author  hoelzl 
Fri, 04 Feb 2011 14:16:48 +0100  
changeset 41704  8c539202f854 
parent 41689  3e39b0e730d6 
child 41706  a207a858d1f6 
permissions  rwrr 
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(* Author: Robert Himmelmann, TU Muenchen *) 
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header {* Lebsegue measure *} 
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theory Lebesgue_Measure 

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imports Product_Measure 
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begin 
6 

7 
subsection {* Standard Cubes *} 

8 

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definition cube :: "nat \<Rightarrow> 'a::ordered_euclidean_space set" where 
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"cube n \<equiv> {\<chi>\<chi> i.  real n .. \<chi>\<chi> i. real n}" 

11 

12 
lemma cube_closed[intro]: "closed (cube n)" 

13 
unfolding cube_def by auto 

14 

15 
lemma cube_subset[intro]: "n \<le> N \<Longrightarrow> cube n \<subseteq> cube N" 

16 
by (fastsimp simp: eucl_le[where 'a='a] cube_def) 

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lemma cube_subset_iff: 
19 
"cube n \<subseteq> cube N \<longleftrightarrow> n \<le> N" 

20 
proof 

21 
assume subset: "cube n \<subseteq> (cube N::'a set)" 

22 
then have "((\<chi>\<chi> i. real n)::'a) \<in> cube N" 

23 
using DIM_positive[where 'a='a] 

24 
by (fastsimp simp: cube_def eucl_le[where 'a='a]) 

25 
then show "n \<le> N" 

26 
by (fastsimp simp: cube_def eucl_le[where 'a='a]) 

27 
next 

28 
assume "n \<le> N" then show "cube n \<subseteq> (cube N::'a set)" by (rule cube_subset) 

29 
qed 

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31 
lemma ball_subset_cube:"ball (0::'a::ordered_euclidean_space) (real n) \<subseteq> cube n" 

32 
unfolding cube_def subset_eq mem_interval apply safe unfolding euclidean_lambda_beta' 

33 
proof fix x::'a and i assume as:"x\<in>ball 0 (real n)" "i<DIM('a)" 

34 
thus " real n \<le> x $$ i" "real n \<ge> x $$ i" 

35 
using component_le_norm[of x i] by(auto simp: dist_norm) 

36 
qed 

37 

38 
lemma mem_big_cube: obtains n where "x \<in> cube n" 

39 
proof from real_arch_lt[of "norm x"] guess n .. 

40 
thus ?thesis applyapply(rule that[where n=n]) 

41 
apply(rule ball_subset_cube[unfolded subset_eq,rule_format]) 

42 
by (auto simp add:dist_norm) 

43 
qed 

44 

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lemma cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)" 
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unfolding cube_def_raw subset_eq apply safe unfolding mem_interval by auto 
41654  47 

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lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)" 
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unfolding Pi_def by auto 
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definition lebesgue :: "'a::ordered_euclidean_space measure_space" where 
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"lebesgue = \<lparr> space = UNIV, 
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sets = {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n}, 
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measure = \<lambda>A. SUP n. Real (integral (cube n) (indicator A)) \<rparr>" 
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41654  56 
lemma space_lebesgue[simp]: "space lebesgue = UNIV" 
57 
unfolding lebesgue_def by simp 

58 

59 
lemma lebesgueD: "A \<in> sets lebesgue \<Longrightarrow> (indicator A :: _ \<Rightarrow> real) integrable_on cube n" 

60 
unfolding lebesgue_def by simp 

61 

62 
lemma lebesgueI: "(\<And>n. (indicator A :: _ \<Rightarrow> real) integrable_on cube n) \<Longrightarrow> A \<in> sets lebesgue" 

63 
unfolding lebesgue_def by simp 

64 

65 
lemma absolutely_integrable_on_indicator[simp]: 

66 
fixes A :: "'a::ordered_euclidean_space set" 

67 
shows "((indicator A :: _ \<Rightarrow> real) absolutely_integrable_on X) \<longleftrightarrow> 

68 
(indicator A :: _ \<Rightarrow> real) integrable_on X" 

69 
unfolding absolutely_integrable_on_def by simp 

70 

71 
lemma LIMSEQ_indicator_UN: 

72 
"(\<lambda>k. indicator (\<Union> i<k. A i) x) > (indicator (\<Union>i. A i) x :: real)" 

73 
proof cases 

74 
assume "\<exists>i. x \<in> A i" then guess i .. note i = this 

75 
then have *: "\<And>n. (indicator (\<Union> i<n + Suc i. A i) x :: real) = 1" 

76 
"(indicator (\<Union> i. A i) x :: real) = 1" by (auto simp: indicator_def) 

77 
show ?thesis 

78 
apply (rule LIMSEQ_offset[of _ "Suc i"]) unfolding * by auto 

79 
qed (auto simp: indicator_def) 

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lemma indicator_add: 
82 
"A \<inter> B = {} \<Longrightarrow> (indicator A x::_::monoid_add) + indicator B x = indicator (A \<union> B) x" 

83 
unfolding indicator_def by auto 

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interpretation lebesgue: sigma_algebra lebesgue 
86 
proof (intro sigma_algebra_iff2[THEN iffD2] conjI allI ballI impI lebesgueI) 

87 
fix A n assume A: "A \<in> sets lebesgue" 

88 
have "indicator (space lebesgue  A) = (\<lambda>x. 1  indicator A x :: real)" 

89 
by (auto simp: fun_eq_iff indicator_def) 

90 
then show "(indicator (space lebesgue  A) :: _ \<Rightarrow> real) integrable_on cube n" 

91 
using A by (auto intro!: integrable_sub dest: lebesgueD simp: cube_def) 

92 
next 

93 
fix n show "(indicator {} :: _\<Rightarrow>real) integrable_on cube n" 

94 
by (auto simp: cube_def indicator_def_raw) 

95 
next 

96 
fix A :: "nat \<Rightarrow> 'a set" and n ::nat assume "range A \<subseteq> sets lebesgue" 

97 
then have A: "\<And>i. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n" 

98 
by (auto dest: lebesgueD) 

99 
show "(indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n" (is "?g integrable_on _") 

100 
proof (intro dominated_convergence[where g="?g"] ballI) 

101 
fix k show "(indicator (\<Union>i<k. A i) :: _ \<Rightarrow> real) integrable_on cube n" 

102 
proof (induct k) 

103 
case (Suc k) 

104 
have *: "(\<Union> i<Suc k. A i) = (\<Union> i<k. A i) \<union> A k" 

105 
unfolding lessThan_Suc UN_insert by auto 

106 
have *: "(\<lambda>x. max (indicator (\<Union> i<k. A i) x) (indicator (A k) x) :: real) = 

107 
indicator (\<Union> i<Suc k. A i)" (is "(\<lambda>x. max (?f x) (?g x)) = _") 

108 
by (auto simp: fun_eq_iff * indicator_def) 

109 
show ?case 

110 
using absolutely_integrable_max[of ?f "cube n" ?g] A Suc by (simp add: *) 

111 
qed auto 

112 
qed (auto intro: LIMSEQ_indicator_UN simp: cube_def) 

113 
qed simp 

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interpretation lebesgue: measure_space lebesgue 
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proof 
117 
have *: "indicator {} = (\<lambda>x. 0 :: real)" by (simp add: fun_eq_iff) 

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show "measure lebesgue {} = 0" by (simp add: integral_0 * lebesgue_def) 
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next 
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show "countably_additive lebesgue (measure lebesgue)" 
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proof (intro countably_additive_def[THEN iffD2] allI impI) 
122 
fix A :: "nat \<Rightarrow> 'b set" assume rA: "range A \<subseteq> sets lebesgue" "disjoint_family A" 

123 
then have A[simp, intro]: "\<And>i n. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n" 

124 
by (auto dest: lebesgueD) 

125 
let "?m n i" = "integral (cube n) (indicator (A i) :: _\<Rightarrow>real)" 

126 
let "?M n I" = "integral (cube n) (indicator (\<Union>i\<in>I. A i) :: _\<Rightarrow>real)" 

