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1 (* Title: Complete_Measure.thy |
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2 Author: Robert Himmelmann, Johannes Hoelzl, TU Muenchen |
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3 *) |
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4 theory Complete_Measure |
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5 imports Product_Measure |
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6 begin |
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7 |
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8 locale completeable_measure_space = measure_space |
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9 |
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10 definition (in completeable_measure_space) completion :: "'a algebra" where |
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11 "completion = \<lparr> space = space M, |
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12 sets = { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets \<and> N \<subseteq> N' } \<rparr>" |
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13 |
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14 lemma (in completeable_measure_space) space_completion[simp]: |
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15 "space completion = space M" unfolding completion_def by simp |
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16 |
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17 lemma (in completeable_measure_space) sets_completionE: |
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18 assumes "A \<in> sets completion" |
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19 obtains S N N' where "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets" "S \<in> sets M" |
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20 using assms unfolding completion_def by auto |
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21 |
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22 lemma (in completeable_measure_space) sets_completionI: |
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23 assumes "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets" "S \<in> sets M" |
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24 shows "A \<in> sets completion" |
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25 using assms unfolding completion_def by auto |
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26 |
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27 lemma (in completeable_measure_space) sets_completionI_sets[intro]: |
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28 "A \<in> sets M \<Longrightarrow> A \<in> sets completion" |
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29 unfolding completion_def by force |
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30 |
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31 lemma (in completeable_measure_space) null_sets_completion: |
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32 assumes "N' \<in> null_sets" "N \<subseteq> N'" shows "N \<in> sets completion" |
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33 apply(rule sets_completionI[of N "{}" N N']) |
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34 using assms by auto |
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35 |
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36 sublocale completeable_measure_space \<subseteq> completion!: sigma_algebra completion |
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37 proof (unfold sigma_algebra_iff2, safe) |
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38 fix A x assume "A \<in> sets completion" "x \<in> A" |
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39 with sets_into_space show "x \<in> space completion" |
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40 by (auto elim!: sets_completionE) |
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41 next |
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42 fix A assume "A \<in> sets completion" |
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43 from this[THEN sets_completionE] guess S N N' . note A = this |
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44 let ?C = "space completion" |
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45 show "?C - A \<in> sets completion" using A |
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46 by (intro sets_completionI[of _ "(?C - S) \<inter> (?C - N')" "(?C - S) \<inter> N' \<inter> (?C - N)"]) |
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47 auto |
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48 next |
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49 fix A ::"nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets completion" |
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50 then have "\<forall>n. \<exists>S N N'. A n = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets \<and> N \<subseteq> N'" |
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51 unfolding completion_def by (auto simp: image_subset_iff) |
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52 from choice[OF this] guess S .. |
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53 from choice[OF this] guess N .. |
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54 from choice[OF this] guess N' .. |
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55 then show "UNION UNIV A \<in> sets completion" |
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56 using null_sets_UN[of N'] |
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57 by (intro sets_completionI[of _ "UNION UNIV S" "UNION UNIV N" "UNION UNIV N'"]) |
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58 auto |
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59 qed auto |
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60 |
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61 definition (in completeable_measure_space) |
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62 "split_completion A p = (\<exists>N'. A = fst p \<union> snd p \<and> fst p \<inter> snd p = {} \<and> |
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63 fst p \<in> sets M \<and> snd p \<subseteq> N' \<and> N' \<in> null_sets)" |
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64 |
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65 definition (in completeable_measure_space) |
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66 "main_part A = fst (Eps (split_completion A))" |
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67 |
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68 definition (in completeable_measure_space) |
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69 "null_part A = snd (Eps (split_completion A))" |
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70 |
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71 lemma (in completeable_measure_space) split_completion: |
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72 assumes "A \<in> sets completion" |
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73 shows "split_completion A (main_part A, null_part A)" |
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74 unfolding main_part_def null_part_def |
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75 proof (rule someI2_ex) |
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76 from assms[THEN sets_completionE] guess S N N' . note A = this |
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77 let ?P = "(S, N - S)" |
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78 show "\<exists>p. split_completion A p" |
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79 unfolding split_completion_def using A |
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80 proof (intro exI conjI) |
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81 show "A = fst ?P \<union> snd ?P" using A by auto |
