma completionI:
hoelzl@47694:   assumes "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
hoelzl@47694:   shows "A \<in> { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
hoelzl@40859:   using assms by auto
hoelzl@40859: 
hoelzl@47694: lemma completionE:
hoelzl@47694:   assumes "A \<in> { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
hoelzl@47694:   obtains S N N' where "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
hoelzl@47694:   using assms by auto
hoelzl@47694: 
hoelzl@47694: lemma sigma_algebra_completion:
hoelzl@47694:   "sigma_algebra (space M) { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
hoelzl@47694:     (is "sigma_algebra _ ?A")
hoelzl@47694:   unfolding sigma_algebra_iff2
hoelzl@47694: proof (intro conjI ballI allI impI)
hoelzl@47694:   show "?A \<subseteq> Pow (space M)"
immler@50244:     using sets.sets_into_space by auto
hoelzl@40859: next
hoelzl@47694:   show "{} \<in> ?A" by auto
hoelzl@40859: next
hoelzl@47694:   let ?C = "space M"
hoelzl@47694:   fix A assume "A \<in> ?A" from completionE[OF this] guess S N N' .
hoelzl@47694:   then show "space M - A \<in> ?A"
hoelzl@47694:     by (intro completionI[of _ "(?C - S) \<inter> (?C - N')" "(?C - S) \<inter> N' \<inter> (?C - N)"]) auto
hoelzl@47694: next
hoelzl@47694:   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> ?A"
hoelzl@47694:   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'"
hoelzl@47694:     by (auto simp: image_subset_iff)
hoelzl@40859:   from choice[OF this] guess S ..
hoelzl@40859:   from choice[OF this] guess N ..
hoelzl@40859:   from choice[OF this] guess N' ..
hoelzl@47694:   then show "UNION UNIV A \<in> ?A"
hoelzl@40859:     using null_sets_UN[of N']
hoelzl@47694:     by (intro completionI[of _ "UNION UNIV S" "UNION UNIV N" "UNION UNIV N'"]) auto
hoelzl@47694: qed
hoelzl@47694: 
hoelzl@47694: lemma sets_completion:
hoelzl@47694:   "sets (completion M) = { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
hoelzl@47694:   using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_completion] by (simp add: completion_def)
hoelzl@47694: 
hoelzl@47694: lemma sets_completionE:
hoelzl@47694:   assumes "A \<in> sets (completion M)"
hoelzl@47694:   obtains S N N' where "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
hoelzl@47694:   using assms unfolding sets_completion by auto
hoelzl@47694: 
hoelzl@47694: lemma sets_completionI:
hoelzl@47694:   assumes "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
hoelzl@47694:   shows "A \<in> sets (completion M)"
hoelzl@47694:   using assms unfolding sets_completion by auto
hoelzl@47694: 
hoelzl@47694: lemma sets_completionI_sets[intro, simp]:
hoelzl@47694:   "A \<in> sets M \<Longrightarrow> A \<in> sets (completion M)"
hoelzl@47694:   unfolding sets_completion by force
hoelzl@40859: 
hoelzl@47694: lemma null_sets_completion:
hoelzl@47694:   assumes "N' \<in> null_sets M" "N \<subseteq> N'" shows "N \<in> sets (completion M)"
hoelzl@47694:   using assms by (intro sets_completionI[of N "{}" N N']) auto
hoelzl@47694: 
hoelzl@47694: lemma split_completion:
hoelzl@47694:   assumes "A \<in> sets (completion M)"
hoelzl@47694:   shows "split_completion M A (main_part M A, null_part M A)"
hoelzl@47694: proof cases
hoelzl@47694:   assume "A \<in> sets M" then show ?thesis
hoelzl@47694:     by (simp add: split_completion_def[abs_def] main_part_def null_part_def)
hoelzl@47694: next
hoelzl@47694:   assume nA: "A \<notin> sets M"
hoelzl@47694:   show ?thesis
hoelzl@47694:     unfolding main_part_def null_part_def if_not_P[OF nA]
hoelzl@47694:   proof (rule someI2_ex)
hoelzl@47694:     from assms[THEN sets_completionE] guess S N N' . note A = this
hoelzl@47694:     let ?P = "(S, N - S)"
hoelzl@47694:     show "\<exists>p. split_completion M A p"
