author  hoelzl 
Wed, 18 Jun 2014 07:31:12 +0200  
changeset 57275  0ddb5b755cdc 
parent 57235  b0b9a10e4bf4 
child 57447  87429bdecad5 
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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introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
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imports Finite_Product_Measure Bochner_Integration 
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begin 
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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_box by (simp add: setsum_negf ex_in_conv eucl_le[where 'a='a]) 
38656  52 

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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_box 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) 
38656  59 
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) 
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qed 
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lemma cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)" 
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unfolding cube_def cbox_interval[symmetric] subset_box by (simp add: setsum_negf) 
41654  71 

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)" 
50104  75 
(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 cbox_interval[symmetric] 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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prefer p2e before e2p; use measure_unique_Int_stable_vimage;
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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}" 

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shows "sigma_algebra UNIV leb" 

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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) 
47694  107 
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 
47694  113 
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]) 

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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) 

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let ?m = "\<lambda>n i. integral (cube n) (indicator (A i) :: _\<Rightarrow>real)" 
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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" 
160 
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 
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show "(SUP n. \<Sum>i. ereal (?m n i)) = (SUP n. ereal (?M n UNIV))" 
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proof (intro arg_cong[where f="SUPREMUM UNIV"] ext sums_unique[symmetric] sums_ereal[THEN iffD2] sums_def[THEN iffD2]) 
41654  173 
fix n 
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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") 

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

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(auto intro: LIMSEQ_indicator_UN simp: cube_def) 

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moreover 

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{ 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)" 

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by (auto dest!: lebesgueD) 

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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<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" 
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assumes "negligible s" shows "s \<in> sets lebesgue" 

56188  225 
using assms by (force simp: cbox_interval[symmetric] 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) 
56188  234 
(auto simp: cube_def negligible_def cbox_interval[symmetric]) 
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" 
41689
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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 

56188  384 
have integrable: "(indicator A :: _\<Rightarrow>real) integrable_on cbox a b" 
385 
by (intro integrable_on_subcbox has_integral_integrable) (auto intro: *) 

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

47694  387 
using * by (auto intro!: integral_subset_le) 
56188  388 
moreover have "(0::real) \<le> integral (cbox a b) (indicator A)" 
41654  389 
using integrable by (auto intro!: integral_nonneg) 
56188  390 
ultimately have "integral (cbox a b) (indicator A) = (0::real)" 
41654  391 
using integral_unique[OF *] by auto 
56188  392 
then show "(indicator A has_integral (0::real)) (cbox a b)" 
41654  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) 

41981
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399 
fix n :: nat 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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400 
have "indicator UNIV = (\<lambda>x::'a. 1 :: real)" by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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changeset

401 
moreover 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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changeset

402 
{ have "real n \<le> (2 * real n) ^ DIM('a)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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changeset

403 
proof (cases n) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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diff
changeset

404 
case 0 then show ?thesis by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

405 
next 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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diff
changeset

406 
case (Suc n') 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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diff
changeset

407 
have "real n \<le> (2 * real n)^1" by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

408 
also have "(2 * real n)^1 \<le> (2 * real n) ^ DIM('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
diff
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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50418
diff
changeset

410 
by (intro power_increasing) (auto simp: real_of_nat_Suc simp del: DIM_positive) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

411 
finally show ?thesis . 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

412 
qed } 
43920  413 
ultimately show "ereal (real n) \<le> ereal (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

414 
using integral_const DIM_positive[where 'a='a] 
56188  415 
by (auto simp: cube_def content_cbox_cases setprod_constant setsum_negf cbox_interval[symmetric]) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

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" 

56188  426 
unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def [abs_def] cbox_interval[symmetric]) 
41654  427 
from lmeasure_eq_integral[OF this] show ?thesis unfolding integral_indicator_UNIV 
56188  428 
by (simp add: indicator_def [abs_def] cbox_interval[symmetric]) 
40859  429 
qed 
430 

57138
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hoelzl
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57137
diff
changeset

431 
lemma 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
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changeset

432 
fixes a b ::"'a::ordered_euclidean_space" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
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57137
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changeset

433 
shows lmeasure_cbox[simp]: "emeasure lebesgue (cbox a b) = ereal (content (cbox a b))" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
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changeset

434 
proof  
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
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changeset

435 
have "(indicator (UNIV \<inter> {a..b})::_\<Rightarrow>real) integrable_on UNIV" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

436 
unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def [abs_def] cbox_interval[symmetric]) 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
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changeset

437 
from lmeasure_eq_integral[OF this] show ?thesis unfolding integral_indicator_UNIV 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
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57137
diff
changeset

438 
by (simp add: indicator_def [abs_def] cbox_interval[symmetric]) 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

439 
qed 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
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57137
diff
changeset

