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1 (* Title: HOL/Analysis/Binary_Product_Measure.thy |
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2 Author: Johannes Hölzl, TU München |
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3 *) |
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4 |
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5 section \<open>Binary product measures\<close> |
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6 |
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7 theory Binary_Product_Measure |
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8 imports Nonnegative_Lebesgue_Integration |
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9 begin |
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10 |
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11 lemma Pair_vimage_times[simp]: "Pair x -` (A \<times> B) = (if x \<in> A then B else {})" |
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12 by auto |
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13 |
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14 lemma rev_Pair_vimage_times[simp]: "(\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})" |
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15 by auto |
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16 |
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17 subsection "Binary products" |
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18 |
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19 definition pair_measure (infixr "\<Otimes>\<^sub>M" 80) where |
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20 "A \<Otimes>\<^sub>M B = measure_of (space A \<times> space B) |
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21 {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} |
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22 (\<lambda>X. \<integral>\<^sup>+x. (\<integral>\<^sup>+y. indicator X (x,y) \<partial>B) \<partial>A)" |
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23 |
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24 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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25 using sets.space_closed[of A] sets.space_closed[of B] by auto |
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26 |
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27 lemma space_pair_measure: |
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28 "space (A \<Otimes>\<^sub>M B) = space A \<times> space B" |
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29 unfolding pair_measure_def using pair_measure_closed[of A B] |
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30 by (rule space_measure_of) |
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31 |
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32 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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33 by (auto simp: space_pair_measure) |
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34 |
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35 lemma sets_pair_measure: |
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36 "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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37 unfolding pair_measure_def using pair_measure_closed[of A B] |
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38 by (rule sets_measure_of) |
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39 |
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40 lemma sets_pair_measure_cong[measurable_cong, cong]: |
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41 "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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42 unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq) |
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43 |
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44 lemma pair_measureI[intro, simp, measurable]: |
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45 "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)" |
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46 by (auto simp: sets_pair_measure) |
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47 |
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48 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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49 using pair_measureI[of "{x}" M1 "{y}" M2] by simp |
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50 |
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51 lemma measurable_pair_measureI: |
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52 assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2" |
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53 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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54 shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)" |
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55 unfolding pair_measure_def using 1 2 |
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56 by (intro measurable_measure_of) (auto dest: sets.sets_into_space) |
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57 |
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58 lemma measurable_split_replace[measurable (raw)]: |
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59 "(\<lambda>x. f x (fst (g x)) (snd (g x))) \<in> measurable M N \<Longrightarrow> (\<lambda>x. case_prod (f x) (g x)) \<in> measurable M N" |
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60 unfolding split_beta' . |
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61 |
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62 lemma measurable_Pair[measurable (raw)]: |
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63 assumes f: "f \<in> measurable M M1" and g: "g \<in> measurable M M2" |
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64 shows "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)" |
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65 proof (rule measurable_pair_measureI) |
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66 show "(\<lambda>x. (f x, g x)) \<in> space M \<rightarrow> space M1 \<times> space M2" |
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67 using f g by (auto simp: measurable_def) |
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68 fix A B assume *: "A \<in> sets M1" "B \<in> sets M2" |
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69 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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70 by auto |
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71 also have "\<dots> \<in> sets M" |
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72 by (rule sets.Int) (auto intro!: measurable_sets * f g) |
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73 finally show "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M \<in> sets M" . |
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74 qed |
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75 |
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76 lemma measurable_fst[intro!, simp, measurable]: "fst \<in> measurable (M1 \<Otimes>\<^sub>M M2) M1" |
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77 by (auto simp: fst_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times |
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78 measurable_def) |
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79 |
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80 lemma measurable_snd[intro!, simp, measurable]: "snd \<in> measurable (M1 \<Otimes>\<^sub>M M2) M2" |
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81 by (auto simp: snd_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times |
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82 measurable_def) |
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83 |
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84 lemma measurable_Pair_compose_split[measurable_dest]: |
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85 assumes f: "case_prod f \<in> measurable (M1 \<Otimes>\<^sub>M M2) N" |
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86 assumes g: "g \<in> measurable M M1" and h: "h \<in> measurable M M2" |
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87 shows "(\<lambda>x. f (g x) (h x)) \<in> measurable M N" |
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88 using measurable_compose[OF measurable_Pair f, OF g h] by simp |
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89 |
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90 lemma measurable_Pair1_compose[measurable_dest]: |
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91 assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)" |
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92 assumes [measurable]: "h \<in> measurable N M" |
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93 shows "(\<lambda>x. f (h x)) \<in> measurable N M1" |
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94 using measurable_compose[OF f measurable_fst] by simp |
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95 |
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96 lemma measurable_Pair2_compose[measurable_dest]: |
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97 assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)" |
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98 assumes [measurable]: "h \<in> measurable N M" |
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99 shows "(\<lambda>x. g (h x)) \<in> measurable N M2" |
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100 using measurable_compose[OF f measurable_snd] by simp |
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101 |
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102 lemma measurable_pair: |
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103 assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2" |
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104 shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)" |
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105 using measurable_Pair[OF assms] by simp |
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106 |
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107 lemma |
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108 assumes f[measurable]: "f \<in> measurable M (N \<Otimes>\<^sub>M P)" |
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109 shows measurable_fst': "(\<lambda>x. fst (f x)) \<in> measurable M N" |
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110 and measurable_snd': "(\<lambda>x. snd (f x)) \<in> measurable M P" |
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111 by simp_all |
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112 |
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113 lemma |
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114 assumes f[measurable]: "f \<in> measurable M N" |
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115 shows measurable_fst'': "(\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^sub>M P) N" |
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116 and measurable_snd'': "(\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^sub>M M) N" |
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117 by simp_all |
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118 |
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119 lemma sets_pair_in_sets: |
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120 assumes "\<And>a b. a \<in> sets A \<Longrightarrow> b \<in> sets B \<Longrightarrow> a \<times> b \<in> sets N" |
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121 shows "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets N" |
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122 unfolding sets_pair_measure |
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123 by (intro sets.sigma_sets_subset') (auto intro!: assms) |
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124 |