hoelzl@47694:       unfolding split_completion_def if_not_P[OF nA] using A
hoelzl@47694:     proof (intro exI conjI)
hoelzl@47694:       show "A = fst ?P \<union> snd ?P" using A by auto
hoelzl@47694:       show "snd ?P \<subseteq> N'" using A by auto
hoelzl@47694:    qed auto
hoelzl@40859:   qed auto
hoelzl@47694: qed
hoelzl@40859: 
hoelzl@47694: lemma
hoelzl@47694:   assumes "S \<in> sets (completion M)"
hoelzl@47694:   shows main_part_sets[intro, simp]: "main_part M S \<in> sets M"
hoelzl@47694:     and main_part_null_part_Un[simp]: "main_part M S \<union> null_part M S = S"
hoelzl@47694:     and main_part_null_part_Int[simp]: "main_part M S \<inter> null_part M S = {}"
hoelzl@47694:   using split_completion[OF assms]
hoelzl@47694:   by (auto simp: split_completion_def split: split_if_asm)
hoelzl@40859: 
hoelzl@47694: lemma main_part[simp]: "S \<in> sets M \<Longrightarrow> main_part M S = S"
hoelzl@47694:   using split_completion[of S M]
hoelzl@47694:   by (auto simp: split_completion_def split: split_if_asm)
hoelzl@40859: 
hoelzl@47694: lemma null_part:
hoelzl@47694:   assumes "S \<in> sets (completion M)" shows "\<exists>N. N\<in>null_sets M \<and> null_part M S \<subseteq> N"
hoelzl@47694:   using split_completion[OF assms] by (auto simp: split_completion_def split: split_if_asm)
hoelzl@47694: 
hoelzl@47694: lemma null_part_sets[intro, simp]:
hoelzl@47694:   assumes "S \<in> sets M" shows "null_part M S \<in> sets M" "emeasure M (null_part M S) = 0"
hoelzl@40859: proof -
hoelzl@47694:   have S: "S \<in> sets (completion M)" using assms by auto
hoelzl@47694:   have "S - main_part M S \<in> sets M" using assms by auto
hoelzl@40859:   moreover
hoelzl@40859:   from main_part_null_part_Un[OF S] main_part_null_part_Int[OF S]
hoelzl@47694:   have "S - main_part M S = null_part M S" by auto
hoelzl@47694:   ultimately show sets: "null_part M S \<in> sets M" by auto
hoelzl@40859:   from null_part[OF S] guess N ..
hoelzl@47694:   with emeasure_eq_0[of N _ "null_part M S"] sets
hoelzl@47694:   show "emeasure M (null_part M S) = 0" by auto
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@47694: lemma emeasure_main_part_UN:
hoelzl@40859:   fixes S :: "nat \<Rightarrow> 'a set"
hoelzl@47694:   assumes "range S \<subseteq> sets (completion M)"
hoelzl@47694:   shows "emeasure M (main_part M (\<Union>i. (S i))) = emeasure M (\<Union>i. main_part M (S i))"
hoelzl@40859: proof -
hoelzl@47694:   have S: "\<And>i. S i \<in> sets (completion M)" using assms by auto
hoelzl@47694:   then have UN: "(\<Union>i. S i) \<in> sets (completion M)" by auto
hoelzl@47694:   have "\<forall>i. \<exists>N. N \<in> null_sets M \<and> null_part M (S i) \<subseteq> N"
hoelzl@40859:     using null_part[OF S] by auto
hoelzl@40859:   from choice[OF this] guess N .. note N = this
hoelzl@47694:   then have UN_N: "(\<Union>i. N i) \<in> null_sets M" by (intro null_sets_UN) auto
hoelzl@47694:   have "(\<Union>i. S i) \<in> sets (completion M)" using S by auto
hoelzl@40859:   from null_part[OF this] guess N' .. note N' = this
hoelzl@40859:   let ?N = "(\<Union>i. N i) \<union> N'"
hoelzl@47694:   have null_set: "?N \<in> null_sets M" using N' UN_N by (intro null_sets.Un) auto
hoelzl@47694:   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"
hoelzl@40859:     using N' by auto
hoelzl@47694:   also have "\<dots> = (\<Union>i. main_part M (S i) \<union> null_part M (S i)) \<union> ?N"
hoelzl@40859:     unfolding main_part_null_part_Un[OF S] main_part_null_part_Un[OF UN] by auto
hoelzl@47694:   also have "\<dots> = (\<Union>i. main_part M (S i)) \<union> ?N"
hoelzl@40859:     using N by auto
hoelzl@47694:   finally have *: "main_part M (\<Union>i. S i) \<union> ?N = (\<Union>i. main_part M (S i)) \<union> ?N" .