127 
have nn[simp, intro]: "\<And>i n. 0 \<le> ?m n i" by (auto intro!: integral_nonneg) 

128 
assume "(\<Union>i. A i) \<in> sets lebesgue" 

129 
then have UN_A[simp, intro]: "\<And>i n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n" 

130 
by (auto dest: lebesgueD) 

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show "(\<Sum>\<^isub>\<infinity>n. measure lebesgue (A n)) = measure lebesgue (\<Union>i. A i)" 
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proof (simp add: lebesgue_def, subst psuminf_SUP_eq) 
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fix n i show "Real (?m n i) \<le> Real (?m (Suc n) i)" 
134 
using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le) 

135 
next 

136 
show "(SUP n. \<Sum>\<^isub>\<infinity>i. Real (?m n i)) = (SUP n. Real (?M n UNIV))" 

137 
unfolding psuminf_def 

138 
proof (subst setsum_Real, (intro arg_cong[where f="SUPR UNIV"] ext ballI nn SUP_eq_LIMSEQ[THEN iffD2])+) 

139 
fix n :: nat show "mono (\<lambda>m. \<Sum>x<m. ?m n x)" 

140 
proof (intro mono_iff_le_Suc[THEN iffD2] allI) 

141 
fix m show "(\<Sum>x<m. ?m n x) \<le> (\<Sum>x<Suc m. ?m n x)" 

142 
using nn[of n m] by auto 

143 
qed 

144 
show "0 \<le> ?M n UNIV" 

145 
using UN_A by (auto intro!: integral_nonneg) 

146 
fix m show "0 \<le> (\<Sum>x<m. ?m n x)" by (auto intro!: setsum_nonneg) 

147 
next 

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fix n 

149 
have "\<And>m. (UNION {..<m} A) \<in> sets lebesgue" using rA by auto 

150 
from lebesgueD[OF this] 

151 
have "(\<lambda>m. ?M n {..< m}) > ?M n UNIV" 

152 
(is "(\<lambda>m. integral _ (?A m)) > ?I") 

153 
by (intro dominated_convergence(2)[where f="?A" and h="\<lambda>x. 1::real"]) 

154 
(auto intro: LIMSEQ_indicator_UN simp: cube_def) 

155 
moreover 

156 
{ fix m have *: "(\<Sum>x<m. ?m n x) = ?M n {..< m}" 

157 
proof (induct m) 

158 
case (Suc m) 

159 
have "(\<Union>i<m. A i) \<in> sets lebesgue" using rA by auto 

160 
then have "(indicator (\<Union>i<m. A i) :: _\<Rightarrow>real) integrable_on (cube n)" 

161 
by (auto dest!: lebesgueD) 

162 
moreover 

163 
have "(\<Union>i<m. A i) \<inter> A m = {}" 

164 
using rA(2)[unfolded disjoint_family_on_def, THEN bspec, of m] 

165 
by auto 

166 
then have "\<And>x. indicator (\<Union>i<Suc m. A i) x = 

167 
indicator (\<Union>i<m. A i) x + (indicator (A m) x :: real)" 

168 
by (auto simp: indicator_add lessThan_Suc ac_simps) 

169 
ultimately show ?case 

170 
using Suc A by (simp add: integral_add[symmetric]) 

171 
qed auto } 

172 
ultimately show "(\<lambda>m. \<Sum>x<m. ?m n x) > ?M n UNIV" 

173 
by simp 

174 
qed 

175 
qed 

176 
qed 

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qed 
178 

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lemma has_integral_interval_cube: 
180 
fixes a b :: "'a::ordered_euclidean_space" 

181 
shows "(indicator {a .. b} has_integral 

182 
content ({\<chi>\<chi> i. max ( real n) (a $$ i) .. \<chi>\<chi> i. min (real n) (b $$ i)} :: 'a set)) (cube n)" 

183 
(is "(?I has_integral content ?R) (cube n)") 

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proof  
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let "{?N .. ?P}" = ?R 
186 
have [simp]: "(\<lambda>x. if x \<in> cube n then ?I x else 0) = indicator ?R" 

187 
by (auto simp: indicator_def cube_def fun_eq_iff eucl_le[where 'a='a]) 

188 
have "(?I has_integral content ?R) (cube n) \<longleftrightarrow> (indicator ?R has_integral content ?R) UNIV" 

189 
unfolding has_integral_restrict_univ[where s="cube n", symmetric] by simp 

190 
also have "\<dots> \<longleftrightarrow> ((\<lambda>x. 1) has_integral content ?R) ?R" 

191 
unfolding indicator_def_raw has_integral_restrict_univ .. 

192 
finally show ?thesis 

193 
using has_integral_const[of "1::real" "?N" "?P"] by simp 

40859  194 
qed 
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lemma lebesgueI_borel[intro, simp]: 
197 
fixes s::"'a::ordered_euclidean_space set" 

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assumes "s \<in> sets borel" shows "s \<in> sets lebesgue" 
41654  199 
proof  
200 
let ?S = "range (\<lambda>(a, b). {a .. (b :: 'a\<Colon>ordered_euclidean_space)})" 

201 
have *:"?S \<subseteq> sets lebesgue" 

202 
proof (safe intro!: lebesgueI) 

203 
fix n :: nat and a b :: 'a 

204 
let ?N = "\<chi>\<chi> i. max ( real n) (a $$ i)" 

205 
let ?P = "\<chi>\<chi> i. min (real n) (b $$ i)" 

206 
show "(indicator {a..b} :: 'a\<Rightarrow>real) integrable_on cube n" 

207 
unfolding integrable_on_def 

208 
using has_integral_interval_cube[of a b] by auto 

209 
qed 

40859  210 
have "s \<in> sigma_sets UNIV ?S" using assms 
211 
unfolding borel_eq_atLeastAtMost by (simp add: sigma_def) 

212 
thus ?thesis 

213 
using lebesgue.sigma_subset[of "\<lparr> space = UNIV, sets = ?S\<rparr>", simplified, OF *] 

214 
by (auto simp: sigma_def) 

38656  215 
qed 
216 

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lemma lebesgueI_negligible[dest]: fixes s::"'a::ordered_euclidean_space set" 
218 
assumes "negligible s" shows "s \<in> sets lebesgue" 

41654  219 
using assms by (force simp: cube_def integrable_on_def negligible_def intro!: lebesgueI) 
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lemma lmeasure_eq_0: 
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fixes S :: "'a::ordered_euclidean_space set" assumes "negligible S" shows "lebesgue.\<mu> S = 0" 
40859  223 
proof  
41654  224 
have "\<And>n. integral (cube n) (indicator S :: 'a\<Rightarrow>real) = 0" 
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unfolding lebesgue_integral_def using assms 
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by (intro integral_unique some1_equality ex_ex1I) 
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(auto simp: cube_def negligible_def) 
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then show ?thesis by (auto simp: lebesgue_def) 
40859  229 
qed 
230 

231 
lemma lmeasure_iff_LIMSEQ: 

232 
assumes "A \<in> sets lebesgue" "0 \<le> m" 

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shows "lebesgue.\<mu> A = Real m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) > m" 
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proof (simp add: lebesgue_def, intro SUP_eq_LIMSEQ) 
41654  235 
show "mono (\<lambda>n. integral (cube n) (indicator A::_=>real))" 
236 
using cube_subset assms by (intro monoI integral_subset_le) (auto dest!: lebesgueD) 

237 
fix n show "0 \<le> integral (cube n) (indicator A::_=>real)" 

238 
using assms by (auto dest!: lebesgueD intro!: integral_nonneg) 

239 
qed fact 

38656  240 

41654  241 
lemma has_integral_indicator_UNIV: 
242 
fixes s A :: "'a::ordered_euclidean_space set" and x :: real 

243 
shows "((indicator (s \<inter> A) :: 'a\<Rightarrow>real) has_integral x) UNIV = ((indicator s :: _\<Rightarrow>real) has_integral x) A" 

244 
proof  

245 
have "(\<lambda>x. if x \<in> A then indicator s x else 0) = (indicator (s \<inter> A) :: _\<Rightarrow>real)" 