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82 show "snd ?P \<subseteq> N'" using A by auto |
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83 qed auto |
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84 qed auto |
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85 |
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86 lemma (in completeable_measure_space) |
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87 assumes "S \<in> sets completion" |
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88 shows main_part_sets[intro, simp]: "main_part S \<in> sets M" |
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89 and main_part_null_part_Un[simp]: "main_part S \<union> null_part S = S" |
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90 and main_part_null_part_Int[simp]: "main_part S \<inter> null_part S = {}" |
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91 using split_completion[OF assms] by (auto simp: split_completion_def) |
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92 |
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93 lemma (in completeable_measure_space) null_part: |
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94 assumes "S \<in> sets completion" shows "\<exists>N. N\<in>null_sets \<and> null_part S \<subseteq> N" |
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95 using split_completion[OF assms] by (auto simp: split_completion_def) |
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96 |
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97 lemma (in completeable_measure_space) null_part_sets[intro, simp]: |
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98 assumes "S \<in> sets M" shows "null_part S \<in> sets M" "\<mu> (null_part S) = 0" |
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99 proof - |
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100 have S: "S \<in> sets completion" using assms by auto |
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101 have "S - main_part S \<in> sets M" using assms by auto |
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102 moreover |
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103 from main_part_null_part_Un[OF S] main_part_null_part_Int[OF S] |
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104 have "S - main_part S = null_part S" by auto |
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105 ultimately show sets: "null_part S \<in> sets M" by auto |
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106 from null_part[OF S] guess N .. |
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107 with measure_eq_0[of N "null_part S"] sets |
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108 show "\<mu> (null_part S) = 0" by auto |
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109 qed |
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110 |
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111 definition (in completeable_measure_space) "\<mu>' A = \<mu> (main_part A)" |
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112 |
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113 lemma (in completeable_measure_space) \<mu>'_set[simp]: |
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114 assumes "S \<in> sets M" shows "\<mu>' S = \<mu> S" |
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115 proof - |
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116 have S: "S \<in> sets completion" using assms by auto |
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117 then have "\<mu> S = \<mu> (main_part S \<union> null_part S)" by simp |
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118 also have "\<dots> = \<mu> (main_part S)" |
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119 using S assms measure_additive[of "main_part S" "null_part S"] |
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120 by (auto simp: measure_additive) |
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121 finally show ?thesis unfolding \<mu>'_def by simp |
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122 qed |
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123 |
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124 lemma (in completeable_measure_space) sets_completionI_sub: |
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125 assumes N: "N' \<in> null_sets" "N \<subseteq> N'" |
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126 shows "N \<in> sets completion" |
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127 using assms by (intro sets_completionI[of _ "{}" N N']) auto |
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128 |
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129 lemma (in completeable_measure_space) \<mu>_main_part_UN: |
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130 fixes S :: "nat \<Rightarrow> 'a set" |
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131 assumes "range S \<subseteq> sets completion" |
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132 shows "\<mu>' (\<Union>i. (S i)) = \<mu> (\<Union>i. main_part (S i))" |
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133 proof - |
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134 have S: "\<And>i. S i \<in> sets completion" using assms by auto |
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135 then have UN: "(\<Union>i. S i) \<in> sets completion" by auto |
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136 have "\<forall>i. \<exists>N. N \<in> null_sets \<and> null_part (S i) \<subseteq> N" |
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137 using null_part[OF S] by auto |
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138 from choice[OF this] guess N .. note N = this |
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139 then have UN_N: "(\<Union>i. N i) \<in> null_sets" by (intro null_sets_UN) auto |
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140 have "(\<Union>i. S i) \<in> sets completion" using S by auto |
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141 from null_part[OF this] guess N' .. note N' = this |
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142 let ?N = "(\<Union>i. N i) \<union> N'" |
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143 have null_set: "?N \<in> null_sets" using N' UN_N by (intro null_sets_Un) auto |
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144 have "main_part (\<Union>i. S i) \<union> ?N = (main_part (\<Union>i. S i) \<union> null_part (\<Union>i. S i)) \<union> ?N" |
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145 using N' by auto |
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146 also have "\<dots> = (\<Union>i. main_part (S i) \<union> null_part (S i)) \<union> ?N" |
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147 unfolding main_part_null_part_Un[OF S] main_part_null_part_Un[OF UN] by auto |
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148 also have "\<dots> = (\<Union>i. main_part (S i)) \<union> ?N" |
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149 using N by auto |
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150 finally have *: "main_part (\<Union>i. S i) \<union> ?N = (\<Union>i. main_part (S i)) \<union> ?N" . |
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151 have "\<mu> (main_part (\<Union>i. S i)) = \<mu> (main_part (\<Union>i. S i) \<union> ?N)" |
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152 using null_set UN by (intro measure_Un_null_set[symmetric]) auto |
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153 also have "\<dots> = \<mu> ((\<Union>i. main_part (S i)) \<union> ?N)" |
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154 unfolding * .. |
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155 also have "\<dots> = \<mu> (\<Union>i. main_part (S i))" |
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156 using null_set S by (intro measure_Un_null_set) auto |
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157 finally show ?thesis unfolding \<mu>'_def . |
