1 theory Equivalence_Lebesgue_Henstock_Integration |
1 theory Equivalence_Lebesgue_Henstock_Integration |
2 imports Lebesgue_Measure Henstock_Kurzweil_Integration |
2 imports Lebesgue_Measure Henstock_Kurzweil_Integration Complete_Measure |
3 begin |
3 begin |
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4 |
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5 lemma le_left_mono: "x \<le> y \<Longrightarrow> y \<le> a \<longrightarrow> x \<le> (a::'a::preorder)" |
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6 by (auto intro: order_trans) |
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7 |
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8 lemma ball_trans: |
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9 assumes "y \<in> ball z q" "r + q \<le> s" shows "ball y r \<subseteq> ball z s" |
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10 proof safe |
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11 fix x assume x: "x \<in> ball y r" |
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12 have "dist z x \<le> dist z y + dist y x" |
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13 by (rule dist_triangle) |
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14 also have "\<dots> < s" |
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15 using assms x by auto |
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16 finally show "x \<in> ball z s" |
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17 by simp |
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18 qed |
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19 |
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20 abbreviation lebesgue :: "'a::euclidean_space measure" |
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21 where "lebesgue \<equiv> completion lborel" |
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22 |
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23 abbreviation lebesgue_on :: "'a set \<Rightarrow> 'a::euclidean_space measure" |
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24 where "lebesgue_on \<Omega> \<equiv> restrict_space (completion lborel) \<Omega>" |
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25 |
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26 lemma has_integral_implies_lebesgue_measurable_cbox: |
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27 fixes f :: "'a :: euclidean_space \<Rightarrow> real" |
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28 assumes f: "(f has_integral I) (cbox x y)" |
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29 shows "f \<in> lebesgue_on (cbox x y) \<rightarrow>\<^sub>M borel" |
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30 proof (rule cld_measure.borel_measurable_cld) |
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31 let ?L = "lebesgue_on (cbox x y)" |
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32 let ?\<mu> = "emeasure ?L" |
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33 let ?\<mu>' = "outer_measure_of ?L" |
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34 interpret L: finite_measure ?L |
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35 proof |
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36 show "?\<mu> (space ?L) \<noteq> \<infinity>" |
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37 by (simp add: emeasure_restrict_space space_restrict_space emeasure_lborel_cbox_eq) |
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38 qed |
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39 |
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40 show "cld_measure ?L" |
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41 proof |
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42 fix B A assume "B \<subseteq> A" "A \<in> null_sets ?L" |
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43 then show "B \<in> sets ?L" |
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44 using null_sets_completion_subset[OF \<open>B \<subseteq> A\<close>, of lborel] |
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45 by (auto simp add: null_sets_restrict_space sets_restrict_space_iff intro: ) |
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46 next |
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47 fix A assume "A \<subseteq> space ?L" "\<And>B. B \<in> sets ?L \<Longrightarrow> ?\<mu> B < \<infinity> \<Longrightarrow> A \<inter> B \<in> sets ?L" |
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48 from this(1) this(2)[of "space ?L"] show "A \<in> sets ?L" |
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49 by (auto simp: Int_absorb2 less_top[symmetric]) |
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50 qed auto |
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51 then interpret cld_measure ?L |
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52 . |
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53 |
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54 have content_eq_L: "A \<in> sets borel \<Longrightarrow> A \<subseteq> cbox x y \<Longrightarrow> content A = measure ?L A" for A |
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55 by (subst measure_restrict_space) (auto simp: measure_def) |
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56 |
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57 fix E and a b :: real assume "E \<in> sets ?L" "a < b" "0 < ?\<mu> E" "?\<mu> E < \<infinity>" |
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58 then obtain M :: real where "?\<mu> E = M" "0 < M" |
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59 by (cases "?\<mu> E") auto |
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60 define e where "e = M / (4 + 2 / (b - a))" |