246 
by (auto simp: fun_eq_iff indicator_def) 

247 
then show ?thesis 

248 
unfolding has_integral_restrict_univ[where s=A, symmetric] by simp 

40859  249 
qed 
38656  250 

41654  251 
lemma 
252 
fixes s a :: "'a::ordered_euclidean_space set" 

253 
shows integral_indicator_UNIV: 

254 
"integral UNIV (indicator (s \<inter> A) :: 'a\<Rightarrow>real) = integral A (indicator s :: _\<Rightarrow>real)" 

255 
and integrable_indicator_UNIV: 

256 
"(indicator (s \<inter> A) :: 'a\<Rightarrow>real) integrable_on UNIV \<longleftrightarrow> (indicator s :: 'a\<Rightarrow>real) integrable_on A" 

257 
unfolding integral_def integrable_on_def has_integral_indicator_UNIV by auto 

258 

259 
lemma lmeasure_finite_has_integral: 

260 
fixes s :: "'a::ordered_euclidean_space set" 

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assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s = Real m" "0 \<le> m" 
41654  262 
shows "(indicator s has_integral m) UNIV" 
263 
proof  

264 
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real" 

265 
have **: "(?I s) integrable_on UNIV \<and> (\<lambda>k. integral UNIV (?I (s \<inter> cube k))) > integral UNIV (?I s)" 

266 
proof (intro monotone_convergence_increasing allI ballI) 

267 
have LIMSEQ: "(\<lambda>n. integral (cube n) (?I s)) > m" 

268 
using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1, 3)] . 

269 
{ fix n have "integral (cube n) (?I s) \<le> m" 

270 
using cube_subset assms 

271 
by (intro incseq_le[where L=m] LIMSEQ incseq_def[THEN iffD2] integral_subset_le allI impI) 

272 
(auto dest!: lebesgueD) } 

273 
moreover 

274 
{ fix n have "0 \<le> integral (cube n) (?I s)" 

275 
using assms by (auto dest!: lebesgueD intro!: integral_nonneg) } 

276 
ultimately 

277 
show "bounded {integral UNIV (?I (s \<inter> cube k)) k. True}" 

278 
unfolding bounded_def 

279 
apply (rule_tac exI[of _ 0]) 

280 
apply (rule_tac exI[of _ m]) 

281 
by (auto simp: dist_real_def integral_indicator_UNIV) 

282 
fix k show "?I (s \<inter> cube k) integrable_on UNIV" 

283 
unfolding integrable_indicator_UNIV using assms by (auto dest!: lebesgueD) 

284 
fix x show "?I (s \<inter> cube k) x \<le> ?I (s \<inter> cube (Suc k)) x" 

285 
using cube_subset[of k "Suc k"] by (auto simp: indicator_def) 

286 
next 

287 
fix x :: 'a 

288 
from mem_big_cube obtain k where k: "x \<in> cube k" . 

289 
{ fix n have "?I (s \<inter> cube (n + k)) x = ?I s x" 

290 
using k cube_subset[of k "n + k"] by (auto simp: indicator_def) } 

291 
note * = this 

292 
show "(\<lambda>k. ?I (s \<inter> cube k) x) > ?I s x" 

293 
by (rule LIMSEQ_offset[where k=k]) (auto simp: *) 

294 
qed 

295 
note ** = conjunctD2[OF this] 

296 
have m: "m = integral UNIV (?I s)" 

297 
apply (intro LIMSEQ_unique[OF _ **(2)]) 

298 
using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1,3)] integral_indicator_UNIV . 

299 
show ?thesis 

300 
unfolding m by (intro integrable_integral **) 

38656  301 
qed 
302 

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lemma lmeasure_finite_integrable: assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s \<noteq> \<omega>" 
41654  304 
shows "(indicator s :: _ \<Rightarrow> real) integrable_on UNIV" 
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305 
proof (cases "lebesgue.\<mu> s") 
41654  306 
case (preal m) from lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this] 
307 
show ?thesis unfolding integrable_on_def by auto 

40859  308 
qed (insert assms, auto) 
38656  309 

41654  310 
lemma has_integral_lebesgue: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV" 
311 
shows "s \<in> sets lebesgue" 

312 
proof (intro lebesgueI) 

313 
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real" 

314 
fix n show "(?I s) integrable_on cube n" unfolding cube_def 

315 
proof (intro integrable_on_subinterval) 

316 
show "(?I s) integrable_on UNIV" 

317 
unfolding integrable_on_def using assms by auto 

318 
qed auto 

38656  319 
qed 
320 

41654  321 
lemma has_integral_lmeasure: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV" 
41689
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322 
shows "lebesgue.\<mu> s = Real m" 
41654  323 
proof (intro lmeasure_iff_LIMSEQ[THEN iffD2]) 
324 
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real" 

325 
show "s \<in> sets lebesgue" using has_integral_lebesgue[OF assms] . 

326 
show "0 \<le> m" using assms by (rule has_integral_nonneg) auto 

327 
have "(\<lambda>n. integral UNIV (?I (s \<inter> cube n))) > integral UNIV (?I s)" 

328 
proof (intro dominated_convergence(2) ballI) 

329 
show "(?I s) integrable_on UNIV" unfolding integrable_on_def using assms by auto 

330 
fix n show "?I (s \<inter> cube n) integrable_on UNIV" 

331 
unfolding integrable_indicator_UNIV using `s \<in> sets lebesgue` by (auto dest: lebesgueD) 

332 
fix x show "norm (?I (s \<inter> cube n) x) \<le> ?I s x" by (auto simp: indicator_def) 

333 
next 

334 
fix x :: 'a 

335 
from mem_big_cube obtain k where k: "x \<in> cube k" . 

336 
{ fix n have "?I (s \<inter> cube (n + k)) x = ?I s x" 

337 
using k cube_subset[of k "n + k"] by (auto simp: indicator_def) } 

338 
note * = this 

339 
show "(\<lambda>k. ?I (s \<inter> cube k) x) > ?I s x" 

340 
by (rule LIMSEQ_offset[where k=k]) (auto simp: *) 

341 
qed 

342 
then show "(\<lambda>n. integral (cube n) (?I s)) > m" 

343 
unfolding integral_unique[OF assms] integral_indicator_UNIV by simp 

344 
qed 

345 

346 
lemma has_integral_iff_lmeasure: 

41689
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347 
"(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m)" 
40859  348 
proof 
41654  349 
assume "(indicator A has_integral m) UNIV" 
350 
with has_integral_lmeasure[OF this] has_integral_lebesgue[OF this] 

41689
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parents:
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351 
show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m" 
41654  352 
by (auto intro: has_integral_nonneg) 
40859  353 
next 
41689
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changeset

354 
assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m" 
41654  355 
then show "(indicator A has_integral m) UNIV" by (intro lmeasure_finite_has_integral) auto 
38656  356 
qed 
357 

41654  358 
lemma lmeasure_eq_integral: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" 
41689
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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parents:
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changeset

359 
shows "lebesgue.\<mu> s = Real (integral UNIV (indicator s))" 
41654  360 
using assms unfolding integrable_on_def 
361 
proof safe 

362 
fix y :: real assume "(indicator s has_integral y) UNIV" 

363 
from this[unfolded has_integral_iff_lmeasure] integral_unique[OF this] 

41689
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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parents:
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diff
changeset

364 
show "lebesgue.\<mu> s = Real (integral UNIV (indicator s))" by simp 
40859  365 
qed 
38656  366 

367 
lemma lebesgue_simple_function_indicator: 

41023
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it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

368 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal" 
41689
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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diff
changeset

369 
assumes f:"simple_function lebesgue f" 
38656  370 
shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f ` {y}) x))" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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parents:
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diff
changeset

371 
by (rule, subst lebesgue.simple_function_indicator_representation[OF f]) auto 
38656  372 

41654  373 
lemma integral_eq_lmeasure: 
41689
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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changeset

374 
"(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (lebesgue.\<mu> s)" 
41654  375 
by (subst lmeasure_eq_integral) (auto intro!: integral_nonneg) 
38656  376 