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125 lemma sets_pair_eq_sets_fst_snd: |
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126 "sets (A \<Otimes>\<^sub>M B) = sets (Sup {vimage_algebra (space A \<times> space B) fst A, vimage_algebra (space A \<times> space B) snd B})" |
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127 (is "?P = sets (Sup {?fst, ?snd})") |
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128 proof - |
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129 { fix a b assume ab: "a \<in> sets A" "b \<in> sets B" |
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130 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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131 by (auto dest: sets.sets_into_space) |
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132 also have "\<dots> \<in> sets (Sup {?fst, ?snd})" |
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133 apply (rule sets.Int) |
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134 apply (rule in_sets_Sup) |
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135 apply auto [] |
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136 apply (rule insertI1) |
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137 apply (auto intro: ab in_vimage_algebra) [] |
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138 apply (rule in_sets_Sup) |
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139 apply auto [] |
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140 apply (rule insertI2) |
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141 apply (auto intro: ab in_vimage_algebra) |
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142 done |
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143 finally have "a \<times> b \<in> sets (Sup {?fst, ?snd})" . } |
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144 moreover have "sets ?fst \<subseteq> sets (A \<Otimes>\<^sub>M B)" |
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145 by (rule sets_image_in_sets) (auto simp: space_pair_measure[symmetric]) |
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146 moreover have "sets ?snd \<subseteq> sets (A \<Otimes>\<^sub>M B)" |
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147 by (rule sets_image_in_sets) (auto simp: space_pair_measure) |
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148 ultimately show ?thesis |
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149 apply (intro antisym[of "sets A" for A] sets_Sup_in_sets sets_pair_in_sets) |
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150 apply simp |
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151 apply simp |
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152 apply simp |
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153 apply (elim disjE) |
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154 apply (simp add: space_pair_measure) |
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155 apply (simp add: space_pair_measure) |
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156 apply (auto simp add: space_pair_measure) |
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157 done |
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158 qed |
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159 |
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160 lemma measurable_pair_iff: |
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161 "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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162 by (auto intro: measurable_pair[of f M M1 M2]) |
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163 |
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164 lemma measurable_split_conv: |
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165 "(\<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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166 by (intro arg_cong2[where f="op \<in>"]) auto |
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167 |
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168 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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169 by (auto intro!: measurable_Pair simp: measurable_split_conv) |
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170 |
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171 lemma measurable_pair_swap: |
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172 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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173 using measurable_comp[OF measurable_Pair f] by (auto simp: measurable_split_conv comp_def) |
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174 |
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175 lemma measurable_pair_swap_iff: |
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176 "f \<in> measurable (M2 \<Otimes>\<^sub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" |
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177 by (auto dest: measurable_pair_swap) |
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178 |
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179 lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^sub>M M2)" |
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180 by simp |
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181 |
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182 lemma sets_Pair1[measurable (raw)]: |
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183 assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "Pair x -` A \<in> sets M2" |
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184 proof - |
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185 have "Pair x -` A = (if x \<in> space M1 then Pair x -` A \<inter> space M2 else {})" |
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186 using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure) |
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187 also have "\<dots> \<in> sets M2" |
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188 using A by (auto simp add: measurable_Pair1' intro!: measurable_sets split: if_split_asm) |
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189 finally show ?thesis . |
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190 qed |
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191 |
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192 lemma measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^sub>M M2)" |
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193 by (auto intro!: measurable_Pair) |
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194 |
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195 lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>x. (x, y)) -` A \<in> sets M1" |
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196 proof - |
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197 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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198 using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure) |
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199 also have "\<dots> \<in> sets M1" |
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200 using A by (auto simp add: measurable_Pair2' intro!: measurable_sets split: if_split_asm) |
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201 finally show ?thesis . |
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202 qed |
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203 |
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204 lemma measurable_Pair2: |
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205 assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and x: "x \<in> space M1" |
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206 shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M" |
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207 using measurable_comp[OF measurable_Pair1' f, OF x] |
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208 by (simp add: comp_def) |
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209 |
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210 lemma measurable_Pair1: |
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211 assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and y: "y \<in> space M2" |
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212 shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M" |
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213 using measurable_comp[OF measurable_Pair2' f, OF y] |
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214 by (simp add: comp_def) |
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215 |
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216 lemma Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}" |
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217 unfolding Int_stable_def |
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218 by safe (auto simp add: times_Int_times) |
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219 |
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220 lemma (in finite_measure) finite_measure_cut_measurable: |
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221 assumes [measurable]: "Q \<in> sets (N \<Otimes>\<^sub>M M)" |
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222 shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N" |
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223 (is "?s Q \<in> _") |
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224 using Int_stable_pair_measure_generator pair_measure_closed assms |
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225 unfolding sets_pair_measure |
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226 proof (induct rule: sigma_sets_induct_disjoint) |
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227 case (compl A) |
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228 with sets.sets_into_space have "\<And>x. emeasure M (Pair x -` ((space N \<times> space M) - A)) = |
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229 (if x \<in> space N then emeasure M (space M) - ?s A x else 0)" |
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230 unfolding sets_pair_measure[symmetric] |
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231 by (auto intro!: emeasure_compl simp: vimage_Diff sets_Pair1) |
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232 with compl sets.top show ?case |
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233 by (auto intro!: measurable_If simp: space_pair_measure) |
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234 next |
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235 case (union F) |
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236 then have "\<And>x. emeasure M (Pair x -` (\<Union>i. F i)) = (\<Sum>i. ?s (F i) x)" |
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237 by (simp add: suminf_emeasure disjoint_family_on_vimageI subset_eq vimage_UN sets_pair_measure[symmetric]) |
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238 with union show ?case |
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239 unfolding sets_pair_measure[symmetric] by simp |
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240 qed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If) |
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241 |
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242 lemma (in sigma_finite_measure) measurable_emeasure_Pair: |
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243 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> _") |
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244 proof - |
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245 from sigma_finite_disjoint guess F . note F = this |
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246 then have F_sets: "\<And>i. F i \<in> sets M" by auto |
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247 let ?C = "\<lambda>x i. F i \<inter> Pair x -` Q" |
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248 { fix i |
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249 have [simp]: "space N \<times> F i \<inter> space N \<times> space M = space N \<times> F i" |