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61 from \<open>a < b\<close> \<open>0<M\<close> have "0 < e" |
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62 by (auto intro!: divide_pos_pos simp: field_simps e_def) |
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63 |
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64 have "e < M / (3 + 2 / (b - a))" |
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65 using \<open>a < b\<close> \<open>0 < M\<close> |
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66 unfolding e_def by (intro divide_strict_left_mono add_strict_right_mono mult_pos_pos) (auto simp: field_simps) |
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67 then have "2 * e < (b - a) * (M - e * 3)" |
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68 using \<open>0<M\<close> \<open>0 < e\<close> \<open>a < b\<close> by (simp add: field_simps) |
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69 |
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70 have e_less_M: "e < M / 1" |
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71 unfolding e_def using \<open>a < b\<close> \<open>0<M\<close> by (intro divide_strict_left_mono) (auto simp: field_simps) |
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72 |
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73 obtain d |
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74 where "gauge d" |
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75 and integral_f: "\<forall>p. p tagged_division_of cbox x y \<and> d fine p \<longrightarrow> |
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76 norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - I) < e" |
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77 using \<open>0<e\<close> f unfolding has_integral by auto |
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78 |
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79 define C where "C X m = X \<inter> {x. ball x (1/Suc m) \<subseteq> d x}" for X m |
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80 have "incseq (C X)" for X |
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81 unfolding C_def [abs_def] |
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82 by (intro monoI Collect_mono conj_mono imp_refl le_left_mono subset_ball divide_left_mono Int_mono) auto |
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83 |
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84 { fix X assume "X \<subseteq> space ?L" and eq: "?\<mu>' X = ?\<mu> E" |
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85 have "(SUP m. outer_measure_of ?L (C X m)) = outer_measure_of ?L (\<Union>m. C X m)" |
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86 using \<open>X \<subseteq> space ?L\<close> by (intro SUP_outer_measure_of_incseq \<open>incseq (C X)\<close>) (auto simp: C_def) |
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87 also have "(\<Union>m. C X m) = X" |
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88 proof - |
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89 { fix x |
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90 obtain e where "0 < e" "ball x e \<subseteq> d x" |
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91 using gaugeD[OF \<open>gauge d\<close>, of x] unfolding open_contains_ball by auto |
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92 moreover |
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93 obtain n where "1 / (1 + real n) < e" |
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94 using reals_Archimedean[OF \<open>0<e\<close>] by (auto simp: inverse_eq_divide) |
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95 then have "ball x (1 / (1 + real n)) \<subseteq> ball x e" |
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96 by (intro subset_ball) auto |
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97 ultimately have "\<exists>n. ball x (1 / (1 + real n)) \<subseteq> d x" |
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98 by blast } |
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99 then show ?thesis |
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100 by (auto simp: C_def) |
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101 qed |
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102 finally have "(SUP m. outer_measure_of ?L (C X m)) = ?\<mu> E" |
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103 using eq by auto |
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104 also have "\<dots> > M - e" |
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105 using \<open>0 < M\<close> \<open>?\<mu> E = M\<close> \<open>0<e\<close> by (auto intro!: ennreal_lessI) |
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106 finally have "\<exists>m. M - e < outer_measure_of ?L (C X m)" |
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107 unfolding less_SUP_iff by auto } |
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108 note C = this |
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109 |
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110 let ?E = "{x\<in>E. f x \<le> a}" and ?F = "{x\<in>E. b \<le> f x}" |
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111 |
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112 have "\<not> (?\<mu>' ?E = ?\<mu> E \<and> ?\<mu>' ?F = ?\<mu> E)" |
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113 proof |
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114 assume eq: "?\<mu>' ?E = ?\<mu> E \<and> ?\<mu>' ?F = ?\<mu> E" |
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115 with C[of ?E] C[of ?F] \<open>E \<in> sets ?L\<close>[THEN sets.sets_into_space] obtain ma mb |