41689
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hoelzl
parents:
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diff
changeset

377 
lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "lebesgue.\<mu> s \<noteq> \<omega>" 
41654  378 
using lmeasure_eq_integral[OF assms] by auto 
38656  379 

40859  380 
lemma negligible_iff_lebesgue_null_sets: 
381 
"negligible A \<longleftrightarrow> A \<in> lebesgue.null_sets" 

382 
proof 

383 
assume "negligible A" 

384 
from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0] 

385 
show "A \<in> lebesgue.null_sets" by auto 

386 
next 

387 
assume A: "A \<in> lebesgue.null_sets" 

41654  388 
then have *:"((indicator A) has_integral (0::real)) UNIV" using lmeasure_finite_has_integral[of A] by auto 
389 
show "negligible A" unfolding negligible_def 

390 
proof (intro allI) 

391 
fix a b :: 'a 

392 
have integrable: "(indicator A :: _\<Rightarrow>real) integrable_on {a..b}" 

393 
by (intro integrable_on_subinterval has_integral_integrable) (auto intro: *) 

394 
then have "integral {a..b} (indicator A) \<le> (integral UNIV (indicator A) :: real)" 

395 
using * by (auto intro!: integral_subset_le has_integral_integrable) 

396 
moreover have "(0::real) \<le> integral {a..b} (indicator A)" 

397 
using integrable by (auto intro!: integral_nonneg) 

398 
ultimately have "integral {a..b} (indicator A) = (0::real)" 

399 
using integral_unique[OF *] by auto 

400 
then show "(indicator A has_integral (0::real)) {a..b}" 

401 
using integrable_integral[OF integrable] by simp 

402 
qed 

403 
qed 

404 

405 
lemma integral_const[simp]: 

406 
fixes a b :: "'a::ordered_euclidean_space" 

407 
shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c" 

408 
by (rule integral_unique) (rule has_integral_const) 

409 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

410 
lemma lmeasure_UNIV[intro]: "lebesgue.\<mu> (UNIV::'a::ordered_euclidean_space set) = \<omega>" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

411 
proof (simp add: lebesgue_def SUP_\<omega>, intro allI impI) 
41654  412 
fix x assume "x < \<omega>" 
413 
then obtain r where r: "x = Real r" "0 \<le> r" by (cases x) auto 

414 
then obtain n where n: "r < of_nat n" using ex_less_of_nat by auto 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

415 
show "\<exists>i. x < Real (integral (cube i) (indicator UNIV::'a\<Rightarrow>real))" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

416 
proof (intro exI[of _ n]) 
41654  417 
have [simp]: "indicator UNIV = (\<lambda>x. 1)" by (auto simp: fun_eq_iff) 
418 
{ fix m :: nat assume "0 < m" then have "real n \<le> (\<Prod>x<m. 2 * real n)" 

419 
proof (induct m) 

420 
case (Suc m) 

421 
show ?case 

422 
proof cases 

423 
assume "m = 0" then show ?thesis by (simp add: lessThan_Suc) 

424 
next 

425 
assume "m \<noteq> 0" then have "real n \<le> (\<Prod>x<m. 2 * real n)" using Suc by auto 

426 
then show ?thesis 

427 
by (auto simp: lessThan_Suc field_simps mult_le_cancel_left1) 

428 
qed 

429 
qed auto } note this[OF DIM_positive[where 'a='a], simp] 

430 
then have [simp]: "0 \<le> (\<Prod>x<DIM('a). 2 * real n)" using real_of_nat_ge_zero by arith 

431 
have "x < Real (of_nat n)" using n r by auto 

432 
also have "Real (of_nat n) \<le> Real (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))" 

433 
by (auto simp: real_eq_of_nat[symmetric] cube_def content_closed_interval_cases) 

434 
finally show "x < Real (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))" . 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

435 
qed 
40859  436 
qed 
437 

438 
lemma 

439 
fixes a b ::"'a::ordered_euclidean_space" 

41689
3e39b0e730d6
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hoelzl
parents:
41661
diff
changeset

440 
shows lmeasure_atLeastAtMost[simp]: "lebesgue.\<mu> {a..b} = Real (content {a..b})" 
41654  441 
proof  
442 
have "(indicator (UNIV \<inter> {a..b})::_\<Rightarrow>real) integrable_on UNIV" 

443 
unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def_raw) 

444 
from lmeasure_eq_integral[OF this] show ?thesis unfolding integral_indicator_UNIV 

445 
by (simp add: indicator_def_raw) 

40859  446 
qed 
447 

448 
lemma atLeastAtMost_singleton_euclidean[simp]: 

449 
fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}" 

450 
by (force simp: eucl_le[where 'a='a] euclidean_eq[where 'a='a]) 

451 

452 
lemma content_singleton[simp]: "content {a} = 0" 

453 
proof  

454 
have "content {a .. a} = 0" 

455 
by (subst content_closed_interval) auto 

456 
then show ?thesis by simp 

457 
qed 

458 

459 
lemma lmeasure_singleton[simp]: 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

460 
fixes a :: "'a::ordered_euclidean_space" shows "lebesgue.\<mu> {a} = 0" 
41654  461 
using lmeasure_atLeastAtMost[of a a] by simp 
40859  462 

463 
declare content_real[simp] 

464 

465 
lemma 

466 
fixes a b :: real 

467 
shows lmeasure_real_greaterThanAtMost[simp]: 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

468 
"lebesgue.\<mu> {a <.. b} = Real (if a \<le> b then b  a else 0)" 
40859  469 
proof cases 
470 
assume "a < b" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

471 
then have "lebesgue.\<mu> {a <.. b} = lebesgue.\<mu> {a .. b}  lebesgue.\<mu> {a}" 
41654  472 
by (subst lebesgue.measure_Diff[symmetric]) 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

473 
(auto intro!: arg_cong[where f=lebesgue.\<mu>]) 
40859  474 
then show ?thesis by auto 
475 
qed auto 

476 

477 
lemma 

478 
fixes a b :: real 

479 
shows lmeasure_real_atLeastLessThan[simp]: 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

480 
"lebesgue.\<mu> {a ..< b} = Real (if a \<le> b then b  a else 0)" 
40859  481 
proof cases 
482 
assume "a < b" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

483 
then have "lebesgue.\<mu> {a ..< b} = lebesgue.\<mu> {a .. b}  lebesgue.\<mu> {b}" 
41654  484 
by (subst lebesgue.measure_Diff[symmetric]) 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

485 
(auto intro!: arg_cong[where f=lebesgue.\<mu>]) 
41654  486 
then show ?thesis by auto 
487 
qed auto 

488 

489 
lemma 

490 
fixes a b :: real 

491 
shows lmeasure_real_greaterThanLessThan[simp]: 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

492 
"lebesgue.\<mu> {a <..< b} = Real (if a \<le> b then b  a else 0)" 
41654  493 
proof cases 
494 
assume "a < b" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

495 
then have "lebesgue.\<mu> {a <..< b} = lebesgue.\<mu> {a <.. b}  lebesgue.\<mu> {b}" 
41654  496 
by (subst lebesgue.measure_Diff[symmetric]) 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

497 
(auto intro!: arg_cong[where f=lebesgue.\<mu>]) 
40859  498 
then show ?thesis by auto 
499 
qed auto 

500 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

501 
definition "lborel = lebesgue \<lparr> sets := sets borel \<rparr>" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

502 

3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

503 
lemma 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

504 
shows space_lborel[simp]: "space lborel = UNIV" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

505 
and sets_lborel[simp]: "sets lborel = sets borel" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

506 
and measure_lborel[simp]: "measure lborel = lebesgue.\<mu>" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

507 
and measurable_lborel[simp]: "measurable lborel = measurable borel" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

508 
by (simp_all add: measurable_def_raw lborel_def) 
40859  509 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

510 
interpretation lborel: measure_space lborel 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

511 
where "space lborel = UNIV" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

512 
and "sets lborel = sets borel" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

513 
and "measure lborel = lebesgue.\<mu>" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

514 
and "measurable lborel = measurable borel" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