440 

40859  441 
lemma lmeasure_singleton[simp]: 
47694  442 
fixes a :: "'a::ordered_euclidean_space" shows "emeasure lebesgue {a} = 0" 
41654  443 
using lmeasure_atLeastAtMost[of a a] by simp 
40859  444 

49777  445 
lemma AE_lebesgue_singleton: 
446 
fixes a :: "'a::ordered_euclidean_space" shows "AE x in lebesgue. x \<noteq> a" 

447 
by (rule AE_I[where N="{a}"]) auto 

448 

40859  449 
declare content_real[simp] 
450 

451 
lemma 

452 
fixes a b :: real 

453 
shows lmeasure_real_greaterThanAtMost[simp]: 

47694  454 
"emeasure lebesgue {a <.. b} = ereal (if a \<le> b then b  a else 0)" 
49777  455 
proof  
456 
have "emeasure lebesgue {a <.. b} = emeasure lebesgue {a .. b}" 

457 
using AE_lebesgue_singleton[of a] 

458 
by (intro emeasure_eq_AE) auto 

40859  459 
then show ?thesis by auto 
49777  460 
qed 
40859  461 

462 
lemma 

463 
fixes a b :: real 

464 
shows lmeasure_real_atLeastLessThan[simp]: 

47694  465 
"emeasure lebesgue {a ..< b} = ereal (if a \<le> b then b  a else 0)" 
49777  466 
proof  
467 
have "emeasure lebesgue {a ..< b} = emeasure lebesgue {a .. b}" 

468 
using AE_lebesgue_singleton[of b] 

469 
by (intro emeasure_eq_AE) auto 

41654  470 
then show ?thesis by auto 
49777  471 
qed 
41654  472 

473 
lemma 

474 
fixes a b :: real 

475 
shows lmeasure_real_greaterThanLessThan[simp]: 

47694  476 
"emeasure lebesgue {a <..< b} = ereal (if a \<le> b then b  a else 0)" 
49777  477 
proof  
478 
have "emeasure lebesgue {a <..< b} = emeasure lebesgue {a .. b}" 

479 
using AE_lebesgue_singleton[of a] AE_lebesgue_singleton[of b] 

480 
by (intro emeasure_eq_AE) auto 

40859  481 
then show ?thesis by auto 
49777  482 
qed 
40859  483 

41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset

484 
subsection {* LebesgueBorel measure *} 
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset

485 

47694  486 
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
parents:
41661
diff
changeset

487 

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

488 
lemma 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

489 
shows space_lborel[simp]: "space lborel = UNIV" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

490 
and sets_lborel[simp]: "sets lborel = sets borel" 
47694  491 
and measurable_lborel1[simp]: "measurable lborel = measurable borel" 
492 
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

493 
using sets.sigma_sets_eq[of borel] 
47694  494 
by (auto simp add: lborel_def measurable_def[abs_def]) 
40859  495 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
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56218
diff
changeset

496 
(* TODO: switch these rules! *) 
47694  497 
lemma emeasure_lborel[simp]: "A \<in> sets borel \<Longrightarrow> emeasure lborel A = emeasure lebesgue A" 
498 
by (rule emeasure_measure_of[OF lborel_def]) 

499 
(auto simp: positive_def emeasure_nonneg countably_additive_def intro!: suminf_emeasure) 

40859  500 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

501 
lemma measure_lborel[simp]: "A \<in> sets borel \<Longrightarrow> measure lborel A = measure lebesgue A" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

502 
unfolding measure_def by simp 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

503 

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

504 
interpretation lborel: sigma_finite_measure lborel 
47694  505 
proof (default, intro conjI exI[of _ "\<lambda>n. cube n"]) 
506 
show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed) 

507 
{ fix x :: 'a have "\<exists>n. x\<in>cube n" using mem_big_cube by auto } 

508 
then show "(\<Union>i. cube i) = (space lborel :: 'a set)" using mem_big_cube by auto 

509 
show "\<forall>i. emeasure lborel (cube i) \<noteq> \<infinity>" by (simp add: cube_def) 

510 
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

511 

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

512 
interpretation lebesgue: sigma_finite_measure lebesgue 
40859  513 
proof 
47694  514 
from lborel.sigma_finite guess A :: "nat \<Rightarrow> 'a set" .. 
515 
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>)" 

516 
by (intro exI[of _ A]) (auto simp: subset_eq) 

40859  517 
qed 
518 

57235
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57166
diff
changeset

519 
interpretation lborel_pair: pair_sigma_finite lborel lborel .. 
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57166
diff
changeset

520 

49777  521 
lemma Int_stable_atLeastAtMost: 
522 
fixes x::"'a::ordered_euclidean_space" 

523 
shows "Int_stable (range (\<lambda>(a, b::'a). {a..b}))" 

