32 lemma sets_pair_measure: |
32 lemma sets_pair_measure: |
33 "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}" |
33 "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}" |
34 unfolding pair_measure_def using pair_measure_closed[of A B] |
34 unfolding pair_measure_def using pair_measure_closed[of A B] |
35 by (rule sets_measure_of) |
35 by (rule sets_measure_of) |
36 |
36 |
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37 lemma sets_pair_in_sets: |
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38 assumes N: "space A \<times> space B = space N" |
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39 assumes "\<And>a b. a \<in> sets A \<Longrightarrow> b \<in> sets B \<Longrightarrow> a \<times> b \<in> sets N" |
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40 shows "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets N" |
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41 using assms by (auto intro!: sets.sigma_sets_subset simp: sets_pair_measure N) |
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42 |
37 lemma sets_pair_measure_cong[cong]: |
43 lemma sets_pair_measure_cong[cong]: |
38 "sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>M M2')" |
44 "sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>M M2')" |
39 unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq) |
45 unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq) |
40 |
46 |
41 lemma pair_measureI[intro, simp, measurable]: |
47 lemma pair_measureI[intro, simp, measurable]: |
42 "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)" |
48 "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)" |
43 by (auto simp: sets_pair_measure) |
49 by (auto simp: sets_pair_measure) |
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50 |
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51 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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52 using pair_measureI[of "{x}" M1 "{y}" M2] by simp |
44 |
53 |
45 lemma measurable_pair_measureI: |
54 lemma measurable_pair_measureI: |
46 assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2" |
55 assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2" |
47 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" |
56 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" |
48 shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)" |
57 shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)" |
95 lemma |
104 lemma |
96 assumes f[measurable]: "f \<in> measurable M N" |
105 assumes f[measurable]: "f \<in> measurable M N" |
97 shows measurable_fst'': "(\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^sub>M P) N" |
106 shows measurable_fst'': "(\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^sub>M P) N" |
98 and measurable_snd'': "(\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^sub>M M) N" |
107 and measurable_snd'': "(\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^sub>M M) N" |
99 by simp_all |
108 by simp_all |
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109 |
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110 lemma sets_pair_eq_sets_fst_snd: |
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111 "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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112 (is "?P = sets (Sup_sigma {?fst, ?snd})") |
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113 proof - |
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114 { fix a b assume ab: "a \<in> sets A" "b \<in> sets B" |
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115 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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116 by (auto dest: sets.sets_into_space) |
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117 also have "\<dots> \<in> sets (Sup_sigma {?fst, ?snd})" |
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118 using ab by (auto intro: in_Sup_sigma in_vimage_algebra) |
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119 finally have "a \<times> b \<in> sets (Sup_sigma {?fst, ?snd})" . } |
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120 moreover have "sets ?fst \<subseteq> sets (A \<Otimes>\<^sub>M B)" |
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121 by (rule sets_image_in_sets) (auto simp: space_pair_measure[symmetric]) |
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122 moreover have "sets ?snd \<subseteq> sets (A \<Otimes>\<^sub>M B)" |
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123 by (rule sets_image_in_sets) (auto simp: space_pair_measure) |
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124 ultimately show ?thesis |
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125 by (intro antisym[of "sets A" for A] sets_Sup_in_sets sets_pair_in_sets ) |
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126 (auto simp add: space_Sup_sigma space_pair_measure) |
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127 qed |
100 |
128 |
101 lemma measurable_pair_iff: |
129 lemma measurable_pair_iff: |
102 "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" |
130 "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" |
103 by (auto intro: measurable_pair[of f M M1 M2]) |
131 by (auto intro: measurable_pair[of f M M1 M2]) |
104 |
132 |