author | hoelzl |
Thu, 16 Jul 2015 10:48:20 +0200 | |
changeset 60727 | 53697011b03a |
parent 60066 | 14efa7f4ee7b |
child 61169 | 4de9ff3ea29a |
permissions | -rw-r--r-- |
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(* Title: HOL/Probability/Binary_Product_Measure.thy |
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Author: Johannes Hölzl, TU München |
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*) |
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section {*Binary product measures*} |
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|
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theory Binary_Product_Measure |
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imports Nonnegative_Lebesgue_Integration |
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begin |
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lemma Pair_vimage_times[simp]: "Pair x -` (A \<times> B) = (if x \<in> A then B else {})" |
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by auto |
13 |
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lemma rev_Pair_vimage_times[simp]: "(\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})" |
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by auto |
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subsection "Binary products" |
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|
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definition pair_measure (infixr "\<Otimes>\<^sub>M" 80) where |
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"A \<Otimes>\<^sub>M B = measure_of (space A \<times> space B) |
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{a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} |
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(\<lambda>X. \<integral>\<^sup>+x. (\<integral>\<^sup>+y. indicator X (x,y) \<partial>B) \<partial>A)" |
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lemma pair_measure_closed: "{a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} \<subseteq> Pow (space A \<times> space B)" |
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using sets.space_closed[of A] sets.space_closed[of B] by auto |
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|
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lemma space_pair_measure: |
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"space (A \<Otimes>\<^sub>M B) = space A \<times> space B" |
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unfolding pair_measure_def using pair_measure_closed[of A B] |
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by (rule space_measure_of) |
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lemma SIGMA_Collect_eq: "(SIGMA x:space M. {y\<in>space N. P x y}) = {x\<in>space (M \<Otimes>\<^sub>M N). P (fst x) (snd x)}" |
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by (auto simp: space_pair_measure) |
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lemma sets_pair_measure: |
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"sets (A \<Otimes>\<^sub>M B) = sigma_sets (space A \<times> space B) {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}" |
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unfolding pair_measure_def using pair_measure_closed[of A B] |
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by (rule sets_measure_of) |
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lemma sets_pair_in_sets: |
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assumes N: "space A \<times> space B = space N" |
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assumes "\<And>a b. a \<in> sets A \<Longrightarrow> b \<in> sets B \<Longrightarrow> a \<times> b \<in> sets N" |
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shows "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets N" |
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using assms by (auto intro!: sets.sigma_sets_subset simp: sets_pair_measure N) |
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lemma sets_pair_measure_cong[measurable_cong, cong]: |
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"sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>M M2')" |
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unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq) |
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lemma pair_measureI[intro, simp, measurable]: |
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"x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)" |
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by (auto simp: sets_pair_measure) |
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lemma sets_Pair: "{x} \<in> sets M1 \<Longrightarrow> {y} \<in> sets M2 \<Longrightarrow> {(x, y)} \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
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using pair_measureI[of "{x}" M1 "{y}" M2] by simp |
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lemma measurable_pair_measureI: |
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assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2" |
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assumes 2: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> f -` (A \<times> B) \<inter> space M \<in> sets M" |
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shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)" |
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unfolding pair_measure_def using 1 2 |
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by (intro measurable_measure_of) (auto dest: sets.sets_into_space) |
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lemma measurable_split_replace[measurable (raw)]: |
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"(\<lambda>x. f x (fst (g x)) (snd (g x))) \<in> measurable M N \<Longrightarrow> (\<lambda>x. split (f x) (g x)) \<in> measurable M N" |
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unfolding split_beta' . |
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lemma measurable_Pair[measurable (raw)]: |
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assumes f: "f \<in> measurable M M1" and g: "g \<in> measurable M M2" |
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shows "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)" |
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proof (rule measurable_pair_measureI) |
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show "(\<lambda>x. (f x, g x)) \<in> space M \<rightarrow> space M1 \<times> space M2" |
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using f g by (auto simp: measurable_def) |
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fix A B assume *: "A \<in> sets M1" "B \<in> sets M2" |
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have "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)" |
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by auto |
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also have "\<dots> \<in> sets M" |
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by (rule sets.Int) (auto intro!: measurable_sets * f g) |
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finally show "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M \<in> sets M" . |
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qed |
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lemma measurable_fst[intro!, simp, measurable]: "fst \<in> measurable (M1 \<Otimes>\<^sub>M M2) M1" |
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by (auto simp: fst_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times |
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measurable_def) |
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lemma measurable_snd[intro!, simp, measurable]: "snd \<in> measurable (M1 \<Otimes>\<^sub>M M2) M2" |
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by (auto simp: snd_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times |
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measurable_def) |
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lemma measurable_Pair_compose_split[measurable_dest]: |
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assumes f: "split f \<in> measurable (M1 \<Otimes>\<^sub>M M2) N" |
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assumes g: "g \<in> measurable M M1" and h: "h \<in> measurable M M2" |
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shows "(\<lambda>x. f (g x) (h x)) \<in> measurable M N" |
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using measurable_compose[OF measurable_Pair f, OF g h] by simp |
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|
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lemma measurable_Pair1_compose[measurable_dest]: |
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assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)" |
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assumes [measurable]: "h \<in> measurable N M" |
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shows "(\<lambda>x. f (h x)) \<in> measurable N M1" |
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using measurable_compose[OF f measurable_fst] by simp |
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|
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lemma measurable_Pair2_compose[measurable_dest]: |
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assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)" |
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assumes [measurable]: "h \<in> measurable N M" |
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shows "(\<lambda>x. g (h x)) \<in> measurable N M2" |
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using measurable_compose[OF f measurable_snd] by simp |
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|
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lemma measurable_pair: |
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assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2" |
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shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)" |
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using measurable_Pair[OF assms] by simp |
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lemma |
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assumes f[measurable]: "f \<in> measurable M (N \<Otimes>\<^sub>M P)" |
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shows measurable_fst': "(\<lambda>x. fst (f x)) \<in> measurable M N" |
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and measurable_snd': "(\<lambda>x. snd (f x)) \<in> measurable M P" |
|
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by simp_all |
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lemma |
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assumes f[measurable]: "f \<in> measurable M N" |
|
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shows measurable_fst'': "(\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^sub>M P) N" |
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and measurable_snd'': "(\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^sub>M M) N" |
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by simp_all |
47694 | 124 |
|
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lemma sets_pair_eq_sets_fst_snd: |
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"sets (A \<Otimes>\<^sub>M B) = sets (Sup_sigma {vimage_algebra (space A \<times> space B) fst A, vimage_algebra (space A \<times> space B) snd B})" |
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(is "?P = sets (Sup_sigma {?fst, ?snd})") |
|