hoelzl@47694:   have "emeasure M (main_part M (\<Union>i. S i)) = emeasure M (main_part M (\<Union>i. S i) \<union> ?N)"
hoelzl@47694:     using null_set UN by (intro emeasure_Un_null_set[symmetric]) auto
hoelzl@47694:   also have "\<dots> = emeasure M ((\<Union>i. main_part M (S i)) \<union> ?N)"
hoelzl@40859:     unfolding * ..
hoelzl@47694:   also have "\<dots> = emeasure M (\<Union>i. main_part M (S i))"
hoelzl@47694:     using null_set S by (intro emeasure_Un_null_set) auto
hoelzl@41689:   finally show ?thesis .
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@47694: lemma emeasure_completion[simp]:
hoelzl@47694:   assumes S: "S \<in> sets (completion M)" shows "emeasure (completion M) S = emeasure M (main_part M S)"
hoelzl@47694: proof (subst emeasure_measure_of[OF completion_def completion_into_space])
hoelzl@47694:   let ?\<mu> = "emeasure M \<circ> main_part M"
hoelzl@47694:   show "S \<in> sets (completion M)" "?\<mu> S = emeasure M (main_part M S) " using S by simp_all
hoelzl@47694:   show "positive (sets (completion M)) ?\<mu>"
hoelzl@47694:     by (simp add: positive_def emeasure_nonneg)
hoelzl@47694:   show "countably_additive (sets (completion M)) ?\<mu>"
hoelzl@47694:   proof (intro countably_additiveI)
hoelzl@47694:     fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets (completion M)" "disjoint_family A"
hoelzl@47694:     have "disjoint_family (\<lambda>i. main_part M (A i))"
hoelzl@47694:     proof (intro disjoint_family_on_bisimulation[OF A(2)])
hoelzl@47694:       fix n m assume "A n \<inter> A m = {}"
hoelzl@47694:       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)) = {}"
hoelzl@47694:         using A by (subst (1 2) main_part_null_part_Un) auto
hoelzl@47694:       then show "main_part M (A n) \<inter> main_part M (A m) = {}" by auto
hoelzl@47694:     qed
hoelzl@47694:     then have "(\<Sum>n. emeasure M (main_part M (A n))) = emeasure M (\<Union>i. main_part M (A i))"
hoelzl@47694:       using A by (auto intro!: suminf_emeasure)
hoelzl@47694:     then show "(\<Sum>n. ?\<mu> (A n)) = ?\<mu> (UNION UNIV A)"
hoelzl@47694:       by (simp add: completion_def emeasure_main_part_UN[OF A(1)])
hoelzl@47694:   qed
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@47694: lemma emeasure_completion_UN:
hoelzl@47694:   "range S \<subseteq> sets (completion M) \<Longrightarrow>
hoelzl@47694:     emeasure (completion M) (\<Union>i::nat. (S i)) = emeasure M (\<Union>i. main_part M (S i))"
hoelzl@47694:   by (subst emeasure_completion) (auto simp add: emeasure_main_part_UN)
hoelzl@40859: 
hoelzl@47694: lemma emeasure_completion_Un:
hoelzl@47694:   assumes S: "S \<in> sets (completion M)" and T: "T \<in> sets (completion M)"
hoelzl@47694:   shows "emeasure (completion M) (S \<union> T) = emeasure M (main_part M S \<union> main_part M T)"
hoelzl@47694: proof (subst emeasure_completion)
hoelzl@47694:   have UN: "(\<Union>i. binary (main_part M S) (main_part M T) i) = (\<Union>i. main_part M (binary S T i))"
hoelzl@47694:     unfolding binary_def by (auto split: split_if_asm)
hoelzl@47694:   show "emeasure M (main_part M (S \<union> T)) = emeasure M (main_part M S \<union> main_part M T)"
hoelzl@47694:     using emeasure_main_part_UN[of "binary S T" M] assms