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116 where "M - e < outer_measure_of ?L (C ?E ma)" "M - e < outer_measure_of ?L (C ?F mb)" |
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117 by auto |
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118 moreover define m where "m = max ma mb" |
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119 ultimately have M_minus_e: "M - e < outer_measure_of ?L (C ?E m)" "M - e < outer_measure_of ?L (C ?F m)" |
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120 using |
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121 incseqD[OF \<open>incseq (C ?E)\<close>, of ma m, THEN outer_measure_of_mono] |
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122 incseqD[OF \<open>incseq (C ?F)\<close>, of mb m, THEN outer_measure_of_mono] |
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123 by (auto intro: less_le_trans) |
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124 define d' where "d' x = d x \<inter> ball x (1 / (3 * Suc m))" for x |
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125 have "gauge d'" |
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126 unfolding d'_def by (intro gauge_inter \<open>gauge d\<close> gauge_ball) auto |
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127 then obtain p where p: "p tagged_division_of cbox x y" "d' fine p" |
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128 by (rule fine_division_exists) |
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129 then have "d fine p" |
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130 unfolding d'_def[abs_def] fine_def by auto |
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131 |
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132 define s where "s = {(x::'a, k). k \<inter> (C ?E m) \<noteq> {} \<and> k \<inter> (C ?F m) \<noteq> {}}" |
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133 define T where "T E k = (SOME x. x \<in> k \<inter> C E m)" for E k |
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134 let ?A = "(\<lambda>(x, k). (T ?E k, k)) ` (p \<inter> s) \<union> (p - s)" |
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135 let ?B = "(\<lambda>(x, k). (T ?F k, k)) ` (p \<inter> s) \<union> (p - s)" |
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136 |
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137 { fix X assume X_eq: "X = ?E \<or> X = ?F" |
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138 let ?T = "(\<lambda>(x, k). (T X k, k))" |
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139 let ?p = "?T ` (p \<inter> s) \<union> (p - s)" |
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140 |
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141 have in_s: "(x, k) \<in> s \<Longrightarrow> T X k \<in> k \<inter> C X m" for x k |
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142 using someI_ex[of "\<lambda>x. x \<in> k \<inter> C X m"] X_eq unfolding ex_in_conv by (auto simp: T_def s_def) |
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143 |
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144 { fix x k assume "(x, k) \<in> p" "(x, k) \<in> s" |
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145 have k: "k \<subseteq> ball x (1 / (3 * Suc m))" |
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146 using \<open>d' fine p\<close>[THEN fineD, OF \<open>(x, k) \<in> p\<close>] by (auto simp: d'_def) |
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147 then have "x \<in> ball (T X k) (1 / (3 * Suc m))" |
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148 using in_s[OF \<open>(x, k) \<in> s\<close>] by (auto simp: C_def subset_eq dist_commute) |
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149 then have "ball x (1 / (3 * Suc m)) \<subseteq> ball (T X k) (1 / Suc m)" |
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150 by (rule ball_trans) (auto simp: divide_simps) |
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151 with k in_s[OF \<open>(x, k) \<in> s\<close>] have "k \<subseteq> d (T X k)" |
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152 by (auto simp: C_def) } |
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153 then have "d fine ?p" |
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154 using \<open>d fine p\<close> by (auto intro!: fineI) |
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155 moreover |
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156 have "?p tagged_division_of cbox x y" |
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157 proof (rule tagged_division_ofI) |
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158 show "finite ?p" |
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159 using p(1) by auto |
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160 next |
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161 fix z k assume *: "(z, k) \<in> ?p" |
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162 then consider "(z, k) \<in> p" "(z, k) \<notin> s" |
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163 | x' where "(x', k) \<in> p" "(x', k) \<in> s" "z = T X k" |
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164 by (auto simp: T_def) |
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165 then have "z \<in> k \<and> k \<subseteq> cbox x y \<and> (\<exists>a b. k = cbox a b)" |
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166 using p(1) by cases (auto dest: in_s) |
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167 then show "z \<in> k" "k \<subseteq> cbox x y" "\<exists>a b. k = cbox a b" |
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168 by auto |
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169 next |
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170 fix z k z' k' assume "(z, k) \<in> ?p" "(z', k') \<in> ?p" "(z, k) \<noteq> (z', k')" |
