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1 (* Title: HOL/Analysis/Radon_Nikodym.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>Radon-Nikod{\'y}m derivative\<close> |
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6 |
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7 theory Radon_Nikodym |
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8 imports Bochner_Integration |
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9 begin |
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10 |
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11 definition diff_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure" |
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12 where |
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13 "diff_measure M N = measure_of (space M) (sets M) (\<lambda>A. emeasure M A - emeasure N A)" |
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14 |
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15 lemma |
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16 shows space_diff_measure[simp]: "space (diff_measure M N) = space M" |
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17 and sets_diff_measure[simp]: "sets (diff_measure M N) = sets M" |
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18 by (auto simp: diff_measure_def) |
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19 |
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20 lemma emeasure_diff_measure: |
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21 assumes fin: "finite_measure M" "finite_measure N" and sets_eq: "sets M = sets N" |
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22 assumes pos: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure N A \<le> emeasure M A" and A: "A \<in> sets M" |
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23 shows "emeasure (diff_measure M N) A = emeasure M A - emeasure N A" (is "_ = ?\<mu> A") |
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24 unfolding diff_measure_def |
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25 proof (rule emeasure_measure_of_sigma) |
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26 show "sigma_algebra (space M) (sets M)" .. |
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27 show "positive (sets M) ?\<mu>" |
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28 using pos by (simp add: positive_def) |
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29 show "countably_additive (sets M) ?\<mu>" |
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30 proof (rule countably_additiveI) |
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31 fix A :: "nat \<Rightarrow> _" assume A: "range A \<subseteq> sets M" and "disjoint_family A" |
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32 then have suminf: |
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33 "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)" |
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34 "(\<Sum>i. emeasure N (A i)) = emeasure N (\<Union>i. A i)" |
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35 by (simp_all add: suminf_emeasure sets_eq) |
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36 with A have "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) = |
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37 (\<Sum>i. emeasure M (A i)) - (\<Sum>i. emeasure N (A i))" |
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38 using fin pos[of "A _"] |
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39 by (intro ennreal_suminf_minus) |
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40 (auto simp: sets_eq finite_measure.emeasure_eq_measure suminf_emeasure) |
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41 then show "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) = |
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42 emeasure M (\<Union>i. A i) - emeasure N (\<Union>i. A i) " |
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43 by (simp add: suminf) |
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44 qed |
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45 qed fact |
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46 |
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47 lemma (in sigma_finite_measure) Ex_finite_integrable_function: |
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48 "\<exists>h\<in>borel_measurable M. integral\<^sup>N M h \<noteq> \<infinity> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>)" |
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49 proof - |
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50 obtain A :: "nat \<Rightarrow> 'a set" where |
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51 range[measurable]: "range A \<subseteq> sets M" and |
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52 space: "(\<Union>i. A i) = space M" and |
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53 measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" and |
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54 disjoint: "disjoint_family A" |
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55 using sigma_finite_disjoint by blast |
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56 let ?B = "\<lambda>i. 2^Suc i * emeasure M (A i)" |
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57 have [measurable]: "\<And>i. A i \<in> sets M" |
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58 using range by fastforce+ |
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59 have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)" |
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60 proof |
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61 fix i show "\<exists>x. 0 < x \<and> x < inverse (?B i)" |
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62 using measure[of i] |
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63 by (auto intro!: dense simp: ennreal_inverse_positive ennreal_mult_eq_top_iff power_eq_top_ennreal) |
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64 qed |
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65 from choice[OF this] obtain n where n: "\<And>i. 0 < n i" |
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66 "\<And>i. n i < inverse (2^Suc i * emeasure M (A i))" by auto |
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67 { fix i have "0 \<le> n i" using n(1)[of i] by auto } note pos = this |
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68 let ?h = "\<lambda>x. \<Sum>i. n i * indicator (A i) x" |
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69 show ?thesis |
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70 proof (safe intro!: bexI[of _ ?h] del: notI) |
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71 have "integral\<^sup>N M ?h = (\<Sum>i. n i * emeasure M (A i))" using pos |
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72 by (simp add: nn_integral_suminf nn_integral_cmult_indicator) |
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73 also have "\<dots> \<le> (\<Sum>i. ennreal ((1/2)^Suc i))" |
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74 proof (intro suminf_le allI) |
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75 fix N |
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76 have "n N * emeasure M (A N) \<le> inverse (2^Suc N * emeasure M (A N)) * emeasure M (A N)" |
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77 using n[of N] by (intro mult_right_mono) auto |
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78 also have "\<dots> = (1/2)^Suc N * (inverse (emeasure M (A N)) * emeasure M (A N))" |
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79 using measure[of N] |
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80 by (simp add: ennreal_inverse_power divide_ennreal_def ennreal_inverse_mult |
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81 power_eq_top_ennreal less_top[symmetric] mult_ac |
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82 del: power_Suc) |
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83 also have "\<dots> \<le> inverse (ennreal 2) ^ Suc N" |
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84 using measure[of N] |
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85 by (cases "emeasure M (A N)"; cases "emeasure M (A N) = 0") |
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86 (auto simp: inverse_ennreal ennreal_mult[symmetric] divide_ennreal_def simp del: power_Suc) |
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87 also have "\<dots> = ennreal (inverse 2 ^ Suc N)" |
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88 by (subst ennreal_power[symmetric], simp) (simp add: inverse_ennreal) |
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89 finally show "n N * emeasure M (A N) \<le> ennreal ((1/2)^Suc N)" |
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90 by simp |
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91 qed auto |
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92 also have "\<dots> < top" |
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93 unfolding less_top[symmetric] |
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94 by (rule ennreal_suminf_neq_top) |
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95 (auto simp: summable_geometric summable_Suc_iff simp del: power_Suc) |
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96 finally show "integral\<^sup>N M ?h \<noteq> \<infinity>" |
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97 by (auto simp: top_unique) |
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98 next |
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99 { fix x assume "x \<in> space M" |
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100 then obtain i where "x \<in> A i" using space[symmetric] by auto |
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101 with disjoint n have "?h x = n i" |
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102 by (auto intro!: suminf_cmult_indicator intro: less_imp_le) |
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103 then show "0 < ?h x" and "?h x < \<infinity>" using n[of i] by (auto simp: less_top[symmetric]) } |
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104 note pos = this |
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105 qed measurable |
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106 qed |
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107 |
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108 subsection "Absolutely continuous" |
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109 |
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110 definition absolutely_continuous :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where |
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111 "absolutely_continuous M N \<longleftrightarrow> null_sets M \<subseteq> null_sets N" |
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112 |
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113 lemma absolutely_continuousI_count_space: "absolutely_continuous (count_space A) M" |
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114 unfolding absolutely_continuous_def by (auto simp: null_sets_count_space) |
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115 |
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116 lemma absolutely_continuousI_density: |
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117 "f \<in> borel_measurable M \<Longrightarrow> absolutely_continuous M (density M f)" |
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118 by (force simp add: absolutely_continuous_def null_sets_density_iff dest: AE_not_in) |
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119 |
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120 lemma absolutely_continuousI_point_measure_finite: |
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121 "(\<And>x. \<lbrakk> x \<in> A ; f x \<le> 0 \<rbrakk> \<Longrightarrow> g x \<le> 0) \<Longrightarrow> absolutely_continuous (point_measure A f) (point_measure A g)" |
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122 unfolding absolutely_continuous_def by (force simp: null_sets_point_measure_iff) |
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123 |
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124 lemma absolutely_continuousD: |
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125 "absolutely_continuous M N \<Longrightarrow> A \<in> sets M \<Longrightarrow> emeasure M A = 0 \<Longrightarrow> emeasure N A = 0" |