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250 using F sets.sets_into_space by auto |
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251 let ?R = "density M (indicator (F i))" |
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252 have "finite_measure ?R" |
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253 using F by (intro finite_measureI) (auto simp: emeasure_restricted subset_eq) |
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254 then have "(\<lambda>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q))) \<in> borel_measurable N" |
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255 by (rule finite_measure.finite_measure_cut_measurable) (auto intro: Q) |
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256 moreover have "\<And>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q)) |
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257 = emeasure M (F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q))" |
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258 using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1) |
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259 moreover have "\<And>x. F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q) = ?C x i" |
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260 using sets.sets_into_space[OF Q] by (auto simp: space_pair_measure) |
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261 ultimately have "(\<lambda>x. emeasure M (?C x i)) \<in> borel_measurable N" |
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262 by simp } |
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263 moreover |
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264 { fix x |
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265 have "(\<Sum>i. emeasure M (?C x i)) = emeasure M (\<Union>i. ?C x i)" |
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266 proof (intro suminf_emeasure) |
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267 show "range (?C x) \<subseteq> sets M" |
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268 using F \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close> by (auto intro!: sets_Pair1) |
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269 have "disjoint_family F" using F by auto |
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270 show "disjoint_family (?C x)" |
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271 by (rule disjoint_family_on_bisimulation[OF \<open>disjoint_family F\<close>]) auto |
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272 qed |
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273 also have "(\<Union>i. ?C x i) = Pair x -` Q" |
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274 using F sets.sets_into_space[OF \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close>] |
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275 by (auto simp: space_pair_measure) |
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276 finally have "emeasure M (Pair x -` Q) = (\<Sum>i. emeasure M (?C x i))" |
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277 by simp } |
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278 ultimately show ?thesis using \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close> F_sets |
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279 by auto |
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280 qed |
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281 |
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282 lemma (in sigma_finite_measure) measurable_emeasure[measurable (raw)]: |
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283 assumes space: "\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M" |
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284 assumes A: "{x\<in>space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>M M)" |
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285 shows "(\<lambda>x. emeasure M (A x)) \<in> borel_measurable N" |
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286 proof - |
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287 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" |
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288 by (auto simp: space_pair_measure) |
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289 with measurable_emeasure_Pair[OF A] show ?thesis |
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290 by (auto cong: measurable_cong) |
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291 qed |
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292 |
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293 lemma (in sigma_finite_measure) emeasure_pair_measure: |
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294 assumes "X \<in> sets (N \<Otimes>\<^sub>M M)" |
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295 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") |
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296 proof (rule emeasure_measure_of[OF pair_measure_def]) |
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297 show "positive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>" |
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298 by (auto simp: positive_def) |
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299 have eq[simp]: "\<And>A x y. indicator A (x, y) = indicator (Pair x -` A) y" |
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300 by (auto simp: indicator_def) |
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301 show "countably_additive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>" |
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302 proof (rule countably_additiveI) |
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303 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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304 from F have *: "\<And>i. F i \<in> sets (N \<Otimes>\<^sub>M M)" by auto |
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305 moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)" |
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306 by (intro disjoint_family_on_bisimulation[OF F(2)]) auto |
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307 moreover have "\<And>x. range (\<lambda>i. Pair x -` F i) \<subseteq> sets M" |
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308 using F by (auto simp: sets_Pair1) |
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309 ultimately show "(\<Sum>n. ?\<mu> (F n)) = ?\<mu> (\<Union>i. F i)" |
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310 by (auto simp add: nn_integral_suminf[symmetric] vimage_UN suminf_emeasure |
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311 intro!: nn_integral_cong nn_integral_indicator[symmetric]) |
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312 qed |
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313 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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314 using sets.space_closed[of N] sets.space_closed[of M] by auto |
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315 qed fact |
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316 |
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317 lemma (in sigma_finite_measure) emeasure_pair_measure_alt: |
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318 assumes X: "X \<in> sets (N \<Otimes>\<^sub>M M)" |
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319 shows "emeasure (N \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+x. emeasure M (Pair x -` X) \<partial>N)" |
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320 proof - |
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321 have [simp]: "\<And>x y. indicator X (x, y) = indicator (Pair x -` X) y" |
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322 by (auto simp: indicator_def) |
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323 show ?thesis |
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324 using X by (auto intro!: nn_integral_cong simp: emeasure_pair_measure sets_Pair1) |
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325 qed |
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326 |
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327 lemma (in sigma_finite_measure) emeasure_pair_measure_Times: |
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328 assumes A: "A \<in> sets N" and B: "B \<in> sets M" |
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329 shows "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B" |
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330 proof - |
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331 have "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = (\<integral>\<^sup>+x. emeasure M B * indicator A x \<partial>N)" |
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332 using A B by (auto intro!: nn_integral_cong simp: emeasure_pair_measure_alt) |
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333 also have "\<dots> = emeasure M B * emeasure N A" |
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334 using A by (simp add: nn_integral_cmult_indicator) |
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335 finally show ?thesis |
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336 by (simp add: ac_simps) |
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337 qed |
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338 |
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339 subsection \<open>Binary products of $\sigma$-finite emeasure spaces\<close> |
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340 |
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341 locale pair_sigma_finite = M1?: sigma_finite_measure M1 + M2?: sigma_finite_measure M2 |
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342 for M1 :: "'a measure" and M2 :: "'b measure" |
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343 |
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344 lemma (in pair_sigma_finite) measurable_emeasure_Pair1: |
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345 "Q \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1" |
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346 using M2.measurable_emeasure_Pair . |
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347 |
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348 lemma (in pair_sigma_finite) measurable_emeasure_Pair2: |
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349 assumes Q: "Q \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2" |
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350 proof - |
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351 have "(\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)" |
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352 using Q measurable_pair_swap' by (auto intro: measurable_sets) |
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353 note M1.measurable_emeasure_Pair[OF this] |
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354 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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355 using Q[THEN sets.sets_into_space] by (auto simp: space_pair_measure) |
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356 ultimately show ?thesis by simp |
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357 qed |
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358 |
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359 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator: |
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360 defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}" |
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361 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> |
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362 (\<forall>i. emeasure (M1 \<Otimes>\<^sub>M M2) (F i) \<noteq> \<infinity>)" |
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363 proof - |
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364 from M1.sigma_finite_incseq guess F1 . note F1 = this |
