author  hoelzl 
Fri, 14 Dec 2012 15:46:01 +0100  
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parent 50418  bd68cf816dd3 
child 51000  c9adb50f74ad 
permissions  rwrr 
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(* Title: HOL/Probability/Lebesgue_Measure.thy 
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Author: Johannes Hölzl, TU München 

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Author: Robert Himmelmann, TU München 

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*) 

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header {* Lebsegue measure *} 
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theory Lebesgue_Measure 
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imports Finite_Product_Measure 
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begin 
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50104  12 
lemma absolutely_integrable_on_indicator[simp]: 
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fixes A :: "'a::ordered_euclidean_space set" 

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

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

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unfolding absolutely_integrable_on_def by simp 

49777  17 

50104  18 
lemma has_integral_indicator_UNIV: 
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fixes s A :: "'a::ordered_euclidean_space set" and x :: real 

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

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proof  

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

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by (auto simp: fun_eq_iff indicator_def) 

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then show ?thesis 

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

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qed 

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lemma 

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

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shows integral_indicator_UNIV: 

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

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and integrable_indicator_UNIV: 

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

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unfolding integral_def integrable_on_def has_integral_indicator_UNIV by auto 

47694  35 

38656  36 
subsection {* Standard Cubes *} 
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40859  38 
definition cube :: "nat \<Rightarrow> 'a::ordered_euclidean_space set" where 
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"cube n \<equiv> {\<Sum>i\<in>Basis.  n *\<^sub>R i .. \<Sum>i\<in>Basis. n *\<^sub>R i}" 
40859  40 

49777  41 
lemma borel_cube[intro]: "cube n \<in> sets borel" 
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unfolding cube_def by auto 

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40859  44 
lemma cube_closed[intro]: "closed (cube n)" 
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unfolding cube_def by auto 

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

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by (fastforce simp: eucl_le[where 'a='a] cube_def setsum_negf) 
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lemma cube_subset_iff: "cube n \<subseteq> cube N \<longleftrightarrow> n \<le> N" 
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unfolding cube_def subset_interval by (simp add: setsum_negf ex_in_conv) 
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lemma ball_subset_cube: "ball (0::'a::ordered_euclidean_space) (real n) \<subseteq> cube n" 
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apply (simp add: cube_def subset_eq mem_interval setsum_negf eucl_le[where 'a='a]) 
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proof safe 
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fix x i :: 'a assume x: "x \<in> ball 0 (real n)" and i: "i \<in> Basis" 
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thus " real n \<le> x \<bullet> i" "real n \<ge> x \<bullet> i" 
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using Basis_le_norm[OF i, of x] by(auto simp: dist_norm) 
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qed 
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lemma mem_big_cube: obtains n where "x \<in> cube n" 

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proof  
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from reals_Archimedean2[of "norm x"] guess n .. 
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with ball_subset_cube[unfolded subset_eq, of n] 
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show ?thesis 
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by (intro that[where n=n]) (auto simp add: dist_norm) 
38656  67 
qed 
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lemma cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)" 
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unfolding cube_def subset_interval by (simp add: setsum_negf) 
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50104  72 
lemma has_integral_interval_cube: 
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fixes a b :: "'a::ordered_euclidean_space" 

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shows "(indicator {a .. b} has_integral content ({a .. b} \<inter> cube n)) (cube n)" 
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(is "(?I has_integral content ?R) (cube n)") 
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proof  

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have [simp]: "(\<lambda>x. if x \<in> cube n then ?I x else 0) = indicator ?R" 

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

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

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

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also have "\<dots> \<longleftrightarrow> ((\<lambda>x. 1::real) has_integral content ?R *\<^sub>R 1) ?R" 
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unfolding indicator_def [abs_def] has_integral_restrict_univ real_scaleR_def mult_1_right .. 
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also have "((\<lambda>x. 1) has_integral content ?R *\<^sub>R 1) ?R" 
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unfolding cube_def inter_interval by (rule has_integral_const) 
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finally show ?thesis . 
50104  86 
qed 
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subsection {* Lebesgue measure *} 
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47694  90 
definition lebesgue :: "'a::ordered_euclidean_space measure" where 
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"lebesgue = measure_of UNIV {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n} 

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(\<lambda>A. SUP n. ereal (integral (cube n) (indicator A)))" 

41661  93 

41654  94 
lemma space_lebesgue[simp]: "space lebesgue = UNIV" 
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unfolding lebesgue_def by simp 

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

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unfolding lebesgue_def by simp 

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47694  100 
lemma sigma_algebra_lebesgue: 
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defines "leb \<equiv> {A. \<forall>n. (indicator A :: 'a::ordered_euclidean_space \<Rightarrow> real) integrable_on cube n}" 

102 
shows "sigma_algebra UNIV leb" 

103 
proof (safe intro!: sigma_algebra_iff2[THEN iffD2]) 

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fix A assume A: "A \<in> leb" 

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moreover have "indicator (UNIV  A) = (\<lambda>x. 1  indicator A x :: real)" 

41654  106 
by (auto simp: fun_eq_iff indicator_def) 
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ultimately show "UNIV  A \<in> leb" 
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using A by (auto intro!: integrable_sub simp: cube_def leb_def) 

41654  109 
next 
47694  110 
fix n show "{} \<in> leb" 
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by (auto simp: cube_def indicator_def[abs_def] leb_def) 

41654  112 
next 
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fix A :: "nat \<Rightarrow> _" assume A: "range A \<subseteq> leb" 
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have "\<forall>n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n" (is "\<forall>n. ?g integrable_on _") 

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

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

41654  117 
proof (induct k) 
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case (Suc k) 

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

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unfolding lessThan_Suc UN_insert by auto 

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

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

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by (auto simp: fun_eq_iff * indicator_def) 

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show ?case 

47694  125 
using absolutely_integrable_max[of ?f "cube n" ?g] A Suc 
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by (simp add: * leb_def subset_eq) 

41654  127 
qed auto 
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qed (auto intro: LIMSEQ_indicator_UN simp: cube_def) 

47694  129 
then show "(\<Union>i. A i) \<in> leb" by (auto simp: leb_def) 
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qed simp 
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47694  132 
lemma sets_lebesgue: "sets lebesgue = {A. \<forall>n. (indicator A :: _ \<Rightarrow> real) integrable_on cube n}" 
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unfolding lebesgue_def sigma_algebra.sets_measure_of_eq[OF sigma_algebra_lebesgue] .. 

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

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unfolding sets_lebesgue by simp 

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lemma emeasure_lebesgue: 
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assumes "A \<in> sets lebesgue" 
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shows "emeasure lebesgue A = (SUP n. ereal (integral (cube n) (indicator A)))" 

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(is "_ = ?\<mu> A") 

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proof (rule emeasure_measure_of[OF lebesgue_def]) 

41654  143 
have *: "indicator {} = (\<lambda>x. 0 :: real)" by (simp add: fun_eq_iff) 
47694  144 
show "positive (sets lebesgue) ?\<mu>" 
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proof (unfold positive_def, intro conjI ballI) 

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show "?\<mu> {} = 0" by (simp add: integral_0 *) 

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fix A :: "'a set" assume "A \<in> sets lebesgue" then show "0 \<le> ?\<mu> A" 

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by (auto intro!: SUP_upper2 Integration.integral_nonneg simp: sets_lebesgue) 

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qed 
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next 
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show "countably_additive (sets lebesgue) ?\<mu>" 
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proof (intro countably_additive_def[THEN iffD2] allI impI) 
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fix A :: "nat \<Rightarrow> 'a set" assume rA: "range A \<subseteq> sets lebesgue" "disjoint_family A" 
41654  154 
then have A[simp, intro]: "\<And>i n. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n" 
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by (auto dest: lebesgueD) 

46731  156 
let ?m = "\<lambda>n i. integral (cube n) (indicator (A i) :: _\<Rightarrow>real)" 
157 
let ?M = "\<lambda>n I. integral (cube n) (indicator (\<Union>i\<in>I. A i) :: _\<Rightarrow>real)" 

47694  158 
have nn[simp, intro]: "\<And>i n. 0 \<le> ?m n i" by (auto intro!: Integration.integral_nonneg) 
41654  159 
assume "(\<Union>i. A i) \<in> sets lebesgue" 
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then have UN_A[simp, intro]: "\<And>i n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n" 

47694  161 
by (auto simp: sets_lebesgue) 
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show "(\<Sum>n. ?\<mu> (A n)) = ?\<mu> (\<Union>i. A i)" 

49777  163 
proof (subst suminf_SUP_eq, safe intro!: incseq_SucI) 
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fix i n show "ereal (?m n i) \<le> ereal (?m (Suc n) i)" 
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using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le incseq_SucI) 
41654  166 
next 
43920  167 
fix i n show "0 \<le> ereal (?m n i)" 
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using rA unfolding lebesgue_def 
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by (auto intro!: SUP_upper2 integral_nonneg) 
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next 
43920  171 
show "(SUP n. \<Sum>i. ereal (?m n i)) = (SUP n. ereal (?M n UNIV))" 
172 
proof (intro arg_cong[where f="SUPR UNIV"] ext sums_unique[symmetric] sums_ereal[THEN iffD2] sums_def[THEN iffD2]) 