56188  524 
by (auto simp: inter_interval Int_stable_def cbox_interval[symmetric]) 
49777  525 

526 
lemma lborel_eqI: 

527 
fixes M :: "'a::ordered_euclidean_space measure" 

528 
assumes emeasure_eq: "\<And>a b. emeasure M {a .. b} = content {a .. b}" 

529 
assumes sets_eq: "sets M = sets borel" 

530 
shows "lborel = M" 

531 
proof (rule measure_eqI_generator_eq[OF Int_stable_atLeastAtMost]) 

53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

532 
let ?P = "\<Pi>\<^sub>M i\<in>{..<DIM('a::ordered_euclidean_space)}. lborel" 
49777  533 
let ?E = "range (\<lambda>(a, b). {a..b} :: 'a set)" 
534 
show "?E \<subseteq> Pow UNIV" "sets lborel = sigma_sets UNIV ?E" "sets M = sigma_sets UNIV ?E" 

535 
by (simp_all add: borel_eq_atLeastAtMost sets_eq) 

536 

537 
show "range cube \<subseteq> ?E" unfolding cube_def [abs_def] by auto 

538 
{ fix x :: 'a have "\<exists>n. x \<in> cube n" using mem_big_cube[of x] by fastforce } 

539 
then show "(\<Union>i. cube i :: 'a set) = UNIV" by auto 

540 

541 
{ fix i show "emeasure lborel (cube i) \<noteq> \<infinity>" unfolding cube_def by auto } 

542 
{ fix X assume "X \<in> ?E" then show "emeasure lborel X = emeasure M X" 

543 
by (auto simp: emeasure_eq) } 

544 
qed 

545 

56993
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introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
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diff
changeset

546 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
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diff
changeset

547 
(* GENEREALIZE to euclidean_spaces *) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
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parents:
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548 
lemma lborel_real_affine: 
49777  549 
fixes c :: real assumes "c \<noteq> 0" 
550 
shows "lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. \<bar>c\<bar>)" (is "_ = ?D") 

551 
proof (rule lborel_eqI) 

552 
fix a b show "emeasure ?D {a..b} = content {a .. b}" 

553 
proof cases 

554 
assume "0 < c" 

555 
then have "(\<lambda>x. t + c * x) ` {a..b} = {(a  t) / c .. (b  t) / c}" 

556 
by (auto simp: field_simps) 

557 
with `0 < c` show ?thesis 

558 
by (cases "a \<le> b") 

56996  559 
(auto simp: field_simps emeasure_density nn_integral_distr nn_integral_cmult 
49777  560 
borel_measurable_indicator' emeasure_distr) 
561 
next 

562 
assume "\<not> 0 < c" with `c \<noteq> 0` have "c < 0" by auto 

563 
then have *: "(\<lambda>x. t + c * x) ` {a..b} = {(b  t) / c .. (a  t) / c}" 

564 
by (auto simp: field_simps) 

565 
with `c < 0` show ?thesis 

566 
by (cases "a \<le> b") 

56996  567 
(auto simp: field_simps emeasure_density nn_integral_distr 
568 
nn_integral_cmult borel_measurable_indicator' emeasure_distr) 

49777  569 
qed 
570 
qed simp 

571 

56996  572 
lemma nn_integral_real_affine: 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
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diff
changeset

573 
fixes c :: real assumes [measurable]: "f \<in> borel_measurable borel" and c: "c \<noteq> 0" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
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diff
changeset

574 
shows "(\<integral>\<^sup>+x. f x \<partial>lborel) = \<bar>c\<bar> * (\<integral>\<^sup>+x. f (t + c * x) \<partial>lborel)" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

575 
by (subst lborel_real_affine[OF c, of t]) 
56996  576 
(simp add: nn_integral_density nn_integral_distr nn_integral_cmult) 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

577 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

578 
lemma lborel_integrable_real_affine: 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

579 
fixes f :: "real \<Rightarrow> _ :: {banach, second_countable_topology}" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

580 
assumes f: "integrable lborel f" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

581 
shows "c \<noteq> 0 \<Longrightarrow> integrable lborel (\<lambda>x. f (t + c * x))" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
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diff
changeset

582 
using f f[THEN borel_measurable_integrable] unfolding integrable_iff_bounded 
56996  583 
by (subst (asm) nn_integral_real_affine[where c=c and t=t]) auto 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

584 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

585 
lemma lborel_integrable_real_affine_iff: 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
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diff
changeset

586 
fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

587 
shows "c \<noteq> 0 \<Longrightarrow> integrable lborel (\<lambda>x. f (t + c * x)) \<longleftrightarrow> integrable lborel f" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

588 
using 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

589 
lborel_integrable_real_affine[of f c t] 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
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diff
changeset