515 
proof  
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

516 
show "measure_space lborel" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

517 
proof 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

518 
show "countably_additive lborel (measure lborel)" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

519 
using lebesgue.ca unfolding countably_additive_def lborel_def 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

520 
apply safe apply (erule_tac x=A in allE) by auto 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

521 
qed (auto simp: lborel_def) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

522 
qed simp_all 
40859  523 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

524 
interpretation lborel: sigma_finite_measure lborel 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

525 
where "space lborel = UNIV" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

526 
and "sets lborel = sets borel" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

527 
and "measure lborel = lebesgue.\<mu>" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

528 
and "measurable lborel = measurable borel" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

529 
proof  
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

530 
show "sigma_finite_measure lborel" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

531 
proof (default, intro conjI exI[of _ "\<lambda>n. cube n"]) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

532 
show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

533 
{ fix x have "\<exists>n. x\<in>cube n" using mem_big_cube by auto } 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

534 
thus "(\<Union>i. cube i) = space lborel" by auto 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

535 
show "\<forall>i. measure lborel (cube i) \<noteq> \<omega>" by (simp add: cube_def) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

536 
qed 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

537 
qed simp_all 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

538 

3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

539 
interpretation lebesgue: sigma_finite_measure lebesgue 
40859  540 
proof 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

541 
from lborel.sigma_finite guess A .. 
40859  542 
moreover then have "range A \<subseteq> sets lebesgue" using lebesgueI_borel by blast 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

543 
ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lebesgue.\<mu> (A i) \<noteq> \<omega>)" 
40859  544 
by auto 
545 
qed 

546 

547 
lemma simple_function_has_integral: 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

548 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

549 
assumes f:"simple_function lebesgue f" 
40859  550 
and f':"\<forall>x. f x \<noteq> \<omega>" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

551 
and om:"\<forall>x\<in>range f. lebesgue.\<mu> (f ` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

552 
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

553 
unfolding simple_integral_def 
40859  554 
apply(subst lebesgue_simple_function_indicator[OF f]) 
41654  555 
proof  
556 
case goal1 

40859  557 
have *:"\<And>x. \<forall>y\<in>range f. y * indicator (f ` {y}) x \<noteq> \<omega>" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

558 
"\<forall>x\<in>range f. x * lebesgue.\<mu> (f ` {x} \<inter> UNIV) \<noteq> \<omega>" 
40859  559 
using f' om unfolding indicator_def by auto 
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

560 
show ?case unfolding space_lebesgue real_of_pextreal_setsum'[OF *(1),THEN sym] 
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

561 
unfolding real_of_pextreal_setsum'[OF *(2),THEN sym] 
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

562 
unfolding real_of_pextreal_setsum space_lebesgue 
40859  563 
apply(rule has_integral_setsum) 
564 
proof safe show "finite (range f)" using f by (auto dest: lebesgue.simple_functionD) 

565 
fix y::'a show "((\<lambda>x. real (f y * indicator (f ` {f y}) x)) has_integral 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

566 
real (f y * lebesgue.\<mu> (f ` {f y} \<inter> UNIV))) UNIV" 
40859  567 
proof(cases "f y = 0") case False 
41654  568 
have mea:"(indicator (f ` {f y}) ::_\<Rightarrow>real) integrable_on UNIV" 
569 
apply(rule lmeasure_finite_integrable) 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

570 
using assms unfolding simple_function_def using False by auto 
41654  571 
have *:"\<And>x. real (indicator (f ` {f y}) x::pextreal) = (indicator (f ` {f y}) x)" 
572 
by (auto simp: indicator_def) 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

573 
show ?thesis unfolding real_of_pextreal_mult[THEN sym] 
40859  574 
apply(rule has_integral_cmul[where 'b=real, unfolded real_scaleR_def]) 
41654  575 
unfolding Int_UNIV_right lmeasure_eq_integral[OF mea,THEN sym] 
576 
unfolding integral_eq_lmeasure[OF mea, symmetric] * 

577 
apply(rule integrable_integral) using mea . 

40859  578 
qed auto 
41654  579 
qed 
580 
qed 

40859  581 

582 
lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s" 

583 
unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI) 

584 
using assms by auto 

585 

586 
lemma simple_function_has_integral': 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

587 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

588 
assumes f:"simple_function lebesgue f" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

589 
and i: "integral\<^isup>S lebesgue f \<noteq> \<omega>" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

590 
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV" 
40859  591 
proof let ?f = "\<lambda>x. if f x = \<omega> then 0 else f x" 
592 
{ fix x have "real (f x) = real (?f x)" by (cases "f x") auto } note * = this 

593 
have **:"{x. f x \<noteq> ?f x} = f ` {\<omega>}" by auto 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

594 
have **:"lebesgue.\<mu> {x\<in>space lebesgue. f x \<noteq> ?f x} = 0" 
40859  595 
using lebesgue.simple_integral_omega[OF assms] by(auto simp add:**) 
596 
show ?thesis apply(subst lebesgue.simple_integral_cong'[OF f _ **]) 

597 
apply(rule lebesgue.simple_function_compose1[OF f]) 

598 
unfolding * defer apply(rule simple_function_has_integral) 

599 
proof 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

600 
show "simple_function lebesgue ?f" 
40859  601 
using lebesgue.simple_function_compose1[OF f] . 
602 
show "\<forall>x. ?f x \<noteq> \<omega>" by auto 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

603 
show "\<forall>x\<in>range ?f. lebesgue.\<mu> (?f ` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0" 
40859  604 
proof (safe, simp, safe, rule ccontr) 
605 
fix y assume "f y \<noteq> \<omega>" "f y \<noteq> 0" 

606 
hence "(\<lambda>x. if f x = \<omega> then 0 else f x) ` {if f y = \<omega> then 0 else f y} = f ` {f y}" 

607 
by (auto split: split_if_asm) 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

608 
moreover assume "lebesgue.\<mu> ((\<lambda>x. if f x = \<omega> then 0 else f x) ` {if f y = \<omega> then 0 else f y}) = \<omega>" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

609 
ultimately have "lebesgue.\<mu> (f ` {f y}) = \<omega>" by simp 
40859  610 
moreover 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

611 
have "f y * lebesgue.\<mu> (f ` {f y}) \<noteq> \<omega>" using i f 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

612 
unfolding simple_integral_def setsum_\<omega> simple_function_def 
40859  613 
by auto 
614 
ultimately have "f y = 0" by (auto split: split_if_asm) 

615 
then show False using `f y \<noteq> 0` by simp 

616 
qed 

617 
qed 

618 
qed 

619 

620 
lemma (in measure_space) positive_integral_monotone_convergence: 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

621 
fixes f::"nat \<Rightarrow> 'a \<Rightarrow> pextreal" 
40859  622 
assumes i: "\<And>i. f i \<in> borel_measurable M" and mono: "\<And>x. mono (\<lambda>n. f n x)" 
623 
and lim: "\<And>x. (\<lambda>i. f i x) > u x" 

624 
shows "u \<in> borel_measurable M" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

625 
and "(\<lambda>i. integral\<^isup>P M (f i)) > integral\<^isup>P M u" (is ?ilim) 
40859  626 
proof  
627 
from positive_integral_isoton[unfolded isoton_fun_expand isoton_iff_Lim_mono, of f u] 

628 
show ?ilim using mono lim i by auto 

41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset

629 
have "\<And>x. (SUP i. f i x) = u x" using mono lim SUP_Lim_pextreal 
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset

630 
unfolding fun_eq_iff mono_def by auto 
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset

631 
moreover have "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M" 
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset

632 
using i by auto 
40859  633 
ultimately show "u \<in> borel_measurable M" by simp 
634 
qed 

635 

636 
lemma positive_integral_has_integral: 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

637 
fixes f::"'a::ordered_euclidean_space => pextreal" 
40859  638 
assumes f:"f \<in> borel_measurable lebesgue" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

639 
and int_om:"integral\<^isup>P lebesgue f \<noteq> \<omega>" 
40859  640 
and f_om:"\<forall>x. f x \<noteq> \<omega>" (* TODO: remove this *) 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