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proof - |
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{ fix a b assume ab: "a \<in> sets A" "b \<in> sets B" |
|
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then have "a \<times> b = (fst -` a \<inter> (space A \<times> space B)) \<inter> (snd -` b \<inter> (space A \<times> space B))" |
|
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by (auto dest: sets.sets_into_space) |
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also have "\<dots> \<in> sets (Sup_sigma {?fst, ?snd})" |
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using ab by (auto intro: in_Sup_sigma in_vimage_algebra) |
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finally have "a \<times> b \<in> sets (Sup_sigma {?fst, ?snd})" . } |
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moreover have "sets ?fst \<subseteq> sets (A \<Otimes>\<^sub>M B)" |
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by (rule sets_image_in_sets) (auto simp: space_pair_measure[symmetric]) |
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moreover have "sets ?snd \<subseteq> sets (A \<Otimes>\<^sub>M B)" |
|
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by (rule sets_image_in_sets) (auto simp: space_pair_measure) |
|
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ultimately show ?thesis |
|
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by (intro antisym[of "sets A" for A] sets_Sup_in_sets sets_pair_in_sets ) |
|
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(auto simp add: space_Sup_sigma space_pair_measure) |
|
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qed |
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||
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lemma measurable_pair_iff: |
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"f \<in> measurable M (M1 \<Otimes>\<^sub>M M2) \<longleftrightarrow> (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2" |
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by (auto intro: measurable_pair[of f M M1 M2]) |
40859 | 147 |
|
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lemma measurable_split_conv: |
149 |
"(\<lambda>(x, y). f x y) \<in> measurable A B \<longleftrightarrow> (\<lambda>x. f (fst x) (snd x)) \<in> measurable A B" |
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by (intro arg_cong2[where f="op \<in>"]) auto |
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40859 | 151 |
|
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lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) (M2 \<Otimes>\<^sub>M M1)" |
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by (auto intro!: measurable_Pair simp: measurable_split_conv) |
47694 | 154 |
|
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lemma measurable_pair_swap: |
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assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^sub>M M1) M" |
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using measurable_comp[OF measurable_Pair f] by (auto simp: measurable_split_conv comp_def) |
40859 | 158 |
|
47694 | 159 |
lemma measurable_pair_swap_iff: |
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"f \<in> measurable (M2 \<Otimes>\<^sub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" |
50003 | 161 |
by (auto dest: measurable_pair_swap) |
49776 | 162 |
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lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^sub>M M2)" |
50003 | 164 |
by simp |
40859 | 165 |
|
50003 | 166 |
lemma sets_Pair1[measurable (raw)]: |
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167 |
assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "Pair x -` A \<in> sets M2" |
40859 | 168 |
proof - |
47694 | 169 |
have "Pair x -` A = (if x \<in> space M1 then Pair x -` A \<inter> space M2 else {})" |
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170 |
using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure) |
47694 | 171 |
also have "\<dots> \<in> sets M2" |
172 |
using A by (auto simp add: measurable_Pair1' intro!: measurable_sets split: split_if_asm) |
|
173 |
finally show ?thesis . |
|
40859 | 174 |
qed |
175 |
||
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176 |
lemma measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^sub>M M2)" |
49776 | 177 |
by (auto intro!: measurable_Pair) |
40859 | 178 |
|
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179 |
lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>x. (x, y)) -` A \<in> sets M1" |
47694 | 180 |
proof - |
181 |
have "(\<lambda>x. (x, y)) -` A = (if y \<in> space M2 then (\<lambda>x. (x, y)) -` A \<inter> space M1 else {})" |
|
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182 |
using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure) |
47694 | 183 |
also have "\<dots> \<in> sets M1" |
184 |
using A by (auto simp add: measurable_Pair2' intro!: measurable_sets split: split_if_asm) |
|
185 |
finally show ?thesis . |
|
40859 | 186 |
qed |
187 |
||
47694 | 188 |
lemma measurable_Pair2: |
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189 |
assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and x: "x \<in> space M1" |
47694 | 190 |
shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M" |
191 |
using measurable_comp[OF measurable_Pair1' f, OF x] |
|
192 |
by (simp add: comp_def) |
|
193 |
||
194 |
lemma measurable_Pair1: |
|
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195 |
assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and y: "y \<in> space M2" |
40859 | 196 |
shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M" |
47694 | 197 |
using measurable_comp[OF measurable_Pair2' f, OF y] |
198 |
by (simp add: comp_def) |
|
40859 | 199 |
|
47694 | 200 |
lemma Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}" |
40859 | 201 |
unfolding Int_stable_def |
47694 | 202 |
by safe (auto simp add: times_Int_times) |
40859 | 203 |
|
49776 | 204 |
lemma (in finite_measure) finite_measure_cut_measurable: |
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205 |
assumes [measurable]: "Q \<in> sets (N \<Otimes>\<^sub>M M)" |
49776 | 206 |
shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N" |
40859 | 207 |
(is "?s Q \<in> _") |
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208 |
using Int_stable_pair_measure_generator pair_measure_closed assms |
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|
209 |
unfolding sets_pair_measure |
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210 |
proof (induct rule: sigma_sets_induct_disjoint) |
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211 |
case (compl A) |
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212 |
with sets.sets_into_space have "\<And>x. emeasure M (Pair x -` ((space N \<times> space M) - A)) = |
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213 |
(if x \<in> space N then emeasure M (space M) - ?s A x else 0)" |
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214 |
unfolding sets_pair_measure[symmetric] |
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215 |
by (auto intro!: emeasure_compl simp: vimage_Diff sets_Pair1) |
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216 |
with compl sets.top show ?case |
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217 |
by (auto intro!: measurable_If simp: space_pair_measure) |
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218 |
next |
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|
219 |
case (union F) |
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220 |
then have "\<And>x. emeasure M (Pair x -` (\<Union>i. F i)) = (\<Sum>i. ?s (F i) x)" |
60727 | 221 |
by (simp add: suminf_emeasure disjoint_family_on_vimageI subset_eq vimage_UN sets_pair_measure[symmetric]) |
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222 |
with union show ?case |
50003 | 223 |
unfolding sets_pair_measure[symmetric] by simp |
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224 |
qed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If) |
49776 | 225 |
|
226 |
lemma (in sigma_finite_measure) measurable_emeasure_Pair: |
|
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227 |
assumes Q: "Q \<in> sets (N \<Otimes>\<^sub>M M)" shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N" (is "?s Q \<in> _") |
49776 | 228 |
proof - |
229 |
from sigma_finite_disjoint guess F . note F = this |
|
230 |
then have F_sets: "\<And>i. F i \<in> sets M" by auto |
|
231 |
let ?C = "\<lambda>x i. F i \<inter> Pair x -` Q" |
|
232 |
{ fix i |
|
233 |
have [simp]: "space N \<times> F i \<inter> space N \<times> space M = space N \<times> F i" |
|
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234 |
using F sets.sets_into_space by auto |
49776 | 235 |
let ?R = "density M (indicator (F i))" |
236 |
have "finite_measure ?R" |
|
237 |
using F by (intro finite_measureI) (auto simp: emeasure_restricted subset_eq) |
|
238 |
then have "(\<lambda>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q))) \<in> borel_measurable N" |
|
239 |
by (rule finite_measure.finite_measure_cut_measurable) (auto intro: Q) |
|
240 |
moreover have "\<And>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q)) |
|
241 |
= emeasure M (F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q))" |
|
242 |
using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1) |
|
243 |
moreover have "\<And>x. F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q) = ?C x i" |
|
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244 |
using sets.sets_into_space[OF Q] by (auto simp: space_pair_measure) |
49776 | 245 |
ultimately have "(\<lambda>x. emeasure M (?C x i)) \<in> borel_measurable N" |
246 |
by simp } |
|
247 |
moreover |
|
248 |
{ fix x |
|
249 |
have "(\<Sum>i. emeasure M (?C x i)) = emeasure M (\<Union>i. ?C x i)" |
|
250 |
proof (intro suminf_emeasure) |
|
251 |
show "range (?C x) \<subseteq> sets M" |
|
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252 |
using F `Q \<in> sets (N \<Otimes>\<^sub>M M)` by (auto intro!: sets_Pair1) |
49776 | 253 |
have "disjoint_family F" using F by auto |
254 |
show "disjoint_family (?C x)" |
|
255 |
by (rule disjoint_family_on_bisimulation[OF `disjoint_family F`]) auto |
|
256 |
qed |
|
257 |
also have "(\<Union>i. ?C x i) = Pair x -` Q" |
|
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258 |
using F sets.sets_into_space[OF `Q \<in> sets (N \<Otimes>\<^sub>M M)`] |
49776 | 259 |
by (auto simp: space_pair_measure) |
260 |
finally have "emeasure M (Pair x -` Q) = (\<Sum>i. emeasure M (?C x i))" |
|
261 |
by simp } |
|
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262 |
ultimately show ?thesis using `Q \<in> sets (N \<Otimes>\<^sub>M M)` F_sets |
49776 | 263 |
by auto |
264 |
qed |
|
265 |
||
50003 | 266 |
lemma (in sigma_finite_measure) measurable_emeasure[measurable (raw)]: |
267 |
assumes space: "\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M" |
|
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268 |
assumes A: "{x\<in>space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>M M)" |
50003 | 269 |
shows "(\<lambda>x. emeasure M (A x)) \<in> borel_measurable N" |
270 |
proof - |
|
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|
271 |
from space have "\<And>x. x \<in> space N \<Longrightarrow> Pair x -` {x \<in> space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} = A x" |
50003 | 272 |
by (auto simp: space_pair_measure) |
273 |
with measurable_emeasure_Pair[OF A] show ?thesis |
|
274 |
by (auto cong: measurable_cong) |
|
275 |
qed |
|
276 |
||
49776 | 277 |
lemma (in sigma_finite_measure) emeasure_pair_measure: |
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278 |
assumes "X \<in> sets (N \<Otimes>\<^sub>M M)" |