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365 from M2.sigma_finite_incseq guess F2 . note F2 = this |
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366 from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto |
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367 let ?F = "\<lambda>i. F1 i \<times> F2 i" |
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368 show ?thesis |
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369 proof (intro exI[of _ ?F] conjI allI) |
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370 show "range ?F \<subseteq> E" using F1 F2 by (auto simp: E_def) (metis range_subsetD) |
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371 next |
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372 have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)" |
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373 proof (intro subsetI) |
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374 fix x assume "x \<in> space M1 \<times> space M2" |
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375 then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j" |
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376 by (auto simp: space) |
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377 then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)" |
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378 using \<open>incseq F1\<close> \<open>incseq F2\<close> unfolding incseq_def |
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379 by (force split: split_max)+ |
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380 then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)" |
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381 by (intro SigmaI) (auto simp add: max.commute) |
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382 then show "x \<in> (\<Union>i. ?F i)" by auto |
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383 qed |
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384 then show "(\<Union>i. ?F i) = space M1 \<times> space M2" |
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385 using space by (auto simp: space) |
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386 next |
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387 fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)" |
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388 using \<open>incseq F1\<close> \<open>incseq F2\<close> unfolding incseq_Suc_iff by auto |
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389 next |
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390 fix i |
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391 from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto |
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392 with F1 F2 show "emeasure (M1 \<Otimes>\<^sub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>" |
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393 by (auto simp add: emeasure_pair_measure_Times ennreal_mult_eq_top_iff) |
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394 qed |
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395 qed |
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396 |
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397 sublocale pair_sigma_finite \<subseteq> P?: sigma_finite_measure "M1 \<Otimes>\<^sub>M M2" |
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398 proof |
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399 from M1.sigma_finite_countable guess F1 .. |
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400 moreover from M2.sigma_finite_countable guess F2 .. |
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401 ultimately show |
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402 "\<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>)" |
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403 by (intro exI[of _ "(\<lambda>(a, b). a \<times> b) ` (F1 \<times> F2)"] conjI) |
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404 (auto simp: M2.emeasure_pair_measure_Times space_pair_measure set_eq_iff subset_eq ennreal_mult_eq_top_iff) |
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405 qed |
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406 |
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407 lemma sigma_finite_pair_measure: |
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408 assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B" |
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409 shows "sigma_finite_measure (A \<Otimes>\<^sub>M B)" |
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410 proof - |
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411 interpret A: sigma_finite_measure A by fact |
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412 interpret B: sigma_finite_measure B by fact |
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413 interpret AB: pair_sigma_finite A B .. |
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414 show ?thesis .. |
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415 qed |
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416 |
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417 lemma sets_pair_swap: |
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418 assumes "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
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419 shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)" |
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420 using measurable_pair_swap' assms by (rule measurable_sets) |
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421 |
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422 lemma (in pair_sigma_finite) distr_pair_swap: |
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423 "M1 \<Otimes>\<^sub>M M2 = distr (M2 \<Otimes>\<^sub>M M1) (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D") |
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424 proof - |
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425 from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this |
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426 let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}" |
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427 show ?thesis |
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428 proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]]) |
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429 show "?E \<subseteq> Pow (space ?P)" |
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430 using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure) |
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431 show "sets ?P = sigma_sets (space ?P) ?E" |
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432 by (simp add: sets_pair_measure space_pair_measure) |
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433 then show "sets ?D = sigma_sets (space ?P) ?E" |
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434 by simp |
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435 next |
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436 show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>" |
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437 using F by (auto simp: space_pair_measure) |
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438 next |
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439 fix X assume "X \<in> ?E" |
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440 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 |
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441 have "(\<lambda>(y, x). (x, y)) -` X \<inter> space (M2 \<Otimes>\<^sub>M M1) = B \<times> A" |
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442 using sets.sets_into_space[OF A] sets.sets_into_space[OF B] by (auto simp: space_pair_measure) |
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443 with A B show "emeasure (M1 \<Otimes>\<^sub>M M2) X = emeasure ?D X" |
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444 by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_pair_measure_Times emeasure_distr |
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445 measurable_pair_swap' ac_simps) |
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446 qed |
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447 qed |
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448 |
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449 lemma (in pair_sigma_finite) emeasure_pair_measure_alt2: |
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450 assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
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451 shows "emeasure (M1 \<Otimes>\<^sub>M M2) A = (\<integral>\<^sup>+y. emeasure M1 ((\<lambda>x. (x, y)) -` A) \<partial>M2)" |
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452 (is "_ = ?\<nu> A") |
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453 proof - |
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454 have [simp]: "\<And>y. (Pair y -` ((\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1))) = (\<lambda>x. (x, y)) -` A" |
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455 using sets.sets_into_space[OF A] by (auto simp: space_pair_measure) |
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456 show ?thesis using A |
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457 by (subst distr_pair_swap) |
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458 (simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap'] |
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459 M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A]) |
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460 qed |
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461 |
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462 lemma (in pair_sigma_finite) AE_pair: |
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463 assumes "AE x in (M1 \<Otimes>\<^sub>M M2). Q x" |
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464 shows "AE x in M1. (AE y in M2. Q (x, y))" |
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465 proof - |
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466 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" |
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467 using assms unfolding eventually_ae_filter by auto |
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468 show ?thesis |
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469 proof (rule AE_I) |
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470 from N measurable_emeasure_Pair1[OF \<open>N \<in> sets (M1 \<Otimes>\<^sub>M M2)\<close>] |
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471 show "emeasure M1 {x\<in>space M1. emeasure M2 (Pair x -` N) \<noteq> 0} = 0" |
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472 by (auto simp: M2.emeasure_pair_measure_alt nn_integral_0_iff) |
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473 show "{x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0} \<in> sets M1" |
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474 by (intro borel_measurable_eq measurable_emeasure_Pair1 N sets.sets_Collect_neg N) simp |
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475 { fix x assume "x \<in> space M1" "emeasure M2 (Pair x -` N) = 0" |
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476 have "AE y in M2. Q (x, y)" |
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477 proof (rule AE_I) |
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478 show "emeasure M2 (Pair x -` N) = 0" by fact |
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479 show "Pair x -` N \<in> sets M2" using N(1) by (rule sets_Pair1) |
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480 show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N" |
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481 using N \<open>x \<in> space M1\<close> unfolding space_pair_measure by auto |
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482 qed } |
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483 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}" |