41654  173 
fix n 
174 
have "\<And>m. (UNION {..<m} A) \<in> sets lebesgue" using rA by auto 

175 
from lebesgueD[OF this] 

176 
have "(\<lambda>m. ?M n {..< m}) > ?M n UNIV" 

177 
(is "(\<lambda>m. integral _ (?A m)) > ?I") 

178 
by (intro dominated_convergence(2)[where f="?A" and h="\<lambda>x. 1::real"]) 

179 
(auto intro: LIMSEQ_indicator_UN simp: cube_def) 

180 
moreover 

181 
{ fix m have *: "(\<Sum>x<m. ?m n x) = ?M n {..< m}" 

182 
proof (induct m) 

183 
case (Suc m) 

184 
have "(\<Union>i<m. A i) \<in> sets lebesgue" using rA by auto 

185 
then have "(indicator (\<Union>i<m. A i) :: _\<Rightarrow>real) integrable_on (cube n)" 

186 
by (auto dest!: lebesgueD) 

187 
moreover 

188 
have "(\<Union>i<m. A i) \<inter> A m = {}" 

189 
using rA(2)[unfolded disjoint_family_on_def, THEN bspec, of m] 

190 
by auto 

191 
then have "\<And>x. indicator (\<Union>i<Suc m. A i) x = 

192 
indicator (\<Union>i<m. A i) x + (indicator (A m) x :: real)" 

193 
by (auto simp: indicator_add lessThan_Suc ac_simps) 

194 
ultimately show ?case 

47694  195 
using Suc A by (simp add: Integration.integral_add[symmetric]) 
41654  196 
qed auto } 
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ultimately show "(\<lambda>m. \<Sum>x = 0..<m. ?m n x) > ?M n UNIV" 
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by (simp add: atLeast0LessThan) 
41654  199 
qed 
200 
qed 

201 
qed 

47694  202 
qed (auto, fact) 
40859  203 

41654  204 
lemma lebesgueI_borel[intro, simp]: 
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fixes s::"'a::ordered_euclidean_space set" 

40859  206 
assumes "s \<in> sets borel" shows "s \<in> sets lebesgue" 
41654  207 
proof  
47694  208 
have "s \<in> sigma_sets (space lebesgue) (range (\<lambda>(a, b). {a .. (b :: 'a\<Colon>ordered_euclidean_space)}))" 
209 
using assms by (simp add: borel_eq_atLeastAtMost) 

210 
also have "\<dots> \<subseteq> sets lebesgue" 

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proof (safe intro!: sets.sigma_sets_subset lebesgueI) 
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fix n :: nat and a b :: 'a 
213 
show "(indicator {a..b} :: 'a\<Rightarrow>real) integrable_on cube n" 

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unfolding integrable_on_def using has_integral_interval_cube[of a b] by auto 
41654  215 
qed 
47694  216 
finally show ?thesis . 
38656  217 
qed 
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lemma borel_measurable_lebesgueI: 
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"f \<in> borel_measurable borel \<Longrightarrow> f \<in> borel_measurable lebesgue" 
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unfolding measurable_def by simp 
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40859  223 
lemma lebesgueI_negligible[dest]: fixes s::"'a::ordered_euclidean_space set" 
224 
assumes "negligible s" shows "s \<in> sets lebesgue" 

41654  225 
using assms by (force simp: cube_def integrable_on_def negligible_def intro!: lebesgueI) 
38656  226 

41654  227 
lemma lmeasure_eq_0: 
47694  228 
fixes S :: "'a::ordered_euclidean_space set" 
229 
assumes "negligible S" shows "emeasure lebesgue S = 0" 

40859  230 
proof  
41654  231 
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) 
47694  235 
then show ?thesis 
236 
using assms by (simp add: emeasure_lebesgue lebesgueI_negligible) 

40859  237 
qed 
238 

239 
lemma lmeasure_iff_LIMSEQ: 

47694  240 
assumes A: "A \<in> sets lebesgue" and "0 \<le> m" 
241 
shows "emeasure lebesgue A = ereal m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) > m" 

242 
proof (subst emeasure_lebesgue[OF A], intro SUP_eq_LIMSEQ) 

41654  243 
show "mono (\<lambda>n. integral (cube n) (indicator A::_=>real))" 
244 
using cube_subset assms by (intro monoI integral_subset_le) (auto dest!: lebesgueD) 

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qed 
38656  246 

41654  247 
lemma lmeasure_finite_has_integral: 
248 
fixes s :: "'a::ordered_euclidean_space set" 

49777  249 
assumes "s \<in> sets lebesgue" "emeasure lebesgue s = ereal m" 
41654  250 
shows "(indicator s has_integral m) UNIV" 
251 
proof  

252 
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real" 

49777  253 
have "0 \<le> m" 
254 
using emeasure_nonneg[of lebesgue s] `emeasure lebesgue s = ereal m` by simp 

41654  255 
have **: "(?I s) integrable_on UNIV \<and> (\<lambda>k. integral UNIV (?I (s \<inter> cube k))) > integral UNIV (?I s)" 
256 
proof (intro monotone_convergence_increasing allI ballI) 

257 
have LIMSEQ: "(\<lambda>n. integral (cube n) (?I s)) > m" 

49777  258 
using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1) `0 \<le> m`] . 
41654  259 
{ fix n have "integral (cube n) (?I s) \<le> m" 
260 
using cube_subset assms 

261 
by (intro incseq_le[where L=m] LIMSEQ incseq_def[THEN iffD2] integral_subset_le allI impI) 

262 
(auto dest!: lebesgueD) } 

263 
moreover 

264 
{ fix n have "0 \<le> integral (cube n) (?I s)" 

47694  265 
using assms by (auto dest!: lebesgueD intro!: Integration.integral_nonneg) } 
41654  266 
ultimately 
267 
show "bounded {integral UNIV (?I (s \<inter> cube k)) k. True}" 

268 
unfolding bounded_def 

269 
apply (rule_tac exI[of _ 0]) 

270 
apply (rule_tac exI[of _ m]) 

271 
by (auto simp: dist_real_def integral_indicator_UNIV) 

272 
fix k show "?I (s \<inter> cube k) integrable_on UNIV" 

273 
unfolding integrable_indicator_UNIV using assms by (auto dest!: lebesgueD) 

274 
fix x show "?I (s \<inter> cube k) x \<le> ?I (s \<inter> cube (Suc k)) x" 

275 
using cube_subset[of k "Suc k"] by (auto simp: indicator_def) 

276 
next 

277 
fix x :: 'a 

278 
from mem_big_cube obtain k where k: "x \<in> cube k" . 

279 
{ fix n have "?I (s \<inter> cube (n + k)) x = ?I s x" 

280 
using k cube_subset[of k "n + k"] by (auto simp: indicator_def) } 

281 
note * = this 

282 
show "(\<lambda>k. ?I (s \<inter> cube k) x) > ?I s x" 

283 
by (rule LIMSEQ_offset[where k=k]) (auto simp: *) 

284 
qed 

285 
note ** = conjunctD2[OF this] 

286 
have m: "m = integral UNIV (?I s)" 

287 
apply (intro LIMSEQ_unique[OF _ **(2)]) 

49777  288 
using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1) `0 \<le> m`] integral_indicator_UNIV . 
41654  289 
show ?thesis 
290 
unfolding m by (intro integrable_integral **) 

38656  291 
qed 
292 

47694  293 
lemma lmeasure_finite_integrable: assumes s: "s \<in> sets lebesgue" and "emeasure lebesgue s \<noteq> \<infinity>" 
41654  294 
shows "(indicator s :: _ \<Rightarrow> real) integrable_on UNIV" 
47694  295 
proof (cases "emeasure lebesgue s") 
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296 
case (real m) 
47694  297 
with lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this] emeasure_nonneg[of lebesgue s] 
41654  298 
show ?thesis unfolding integrable_on_def by auto 
47694  299 
qed (insert assms emeasure_nonneg[of lebesgue s], auto) 
38656  300 

41654  301 
lemma has_integral_lebesgue: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV" 
302 
shows "s \<in> sets lebesgue" 

303 
proof (intro lebesgueI) 

304 
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real" 

305 
fix n show "(?I s) integrable_on cube n" unfolding cube_def 

306 
proof (intro integrable_on_subinterval) 

307 
show "(?I s) integrable_on UNIV" 

308 
unfolding integrable_on_def using assms by auto 

309 
qed auto 

38656  310 
qed 
311 

41654  312 
lemma has_integral_lmeasure: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV" 
47694  313 
shows "emeasure lebesgue s = ereal m" 
41654  314 
proof (intro lmeasure_iff_LIMSEQ[THEN iffD2]) 
315 
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real" 

316 
show "s \<in> sets lebesgue" using has_integral_lebesgue[OF assms] . 

317 
show "0 \<le> m" using assms by (rule has_integral_nonneg) auto 

318 
have "(\<lambda>n. integral UNIV (?I (s \<inter> cube n))) > integral UNIV (?I s)" 

319 
proof (intro dominated_convergence(2) ballI) 

320 
show "(?I s) integrable_on UNIV" unfolding integrable_on_def using assms by auto 

321 
fix n show "?I (s \<inter> cube n) integrable_on UNIV" 

322 
unfolding integrable_indicator_UNIV using `s \<in> sets lebesgue` by (auto dest: lebesgueD) 

323 
fix x show "norm (?I (s \<inter> cube n) x) \<le> ?I s x" by (auto simp: indicator_def) 

324 
next 

325 
fix x :: 'a 

326 
from mem_big_cube obtain k where k: "x \<in> cube k" . 