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|
279 |
shows "emeasure (N \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator X (x, y) \<partial>M \<partial>N)" (is "_ = ?\<mu> X") |
49776 | 280 |
proof (rule emeasure_measure_of[OF pair_measure_def]) |
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|
281 |
show "positive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>" |
56996 | 282 |
by (auto simp: positive_def nn_integral_nonneg) |
49776 | 283 |
have eq[simp]: "\<And>A x y. indicator A (x, y) = indicator (Pair x -` A) y" |
284 |
by (auto simp: indicator_def) |
|
53015
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|
285 |
show "countably_additive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>" |
49776 | 286 |
proof (rule countably_additiveI) |
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|
287 |
fix F :: "nat \<Rightarrow> ('b \<times> 'a) set" assume F: "range F \<subseteq> sets (N \<Otimes>\<^sub>M M)" "disjoint_family F" |
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|
288 |
from F have *: "\<And>i. F i \<in> sets (N \<Otimes>\<^sub>M M)" by auto |
49776 | 289 |
moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)" |
290 |
by (intro disjoint_family_on_bisimulation[OF F(2)]) auto |
|
291 |
moreover have "\<And>x. range (\<lambda>i. Pair x -` F i) \<subseteq> sets M" |
|
292 |
using F by (auto simp: sets_Pair1) |
|
293 |
ultimately show "(\<Sum>n. ?\<mu> (F n)) = ?\<mu> (\<Union>i. F i)" |
|
59353
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|
294 |
by (auto simp add: nn_integral_suminf[symmetric] vimage_UN suminf_emeasure emeasure_nonneg |
56996 | 295 |
intro!: nn_integral_cong nn_integral_indicator[symmetric]) |
49776 | 296 |
qed |
297 |
show "{a \<times> b |a b. a \<in> sets N \<and> b \<in> sets M} \<subseteq> Pow (space N \<times> space M)" |
|
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|
298 |
using sets.space_closed[of N] sets.space_closed[of M] by auto |
49776 | 299 |
qed fact |
300 |
||
301 |
lemma (in sigma_finite_measure) emeasure_pair_measure_alt: |
|
53015
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|
302 |
assumes X: "X \<in> sets (N \<Otimes>\<^sub>M M)" |
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changeset
|
303 |
shows "emeasure (N \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+x. emeasure M (Pair x -` X) \<partial>N)" |
49776 | 304 |
proof - |
305 |
have [simp]: "\<And>x y. indicator X (x, y) = indicator (Pair x -` X) y" |
|
306 |
by (auto simp: indicator_def) |
|
307 |
show ?thesis |
|
56996 | 308 |
using X by (auto intro!: nn_integral_cong simp: emeasure_pair_measure sets_Pair1) |
49776 | 309 |
qed |
310 |
||
311 |
lemma (in sigma_finite_measure) emeasure_pair_measure_Times: |
|
312 |
assumes A: "A \<in> sets N" and B: "B \<in> sets M" |
|
53015
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|
313 |
shows "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B" |
49776 | 314 |
proof - |
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|
315 |
have "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = (\<integral>\<^sup>+x. emeasure M B * indicator A x \<partial>N)" |
56996 | 316 |
using A B by (auto intro!: nn_integral_cong simp: emeasure_pair_measure_alt) |
49776 | 317 |
also have "\<dots> = emeasure M B * emeasure N A" |
56996 | 318 |
using A by (simp add: emeasure_nonneg nn_integral_cmult_indicator) |
49776 | 319 |
finally show ?thesis |
320 |
by (simp add: ac_simps) |
|
40859 | 321 |
qed |
322 |
||
47694 | 323 |
subsection {* Binary products of $\sigma$-finite emeasure spaces *} |
40859 | 324 |
|
47694 | 325 |
locale pair_sigma_finite = M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2 |
326 |
for M1 :: "'a measure" and M2 :: "'b measure" |
|
40859 | 327 |
|
47694 | 328 |
lemma (in pair_sigma_finite) measurable_emeasure_Pair1: |
53015
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329 |
"Q \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1" |
49776 | 330 |
using M2.measurable_emeasure_Pair . |
40859 | 331 |
|
47694 | 332 |
lemma (in pair_sigma_finite) measurable_emeasure_Pair2: |
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|
333 |
assumes Q: "Q \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2" |
40859 | 334 |
proof - |
53015
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|
335 |
have "(\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)" |
47694 | 336 |
using Q measurable_pair_swap' by (auto intro: measurable_sets) |
49776 | 337 |
note M1.measurable_emeasure_Pair[OF this] |
53015
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|
338 |
moreover have "\<And>y. Pair y -` ((\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1)) = (\<lambda>x. (x, y)) -` Q" |
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|
339 |
using Q[THEN sets.sets_into_space] by (auto simp: space_pair_measure) |
47694 | 340 |
ultimately show ?thesis by simp |
39088
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|
341 |
qed |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
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changeset
|
342 |
|
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|
343 |
lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator: |
47694 | 344 |
defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}" |
345 |
shows "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> E \<and> incseq F \<and> (\<Union>i. F i) = space M1 \<times> space M2 \<and> |
|
53015
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|
346 |
(\<forall>i. emeasure (M1 \<Otimes>\<^sub>M M2) (F i) \<noteq> \<infinity>)" |
40859 | 347 |
proof - |
47694 | 348 |
from M1.sigma_finite_incseq guess F1 . note F1 = this |
349 |
from M2.sigma_finite_incseq guess F2 . note F2 = this |
|
350 |
from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto |
|
40859 | 351 |
let ?F = "\<lambda>i. F1 i \<times> F2 i" |
47694 | 352 |
show ?thesis |
40859 | 353 |
proof (intro exI[of _ ?F] conjI allI) |
47694 | 354 |
show "range ?F \<subseteq> E" using F1 F2 by (auto simp: E_def) (metis range_subsetD) |
40859 | 355 |
next |
356 |
have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)" |
|
357 |
proof (intro subsetI) |
|
358 |
fix x assume "x \<in> space M1 \<times> space M2" |
|
359 |
then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j" |
|
360 |
by (auto simp: space) |
|
361 |
then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
362 |
using `incseq F1` `incseq F2` unfolding incseq_def |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
363 |
by (force split: split_max)+ |
40859 | 364 |
then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)" |
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
53374
diff
changeset
|
365 |
by (intro SigmaI) (auto simp add: max.commute) |
40859 | 366 |
then show "x \<in> (\<Union>i. ?F i)" by auto |
367 |
qed |
|
47694 | 368 |
then show "(\<Union>i. ?F i) = space M1 \<times> space M2" |
369 |
using space by (auto simp: space) |
|
40859 | 370 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
371 |
fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
372 |
using `incseq F1` `incseq F2` unfolding incseq_Suc_iff by auto |
40859 | 373 |
next |
374 |
fix i |
|
375 |
from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto |
|
47694 | 376 |
with F1 F2 emeasure_nonneg[of M1 "F1 i"] emeasure_nonneg[of M2 "F2 i"] |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
377 |
show "emeasure (M1 \<Otimes>\<^sub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>" |
47694 | 378 |
by (auto simp add: emeasure_pair_measure_Times) |
379 |
qed |
|
380 |
qed |
|
381 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
382 |
sublocale pair_sigma_finite \<subseteq> P: sigma_finite_measure "M1 \<Otimes>\<^sub>M M2" |
47694 | 383 |
proof |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57235
diff
changeset
|
384 |
from M1.sigma_finite_countable guess F1 .. |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57235
diff
changeset
|
385 |
moreover from M2.sigma_finite_countable guess F2 .. |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57235
diff
changeset
|
386 |
ultimately show |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57235
diff
changeset
|
387 |
"\<exists>A. countable A \<and> A \<subseteq> sets (M1 \<Otimes>\<^sub>M M2) \<and> \<Union>A = space (M1 \<Otimes>\<^sub>M M2) \<and> (\<forall>a\<in>A. emeasure (M1 \<Otimes>\<^sub>M M2) a \<noteq> \<infinity>)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57235
diff
changeset
|
388 |
by (intro exI[of _ "(\<lambda>(a, b). a \<times> b) ` (F1 \<times> F2)"] conjI) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57235
diff
changeset
|
389 |
(auto simp: M2.emeasure_pair_measure_Times space_pair_measure set_eq_iff subset_eq |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57235
diff
changeset
|
390 |
dest: sets.sets_into_space) |
40859 | 391 |
qed |
392 |
||
47694 | 393 |
lemma sigma_finite_pair_measure: |
394 |
assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
395 |
shows "sigma_finite_measure (A \<Otimes>\<^sub>M B)" |
47694 | 396 |
proof - |
397 |
interpret A: sigma_finite_measure A by fact |
|
398 |
interpret B: sigma_finite_measure B by fact |
|
399 |
interpret AB: pair_sigma_finite A B .. |
|
400 |
show ?thesis .. |
|
40859 | 401 |
qed |
39088
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
402 |
|
47694 | 403 |
lemma sets_pair_swap: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
404 |
assumes "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
405 |
shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)" |
47694 | 406 |
using measurable_pair_swap' assms by (rule measurable_sets) |
41661 | 407 |
|
47694 | 408 |
lemma (in pair_sigma_finite) distr_pair_swap: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
409 |
"M1 \<Otimes>\<^sub>M M2 = distr (M2 \<Otimes>\<^sub>M M1) (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D") |
40859 | 410 |
proof - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
411 |
from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this |
47694 | 412 |
let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}" |
413 |
show ?thesis |
|
414 |
proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]]) |
|
415 |
show "?E \<subseteq> Pow (space ?P)" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
416 |
using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure) |
47694 | 417 |
show "sets ?P = sigma_sets (space ?P) ?E" |
418 |
by (simp add: sets_pair_measure space_pair_measure) |
|
419 |
then show "sets ?D = sigma_sets (space ?P) ?E" |
|
420 |
by simp |
|
421 |
next |
|
49784
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49776
diff
changeset
|
422 |
show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>" |
47694 | 423 |
using F by (auto simp: space_pair_measure) |
424 |
next |
|
425 |
fix X assume "X \<in> ?E" |
|
426 |
then obtain A B where X[simp]: "X = A \<times> B" and A: "A \<in> sets M1" and B: "B \<in> sets M2" by auto |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
427 |
have "(\<lambda>(y, x). (x, y)) -` X \<inter> space (M2 \<Otimes>\<^sub>M M1) = B \<times> A" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
428 |
using sets.sets_into_space[OF A] sets.sets_into_space[OF B] by (auto simp: space_pair_measure) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
429 |
with A B show "emeasure (M1 \<Otimes>\<^sub>M M2) X = emeasure ?D X" |
49776 | 430 |
by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_pair_measure_Times emeasure_distr |
47694 | 431 |
measurable_pair_swap' ac_simps) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
432 |
qed |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
433 |
qed |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