590 
lborel_integrable_real_affine[of "\<lambda>x. f (t + c * x)" "1/c" "t/c"] 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

591 
by (auto simp add: field_simps) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

592 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

593 
lemma lborel_integral_real_affine: 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

594 
fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}" and c :: real 
57166
5cfcc616d485
use 0 as integralvalue for nonintegrable functions, simplify a couple of rewrite rules
hoelzl
parents:
57138
diff
changeset

595 
assumes c: "c \<noteq> 0" shows "(\<integral>x. f x \<partial> lborel) = \<bar>c\<bar> *\<^sub>R (\<integral>x. f (t + c * x) \<partial>lborel)" 
5cfcc616d485
use 0 as integralvalue for nonintegrable functions, simplify a couple of rewrite rules
hoelzl
parents:
57138
diff
changeset

596 
proof cases 
5cfcc616d485
use 0 as integralvalue for nonintegrable functions, simplify a couple of rewrite rules
hoelzl
parents:
57138
diff
changeset

597 
assume f[measurable]: "integrable lborel f" then show ?thesis 
5cfcc616d485
use 0 as integralvalue for nonintegrable functions, simplify a couple of rewrite rules
hoelzl
parents:
57138
diff
changeset

598 
using c f f[THEN borel_measurable_integrable] f[THEN lborel_integrable_real_affine, of c t] 
5cfcc616d485
use 0 as integralvalue for nonintegrable functions, simplify a couple of rewrite rules
hoelzl
parents:
57138
diff
changeset

599 
by (subst lborel_real_affine[OF c, of t]) (simp add: integral_density integral_distr) 
5cfcc616d485
use 0 as integralvalue for nonintegrable functions, simplify a couple of rewrite rules
hoelzl
parents:
57138
diff
changeset

600 
next 
5cfcc616d485
use 0 as integralvalue for nonintegrable functions, simplify a couple of rewrite rules
hoelzl
parents:
57138
diff
changeset

601 
assume "\<not> integrable lborel f" with c show ?thesis 
5cfcc616d485
use 0 as integralvalue for nonintegrable functions, simplify a couple of rewrite rules
hoelzl
parents:
57138
diff
changeset

602 
by (simp add: lborel_integrable_real_affine_iff not_integrable_integral_eq) 
5cfcc616d485
use 0 as integralvalue for nonintegrable functions, simplify a couple of rewrite rules
hoelzl
parents:
57138
diff
changeset

603 
qed 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

604 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

605 
lemma divideR_right: 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

606 
fixes x y :: "'a::real_normed_vector" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

607 
shows "r \<noteq> 0 \<Longrightarrow> y = x /\<^sub>R r \<longleftrightarrow> r *\<^sub>R y = x" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

608 
using scaleR_cancel_left[of r y "x /\<^sub>R r"] by simp 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

609 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

610 
lemma integrable_on_cmult_iff2: 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

611 
fixes c :: real 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

612 
shows "(\<lambda>x. c * f x) integrable_on s \<longleftrightarrow> c = 0 \<or> f integrable_on s" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

613 
using integrable_cmul[of "\<lambda>x. c * f x" s "1 / c"] integrable_cmul[of f s c] 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

614 
by (cases "c = 0") auto 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

615 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

616 
lemma lborel_has_bochner_integral_real_affine_iff: 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

617 
fixes x :: "'a :: {banach, second_countable_topology}" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

618 
shows "c \<noteq> 0 \<Longrightarrow> 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

619 
has_bochner_integral lborel f x \<longleftrightarrow> 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

620 
has_bochner_integral lborel (\<lambda>x. f (t + c * x)) (x /\<^sub>R \<bar>c\<bar>)" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

621 
unfolding has_bochner_integral_iff lborel_integrable_real_affine_iff 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

622 
by (simp_all add: lborel_integral_real_affine[symmetric] divideR_right cong: conj_cong) 
49777  623 

41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset

624 
subsection {* Lebesgue integrable implies Gauge integrable *} 
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset

625 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

626 
lemma has_integral_scaleR_left: 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

627 
"(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x *\<^sub>R c) has_integral (y *\<^sub>R c)) s" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

628 
using has_integral_linear[OF _ bounded_linear_scaleR_left] by (simp add: comp_def) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

629 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

630 
lemma has_integral_mult_left: 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

631 
fixes c :: "_ :: {real_normed_algebra}" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

632 
shows "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x * c) has_integral (y * c)) s" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

633 
using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

634 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

635 
(* GENERALIZE Integration.dominated_convergence, then generalize the following theorems *) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

636 
lemma has_integral_dominated_convergence: 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

637 
fixes f :: "nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

638 
assumes "\<And>k. (f k has_integral y k) s" "h integrable_on s" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