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484 by auto |
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485 qed |
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486 qed |
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487 |
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488 lemma (in pair_sigma_finite) AE_pair_measure: |
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489 assumes "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
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490 assumes ae: "AE x in M1. AE y in M2. P (x, y)" |
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491 shows "AE x in M1 \<Otimes>\<^sub>M M2. P x" |
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492 proof (subst AE_iff_measurable[OF _ refl]) |
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493 show "{x\<in>space (M1 \<Otimes>\<^sub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
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494 by (rule sets.sets_Collect) fact |
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495 then have "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} = |
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496 (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} (x, y) \<partial>M2 \<partial>M1)" |
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497 by (simp add: M2.emeasure_pair_measure) |
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498 also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. 0 \<partial>M2 \<partial>M1)" |
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499 using ae |
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500 apply (safe intro!: nn_integral_cong_AE) |
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501 apply (intro AE_I2) |
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502 apply (safe intro!: nn_integral_cong_AE) |
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503 apply auto |
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504 done |
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505 finally show "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} = 0" by simp |
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506 qed |
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507 |
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508 lemma (in pair_sigma_finite) AE_pair_iff: |
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509 "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow> |
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510 (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))" |
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511 using AE_pair[of "\<lambda>x. P (fst x) (snd x)"] AE_pair_measure[of "\<lambda>x. P (fst x) (snd x)"] by auto |
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512 |
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513 lemma (in pair_sigma_finite) AE_commute: |
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514 assumes P: "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
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515 shows "(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE y in M2. AE x in M1. P x y)" |
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516 proof - |
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517 interpret Q: pair_sigma_finite M2 M1 .. |
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518 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" |
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519 by auto |
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520 have "{x \<in> space (M2 \<Otimes>\<^sub>M M1). P (snd x) (fst x)} = |
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521 (\<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)" |
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522 by (auto simp: space_pair_measure) |
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523 also have "\<dots> \<in> sets (M2 \<Otimes>\<^sub>M M1)" |
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524 by (intro sets_pair_swap P) |
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525 finally show ?thesis |
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526 apply (subst AE_pair_iff[OF P]) |
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527 apply (subst distr_pair_swap) |
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528 apply (subst AE_distr_iff[OF measurable_pair_swap' P]) |
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529 apply (subst Q.AE_pair_iff) |
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530 apply simp_all |
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531 done |
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532 qed |
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533 |
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534 subsection "Fubinis theorem" |
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535 |
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536 lemma measurable_compose_Pair1: |
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537 "x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^sub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L" |
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538 by simp |
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539 |
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540 lemma (in sigma_finite_measure) borel_measurable_nn_integral_fst: |
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541 assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" |
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542 shows "(\<lambda>x. \<integral>\<^sup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1" |
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543 using f proof induct |
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544 case (cong u v) |
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545 then have "\<And>w x. w \<in> space M1 \<Longrightarrow> x \<in> space M \<Longrightarrow> u (w, x) = v (w, x)" |
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546 by (auto simp: space_pair_measure) |
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547 show ?case |
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548 apply (subst measurable_cong) |
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549 apply (rule nn_integral_cong) |
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550 apply fact+ |
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551 done |
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552 next |
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553 case (set Q) |
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554 have [simp]: "\<And>x y. indicator Q (x, y) = indicator (Pair x -` Q) y" |
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555 by (auto simp: indicator_def) |
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556 have "\<And>x. x \<in> space M1 \<Longrightarrow> emeasure M (Pair x -` Q) = \<integral>\<^sup>+ y. indicator Q (x, y) \<partial>M" |
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557 by (simp add: sets_Pair1[OF set]) |
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558 from this measurable_emeasure_Pair[OF set] show ?case |
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559 by (rule measurable_cong[THEN iffD1]) |
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560 qed (simp_all add: nn_integral_add nn_integral_cmult measurable_compose_Pair1 |
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561 nn_integral_monotone_convergence_SUP incseq_def le_fun_def |
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562 cong: measurable_cong) |
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563 |
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564 lemma (in sigma_finite_measure) nn_integral_fst: |
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565 assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" |
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566 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 = _") |
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567 using f proof induct |
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568 case (cong u v) |
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569 then have "?I u = ?I v" |
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570 by (intro nn_integral_cong) (auto simp: space_pair_measure) |
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571 with cong show ?case |
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572 by (simp cong: nn_integral_cong) |
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573 qed (simp_all add: emeasure_pair_measure nn_integral_cmult nn_integral_add |
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574 nn_integral_monotone_convergence_SUP measurable_compose_Pair1 |
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575 borel_measurable_nn_integral_fst nn_integral_mono incseq_def le_fun_def |
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576 cong: nn_integral_cong) |
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577 |
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578 lemma (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]: |
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579 "case_prod f \<in> borel_measurable (N \<Otimes>\<^sub>M M) \<Longrightarrow> (\<lambda>x. \<integral>\<^sup>+ y. f x y \<partial>M) \<in> borel_measurable N" |
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580 using borel_measurable_nn_integral_fst[of "case_prod f" N] by simp |
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581 |
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582 lemma (in pair_sigma_finite) nn_integral_snd: |
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583 assumes f[measurable]: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" |
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584 shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f" |
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585 proof - |
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586 note measurable_pair_swap[OF f] |
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587 from M1.nn_integral_fst[OF this] |
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588 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))" |
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589 by simp |
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590 also have "(\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1)) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f" |
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591 by (subst distr_pair_swap) (auto simp add: nn_integral_distr intro!: nn_integral_cong) |
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592 finally show ?thesis . |
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593 qed |
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594 |
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595 lemma (in pair_sigma_finite) Fubini: |
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596 assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" |
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597 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)" |
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598 unfolding nn_integral_snd[OF assms] M2.nn_integral_fst[OF assms] .. |
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599 |
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600 lemma (in pair_sigma_finite) Fubini': |
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601 assumes f: "case_prod f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" |
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602 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)" |
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603 using Fubini[OF f] by simp |
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604 |
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605 subsection \<open>Products on counting spaces, densities and distributions\<close> |