434 |
|
47694 | 435 |
lemma (in pair_sigma_finite) emeasure_pair_measure_alt2: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
436 |
assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
437 |
shows "emeasure (M1 \<Otimes>\<^sub>M M2) A = (\<integral>\<^sup>+y. emeasure M1 ((\<lambda>x. (x, y)) -` A) \<partial>M2)" |
47694 | 438 |
(is "_ = ?\<nu> A") |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
439 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
440 |
have [simp]: "\<And>y. (Pair y -` ((\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1))) = (\<lambda>x. (x, y)) -` A" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
441 |
using sets.sets_into_space[OF A] by (auto simp: space_pair_measure) |
47694 | 442 |
show ?thesis using A |
443 |
by (subst distr_pair_swap) |
|
444 |
(simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap'] |
|
49776 | 445 |
M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A]) |
446 |
qed |
|
447 |
||
448 |
lemma (in pair_sigma_finite) AE_pair: |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
449 |
assumes "AE x in (M1 \<Otimes>\<^sub>M M2). Q x" |
49776 | 450 |
shows "AE x in M1. (AE y in M2. Q (x, y))" |
451 |
proof - |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
452 |
obtain N where N: "N \<in> sets (M1 \<Otimes>\<^sub>M M2)" "emeasure (M1 \<Otimes>\<^sub>M M2) N = 0" "{x\<in>space (M1 \<Otimes>\<^sub>M M2). \<not> Q x} \<subseteq> N" |
49776 | 453 |
using assms unfolding eventually_ae_filter by auto |
454 |
show ?thesis |
|
455 |
proof (rule AE_I) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
456 |
from N measurable_emeasure_Pair1[OF `N \<in> sets (M1 \<Otimes>\<^sub>M M2)`] |
49776 | 457 |
show "emeasure M1 {x\<in>space M1. emeasure M2 (Pair x -` N) \<noteq> 0} = 0" |
56996 | 458 |
by (auto simp: M2.emeasure_pair_measure_alt nn_integral_0_iff emeasure_nonneg) |
49776 | 459 |
show "{x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0} \<in> sets M1" |
460 |
by (intro borel_measurable_ereal_neq_const measurable_emeasure_Pair1 N) |
|
461 |
{ fix x assume "x \<in> space M1" "emeasure M2 (Pair x -` N) = 0" |
|
462 |
have "AE y in M2. Q (x, y)" |
|
463 |
proof (rule AE_I) |
|
464 |
show "emeasure M2 (Pair x -` N) = 0" by fact |
|
465 |
show "Pair x -` N \<in> sets M2" using N(1) by (rule sets_Pair1) |
|
466 |
show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N" |
|
467 |
using N `x \<in> space M1` unfolding space_pair_measure by auto |
|
468 |
qed } |
|
469 |
then show "{x \<in> space M1. \<not> (AE y in M2. Q (x, y))} \<subseteq> {x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0}" |
|
470 |
by auto |
|
471 |
qed |
|
472 |
qed |
|
473 |
||
474 |
lemma (in pair_sigma_finite) AE_pair_measure: |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
475 |
assumes "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
49776 | 476 |
assumes ae: "AE x in M1. AE y in M2. P (x, y)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
477 |
shows "AE x in M1 \<Otimes>\<^sub>M M2. P x" |
49776 | 478 |
proof (subst AE_iff_measurable[OF _ refl]) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
479 |
show "{x\<in>space (M1 \<Otimes>\<^sub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
480 |
by (rule sets.sets_Collect) fact |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
481 |
then have "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
482 |
(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} (x, y) \<partial>M2 \<partial>M1)" |
49776 | 483 |
by (simp add: M2.emeasure_pair_measure) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
484 |
also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. 0 \<partial>M2 \<partial>M1)" |
49776 | 485 |
using ae |
56996 | 486 |
apply (safe intro!: nn_integral_cong_AE) |
49776 | 487 |
apply (intro AE_I2) |
56996 | 488 |
apply (safe intro!: nn_integral_cong_AE) |
49776 | 489 |
apply auto |
490 |
done |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
491 |
finally show "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} = 0" by simp |
49776 | 492 |
qed |
493 |
||
494 |
lemma (in pair_sigma_finite) AE_pair_iff: |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
495 |
"{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow> |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
496 |
(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE x in (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x))" |
49776 | 497 |
using AE_pair[of "\<lambda>x. P (fst x) (snd x)"] AE_pair_measure[of "\<lambda>x. P (fst x) (snd x)"] by auto |
498 |
||
499 |
lemma (in pair_sigma_finite) AE_commute: |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
500 |
assumes P: "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
49776 | 501 |
shows "(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE y in M2. AE x in M1. P x y)" |
502 |
proof - |
|
503 |
interpret Q: pair_sigma_finite M2 M1 .. |
|
504 |
have [simp]: "\<And>x. (fst (case x of (x, y) \<Rightarrow> (y, x))) = snd x" "\<And>x. (snd (case x of (x, y) \<Rightarrow> (y, x))) = fst x" |
|
505 |
by auto |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
506 |
have "{x \<in> space (M2 \<Otimes>\<^sub>M M1). P (snd x) (fst x)} = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
507 |
(\<lambda>(x, y). (y, x)) -` {x \<in> space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<inter> space (M2 \<Otimes>\<^sub>M M1)" |
49776 | 508 |
by (auto simp: space_pair_measure) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
509 |
also have "\<dots> \<in> sets (M2 \<Otimes>\<^sub>M M1)" |
49776 | 510 |
by (intro sets_pair_swap P) |
511 |
finally show ?thesis |
|
512 |
apply (subst AE_pair_iff[OF P]) |
|
513 |
apply (subst distr_pair_swap) |
|
514 |
apply (subst AE_distr_iff[OF measurable_pair_swap' P]) |
|
515 |
apply (subst Q.AE_pair_iff) |
|
516 |
apply simp_all |
|
517 |
done |
|
40859 | 518 |
qed |
519 |
||
56994 | 520 |
subsection "Fubinis theorem" |
40859 | 521 |
|
49800 | 522 |
lemma measurable_compose_Pair1: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
523 |
"x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^sub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L" |
50003 | 524 |
by simp |
49800 | 525 |
|
56996 | 526 |
lemma (in sigma_finite_measure) borel_measurable_nn_integral_fst': |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
527 |
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" "\<And>x. 0 \<le> f x" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
528 |
shows "(\<lambda>x. \<integral>\<^sup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1" |
49800 | 529 |
using f proof induct |
530 |
case (cong u v) |
|
49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigma-finite
hoelzl
parents:
49825
diff
changeset
|
531 |
then have "\<And>w x. w \<in> space M1 \<Longrightarrow> x \<in> space M \<Longrightarrow> u (w, x) = v (w, x)" |
49800 | 532 |
by (auto simp: space_pair_measure) |
533 |
show ?case |
|
534 |
apply (subst measurable_cong) |
|
56996 | 535 |
apply (rule nn_integral_cong) |
49800 | 536 |
apply fact+ |
537 |
done |
|
538 |
next |
|
539 |
case (set Q) |
|
540 |
have [simp]: "\<And>x y. indicator Q (x, y) = indicator (Pair x -` Q) y" |
|
541 |
by (auto simp: indicator_def) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
542 |
have "\<And>x. x \<in> space M1 \<Longrightarrow> emeasure M (Pair x -` Q) = \<integral>\<^sup>+ y. indicator Q (x, y) \<partial>M" |
49800 | 543 |
by (simp add: sets_Pair1[OF set]) |
49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigma-finite
hoelzl
parents:
49825
diff
changeset
|
544 |
from this measurable_emeasure_Pair[OF set] show ?case |
49800 | 545 |
by (rule measurable_cong[THEN iffD1]) |
56996 | 546 |
qed (simp_all add: nn_integral_add nn_integral_cmult measurable_compose_Pair1 |
547 |
nn_integral_monotone_convergence_SUP incseq_def le_fun_def |
|
49800 | 548 |
cong: measurable_cong) |
549 |
||
56996 | 550 |
lemma (in sigma_finite_measure) nn_integral_fst': |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
551 |
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" "\<And>x. 0 \<le> f x" |
56996 | 552 |
shows "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>M \<partial>M1) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M) f" (is "?I f = _") |
49800 | 553 |
using f proof induct |
554 |
case (cong u v) |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
555 |
then have "?I u = ?I v" |
56996 | 556 |
by (intro nn_integral_cong) (auto simp: space_pair_measure) |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
557 |
with cong show ?case |
56996 | 558 |
by (simp cong: nn_integral_cong) |
559 |
qed (simp_all add: emeasure_pair_measure nn_integral_cmult nn_integral_add |
|
560 |
nn_integral_monotone_convergence_SUP |
|
561 |
measurable_compose_Pair1 nn_integral_nonneg |
|
562 |
borel_measurable_nn_integral_fst' nn_integral_mono incseq_def le_fun_def |
|
563 |
cong: nn_integral_cong) |
|
40859 | 564 |
|
56996 | 565 |
lemma (in sigma_finite_measure) nn_integral_fst: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
566 |
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" |
56996 | 567 |
shows "(\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x, y) \<partial>M) \<partial>M1) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M) f" |
49800 | 568 |
using f |
56996 | 569 |
borel_measurable_nn_integral_fst'[of "\<lambda>x. max 0 (f x)"] |
570 |
nn_integral_fst'[of "\<lambda>x. max 0 (f x)"] |
|
571 |
unfolding nn_integral_max_0 by auto |
|
40859 | 572 |
|
56996 | 573 |
lemma (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
574 |
"split f \<in> borel_measurable (N \<Otimes>\<^sub>M M) \<Longrightarrow> (\<lambda>x. \<integral>\<^sup>+ y. f x y \<partial>M) \<in> borel_measurable N" |
56996 | 575 |
using borel_measurable_nn_integral_fst'[of "\<lambda>x. max 0 (split f x)" N] |
576 |
by (simp add: nn_integral_max_0) |
|
50003 | 577 |
|
56996 | 578 |
lemma (in pair_sigma_finite) nn_integral_snd: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
579 |
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" |
56996 | 580 |
shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f" |
41661 | 581 |
proof - |
47694 | 582 |
note measurable_pair_swap[OF f] |
56996 | 583 |
from M1.nn_integral_fst[OF this] |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
584 |
have "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1))" |
40859 | 585 |
by simp |
56996 | 586 |
also have "(\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1)) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f" |
47694 | 587 |
by (subst distr_pair_swap) |
56996 | 588 |
(auto simp: nn_integral_distr[OF measurable_pair_swap' f] intro!: nn_integral_cong) |
40859 | 589 |
finally show ?thesis . |
590 |
qed |
|
591 |
||
592 |
lemma (in pair_sigma_finite) Fubini: |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
593 |
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
594 |
shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x, y) \<partial>M2) \<partial>M1)" |
56996 | 595 |
unfolding nn_integral_snd[OF assms] M2.nn_integral_fst[OF assms] .. |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
596 |
|
57235
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
597 |
lemma (in pair_sigma_finite) Fubini': |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
598 |
assumes f: "split f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
599 |
shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f x y \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f x y \<partial>M2) \<partial>M1)" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
600 |
using Fubini[OF f] by simp |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