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171 with tagged_division_ofD(5)[OF p(1), of _ k _ k'] |
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172 show "interior k \<inter> interior k' = {}" |
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173 by (auto simp: T_def dest: in_s) |
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174 next |
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175 have "{k. \<exists>x. (x, k) \<in> ?p} = {k. \<exists>x. (x, k) \<in> p}" |
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176 by (auto simp: T_def image_iff Bex_def) |
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177 then show "\<Union>{k. \<exists>x. (x, k) \<in> ?p} = cbox x y" |
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178 using p(1) by auto |
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179 qed |
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180 ultimately have I: "norm ((\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) - I) < e" |
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181 using integral_f by auto |
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182 |
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183 have "(\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) = |
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184 (\<Sum>(x, k)\<in>?T ` (p \<inter> s). content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p - s. content k *\<^sub>R f x)" |
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185 using p(1)[THEN tagged_division_ofD(1)] |
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186 by (safe intro!: setsum.union_inter_neutral) (auto simp: s_def T_def) |
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187 also have "(\<Sum>(x, k)\<in>?T ` (p \<inter> s). content k *\<^sub>R f x) = (\<Sum>(x, k)\<in>p \<inter> s. content k *\<^sub>R f (T X k))" |
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188 proof (subst setsum.reindex_nontrivial, safe) |
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189 fix x1 x2 k assume 1: "(x1, k) \<in> p" "(x1, k) \<in> s" and 2: "(x2, k) \<in> p" "(x2, k) \<in> s" |
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190 and eq: "content k *\<^sub>R f (T X k) \<noteq> 0" |
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191 with tagged_division_ofD(5)[OF p(1), of x1 k x2 k] tagged_division_ofD(4)[OF p(1), of x1 k] |
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192 show "x1 = x2" |
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193 by (auto simp: content_eq_0_interior) |
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194 qed (use p in \<open>auto intro!: setsum.cong\<close>) |
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195 finally have eq: "(\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) = |
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196 (\<Sum>(x, k)\<in>p \<inter> s. content k *\<^sub>R f (T X k)) + (\<Sum>(x, k)\<in>p - s. content k *\<^sub>R f x)" . |
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197 |
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198 have in_T: "(x, k) \<in> s \<Longrightarrow> T X k \<in> X" for x k |
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199 using in_s[of x k] by (auto simp: C_def) |
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200 |
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201 note I eq in_T } |
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202 note parts = this |
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203 |
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204 have p_in_L: "(x, k) \<in> p \<Longrightarrow> k \<in> sets ?L" for x k |
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205 using tagged_division_ofD(3, 4)[OF p(1), of x k] by (auto simp: sets_restrict_space) |
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206 |
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207 have [simp]: "finite p" |
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208 using tagged_division_ofD(1)[OF p(1)] . |
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209 |
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210 have "(M - 3*e) * (b - a) \<le> (\<Sum>(x, k)\<in>p \<inter> s. content k) * (b - a)" |
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211 proof (intro mult_right_mono) |
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212 have fin: "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) < \<infinity>" for X |
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213 using \<open>?\<mu> E < \<infinity>\<close> by (rule le_less_trans[rotated]) (auto intro!: emeasure_mono \<open>E \<in> sets ?L\<close>) |
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214 have sets: "(E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) \<in> sets ?L" for X |
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215 using tagged_division_ofD(1)[OF p(1)] by (intro sets.Diff \<open>E \<in> sets ?L\<close> sets.finite_Union sets.Int) (auto intro: p_in_L) |
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216 { fix X assume "X \<subseteq> E" "M - e < ?\<mu>' (C X m)" |
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217 have "M - e \<le> ?\<mu>' (C X m)" |
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218 by (rule less_imp_le) fact |
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219 also have "\<dots> \<le> ?\<mu>' (E - (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}))" |
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220 proof (intro outer_measure_of_mono subsetI) |
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221 fix v assume "v \<in> C X m" |