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606 |
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607 lemma sigma_prod: |
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608 assumes X_cover: "\<exists>E\<subseteq>A. countable E \<and> X = \<Union>E" and A: "A \<subseteq> Pow X" |
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609 assumes Y_cover: "\<exists>E\<subseteq>B. countable E \<and> Y = \<Union>E" and B: "B \<subseteq> Pow Y" |
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610 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}" |
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611 (is "?P = ?S") |
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612 proof (rule measure_eqI) |
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613 have [simp]: "snd \<in> X \<times> Y \<rightarrow> Y" "fst \<in> X \<times> Y \<rightarrow> X" |
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614 by auto |
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615 let ?XY = "{{fst -` a \<inter> X \<times> Y | a. a \<in> A}, {snd -` b \<inter> X \<times> Y | b. b \<in> B}}" |
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616 have "sets ?P = sets (SUP xy:?XY. sigma (X \<times> Y) xy)" |
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617 by (simp add: vimage_algebra_sigma sets_pair_eq_sets_fst_snd A B) |
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618 also have "\<dots> = sets (sigma (X \<times> Y) (\<Union>?XY))" |
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619 by (intro Sup_sigma arg_cong[where f=sets]) auto |
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620 also have "\<dots> = sets ?S" |
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621 proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI) |
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622 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)" |
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623 using A B by auto |
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624 next |
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625 interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}" |
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626 using A B by (intro sigma_algebra_sigma_sets) auto |
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627 fix Z assume "Z \<in> \<Union>?XY" |
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628 then show "Z \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}" |
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629 proof safe |
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630 fix a assume "a \<in> A" |
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631 from Y_cover obtain E where E: "E \<subseteq> B" "countable E" and "Y = \<Union>E" |
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632 by auto |
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633 with \<open>a \<in> A\<close> A have eq: "fst -` a \<inter> X \<times> Y = (\<Union>e\<in>E. a \<times> e)" |
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634 by auto |
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635 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}" |
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636 using \<open>a \<in> A\<close> E unfolding eq by (auto intro!: XY.countable_UN') |
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637 next |
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638 fix b assume "b \<in> B" |
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639 from X_cover obtain E where E: "E \<subseteq> A" "countable E" and "X = \<Union>E" |
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640 by auto |
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641 with \<open>b \<in> B\<close> B have eq: "snd -` b \<inter> X \<times> Y = (\<Union>e\<in>E. e \<times> b)" |
|
642 by auto |
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643 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}" |
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644 using \<open>b \<in> B\<close> E unfolding eq by (auto intro!: XY.countable_UN') |
|
645 qed |
|
646 next |
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647 fix Z assume "Z \<in> {a \<times> b |a b. a \<in> A \<and> b \<in> B}" |
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648 then obtain a b where "Z = a \<times> b" and ab: "a \<in> A" "b \<in> B" |
|
649 by auto |
|
650 then have Z: "Z = (fst -` a \<inter> X \<times> Y) \<inter> (snd -` b \<inter> X \<times> Y)" |
|
651 using A B by auto |
|
652 interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) (\<Union>?XY)" |
|
653 by (intro sigma_algebra_sigma_sets) auto |
|
654 show "Z \<in> sigma_sets (X \<times> Y) (\<Union>?XY)" |
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655 unfolding Z by (rule XY.Int) (blast intro: ab)+ |
|
656 qed |
|
657 finally show "sets ?P = sets ?S" . |
|
658 next |
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659 interpret finite_measure "sigma X A" for X A |
|
660 proof qed (simp add: emeasure_sigma) |
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661 fix A assume "A \<in> sets ?P" then show "emeasure ?P A = emeasure ?S A" |
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662 by (simp add: emeasure_pair_measure_alt emeasure_sigma) |
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663 qed |
|
664 |
|
665 lemma sigma_sets_pair_measure_generator_finite: |
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666 assumes "finite A" and "finite B" |
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667 shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)" |
|
668 (is "sigma_sets ?prod ?sets = _") |
|
669 proof safe |
|
670 have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product) |
|
671 fix x assume subset: "x \<subseteq> A \<times> B" |
|
672 hence "finite x" using fin by (rule finite_subset) |
|
673 from this subset show "x \<in> sigma_sets ?prod ?sets" |
|
674 proof (induct x) |
|
675 case empty show ?case by (rule sigma_sets.Empty) |
|
676 next |
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677 case (insert a x) |
|
678 hence "{a} \<in> sigma_sets ?prod ?sets" by auto |
|
679 moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto |
|
680 ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un) |
|
681 qed |
|
682 next |
|
683 fix x a b |
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684 assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x" |
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685 from sigma_sets_into_sp[OF _ this(1)] this(2) |
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686 show "a \<in> A" and "b \<in> B" by auto |
|
687 qed |
|
688 |
|
689 lemma borel_prod: |
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690 "(borel \<Otimes>\<^sub>M borel) = (borel :: ('a::second_countable_topology \<times> 'b::second_countable_topology) measure)" |
|
691 (is "?P = ?B") |
|
692 proof - |
|
693 have "?B = sigma UNIV {A \<times> B | A B. open A \<and> open B}" |
|
694 by (rule second_countable_borel_measurable[OF open_prod_generated]) |
|
695 also have "\<dots> = ?P" |
|
696 unfolding borel_def |
|
697 by (subst sigma_prod) (auto intro!: exI[of _ "{UNIV}"]) |
|
698 finally show ?thesis .. |
|
699 qed |
|
700 |
|
701 lemma pair_measure_count_space: |
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702 assumes A: "finite A" and B: "finite B" |
|
703 shows "count_space A \<Otimes>\<^sub>M count_space B = count_space (A \<times> B)" (is "?P = ?C") |
|
704 proof (rule measure_eqI) |
|
705 interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact |
|
706 interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact |
|
707 interpret P: pair_sigma_finite "count_space A" "count_space B" .. |
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708 show eq: "sets ?P = sets ?C" |
|
709 by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B) |
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710 fix X assume X: "X \<in> sets ?P" |
|
711 with eq have X_subset: "X \<subseteq> A \<times> B" by simp |
|
712 with A B have fin_Pair: "\<And>x. finite (Pair x -` X)" |
|
713 by (intro finite_subset[OF _ B]) auto |
|
714 have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B) |
|
715 have pos_card: "(0::ennreal) < of_nat (card (Pair x -` X)) \<longleftrightarrow> Pair x -` X \<noteq> {}" for x |
|
716 by (auto simp: card_eq_0_iff fin_Pair) blast |
|
717 |
|
718 show "emeasure ?P X = emeasure ?C X" |
|
719 using X_subset A fin_Pair fin_X |
|
720 apply (subst B.emeasure_pair_measure_alt[OF X]) |
|
721 apply (subst emeasure_count_space) |
|
722 apply (auto simp add: emeasure_count_space nn_integral_count_space |
|
723 pos_card of_nat_setsum[symmetric] card_SigmaI[symmetric] |
|
724 simp del: of_nat_setsum card_SigmaI |
|
725 intro!: arg_cong[where f=card]) |
|
726 done |
|
727 qed |
|
728 |
|
729 |
|
730 lemma emeasure_prod_count_space: |
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731 assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M M)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)") |
|
732 shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator A (x, y) \<partial>?B \<partial>?A)" |
|
733 by (rule emeasure_measure_of[OF pair_measure_def]) |
|
734 (auto simp: countably_additive_def positive_def suminf_indicator A |
|
735 nn_integral_suminf[symmetric] dest: sets.sets_into_space) |
|
736 |
|
737 lemma emeasure_prod_count_space_single[simp]: "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) {x} = 1" |
|
738 proof - |
|
739 have [simp]: "\<And>a b x y. indicator {(a, b)} (x, y) = (indicator {a} x * indicator {b} y::ennreal)" |
|
740 by (auto split: split_indicator) |
|
741 show ?thesis |
|
742 by (cases x) (auto simp: emeasure_prod_count_space nn_integral_cmult sets_Pair) |
|
743 qed |
|
744 |
|
745 lemma emeasure_count_space_prod_eq: |
|
746 fixes A :: "('a \<times> 'b) set" |
|
747 assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M count_space UNIV)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)") |
|
748 shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A" |
|
749 proof - |
|
750 { fix A :: "('a \<times> 'b) set" assume "countable A" |
|
751 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)" |
|
752 by (intro emeasure_UN_countable) (auto simp: sets_Pair disjoint_family_on_def) |
|
753 also have "\<dots> = (\<integral>\<^sup>+a. indicator A a \<partial>count_space UNIV)" |
|
754 by (subst nn_integral_count_space_indicator) auto |
|
755 finally have "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A" |
|
756 by simp } |
|
757 note * = this |
|
758 |
|
759 show ?thesis |
|
760 proof cases |
|
761 assume "finite A" then show ?thesis |
|
762 by (intro * countable_finite) |
|
763 next |
|
764 assume "infinite A" |
|
765 then obtain C where "countable C" and "infinite C" and "C \<subseteq> A" |
|
766 by (auto dest: infinite_countable_subset') |
|
767 with A have "emeasure (?A \<Otimes>\<^sub>M ?B) C \<le> emeasure (?A \<Otimes>\<^sub>M ?B) A" |
|
768 by (intro emeasure_mono) auto |
|
769 also have "emeasure (?A \<Otimes>\<^sub>M ?B) C = emeasure (count_space UNIV) C" |
|
770 using \<open>countable C\<close> by (rule *) |
|
771 finally show ?thesis |
|
772 using \<open>infinite C\<close> \<open>infinite A\<close> by (simp add: top_unique) |
|
773 qed |
|
774 qed |
|
775 |
|
776 lemma nn_integral_count_space_prod_eq: |
|
777 "nn_integral (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) f = nn_integral (count_space UNIV) f" |
|
778 (is "nn_integral ?P f = _") |
|
779 proof cases |
|
780 assume cntbl: "countable {x. f x \<noteq> 0}" |
|
781 have [simp]: "\<And>x. card ({x} \<inter> {x. f x \<noteq> 0}) = (indicator {x. f x \<noteq> 0} x::ennreal)" |
|
782 by (auto split: split_indicator) |
|
783 have [measurable]: "\<And>y. (\<lambda>x. indicator {y} x) \<in> borel_measurable ?P" |
|
784 by (rule measurable_discrete_difference[of "\<lambda>x. 0" _ borel "{y}" "\<lambda>x. indicator {y} x" for y]) |
|
785 (auto intro: sets_Pair) |
|
786 |
|
787 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)" |
|