601 |
|
56994 | 602 |
subsection {* Products on counting spaces, densities and distributions *} |
40859 | 603 |
|
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
604 |
lemma sigma_prod: |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
605 |
assumes X_cover: "\<exists>E\<subseteq>A. countable E \<and> X = \<Union>E" and A: "A \<subseteq> Pow X" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
606 |
assumes Y_cover: "\<exists>E\<subseteq>B. countable E \<and> Y = \<Union>E" and B: "B \<subseteq> Pow Y" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
607 |
shows "sigma X A \<Otimes>\<^sub>M sigma Y B = sigma (X \<times> Y) {a \<times> b | a b. a \<in> A \<and> b \<in> B}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
608 |
(is "?P = ?S") |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
609 |
proof (rule measure_eqI) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
610 |
have [simp]: "snd \<in> X \<times> Y \<rightarrow> Y" "fst \<in> X \<times> Y \<rightarrow> X" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
611 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
612 |
let ?XY = "{{fst -` a \<inter> X \<times> Y | a. a \<in> A}, {snd -` b \<inter> X \<times> Y | b. b \<in> B}}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
613 |
have "sets ?P = |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
614 |
sets (\<Squnion>\<^sub>\<sigma> xy\<in>?XY. sigma (X \<times> Y) xy)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
615 |
by (simp add: vimage_algebra_sigma sets_pair_eq_sets_fst_snd A B) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
616 |
also have "\<dots> = sets (sigma (X \<times> Y) (\<Union>?XY))" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
617 |
by (intro Sup_sigma_sigma arg_cong[where f=sets]) auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
618 |
also have "\<dots> = sets ?S" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
619 |
proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
620 |
show "\<Union>?XY \<subseteq> Pow (X \<times> Y)" "{a \<times> b |a b. a \<in> A \<and> b \<in> B} \<subseteq> Pow (X \<times> Y)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
621 |
using A B by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
622 |
next |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
623 |
interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
624 |
using A B by (intro sigma_algebra_sigma_sets) auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
625 |
fix Z assume "Z \<in> \<Union>?XY" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
626 |
then show "Z \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
627 |
proof safe |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
628 |
fix a assume "a \<in> A" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
629 |
from Y_cover obtain E where E: "E \<subseteq> B" "countable E" and "Y = \<Union>E" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
630 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
631 |
with `a \<in> A` A have eq: "fst -` a \<inter> X \<times> Y = (\<Union>e\<in>E. a \<times> e)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
632 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
633 |
show "fst -` a \<inter> X \<times> Y \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
634 |
using `a \<in> A` E unfolding eq by (auto intro!: XY.countable_UN') |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
635 |
next |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
636 |
fix b assume "b \<in> B" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
637 |
from X_cover obtain E where E: "E \<subseteq> A" "countable E" and "X = \<Union>E" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
638 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
639 |
with `b \<in> B` B have eq: "snd -` b \<inter> X \<times> Y = (\<Union>e\<in>E. e \<times> b)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
640 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
641 |
show "snd -` b \<inter> X \<times> Y \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
642 |
using `b \<in> B` E unfolding eq by (auto intro!: XY.countable_UN') |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
643 |
qed |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
644 |
next |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
645 |
fix Z assume "Z \<in> {a \<times> b |a b. a \<in> A \<and> b \<in> B}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
646 |
then obtain a b where "Z = a \<times> b" and ab: "a \<in> A" "b \<in> B" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
647 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
648 |
then have Z: "Z = (fst -` a \<inter> X \<times> Y) \<inter> (snd -` b \<inter> X \<times> Y)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
649 |
using A B by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
650 |
interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) (\<Union>?XY)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
651 |
by (intro sigma_algebra_sigma_sets) auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
652 |
show "Z \<in> sigma_sets (X \<times> Y) (\<Union>?XY)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
653 |
unfolding Z by (rule XY.Int) (blast intro: ab)+ |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
654 |
qed |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
655 |
finally show "sets ?P = sets ?S" . |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
656 |
next |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
657 |
interpret finite_measure "sigma X A" for X A |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
658 |
proof qed (simp add: emeasure_sigma) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
659 |
fix A assume "A \<in> sets ?P" then show "emeasure ?P A = emeasure ?S A" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
660 |
by (simp add: emeasure_pair_measure_alt emeasure_sigma) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
661 |
qed |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
662 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
663 |
lemma sigma_sets_pair_measure_generator_finite: |
38656 | 664 |
assumes "finite A" and "finite B" |
47694 | 665 |
shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)" |
40859 | 666 |
(is "sigma_sets ?prod ?sets = _") |
38656 | 667 |
proof safe |
668 |
have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product) |
|
669 |
fix x assume subset: "x \<subseteq> A \<times> B" |
|
670 |
hence "finite x" using fin by (rule finite_subset) |
|
40859 | 671 |
from this subset show "x \<in> sigma_sets ?prod ?sets" |
38656 | 672 |
proof (induct x) |
673 |
case empty show ?case by (rule sigma_sets.Empty) |
|
674 |
next |
|
675 |
case (insert a x) |
|
47694 | 676 |
hence "{a} \<in> sigma_sets ?prod ?sets" by auto |
38656 | 677 |
moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto |
678 |
ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un) |
|
679 |
qed |
|
680 |
next |
|
681 |
fix x a b |
|
40859 | 682 |
assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x" |
38656 | 683 |
from sigma_sets_into_sp[OF _ this(1)] this(2) |
40859 | 684 |
show "a \<in> A" and "b \<in> B" by auto |
35833 | 685 |
qed |
686 |
||
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
687 |
lemma borel_prod: |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
688 |
"(borel \<Otimes>\<^sub>M borel) = (borel :: ('a::second_countable_topology \<times> 'b::second_countable_topology) measure)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
689 |
(is "?P = ?B") |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
690 |
proof - |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
691 |
have "?B = sigma UNIV {A \<times> B | A B. open A \<and> open B}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
692 |
by (rule second_countable_borel_measurable[OF open_prod_generated]) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
693 |
also have "\<dots> = ?P" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
694 |
unfolding borel_def |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
695 |
by (subst sigma_prod) (auto intro!: exI[of _ "{UNIV}"]) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
696 |
finally show ?thesis .. |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
697 |
qed |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
698 |
|
47694 | 699 |
lemma pair_measure_count_space: |
700 |
assumes A: "finite A" and B: "finite B" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
701 |
shows "count_space A \<Otimes>\<^sub>M count_space B = count_space (A \<times> B)" (is "?P = ?C") |
47694 | 702 |
proof (rule measure_eqI) |
703 |
interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact |
|
704 |
interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact |
|
705 |
interpret P: pair_sigma_finite "count_space A" "count_space B" by default |
|
706 |
show eq: "sets ?P = sets ?C" |
|
707 |
by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B) |
|
708 |
fix X assume X: "X \<in> sets ?P" |
|
709 |
with eq have X_subset: "X \<subseteq> A \<times> B" by simp |
|
710 |
with A B have fin_Pair: "\<And>x. finite (Pair x -` X)" |
|
711 |
by (intro finite_subset[OF _ B]) auto |
|
712 |
have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B) |
|
713 |
show "emeasure ?P X = emeasure ?C X" |
|
49776 | 714 |
apply (subst B.emeasure_pair_measure_alt[OF X]) |
47694 | 715 |
apply (subst emeasure_count_space) |
716 |
using X_subset apply auto [] |
|
717 |
apply (simp add: fin_Pair emeasure_count_space X_subset fin_X) |
|
56996 | 718 |
apply (subst nn_integral_count_space) |
47694 | 719 |
using A apply simp |
720 |
apply (simp del: real_of_nat_setsum add: real_of_nat_setsum[symmetric]) |
|
721 |
apply (subst card_gt_0_iff) |
|
722 |
apply (simp add: fin_Pair) |
|
723 |
apply (subst card_SigmaI[symmetric]) |
|
724 |
using A apply simp |
|
725 |
using fin_Pair apply simp |
|
726 |
using X_subset apply (auto intro!: arg_cong[where f=card]) |
|
727 |
done |
|
45777
c36637603821
remove unnecessary sublocale instantiations in HOL-Probability (for clarity and speedup); remove Infinite_Product_Measure.product_prob_space which was a duplicate of Probability_Measure.product_prob_space
hoelzl
parents:
44890
diff
changeset
|
728 |
qed |
35833 | 729 |
|
59426
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
730 |
|
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
731 |
lemma emeasure_prod_count_space: |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
732 |
assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M M)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)") |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
733 |
shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator A (x, y) \<partial>?B \<partial>?A)" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
734 |
by (rule emeasure_measure_of[OF pair_measure_def]) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
735 |
(auto simp: countably_additive_def positive_def suminf_indicator nn_integral_nonneg A |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
736 |
nn_integral_suminf[symmetric] dest: sets.sets_into_space) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
737 |
|
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
738 |
lemma emeasure_prod_count_space_single[simp]: "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) {x} = 1" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
739 |
proof - |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
740 |
have [simp]: "\<And>a b x y. indicator {(a, b)} (x, y) = (indicator {a} x * indicator {b} y::ereal)" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