641 
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>P lebesgue f))) UNIV" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

642 
proof let ?i = "integral\<^isup>P lebesgue f" 
40859  643 
from lebesgue.borel_measurable_implies_simple_function_sequence[OF f] 
644 
guess u .. note conjunctD2[OF this,rule_format] note u = conjunctD2[OF this(1)] this(2) 

645 
let ?u = "\<lambda>i x. real (u i x)" and ?f = "\<lambda>x. real (f x)" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

646 
have u_simple:"\<And>k. integral\<^isup>S lebesgue (u k) = integral\<^isup>P lebesgue (u k)" 
40859  647 
apply(subst lebesgue.positive_integral_eq_simple_integral[THEN sym,OF u(1)]) .. 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

648 
have int_u_le:"\<And>k. integral\<^isup>S lebesgue (u k) \<le> integral\<^isup>P lebesgue f" 
40859  649 
unfolding u_simple apply(rule lebesgue.positive_integral_mono) 
650 
using isoton_Sup[OF u(3)] unfolding le_fun_def by auto 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

651 
have u_int_om:"\<And>i. integral\<^isup>S lebesgue (u i) \<noteq> \<omega>" 
40859  652 
proof case goal1 thus ?case using int_u_le[of i] int_om by auto qed 
653 

654 
note u_int = simple_function_has_integral'[OF u(1) this] 

655 
have "(\<lambda>x. real (f x)) integrable_on UNIV \<and> 

656 
(\<lambda>k. Integration.integral UNIV (\<lambda>x. real (u k x))) > Integration.integral UNIV (\<lambda>x. real (f x))" 

657 
apply(rule monotone_convergence_increasing) apply(rule,rule,rule u_int) 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

658 
proof safe case goal1 show ?case apply(rule real_of_pextreal_mono) using u(2,3) by auto 
40859  659 
next case goal2 show ?case using u(3) apply(subst lim_Real[THEN sym]) 
660 
prefer 3 apply(subst Real_real') defer apply(subst Real_real') 

661 
using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] using f_om u by auto 

662 
next case goal3 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

663 
show ?case apply(rule bounded_realI[where B="real (integral\<^isup>P lebesgue f)"]) 
40859  664 
apply safe apply(subst abs_of_nonneg) apply(rule integral_nonneg,rule) apply(rule u_int) 
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

665 
unfolding integral_unique[OF u_int] defer apply(rule real_of_pextreal_mono[OF _ int_u_le]) 
40859  666 
using u int_om by auto 
667 
qed note int = conjunctD2[OF this] 

668 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

669 
have "(\<lambda>i. integral\<^isup>S lebesgue (u i)) > ?i" unfolding u_simple 
40859  670 
apply(rule lebesgue.positive_integral_monotone_convergence(2)) 
671 
apply(rule lebesgue.borel_measurable_simple_function[OF u(1)]) 

672 
using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] by auto 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

673 
hence "(\<lambda>i. real (integral\<^isup>S lebesgue (u i))) > real ?i" apply 
40859  674 
apply(subst lim_Real[THEN sym]) prefer 3 
675 
apply(subst Real_real') defer apply(subst Real_real') 

676 
using u f_om int_om u_int_om by auto 

677 
note * = LIMSEQ_unique[OF this int(2)[unfolded integral_unique[OF u_int]]] 

678 
show ?thesis unfolding * by(rule integrable_integral[OF int(1)]) 

679 
qed 

680 

681 
lemma lebesgue_integral_has_integral: 

682 
fixes f::"'a::ordered_euclidean_space => real" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

683 
assumes f:"integrable lebesgue f" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

684 
shows "(f has_integral (integral\<^isup>L lebesgue f)) UNIV" 
40859  685 
proof let ?n = "\<lambda>x.  min (f x) 0" and ?p = "\<lambda>x. max (f x) 0" 
686 
have *:"f = (\<lambda>x. ?p x  ?n x)" apply rule by auto 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

687 
note f = integrableD[OF f] 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

688 
show ?thesis unfolding lebesgue_integral_def apply(subst *) 
40859  689 
proof(rule has_integral_sub) case goal1 
690 
have *:"\<forall>x. Real (f x) \<noteq> \<omega>" by auto 

691 
note lebesgue.borel_measurable_Real[OF f(1)] 

692 
from positive_integral_has_integral[OF this f(2) *] 

693 
show ?case unfolding real_Real_max . 

694 
next case goal2 

695 
have *:"\<forall>x. Real ( f x) \<noteq> \<omega>" by auto 

696 
note lebesgue.borel_measurable_uminus[OF f(1)] 

697 
note lebesgue.borel_measurable_Real[OF this] 

698 
from positive_integral_has_integral[OF this f(3) *] 

699 
show ?case unfolding real_Real_max minus_min_eq_max by auto 

700 
qed 

701 
qed 

702 

41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

703 
lemma lebesgue_positive_integral_eq_borel: 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

704 
"f \<in> borel_measurable borel \<Longrightarrow> integral\<^isup>P lebesgue f = integral\<^isup>P lborel f" 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

705 
by (auto intro!: lebesgue.positive_integral_subalgebra[symmetric]) default 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

706 

2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

707 
lemma lebesgue_integral_eq_borel: 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

708 
assumes "f \<in> borel_measurable borel" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

709 
shows "integrable lebesgue f \<longleftrightarrow> integrable lborel f" (is ?P) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

710 
and "integral\<^isup>L lebesgue f = integral\<^isup>L lborel f" (is ?I) 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

711 
proof  
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

712 
have *: "sigma_algebra lborel" by default 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

713 
have "sets lborel \<subseteq> sets lebesgue" by auto 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

714 
from lebesgue.integral_subalgebra[of f lborel, OF _ this _ _ *] assms 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

715 
show ?P ?I by auto 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

716 
qed 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

717 

2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

718 
lemma borel_integral_has_integral: 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

719 
fixes f::"'a::ordered_euclidean_space => real" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

720 
assumes f:"integrable lborel f" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

721 
shows "(f has_integral (integral\<^isup>L lborel f)) UNIV" 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

722 
proof  
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

723 
have borel: "f \<in> borel_measurable borel" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

724 
using f unfolding integrable_def by auto 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

725 
from f show ?thesis 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

726 
using lebesgue_integral_has_integral[of f] 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

727 
unfolding lebesgue_integral_eq_borel[OF borel] by simp 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

728 
qed 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

729 

40859  730 
lemma continuous_on_imp_borel_measurable: 
731 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space" 

732 
assumes "continuous_on UNIV f" 

41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

733 
shows "f \<in> borel_measurable borel" 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

734 
apply(rule borel.borel_measurableI) 
40859  735 
using continuous_open_preimage[OF assms] unfolding vimage_def by auto 
736 

737 
lemma (in measure_space) integral_monotone_convergence_pos': 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

738 
assumes i: "\<And>i. integrable M (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)" 
40859  739 
and pos: "\<And>x i. 0 \<le> f i x" 
740 
and lim: "\<And>x. (\<lambda>i. f i x) > u x" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

741 
and ilim: "(\<lambda>i. integral\<^isup>L M (f i)) > x" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

742 
shows "integrable M u \<and> integral\<^isup>L M u = x" 
40859  743 
using integral_monotone_convergence_pos[OF assms] by auto 
744 

745 
definition e2p :: "'a::ordered_euclidean_space \<Rightarrow> (nat \<Rightarrow> real)" where 

746 
"e2p x = (\<lambda>i\<in>{..<DIM('a)}. x$$i)" 

747 

748 
definition p2e :: "(nat \<Rightarrow> real) \<Rightarrow> 'a::ordered_euclidean_space" where 

749 
"p2e x = (\<chi>\<chi> i. x i)" 

750 

41095  751 
lemma e2p_p2e[simp]: 
752 
"x \<in> extensional {..<DIM('a)} \<Longrightarrow> e2p (p2e x::'a::ordered_euclidean_space) = x" 

753 
by (auto simp: fun_eq_iff extensional_def p2e_def e2p_def) 