788 by (auto simp add: nn_integral_cmult nn_integral_indicator' intro!: nn_integral_cong split: split_indicator) |
|
789 also have "\<dots> = (\<integral>\<^sup>+x. \<integral>\<^sup>+x'. f x' * indicator {x'} x \<partial>count_space {x. f x \<noteq> 0} \<partial>?P)" |
|
790 by (auto intro!: nn_integral_cong split: split_indicator) |
|
791 also have "\<dots> = (\<integral>\<^sup>+x'. \<integral>\<^sup>+x. f x' * indicator {x'} x \<partial>?P \<partial>count_space {x. f x \<noteq> 0})" |
|
792 by (intro nn_integral_count_space_nn_integral cntbl) auto |
|
793 also have "\<dots> = (\<integral>\<^sup>+x'. f x' \<partial>count_space {x. f x \<noteq> 0})" |
|
794 by (intro nn_integral_cong) (auto simp: nn_integral_cmult sets_Pair) |
|
795 finally show ?thesis |
|
796 by (auto simp add: nn_integral_count_space_indicator intro!: nn_integral_cong split: split_indicator) |
|
797 next |
|
798 { fix x assume "f x \<noteq> 0" |
|
799 then have "(\<exists>r\<ge>0. 0 < r \<and> f x = ennreal r) \<or> f x = \<infinity>" |
|
800 by (cases "f x" rule: ennreal_cases) (auto simp: less_le) |
|
801 then have "\<exists>n. ennreal (1 / real (Suc n)) \<le> f x" |
|
802 by (auto elim!: nat_approx_posE intro!: less_imp_le) } |
|
803 note * = this |
|
804 |
|
805 assume cntbl: "uncountable {x. f x \<noteq> 0}" |
|
806 also have "{x. f x \<noteq> 0} = (\<Union>n. {x. 1/Suc n \<le> f x})" |
|
807 using * by auto |
|
808 finally obtain n where "infinite {x. 1/Suc n \<le> f x}" |
|
809 by (meson countableI_type countable_UN uncountable_infinite) |
|
810 then obtain C where C: "C \<subseteq> {x. 1/Suc n \<le> f x}" and "countable C" "infinite C" |
|
811 by (metis infinite_countable_subset') |
|
812 |
|
813 have [measurable]: "C \<in> sets ?P" |
|
814 using sets.countable[OF _ \<open>countable C\<close>, of ?P] by (auto simp: sets_Pair) |
|
815 |
|
816 have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>?P) \<le> nn_integral ?P f" |
|
817 using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric]) |
|
818 moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>?P) = \<infinity>" |
|
819 using \<open>infinite C\<close> by (simp add: nn_integral_cmult emeasure_count_space_prod_eq ennreal_mult_top) |
|
820 moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>count_space UNIV) \<le> nn_integral (count_space UNIV) f" |
|
821 using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric]) |
|
822 moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>count_space UNIV) = \<infinity>" |
|
823 using \<open>infinite C\<close> by (simp add: nn_integral_cmult ennreal_mult_top) |
|
824 ultimately show ?thesis |
|
825 by (simp add: top_unique) |
|
826 qed |
|
827 |
|
828 lemma pair_measure_density: |
|
829 assumes f: "f \<in> borel_measurable M1" |
|
830 assumes g: "g \<in> borel_measurable M2" |
|
831 assumes "sigma_finite_measure M2" "sigma_finite_measure (density M2 g)" |
|
832 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") |
|
833 proof (rule measure_eqI) |
|
834 interpret M2: sigma_finite_measure M2 by fact |
|
835 interpret D2: sigma_finite_measure "density M2 g" by fact |
|
836 |
|
837 fix A assume A: "A \<in> sets ?L" |
|
838 with f g have "(\<integral>\<^sup>+ x. f x * \<integral>\<^sup>+ y. g y * indicator A (x, y) \<partial>M2 \<partial>M1) = |
|
839 (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f x * g y * indicator A (x, y) \<partial>M2 \<partial>M1)" |
|
840 by (intro nn_integral_cong_AE) |
|
841 (auto simp add: nn_integral_cmult[symmetric] ac_simps) |
|
842 with A f g show "emeasure ?L A = emeasure ?R A" |
|
843 by (simp add: D2.emeasure_pair_measure emeasure_density nn_integral_density |
|
844 M2.nn_integral_fst[symmetric] |
|
845 cong: nn_integral_cong) |
|
846 qed simp |
|
847 |
|
848 lemma sigma_finite_measure_distr: |
|
849 assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N" |
|
850 shows "sigma_finite_measure M" |
|
851 proof - |
|
852 interpret sigma_finite_measure "distr M N f" by fact |
|
853 from sigma_finite_countable guess A .. note A = this |
|
854 show ?thesis |
|
855 proof |
|
856 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>)" |
|
857 using A f |
|
858 by (intro exI[of _ "(\<lambda>a. f -` a \<inter> space M) ` A"]) |
|
859 (auto simp: emeasure_distr set_eq_iff subset_eq intro: measurable_space) |
|
860 qed |
|
861 qed |
|
862 |
|
863 lemma pair_measure_distr: |
|
864 assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T" |
|
865 assumes "sigma_finite_measure (distr N T g)" |
|
866 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") |
|
867 proof (rule measure_eqI) |
|
868 interpret T: sigma_finite_measure "distr N T g" by fact |
|
869 interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+ |
|
870 |
|
871 fix A assume A: "A \<in> sets ?P" |
|
872 with f g show "emeasure ?P A = emeasure ?D A" |
|
873 by (auto simp add: N.emeasure_pair_measure_alt space_pair_measure emeasure_distr |
|
874 T.emeasure_pair_measure_alt nn_integral_distr |
|
875 intro!: nn_integral_cong arg_cong[where f="emeasure N"]) |
|
876 qed simp |
|
877 |
|
878 lemma pair_measure_eqI: |
|
879 assumes "sigma_finite_measure M1" "sigma_finite_measure M2" |
|
880 assumes sets: "sets (M1 \<Otimes>\<^sub>M M2) = sets M" |
|
881 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)" |
|
882 shows "M1 \<Otimes>\<^sub>M M2 = M" |
|
883 proof - |
|
884 interpret M1: sigma_finite_measure M1 by fact |
|
885 interpret M2: sigma_finite_measure M2 by fact |
|
886 interpret pair_sigma_finite M1 M2 .. |
|
887 from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this |
|
888 let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}" |
|
889 let ?P = "M1 \<Otimes>\<^sub>M M2" |
|
890 show ?thesis |
|
891 proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]]) |
|
892 show "?E \<subseteq> Pow (space ?P)" |
|
893 using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure) |
|
894 show "sets ?P = sigma_sets (space ?P) ?E" |
|
895 by (simp add: sets_pair_measure space_pair_measure) |
|
896 then show "sets M = sigma_sets (space ?P) ?E" |
|
897 using sets[symmetric] by simp |
|
898 next |
|
899 show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>" |
|
900 using F by (auto simp: space_pair_measure) |
|
901 next |
|
902 fix X assume "X \<in> ?E" |
|
903 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 |
|
904 then have "emeasure ?P X = emeasure M1 A * emeasure M2 B" |
|
905 by (simp add: M2.emeasure_pair_measure_Times) |
|
906 also have "\<dots> = emeasure M (A \<times> B)" |
|
907 using A B emeasure by auto |
|
908 finally show "emeasure ?P X = emeasure M X" |
|
909 by simp |
|
910 qed |
|
911 qed |
|
912 |
|
913 lemma sets_pair_countable: |
|
914 assumes "countable S1" "countable S2" |
|
915 assumes M: "sets M = Pow S1" and N: "sets N = Pow S2" |
|
916 shows "sets (M \<Otimes>\<^sub>M N) = Pow (S1 \<times> S2)" |
|
917 proof auto |
|
918 fix x a b assume x: "x \<in> sets (M \<Otimes>\<^sub>M N)" "(a, b) \<in> x" |
|
919 from sets.sets_into_space[OF x(1)] x(2) |
|
920 sets_eq_imp_space_eq[of N "count_space S2"] sets_eq_imp_space_eq[of M "count_space S1"] M N |
|
921 show "a \<in> S1" "b \<in> S2" |
|
922 by (auto simp: space_pair_measure) |
|
923 next |
|
924 fix X assume X: "X \<subseteq> S1 \<times> S2" |
|
925 then have "countable X" |
|
926 by (metis countable_subset \<open>countable S1\<close> \<open>countable S2\<close> countable_SIGMA) |
|
927 have "X = (\<Union>(a, b)\<in>X. {a} \<times> {b})" by auto |
|
928 also have "\<dots> \<in> sets (M \<Otimes>\<^sub>M N)" |
|
929 using X |
|
930 by (safe intro!: sets.countable_UN' \<open>countable X\<close> subsetI pair_measureI) (auto simp: M N) |
|
931 finally show "X \<in> sets (M \<Otimes>\<^sub>M N)" . |
|
932 qed |
|
933 |
|
934 lemma pair_measure_countable: |
|
935 assumes "countable S1" "countable S2" |
|
936 shows "count_space S1 \<Otimes>\<^sub>M count_space S2 = count_space (S1 \<times> S2)" |
|
937 proof (rule pair_measure_eqI) |
|
938 show "sigma_finite_measure (count_space S1)" "sigma_finite_measure (count_space S2)" |
|
939 using assms by (auto intro!: sigma_finite_measure_count_space_countable) |
|
940 show "sets (count_space S1 \<Otimes>\<^sub>M count_space S2) = sets (count_space (S1 \<times> S2))" |
|
941 by (subst sets_pair_countable[OF assms]) auto |
|
942 next |
|
943 fix A B assume "A \<in> sets (count_space S1)" "B \<in> sets (count_space S2)" |
|
944 then show "emeasure (count_space S1) A * emeasure (count_space S2) B = |
|
945 emeasure (count_space (S1 \<times> S2)) (A \<times> B)" |
|
946 by (subst (1 2 3) emeasure_count_space) (auto simp: finite_cartesian_product_iff ennreal_mult_top ennreal_top_mult) |
|
947 qed |
|
948 |
|
949 lemma nn_integral_fst_count_space: |
|
950 "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f" |
|
951 (is "?lhs = ?rhs") |
|
952 proof(cases) |
|
953 assume *: "countable {xy. f xy \<noteq> 0}" |
|
954 let ?A = "fst ` {xy. f xy \<noteq> 0}" |
|
955 let ?B = "snd ` {xy. f xy \<noteq> 0}" |
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956 from * have [simp]: "countable ?A" "countable ?B" by(rule countable_image)+ |
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957 have "?lhs = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space ?A)" |
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958 by(rule nn_integral_count_space_eq) |
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959 (auto simp add: nn_integral_0_iff_AE AE_count_space not_le intro: rev_image_eqI) |
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960 also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space ?B \<partial>count_space ?A)" |
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961 by(intro nn_integral_count_space_eq nn_integral_cong)(auto intro: rev_image_eqI) |
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962 also have "\<dots> = (\<integral>\<^sup>+ xy. f xy \<partial>count_space (?A \<times> ?B))" |
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963 by(subst sigma_finite_measure.nn_integral_fst) |
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964 (simp_all add: sigma_finite_measure_count_space_countable pair_measure_countable) |
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965 also have "\<dots> = ?rhs" |
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966 by(rule nn_integral_count_space_eq)(auto intro: rev_image_eqI) |
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967 finally show ?thesis . |
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968 next |
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969 { fix xy assume "f xy \<noteq> 0" |
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970 then have "(\<exists>r\<ge>0. 0 < r \<and> f xy = ennreal r) \<or> f xy = \<infinity>" |
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971 by (cases "f xy" rule: ennreal_cases) (auto simp: less_le) |
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972 then have "\<exists>n. ennreal (1 / real (Suc n)) \<le> f xy" |
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973 by (auto elim!: nat_approx_posE intro!: less_imp_le) } |
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974 note * = this |
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975 |
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976 assume cntbl: "uncountable {xy. f xy \<noteq> 0}" |
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977 also have "{xy. f xy \<noteq> 0} = (\<Union>n. {xy. 1/Suc n \<le> f xy})" |
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978 using * by auto |
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979 finally obtain n where "infinite {xy. 1/Suc n \<le> f xy}" |
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980 by (meson countableI_type countable_UN uncountable_infinite) |
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981 then obtain C where C: "C \<subseteq> {xy. 1/Suc n \<le> f xy}" and "countable C" "infinite C" |
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982 by (metis infinite_countable_subset') |
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983 |
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984 have "\<infinity> = (\<integral>\<^sup>+ xy. ennreal (1 / Suc n) * indicator C xy \<partial>count_space UNIV)" |