741 |
by (auto split: split_indicator) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
742 |
show ?thesis |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
743 |
by (cases x) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
744 |
(auto simp: emeasure_prod_count_space nn_integral_cmult sets_Pair nn_integral_max_0 one_ereal_def[symmetric]) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
745 |
qed |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
746 |
|
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
747 |
lemma emeasure_count_space_prod_eq: |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
748 |
fixes A :: "('a \<times> 'b) set" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
749 |
assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M count_space UNIV)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)") |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
750 |
shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
751 |
proof - |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
752 |
{ fix A :: "('a \<times> 'b) set" assume "countable A" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
753 |
then have "emeasure (?A \<Otimes>\<^sub>M ?B) (\<Union>a\<in>A. {a}) = (\<integral>\<^sup>+a. emeasure (?A \<Otimes>\<^sub>M ?B) {a} \<partial>count_space A)" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
754 |
by (intro emeasure_UN_countable) (auto simp: sets_Pair disjoint_family_on_def) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
755 |
also have "\<dots> = (\<integral>\<^sup>+a. indicator A a \<partial>count_space UNIV)" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
756 |
by (subst nn_integral_count_space_indicator) auto |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
757 |
finally have "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
758 |
by simp } |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
759 |
note * = this |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
760 |
|
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
761 |
show ?thesis |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
762 |
proof cases |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
763 |
assume "finite A" then show ?thesis |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
764 |
by (intro * countable_finite) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
765 |
next |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
766 |
assume "infinite A" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
767 |
then obtain C where "countable C" and "infinite C" and "C \<subseteq> A" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
768 |
by (auto dest: infinite_countable_subset') |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
769 |
with A have "emeasure (?A \<Otimes>\<^sub>M ?B) C \<le> emeasure (?A \<Otimes>\<^sub>M ?B) A" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
770 |
by (intro emeasure_mono) auto |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
771 |
also have "emeasure (?A \<Otimes>\<^sub>M ?B) C = emeasure (count_space UNIV) C" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
772 |
using `countable C` by (rule *) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
773 |
finally show ?thesis |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
774 |
using `infinite C` `infinite A` by simp |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
775 |
qed |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
776 |
qed |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
777 |
|
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
778 |
lemma nn_intergal_count_space_prod_eq': |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
779 |
assumes [simp]: "\<And>x. 0 \<le> f x" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
780 |
shows "nn_integral (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) f = nn_integral (count_space UNIV) f" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
781 |
(is "nn_integral ?P f = _") |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
782 |
proof cases |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
783 |
assume cntbl: "countable {x. f x \<noteq> 0}" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
784 |
have [simp]: "\<And>x. ereal (real (card ({x} \<inter> {x. f x \<noteq> 0}))) = indicator {x. f x \<noteq> 0} x" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
785 |
by (auto split: split_indicator) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
786 |
have [measurable]: "\<And>y. (\<lambda>x. indicator {y} x) \<in> borel_measurable ?P" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
787 |
by (rule measurable_discrete_difference[of "\<lambda>x. 0" _ borel "{y}" "\<lambda>x. indicator {y} x" for y]) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
788 |
(auto intro: sets_Pair) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
789 |
|
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
790 |
have "(\<integral>\<^sup>+x. f x \<partial>?P) = (\<integral>\<^sup>+x. \<integral>\<^sup>+x'. f x * indicator {x} x' \<partial>count_space {x. f x \<noteq> 0} \<partial>?P)" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
791 |
by (auto simp add: nn_integral_cmult nn_integral_indicator' intro!: nn_integral_cong split: split_indicator) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
792 |
also have "\<dots> = (\<integral>\<^sup>+x. \<integral>\<^sup>+x'. f x' * indicator {x'} x \<partial>count_space {x. f x \<noteq> 0} \<partial>?P)" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
793 |
by (auto intro!: nn_integral_cong split: split_indicator) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
794 |
also have "\<dots> = (\<integral>\<^sup>+x'. \<integral>\<^sup>+x. f x' * indicator {x'} x \<partial>?P \<partial>count_space {x. f x \<noteq> 0})" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
795 |
by (intro nn_integral_count_space_nn_integral cntbl) auto |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
796 |
also have "\<dots> = (\<integral>\<^sup>+x'. f x' \<partial>count_space {x. f x \<noteq> 0})" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
797 |
by (intro nn_integral_cong) (auto simp: nn_integral_cmult sets_Pair) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
798 |
finally show ?thesis |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
799 |
by (auto simp add: nn_integral_count_space_indicator intro!: nn_integral_cong split: split_indicator) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
800 |
next |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
801 |
{ fix x assume "f x \<noteq> 0" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
802 |
with `0 \<le> f x` have "(\<exists>r. 0 < r \<and> f x = ereal r) \<or> f x = \<infinity>" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
803 |
by (cases "f x") (auto simp: less_le) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
804 |
then have "\<exists>n. ereal (1 / real (Suc n)) \<le> f x" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
805 |
by (auto elim!: nat_approx_posE intro!: less_imp_le) } |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
806 |
note * = this |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
807 |
|
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
808 |
assume cntbl: "uncountable {x. f x \<noteq> 0}" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
809 |
also have "{x. f x \<noteq> 0} = (\<Union>n. {x. 1/Suc n \<le> f x})" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
810 |
using * by auto |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
811 |
finally obtain n where "infinite {x. 1/Suc n \<le> f x}" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
812 |
by (meson countableI_type countable_UN uncountable_infinite) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
813 |
then obtain C where C: "C \<subseteq> {x. 1/Suc n \<le> f x}" and "countable C" "infinite C" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
814 |
by (metis infinite_countable_subset') |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
815 |
|
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
816 |
have [measurable]: "C \<in> sets ?P" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
817 |
using sets.countable[OF _ `countable C`, of ?P] by (auto simp: sets_Pair) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
818 |
|
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
819 |
have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>?P) \<le> nn_integral ?P f" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
820 |
using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric]) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
821 |
moreover have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>?P) = \<infinity>" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
822 |
using `infinite C` by (simp add: nn_integral_cmult emeasure_count_space_prod_eq) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
823 |
moreover have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>count_space UNIV) \<le> nn_integral (count_space UNIV) f" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
824 |
using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric]) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
825 |
moreover have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>count_space UNIV) = \<infinity>" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
826 |
using `infinite C` by (simp add: nn_integral_cmult) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
827 |
ultimately show ?thesis |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
828 |
by simp |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
829 |
qed |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
830 |
|
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
831 |
lemma nn_intergal_count_space_prod_eq: |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
832 |
"nn_integral (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) f = nn_integral (count_space UNIV) f" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
833 |
by (subst (1 2) nn_integral_max_0[symmetric]) (auto intro!: nn_intergal_count_space_prod_eq') |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59353
diff
changeset
|
834 |
|
47694 | 835 |
lemma pair_measure_density: |
836 |
assumes f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x" |
|
837 |
assumes g: "g \<in> borel_measurable M2" "AE x in M2. 0 \<le> g x" |
|
50003 | 838 |
assumes "sigma_finite_measure M2" "sigma_finite_measure (density M2 g)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
839 |
shows "density M1 f \<Otimes>\<^sub>M density M2 g = density (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x,y). f x * g y)" (is "?L = ?R") |
47694 | 840 |
proof (rule measure_eqI) |
841 |
interpret M2: sigma_finite_measure M2 by fact |
|
842 |
interpret D2: sigma_finite_measure "density M2 g" by fact |
|
843 |
||
844 |
fix A assume A: "A \<in> sets ?L" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
845 |
with f g have "(\<integral>\<^sup>+ x. f x * \<integral>\<^sup>+ y. g y * indicator A (x, y) \<partial>M2 \<partial>M1) = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
846 |
(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f x * g y * indicator A (x, y) \<partial>M2 \<partial>M1)" |
56996 | 847 |
by (intro nn_integral_cong_AE) |
848 |
(auto simp add: nn_integral_cmult[symmetric] ac_simps) |
|
50003 | 849 |
with A f g show "emeasure ?L A = emeasure ?R A" |
56996 | 850 |
by (simp add: D2.emeasure_pair_measure emeasure_density nn_integral_density |
851 |
M2.nn_integral_fst[symmetric] |