639 
"\<And>k. \<forall>x\<in>s. norm (f k x) \<le> h x" "\<forall>x\<in>s. (\<lambda>k. f k x) > g x" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

640 
and x: "y > x" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

641 
shows "(g has_integral x) s" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

642 
proof  
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

643 
have int_f: "\<And>k. (f k) integrable_on s" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

644 
using assms by (auto simp: integrable_on_def) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

645 
have "(g has_integral (integral s g)) s" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

646 
by (intro integrable_integral dominated_convergence[OF int_f assms(2)]) fact+ 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

647 
moreover have "integral s g = x" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

648 
proof (rule LIMSEQ_unique) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

649 
show "(\<lambda>i. integral s (f i)) > x" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

650 
using integral_unique[OF assms(1)] x by simp 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

651 
show "(\<lambda>i. integral s (f i)) > integral s g" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

652 
by (intro dominated_convergence[OF int_f assms(2)]) fact+ 
41654  653 
qed 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

654 
ultimately show ?thesis 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

655 
by simp 
40859  656 
qed 
657 

56996  658 
lemma nn_integral_has_integral: 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

659 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> real" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

660 
assumes f: "f \<in> borel_measurable lebesgue" "\<And>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>lebesgue) = ereal r" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

661 
shows "(f has_integral r) UNIV" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

662 
using f proof (induct arbitrary: r rule: borel_measurable_induct_real) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

663 
case (set A) then show ?case 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

664 
by (auto simp add: ereal_indicator has_integral_iff_lmeasure) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

665 
next 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

666 
case (mult g c) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

667 
then have "ereal c * (\<integral>\<^sup>+ x. g x \<partial>lebesgue) = ereal r" 
56996  668 
by (subst nn_integral_cmult[symmetric]) auto 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

669 
then obtain r' where "(c = 0 \<and> r = 0) \<or> ((\<integral>\<^sup>+ x. ereal (g x) \<partial>lebesgue) = ereal r' \<and> r = c * r')" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

670 
by (cases "\<integral>\<^sup>+ x. ereal (g x) \<partial>lebesgue") (auto split: split_if_asm) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

671 
with mult show ?case 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

672 
by (auto intro!: has_integral_cmult_real) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

673 
next 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

674 
case (add g h) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

675 
moreover 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

676 
then have "(\<integral>\<^sup>+ x. h x + g x \<partial>lebesgue) = (\<integral>\<^sup>+ x. h x \<partial>lebesgue) + (\<integral>\<^sup>+ x. g x \<partial>lebesgue)" 
56996  677 
unfolding plus_ereal.simps[symmetric] by (subst nn_integral_add) auto 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

678 
with add obtain a b where "(\<integral>\<^sup>+ x. h x \<partial>lebesgue) = ereal a" "(\<integral>\<^sup>+ x. g x \<partial>lebesgue) = ereal b" "r = a + b" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

679 
by (cases "\<integral>\<^sup>+ x. h x \<partial>lebesgue" "\<integral>\<^sup>+ x. g x \<partial>lebesgue" rule: ereal2_cases) auto 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

680 
ultimately show ?case 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

681 
by (auto intro!: has_integral_add) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

682 
next 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

683 
case (seq U) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

684 
note seq(1)[measurable] and f[measurable] 
40859  685 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

686 
{ fix i x 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

687 
have "U i x \<le> f x" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

688 
using seq(5) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

689 
apply (rule LIMSEQ_le_const) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

690 
using seq(4) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

691 
apply (auto intro!: exI[of _ i] simp: incseq_def le_fun_def) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

692 
done } 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

693 
note U_le_f = this 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

694 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

695 
{ fix i 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

696 
have "(\<integral>\<^sup>+x. ereal (U i x) \<partial>lebesgue) \<le> (\<integral>\<^sup>+x. ereal (f x) \<partial>lebesgue)" 
56996  697 
using U_le_f by (intro nn_integral_mono) simp 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

698 
then obtain p where "(\<integral>\<^sup>+x. U i x \<partial>lebesgue) = ereal p" "p \<le> r" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

699 
using seq(6) by (cases "\<integral>\<^sup>+x. U i x \<partial>lebesgue") auto 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

700 
moreover then have "0 \<le> p" 
56996  701 
by (metis ereal_less_eq(5) nn_integral_nonneg) 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

702 
moreover note seq 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

703 
ultimately have "\<exists>p. (\<integral>\<^sup>+x. U i x \<partial>lebesgue) = ereal p \<and> 0 \<le> p \<and> p \<le> r \<and> (U i has_integral p) UNIV" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

704 
by auto } 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

705 
then obtain p where p: "\<And>i. (\<integral>\<^sup>+x. ereal (U i x) \<partial>lebesgue) = ereal (p i)" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

706 
and bnd: "\<And>i. p i \<le> r" "\<And>i. 0 \<le> p i" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