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222 then have "v \<in> cbox x y" "v \<in> E" |
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223 using \<open>E \<subseteq> space ?L\<close> \<open>X \<subseteq> E\<close> by (auto simp: space_restrict_space C_def) |
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224 then obtain z k where "(z, k) \<in> p" "v \<in> k" |
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225 using tagged_division_ofD(6)[OF p(1), symmetric] by auto |
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226 then show "v \<in> E - E \<inter> (\<Union>{k\<in>snd`p. k \<inter> C X m = {}})" |
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227 using \<open>v \<in> C X m\<close> \<open>v \<in> E\<close> by auto |
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228 qed |
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229 also have "\<dots> = ?\<mu> E - ?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}})" |
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230 using \<open>E \<in> sets ?L\<close> fin[of X] sets[of X] by (auto intro!: emeasure_Diff) |
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231 finally have "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) \<le> e" |
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232 using \<open>0 < e\<close> e_less_M apply (cases "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}})") |
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233 by (auto simp add: \<open>?\<mu> E = M\<close> ennreal_minus ennreal_le_iff2) |
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234 note this } |
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235 note upper_bound = this |
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236 |
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237 have "?\<mu> (E \<inter> \<Union>(snd`(p - s))) = |
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238 ?\<mu> ((E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?E m = {}}) \<union> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?F m = {}}))" |
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239 by (intro arg_cong[where f="?\<mu>"]) (auto simp: s_def image_def Bex_def) |
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240 also have "\<dots> \<le> ?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?E m = {}}) + ?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?F m = {}})" |
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241 using sets[of ?E] sets[of ?F] M_minus_e by (intro emeasure_subadditive) auto |
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242 also have "\<dots> \<le> e + ennreal e" |
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243 using upper_bound[of ?E] upper_bound[of ?F] M_minus_e by (intro add_mono) auto |
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244 finally have "?\<mu> E - 2*e \<le> ?\<mu> (E - (E \<inter> \<Union>(snd`(p - s))))" |
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245 using \<open>0 < e\<close> \<open>E \<in> sets ?L\<close> tagged_division_ofD(1)[OF p(1)] |
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246 by (subst emeasure_Diff) |
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247 (auto simp: ennreal_plus[symmetric] top_unique simp del: ennreal_plus |
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248 intro!: sets.Int sets.finite_UN ennreal_mono_minus intro: p_in_L) |
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249 also have "\<dots> \<le> ?\<mu> (\<Union>x\<in>p \<inter> s. snd x)" |
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250 proof (safe intro!: emeasure_mono subsetI) |
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251 fix v assume "v \<in> E" and not: "v \<notin> (\<Union>x\<in>p \<inter> s. snd x)" |
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252 then have "v \<in> cbox x y" |
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253 using \<open>E \<subseteq> space ?L\<close> by (auto simp: space_restrict_space) |
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254 then obtain z k where "(z, k) \<in> p" "v \<in> k" |
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255 using tagged_division_ofD(6)[OF p(1), symmetric] by auto |
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256 with not show "v \<in> UNION (p - s) snd" |
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257 by (auto intro!: bexI[of _ "(z, k)"] elim: ballE[of _ _ "(z, k)"]) |
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258 qed (auto intro!: sets.Int sets.finite_UN ennreal_mono_minus intro: p_in_L) |
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259 also have "\<dots> = measure ?L (\<Union>x\<in>p \<inter> s. snd x)" |
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260 by (auto intro!: emeasure_eq_ennreal_measure) |
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261 finally have "M - 2 * e \<le> measure ?L (\<Union>x\<in>p \<inter> s. snd x)" |
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262 unfolding \<open>?\<mu> E = M\<close> using \<open>0 < e\<close> by (simp add: ennreal_minus) |
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263 also have "measure ?L (\<Union>x\<in>p \<inter> s. snd x) = content (\<Union>x\<in>p \<inter> s. snd x)" |
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264 using tagged_division_ofD(1,3,4) [OF p(1)] |
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265 by (intro content_eq_L[symmetric]) |
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266 (fastforce intro!: sets.finite_UN UN_least del: subsetI)+ |
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267 also have "content (\<Union>x\<in>p \<inter> s. snd x) \<le> (\<Sum>k\<in>p \<inter> s. content (snd k))" |