327 
{ fix n have "?I (s \<inter> cube (n + k)) x = ?I s x" 

328 
using k cube_subset[of k "n + k"] by (auto simp: indicator_def) } 

329 
note * = this 

330 
show "(\<lambda>k. ?I (s \<inter> cube k) x) > ?I s x" 

331 
by (rule LIMSEQ_offset[where k=k]) (auto simp: *) 

332 
qed 

333 
then show "(\<lambda>n. integral (cube n) (?I s)) > m" 

334 
unfolding integral_unique[OF assms] integral_indicator_UNIV by simp 

335 
qed 

336 

337 
lemma has_integral_iff_lmeasure: 

49777  338 
"(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> emeasure lebesgue A = ereal m)" 
40859  339 
proof 
41654  340 
assume "(indicator A has_integral m) UNIV" 
341 
with has_integral_lmeasure[OF this] has_integral_lebesgue[OF this] 

49777  342 
show "A \<in> sets lebesgue \<and> emeasure lebesgue A = ereal m" 
41654  343 
by (auto intro: has_integral_nonneg) 
40859  344 
next 
49777  345 
assume "A \<in> sets lebesgue \<and> emeasure lebesgue A = ereal m" 
41654  346 
then show "(indicator A has_integral m) UNIV" by (intro lmeasure_finite_has_integral) auto 
38656  347 
qed 
348 

41654  349 
lemma lmeasure_eq_integral: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" 
47694  350 
shows "emeasure lebesgue s = ereal (integral UNIV (indicator s))" 
41654  351 
using assms unfolding integrable_on_def 
352 
proof safe 

353 
fix y :: real assume "(indicator s has_integral y) UNIV" 

354 
from this[unfolded has_integral_iff_lmeasure] integral_unique[OF this] 

47694  355 
show "emeasure lebesgue s = ereal (integral UNIV (indicator s))" by simp 
40859  356 
qed 
38656  357 

358 
lemma lebesgue_simple_function_indicator: 

43920  359 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal" 
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360 
assumes f:"simple_function lebesgue f" 
38656  361 
shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f ` {y}) x))" 
47694  362 
by (rule, subst simple_function_indicator_representation[OF f]) auto 
38656  363 

41654  364 
lemma integral_eq_lmeasure: 
47694  365 
"(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (emeasure lebesgue s)" 
41654  366 
by (subst lmeasure_eq_integral) (auto intro!: integral_nonneg) 
38656  367 

47694  368 
lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "emeasure lebesgue s \<noteq> \<infinity>" 
41654  369 
using lmeasure_eq_integral[OF assms] by auto 
38656  370 

40859  371 
lemma negligible_iff_lebesgue_null_sets: 
47694  372 
"negligible A \<longleftrightarrow> A \<in> null_sets lebesgue" 
40859  373 
proof 
374 
assume "negligible A" 

375 
from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0] 

47694  376 
show "A \<in> null_sets lebesgue" by auto 
40859  377 
next 
47694  378 
assume A: "A \<in> null_sets lebesgue" 
379 
then have *:"((indicator A) has_integral (0::real)) UNIV" using lmeasure_finite_has_integral[of A] 

380 
by (auto simp: null_sets_def) 

41654  381 
show "negligible A" unfolding negligible_def 
382 
proof (intro allI) 

383 
fix a b :: 'a 

384 
have integrable: "(indicator A :: _\<Rightarrow>real) integrable_on {a..b}" 

385 
by (intro integrable_on_subinterval has_integral_integrable) (auto intro: *) 

386 
then have "integral {a..b} (indicator A) \<le> (integral UNIV (indicator A) :: real)" 

47694  387 
using * by (auto intro!: integral_subset_le) 
41654  388 
moreover have "(0::real) \<le> integral {a..b} (indicator A)" 
389 
using integrable by (auto intro!: integral_nonneg) 

390 
ultimately have "integral {a..b} (indicator A) = (0::real)" 

391 
using integral_unique[OF *] by auto 

392 
then show "(indicator A has_integral (0::real)) {a..b}" 

393 
using integrable_integral[OF integrable] by simp 

394 
qed 

395 
qed 

396 

47694  397 
lemma lmeasure_UNIV[intro]: "emeasure lebesgue (UNIV::'a::ordered_euclidean_space set) = \<infinity>" 
398 
proof (simp add: emeasure_lebesgue, intro SUP_PInfty bexI) 

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399 
fix n :: nat 
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400 
have "indicator UNIV = (\<lambda>x::'a. 1 :: real)" by auto 
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reworked Probability theory: measures are not type restricted to positive extended reals
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401 
moreover 
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402 
{ have "real n \<le> (2 * real n) ^ DIM('a)" 
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403 
proof (cases n) 
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404 
case 0 then show ?thesis by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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changeset

405 
next 
cdf7693bbe08
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changeset

406 
case (Suc n') 
cdf7693bbe08
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407 
have "real n \<le> (2 * real n)^1" by auto 
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changeset

408 
also have "(2 * real n)^1 \<le> (2 * real n) ^ DIM('a)" 
50526
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Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
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changeset

409 
using Suc DIM_positive[where 'a='a] 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
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changeset

410 
by (intro power_increasing) (auto simp: real_of_nat_Suc simp del: DIM_positive) 
41981
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411 
finally show ?thesis . 
cdf7693bbe08
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diff
changeset

412 
qed } 
43920  413 
ultimately show "ereal (real n) \<le> ereal (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))" 
41981
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changeset

414 
using integral_const DIM_positive[where 'a='a] 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
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50418
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changeset

415 
by (auto simp: cube_def content_closed_interval_cases setprod_constant setsum_negf) 
41981
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416 
qed simp 
40859  417 

49777  418 
lemma lmeasure_complete: "A \<subseteq> B \<Longrightarrow> B \<in> null_sets lebesgue \<Longrightarrow> A \<in> null_sets lebesgue" 
419 
unfolding negligible_iff_lebesgue_null_sets[symmetric] by (auto simp: negligible_subset) 

420 

40859  421 
lemma 
422 
fixes a b ::"'a::ordered_euclidean_space" 

47694  423 
shows lmeasure_atLeastAtMost[simp]: "emeasure lebesgue {a..b} = ereal (content {a..b})" 
41654  424 
proof  
425 
have "(indicator (UNIV \<inter> {a..b})::_\<Rightarrow>real) integrable_on UNIV" 

46905  426 
unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def [abs_def]) 
41654  427 
from lmeasure_eq_integral[OF this] show ?thesis unfolding integral_indicator_UNIV 
46905  428 
by (simp add: indicator_def [abs_def]) 
40859  429 
qed 
430 

431 
lemma lmeasure_singleton[simp]: 

47694  432 
fixes a :: "'a::ordered_euclidean_space" shows "emeasure lebesgue {a} = 0" 
41654  433 
using lmeasure_atLeastAtMost[of a a] by simp 
40859  434 

49777  435 
lemma AE_lebesgue_singleton: 
436 
fixes a :: "'a::ordered_euclidean_space" shows "AE x in lebesgue. x \<noteq> a" 

437 
by (rule AE_I[where N="{a}"]) auto 

438 

40859  439 
declare content_real[simp] 
440 

441 
lemma 

442 
fixes a b :: real 

443 
shows lmeasure_real_greaterThanAtMost[simp]: 

47694  444 
"emeasure lebesgue {a <.. b} = ereal (if a \<le> b then b  a else 0)" 
49777  445 
proof  
446 
have "emeasure lebesgue {a <.. b} = emeasure lebesgue {a .. b}" 

447 
using AE_lebesgue_singleton[of a] 

448 
by (intro emeasure_eq_AE) auto 

40859  449 
then show ?thesis by auto 
49777  450 
qed 
40859  451 

452 
lemma 

453 
fixes a b :: real 

454 
shows lmeasure_real_atLeastLessThan[simp]: 

47694  455 
"emeasure lebesgue {a ..< b} = ereal (if a \<le> b then b  a else 0)" 
49777  456 
proof  
457 
have "emeasure lebesgue {a ..< b} = emeasure lebesgue {a .. b}" 

458 
using AE_lebesgue_singleton[of b] 

459 
by (intro emeasure_eq_AE) auto 

41654  460 
then show ?thesis by auto 
49777  461 
qed 
41654  462 

463 
lemma 

464 
fixes a b :: real 

465 
shows lmeasure_real_greaterThanLessThan[simp]: 

47694  466 
"emeasure lebesgue {a <..< b} = ereal (if a \<le> b then b  a else 0)" 
49777  467 
proof  
468 
have "emeasure lebesgue {a <..< b} = emeasure lebesgue {a .. b}" 

469 
using AE_lebesgue_singleton[of a] AE_lebesgue_singleton[of b] 

470 
by (intro emeasure_eq_AE) auto 

40859  471 
then show ?thesis by auto 
49777  472 
qed 
40859  473 

41706
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hoelzl
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41704
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changeset

474 
subsection {* LebesgueBorel measure *} 
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
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475 