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126 by (auto simp: absolutely_continuous_def null_sets_def) |
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127 |
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128 lemma absolutely_continuous_AE: |
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129 assumes sets_eq: "sets M' = sets M" |
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130 and "absolutely_continuous M M'" "AE x in M. P x" |
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131 shows "AE x in M'. P x" |
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132 proof - |
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133 from \<open>AE x in M. P x\<close> obtain N where N: "N \<in> null_sets M" "{x\<in>space M. \<not> P x} \<subseteq> N" |
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134 unfolding eventually_ae_filter by auto |
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135 show "AE x in M'. P x" |
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136 proof (rule AE_I') |
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137 show "{x\<in>space M'. \<not> P x} \<subseteq> N" using sets_eq_imp_space_eq[OF sets_eq] N(2) by simp |
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138 from \<open>absolutely_continuous M M'\<close> show "N \<in> null_sets M'" |
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139 using N unfolding absolutely_continuous_def sets_eq null_sets_def by auto |
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140 qed |
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141 qed |
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142 |
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143 subsection "Existence of the Radon-Nikodym derivative" |
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144 |
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145 lemma (in finite_measure) Radon_Nikodym_finite_measure: |
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146 assumes "finite_measure N" and sets_eq[simp]: "sets N = sets M" |
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147 assumes "absolutely_continuous M N" |
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148 shows "\<exists>f \<in> borel_measurable M. density M f = N" |
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149 proof - |
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150 interpret N: finite_measure N by fact |
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151 define G where "G = {g \<in> borel_measurable M. \<forall>A\<in>sets M. (\<integral>\<^sup>+x. g x * indicator A x \<partial>M) \<le> N A}" |
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152 have [measurable_dest]: "f \<in> G \<Longrightarrow> f \<in> borel_measurable M" |
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153 and G_D: "\<And>A. f \<in> G \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+x. f x * indicator A x \<partial>M) \<le> N A" for f |
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154 by (auto simp: G_def) |
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155 note this[measurable_dest] |
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156 have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto |
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157 hence "G \<noteq> {}" by auto |
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158 { fix f g assume f[measurable]: "f \<in> G" and g[measurable]: "g \<in> G" |
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159 have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def |
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160 proof safe |
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161 let ?A = "{x \<in> space M. f x \<le> g x}" |
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162 have "?A \<in> sets M" using f g unfolding G_def by auto |
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163 fix A assume [measurable]: "A \<in> sets M" |
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164 have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A" |
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165 using sets.sets_into_space[OF \<open>A \<in> sets M\<close>] by auto |
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166 have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x = |
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167 g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x" |
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168 by (auto simp: indicator_def max_def) |
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169 hence "(\<integral>\<^sup>+x. max (g x) (f x) * indicator A x \<partial>M) = |
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170 (\<integral>\<^sup>+x. g x * indicator (?A \<inter> A) x \<partial>M) + |
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171 (\<integral>\<^sup>+x. f x * indicator ((space M - ?A) \<inter> A) x \<partial>M)" |
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172 by (auto cong: nn_integral_cong intro!: nn_integral_add) |
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173 also have "\<dots> \<le> N (?A \<inter> A) + N ((space M - ?A) \<inter> A)" |
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174 using f g unfolding G_def by (auto intro!: add_mono) |
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175 also have "\<dots> = N A" |
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176 using union by (subst plus_emeasure) auto |
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177 finally show "(\<integral>\<^sup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> N A" . |
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178 qed auto } |
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179 note max_in_G = this |
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180 { fix f assume "incseq f" and f: "\<And>i. f i \<in> G" |
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181 then have [measurable]: "\<And>i. f i \<in> borel_measurable M" by (auto simp: G_def) |
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182 have "(\<lambda>x. SUP i. f i x) \<in> G" unfolding G_def |
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183 proof safe |
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184 show "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M" by measurable |
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185 next |
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186 fix A assume "A \<in> sets M" |
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187 have "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) = |
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188 (\<integral>\<^sup>+x. (SUP i. f i x * indicator A x) \<partial>M)" |
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189 by (intro nn_integral_cong) (simp split: split_indicator) |
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190 also have "\<dots> = (SUP i. (\<integral>\<^sup>+x. f i x * indicator A x \<partial>M))" |
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191 using \<open>incseq f\<close> f \<open>A \<in> sets M\<close> |
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192 by (intro nn_integral_monotone_convergence_SUP) |
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193 (auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator) |
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194 finally show "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> N A" |
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195 using f \<open>A \<in> sets M\<close> by (auto intro!: SUP_least simp: G_D) |
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196 qed } |
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197 note SUP_in_G = this |
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198 let ?y = "SUP g : G. integral\<^sup>N M g" |
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199 have y_le: "?y \<le> N (space M)" unfolding G_def |
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200 proof (safe intro!: SUP_least) |
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201 fix g assume "\<forall>A\<in>sets M. (\<integral>\<^sup>+x. g x * indicator A x \<partial>M) \<le> N A" |
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202 from this[THEN bspec, OF sets.top] show "integral\<^sup>N M g \<le> N (space M)" |
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203 by (simp cong: nn_integral_cong) |
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204 qed |
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205 from ennreal_SUP_countable_SUP [OF \<open>G \<noteq> {}\<close>, of "integral\<^sup>N M"] guess ys .. note ys = this |
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206 then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n" |
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207 proof safe |
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208 fix n assume "range ys \<subseteq> integral\<^sup>N M ` G" |
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209 hence "ys n \<in> integral\<^sup>N M ` G" by auto |
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210 thus "\<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n" by auto |
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211 qed |
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212 from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. integral\<^sup>N M (gs n) = ys n" by auto |
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213 hence y_eq: "?y = (SUP i. integral\<^sup>N M (gs i))" using ys by auto |
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214 let ?g = "\<lambda>i x. Max ((\<lambda>n. gs n x) ` {..i})" |
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215 define f where [abs_def]: "f x = (SUP i. ?g i x)" for x |
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216 let ?F = "\<lambda>A x. f x * indicator A x" |
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217 have gs_not_empty: "\<And>i x. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto |
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218 { fix i have "?g i \<in> G" |
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219 proof (induct i) |
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220 case 0 thus ?case by simp fact |
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221 next |
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222 case (Suc i) |
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223 with Suc gs_not_empty \<open>gs (Suc i) \<in> G\<close> show ?case |
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224 by (auto simp add: atMost_Suc intro!: max_in_G) |
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225 qed } |
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226 note g_in_G = this |
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227 have "incseq ?g" using gs_not_empty |
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228 by (auto intro!: incseq_SucI le_funI simp add: atMost_Suc) |
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229 |
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230 from SUP_in_G[OF this g_in_G] have [measurable]: "f \<in> G" unfolding f_def . |
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231 then have [measurable]: "f \<in> borel_measurable M" unfolding G_def by auto |
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232 |
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233 have "integral\<^sup>N M f = (SUP i. integral\<^sup>N M (?g i))" unfolding f_def |
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234 using g_in_G \<open>incseq ?g\<close> by (auto intro!: nn_integral_monotone_convergence_SUP simp: G_def) |
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235 also have "\<dots> = ?y" |
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236 proof (rule antisym) |
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237 show "(SUP i. integral\<^sup>N M (?g i)) \<le> ?y" |
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238 using g_in_G by (auto intro: SUP_mono) |
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239 show "?y \<le> (SUP i. integral\<^sup>N M (?g i))" unfolding y_eq |
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240 by (auto intro!: SUP_mono nn_integral_mono Max_ge) |
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241 qed |
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242 finally have int_f_eq_y: "integral\<^sup>N M f = ?y" . |
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243 |