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158 qed |
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159 |
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160 lemma (in completeable_measure_space) \<mu>_main_part_Un: |
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161 assumes S: "S \<in> sets completion" and T: "T \<in> sets completion" |
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162 shows "\<mu>' (S \<union> T) = \<mu> (main_part S \<union> main_part T)" |
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163 proof - |
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164 have UN: "(\<Union>i. binary (main_part S) (main_part T) i) = (\<Union>i. main_part (binary S T i))" |
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165 unfolding binary_def by (auto split: split_if_asm) |
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166 show ?thesis |
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167 using \<mu>_main_part_UN[of "binary S T"] assms |
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168 unfolding range_binary_eq Un_range_binary UN by auto |
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169 qed |
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170 |
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171 sublocale completeable_measure_space \<subseteq> completion!: measure_space completion \<mu>' |
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172 proof |
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173 show "\<mu>' {} = 0" by auto |
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174 next |
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175 show "countably_additive completion \<mu>'" |
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176 proof (unfold countably_additive_def, intro allI conjI impI) |
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177 fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets completion" "disjoint_family A" |
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178 have "disjoint_family (\<lambda>i. main_part (A i))" |
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179 proof (intro disjoint_family_on_bisimulation[OF A(2)]) |
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180 fix n m assume "A n \<inter> A m = {}" |
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181 then have "(main_part (A n) \<union> null_part (A n)) \<inter> (main_part (A m) \<union> null_part (A m)) = {}" |
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182 using A by (subst (1 2) main_part_null_part_Un) auto |
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183 then show "main_part (A n) \<inter> main_part (A m) = {}" by auto |
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184 qed |
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185 then have "(\<Sum>\<^isub>\<infinity>n. \<mu>' (A n)) = \<mu> (\<Union>i. main_part (A i))" |
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186 unfolding \<mu>'_def using A by (intro measure_countably_additive) auto |
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187 then show "(\<Sum>\<^isub>\<infinity>n. \<mu>' (A n)) = \<mu>' (UNION UNIV A)" |
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188 unfolding \<mu>_main_part_UN[OF A(1)] . |
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189 qed |
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190 qed |
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191 |
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192 lemma (in sigma_algebra) simple_functionD': |
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193 assumes "simple_function f" |
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194 shows "f -` {x} \<inter> space M \<in> sets M" |
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195 proof cases |
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196 assume "x \<in> f`space M" from simple_functionD(2)[OF assms this] show ?thesis . |
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197 next |
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198 assume "x \<notin> f`space M" then have "f -` {x} \<inter> space M = {}" by auto |
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199 then show ?thesis by auto |
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200 qed |
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201 |
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202 lemma (in sigma_algebra) simple_function_If: |
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203 assumes sf: "simple_function f" "simple_function g" and A: "A \<in> sets M" |
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204 shows "simple_function (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function ?IF") |
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205 proof - |
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206 def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M" |
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207 show ?thesis unfolding simple_function_def |
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208 proof safe |
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209 have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto |
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210 from finite_subset[OF this] assms |
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211 show "finite (?IF ` space M)" unfolding simple_function_def by auto |
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212 next |
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213 fix x assume "x \<in> space M" |
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214 then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A |
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215 then ((F (f x) \<inter> A) \<union> (G (f x) - (G (f x) \<inter> A))) |
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216 else ((F (g x) \<inter> A) \<union> (G (g x) - (G (g x) \<inter> A))))" |
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217 using sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def) |
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218 have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M" |
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219 unfolding F_def G_def using sf[THEN simple_functionD'] by auto |
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220 show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto |
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221 qed |
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222 qed |
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223 |
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224 lemma (in measure_space) null_sets_finite_UN: |
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225 assumes "finite S" "\<And>i. i \<in> S \<Longrightarrow> A i \<in> null_sets" |
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226 shows "(\<Union>i\<in>S. A i) \<in> null_sets" |
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227 proof (intro CollectI conjI) |
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228 show "(\<Union>i\<in>S. A i) \<in> sets M" using assms by (intro finite_UN) auto |
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229 have "\<mu> (\<Union>i\<in>S. A i) \<le> (\<Sum>i\<in>S. \<mu> (A i))" |
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230 using assms by (intro measure_finitely_subadditive) auto |
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231 then show "\<mu> (\<Union>i\<in>S. A i) = 0" |
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232 using assms by auto |
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233 qed |
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234 |
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235 lemma (in completeable_measure_space) completion_ex_simple_function: |
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236 assumes f: "completion.simple_function f" |