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268 using p(1) by (auto simp: emeasure_lborel_cbox_eq intro!: measure_subadditive_finite |
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269 dest!: p(1)[THEN tagged_division_ofD(4)]) |
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270 finally show "M - 3 * e \<le> (\<Sum>(x, y)\<in>p \<inter> s. content y)" |
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271 using \<open>0 < e\<close> by (simp add: split_beta) |
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272 qed (use \<open>a < b\<close> in auto) |
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273 also have "\<dots> = (\<Sum>(x, k)\<in>p \<inter> s. content k * (b - a))" |
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274 by (simp add: setsum_distrib_right split_beta') |
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275 also have "\<dots> \<le> (\<Sum>(x, k)\<in>p \<inter> s. content k * (f (T ?F k) - f (T ?E k)))" |
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276 using parts(3) by (auto intro!: setsum_mono mult_left_mono diff_mono) |
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277 also have "\<dots> = (\<Sum>(x, k)\<in>p \<inter> s. content k * f (T ?F k)) - (\<Sum>(x, k)\<in>p \<inter> s. content k * f (T ?E k))" |
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278 by (auto intro!: setsum.cong simp: field_simps setsum_subtractf[symmetric]) |
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279 also have "\<dots> = (\<Sum>(x, k)\<in>?B. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>?A. content k *\<^sub>R f x)" |
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280 by (subst (1 2) parts) auto |
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281 also have "\<dots> \<le> norm ((\<Sum>(x, k)\<in>?B. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>?A. content k *\<^sub>R f x))" |
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282 by auto |
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283 also have "\<dots> \<le> e + e" |
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284 using parts(1)[of ?E] parts(1)[of ?F] by (intro norm_diff_triangle_le[of _ I]) auto |
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285 finally show False |
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286 using \<open>2 * e < (b - a) * (M - e * 3)\<close> by (auto simp: field_simps) |
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287 qed |
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288 moreover have "?\<mu>' ?E \<le> ?\<mu> E" "?\<mu>' ?F \<le> ?\<mu> E" |
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289 unfolding outer_measure_of_eq[OF \<open>E \<in> sets ?L\<close>, symmetric] by (auto intro!: outer_measure_of_mono) |
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290 ultimately show "min (?\<mu>' ?E) (?\<mu>' ?F) < ?\<mu> E" |
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291 unfolding min_less_iff_disj by (auto simp: less_le) |
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292 qed |
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293 |
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294 lemma has_integral_implies_lebesgue_measurable_real: |
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295 fixes f :: "'a :: euclidean_space \<Rightarrow> real" |
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296 assumes f: "(f has_integral I) \<Omega>" |
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297 shows "(\<lambda>x. f x * indicator \<Omega> x) \<in> lebesgue \<rightarrow>\<^sub>M borel" |
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298 proof - |
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299 define B :: "nat \<Rightarrow> 'a set" where "B n = cbox (- real n *\<^sub>R One) (real n *\<^sub>R One)" for n |
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300 show "(\<lambda>x. f x * indicator \<Omega> x) \<in> lebesgue \<rightarrow>\<^sub>M borel" |
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301 proof (rule measurable_piecewise_restrict) |
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302 have "(\<Union>n. box (- real n *\<^sub>R One) (real n *\<^sub>R One)) \<subseteq> UNION UNIV B" |
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303 unfolding B_def by (intro UN_mono box_subset_cbox order_refl) |
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304 then show "countable (range B)" "space lebesgue \<subseteq> UNION UNIV B" |
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305 by (auto simp: B_def UN_box_eq_UNIV) |
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306 next |
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307 fix \<Omega>' assume "\<Omega>' \<in> range B" |
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308 then obtain n where \<Omega>': "\<Omega>' = B n" by auto |
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309 then show "\<Omega>' \<inter> space lebesgue \<in> sets lebesgue" |
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310 by (auto simp: B_def) |
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311 |
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312 have "f integrable_on \<Omega>" |
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313 using f by auto |
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314 then have "(\<lambda>x. f x * indicator \<Omega> x) integrable_on \<Omega>" |
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315 by (auto simp: integrable_on_def cong: has_integral_cong) |
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316 then have "(\<lambda>x. f x * indicator \<Omega> x) integrable_on (\<Omega> \<union> B n)" |