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244 have upper_bound: "\<forall>A\<in>sets M. N A \<le> density M f A" |
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245 proof (rule ccontr) |
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246 assume "\<not> ?thesis" |
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247 then obtain A where A[measurable]: "A \<in> sets M" and f_less_N: "density M f A < N A" |
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248 by (auto simp: not_le) |
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249 then have pos_A: "0 < M A" |
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250 using \<open>absolutely_continuous M N\<close>[THEN absolutely_continuousD, OF A] |
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251 by (auto simp: zero_less_iff_neq_zero) |
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252 |
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253 define b where "b = (N A - density M f A) / M A / 2" |
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254 with f_less_N pos_A have "0 < b" "b \<noteq> top" |
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255 by (auto intro!: diff_gr0_ennreal simp: zero_less_iff_neq_zero diff_eq_0_iff_ennreal ennreal_divide_eq_top_iff) |
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256 |
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257 let ?f = "\<lambda>x. f x + b" |
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258 have "nn_integral M f \<noteq> top" |
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259 using `f \<in> G`[THEN G_D, of "space M"] by (auto simp: top_unique cong: nn_integral_cong) |
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260 with \<open>b \<noteq> top\<close> interpret Mf: finite_measure "density M ?f" |
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261 by (intro finite_measureI) |
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262 (auto simp: field_simps mult_indicator_subset ennreal_mult_eq_top_iff |
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263 emeasure_density nn_integral_cmult_indicator nn_integral_add |
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264 cong: nn_integral_cong) |
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265 |
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266 from unsigned_Hahn_decomposition[of "density M ?f" N A] |
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267 obtain Y where [measurable]: "Y \<in> sets M" and [simp]: "Y \<subseteq> A" |
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268 and Y1: "\<And>C. C \<in> sets M \<Longrightarrow> C \<subseteq> Y \<Longrightarrow> density M ?f C \<le> N C" |
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269 and Y2: "\<And>C. C \<in> sets M \<Longrightarrow> C \<subseteq> A \<Longrightarrow> C \<inter> Y = {} \<Longrightarrow> N C \<le> density M ?f C" |
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270 by auto |
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271 |
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272 let ?f' = "\<lambda>x. f x + b * indicator Y x" |
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273 have "M Y \<noteq> 0" |
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274 proof |
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275 assume "M Y = 0" |
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276 then have "N Y = 0" |
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277 using \<open>absolutely_continuous M N\<close>[THEN absolutely_continuousD, of Y] by auto |
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278 then have "N A = N (A - Y)" |
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279 by (subst emeasure_Diff) auto |
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280 also have "\<dots> \<le> density M ?f (A - Y)" |
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281 by (rule Y2) auto |
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282 also have "\<dots> \<le> density M ?f A - density M ?f Y" |
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283 by (subst emeasure_Diff) auto |
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284 also have "\<dots> \<le> density M ?f A - 0" |
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285 by (intro ennreal_minus_mono) auto |
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286 also have "density M ?f A = b * M A + density M f A" |
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287 by (simp add: emeasure_density field_simps mult_indicator_subset nn_integral_add nn_integral_cmult_indicator) |
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288 also have "\<dots> < N A" |
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289 using f_less_N pos_A |
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290 by (cases "density M f A"; cases "M A"; cases "N A") |
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291 (auto simp: b_def ennreal_less_iff ennreal_minus divide_ennreal ennreal_numeral[symmetric] |
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292 ennreal_plus[symmetric] ennreal_mult[symmetric] field_simps |
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293 simp del: ennreal_numeral ennreal_plus) |
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294 finally show False |
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295 by simp |
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296 qed |
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297 then have "nn_integral M f < nn_integral M ?f'" |
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298 using \<open>0 < b\<close> \<open>nn_integral M f \<noteq> top\<close> |
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299 by (simp add: nn_integral_add nn_integral_cmult_indicator ennreal_zero_less_mult_iff zero_less_iff_neq_zero) |
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300 moreover |
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301 have "?f' \<in> G" |
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302 unfolding G_def |
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303 proof safe |
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304 fix X assume [measurable]: "X \<in> sets M" |
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305 have "(\<integral>\<^sup>+ x. ?f' x * indicator X x \<partial>M) = density M f (X - Y) + density M ?f (X \<inter> Y)" |
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306 by (auto simp add: emeasure_density nn_integral_add[symmetric] split: split_indicator intro!: nn_integral_cong) |
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307 also have "\<dots> \<le> N (X - Y) + N (X \<inter> Y)" |
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308 using G_D[OF \<open>f \<in> G\<close>] by (intro add_mono Y1) (auto simp: emeasure_density) |
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309 also have "\<dots> = N X" |
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310 by (subst plus_emeasure) (auto intro!: arg_cong2[where f=emeasure]) |
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311 finally show "(\<integral>\<^sup>+ x. ?f' x * indicator X x \<partial>M) \<le> N X" . |
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312 qed simp |
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313 then have "nn_integral M ?f' \<le> ?y" |
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314 by (rule SUP_upper) |
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315 ultimately show False |
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316 by (simp add: int_f_eq_y) |
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317 qed |
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318 show ?thesis |
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319 proof (intro bexI[of _ f] measure_eqI conjI antisym) |
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320 fix A assume "A \<in> sets (density M f)" then show "emeasure (density M f) A \<le> emeasure N A" |
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321 by (auto simp: emeasure_density intro!: G_D[OF \<open>f \<in> G\<close>]) |
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322 next |
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323 fix A assume A: "A \<in> sets (density M f)" then show "emeasure N A \<le> emeasure (density M f) A" |
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324 using upper_bound by auto |
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325 qed auto |
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326 qed |
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327 |
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328 lemma (in finite_measure) split_space_into_finite_sets_and_rest: |
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329 assumes ac: "absolutely_continuous M N" and sets_eq[simp]: "sets N = sets M" |
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330 shows "\<exists>B::nat\<Rightarrow>'a set. disjoint_family B \<and> range B \<subseteq> sets M \<and> (\<forall>i. N (B i) \<noteq> \<infinity>) \<and> |
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331 (\<forall>A\<in>sets M. A \<inter> (\<Union>i. B i) = {} \<longrightarrow> (emeasure M A = 0 \<and> N A = 0) \<or> (emeasure M A > 0 \<and> N A = \<infinity>))" |
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332 proof - |
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333 let ?Q = "{Q\<in>sets M. N Q \<noteq> \<infinity>}" |
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334 let ?a = "SUP Q:?Q. emeasure M Q" |
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335 have "{} \<in> ?Q" by auto |
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336 then have Q_not_empty: "?Q \<noteq> {}" by blast |
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337 have "?a \<le> emeasure M (space M)" using sets.sets_into_space |
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338 by (auto intro!: SUP_least emeasure_mono) |
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339 then have "?a \<noteq> \<infinity>" |
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340 using finite_emeasure_space |
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341 by (auto simp: less_top[symmetric] top_unique simp del: SUP_eq_top_iff Sup_eq_top_iff) |
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342 from ennreal_SUP_countable_SUP [OF Q_not_empty, of "emeasure M"] |
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343 obtain Q'' where "range Q'' \<subseteq> emeasure M ` ?Q" and a: "?a = (SUP i::nat. Q'' i)" |
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344 by auto |
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345 then have "\<forall>i. \<exists>Q'. Q'' i = emeasure M Q' \<and> Q' \<in> ?Q" by auto |
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346 from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = emeasure M (Q' i)" "\<And>i. Q' i \<in> ?Q" |
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347 by auto |
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348 then have a_Lim: "?a = (SUP i::nat. emeasure M (Q' i))" using a by simp |
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349 let ?O = "\<lambda>n. \<Union>i\<le>n. Q' i" |
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350 have Union: "(SUP i. emeasure M (?O i)) = emeasure M (\<Union>i. ?O i)" |
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351 proof (rule SUP_emeasure_incseq[of ?O]) |
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352 show "range ?O \<subseteq> sets M" using Q' by auto |
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353 show "incseq ?O" by (fastforce intro!: incseq_SucI) |
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354 qed |
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355 have Q'_sets[measurable]: "\<And>i. Q' i \<in> sets M" using Q' by auto |
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356 have O_sets: "\<And>i. ?O i \<in> sets M" using Q' by auto |
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357 then have O_in_G: "\<And>i. ?O i \<in> ?Q" |
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358 proof (safe del: notI) |
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359 fix i have "Q' ` {..i} \<subseteq> sets M" using Q' by auto |
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360 then have "N (?O i) \<le> (\<Sum>i\<le>i. N (Q' i))" |
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361 by (simp add: emeasure_subadditive_finite) |