40859  754 

41095  755 
lemma p2e_e2p[simp]: 
756 
"p2e (e2p x) = (x::'a::ordered_euclidean_space)" 

757 
by (auto simp: euclidean_eq[where 'a='a] p2e_def e2p_def) 

40859  758 

41661  759 
declare restrict_extensional[intro] 
760 

761 
lemma e2p_extensional[intro]:"e2p (y::'a::ordered_euclidean_space) \<in> extensional {..<DIM('a)}" 

762 
unfolding e2p_def by auto 

763 

764 
lemma e2p_image_vimage: fixes A::"'a::ordered_euclidean_space set" 

765 
shows "e2p ` A = p2e ` A \<inter> extensional {..<DIM('a)}" 

766 
proof(rule set_eqI,rule) 

767 
fix x assume "x \<in> e2p ` A" then guess y unfolding image_iff .. note y=this 

768 
show "x \<in> p2e ` A \<inter> extensional {..<DIM('a)}" 

769 
apply safe apply(rule vimageI[OF _ y(1)]) unfolding y p2e_e2p by auto 

770 
next fix x assume "x \<in> p2e ` A \<inter> extensional {..<DIM('a)}" 

771 
thus "x \<in> e2p ` A" unfolding image_iff apply(rule_tac x="p2e x" in bexI) apply(subst e2p_p2e) by auto 

772 
qed 

773 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

774 
interpretation lborel_product: product_sigma_finite "\<lambda>x. lborel::real measure_space" 
40859  775 
by default 
776 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

777 
interpretation lborel_space: finite_product_sigma_finite "\<lambda>x. lborel::real measure_space" "{..<DIM('a::ordered_euclidean_space)}" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

778 
where "space lborel = UNIV" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

779 
and "sets lborel = sets borel" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

780 
and "measure lborel = lebesgue.\<mu>" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

781 
and "measurable lborel = measurable borel" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

782 
proof  
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

783 
show "finite_product_sigma_finite (\<lambda>x. lborel::real measure_space) {..<DIM('a::ordered_euclidean_space)}" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

784 
by default simp 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

785 
qed simp_all 
40859  786 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

787 
lemma sets_product_borel: 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

788 
assumes [intro]: "finite I" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

789 
shows "sets (\<Pi>\<^isub>M i\<in>I. 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

790 
\<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>) = 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

791 
sets (\<Pi>\<^isub>M i\<in>I. lborel)" (is "sets ?G = _") 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

792 
proof  
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

793 
have "sets ?G = sets (\<Pi>\<^isub>M i\<in>I. 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

794 
sigma \<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>)" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

795 
by (subst sigma_product_algebra_sigma_eq[of I "\<lambda>_ i. {..<real i}" ]) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

796 
(auto intro!: measurable_sigma_sigma isotoneI real_arch_lt 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

797 
simp: product_algebra_def) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

798 
then show ?thesis 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

799 
unfolding lborel_def borel_eq_lessThan lebesgue_def sigma_def by simp 
40859  800 
qed 
801 

41661  802 
lemma measurable_e2p: 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

803 
"e2p \<in> measurable (borel::'a::ordered_euclidean_space algebra) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

804 
(\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

805 
(is "_ \<in> measurable ?E ?P") 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

806 
proof  
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

807 
let ?B = "\<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

808 
let ?G = "product_algebra_generator {..<DIM('a)} (\<lambda>_. ?B)" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

809 
have "e2p \<in> measurable ?E (sigma ?G)" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

810 
proof (rule borel.measurable_sigma) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

811 
show "e2p \<in> space ?E \<rightarrow> space ?G" by (auto simp: e2p_def) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

812 
fix A assume "A \<in> sets ?G" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

813 
then obtain E where A: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. E i)" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

814 
and "E \<in> {..<DIM('a)} \<rightarrow> (range lessThan)" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

815 
by (auto elim!: product_algebraE simp: ) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

816 
then have "\<forall>i\<in>{..<DIM('a)}. \<exists>xs. E i = {..< xs}" by auto 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

817 
from this[THEN bchoice] guess xs .. 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

818 
then have A_eq: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< xs i})" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

819 
using A by auto 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

820 
have "e2p ` A = {..< (\<chi>\<chi> i. xs i) :: 'a}" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

821 
using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def A_eq 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

822 
euclidean_eq[where 'a='a] eucl_less[where 'a='a]) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

823 
then show "e2p ` A \<inter> space ?E \<in> sets ?E" by simp 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

824 
qed (auto simp: product_algebra_generator_def) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

825 
with sets_product_borel[of "{..<DIM('a)}"] show ?thesis 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

826 
unfolding measurable_def product_algebra_def by simp 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

827 
qed 
41661  828 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

829 
lemma measurable_p2e: 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

830 
"p2e \<in> measurable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space)) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

831 
(borel :: 'a::ordered_euclidean_space algebra)" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

832 
(is "p2e \<in> measurable ?P _") 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

833 
unfolding borel_eq_lessThan 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

834 
proof (intro lborel_space.measurable_sigma) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

835 
let ?E = "\<lparr> space = UNIV :: 'a set, sets = range lessThan \<rparr>" 
41095  836 
show "p2e \<in> space ?P \<rightarrow> space ?E" by simp 
837 
fix A assume "A \<in> sets ?E" 

838 
then obtain x where "A = {..<x}" by auto 

839 
then have "p2e ` A \<inter> space ?P = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< x $$ i})" 

840 
using DIM_positive 

841 
by (auto simp: Pi_iff set_eq_iff p2e_def 

842 
euclidean_eq[where 'a='a] eucl_less[where 'a='a]) 

843 
then show "p2e ` A \<inter> space ?P \<in> sets ?P" by auto 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

844 
qed simp 
41095  845 

41704
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

846 
lemma inj_e2p[intro, simp]: "inj e2p" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

847 
proof (intro inj_onI conjI allI impI euclidean_eq[where 'a='a, THEN iffD2]) 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

848 
fix x y ::'a and i assume "e2p x = e2p y" "i < DIM('a)" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

849 
then show "x $$ i= y $$ i" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

850 
by (auto simp: e2p_def restrict_def fun_eq_iff elim!: allE[where x=i]) 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

851 
qed 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

852 

40859  853 
lemma e2p_Int:"e2p ` A \<inter> e2p ` B = e2p ` (A \<inter> B)" (is "?L = ?R") 
41704
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

854 
by (auto simp: image_Int[THEN sym]) 
40859  855 

856 
lemma Int_stable_cuboids: fixes x::"'a::ordered_euclidean_space" 

857 
shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). e2p ` {a..b})\<rparr>" 

858 
unfolding Int_stable_def algebra.select_convs 

859 
proof safe fix a b x y::'a 

860 
have *:"e2p ` {a..b} \<inter> e2p ` {x..y} = 

861 
(\<lambda>(a, b). e2p ` {a..b}) (\<chi>\<chi> i. max (a $$ i) (x $$ i), \<chi>\<chi> i. min (b $$ i) (y $$ i)::'a)" 

862 
unfolding e2p_Int inter_interval by auto 

863 
show "e2p ` {a..b} \<inter> e2p ` {x..y} \<in> range (\<lambda>(a, b). e2p ` {a..b::'a})" unfolding * 

864 
apply(rule range_eqI) .. 