hoelzl@47694:     unfolding range_binary_eq Un_range_binary UN by auto
hoelzl@47694: qed (auto intro: S T)
hoelzl@47694: 
hoelzl@47694: lemma sets_completionI_sub:
hoelzl@47694:   assumes N: "N' \<in> null_sets M" "N \<subseteq> N'"
hoelzl@47694:   shows "N \<in> sets (completion M)"
hoelzl@47694:   using assms by (intro sets_completionI[of _ "{}" N N']) auto
hoelzl@47694: 
hoelzl@47694: lemma completion_ex_simple_function:
hoelzl@47694:   assumes f: "simple_function (completion M) f"
hoelzl@47694:   shows "\<exists>f'. simple_function M f' \<and> (AE x in M. f x = f' x)"
hoelzl@40859: proof -
wenzelm@46731:   let ?F = "\<lambda>x. f -` {x} \<inter> space M"
hoelzl@47694:   have F: "\<And>x. ?F x \<in> sets (completion M)" and fin: "finite (f`space M)"
hoelzl@47694:     using simple_functionD[OF f] simple_functionD[OF f] by simp_all
hoelzl@47694:   have "\<forall>x. \<exists>N. N \<in> null_sets M \<and> null_part M (?F x) \<subseteq> N"
hoelzl@40859:     using F null_part by auto
hoelzl@40859:   from choice[OF this] obtain N where
hoelzl@47694:     N: "\<And>x. null_part M (?F x) \<subseteq> N x" "\<And>x. N x \<in> null_sets M" by auto
wenzelm@46731:   let ?N = "\<Union>x\<in>f`space M. N x"
wenzelm@46731:   let ?f' = "\<lambda>x. if x \<in> ?N then undefined else f x"
hoelzl@47694:   have sets: "?N \<in> null_sets M" using N fin by (intro null_sets.finite_UN) auto
hoelzl@40859:   show ?thesis unfolding simple_function_def
hoelzl@40859:   proof (safe intro!: exI[of _ ?f'])
hoelzl@40859:     have "?f' ` space M \<subseteq> f`space M \<union> {undefined}" by auto
hoelzl@47694:     from finite_subset[OF this] simple_functionD(1)[OF f]
hoelzl@40859:     show "finite (?f' ` space M)" by auto
hoelzl@40859:   next
hoelzl@40859:     fix x assume "x \<in> space M"
hoelzl@40859:     have "?f' -` {?f' x} \<inter> space M =
hoelzl@40859:       (if x \<in> ?N then ?F undefined \<union> ?N
hoelzl@40859:        else if f x = undefined then ?F (f x) \<union> ?N
hoelzl@40859:        else ?F (f x) - ?N)"
immler@50244:       using N(2) sets.sets_into_space by (auto split: split_if_asm simp: null_sets_def)
hoelzl@40859:     moreover { fix y have "?F y \<union> ?N \<in> sets M"
hoelzl@40859:       proof cases
hoelzl@40859:         assume y: "y \<in> f`space M"
hoelzl@47694:         have "?F y \<union> ?N = (main_part M (?F y) \<union> null_part M (?F y)) \<union> ?N"
hoelzl@40859:           using main_part_null_part_Un[OF F] by auto
hoelzl@47694:         also have "\<dots> = main_part M (?F y) \<union> ?N"
hoelzl@40859:           using y N by auto
hoelzl@40859:         finally show ?thesis
hoelzl@40859:           using F sets by auto
hoelzl@40859:       next
hoelzl@40859:         assume "y \<notin> f`space M" then have "?F y = {}" by auto
hoelzl@40859:         then show ?thesis using sets by auto
hoelzl@40859:       qed }
hoelzl@40859:     moreover {
hoelzl@47694:       have "?F (f x) - ?N = main_part M (?F (f x)) \<union> null_part M (?F (f x)) - ?N"
hoelzl@40859:         using main_part_null_part_Un[OF F] by auto
hoelzl@47694:       also have "\<dots> = main_part M (?F (f x)) - ?N"
hoelzl@40859:         using N `x \<in> space M` by auto