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362 also have "\<dots> < \<infinity>" using Q' by (simp add: less_top) |
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363 finally show "N (?O i) \<noteq> \<infinity>" by simp |
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364 qed auto |
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365 have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastforce |
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366 have a_eq: "?a = emeasure M (\<Union>i. ?O i)" unfolding Union[symmetric] |
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367 proof (rule antisym) |
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368 show "?a \<le> (SUP i. emeasure M (?O i))" unfolding a_Lim |
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369 using Q' by (auto intro!: SUP_mono emeasure_mono) |
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370 show "(SUP i. emeasure M (?O i)) \<le> ?a" |
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371 proof (safe intro!: Sup_mono, unfold bex_simps) |
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372 fix i |
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373 have *: "(\<Union>(Q' ` {..i})) = ?O i" by auto |
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374 then show "\<exists>x. (x \<in> sets M \<and> N x \<noteq> \<infinity>) \<and> |
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375 emeasure M (\<Union>(Q' ` {..i})) \<le> emeasure M x" |
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376 using O_in_G[of i] by (auto intro!: exI[of _ "?O i"]) |
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377 qed |
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378 qed |
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379 let ?O_0 = "(\<Union>i. ?O i)" |
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380 have "?O_0 \<in> sets M" using Q' by auto |
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381 have "disjointed Q' i \<in> sets M" for i |
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382 using sets.range_disjointed_sets[of Q' M] using Q'_sets by (auto simp: subset_eq) |
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383 note Q_sets = this |
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384 show ?thesis |
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385 proof (intro bexI exI conjI ballI impI allI) |
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386 show "disjoint_family (disjointed Q')" |
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387 by (rule disjoint_family_disjointed) |
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388 show "range (disjointed Q') \<subseteq> sets M" |
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389 using Q'_sets by (intro sets.range_disjointed_sets) auto |
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390 { fix A assume A: "A \<in> sets M" "A \<inter> (\<Union>i. disjointed Q' i) = {}" |
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391 then have A1: "A \<inter> (\<Union>i. Q' i) = {}" |
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392 unfolding UN_disjointed_eq by auto |
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393 show "emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>" |
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394 proof (rule disjCI, simp) |
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395 assume *: "emeasure M A = 0 \<or> N A \<noteq> top" |
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396 show "emeasure M A = 0 \<and> N A = 0" |
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397 proof (cases "emeasure M A = 0") |
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398 case True |
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399 with ac A have "N A = 0" |
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400 unfolding absolutely_continuous_def by auto |
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401 with True show ?thesis by simp |
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402 next |
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403 case False |
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404 with * have "N A \<noteq> \<infinity>" by auto |
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405 with A have "emeasure M ?O_0 + emeasure M A = emeasure M (?O_0 \<union> A)" |
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406 using Q' A1 by (auto intro!: plus_emeasure simp: set_eq_iff) |
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407 also have "\<dots> = (SUP i. emeasure M (?O i \<union> A))" |
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408 proof (rule SUP_emeasure_incseq[of "\<lambda>i. ?O i \<union> A", symmetric, simplified]) |
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409 show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M" |
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410 using \<open>N A \<noteq> \<infinity>\<close> O_sets A by auto |
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411 qed (fastforce intro!: incseq_SucI) |
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412 also have "\<dots> \<le> ?a" |
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413 proof (safe intro!: SUP_least) |
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414 fix i have "?O i \<union> A \<in> ?Q" |
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415 proof (safe del: notI) |
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416 show "?O i \<union> A \<in> sets M" using O_sets A by auto |
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417 from O_in_G[of i] have "N (?O i \<union> A) \<le> N (?O i) + N A" |
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418 using emeasure_subadditive[of "?O i" N A] A O_sets by auto |
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419 with O_in_G[of i] show "N (?O i \<union> A) \<noteq> \<infinity>" |
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420 using \<open>N A \<noteq> \<infinity>\<close> by (auto simp: top_unique) |
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421 qed |
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422 then show "emeasure M (?O i \<union> A) \<le> ?a" by (rule SUP_upper) |
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423 qed |
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424 finally have "emeasure M A = 0" |
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425 unfolding a_eq using measure_nonneg[of M A] by (simp add: emeasure_eq_measure) |
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426 with \<open>emeasure M A \<noteq> 0\<close> show ?thesis by auto |
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427 qed |
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428 qed } |
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429 { fix i |
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430 have "N (disjointed Q' i) \<le> N (Q' i)" |
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431 by (auto intro!: emeasure_mono simp: disjointed_def) |
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432 then show "N (disjointed Q' i) \<noteq> \<infinity>" |
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433 using Q'(2)[of i] by (auto simp: top_unique) } |
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434 qed |
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435 qed |
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436 |
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437 lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite: |
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438 assumes "absolutely_continuous M N" and sets_eq: "sets N = sets M" |
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439 shows "\<exists>f\<in>borel_measurable M. density M f = N" |
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440 proof - |
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441 from split_space_into_finite_sets_and_rest[OF assms] |
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442 obtain Q :: "nat \<Rightarrow> 'a set" |
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443 where Q: "disjoint_family Q" "range Q \<subseteq> sets M" |
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444 and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<inter> (\<Union>i. Q i) = {} \<Longrightarrow> emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>" |
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445 and Q_fin: "\<And>i. N (Q i) \<noteq> \<infinity>" by force |
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446 from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto |
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447 let ?N = "\<lambda>i. density N (indicator (Q i))" and ?M = "\<lambda>i. density M (indicator (Q i))" |
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448 have "\<forall>i. \<exists>f\<in>borel_measurable (?M i). density (?M i) f = ?N i" |
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449 proof (intro allI finite_measure.Radon_Nikodym_finite_measure) |
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450 fix i |
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451 from Q show "finite_measure (?M i)" |
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452 by (auto intro!: finite_measureI cong: nn_integral_cong |
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453 simp add: emeasure_density subset_eq sets_eq) |
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454 from Q have "emeasure (?N i) (space N) = emeasure N (Q i)" |
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455 by (simp add: sets_eq[symmetric] emeasure_density subset_eq cong: nn_integral_cong) |
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456 with Q_fin show "finite_measure (?N i)" |
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457 by (auto intro!: finite_measureI) |
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458 show "sets (?N i) = sets (?M i)" by (simp add: sets_eq) |
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459 have [measurable]: "\<And>A. A \<in> sets M \<Longrightarrow> A \<in> sets N" by (simp add: sets_eq) |
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460 show "absolutely_continuous (?M i) (?N i)" |
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461 using \<open>absolutely_continuous M N\<close> \<open>Q i \<in> sets M\<close> |
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462 by (auto simp: absolutely_continuous_def null_sets_density_iff sets_eq |
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463 intro!: absolutely_continuous_AE[OF sets_eq]) |
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464 qed |
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465 from choice[OF this[unfolded Bex_def]] |
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466 obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x" |
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467 and f_density: "\<And>i. density (?M i) (f i) = ?N i" |
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468 by force |
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469 { fix A i assume A: "A \<in> sets M" |
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470 with Q borel have "(\<integral>\<^sup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M) = emeasure (density (?M i) (f i)) A" |
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471 by (auto simp add: emeasure_density nn_integral_density subset_eq |
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472 intro!: nn_integral_cong split: split_indicator) |
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473 also have "\<dots> = emeasure N (Q i \<inter> A)" |
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474 using A Q by (simp add: f_density emeasure_restricted subset_eq sets_eq) |
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475 finally have "emeasure N (Q i \<inter> A) = (\<integral>\<^sup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)" .. } |
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476 note integral_eq = this |
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477 let ?f = "\<lambda>x. (\<Sum>i. f i x * indicator (Q i) x) + \<infinity> * indicator (space M - (\<Union>i. Q i)) x" |
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478 show ?thesis |
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479 proof (safe intro!: bexI[of _ ?f]) |