707 
and U_int: "\<And>i.(U i has_integral (p i)) UNIV" by metis 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

708 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

709 
have int_eq: "\<And>i. integral UNIV (U i) = p i" using U_int by (rule integral_unique) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

710 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

711 
have *: "f integrable_on UNIV \<and> (\<lambda>k. integral UNIV (U k)) > integral UNIV f" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

712 
proof (rule monotone_convergence_increasing) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

713 
show "\<forall>k. U k integrable_on UNIV" using U_int by auto 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

714 
show "\<forall>k. \<forall>x\<in>UNIV. U k x \<le> U (Suc k) x" using `incseq U` by (auto simp: incseq_def le_fun_def) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

715 
then show "bounded {integral UNIV (U k) k. True}" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

716 
using bnd int_eq by (auto simp: bounded_real intro!: exI[of _ r]) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

717 
show "\<forall>x\<in>UNIV. (\<lambda>k. U k x) > f x" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

718 
using seq by auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

719 
qed 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

720 
moreover have "(\<lambda>i. (\<integral>\<^sup>+x. U i x \<partial>lebesgue)) > (\<integral>\<^sup>+x. f x \<partial>lebesgue)" 
56996  721 
using seq U_le_f by (intro nn_integral_dominated_convergence[where w=f]) auto 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

722 
ultimately have "integral UNIV f = r" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

723 
by (auto simp add: int_eq p seq intro: LIMSEQ_unique) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

724 
with * show ?case 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

725 
by (simp add: has_integral_integral) 
40859  726 
qed 
727 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

728 
lemma has_integral_integrable_lebesgue_nonneg: 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

729 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> real" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

730 
assumes f: "integrable lebesgue f" "\<And>x. 0 \<le> f x" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

731 
shows "(f has_integral integral\<^sup>L lebesgue f) UNIV" 
56996  732 
proof (rule nn_integral_has_integral) 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

733 
show "(\<integral>\<^sup>+ x. ereal (f x) \<partial>lebesgue) = ereal (integral\<^sup>L lebesgue f)" 
56996  734 
using f by (intro nn_integral_eq_integral) auto 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

735 
qed (insert f, auto) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

736 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

737 
lemma has_integral_lebesgue_integral_lebesgue: 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

738 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> real" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

739 
assumes f: "integrable lebesgue f" 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

740 
shows "(f has_integral (integral\<^sup>L lebesgue f)) UNIV" 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

741 
using f proof induct 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

742 
case (base A c) then show ?case 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

743 
by (auto intro!: has_integral_mult_left simp: has_integral_iff_lmeasure) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

744 
(simp add: emeasure_eq_ereal_measure) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

745 
next 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

746 
case (add f g) then show ?case 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

747 
by (auto intro!: has_integral_add) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

748 
next 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

749 
case (lim f s) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

750 
show ?case 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

751 
proof (rule has_integral_dominated_convergence) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

752 
show "\<And>i. (s i has_integral integral\<^sup>L lebesgue (s i)) UNIV" by fact 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

753 
show "(\<lambda>x. norm (2 * f x)) integrable_on UNIV" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

754 
using lim by (intro has_integral_integrable[OF has_integral_integrable_lebesgue_nonneg]) auto 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

755 
show "\<And>k. \<forall>x\<in>UNIV. norm (s k x) \<le> norm (2 * f x)" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

756 
using lim by (auto simp add: abs_mult) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

757 
show "\<forall>x\<in>UNIV. (\<lambda>k. s k x) > f x" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

758 
using lim by auto 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

759 
show "(\<lambda>k. integral\<^sup>L lebesgue (s k)) > integral\<^sup>L lebesgue f" 
57137  760 
using lim by (intro integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"]) auto 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

761 
qed 
40859  762 
qed 
763 

56996  764 
lemma lebesgue_nn_integral_eq_borel: 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

765 
assumes f: "f \<in> borel_measurable borel" 
56996  766 
shows "integral\<^sup>N lebesgue f = integral\<^sup>N lborel f" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

767 
proof  
56996  768 
from f have "integral\<^sup>N lebesgue (\<lambda>x. max 0 (f x)) = integral\<^sup>N lborel (\<lambda>x. max 0 (f x))" 
769 
by (auto intro!: nn_integral_subalgebra[symmetric]) 

770 
then show ?thesis unfolding nn_integral_max_0 . 