hoelzl@40859:       finally have "?F (f x) - ?N \<in> sets M"
hoelzl@40859:         using F sets by auto }
hoelzl@40859:     ultimately show "?f' -` {?f' x} \<inter> space M \<in> sets M" by auto
hoelzl@40859:   next
hoelzl@47694:     show "AE x in M. f x = ?f' x"
hoelzl@40859:       by (rule AE_I', rule sets) auto
hoelzl@40859:   qed
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@47694: lemma completion_ex_borel_measurable_pos:
hoelzl@43920:   fixes g :: "'a \<Rightarrow> ereal"
hoelzl@47694:   assumes g: "g \<in> borel_measurable (completion M)" and "\<And>x. 0 \<le> g x"
hoelzl@47694:   shows "\<exists>g'\<in>borel_measurable M. (AE x in M. g x = g' x)"
hoelzl@40859: proof -
hoelzl@47694:   from g[THEN borel_measurable_implies_simple_function_sequence'] guess f . note f = this
hoelzl@41981:   from this(1)[THEN completion_ex_simple_function]
hoelzl@47694:   have "\<forall>i. \<exists>f'. simple_function M f' \<and> (AE x in M. f i x = f' x)" ..
hoelzl@40859:   from this[THEN choice] obtain f' where
hoelzl@41689:     sf: "\<And>i. simple_function M (f' i)" and
hoelzl@47694:     AE: "\<forall>i. AE x in M. f i x = f' i x" by auto
hoelzl@40859:   show ?thesis
hoelzl@40859:   proof (intro bexI)
hoelzl@41981:     from AE[unfolded AE_all_countable[symmetric]]
hoelzl@47694:     show "AE x in M. g x = (SUP i. f' i x)" (is "AE x in M. g x = ?f x")
hoelzl@41705:     proof (elim AE_mp, safe intro!: AE_I2)
hoelzl@40859:       fix x assume eq: "\<forall>i. f i x = f' i x"
hoelzl@41981:       moreover have "g x = (SUP i. f i x)"
hoelzl@41981:         unfolding f using `0 \<le> g x` by (auto split: split_max)
hoelzl@41981:       ultimately show "g x = ?f x" by auto
hoelzl@40859:     qed
hoelzl@40859:     show "?f \<in> borel_measurable M"
hoelzl@56949:       using sf[THEN borel_measurable_simple_function] by auto
hoelzl@40859:   qed
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@47694: lemma completion_ex_borel_measurable:
hoelzl@43920:   fixes g :: "'a \<Rightarrow> ereal"
hoelzl@47694:   assumes g: "g \<in> borel_measurable (completion M)"
hoelzl@47694:   shows "\<exists>g'\<in>borel_measurable M. (AE x in M. g x = g' x)"
hoelzl@41981: proof -
hoelzl@47694:   have "(\<lambda>x. max 0 (g x)) \<in> borel_measurable (completion M)" "\<And>x. 0 \<le> max 0 (g x)" using g by auto
hoelzl@41981:   from completion_ex_borel_measurable_pos[OF this] guess g_pos ..
hoelzl@41981:   moreover
hoelzl@47694:   have "(\<lambda>x. max 0 (- g x)) \<in> borel_measurable (completion M)" "\<And>x. 0 \<le> max 0 (- g x)" using g by auto
hoelzl@41981:   from completion_ex_borel_measurable_pos[OF this] guess g_neg ..
hoelzl@41981:   ultimately
hoelzl@41981:   show ?thesis
hoelzl@41981:   proof (safe intro!: bexI[of _ "\<lambda>x. g_pos x - g_neg x"])
hoelzl@47694:     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"
hoelzl@41981:     proof (intro AE_I2 impI)
hoelzl@41981:       fix x assume g: "max 0 (- g x) = g_neg x" "max 0 (g x) = g_pos x"
hoelzl@41981:       show "g x = g_pos x - g_neg x" unfolding g[symmetric]
hoelzl@41981:         by (cases "g x") (auto split: split_max)
hoelzl@41981:     qed
hoelzl@41981:   qed auto
hoelzl@41981: qed
hoelzl@41981: 
hoelzl@40859: end