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480 show "?f \<in> borel_measurable M" using borel Q_sets |
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481 by (auto intro!: measurable_If) |
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482 show "density M ?f = N" |
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483 proof (rule measure_eqI) |
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484 fix A assume "A \<in> sets (density M ?f)" |
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485 then have "A \<in> sets M" by simp |
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486 have Qi: "\<And>i. Q i \<in> sets M" using Q by auto |
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487 have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M" |
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488 "\<And>i. AE x in M. 0 \<le> f i x * indicator (Q i \<inter> A) x" |
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489 using borel Qi \<open>A \<in> sets M\<close> by auto |
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490 have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) + \<infinity> * indicator ((space M - (\<Union>i. Q i)) \<inter> A) x \<partial>M)" |
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491 using borel by (intro nn_integral_cong) (auto simp: indicator_def) |
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492 also have "\<dots> = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)" |
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493 using borel Qi \<open>A \<in> sets M\<close> |
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494 by (subst nn_integral_add) |
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495 (auto simp add: nn_integral_cmult_indicator sets.Int intro!: suminf_0_le) |
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496 also have "\<dots> = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)" |
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497 by (subst integral_eq[OF \<open>A \<in> sets M\<close>], subst nn_integral_suminf) auto |
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498 finally have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)" . |
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499 moreover have "(\<Sum>i. N (Q i \<inter> A)) = N ((\<Union>i. Q i) \<inter> A)" |
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500 using Q Q_sets \<open>A \<in> sets M\<close> |
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501 by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq) |
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502 moreover |
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503 have "(space M - (\<Union>x. Q x)) \<inter> A \<inter> (\<Union>x. Q x) = {}" |
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504 by auto |
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505 then have "\<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A) = N ((space M - (\<Union>i. Q i)) \<inter> A)" |
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506 using in_Q0[of "(space M - (\<Union>i. Q i)) \<inter> A"] \<open>A \<in> sets M\<close> Q by (auto simp: ennreal_top_mult) |
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507 moreover have "(space M - (\<Union>i. Q i)) \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M" |
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508 using Q_sets \<open>A \<in> sets M\<close> by auto |
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509 moreover have "((\<Union>i. Q i) \<inter> A) \<union> ((space M - (\<Union>i. Q i)) \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> ((space M - (\<Union>i. Q i)) \<inter> A) = {}" |
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510 using \<open>A \<in> sets M\<close> sets.sets_into_space by auto |
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511 ultimately have "N A = (\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M)" |
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512 using plus_emeasure[of "(\<Union>i. Q i) \<inter> A" N "(space M - (\<Union>i. Q i)) \<inter> A"] by (simp add: sets_eq) |
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513 with \<open>A \<in> sets M\<close> borel Q show "emeasure (density M ?f) A = N A" |
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514 by (auto simp: subset_eq emeasure_density) |
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515 qed (simp add: sets_eq) |
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516 qed |
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517 qed |
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518 |
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519 lemma (in sigma_finite_measure) Radon_Nikodym: |
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520 assumes ac: "absolutely_continuous M N" assumes sets_eq: "sets N = sets M" |
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521 shows "\<exists>f \<in> borel_measurable M. density M f = N" |
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522 proof - |
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523 from Ex_finite_integrable_function |
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524 obtain h where finite: "integral\<^sup>N M h \<noteq> \<infinity>" and |
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525 borel: "h \<in> borel_measurable M" and |
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526 nn: "\<And>x. 0 \<le> h x" and |
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527 pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and |
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528 "\<And>x. x \<in> space M \<Longrightarrow> h x < \<infinity>" by auto |
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529 let ?T = "\<lambda>A. (\<integral>\<^sup>+x. h x * indicator A x \<partial>M)" |
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530 let ?MT = "density M h" |
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531 from borel finite nn interpret T: finite_measure ?MT |
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532 by (auto intro!: finite_measureI cong: nn_integral_cong simp: emeasure_density) |
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533 have "absolutely_continuous ?MT N" "sets N = sets ?MT" |
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534 proof (unfold absolutely_continuous_def, safe) |
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535 fix A assume "A \<in> null_sets ?MT" |
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536 with borel have "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> h x \<le> 0" |
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537 by (auto simp add: null_sets_density_iff) |
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538 with pos sets.sets_into_space have "AE x in M. x \<notin> A" |
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539 by (elim eventually_mono) (auto simp: not_le[symmetric]) |
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540 then have "A \<in> null_sets M" |
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541 using \<open>A \<in> sets M\<close> by (simp add: AE_iff_null_sets) |
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542 with ac show "A \<in> null_sets N" |
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543 by (auto simp: absolutely_continuous_def) |
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544 qed (auto simp add: sets_eq) |
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545 from T.Radon_Nikodym_finite_measure_infinite[OF this] |
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546 obtain f where f_borel: "f \<in> borel_measurable M" "density ?MT f = N" by auto |
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547 with nn borel show ?thesis |
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548 by (auto intro!: bexI[of _ "\<lambda>x. h x * f x"] simp: density_density_eq) |
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549 qed |
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550 |
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551 subsection \<open>Uniqueness of densities\<close> |
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552 |
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553 lemma finite_density_unique: |
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554 assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
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555 assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x" |
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556 and fin: "integral\<^sup>N M f \<noteq> \<infinity>" |
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557 shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)" |
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558 proof (intro iffI ballI) |
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559 fix A assume eq: "AE x in M. f x = g x" |
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560 with borel show "density M f = density M g" |
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561 by (auto intro: density_cong) |
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562 next |
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563 let ?P = "\<lambda>f A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M" |
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564 assume "density M f = density M g" |
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565 with borel have eq: "\<forall>A\<in>sets M. ?P f A = ?P g A" |
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566 by (simp add: emeasure_density[symmetric]) |
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567 from this[THEN bspec, OF sets.top] fin |
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568 have g_fin: "integral\<^sup>N M g \<noteq> \<infinity>" by (simp cong: nn_integral_cong) |
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569 { fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
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570 and pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x" |
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571 and g_fin: "integral\<^sup>N M g \<noteq> \<infinity>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A" |
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572 let ?N = "{x\<in>space M. g x < f x}" |
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573 have N: "?N \<in> sets M" using borel by simp |
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574 have "?P g ?N \<le> integral\<^sup>N M g" using pos |
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575 by (intro nn_integral_mono_AE) (auto split: split_indicator) |
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576 then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by (auto simp: top_unique) |
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577 have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^sup>+x. f x * indicator ?N x - g x * indicator ?N x \<partial>M)" |
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578 by (auto intro!: nn_integral_cong simp: indicator_def) |
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579 also have "\<dots> = ?P f ?N - ?P g ?N" |
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580 proof (rule nn_integral_diff) |
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581 show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M" |
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582 using borel N by auto |
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583 show "AE x in M. g x * indicator ?N x \<le> f x * indicator ?N x" |
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584 using pos by (auto split: split_indicator) |
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585 qed fact |
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586 also have "\<dots> = 0" |
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587 unfolding eq[THEN bspec, OF N] using Pg_fin by auto |
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588 finally have "AE x in M. f x \<le> g x" |
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589 using pos borel nn_integral_PInf_AE[OF borel(2) g_fin] |
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590 by (subst (asm) nn_integral_0_iff_AE) |
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591 (auto split: split_indicator simp: not_less ennreal_minus_eq_0) } |
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592 from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq |
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593 show "AE x in M. f x = g x" by auto |
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594 qed |
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595 |
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596 lemma (in finite_measure) density_unique_finite_measure: |
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597 assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M" |
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598 assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> f' x" |
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599 assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. f' x * indicator A x \<partial>M)" |
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600 (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A") |
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601 shows "AE x in M. f x = f' x" |
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602 proof - |
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603 let ?D = "\<lambda>f. density M f" |
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604 let ?N = "\<lambda>A. ?P f A" and ?N' = "\<lambda>A. ?P f' A" |
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605 let ?f = "\<lambda>A x. f x * indicator A x" and ?f' = "\<lambda>A x. f' x * indicator A x" |
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606 |
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607 have ac: "absolutely_continuous M (density M f)" "sets (density M f) = sets M" |
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608 using borel by (auto intro!: absolutely_continuousI_density) |
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609 from split_space_into_finite_sets_and_rest[OF this] |
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610 obtain Q :: "nat \<Rightarrow> 'a set" |
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611 where Q: "disjoint_family Q" "range Q \<subseteq> sets M" |
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612 and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<inter> (\<Union>i. Q i) = {} \<Longrightarrow> emeasure M A = 0 \<and> ?D f A = 0 \<or> 0 < emeasure M A \<and> ?D f A = \<infinity>" |
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613 and Q_fin: "\<And>i. ?D f (Q i) \<noteq> \<infinity>" by force |
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614 with borel pos have in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<inter> (\<Union>i. Q i) = {} \<Longrightarrow> emeasure M A = 0 \<and> ?N A = 0 \<or> 0 < emeasure M A \<and> ?N A = \<infinity>" |
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615 and Q_fin: "\<And>i. ?N (Q i) \<noteq> \<infinity>" by (auto simp: emeasure_density subset_eq) |
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616 |
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617 from Q have Q_sets[measurable]: "\<And>i. Q i \<in> sets M" by auto |
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618 let ?D = "{x\<in>space M. f x \<noteq> f' x}" |
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619 have "?D \<in> sets M" using borel by auto |
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620 have *: "\<And>i x A. \<And>y::ennreal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x" |
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621 unfolding indicator_def by auto |
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622 have "\<forall>i. AE x in M. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos |
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623 by (intro finite_density_unique[THEN iffD1] allI) |
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624 (auto intro!: f measure_eqI simp: emeasure_density * subset_eq) |
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625 moreover have "AE x in M. ?f (space M - (\<Union>i. Q i)) x = ?f' (space M - (\<Union>i. Q i)) x" |
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626 proof (rule AE_I') |
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627 { fix f :: "'a \<Rightarrow> ennreal" assume borel: "f \<in> borel_measurable M" |
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628 and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?N A = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)" |
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629 let ?A = "\<lambda>i. (space M - (\<Union>i. Q i)) \<inter> {x \<in> space M. f x < (i::nat)}" |
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630 have "(\<Union>i. ?A i) \<in> null_sets M" |
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631 proof (rule null_sets_UN) |
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632 fix i ::nat have "?A i \<in> sets M" |
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633 using borel by auto |
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634 have "?N (?A i) \<le> (\<integral>\<^sup>+x. (i::ennreal) * indicator (?A i) x \<partial>M)" |
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635 unfolding eq[OF \<open>?A i \<in> sets M\<close>] |
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636 by (auto intro!: nn_integral_mono simp: indicator_def) |
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637 also have "\<dots> = i * emeasure M (?A i)" |
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638 using \<open>?A i \<in> sets M\<close> by (auto intro!: nn_integral_cmult_indicator) |
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639 also have "\<dots> < \<infinity>" using emeasure_real[of "?A i"] by (auto simp: ennreal_mult_less_top of_nat_less_top) |
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640 finally have "?N (?A i) \<noteq> \<infinity>" by simp |
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641 then show "?A i \<in> null_sets M" using in_Q0[OF \<open>?A i \<in> sets M\<close>] \<open>?A i \<in> sets M\<close> by auto |
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642 qed |
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643 also have "(\<Union>i. ?A i) = (space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}" |
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644 by (auto simp: ennreal_Ex_less_of_nat less_top[symmetric]) |
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645 finally have "(space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" by simp } |
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646 from this[OF borel(1) refl] this[OF borel(2) f] |
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647 have "(space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" "(space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets M" by simp_all |
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648 then show "((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> ((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>}) \<in> null_sets M" by (rule null_sets.Un) |
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649 show "{x \<in> space M. ?f (space M - (\<Union>i. Q i)) x \<noteq> ?f' (space M - (\<Union>i. Q i)) x} \<subseteq> |
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650 ((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> ((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>})" by (auto simp: indicator_def) |
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651 qed |
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652 moreover have "AE x in M. (?f (space M - (\<Union>i. Q i)) x = ?f' (space M - (\<Union>i. Q i)) x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow> |
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653 ?f (space M) x = ?f' (space M) x" |
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654 by (auto simp: indicator_def) |
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655 ultimately have "AE x in M. ?f (space M) x = ?f' (space M) x" |
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656 unfolding AE_all_countable[symmetric] |
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657 by eventually_elim (auto split: if_split_asm simp: indicator_def) |
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658 then show "AE x in M. f x = f' x" by auto |
|
659 qed |
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660 |
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661 lemma (in sigma_finite_measure) density_unique: |
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662 assumes f: "f \<in> borel_measurable M" |
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663 assumes f': "f' \<in> borel_measurable M" |
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664 assumes density_eq: "density M f = density M f'" |
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665 shows "AE x in M. f x = f' x" |
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666 proof - |
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667 obtain h where h_borel: "h \<in> borel_measurable M" |
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668 and fin: "integral\<^sup>N M h \<noteq> \<infinity>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<infinity>" "\<And>x. 0 \<le> h x" |
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669 using Ex_finite_integrable_function by auto |
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670 then have h_nn: "AE x in M. 0 \<le> h x" by auto |
|
671 let ?H = "density M h" |
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672 interpret h: finite_measure ?H |
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673 using fin h_borel pos |
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674 by (intro finite_measureI) (simp cong: nn_integral_cong emeasure_density add: fin) |
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675 let ?fM = "density M f" |
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676 let ?f'M = "density M f'" |
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677 { fix A assume "A \<in> sets M" |
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678 then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A" |
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679 using pos(1) sets.sets_into_space by (force simp: indicator_def) |
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680 then have "(\<integral>\<^sup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets M" |
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681 using h_borel \<open>A \<in> sets M\<close> h_nn by (subst nn_integral_0_iff) auto } |
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682 note h_null_sets = this |
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683 { fix A assume "A \<in> sets M" |
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684 have "(\<integral>\<^sup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?fM)" |
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685 using \<open>A \<in> sets M\<close> h_borel h_nn f f' |
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686 by (intro nn_integral_density[symmetric]) auto |
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687 also have "\<dots> = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?f'M)" |
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688 by (simp_all add: density_eq) |
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689 also have "\<dots> = (\<integral>\<^sup>+x. f' x * (h x * indicator A x) \<partial>M)" |
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690 using \<open>A \<in> sets M\<close> h_borel h_nn f f' |
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691 by (intro nn_integral_density) auto |
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692 finally have "(\<integral>\<^sup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * (f' x * indicator A x) \<partial>M)" |
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693 by (simp add: ac_simps) |
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694 then have "(\<integral>\<^sup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^sup>+x. (f' x * indicator A x) \<partial>?H)" |
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695 using \<open>A \<in> sets M\<close> h_borel h_nn f f' |
|
696 by (subst (asm) (1 2) nn_integral_density[symmetric]) auto } |
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697 then have "AE x in ?H. f x = f' x" using h_borel h_nn f f' |
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698 by (intro h.density_unique_finite_measure absolutely_continuous_AE[of M]) auto |