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

771 
qed 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

772 

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

773 
lemma lebesgue_integral_eq_borel: 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

774 
fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}" 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

775 
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

776 
shows "integrable lebesgue f \<longleftrightarrow> integrable lborel f" (is ?P) 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

777 
and "integral\<^sup>L lebesgue f = integral\<^sup>L lborel f" (is ?I) 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

778 
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

779 
have "sets lborel \<subseteq> sets lebesgue" by auto 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

780 
from integral_subalgebra[of f lborel, OF _ this _ _] 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

781 
integrable_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

782 
show ?P ?I by auto 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

783 
qed 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

784 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

785 
lemma has_integral_lebesgue_integral: 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

786 
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

787 
assumes f:"integrable lborel f" 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

788 
shows "(f has_integral (integral\<^sup>L lborel f)) UNIV" 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

789 
proof  
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

790 
have borel: "f \<in> borel_measurable borel" 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

791 
using f unfolding integrable_iff_bounded by auto 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

792 
from f show ?thesis 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

793 
using has_integral_lebesgue_integral_lebesgue[of f] 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

794 
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

795 
qed 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

796 

56996  797 
lemma nn_integral_bound_simple_function: 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

798 
assumes bnd: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" "\<And>x. x \<in> space M \<Longrightarrow> f x < \<infinity>" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

799 
assumes f[measurable]: "simple_function M f" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

800 
assumes supp: "emeasure M {x\<in>space M. f x \<noteq> 0} < \<infinity>" 
56996  801 
shows "nn_integral M f < \<infinity>" 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

802 
proof cases 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

803 
assume "space M = {}" 
56996  804 
then have "nn_integral M f = (\<integral>\<^sup>+x. 0 \<partial>M)" 
805 
by (intro nn_integral_cong) auto 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

806 
then show ?thesis by simp 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

807 
next 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

808 
assume "space M \<noteq> {}" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

809 
with simple_functionD(1)[OF f] bnd have bnd: "0 \<le> Max (f`space M) \<and> Max (f`space M) < \<infinity>" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

810 
by (subst Max_less_iff) (auto simp: Max_ge_iff) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

811 

56996  812 
have "nn_integral M f \<le> (\<integral>\<^sup>+x. Max (f`space M) * indicator {x\<in>space M. f x \<noteq> 0} x \<partial>M)" 
813 
proof (rule nn_integral_mono) 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

814 
fix x assume "x \<in> space M" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

815 
with f show "f x \<le> Max (f ` space M) * indicator {x \<in> space M. f x \<noteq> 0} x" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

816 
by (auto split: split_indicator intro!: Max_ge simple_functionD) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

817 
qed 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

818 
also have "\<dots> < \<infinity>" 
56996  819 
using bnd supp by (subst nn_integral_cmult) auto 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

820 
finally show ?thesis . 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

821 
qed 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

822 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

823 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

824 
lemma 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

825 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 
49777  826 
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

827 
assumes I: "(f has_integral I) UNIV" 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

828 
shows integrable_has_integral_lebesgue_nonneg: "integrable lebesgue f" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

829 
and integral_has_integral_lebesgue_nonneg: "integral\<^sup>L lebesgue f = I" 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

830 
proof  
49777  831 
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

832 
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

833 

56996  834 
have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>lebesgue) = (SUP i. integral\<^sup>N lebesgue (F i))" 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

835 
using F 
56996  836 
by (subst nn_integral_monotone_convergence_SUP[symmetric]) 
837 
(simp_all add: nn_integral_max_0 borel_measurable_simple_function) 

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

838 
also have "\<dots> \<le> ereal I" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

839 
proof (rule SUP_least) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

840 
fix i :: nat 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

841 

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

842 
{ fix z 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

843 
from F(4)[of z] have "F i z \<le> ereal (f z)" 
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54775
diff
changeset

844 
by (metis SUP_upper UNIV_I ereal_max_0 max.absorb2 nonneg) 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

845 
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

846 
by (cases "F i z") simp_all } 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

847 
note F_bound = this 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

848 

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

849 
{ 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

850 
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

851 
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

852 
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

853 
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

854 
proof (rule dominated_convergence(1)) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

855 
fix n :: nat 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

856 
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

857 
using x F(1)[of i] 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

858 
by (intro lebesgueD) (auto simp: simple_function_def) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

859 
then have cube: "?s n integrable_on cube n" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

860 
by (simp add: integrable_on_cmult_iff) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

861 
show "?s n integrable_on UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

862 
by (rule integrable_on_superset[OF _ _ cube]) auto 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

863 
next 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

864 
show "f integrable_on UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

865 
unfolding integrable_on_def using I by auto 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

866 
next 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

867 
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

868 
using nonneg F(5) by (auto split: split_indicator) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

869 
next 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

870 
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

871 
proof 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

872 
fix z :: 'a 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

873 
from mem_big_cube[of z] guess j . 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

874 
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

875 
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

876 
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

877 
by (rule Lim_eventually) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

878 
qed 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

879 
qed 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

880 
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

881 
by (simp add: integrable_on_cmult_iff) } 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

882 
note F_finite = lmeasure_finite[OF this] 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