47694  476 
definition "lborel = measure_of UNIV (sets borel) (emeasure lebesgue)" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
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41661
diff
changeset

477 

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

478 
lemma 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
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41661
diff
changeset

479 
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
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41661
diff
changeset

480 
and sets_lborel[simp]: "sets lborel = sets borel" 
47694  481 
and measurable_lborel1[simp]: "measurable lborel = measurable borel" 
482 
and measurable_lborel2[simp]: "measurable A lborel = measurable A borel" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset

483 
using sets.sigma_sets_eq[of borel] 
47694  484 
by (auto simp add: lborel_def measurable_def[abs_def]) 
40859  485 

47694  486 
lemma emeasure_lborel[simp]: "A \<in> sets borel \<Longrightarrow> emeasure lborel A = emeasure lebesgue A" 
487 
by (rule emeasure_measure_of[OF lborel_def]) 

488 
(auto simp: positive_def emeasure_nonneg countably_additive_def intro!: suminf_emeasure) 

40859  489 

41689
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diff
changeset

490 
interpretation lborel: sigma_finite_measure lborel 
47694  491 
proof (default, intro conjI exI[of _ "\<lambda>n. cube n"]) 
492 
show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed) 

493 
{ fix x :: 'a have "\<exists>n. x\<in>cube n" using mem_big_cube by auto } 

494 
then show "(\<Union>i. cube i) = (space lborel :: 'a set)" using mem_big_cube by auto 

495 
show "\<forall>i. emeasure lborel (cube i) \<noteq> \<infinity>" by (simp add: cube_def) 

496 
qed 

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

497 

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

498 
interpretation lebesgue: sigma_finite_measure lebesgue 
40859  499 
proof 
47694  500 
from lborel.sigma_finite guess A :: "nat \<Rightarrow> 'a set" .. 
501 
then show "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. emeasure lebesgue (A i) \<noteq> \<infinity>)" 

502 
by (intro exI[of _ A]) (auto simp: subset_eq) 

40859  503 
qed 
504 

49777  505 
lemma Int_stable_atLeastAtMost: 
506 
fixes x::"'a::ordered_euclidean_space" 

507 
shows "Int_stable (range (\<lambda>(a, b::'a). {a..b}))" 

508 
by (auto simp: inter_interval Int_stable_def) 

509 

510 
lemma lborel_eqI: 

511 
fixes M :: "'a::ordered_euclidean_space measure" 

512 
assumes emeasure_eq: "\<And>a b. emeasure M {a .. b} = content {a .. b}" 

513 
assumes sets_eq: "sets M = sets borel" 

514 
shows "lborel = M" 

515 
proof (rule measure_eqI_generator_eq[OF Int_stable_atLeastAtMost]) 

516 
let ?P = "\<Pi>\<^isub>M i\<in>{..<DIM('a::ordered_euclidean_space)}. lborel" 

517 
let ?E = "range (\<lambda>(a, b). {a..b} :: 'a set)" 

518 
show "?E \<subseteq> Pow UNIV" "sets lborel = sigma_sets UNIV ?E" "sets M = sigma_sets UNIV ?E" 

519 
by (simp_all add: borel_eq_atLeastAtMost sets_eq) 

520 

521 
show "range cube \<subseteq> ?E" unfolding cube_def [abs_def] by auto 

522 
{ fix x :: 'a have "\<exists>n. x \<in> cube n" using mem_big_cube[of x] by fastforce } 

523 
then show "(\<Union>i. cube i :: 'a set) = UNIV" by auto 

524 

525 
{ fix i show "emeasure lborel (cube i) \<noteq> \<infinity>" unfolding cube_def by auto } 

526 
{ fix X assume "X \<in> ?E" then show "emeasure lborel X = emeasure M X" 

527 
by (auto simp: emeasure_eq) } 

528 
qed 

529 

530 
lemma lebesgue_real_affine: 

531 
fixes c :: real assumes "c \<noteq> 0" 

532 
shows "lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. \<bar>c\<bar>)" (is "_ = ?D") 

533 
proof (rule lborel_eqI) 

534 
fix a b show "emeasure ?D {a..b} = content {a .. b}" 

535 
proof cases 

536 
assume "0 < c" 

537 
then have "(\<lambda>x. t + c * x) ` {a..b} = {(a  t) / c .. (b  t) / c}" 

538 
by (auto simp: field_simps) 

539 
with `0 < c` show ?thesis 

540 
by (cases "a \<le> b") 

541 
(auto simp: field_simps emeasure_density positive_integral_distr positive_integral_cmult 

542 
borel_measurable_indicator' emeasure_distr) 

543 
next 

544 
assume "\<not> 0 < c" with `c \<noteq> 0` have "c < 0" by auto 

545 
then have *: "(\<lambda>x. t + c * x) ` {a..b} = {(b  t) / c .. (a  t) / c}" 

546 
by (auto simp: field_simps) 

547 
with `c < 0` show ?thesis 

548 
by (cases "a \<le> b") 

549 
(auto simp: field_simps emeasure_density positive_integral_distr 

550 
positive_integral_cmult borel_measurable_indicator' emeasure_distr) 

551 
qed 

552 
qed simp 

553 

554 
lemma lebesgue_integral_real_affine: 

555 
fixes c :: real assumes c: "c \<noteq> 0" and f: "f \<in> borel_measurable borel" 

556 
shows "(\<integral> x. f x \<partial> lborel) = \<bar>c\<bar> * (\<integral> x. f (t + c * x) \<partial>lborel)" 

557 
by (subst lebesgue_real_affine[OF c, of t]) 

558 
(simp add: f integral_density integral_distr lebesgue_integral_cmult) 

559 

41706
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hoelzl
parents:
41704
diff
changeset

560 
subsection {* Lebesgue integrable implies Gauge integrable *} 
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset

561 

40859  562 
lemma simple_function_has_integral: 
43920  563 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal" 
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

564 
assumes f:"simple_function lebesgue f" 
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

565 
and f':"range f \<subseteq> {0..<\<infinity>}" 
47694  566 
and om:"\<And>x. x \<in> range f \<Longrightarrow> emeasure lebesgue (f ` {x} \<inter> UNIV) = \<infinity> \<Longrightarrow> x = 0" 
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

567 
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

568 
unfolding simple_integral_def space_lebesgue 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

569 
proof (subst lebesgue_simple_function_indicator) 
47694  570 
let ?M = "\<lambda>x. emeasure lebesgue (f ` {x} \<inter> UNIV)" 
46731  571 
let ?F = "\<lambda>x. indicator (f ` {x})" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

572 
{ fix x y assume "y \<in> range f" 
43920  573 
from subsetD[OF f' this] have "y * ?F y x = ereal (real y * ?F y x)" 
574 
by (cases rule: ereal2_cases[of y "?F y x"]) 

575 
(auto simp: indicator_def one_ereal_def split: split_if_asm) } 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

576 
moreover 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

577 
{ fix x assume x: "x\<in>range f" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

578 
have "x * ?M x = real x * real (?M x)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

579 
proof cases 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

580 
assume "x \<noteq> 0" with om[OF x] have "?M x \<noteq> \<infinity>" by auto 
47694  581 
with subsetD[OF f' x] f[THEN simple_functionD(2)] show ?thesis 
43920  582 
by (cases rule: ereal2_cases[of x "?M x"]) auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

583 
qed simp } 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

584 
ultimately 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

585 
have "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV \<longleftrightarrow> 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

586 
((\<lambda>x. \<Sum>y\<in>range f. real y * ?F y x) has_integral (\<Sum>x\<in>range f. real x * real (?M x))) UNIV" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

587 
by simp 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

588 
also have \<dots> 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

589 
proof (intro has_integral_setsum has_integral_cmult_real lmeasure_finite_has_integral 
47694  590 
real_of_ereal_pos emeasure_nonneg ballI) 
591 
show *: "finite (range f)" "\<And>y. f ` {y} \<in> sets lebesgue" 

592 
using simple_functionD[OF f] by auto 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

593 
fix y assume "real y \<noteq> 0" "y \<in> range f" 
47694  594 
with * om[OF this(2)] show "emeasure lebesgue (f ` {y}) = ereal (real (?M y))" 
43920  595 
by (auto simp: ereal_real) 
41654  596 
qed 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

597 
finally show "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV" . 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

598 
qed fact 
40859  599 

600 
lemma simple_function_has_integral': 

43920  601 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

602 
assumes f: "simple_function lebesgue f" "\<And>x. 0 \<le> f x" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

603 
and i: "integral\<^isup>S lebesgue f \<noteq> \<infinity>" 
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

604 
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

605 
proof  
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

606 
let ?f = "\<lambda>x. if x \<in> f ` {\<infinity>} then 0 else f x" 
47694  607 
note f(1)[THEN simple_functionD(2)] 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

608 
then have [simp, intro]: "\<And>X. f ` X \<in> sets lebesgue" by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

609 
have f': "simple_function lebesgue ?f" 
47694  610 
using f by (intro simple_function_If_set) auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

611 
have rng: "range ?f \<subseteq> {0..<\<infinity>}" using f by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

612 
have "AE x in lebesgue. f x = ?f x" 
47694  613 
using simple_integral_PInf[OF f i] 
614 
by (intro AE_I[where N="f ` {\<infinity>} \<inter> space lebesgue"]) auto 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