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985 using \<open>infinite C\<close> by(simp add: nn_integral_cmult ennreal_mult_top) |
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986 also have "\<dots> \<le> ?rhs" using C |
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987 by(intro nn_integral_mono)(auto split: split_indicator) |
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988 finally have "?rhs = \<infinity>" by (simp add: top_unique) |
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989 moreover have "?lhs = \<infinity>" |
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990 proof(cases "finite (fst ` C)") |
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991 case True |
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992 then obtain x C' where x: "x \<in> fst ` C" |
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993 and C': "C' = fst -` {x} \<inter> C" |
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994 and "infinite C'" |
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995 using \<open>infinite C\<close> by(auto elim!: inf_img_fin_domE') |
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996 from x C C' have **: "C' \<subseteq> {xy. 1 / Suc n \<le> f xy}" by auto |
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997 |
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998 from C' \<open>infinite C'\<close> have "infinite (snd ` C')" |
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999 by(auto dest!: finite_imageD simp add: inj_on_def) |
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1000 then have "\<infinity> = (\<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator (snd ` C') y \<partial>count_space UNIV)" |
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1001 by(simp add: nn_integral_cmult ennreal_mult_top) |
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1002 also have "\<dots> = (\<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV)" |
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1003 by(rule nn_integral_cong)(force split: split_indicator intro: rev_image_eqI simp add: C') |
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1004 also have "\<dots> = (\<integral>\<^sup>+ x'. (\<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV) * indicator {x} x' \<partial>count_space UNIV)" |
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1005 by(simp add: one_ereal_def[symmetric]) |
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1006 also have "\<dots> \<le> (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV \<partial>count_space UNIV)" |
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1007 by(rule nn_integral_mono)(simp split: split_indicator) |
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1008 also have "\<dots> \<le> ?lhs" using ** |
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1009 by(intro nn_integral_mono)(auto split: split_indicator) |
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1010 finally show ?thesis by (simp add: top_unique) |
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1011 next |
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1012 case False |
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1013 define C' where "C' = fst ` C" |
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1014 have "\<infinity> = \<integral>\<^sup>+ x. ennreal (1 / Suc n) * indicator C' x \<partial>count_space UNIV" |
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1015 using C'_def False by(simp add: nn_integral_cmult ennreal_mult_top) |
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1016 also have "\<dots> = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C' x * indicator {SOME y. (x, y) \<in> C} y \<partial>count_space UNIV \<partial>count_space UNIV" |
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1017 by(auto simp add: one_ereal_def[symmetric] max_def intro: nn_integral_cong) |
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1018 also have "\<dots> \<le> \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C (x, y) \<partial>count_space UNIV \<partial>count_space UNIV" |
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1019 by(intro nn_integral_mono)(auto simp add: C'_def split: split_indicator intro: someI) |
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1020 also have "\<dots> \<le> ?lhs" using C |
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1021 by(intro nn_integral_mono)(auto split: split_indicator) |
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1022 finally show ?thesis by (simp add: top_unique) |
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1023 qed |
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1024 ultimately show ?thesis by simp |
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1025 qed |
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1026 |
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1027 lemma nn_integral_snd_count_space: |
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1028 "(\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f" |
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1029 (is "?lhs = ?rhs") |
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1030 proof - |
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1031 have "?lhs = (\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. (\<lambda>(y, x). f (x, y)) (y, x) \<partial>count_space UNIV \<partial>count_space UNIV)" |
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1032 by(simp) |
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1033 also have "\<dots> = \<integral>\<^sup>+ yx. (\<lambda>(y, x). f (x, y)) yx \<partial>count_space UNIV" |
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1034 by(rule nn_integral_fst_count_space) |
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1035 also have "\<dots> = \<integral>\<^sup>+ xy. f xy \<partial>count_space ((\<lambda>(x, y). (y, x)) ` UNIV)" |
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1036 by(subst nn_integral_bij_count_space[OF inj_on_imp_bij_betw, symmetric]) |
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1037 (simp_all add: inj_on_def split_def) |
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1038 also have "\<dots> = ?rhs" by(rule nn_integral_count_space_eq) auto |
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1039 finally show ?thesis . |
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1040 qed |
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1041 |
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1042 lemma measurable_pair_measure_countable1: |
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1043 assumes "countable A" |
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1044 and [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N K" |
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1045 shows "f \<in> measurable (count_space A \<Otimes>\<^sub>M N) K" |
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1046 using _ _ assms(1) |
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1047 by(rule measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A, simplified])simp_all |
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1048 |
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1049 subsection \<open>Product of Borel spaces\<close> |
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1050 |
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1051 lemma borel_Times: |
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1052 fixes A :: "'a::topological_space set" and B :: "'b::topological_space set" |
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1053 assumes A: "A \<in> sets borel" and B: "B \<in> sets borel" |
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1054 shows "A \<times> B \<in> sets borel" |
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1055 proof - |
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1056 have "A \<times> B = (A\<times>UNIV) \<inter> (UNIV \<times> B)" |
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1057 by auto |
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1058 moreover |
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1059 { have "A \<in> sigma_sets UNIV {S. open S}" using A by (simp add: sets_borel) |
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1060 then have "A\<times>UNIV \<in> sets borel" |
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1061 proof (induct A) |
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1062 case (Basic S) then show ?case |
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1063 by (auto intro!: borel_open open_Times) |
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1064 next |
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1065 case (Compl A) |
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1066 moreover have *: "(UNIV - A) \<times> UNIV = UNIV - (A \<times> UNIV)" |
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1067 by auto |
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1068 ultimately show ?case |
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1069 unfolding * by auto |
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1070 next |
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1071 case (Union A) |
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1072 moreover have *: "(UNION UNIV A) \<times> UNIV = UNION UNIV (\<lambda>i. A i \<times> UNIV)" |
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1073 by auto |
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1074 ultimately show ?case |
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1075 unfolding * by auto |
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1076 qed simp } |
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1077 moreover |
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1078 { have "B \<in> sigma_sets UNIV {S. open S}" using B by (simp add: sets_borel) |
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1079 then have "UNIV\<times>B \<in> sets borel" |
|
1080 proof (induct B) |
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1081 case (Basic S) then show ?case |
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1082 by (auto intro!: borel_open open_Times) |
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1083 next |
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1084 case (Compl B) |
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1085 moreover have *: "UNIV \<times> (UNIV - B) = UNIV - (UNIV \<times> B)" |
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1086 by auto |
|
1087 ultimately show ?case |
|
1088 unfolding * by auto |
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1089 next |
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1090 case (Union B) |
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1091 moreover have *: "UNIV \<times> (UNION UNIV B) = UNION UNIV (\<lambda>i. UNIV \<times> B i)" |
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1092 by auto |
|
1093 ultimately show ?case |
|
1094 unfolding * by auto |
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1095 qed simp } |
|
1096 ultimately show ?thesis |
|
1097 by auto |
|
1098 qed |
|
1099 |
|
1100 lemma finite_measure_pair_measure: |
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1101 assumes "finite_measure M" "finite_measure N" |
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1102 shows "finite_measure (N \<Otimes>\<^sub>M M)" |
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1103 proof (rule finite_measureI) |
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1104 interpret M: finite_measure M by fact |
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1105 interpret N: finite_measure N by fact |
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1106 show "emeasure (N \<Otimes>\<^sub>M M) (space (N \<Otimes>\<^sub>M M)) \<noteq> \<infinity>" |
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1107 by (auto simp: space_pair_measure M.emeasure_pair_measure_Times ennreal_mult_eq_top_iff) |
|
1108 qed |
|
1109 |
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1110 end |