|
852 |
cong: nn_integral_cong) |
|
47694 | 853 |
qed simp |
854 |
||
855 |
lemma sigma_finite_measure_distr: |
|
856 |
assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N" |
|
857 |
shows "sigma_finite_measure M" |
|
40859 | 858 |
proof - |
47694 | 859 |
interpret sigma_finite_measure "distr M N f" by fact |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57235
diff
changeset
|
860 |
from sigma_finite_countable guess A .. note A = this |
47694 | 861 |
show ?thesis |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57235
diff
changeset
|
862 |
proof |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57235
diff
changeset
|
863 |
show "\<exists>A. countable A \<and> A \<subseteq> sets M \<and> \<Union>A = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57235
diff
changeset
|
864 |
using A f |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57235
diff
changeset
|
865 |
by (intro exI[of _ "(\<lambda>a. f -` a \<inter> space M) ` A"]) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57235
diff
changeset
|
866 |
(auto simp: emeasure_distr set_eq_iff subset_eq intro: measurable_space) |
47694 | 867 |
qed |
38656 | 868 |
qed |
869 |
||
47694 | 870 |
lemma pair_measure_distr: |
871 |
assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T" |
|
50003 | 872 |
assumes "sigma_finite_measure (distr N T g)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
873 |
shows "distr M S f \<Otimes>\<^sub>M distr N T g = distr (M \<Otimes>\<^sub>M N) (S \<Otimes>\<^sub>M T) (\<lambda>(x, y). (f x, g y))" (is "?P = ?D") |
47694 | 874 |
proof (rule measure_eqI) |
875 |
interpret T: sigma_finite_measure "distr N T g" by fact |
|
876 |
interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+ |
|
50003 | 877 |
|
47694 | 878 |
fix A assume A: "A \<in> sets ?P" |
50003 | 879 |
with f g show "emeasure ?P A = emeasure ?D A" |
880 |
by (auto simp add: N.emeasure_pair_measure_alt space_pair_measure emeasure_distr |
|
56996 | 881 |
T.emeasure_pair_measure_alt nn_integral_distr |
882 |
intro!: nn_integral_cong arg_cong[where f="emeasure N"]) |
|
50003 | 883 |
qed simp |
39097 | 884 |
|
50104 | 885 |
lemma pair_measure_eqI: |
886 |
assumes "sigma_finite_measure M1" "sigma_finite_measure M2" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
887 |
assumes sets: "sets (M1 \<Otimes>\<^sub>M M2) = sets M" |
50104 | 888 |
assumes emeasure: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> emeasure M1 A * emeasure M2 B = emeasure M (A \<times> B)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
889 |
shows "M1 \<Otimes>\<^sub>M M2 = M" |
50104 | 890 |
proof - |
891 |
interpret M1: sigma_finite_measure M1 by fact |
|
892 |
interpret M2: sigma_finite_measure M2 by fact |
|
893 |
interpret pair_sigma_finite M1 M2 by default |
|
894 |
from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this |
|
895 |
let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
896 |
let ?P = "M1 \<Otimes>\<^sub>M M2" |
50104 | 897 |
show ?thesis |
898 |
proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]]) |
|
899 |
show "?E \<subseteq> Pow (space ?P)" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
900 |
using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure) |
50104 | 901 |
show "sets ?P = sigma_sets (space ?P) ?E" |
902 |
by (simp add: sets_pair_measure space_pair_measure) |
|
903 |
then show "sets M = sigma_sets (space ?P) ?E" |
|
904 |
using sets[symmetric] by simp |
|
905 |
next |
|
906 |
show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>" |
|
907 |
using F by (auto simp: space_pair_measure) |
|
908 |
next |
|
909 |
fix X assume "X \<in> ?E" |
|
910 |
then obtain A B where X[simp]: "X = A \<times> B" and A: "A \<in> sets M1" and B: "B \<in> sets M2" by auto |
|
911 |
then have "emeasure ?P X = emeasure M1 A * emeasure M2 B" |
|
912 |
by (simp add: M2.emeasure_pair_measure_Times) |
|
913 |
also have "\<dots> = emeasure M (A \<times> B)" |
|
914 |
using A B emeasure by auto |
|
915 |
finally show "emeasure ?P X = emeasure M X" |
|
916 |
by simp |
|
917 |
qed |
|
918 |
qed |
|
57025 | 919 |
|
920 |
lemma sets_pair_countable: |
|
921 |
assumes "countable S1" "countable S2" |
|
922 |
assumes M: "sets M = Pow S1" and N: "sets N = Pow S2" |
|
923 |
shows "sets (M \<Otimes>\<^sub>M N) = Pow (S1 \<times> S2)" |
|
924 |
proof auto |
|
925 |
fix x a b assume x: "x \<in> sets (M \<Otimes>\<^sub>M N)" "(a, b) \<in> x" |
|
926 |
from sets.sets_into_space[OF x(1)] x(2) |
|
927 |
sets_eq_imp_space_eq[of N "count_space S2"] sets_eq_imp_space_eq[of M "count_space S1"] M N |
|
928 |
show "a \<in> S1" "b \<in> S2" |
|
929 |
by (auto simp: space_pair_measure) |
|
930 |
next |
|
931 |
fix X assume X: "X \<subseteq> S1 \<times> S2" |
|
932 |
then have "countable X" |
|
933 |
by (metis countable_subset `countable S1` `countable S2` countable_SIGMA) |
|
934 |
have "X = (\<Union>(a, b)\<in>X. {a} \<times> {b})" by auto |
|
935 |
also have "\<dots> \<in> sets (M \<Otimes>\<^sub>M N)" |
|
936 |
using X |
|
937 |
by (safe intro!: sets.countable_UN' `countable X` subsetI pair_measureI) (auto simp: M N) |
|
938 |
finally show "X \<in> sets (M \<Otimes>\<^sub>M N)" . |
|
939 |
qed |
|
940 |
||
941 |
lemma pair_measure_countable: |
|
942 |
assumes "countable S1" "countable S2" |
|
943 |
shows "count_space S1 \<Otimes>\<^sub>M count_space S2 = count_space (S1 \<times> S2)" |
|
944 |
proof (rule pair_measure_eqI) |
|
945 |
show "sigma_finite_measure (count_space S1)" "sigma_finite_measure (count_space S2)" |
|
946 |
using assms by (auto intro!: sigma_finite_measure_count_space_countable) |
|
947 |
show "sets (count_space S1 \<Otimes>\<^sub>M count_space S2) = sets (count_space (S1 \<times> S2))" |
|
948 |
by (subst sets_pair_countable[OF assms]) auto |
|
949 |
next |
|
950 |
fix A B assume "A \<in> sets (count_space S1)" "B \<in> sets (count_space S2)" |
|
951 |
then show "emeasure (count_space S1) A * emeasure (count_space S2) B = |
|
952 |
emeasure (count_space (S1 \<times> S2)) (A \<times> B)" |
|
953 |
by (subst (1 2 3) emeasure_count_space) (auto simp: finite_cartesian_product_iff) |
|
954 |
qed |
|
50104 | 955 |
|
59489
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
956 |
lemma nn_integral_fst_count_space': |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
957 |
assumes nonneg: "\<And>xy. 0 \<le> f xy" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
958 |
shows "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
959 |
(is "?lhs = ?rhs") |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
960 |
proof(cases) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
961 |
assume *: "countable {xy. f xy \<noteq> 0}" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
962 |
let ?A = "fst ` {xy. f xy \<noteq> 0}" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
963 |
let ?B = "snd ` {xy. f xy \<noteq> 0}" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
964 |
from * have [simp]: "countable ?A" "countable ?B" by(rule countable_image)+ |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
965 |
from nonneg have f_neq_0: "\<And>xy. f xy \<noteq> 0 \<longleftrightarrow> f xy > 0" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
966 |
by(auto simp add: order.order_iff_strict) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
967 |
have "?lhs = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space ?A)" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
968 |
by(rule nn_integral_count_space_eq) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
969 |
(auto simp add: f_neq_0 nn_integral_0_iff_AE AE_count_space not_le intro: rev_image_eqI) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
970 |
also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space ?B \<partial>count_space ?A)" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
971 |
by(intro nn_integral_count_space_eq nn_integral_cong)(auto intro: rev_image_eqI) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
972 |
also have "\<dots> = (\<integral>\<^sup>+ xy. f xy \<partial>count_space (?A \<times> ?B))" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
973 |
by(subst sigma_finite_measure.nn_integral_fst) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
974 |
(simp_all add: sigma_finite_measure_count_space_countable pair_measure_countable) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
975 |
also have "\<dots> = ?rhs" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
976 |
by(rule nn_integral_count_space_eq)(auto intro: rev_image_eqI) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
977 |
finally show ?thesis . |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
978 |
next |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
979 |
{ fix xy assume "f xy \<noteq> 0" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
980 |
with `0 \<le> f xy` have "(\<exists>r. 0 < r \<and> f xy = ereal r) \<or> f xy = \<infinity>" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
981 |
by (cases "f xy") (auto simp: less_le) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
982 |
then have "\<exists>n. ereal (1 / real (Suc n)) \<le> f xy" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
983 |
by (auto elim!: nat_approx_posE intro!: less_imp_le) } |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
984 |
note * = this |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
985 |
|
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
986 |
assume cntbl: "uncountable {xy. f xy \<noteq> 0}" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
987 |
also have "{xy. f xy \<noteq> 0} = (\<Union>n. {xy. 1/Suc n \<le> f xy})" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
988 |
using * by auto |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
989 |
finally obtain n where "infinite {xy. 1/Suc n \<le> f xy}" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
990 |
by (meson countableI_type countable_UN uncountable_infinite) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
991 |
then obtain C where C: "C \<subseteq> {xy. 1/Suc n \<le> f xy}" and "countable C" "infinite C" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
992 |
by (metis infinite_countable_subset') |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
993 |
|
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
994 |
have "\<infinity> = (\<integral>\<^sup>+ xy. ereal (1 / Suc n) * indicator C xy \<partial>count_space UNIV)" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
995 |
using \<open>infinite C\<close> by(simp add: nn_integral_cmult) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
996 |
also have "\<dots> \<le> ?rhs" using C |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
997 |
by(intro nn_integral_mono)(auto split: split_indicator simp add: nonneg) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
998 |
finally have "?rhs = \<infinity>" by simp |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
999 |
moreover have "?lhs = \<infinity>" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1000 |
proof(cases "finite (fst ` C)") |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1001 |
case True |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1002 |
then obtain x C' where x: "x \<in> fst ` C" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1003 |
and C': "C' = fst -` {x} \<inter> C" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1004 |
and "infinite C'" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1005 |
using \<open>infinite C\<close> by(auto elim!: inf_img_fin_domE') |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1006 |
from x C C' have **: "C' \<subseteq> {xy. 1 / Suc n \<le> f xy}" by auto |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1007 |
|
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1008 |