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699 with AE_space[of M] pos show "AE x in M. f x = f' x" |
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700 unfolding AE_density[OF h_borel] by auto |
|
701 qed |
|
702 |
|
703 lemma (in sigma_finite_measure) density_unique_iff: |
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704 assumes f: "f \<in> borel_measurable M" and f': "f' \<in> borel_measurable M" |
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705 shows "density M f = density M f' \<longleftrightarrow> (AE x in M. f x = f' x)" |
|
706 using density_unique[OF assms] density_cong[OF f f'] by auto |
|
707 |
|
708 lemma sigma_finite_density_unique: |
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709 assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
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710 and fin: "sigma_finite_measure (density M f)" |
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711 shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)" |
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712 proof |
|
713 assume "AE x in M. f x = g x" with borel show "density M f = density M g" |
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714 by (auto intro: density_cong) |
|
715 next |
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716 assume eq: "density M f = density M g" |
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717 interpret f: sigma_finite_measure "density M f" by fact |
|
718 from f.sigma_finite_incseq guess A . note cover = this |
|
719 |
|
720 have "AE x in M. \<forall>i. x \<in> A i \<longrightarrow> f x = g x" |
|
721 unfolding AE_all_countable |
|
722 proof |
|
723 fix i |
|
724 have "density (density M f) (indicator (A i)) = density (density M g) (indicator (A i))" |
|
725 unfolding eq .. |
|
726 moreover have "(\<integral>\<^sup>+x. f x * indicator (A i) x \<partial>M) \<noteq> \<infinity>" |
|
727 using cover(1) cover(3)[of i] borel by (auto simp: emeasure_density subset_eq) |
|
728 ultimately have "AE x in M. f x * indicator (A i) x = g x * indicator (A i) x" |
|
729 using borel cover(1) |
|
730 by (intro finite_density_unique[THEN iffD1]) (auto simp: density_density_eq subset_eq) |
|
731 then show "AE x in M. x \<in> A i \<longrightarrow> f x = g x" |
|
732 by auto |
|
733 qed |
|
734 with AE_space show "AE x in M. f x = g x" |
|
735 apply eventually_elim |
|
736 using cover(2)[symmetric] |
|
737 apply auto |
|
738 done |
|
739 qed |
|
740 |
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741 lemma (in sigma_finite_measure) sigma_finite_iff_density_finite': |
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742 assumes f: "f \<in> borel_measurable M" |
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743 shows "sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)" |
|
744 (is "sigma_finite_measure ?N \<longleftrightarrow> _") |
|
745 proof |
|
746 assume "sigma_finite_measure ?N" |
|
747 then interpret N: sigma_finite_measure ?N . |
|
748 from N.Ex_finite_integrable_function obtain h where |
|
749 h: "h \<in> borel_measurable M" "integral\<^sup>N ?N h \<noteq> \<infinity>" and |
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750 fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>" |
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751 by auto |
|
752 have "AE x in M. f x * h x \<noteq> \<infinity>" |
|
753 proof (rule AE_I') |
|
754 have "integral\<^sup>N ?N h = (\<integral>\<^sup>+x. f x * h x \<partial>M)" |
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755 using f h by (auto intro!: nn_integral_density) |
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756 then have "(\<integral>\<^sup>+x. f x * h x \<partial>M) \<noteq> \<infinity>" |
|
757 using h(2) by simp |
|
758 then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets M" |
|
759 using f h(1) by (auto intro!: nn_integral_PInf[unfolded infinity_ennreal_def] borel_measurable_vimage) |
|
760 qed auto |
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761 then show "AE x in M. f x \<noteq> \<infinity>" |
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762 using fin by (auto elim!: AE_Ball_mp simp: less_top ennreal_mult_less_top) |
|
763 next |
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764 assume AE: "AE x in M. f x \<noteq> \<infinity>" |
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765 from sigma_finite guess Q . note Q = this |
|
766 define A where "A i = |
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767 f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. ennreal(of_nat (Suc n))}) \<inter> space M" for i |
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768 { fix i j have "A i \<inter> Q j \<in> sets M" |
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769 unfolding A_def using f Q |
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770 apply (rule_tac sets.Int) |
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771 by (cases i) (auto intro: measurable_sets[OF f(1)]) } |
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772 note A_in_sets = this |
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773 |
|
774 show "sigma_finite_measure ?N" |
|
775 proof (standard, intro exI conjI ballI) |
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776 show "countable (range (\<lambda>(i, j). A i \<inter> Q j))" |
|
777 by auto |
|
778 show "range (\<lambda>(i, j). A i \<inter> Q j) \<subseteq> sets (density M f)" |
|
779 using A_in_sets by auto |
|
780 next |
|
781 have "\<Union>range (\<lambda>(i, j). A i \<inter> Q j) = (\<Union>i j. A i \<inter> Q j)" |
|
782 by auto |
|
783 also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto |
|
784 also have "(\<Union>i. A i) = space M" |
|
785 proof safe |
|
786 fix x assume x: "x \<in> space M" |
|
787 show "x \<in> (\<Union>i. A i)" |
|
788 proof (cases "f x" rule: ennreal_cases) |
|
789 case top with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0]) |
|
790 next |
|
791 case (real r) |
|
792 with ennreal_Ex_less_of_nat[of "f x"] obtain n :: nat where "f x < n" |
|
793 by auto |
|
794 also have "n < (Suc n :: ennreal)" |
|
795 by simp |
|
796 finally show ?thesis |
|
797 using x real by (auto simp: A_def ennreal_of_nat_eq_real_of_nat intro!: exI[of _ "Suc n"]) |
|
798 qed |
|
799 qed (auto simp: A_def) |
|
800 finally show "\<Union>range (\<lambda>(i, j). A i \<inter> Q j) = space ?N" by simp |
|
801 next |
|
802 fix X assume "X \<in> range (\<lambda>(i, j). A i \<inter> Q j)" |
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803 then obtain i j where [simp]:"X = A i \<inter> Q j" by auto |
|
804 have "(\<integral>\<^sup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>" |
|
805 proof (cases i) |
|
806 case 0 |
|
807 have "AE x in M. f x * indicator (A i \<inter> Q j) x = 0" |
|
808 using AE by (auto simp: A_def \<open>i = 0\<close>) |
|
809 from nn_integral_cong_AE[OF this] show ?thesis by simp |
|
810 next |
|
811 case (Suc n) |
|
812 then have "(\<integral>\<^sup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le> |
|
813 (\<integral>\<^sup>+x. (Suc n :: ennreal) * indicator (Q j) x \<partial>M)" |
|
814 by (auto intro!: nn_integral_mono simp: indicator_def A_def ennreal_of_nat_eq_real_of_nat) |
|
815 also have "\<dots> = Suc n * emeasure M (Q j)" |
|
816 using Q by (auto intro!: nn_integral_cmult_indicator) |
|
817 also have "\<dots> < \<infinity>" |
|
818 using Q by (auto simp: ennreal_mult_less_top less_top of_nat_less_top) |
|
819 finally show ?thesis by simp |
|
820 qed |
|
821 then show "emeasure ?N X \<noteq> \<infinity>" |
|
822 using A_in_sets Q f by (auto simp: emeasure_density) |
|
823 qed |
|
824 qed |
|
825 |
|
826 lemma (in sigma_finite_measure) sigma_finite_iff_density_finite: |
|
827 "f \<in> borel_measurable M \<Longrightarrow> sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)" |
|
828 by (subst sigma_finite_iff_density_finite') |
|
829 (auto simp: max_def intro!: measurable_If) |
|
830 |
|
831 subsection \<open>Radon-Nikodym derivative\<close> |
|
832 |
|
833 definition RN_deriv :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a \<Rightarrow> ennreal" where |
|
834 "RN_deriv M N = |
|
835 (if \<exists>f. f \<in> borel_measurable M \<and> density M f = N |
|
836 then SOME f. f \<in> borel_measurable M \<and> density M f = N |
|
837 else (\<lambda>_. 0))" |
|
838 |
|
839 lemma RN_derivI: |
|
840 assumes "f \<in> borel_measurable M" "density M f = N" |
|
841 shows "density M (RN_deriv M N) = N" |
|
842 proof - |
|
843 have *: "\<exists>f. f \<in> borel_measurable M \<and> density M f = N" |
|
844 using assms by auto |
|
845 then have "density M (SOME f. f \<in> borel_measurable M \<and> density M f = N) = N" |
|
846 by (rule someI2_ex) auto |
|
847 with * show ?thesis |
|
848 by (auto simp: RN_deriv_def) |
|
849 qed |
|
850 |
|
851 lemma borel_measurable_RN_deriv[measurable]: "RN_deriv M N \<in> borel_measurable M" |
|
852 proof - |
|
853 { assume ex: "\<exists>f. f \<in> borel_measurable M \<and> density M f = N" |
|
854 have 1: "(SOME f. f \<in> borel_measurable M \<and> density M f = N) \<in> borel_measurable M" |
|
855 using ex by (rule someI2_ex) auto } |
|
856 from this show ?thesis |
|
857 by (auto simp: RN_deriv_def) |
|
858 qed |
|
859 |
|
860 lemma density_RN_deriv_density: |
|
861 assumes f: "f \<in> borel_measurable M" |
|
862 shows "density M (RN_deriv M (density M f)) = density M f" |
|
863 by (rule RN_derivI[OF f]) simp |
|
864 |
|
865 lemma (in sigma_finite_measure) density_RN_deriv: |
|
866 "absolutely_continuous M N \<Longrightarrow> sets N = sets M \<Longrightarrow> density M (RN_deriv M N) = N" |
|
867 by (metis RN_derivI Radon_Nikodym) |
|
868 |
|
869 lemma (in sigma_finite_measure) RN_deriv_nn_integral: |
|
870 assumes N: "absolutely_continuous M N" "sets N = sets M" |
|
871 and f: "f \<in> borel_measurable M" |
|
872 shows "integral\<^sup>N N f = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)" |
|
873 proof - |
|
874 have "integral\<^sup>N N f = integral\<^sup>N (density M (RN_deriv M N)) f" |
|
875 using N by (simp add: density_RN_deriv) |
|
876 also have "\<dots> = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)" |
|
877 using f by (simp add: nn_integral_density) |
|
878 finally show ?thesis by simp |
|
879 qed |
|
880 |
|
881 lemma null_setsD_AE: "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N" |
|
882 using AE_iff_null_sets[of N M] by auto |
|
883 |
|
884 lemma (in sigma_finite_measure) RN_deriv_unique: |
|
885 assumes f: "f \<in> borel_measurable M" |
|
886 and eq: "density M f = N" |
|
887 shows "AE x in M. f x = RN_deriv M N x" |
|
888 unfolding eq[symmetric] |
|
889 by (intro density_unique_iff[THEN iffD1] f borel_measurable_RN_deriv |
|
890 density_RN_deriv_density[symmetric]) |
|
891 |
|
892 lemma RN_deriv_unique_sigma_finite: |
|
893 assumes f: "f \<in> borel_measurable M" |
|
894 and eq: "density M f = N" and fin: "sigma_finite_measure N" |
|
895 shows "AE x in M. f x = RN_deriv M N x" |
|
896 using fin unfolding eq[symmetric] |
|
897 by (intro sigma_finite_density_unique[THEN iffD1] f borel_measurable_RN_deriv |
|
898 density_RN_deriv_density[symmetric]) |
|
899 |
|
900 lemma (in sigma_finite_measure) RN_deriv_distr: |
|
901 fixes T :: "'a \<Rightarrow> 'b" |
|
902 assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M" |
|
903 and inv: "\<forall>x\<in>space M. T' (T x) = x" |
|
904 and ac[simp]: "absolutely_continuous (distr M M' T) (distr N M' T)" |
|
905 and N: "sets N = sets M" |
|
906 shows "AE x in M. RN_deriv (distr M M' T) (distr N M' T) (T x) = RN_deriv M N x" |
|
907 proof (rule RN_deriv_unique) |
|
908 have [simp]: "sets N = sets M" by fact |
|
909 note sets_eq_imp_space_eq[OF N, simp] |
|
910 have measurable_N[simp]: "\<And>M'. measurable N M' = measurable M M'" by (auto simp: measurable_def) |
|
911 { fix A assume "A \<in> sets M" |
|
912 with inv T T' sets.sets_into_space[OF this] |
|
913 have "T -` T' -` A \<inter> T -` space M' \<inter> space M = A" |
|
914 by (auto simp: measurable_def) } |
|
915 note eq = this[simp] |
|
916 { fix A assume "A \<in> sets M" |
|
917 with inv T T' sets.sets_into_space[OF this] |
|
918 have "(T' \<circ> T) -` A \<inter> space M = A" |
|
919 by (auto simp: measurable_def) } |
|
920 note eq2 = this[simp] |
|
921 let ?M' = "distr M M' T" and ?N' = "distr N M' T" |
|
922 interpret M': sigma_finite_measure ?M' |
|
923 proof |
|
924 from sigma_finite_countable guess F .. note F = this |
|
925 show "\<exists>A. countable A \<and> A \<subseteq> sets (distr M M' T) \<and> \<Union>A = space (distr M M' T) \<and> (\<forall>a\<in>A. emeasure (distr M M' T) a \<noteq> \<infinity>)" |
|