615 
from f(1) f' this have eq: "integral\<^isup>S lebesgue f = integral\<^isup>S lebesgue ?f" 
47694  616 
by (rule simple_integral_cong_AE) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

617 
have real_eq: "\<And>x. real (f x) = real (?f x)" by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

618 

cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

619 
show ?thesis 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

620 
unfolding eq real_eq 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

621 
proof (rule simple_function_has_integral[OF f' rng]) 
47694  622 
fix x assume x: "x \<in> range ?f" and inf: "emeasure lebesgue (?f ` {x} \<inter> UNIV) = \<infinity>" 
623 
have "x * emeasure lebesgue (?f ` {x} \<inter> UNIV) = (\<integral>\<^isup>S y. x * indicator (?f ` {x}) y \<partial>lebesgue)" 

624 
using f'[THEN simple_functionD(2)] 

625 
by (simp add: simple_integral_cmult_indicator) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

626 
also have "\<dots> \<le> integral\<^isup>S lebesgue f" 
47694  627 
using f'[THEN simple_functionD(2)] f 
628 
by (intro simple_integral_mono simple_function_mult simple_function_indicator) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

629 
(auto split: split_indicator) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

630 
finally show "x = 0" unfolding inf using i subsetD[OF rng x] by (auto split: split_if_asm) 
40859  631 
qed 
632 
qed 

633 

634 
lemma positive_integral_has_integral: 

43920  635 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> ereal" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

636 
assumes f: "f \<in> borel_measurable lebesgue" "range f \<subseteq> {0..<\<infinity>}" "integral\<^isup>P lebesgue f \<noteq> \<infinity>" 
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

637 
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>P lebesgue f))) UNIV" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

638 
proof  
47694  639 
from borel_measurable_implies_simple_function_sequence'[OF f(1)] 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

640 
guess u . note u = this 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

641 
have SUP_eq: "\<And>x. (SUP i. u i x) = f x" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

642 
using u(4) f(2)[THEN subsetD] by (auto split: split_max) 
46731  643 
let ?u = "\<lambda>i x. real (u i x)" 
47694  644 
note u_eq = positive_integral_eq_simple_integral[OF u(1,5), symmetric] 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

645 
{ fix i 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

646 
note u_eq 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

647 
also have "integral\<^isup>P lebesgue (u i) \<le> (\<integral>\<^isup>+x. max 0 (f x) \<partial>lebesgue)" 
47694  648 
by (intro positive_integral_mono) (auto intro: SUP_upper simp: u(4)[symmetric]) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

649 
finally have "integral\<^isup>S lebesgue (u i) \<noteq> \<infinity>" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

650 
unfolding positive_integral_max_0 using f by auto } 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

651 
note u_fin = this 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

652 
then have u_int: "\<And>i. (?u i has_integral real (integral\<^isup>S lebesgue (u i))) UNIV" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

653 
by (rule simple_function_has_integral'[OF u(1,5)]) 
43920  654 
have "\<forall>x. \<exists>r\<ge>0. f x = ereal r" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

655 
proof 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

656 
fix x from f(2) have "0 \<le> f x" "f x \<noteq> \<infinity>" by (auto simp: subset_eq) 
43920  657 
then show "\<exists>r\<ge>0. f x = ereal r" by (cases "f x") auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

658 
qed 
43920  659 
from choice[OF this] obtain f' where f': "f = (\<lambda>x. ereal (f' x))" "\<And>x. 0 \<le> f' x" by auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

660 

43920  661 
have "\<forall>i. \<exists>r. \<forall>x. 0 \<le> r x \<and> u i x = ereal (r x)" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

662 
proof 
43920  663 
fix i show "\<exists>r. \<forall>x. 0 \<le> r x \<and> u i x = ereal (r x)" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

664 
proof (intro choice allI) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

665 
fix i x have "u i x \<noteq> \<infinity>" using u(3)[of i] by (auto simp: image_iff) metis 
43920  666 
then show "\<exists>r\<ge>0. u i x = ereal r" using u(5)[of i x] by (cases "u i x") auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

667 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

668 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

669 
from choice[OF this] obtain u' where 
43920  670 
u': "u = (\<lambda>i x. ereal (u' i x))" "\<And>i x. 0 \<le> u' i x" by (auto simp: fun_eq_iff) 
40859  671 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

672 
have convergent: "f' integrable_on UNIV \<and> 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

673 
(\<lambda>k. integral UNIV (u' k)) > integral UNIV f'" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

674 
proof (intro monotone_convergence_increasing allI ballI) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

675 
show int: "\<And>k. (u' k) integrable_on UNIV" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

676 
using u_int unfolding integrable_on_def u' by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

677 
show "\<And>k x. u' k x \<le> u' (Suc k) x" using u(2,3,5) 
43920  678 
by (auto simp: incseq_Suc_iff le_fun_def image_iff u' intro!: real_of_ereal_positive_mono) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

679 
show "\<And>x. (\<lambda>k. u' k x) > f' x" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

680 
using SUP_eq u(2) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

681 
by (intro SUP_eq_LIMSEQ[THEN iffD1]) (auto simp: u' f' incseq_mono incseq_Suc_iff le_fun_def) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

682 
show "bounded {integral UNIV (u' k)k. True}" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

683 
proof (safe intro!: bounded_realI) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

684 
fix k 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

685 
have "\<bar>integral UNIV (u' k)\<bar> = integral UNIV (u' k)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

686 
by (intro abs_of_nonneg integral_nonneg int ballI u') 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

687 
also have "\<dots> = real (integral\<^isup>S lebesgue (u k))" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

688 
using u_int[THEN integral_unique] by (simp add: u') 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

689 
also have "\<dots> = real (integral\<^isup>P lebesgue (u k))" 
47694  690 
using positive_integral_eq_simple_integral[OF u(1,5)] by simp 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

691 
also have "\<dots> \<le> real (integral\<^isup>P lebesgue f)" using f 
47694  692 
by (auto intro!: real_of_ereal_positive_mono positive_integral_positive 
693 
positive_integral_mono SUP_upper simp: SUP_eq[symmetric]) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

694 
finally show "\<bar>integral UNIV (u' k)\<bar> \<le> real (integral\<^isup>P lebesgue f)" . 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

695 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

696 
qed 
40859  697 

43920  698 
have "integral\<^isup>P lebesgue f = ereal (integral UNIV f')" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

699 
proof (rule tendsto_unique[OF trivial_limit_sequentially]) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

700 
have "(\<lambda>i. integral\<^isup>S lebesgue (u i)) > (SUP i. integral\<^isup>P lebesgue (u i))" 
47694  701 
unfolding u_eq by (intro LIMSEQ_ereal_SUPR incseq_positive_integral u) 
702 
also note positive_integral_monotone_convergence_SUP 

703 
[OF u(2) borel_measurable_simple_function[OF u(1)] u(5), symmetric] 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

704 
finally show "(\<lambda>k. integral\<^isup>S lebesgue (u k)) > integral\<^isup>P lebesgue f" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

705 
unfolding SUP_eq . 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

706 

cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

707 
{ fix k 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

708 
have "0 \<le> integral\<^isup>S lebesgue (u k)" 
47694  709 
using u by (auto intro!: simple_integral_positive) 
43920  710 
then have "integral\<^isup>S lebesgue (u k) = ereal (real (integral\<^isup>S lebesgue (u k)))" 
711 
using u_fin by (auto simp: ereal_real) } 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

712 
note * = this 
43920  713 
show "(\<lambda>k. integral\<^isup>S lebesgue (u k)) > ereal (integral UNIV f')" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

714 
using convergent using u_int[THEN integral_unique, symmetric] 
47694  715 
by (subst *) (simp add: u') 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

716 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

717 
then show ?thesis using convergent by (simp add: f' integrable_integral) 
40859  718 
qed 
719 

720 
lemma lebesgue_integral_has_integral: 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

721 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

722 
assumes f: "integrable lebesgue f" 
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

723 
shows "(f has_integral (integral\<^isup>L lebesgue f)) UNIV" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

724 
proof  
43920  725 
let ?n = "\<lambda>x. real (ereal (max 0 ( f x)))" and ?p = "\<lambda>x. real (ereal (max 0 (f x)))" 
726 
have *: "f = (\<lambda>x. ?p x  ?n x)" by (auto simp del: ereal_max) 

47694  727 
{ fix f :: "'a \<Rightarrow> real" have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue) = (\<integral>\<^isup>+ x. ereal (max 0 (f x)) \<partial>lebesgue)" 
728 
by (intro positive_integral_cong_pos) (auto split: split_max) } 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

729 
note eq = this 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

730 
show ?thesis 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

731 
unfolding lebesgue_integral_def 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

732 
apply (subst *) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

733 
apply (rule has_integral_sub) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

734 
unfolding eq[of f] eq[of "\<lambda>x.  f x"] 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

735 
apply (safe intro!: positive_integral_has_integral) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

736 
using integrableD[OF f] 
43920  737 
by (auto simp: zero_ereal_def[symmetric] positive_integral_max_0 split: split_max 
47694  738 
intro!: measurable_If) 
40859  739 
qed 
740 

47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

741 
lemma lebesgue_simple_integral_eq_borel: 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