865 
qed 

866 

867 
lemma Int_stable_cuboids': fixes x::"'a::ordered_euclidean_space" 

868 
shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). {a..b})\<rparr>" 

869 
unfolding Int_stable_def algebra.select_convs 

870 
apply safe unfolding inter_interval by auto 

871 

872 
lemma lmeasure_measure_eq_borel_prod: 

873 
fixes A :: "('a::ordered_euclidean_space) set" 

874 
assumes "A \<in> sets borel" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

875 
shows "lebesgue.\<mu> A = lborel_space.\<mu> TYPE('a) (e2p ` A)" (is "_ = ?m A") 
40859  876 
proof (rule measure_unique_Int_stable[where X=A and A=cube]) 
877 
show "Int_stable \<lparr> space = UNIV :: 'a set, sets = range (\<lambda>(a,b). {a..b}) \<rparr>" 

878 
(is "Int_stable ?E" ) using Int_stable_cuboids' . 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

879 
have [simp]: "sigma ?E = borel" using borel_eq_atLeastAtMost .. 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

880 
show "\<And>i. lebesgue.\<mu> (cube i) \<noteq> \<omega>" unfolding cube_def by auto 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

881 
show "\<And>X. X \<in> sets ?E \<Longrightarrow> lebesgue.\<mu> X = ?m X" 
40859  882 
proof case goal1 then obtain a b where X:"X = {a..b}" by auto 
883 
{ presume *:"X \<noteq> {} \<Longrightarrow> ?case" 

884 
show ?case apply(cases,rule *,assumption) by auto } 

885 
def XX \<equiv> "\<lambda>i. {a $$ i .. b $$ i}" assume "X \<noteq> {}" note X' = this[unfolded X interval_ne_empty] 

886 
have *:"Pi\<^isub>E {..<DIM('a)} XX = e2p ` X" apply(rule set_eqI) 

887 
proof fix x assume "x \<in> Pi\<^isub>E {..<DIM('a)} XX" 

888 
thus "x \<in> e2p ` X" unfolding image_iff apply(rule_tac x="\<chi>\<chi> i. x i" in bexI) 

889 
unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by rule auto 

890 
next fix x assume "x \<in> e2p ` X" then guess y unfolding image_iff .. note y = this 

891 
show "x \<in> Pi\<^isub>E {..<DIM('a)} XX" unfolding y using y(1) 

892 
unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by auto 

893 
qed 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

894 
have "lebesgue.\<mu> X = (\<Prod>x<DIM('a). Real (b $$ x  a $$ x))" using X' apply unfolding X 
40859  895 
unfolding lmeasure_atLeastAtMost content_closed_interval apply(subst Real_setprod) by auto 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

896 
also have "... = (\<Prod>i<DIM('a). lebesgue.\<mu> (XX i))" apply(rule setprod_cong2) 
40859  897 
unfolding XX_def lmeasure_atLeastAtMost apply(subst content_real) using X' by auto 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

898 
also have "... = ?m X" unfolding *[THEN sym] 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

899 
apply(rule lborel_space.measure_times[symmetric]) unfolding XX_def by auto 
40859  900 
finally show ?case . 
901 
qed 

902 

903 
show "range cube \<subseteq> sets \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>" 

904 
unfolding cube_def_raw by auto 

905 
have "\<And>x. \<exists>xa. x \<in> cube xa" apply(rule_tac x=x in mem_big_cube) by fastsimp 

906 
thus "cube \<up> space \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>" 

907 
applyapply(rule isotoneI) apply(rule cube_subset_Suc) by auto 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

908 
show "A \<in> sets (sigma ?E)" using assms by simp 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

909 
have "measure_space lborel" by default 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

910 
then show "measure_space \<lparr> space = space ?E, sets = sets (sigma ?E), measure = measure lebesgue\<rparr>" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

911 
unfolding lebesgue_def lborel_def by simp 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

912 
let ?M = "\<lparr> space = space ?E, sets = sets (sigma ?E), measure = ?m \<rparr>" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

913 
show "measure_space ?M" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

914 
proof (rule lborel_space.measure_space_vimage) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

915 
show "sigma_algebra ?M" by (rule lborel.sigma_algebra_cong) auto 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

916 
show "p2e \<in> measurable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel) ?M" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

917 
using measurable_p2e unfolding measurable_def by auto 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

918 
fix A :: "'a set" assume "A \<in> sets ?M" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

919 
show "measure ?M A = lborel_space.\<mu> TYPE('a) (p2e ` A \<inter> space (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel))" 
41661  920 
by (simp add: e2p_image_vimage) 
921 
qed 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

922 
qed simp 
40859  923 

41661  924 
lemma range_e2p:"range (e2p::'a::ordered_euclidean_space \<Rightarrow> _) = extensional {..<DIM('a)}" 
925 
unfolding e2p_def_raw 

926 
apply auto 

927 
by (rule_tac x="\<chi>\<chi> i. x i" in image_eqI) (auto simp: fun_eq_iff extensional_def) 

40859  928 

929 
lemma borel_fubini_positiv_integral: 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

930 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> pextreal" 
40859  931 
assumes f: "f \<in> borel_measurable borel" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

932 
shows "integral\<^isup>P lborel f = \<integral>\<^isup>+x. f (p2e x) \<partial>(lborel_space.P TYPE('a))" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

933 
proof (rule lborel.positive_integral_vimage[symmetric, of _ "e2p :: 'a \<Rightarrow> _" "(\<lambda>x. f (p2e x))", unfolded p2e_e2p]) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

934 
show "(e2p :: 'a \<Rightarrow> _) \<in> measurable borel (lborel_space.P TYPE('a))" by (rule measurable_e2p) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

935 
show "sigma_algebra (lborel_space.P TYPE('a))" by default 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

936 
from measurable_comp[OF measurable_p2e f] 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

937 
show "(\<lambda>x. f (p2e x)) \<in> borel_measurable (lborel_space.P TYPE('a))" by (simp add: comp_def) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

938 
let "?L A" = "lebesgue.\<mu> ((e2p::'a \<Rightarrow> (nat \<Rightarrow> real)) ` A \<inter> UNIV)" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

939 
fix A :: "(nat \<Rightarrow> real) set" assume "A \<in> sets (lborel_space.P TYPE('a))" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

940 
then have A: "(e2p::'a \<Rightarrow> (nat \<Rightarrow> real)) ` A \<inter> space borel \<in> sets borel" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

941 
by (rule measurable_sets[OF measurable_e2p]) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

942 
have [simp]: "A \<inter> extensional {..<DIM('a)} = A" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

943 
using `A \<in> sets (lborel_space.P TYPE('a))`[THEN lborel_space.sets_into_space] by auto 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

944 
show "lborel_space.\<mu> TYPE('a) A = ?L A" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

945 
using lmeasure_measure_eq_borel_prod[OF A] by (simp add: range_e2p) 
40859  946 
qed 
947 

41704
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

948 
lemma borel_fubini_integrable: 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

949 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

950 
shows "integrable lborel f \<longleftrightarrow> 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

951 
integrable (lborel_space.P TYPE('a)) (\<lambda>x. f (p2e x))" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

952 
(is "_ \<longleftrightarrow> integrable ?B ?f") 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

953 
proof 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

954 
assume "integrable lborel f" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

955 
moreover then have f: "f \<in> borel_measurable borel" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

956 
by auto 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

957 
moreover with measurable_p2e 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

958 
have "f \<circ> p2e \<in> borel_measurable ?B" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

959 
by (rule measurable_comp) 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

960 
ultimately show "integrable ?B ?f" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

961 
by (simp add: comp_def borel_fubini_positiv_integral integrable_def) 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

962 
next 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

963 
assume "integrable ?B ?f" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

964 
moreover then 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

965 
have "?f \<circ> e2p \<in> borel_measurable (borel::'a algebra)" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

966 
by (auto intro!: measurable_e2p measurable_comp) 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

967 
then have "f \<in> borel_measurable borel" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

968 
by (simp cong: measurable_cong) 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

969 
ultimately show "integrable lborel f" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

970 
by (simp add: comp_def borel_fubini_positiv_integral integrable_def) 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

971 
qed 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

972 

40859  973 
lemma borel_fubini: 
974 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 

975 
assumes f: "f \<in> borel_measurable borel" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

976 
shows "integral\<^isup>L lborel f = \<integral>x. f (p2e x) \<partial>(lborel_space.P TYPE('a))" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

977 
proof  
40859  978 
have 1:"(\<lambda>x. Real (f x)) \<in> borel_measurable borel" using f by auto 
979 
have 2:"(\<lambda>x. Real ( f x)) \<in> borel_measurable borel" using f by auto 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

980 
show ?thesis unfolding lebesgue_integral_def 
40859  981 
unfolding borel_fubini_positiv_integral[OF 1] borel_fubini_positiv_integral[OF 2] 
982 
unfolding o_def .. 

38656  983 
qed 
984 

985 
end 