926 proof (intro exI conjI ballI) |
|
927 show *: "(\<lambda>A. T' -` A \<inter> space ?M') ` F \<subseteq> sets ?M'" |
|
928 using F T' by (auto simp: measurable_def) |
|
929 show "\<Union>((\<lambda>A. T' -` A \<inter> space ?M')`F) = space ?M'" |
|
930 using F T'[THEN measurable_space] by (auto simp: set_eq_iff) |
|
931 next |
|
932 fix X assume "X \<in> (\<lambda>A. T' -` A \<inter> space ?M')`F" |
|
933 then obtain A where [simp]: "X = T' -` A \<inter> space ?M'" and "A \<in> F" by auto |
|
934 have "X \<in> sets M'" using F T' \<open>A\<in>F\<close> by auto |
|
935 moreover |
|
936 have Fi: "A \<in> sets M" using F \<open>A\<in>F\<close> by auto |
|
937 ultimately show "emeasure ?M' X \<noteq> \<infinity>" |
|
938 using F T T' \<open>A\<in>F\<close> by (simp add: emeasure_distr) |
|
939 qed (insert F, auto) |
|
940 qed |
|
941 have "(RN_deriv ?M' ?N') \<circ> T \<in> borel_measurable M" |
|
942 using T ac by measurable |
|
943 then show "(\<lambda>x. RN_deriv ?M' ?N' (T x)) \<in> borel_measurable M" |
|
944 by (simp add: comp_def) |
|
945 |
|
946 have "N = distr N M (T' \<circ> T)" |
|
947 by (subst measure_of_of_measure[of N, symmetric]) |
|
948 (auto simp add: distr_def sets.sigma_sets_eq intro!: measure_of_eq sets.space_closed) |
|
949 also have "\<dots> = distr (distr N M' T) M T'" |
|
950 using T T' by (simp add: distr_distr) |
|
951 also have "\<dots> = distr (density (distr M M' T) (RN_deriv (distr M M' T) (distr N M' T))) M T'" |
|
952 using ac by (simp add: M'.density_RN_deriv) |
|
953 also have "\<dots> = density M (RN_deriv (distr M M' T) (distr N M' T) \<circ> T)" |
|
954 by (simp add: distr_density_distr[OF T T', OF inv]) |
|
955 finally show "density M (\<lambda>x. RN_deriv (distr M M' T) (distr N M' T) (T x)) = N" |
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956 by (simp add: comp_def) |
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957 qed |
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958 |
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959 lemma (in sigma_finite_measure) RN_deriv_finite: |
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960 assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M" |
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961 shows "AE x in M. RN_deriv M N x \<noteq> \<infinity>" |
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962 proof - |
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963 interpret N: sigma_finite_measure N by fact |
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964 from N show ?thesis |
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965 using sigma_finite_iff_density_finite[OF borel_measurable_RN_deriv, of N] density_RN_deriv[OF ac] |
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966 by simp |
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967 qed |
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968 |
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969 lemma (in sigma_finite_measure) |
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970 assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M" |
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971 and f: "f \<in> borel_measurable M" |
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972 shows RN_deriv_integrable: "integrable N f \<longleftrightarrow> |
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973 integrable M (\<lambda>x. enn2real (RN_deriv M N x) * f x)" (is ?integrable) |
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974 and RN_deriv_integral: "integral\<^sup>L N f = (\<integral>x. enn2real (RN_deriv M N x) * f x \<partial>M)" (is ?integral) |
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975 proof - |
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976 note ac(2)[simp] and sets_eq_imp_space_eq[OF ac(2), simp] |
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977 interpret N: sigma_finite_measure N by fact |
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978 |
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979 have eq: "density M (RN_deriv M N) = density M (\<lambda>x. enn2real (RN_deriv M N x))" |
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980 proof (rule density_cong) |
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981 from RN_deriv_finite[OF assms(1,2,3)] |
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982 show "AE x in M. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))" |
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983 by eventually_elim (auto simp: less_top) |
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984 qed (insert ac, auto) |
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985 |
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986 show ?integrable |
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987 apply (subst density_RN_deriv[OF ac, symmetric]) |
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988 unfolding eq |
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989 apply (intro integrable_real_density f AE_I2 enn2real_nonneg) |
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990 apply (insert ac, auto) |
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991 done |
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992 |
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993 show ?integral |
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994 apply (subst density_RN_deriv[OF ac, symmetric]) |
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995 unfolding eq |
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996 apply (intro integral_real_density f AE_I2 enn2real_nonneg) |
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997 apply (insert ac, auto) |
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998 done |
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999 qed |
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1000 |
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1001 lemma (in sigma_finite_measure) real_RN_deriv: |
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1002 assumes "finite_measure N" |
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1003 assumes ac: "absolutely_continuous M N" "sets N = sets M" |
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1004 obtains D where "D \<in> borel_measurable M" |
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1005 and "AE x in M. RN_deriv M N x = ennreal (D x)" |
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1006 and "AE x in N. 0 < D x" |
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1007 and "\<And>x. 0 \<le> D x" |
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1008 proof |
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1009 interpret N: finite_measure N by fact |
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1010 |
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1011 note RN = borel_measurable_RN_deriv density_RN_deriv[OF ac] |
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1012 |
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1013 let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M N x = t}" |
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1014 |
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1015 show "(\<lambda>x. enn2real (RN_deriv M N x)) \<in> borel_measurable M" |
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1016 using RN by auto |
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1017 |
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1018 have "N (?RN \<infinity>) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN \<infinity>) x \<partial>M)" |
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1019 using RN(1) by (subst RN(2)[symmetric]) (auto simp: emeasure_density) |
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1020 also have "\<dots> = (\<integral>\<^sup>+ x. \<infinity> * indicator (?RN \<infinity>) x \<partial>M)" |
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1021 by (intro nn_integral_cong) (auto simp: indicator_def) |
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1022 also have "\<dots> = \<infinity> * emeasure M (?RN \<infinity>)" |
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1023 using RN by (intro nn_integral_cmult_indicator) auto |
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1024 finally have eq: "N (?RN \<infinity>) = \<infinity> * emeasure M (?RN \<infinity>)" . |
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1025 moreover |
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1026 have "emeasure M (?RN \<infinity>) = 0" |
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1027 proof (rule ccontr) |
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1028 assume "emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>} \<noteq> 0" |
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1029 then have "0 < emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}" |
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1030 by (auto simp: zero_less_iff_neq_zero) |
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1031 with eq have "N (?RN \<infinity>) = \<infinity>" by (simp add: ennreal_mult_eq_top_iff) |
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1032 with N.emeasure_finite[of "?RN \<infinity>"] RN show False by auto |
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1033 qed |
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1034 ultimately have "AE x in M. RN_deriv M N x < \<infinity>" |
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1035 using RN by (intro AE_iff_measurable[THEN iffD2]) (auto simp: less_top[symmetric]) |
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1036 then show "AE x in M. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))" |
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1037 by auto |
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1038 then have eq: "AE x in N. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))" |
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1039 using ac absolutely_continuous_AE by auto |
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1040 |
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1041 |
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1042 have "N (?RN 0) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN 0) x \<partial>M)" |
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1043 by (subst RN(2)[symmetric]) (auto simp: emeasure_density) |
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1044 also have "\<dots> = (\<integral>\<^sup>+ x. 0 \<partial>M)" |
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1045 by (intro nn_integral_cong) (auto simp: indicator_def) |
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1046 finally have "AE x in N. RN_deriv M N x \<noteq> 0" |
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1047 using RN by (subst AE_iff_measurable[OF _ refl]) (auto simp: ac cong: sets_eq_imp_space_eq) |
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1048 with eq show "AE x in N. 0 < enn2real (RN_deriv M N x)" |
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1049 by (auto simp: enn2real_positive_iff less_top[symmetric] zero_less_iff_neq_zero) |
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1050 qed (rule enn2real_nonneg) |
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1051 |
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1052 lemma (in sigma_finite_measure) RN_deriv_singleton: |
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1053 assumes ac: "absolutely_continuous M N" "sets N = sets M" |
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1054 and x: "{x} \<in> sets M" |
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1055 shows "N {x} = RN_deriv M N x * emeasure M {x}" |
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1056 proof - |
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1057 from \<open>{x} \<in> sets M\<close> |
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1058 have "density M (RN_deriv M N) {x} = (\<integral>\<^sup>+w. RN_deriv M N x * indicator {x} w \<partial>M)" |
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1059 by (auto simp: indicator_def emeasure_density intro!: nn_integral_cong) |
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1060 with x density_RN_deriv[OF ac] show ?thesis |
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1061 by (auto simp: max_def) |
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1062 qed |
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1063 |
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1064 end |