742 
assumes f: "f \<in> borel_measurable borel" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

743 
shows "integral\<^isup>S lebesgue f = integral\<^isup>S lborel f" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

744 
using f[THEN measurable_sets] 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

745 
by (auto intro!: setsum_cong arg_cong2[where f="op *"] emeasure_lborel[symmetric] 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

746 
simp: simple_integral_def) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

747 

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

748 
lemma lebesgue_positive_integral_eq_borel: 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

749 
assumes f: "f \<in> borel_measurable borel" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

750 
shows "integral\<^isup>P lebesgue f = integral\<^isup>P lborel f" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

751 
proof  
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

752 
from f have "integral\<^isup>P lebesgue (\<lambda>x. max 0 (f x)) = integral\<^isup>P lborel (\<lambda>x. max 0 (f x))" 
47694  753 
by (auto intro!: positive_integral_subalgebra[symmetric]) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

754 
then show ?thesis unfolding positive_integral_max_0 . 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

755 
qed 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

756 

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

757 
lemma lebesgue_integral_eq_borel: 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

758 
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

759 
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

760 
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

761 
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

762 
have "sets lborel \<subseteq> sets lebesgue" by auto 
47694  763 
from 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

764 
show ?P ?I by auto 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

765 
qed 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

766 

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

767 
lemma borel_integral_has_integral: 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

768 
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

769 
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

770 
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

771 
proof  
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

772 
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

773 
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

774 
from f show ?thesis 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

775 
using lebesgue_integral_has_integral[of f] 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

776 
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

777 
qed 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

778 

49777  779 
lemma positive_integral_lebesgue_has_integral: 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

780 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 
49777  781 
assumes f_borel: "f \<in> borel_measurable lebesgue" and nonneg: "\<And>x. 0 \<le> f x" 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

782 
assumes I: "(f has_integral I) UNIV" 
49777  783 
shows "(\<integral>\<^isup>+x. f x \<partial>lebesgue) = I" 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

784 
proof  
49777  785 
from f_borel have "(\<lambda>x. ereal (f x)) \<in> borel_measurable lebesgue" by auto 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

786 
from borel_measurable_implies_simple_function_sequence'[OF this] guess F . note F = this 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

787 

49777  788 
have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue) = (SUP i. integral\<^isup>S lebesgue (F i))" 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

789 
using F 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

790 
by (subst positive_integral_monotone_convergence_simple) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

791 
(simp_all add: positive_integral_max_0 simple_function_def) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

792 
also have "\<dots> \<le> ereal I" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

793 
proof (rule SUP_least) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

794 
fix i :: nat 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

795 

5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

796 
{ fix z 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

797 
from F(4)[of z] have "F i z \<le> ereal (f z)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

798 
by (metis SUP_upper UNIV_I ereal_max_0 min_max.sup_absorb2 nonneg) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

799 
with F(5)[of i z] have "real (F i z) \<le> f z" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

800 
by (cases "F i z") simp_all } 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

801 
note F_bound = this 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

802 

5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

803 
{ fix x :: ereal assume x: "x \<noteq> 0" "x \<in> range (F i)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

804 
with F(3,5)[of i] have [simp]: "real x \<noteq> 0" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

805 
by (metis image_iff order_eq_iff real_of_ereal_le_0) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

806 
let ?s = "(\<lambda>n z. real x * indicator (F i ` {x} \<inter> cube n) z) :: nat \<Rightarrow> 'a \<Rightarrow> real" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

807 
have "(\<lambda>z::'a. real x * indicator (F i ` {x}) z) integrable_on UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

808 
proof (rule dominated_convergence(1)) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

809 
fix n :: nat 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

810 
have "(\<lambda>z. indicator (F i ` {x} \<inter> cube n) z :: real) integrable_on cube n" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

811 
using x F(1)[of i] 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

812 
by (intro lebesgueD) (auto simp: simple_function_def) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

813 
then have cube: "?s n integrable_on cube n" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

814 
by (simp add: integrable_on_cmult_iff) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

815 
show "?s n integrable_on UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

816 
by (rule integrable_on_superset[OF _ _ cube]) auto 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

817 
next 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

818 
show "f integrable_on UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

819 
unfolding integrable_on_def using I by auto 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

820 
next 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

821 
fix n from F_bound show "\<forall>x\<in>UNIV. norm (?s n x) \<le> f x" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

822 
using nonneg F(5) by (auto split: split_indicator) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

823 
next 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

824 
show "\<forall>z\<in>UNIV. (\<lambda>n. ?s n z) > real x * indicator (F i ` {x}) z" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

825 
proof 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

826 
fix z :: 'a 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

827 
from mem_big_cube[of z] guess j . 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

828 
then have x: "eventually (\<lambda>n. ?s n z = real x * indicator (F i ` {x}) z) sequentially" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

829 
by (auto intro!: eventually_sequentiallyI[where c=j] dest!: cube_subset split: split_indicator) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

830 
then show "(\<lambda>n. ?s n z) > real x * indicator (F i ` {x}) z" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

831 
by (rule Lim_eventually) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

832 
qed 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

833 
qed 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

834 
then have "(indicator (F i ` {x}) :: 'a \<Rightarrow> real) integrable_on UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

835 
by (simp add: integrable_on_cmult_iff) } 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

836 
note F_finite = lmeasure_finite[OF this] 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

837 

5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

838 
have "((\<lambda>x. real (F i x)) has_integral real (integral\<^isup>S lebesgue (F i))) UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

839 
proof (rule simple_function_has_integral[of "F i"]) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

840 
show "simple_function lebesgue (F i)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

841 
using F(1) by (simp add: simple_function_def) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

842 
show "range (F i) \<subseteq> {0..<\<infinity>}" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

843 
using F(3,5)[of i] by (auto simp: image_iff) metis 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

844 
next 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

845 
fix x assume "x \<in> range (F i)" "emeasure lebesgue (F i ` {x} \<inter> UNIV) = \<infinity>" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

846 
with F_finite[of x] show "x = 0" by auto 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

847 
qed 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

848 
from this I have "real (integral\<^isup>S lebesgue (F i)) \<le> I" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

849 
by (rule has_integral_le) (intro ballI F_bound) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

850 
moreover 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

851 
{ fix x assume x: "x \<in> range (F i)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

852 
with F(3,5)[of i] have "x = 0 \<or> (0 < x \<and> x \<noteq> \<infinity>)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

853 
by (auto simp: image_iff le_less) metis 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

854 
with F_finite[OF _ x] x have "x * emeasure lebesgue (F i ` {x} \<inter> UNIV) \<noteq> \<infinity>" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

855 
by auto } 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

856 
then have "integral\<^isup>S lebesgue (F i) \<noteq> \<infinity>" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

857 
unfolding simple_integral_def setsum_Pinfty space_lebesgue by blast 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

858 
moreover have "0 \<le> integral\<^isup>S lebesgue (F i)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

859 
using F(1,5) by (intro simple_integral_positive) (auto simp: simple_function_def) 
49777  860 
ultimately show "integral\<^isup>S lebesgue (F i) \<le> ereal I" 
861 
by (cases "integral\<^isup>S lebesgue (F i)") auto 

47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

862 
qed 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

863 
also have "\<dots> < \<infinity>" by simp 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

864 
finally have finite: "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue) \<noteq> \<infinity>" by simp 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

865 
have borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable lebesgue" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

866 
using f_borel by (auto intro: borel_measurable_lebesgueI) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

867 
from positive_integral_has_integral[OF borel _ finite] 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

868 
have "(f has_integral real (\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue)) UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

869 
using nonneg by (simp add: subset_eq) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

870 
with I have "I = real (\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

871 
by (rule has_integral_unique) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

872 
with finite positive_integral_positive[of _ "\<lambda>x. ereal (f x)"] show ?thesis 
49777  873 
by (cases "\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue") auto 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

874 
qed 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

875 

49777  876 
lemma has_integral_iff_positive_integral_lebesgue: 
877 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 

878 
assumes f: "f \<in> borel_measurable lebesgue" "\<And>x. 0 \<le> f x" 

879 
shows "(f has_integral I) UNIV \<longleftrightarrow> integral\<^isup>P lebesgue f = I" 

880 
using f positive_integral_lebesgue_has_integral[of f I] positive_integral_has_integral[of f] 

881 
by (auto simp: subset_eq) 

882 

883 
lemma has_integral_iff_positive_integral_lborel: 

47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

884 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

885 
assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

886 
shows "(f has_integral I) UNIV \<longleftrightarrow> integral\<^isup>P lborel f = I" 
49777  887 
using assms 
888 
by (subst has_integral_iff_positive_integral_lebesgue) 

889 
(auto simp: borel_measurable_lebesgueI lebesgue_positive_integral_eq_borel) 

890 

891 
subsection {* Equivalence between product spaces and euclidean spaces *} 

892 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

893 
definition e2p :: "'a::ordered_euclidean_space \<Rightarrow> ('a \<Rightarrow> real)" where 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

894 
"e2p x = (\<lambda>i\<in>Basis. x \<bullet> i)" 
49777  895 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

896 
definition p2e :: "('a \<Rightarrow> real) \<Rightarrow> 'a::ordered_euclidean_space" where 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