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237 shows "\<exists>f'. simple_function f' \<and> (AE x. f x = f' x)" |
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238 proof - |
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239 let "?F x" = "f -` {x} \<inter> space M" |
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240 have F: "\<And>x. ?F x \<in> sets completion" and fin: "finite (f`space M)" |
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241 using completion.simple_functionD'[OF f] |
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242 completion.simple_functionD[OF f] by simp_all |
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243 have "\<forall>x. \<exists>N. N \<in> null_sets \<and> null_part (?F x) \<subseteq> N" |
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244 using F null_part by auto |
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245 from choice[OF this] obtain N where |
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246 N: "\<And>x. null_part (?F x) \<subseteq> N x" "\<And>x. N x \<in> null_sets" by auto |
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247 let ?N = "\<Union>x\<in>f`space M. N x" let "?f' x" = "if x \<in> ?N then undefined else f x" |
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248 have sets: "?N \<in> null_sets" using N fin by (intro null_sets_finite_UN) auto |
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249 show ?thesis unfolding simple_function_def |
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250 proof (safe intro!: exI[of _ ?f']) |
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251 have "?f' ` space M \<subseteq> f`space M \<union> {undefined}" by auto |
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252 from finite_subset[OF this] completion.simple_functionD(1)[OF f] |
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253 show "finite (?f' ` space M)" by auto |
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254 next |
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255 fix x assume "x \<in> space M" |
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256 have "?f' -` {?f' x} \<inter> space M = |
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257 (if x \<in> ?N then ?F undefined \<union> ?N |
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258 else if f x = undefined then ?F (f x) \<union> ?N |
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259 else ?F (f x) - ?N)" |
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260 using N(2) sets_into_space by (auto split: split_if_asm) |
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261 moreover { fix y have "?F y \<union> ?N \<in> sets M" |
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262 proof cases |
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263 assume y: "y \<in> f`space M" |
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264 have "?F y \<union> ?N = (main_part (?F y) \<union> null_part (?F y)) \<union> ?N" |
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265 using main_part_null_part_Un[OF F] by auto |
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266 also have "\<dots> = main_part (?F y) \<union> ?N" |
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267 using y N by auto |
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268 finally show ?thesis |
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269 using F sets by auto |
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270 next |
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271 assume "y \<notin> f`space M" then have "?F y = {}" by auto |
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272 then show ?thesis using sets by auto |
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273 qed } |
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274 moreover { |
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275 have "?F (f x) - ?N = main_part (?F (f x)) \<union> null_part (?F (f x)) - ?N" |
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276 using main_part_null_part_Un[OF F] by auto |
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277 also have "\<dots> = main_part (?F (f x)) - ?N" |
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278 using N `x \<in> space M` by auto |
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279 finally have "?F (f x) - ?N \<in> sets M" |
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280 using F sets by auto } |
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281 ultimately show "?f' -` {?f' x} \<inter> space M \<in> sets M" by auto |
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282 next |
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283 show "AE x. f x = ?f' x" |
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284 by (rule AE_I', rule sets) auto |
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285 qed |
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286 qed |
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287 |
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288 lemma (in completeable_measure_space) completion_ex_borel_measurable: |
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289 fixes g :: "'a \<Rightarrow> pinfreal" |
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290 assumes g: "g \<in> borel_measurable completion" |
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291 shows "\<exists>g'\<in>borel_measurable M. (AE x. g x = g' x)" |
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292 proof - |
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293 from g[THEN completion.borel_measurable_implies_simple_function_sequence] |
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294 obtain f where "\<And>i. completion.simple_function (f i)" "f \<up> g" by auto |
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295 then have "\<forall>i. \<exists>f'. simple_function f' \<and> (AE x. f i x = f' x)" |
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296 using completion_ex_simple_function by auto |
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297 from this[THEN choice] obtain f' where |
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298 sf: "\<And>i. simple_function (f' i)" and |
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299 AE: "\<forall>i. AE x. f i x = f' i x" by auto |
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300 show ?thesis |
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301 proof (intro bexI) |
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302 from AE[unfolded all_AE_countable] |
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303 show "AE x. g x = (SUP i. f' i) x" (is "AE x. g x = ?f x") |
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304 proof (rule AE_mp, safe intro!: AE_cong) |
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305 fix x assume eq: "\<forall>i. f i x = f' i x" |
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306 have "g x = (SUP i. f i x)" |
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307 using `f \<up> g` unfolding isoton_def SUPR_fun_expand by auto |
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308 then show "g x = ?f x" |
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309 using eq unfolding SUPR_fun_expand by auto |
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310 qed |
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311 show "?f \<in> borel_measurable M" |
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312 using sf by (auto intro!: borel_measurable_SUP |
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313 intro: borel_measurable_simple_function) |
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314 qed |
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315 qed |
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316 |
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317 end |