from C' \<open>infinite C'\<close> have "infinite (snd ` C')" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1009 |
by(auto dest!: finite_imageD simp add: inj_on_def) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1010 |
then have "\<infinity> = (\<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator (snd ` C') y \<partial>count_space UNIV)" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1011 |
by(simp add: nn_integral_cmult) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1012 |
also have "\<dots> = (\<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV)" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1013 |
by(rule nn_integral_cong)(force split: split_indicator intro: rev_image_eqI simp add: C') |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1014 |
also have "\<dots> = (\<integral>\<^sup>+ x'. (\<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV) * indicator {x} x' \<partial>count_space UNIV)" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1015 |
by(simp add: one_ereal_def[symmetric] nn_integral_nonneg nn_integral_cmult_indicator) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1016 |
also have "\<dots> \<le> (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV \<partial>count_space UNIV)" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1017 |
by(rule nn_integral_mono)(simp split: split_indicator add: nn_integral_nonneg) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1018 |
also have "\<dots> \<le> ?lhs" using ** |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1019 |
by(intro nn_integral_mono)(auto split: split_indicator simp add: nonneg) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1020 |
finally show ?thesis by simp |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1021 |
next |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1022 |
case False |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1023 |
def C' \<equiv> "fst ` C" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1024 |
have "\<infinity> = \<integral>\<^sup>+ x. ereal (1 / Suc n) * indicator C' x \<partial>count_space UNIV" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1025 |
using C'_def False by(simp add: nn_integral_cmult) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1026 |
also have "\<dots> = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C' x * indicator {SOME y. (x, y) \<in> C} y \<partial>count_space UNIV \<partial>count_space UNIV" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1027 |
by(auto simp add: one_ereal_def[symmetric] nn_integral_cmult_indicator intro: nn_integral_cong) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1028 |
also have "\<dots> \<le> \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C (x, y) \<partial>count_space UNIV \<partial>count_space UNIV" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1029 |
by(intro nn_integral_mono)(auto simp add: C'_def split: split_indicator intro: someI) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1030 |
also have "\<dots> \<le> ?lhs" using C |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1031 |
by(intro nn_integral_mono)(auto split: split_indicator simp add: nonneg) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1032 |
finally show ?thesis by simp |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1033 |
qed |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1034 |
ultimately show ?thesis by simp |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1035 |
qed |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1036 |
|
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1037 |
lemma nn_integral_fst_count_space: |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1038 |
"(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f" |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1039 |
by(subst (2 3) nn_integral_max_0[symmetric])(rule nn_integral_fst_count_space', simp) |
fd5d23cc0e97
nn_integral can be split over arbitrary product count_spaces
Andreas Lochbihler
parents:
59426
diff
changeset
|
1040 |
|
59491
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1041 |
lemma nn_integral_snd_count_space: |
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1042 |
"(\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f" |
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1043 |
(is "?lhs = ?rhs") |
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1044 |
proof - |
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1045 |
have "?lhs = (\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. (\<lambda>(y, x). f (x, y)) (y, x) \<partial>count_space UNIV \<partial>count_space UNIV)" |
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1046 |
by(simp) |
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1047 |
also have "\<dots> = \<integral>\<^sup>+ yx. (\<lambda>(y, x). f (x, y)) yx \<partial>count_space UNIV" |
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1048 |
by(rule nn_integral_fst_count_space) |
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1049 |
also have "\<dots> = \<integral>\<^sup>+ xy. f xy \<partial>count_space ((\<lambda>(x, y). (y, x)) ` UNIV)" |
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1050 |
by(subst nn_integral_bij_count_space[OF inj_on_imp_bij_betw, symmetric]) |
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1051 |
(simp_all add: inj_on_def split_def) |
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1052 |
also have "\<dots> = ?rhs" by(rule nn_integral_count_space_eq) auto |
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1053 |
finally show ?thesis . |
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1054 |
qed |
40f570f9a284
add another lemma to split nn_integral over product count_space
Andreas Lochbihler
parents:
59489
diff
changeset
|
1055 |
|
60066 | 1056 |
lemma measurable_pair_measure_countable1: |
1057 |
assumes "countable A" |
|
1058 |
and [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N K" |
|
1059 |
shows "f \<in> measurable (count_space A \<Otimes>\<^sub>M N) K" |
|
1060 |
using _ _ assms(1) |
|
1061 |
by(rule measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A, simplified])simp_all |
|
1062 |
||
57235
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1063 |
subsection {* Product of Borel spaces *} |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1064 |
|
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1065 |
lemma borel_Times: |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1066 |
fixes A :: "'a::topological_space set" and B :: "'b::topological_space set" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1067 |
assumes A: "A \<in> sets borel" and B: "B \<in> sets borel" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1068 |
shows "A \<times> B \<in> sets borel" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1069 |
proof - |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1070 |
have "A \<times> B = (A\<times>UNIV) \<inter> (UNIV \<times> B)" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1071 |
by auto |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1072 |
moreover |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1073 |
{ have "A \<in> sigma_sets UNIV {S. open S}" using A by (simp add: sets_borel) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1074 |
then have "A\<times>UNIV \<in> sets borel" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1075 |
proof (induct A) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1076 |
case (Basic S) then show ?case |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1077 |
by (auto intro!: borel_open open_Times) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1078 |
next |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1079 |
case (Compl A) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1080 |
moreover have *: "(UNIV - A) \<times> UNIV = UNIV - (A \<times> UNIV)" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1081 |
by auto |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1082 |
ultimately show ?case |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1083 |
unfolding * by auto |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1084 |
next |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1085 |
case (Union A) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1086 |
moreover have *: "(UNION UNIV A) \<times> UNIV = UNION UNIV (\<lambda>i. A i \<times> UNIV)" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1087 |
by auto |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1088 |
ultimately show ?case |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1089 |
unfolding * by auto |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1090 |
qed simp } |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1091 |
moreover |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1092 |
{ have "B \<in> sigma_sets UNIV {S. open S}" using B by (simp add: sets_borel) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1093 |
then have "UNIV\<times>B \<in> sets borel" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1094 |
proof (induct B) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1095 |
case (Basic S) then show ?case |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1096 |
by (auto intro!: borel_open open_Times) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1097 |
next |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1098 |
case (Compl B) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1099 |
moreover have *: "UNIV \<times> (UNIV - B) = UNIV - (UNIV \<times> B)" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1100 |
by auto |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1101 |
ultimately show ?case |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1102 |
unfolding * by auto |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1103 |
next |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1104 |
case (Union B) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1105 |
moreover have *: "UNIV \<times> (UNION UNIV B) = UNION UNIV (\<lambda>i. UNIV \<times> B i)" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1106 |
by auto |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1107 |
ultimately show ?case |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1108 |
unfolding * by auto |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1109 |
qed simp } |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1110 |
ultimately show ?thesis |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1111 |
by auto |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1112 |
qed |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1113 |
|
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1114 |
lemma finite_measure_pair_measure: |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1115 |
assumes "finite_measure M" "finite_measure N" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1116 |
shows "finite_measure (N \<Otimes>\<^sub>M M)" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1117 |
proof (rule finite_measureI) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1118 |
interpret M: finite_measure M by fact |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1119 |
interpret N: finite_measure N by fact |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1120 |
show "emeasure (N \<Otimes>\<^sub>M M) (space (N \<Otimes>\<^sub>M M)) \<noteq> \<infinity>" |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1121 |
by (auto simp: space_pair_measure M.emeasure_pair_measure_Times) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1122 |
qed |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57025
diff
changeset
|
1123 |
|
40859 | 1124 |
end |