897 
"p2e x = (\<Sum>i\<in>Basis. x i *\<^sub>R i)" 
49777  898 

899 
lemma e2p_p2e[simp]: 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

900 
"x \<in> extensional Basis \<Longrightarrow> e2p (p2e x::'a::ordered_euclidean_space) = x" 
49777  901 
by (auto simp: fun_eq_iff extensional_def p2e_def e2p_def) 
902 

903 
lemma p2e_e2p[simp]: 

904 
"p2e (e2p x) = (x::'a::ordered_euclidean_space)" 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

905 
by (auto simp: euclidean_eq_iff[where 'a='a] p2e_def e2p_def) 
49777  906 

907 
interpretation lborel_product: product_sigma_finite "\<lambda>x. lborel::real measure" 

908 
by default 

909 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

910 
interpretation lborel_space: finite_product_sigma_finite "\<lambda>x. lborel::real measure" "Basis" 
49777  911 
by default auto 
912 

913 
lemma sets_product_borel: 

914 
assumes I: "finite I" 

915 
shows "sets (\<Pi>\<^isub>M i\<in>I. lborel) = sigma_sets (\<Pi>\<^isub>E i\<in>I. UNIV) { \<Pi>\<^isub>E i\<in>I. {..< x i :: real}  x. True}" (is "_ = ?G") 

916 
proof (subst sigma_prod_algebra_sigma_eq[where S="\<lambda>_ i::nat. {..<real i}" and E="\<lambda>_. range lessThan", OF I]) 

917 
show "sigma_sets (space (Pi\<^isub>M I (\<lambda>i. lborel))) {Pi\<^isub>E I F F. \<forall>i\<in>I. F i \<in> range lessThan} = ?G" 

918 
by (intro arg_cong2[where f=sigma_sets]) (auto simp: space_PiM image_iff bchoice_iff) 

49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49777
diff
changeset

919 
qed (auto simp: borel_eq_lessThan reals_Archimedean2) 
49777  920 

50003  921 
lemma measurable_e2p[measurable]: 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

922 
"e2p \<in> measurable (borel::'a::ordered_euclidean_space measure) (\<Pi>\<^isub>M (i::'a)\<in>Basis. (lborel :: real measure))" 
49777  923 
proof (rule measurable_sigma_sets[OF sets_product_borel]) 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

924 
fix A :: "('a \<Rightarrow> real) set" assume "A \<in> {\<Pi>\<^isub>E (i::'a)\<in>Basis. {..<x i} x. True} " 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

925 
then obtain x where "A = (\<Pi>\<^isub>E (i::'a)\<in>Basis. {..<x i})" by auto 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

926 
then have "e2p ` A = {..< (\<Sum>i\<in>Basis. x i *\<^sub>R i) :: 'a}" 
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50105
diff
changeset

927 
using DIM_positive by (auto simp add: set_eq_iff e2p_def 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

928 
euclidean_eq_iff[where 'a='a] eucl_less[where 'a='a]) 
49777  929 
then show "e2p ` A \<inter> space (borel::'a measure) \<in> sets borel" by simp 
930 
qed (auto simp: e2p_def) 

931 

50003  932 
(* FIXME: conversion in measurable prover *) 
50385  933 
lemma lborelD_Collect[measurable (raw)]: "{x\<in>space borel. P x} \<in> sets borel \<Longrightarrow> {x\<in>space lborel. P x} \<in> sets lborel" by simp 
934 
lemma lborelD[measurable (raw)]: "A \<in> sets borel \<Longrightarrow> A \<in> sets lborel" by simp 

50003  935 

936 
lemma measurable_p2e[measurable]: 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

937 
"p2e \<in> measurable (\<Pi>\<^isub>M (i::'a)\<in>Basis. (lborel :: real measure)) 
49777  938 
(borel :: 'a::ordered_euclidean_space measure)" 
939 
(is "p2e \<in> measurable ?P _") 

940 
proof (safe intro!: borel_measurable_iff_halfspace_le[THEN iffD2]) 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

941 
fix x and i :: 'a 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

942 
let ?A = "{w \<in> space ?P. (p2e w :: 'a) \<bullet> i \<le> x}" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

943 
assume "i \<in> Basis" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

944 
then have "?A = (\<Pi>\<^isub>E j\<in>Basis. if i = j then {.. x} else UNIV)" 
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50105
diff
changeset

945 
using DIM_positive by (auto simp: space_PiM p2e_def PiE_def split: split_if_asm) 
49777  946 
then show "?A \<in> sets ?P" 
947 
by auto 

948 
qed 

949 

950 
lemma lborel_eq_lborel_space: 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

951 
"(lborel :: 'a measure) = distr (\<Pi>\<^isub>M (i::'a::ordered_euclidean_space)\<in>Basis. lborel) borel p2e" 
49777  952 
(is "?B = ?D") 
953 
proof (rule lborel_eqI) 

954 
show "sets ?D = sets borel" by simp 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

955 
let ?P = "(\<Pi>\<^isub>M (i::'a)\<in>Basis. lborel)" 
49777  956 
fix a b :: 'a 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

957 
have *: "p2e ` {a .. b} \<inter> space ?P = (\<Pi>\<^isub>E i\<in>Basis. {a \<bullet> i .. b \<bullet> i})" 
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50105
diff
changeset

958 
by (auto simp: eucl_le[where 'a='a] p2e_def space_PiM PiE_def Pi_iff) 
49777  959 
have "emeasure ?P (p2e ` {a..b} \<inter> space ?P) = content {a..b}" 
960 
proof cases 

961 
assume "{a..b} \<noteq> {}" 

962 
then have "a \<le> b" 

963 
by (simp add: interval_ne_empty eucl_le[where 'a='a]) 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

964 
then have "emeasure lborel {a..b} = (\<Prod>x\<in>Basis. emeasure lborel {a \<bullet> x .. b \<bullet> x})" 
49777  965 
by (auto simp: content_closed_interval eucl_le[where 'a='a] 
966 
intro!: setprod_ereal[symmetric]) 

967 
also have "\<dots> = emeasure ?P (p2e ` {a..b} \<inter> space ?P)" 

968 
unfolding * by (subst lborel_space.measure_times) auto 

969 
finally show ?thesis by simp 

970 
qed simp 

971 
then show "emeasure ?D {a .. b} = content {a .. b}" 

972 
by (simp add: emeasure_distr measurable_p2e) 

973 
qed 

974 

975 
lemma borel_fubini_positiv_integral: 

976 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> ereal" 

977 
assumes f: "f \<in> borel_measurable borel" 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

978 
shows "integral\<^isup>P lborel f = \<integral>\<^isup>+x. f (p2e x) \<partial>(\<Pi>\<^isub>M (i::'a)\<in>Basis. lborel)" 
49777  979 
by (subst lborel_eq_lborel_space) (simp add: positive_integral_distr measurable_p2e f) 
980 

981 
lemma borel_fubini_integrable: 

982 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

983 
shows "integrable lborel f \<longleftrightarrow> integrable (\<Pi>\<^isub>M (i::'a)\<in>Basis. lborel) (\<lambda>x. f (p2e x))" 
49777  984 
(is "_ \<longleftrightarrow> integrable ?B ?f") 
985 
proof 

986 
assume "integrable lborel f" 

987 
moreover then have f: "f \<in> borel_measurable borel" 

988 
by auto 

989 
moreover with measurable_p2e 

990 
have "f \<circ> p2e \<in> borel_measurable ?B" 

991 
by (rule measurable_comp) 

992 
ultimately show "integrable ?B ?f" 

993 
by (simp add: comp_def borel_fubini_positiv_integral integrable_def) 

994 
next 

995 
assume "integrable ?B ?f" 

996 
moreover 

997 
then have "?f \<circ> e2p \<in> borel_measurable (borel::'a measure)" 

998 
by (auto intro!: measurable_e2p) 

999 
then have "f \<in> borel_measurable borel" 

1000 
by (simp cong: measurable_cong) 

1001 
ultimately show "integrable lborel f" 

1002 
by (simp add: borel_fubini_positiv_integral integrable_def) 

1003 
qed 

1004 

1005 
lemma borel_fubini: 

1006 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 

1007 
assumes f: "f \<in> borel_measurable borel" 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

1008 
shows "integral\<^isup>L lborel f = \<integral>x. f (p2e x) \<partial>((\<Pi>\<^isub>M (i::'a)\<in>Basis. lborel))" 
49777  1009 
using f by (simp add: borel_fubini_positiv_integral lebesgue_integral_def) 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1010 