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317 by (rule integrable_on_superset[rotated 2]) auto |
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318 then have "(\<lambda>x. f x * indicator \<Omega> x) integrable_on B n" |
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319 unfolding B_def by (rule integrable_on_subcbox) auto |
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320 then show "(\<lambda>x. f x * indicator \<Omega> x) \<in> lebesgue_on \<Omega>' \<rightarrow>\<^sub>M borel" |
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321 unfolding B_def \<Omega>' by (auto intro: has_integral_implies_lebesgue_measurable_cbox simp: integrable_on_def) |
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322 qed |
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323 qed |
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324 |
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325 lemma has_integral_implies_lebesgue_measurable: |
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326 fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space" |
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327 assumes f: "(f has_integral I) \<Omega>" |
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328 shows "(\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> lebesgue \<rightarrow>\<^sub>M borel" |
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329 proof (intro borel_measurable_euclidean_space[where 'c='b, THEN iffD2] ballI) |
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330 fix i :: "'b" assume "i \<in> Basis" |
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331 have "(\<lambda>x. (f x \<bullet> i) * indicator \<Omega> x) \<in> borel_measurable (completion lborel)" |
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332 using has_integral_linear[OF f bounded_linear_inner_left, of i] |
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333 by (intro has_integral_implies_lebesgue_measurable_real) (auto simp: comp_def) |
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334 then show "(\<lambda>x. indicator \<Omega> x *\<^sub>R f x \<bullet> i) \<in> borel_measurable (completion lborel)" |
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335 by (simp add: ac_simps) |
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336 qed |
4 |
337 |
5 subsection \<open>Equivalence Lebesgue integral on @{const lborel} and HK-integral\<close> |
338 subsection \<open>Equivalence Lebesgue integral on @{const lborel} and HK-integral\<close> |
6 |
339 |
7 lemma has_integral_measure_lborel: |
340 lemma has_integral_measure_lborel: |
8 fixes A :: "'a::euclidean_space set" |
341 fixes A :: "'a::euclidean_space set" |
345 using lim lim(1)[THEN borel_measurable_integrable] |
678 using lim lim(1)[THEN borel_measurable_integrable] |
346 by (intro integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"]) auto |
679 by (intro integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"]) auto |
347 qed |
680 qed |
348 qed |
681 qed |
349 |
682 |
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683 lemma has_integral_AE: |
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684 assumes ae: "AE x in lborel. x \<in> \<Omega> \<longrightarrow> f x = g x" |
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685 shows "(f has_integral x) \<Omega> = (g has_integral x) \<Omega>" |
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686 proof - |
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687 from ae obtain N |
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688 where N: "N \<in> sets borel" "emeasure lborel N = 0" "{x. \<not> (x \<in> \<Omega> \<longrightarrow> f x = g x)} \<subseteq> N" |
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689 by (auto elim!: AE_E) |
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690 then have not_N: "AE x in lborel. x \<notin> N" |
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691 by (simp add: AE_iff_measurable) |
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692 show ?thesis |
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693 proof (rule has_integral_spike_eq[symmetric]) |
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694 show "\<forall>x\<in>\<Omega> - N. f x = g x" using N(3) by auto |
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695 show "negligible N" |
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696 unfolding negligible_def |
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697 proof (intro allI) |
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698 fix a b :: "'a" |
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699 let ?F = "\<lambda>x::'a. if x \<in> cbox a b then indicator N x else 0 :: real" |
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700 have "integrable lborel ?F = integrable lborel (\<lambda>x::'a. 0::real)" |
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701 using not_N N(1) by (intro integrable_cong_AE) auto |
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702 moreover have "(LINT x|lborel. ?F x) = (LINT x::'a|lborel. 0::real)" |
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703 using not_N N(1) by (intro integral_cong_AE) auto |
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704 ultimately have "(?F has_integral 0) UNIV" |
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705 using has_integral_integral_real[of ?F] by simp |
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706 then show "(indicator N has_integral (0::real)) (cbox a b)" |
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707 unfolding has_integral_restrict_univ . |
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708 qed |
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709 qed |