50418
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1011 
lemma integrable_on_borel_integrable: 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1012 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1013 
assumes f_borel: "f \<in> borel_measurable borel" and nonneg: "\<And>x. 0 \<le> f x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1014 
assumes f: "f integrable_on UNIV" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1015 
shows "integrable lborel f" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1016 
proof  
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1017 
have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lborel) \<noteq> \<infinity>" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1018 
using has_integral_iff_positive_integral_lborel[OF f_borel nonneg] f 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1019 
by (auto simp: integrable_on_def) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1020 
moreover have "(\<integral>\<^isup>+ x. ereal ( f x) \<partial>lborel) = 0" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1021 
using f_borel nonneg by (subst positive_integral_0_iff_AE) auto 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1022 
ultimately show ?thesis 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1023 
using f_borel by (auto simp: integrable_def) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1024 
qed 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1025 

bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1026 
subsection {* Fundamental Theorem of Calculus for the Lebesgue integral *} 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1027 

bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1028 
lemma borel_integrable_atLeastAtMost: 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1029 
fixes a b :: real 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1030 
assumes f: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1031 
shows "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)" (is "integrable _ ?f") 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1032 
proof cases 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1033 
assume "a \<le> b" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1034 

bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1035 
from isCont_Lb_Ub[OF `a \<le> b`, of f] f 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1036 
obtain M L where 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1037 
bounds: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> f x \<le> M" "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> L \<le> f x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1038 
by metis 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1039 

bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1040 
show ?thesis 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1041 
proof (rule integrable_bound) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1042 
show "integrable lborel (\<lambda>x. max \<bar>M\<bar> \<bar>L\<bar> * indicator {a..b} x)" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1043 
by (rule integral_cmul_indicator) simp_all 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1044 
show "AE x in lborel. \<bar>?f x\<bar> \<le> max \<bar>M\<bar> \<bar>L\<bar> * indicator {a..b} x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1045 
proof (rule AE_I2) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1046 
fix x show "\<bar>?f x\<bar> \<le> max \<bar>M\<bar> \<bar>L\<bar> * indicator {a..b} x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1047 
using bounds[of x] by (auto split: split_indicator) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1048 
qed 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1049 

bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1050 
let ?g = "\<lambda>x. if x = a then f a else if x = b then f b else if x \<in> {a <..< b} then f x else 0" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1051 
from f have "continuous_on {a <..< b} f" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1052 
by (subst continuous_on_eq_continuous_at) (auto simp add: continuous_isCont) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1053 
then have "?g \<in> borel_measurable borel" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1054 
using borel_measurable_continuous_on_open[of "{a <..< b }" f "\<lambda>x. x" borel 0] 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1055 
by (auto intro!: measurable_If[where P="\<lambda>x. x = a"] measurable_If[where P="\<lambda>x. x = b"]) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1056 
also have "?g = ?f" 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50418
diff
changeset

1057 
using `a \<le> b` by (intro ext) (auto split: split_indicator) 
50418
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1058 
finally show "?f \<in> borel_measurable lborel" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1059 
by simp 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1060 
qed 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1061 
qed simp 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1062 

bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1063 
lemma integral_FTC_atLeastAtMost: 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1064 
fixes a b :: real 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1065 
assumes "a \<le> b" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1066 
and F: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> DERIV F x :> f x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1067 
and f: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1068 
shows "integral\<^isup>L lborel (\<lambda>x. f x * indicator {a .. b} x) = F b  F a" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1069 
proof  
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1070 
let ?f = "\<lambda>x. f x * indicator {a .. b} x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1071 
have "(?f has_integral (\<integral>x. ?f x \<partial>lborel)) UNIV" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1072 
using borel_integrable_atLeastAtMost[OF f] 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1073 
by (rule borel_integral_has_integral) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1074 
moreover 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1075 
have "(f has_integral F b  F a) {a .. b}" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1076 
by (intro fundamental_theorem_of_calculus has_vector_derivative_withinI_DERIV ballI assms) auto 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1077 
then have "(?f has_integral F b  F a) {a .. b}" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1078 
by (subst has_integral_eq_eq[where g=f]) auto 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1079 
then have "(?f has_integral F b  F a) UNIV" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1080 
by (intro has_integral_on_superset[where t=UNIV and s="{a..b}"]) auto 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1081 
ultimately show "integral\<^isup>L lborel ?f = F b  F a" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1082 
by (rule has_integral_unique) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1083 
qed 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1084 

bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1085 
text {* 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1086 

bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1087 
For the positive integral we replace continuity with Borelmeasurability. 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1088 

bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1089 
*} 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1090 

bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1091 
lemma positive_integral_FTC_atLeastAtMost: 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1092 
assumes f_borel: "f \<in> borel_measurable borel" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1093 
assumes f: "\<And>x. x \<in> {a..b} \<Longrightarrow> DERIV F x :> f x" "\<And>x. x \<in> {a..b} \<Longrightarrow> 0 \<le> f x" and "a \<le> b" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1094 
shows "(\<integral>\<^isup>+x. f x * indicator {a .. b} x \<partial>lborel) = F b  F a" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1095 
proof  
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1096 
have i: "(f has_integral F b  F a) {a..b}" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1097 
by (intro fundamental_theorem_of_calculus ballI has_vector_derivative_withinI_DERIV assms) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1098 
have i: "((\<lambda>x. f x * indicator {a..b} x) has_integral F b  F a) {a..b}" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1099 
by (rule has_integral_eq[OF _ i]) auto 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1100 
have i: "((\<lambda>x. f x * indicator {a..b} x) has_integral F b  F a) UNIV" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1101 
by (rule has_integral_on_superset[OF _ _ i]) auto 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1102 
then have "(\<integral>\<^isup>+x. ereal (f x * indicator {a .. b} x) \<partial>lborel) = F b  F a" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1103 
using f f_borel 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1104 
by (subst has_integral_iff_positive_integral_lborel[symmetric]) (auto split: split_indicator) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1105 
also have "(\<integral>\<^isup>+x. ereal (f x * indicator {a .. b} x) \<partial>lborel) = (\<integral>\<^isup>+x. ereal (f x) * indicator {a .. b} x \<partial>lborel)" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1106 
by (auto intro!: positive_integral_cong simp: indicator_def) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1107 
finally show ?thesis by simp 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1108 
qed 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1109 

bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1110 
lemma positive_integral_FTC_atLeast: 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1111 
fixes f :: "real \<Rightarrow> real" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1112 
assumes f_borel: "f \<in> borel_measurable borel" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1113 
assumes f: "\<And>x. a \<le> x \<Longrightarrow> DERIV F x :> f x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1114 
assumes nonneg: "\<And>x. a \<le> x \<Longrightarrow> 0 \<le> f x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1115 
assumes lim: "(F > T) at_top" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1116 
shows "(\<integral>\<^isup>+x. ereal (f x) * indicator {a ..} x \<partial>lborel) = T  F a" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1117 
proof  
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1118 
let ?f = "\<lambda>(i::nat) (x::real). ereal (f x) * indicator {a..a + real i} x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1119 
let ?fR = "\<lambda>x. ereal (f x) * indicator {a ..} x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1120 
have "\<And>x. (SUP i::nat. ?f i x) = ?fR x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1121 
proof (rule SUP_Lim_ereal) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1122 
show "\<And>x. incseq (\<lambda>i. ?f i x)" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1123 
using nonneg by (auto simp: incseq_def le_fun_def split: split_indicator) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1124 

bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1125 
fix x 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1126 
from reals_Archimedean2[of "x  a"] guess n .. 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1127 
then have "eventually (\<lambda>n. ?f n x = ?fR x) sequentially" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1128 
by (auto intro!: eventually_sequentiallyI[where c=n] split: split_indicator) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1129 
then show "(\<lambda>n. ?f n x) > ?fR x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1130 
by (rule Lim_eventually) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1131 
qed 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1132 
then have "integral\<^isup>P lborel ?fR = (\<integral>\<^isup>+ x. (SUP i::nat. ?f i x) \<partial>lborel)" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1133 
by simp 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1134 
also have "\<dots> = (SUP i::nat. (\<integral>\<^isup>+ x. ?f i x \<partial>lborel))" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1135 
proof (rule positive_integral_monotone_convergence_SUP) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1136 
show "incseq ?f" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1137 
using nonneg by (auto simp: incseq_def le_fun_def split: split_indicator) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1138 
show "\<And>i. (?f i) \<in> borel_measurable lborel" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1139 
using f_borel by auto 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1140 
show "\<And>i x. 0 \<le> ?f i x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1141 
using nonneg by (auto split: split_indicator) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1142 
qed 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1143 
also have "\<dots> = (SUP i::nat. F (a + real i)  F a)" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1144 
by (subst positive_integral_FTC_atLeastAtMost[OF f_borel f nonneg]) auto 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1145 
also have "\<dots> = T  F a" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1146 
proof (rule SUP_Lim_ereal) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1147 
show "incseq (\<lambda>n. ereal (F (a + real n)  F a))" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1148 
proof (simp add: incseq_def, safe) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1149 
fix m n :: nat assume "m \<le> n" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1150 
with f nonneg show "F (a + real m) \<le> F (a + real n)" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1151 
by (intro DERIV_nonneg_imp_nondecreasing[where f=F]) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1152 
(simp, metis add_increasing2 order_refl order_trans real_of_nat_ge_zero) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1153 
qed 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1154 
have "(\<lambda>x. F (a + real x)) > T" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1155 
apply (rule filterlim_compose[OF lim filterlim_tendsto_add_at_top]) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1156 
apply (rule LIMSEQ_const_iff[THEN iffD2, OF refl]) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1157 
apply (rule filterlim_real_sequentially) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1158 
done 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1159 
then show "(\<lambda>n. ereal (F (a + real n)  F a)) > ereal (T  F a)" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1160 
unfolding lim_ereal 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1161 
by (intro tendsto_diff) auto 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1162 
qed 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1163 
finally show ?thesis . 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1164 
qed 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1165 

38656  1166 
end 