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710 qed |
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711 |
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712 lemma nn_integral_has_integral_lebesgue: |
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713 fixes f :: "'a::euclidean_space \<Rightarrow> real" |
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714 assumes nonneg: "\<And>x. 0 \<le> f x" and I: "(f has_integral I) \<Omega>" |
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715 shows "integral\<^sup>N lborel (\<lambda>x. indicator \<Omega> x * f x) = I" |
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716 proof - |
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717 from I have "(\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> lebesgue \<rightarrow>\<^sub>M borel" |
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718 by (rule has_integral_implies_lebesgue_measurable) |
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719 then obtain f' :: "'a \<Rightarrow> real" |
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720 where [measurable]: "f' \<in> borel \<rightarrow>\<^sub>M borel" and eq: "AE x in lborel. indicator \<Omega> x * f x = f' x" |
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721 by (auto dest: completion_ex_borel_measurable_real) |
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722 |
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723 from I have "((\<lambda>x. abs (indicator \<Omega> x * f x)) has_integral I) UNIV" |
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724 using nonneg by (simp add: indicator_def if_distrib[of "\<lambda>x. x * f y" for y] cong: if_cong) |
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725 also have "((\<lambda>x. abs (indicator \<Omega> x * f x)) has_integral I) UNIV \<longleftrightarrow> ((\<lambda>x. abs (f' x)) has_integral I) UNIV" |
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726 using eq by (intro has_integral_AE) auto |
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727 finally have "integral\<^sup>N lborel (\<lambda>x. abs (f' x)) = I" |
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728 by (rule nn_integral_has_integral_lborel[rotated 2]) auto |
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729 also have "integral\<^sup>N lborel (\<lambda>x. abs (f' x)) = integral\<^sup>N lborel (\<lambda>x. abs (indicator \<Omega> x * f x))" |
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730 using eq by (intro nn_integral_cong_AE) auto |
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731 finally show ?thesis |
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732 using nonneg by auto |
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733 qed |
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734 |
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735 lemma has_integral_iff_nn_integral_lebesgue: |
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736 assumes f: "\<And>x. 0 \<le> f x" |
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737 shows "(f has_integral r) UNIV \<longleftrightarrow> (f \<in> lebesgue \<rightarrow>\<^sub>M borel \<and> integral\<^sup>N lebesgue f = r \<and> 0 \<le> r)" (is "?I = ?N") |
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738 proof |
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739 assume ?I |
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740 have "0 \<le> r" |
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741 using has_integral_nonneg[OF \<open>?I\<close>] f by auto |
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742 then show ?N |
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743 using nn_integral_has_integral_lebesgue[OF f \<open>?I\<close>] |
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744 has_integral_implies_lebesgue_measurable[OF \<open>?I\<close>] |
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745 by (auto simp: nn_integral_completion) |
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746 next |
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747 assume ?N |
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748 then obtain f' where f': "f' \<in> borel \<rightarrow>\<^sub>M borel" "AE x in lborel. f x = f' x" |
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749 by (auto dest: completion_ex_borel_measurable_real) |
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750 moreover have "(\<integral>\<^sup>+ x. ennreal \<bar>f' x\<bar> \<partial>lborel) = (\<integral>\<^sup>+ x. ennreal \<bar>f x\<bar> \<partial>lborel)" |
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751 using f' by (intro nn_integral_cong_AE) auto |
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752 moreover have "((\<lambda>x. \<bar>f' x\<bar>) has_integral r) UNIV \<longleftrightarrow> ((\<lambda>x. \<bar>f x\<bar>) has_integral r) UNIV" |
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753 using f' by (intro has_integral_AE) auto |
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754 moreover note nn_integral_has_integral[of "\<lambda>x. \<bar>f' x\<bar>" r] \<open>?N\<close> |
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755 ultimately show ?I |
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756 using f by (auto simp: nn_integral_completion) |
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757 qed |
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758 |
350 context |
759 context |
351 fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
760 fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
352 begin |
761 begin |
353 |
762 |
354 lemma has_integral_integral_lborel: |
763 lemma has_integral_integral_lborel: |