883 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

884 
have F_eq: "\<And>x. F i x = ereal (norm (real (F i x)))" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

885 
using F(3,5) by (auto simp: fun_eq_iff ereal_real image_iff eq_commute) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

886 
have F_eq2: "\<And>x. F i x = ereal (real (F i x))" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

887 
using F(3,5) by (auto simp: fun_eq_iff ereal_real image_iff eq_commute) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

888 

e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

889 
have int: "integrable lebesgue (\<lambda>x. real (F i x))" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

890 
unfolding integrable_iff_bounded 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

891 
proof 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

892 
have "(\<integral>\<^sup>+x. F i x \<partial>lebesgue) < \<infinity>" 
56996  893 
proof (rule nn_integral_bound_simple_function) 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

894 
fix x::'a assume "x \<in> space lebesgue" then show "0 \<le> F i x" "F i x < \<infinity>" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

895 
using F by (auto simp: image_iff eq_commute) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

896 
next 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

897 
have eq: "{x \<in> space lebesgue. F i x \<noteq> 0} = (\<Union>x\<in>F i ` space lebesgue  {0}. F i ` {x} \<inter> space lebesgue)" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

898 
by auto 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

899 
show "emeasure lebesgue {x \<in> space lebesgue. F i x \<noteq> 0} < \<infinity>" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

900 
unfolding eq using simple_functionD[OF F(1)] 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

901 
by (subst setsum_emeasure[symmetric]) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

902 
(auto simp: disjoint_family_on_def setsum_Pinfty F_finite) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

903 
qed fact 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

904 
with F_eq show "(\<integral>\<^sup>+x. norm (real (F i x)) \<partial>lebesgue) < \<infinity>" by simp 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

905 
qed (insert F(1), auto intro!: borel_measurable_real_of_ereal dest: borel_measurable_simple_function) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

906 
then have "((\<lambda>x. real (F i x)) has_integral integral\<^sup>L lebesgue (\<lambda>x. real (F i x))) UNIV" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

907 
by (rule has_integral_lebesgue_integral_lebesgue) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

908 
from this I have "integral\<^sup>L lebesgue (\<lambda>x. real (F i x)) \<le> I" 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

909 
by (rule has_integral_le) (intro ballI F_bound) 
56996  910 
moreover have "integral\<^sup>N lebesgue (F i) = integral\<^sup>L lebesgue (\<lambda>x. real (F i x))" 
911 
using int F by (subst nn_integral_eq_integral[symmetric]) (auto simp: F_eq2[symmetric] real_of_ereal_pos) 

912 
ultimately show "integral\<^sup>N lebesgue (F i) \<le> ereal I" 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

913 
by simp 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

914 
qed 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

915 
finally show "integrable lebesgue f" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

916 
using f_borel by (auto simp: integrable_iff_bounded nonneg) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

917 
from has_integral_lebesgue_integral_lebesgue[OF this] I 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

918 
show "integral\<^sup>L lebesgue f = I" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

919 
by (metis has_integral_unique) 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

920 
qed 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

921 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

922 
lemma has_integral_iff_has_bochner_integral_lebesgue_nonneg: 
49777  923 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

924 
shows "f \<in> borel_measurable lebesgue \<Longrightarrow> (\<And>x. 0 \<le> f x) \<Longrightarrow> 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

925 
(f has_integral I) UNIV \<longleftrightarrow> has_bochner_integral lebesgue f I" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

926 
by (metis has_bochner_integral_iff has_integral_unique has_integral_lebesgue_integral_lebesgue 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

927 
integrable_has_integral_lebesgue_nonneg) 
49777  928 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

929 
lemma 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

930 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

931 
assumes "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" "(f has_integral I) UNIV" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

932 
shows integrable_has_integral_nonneg: "integrable lborel f" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

933 
and integral_has_integral_nonneg: "integral\<^sup>L lborel f = I" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

934 
by (metis assms borel_measurable_lebesgueI integrable_has_integral_lebesgue_nonneg lebesgue_integral_eq_borel(1)) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

935 
(metis assms borel_measurable_lebesgueI has_integral_lebesgue_integral has_integral_unique integrable_has_integral_lebesgue_nonneg lebesgue_integral_eq_borel(1)) 
49777  936 

937 
subsection {* Equivalence between product spaces and euclidean spaces *} 

938 

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

939 
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

940 
"e2p x = (\<lambda>i\<in>Basis. x \<bullet> i)" 
49777  941 

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

942 
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

943 
"p2e x = (\<Sum>i\<in>Basis. x i *\<^sub>R i)" 
49777  944 

945 
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

946 
"x \<in> extensional Basis \<Longrightarrow> e2p (p2e x::'a::ordered_euclidean_space) = x" 
49777  947 
by (auto simp: fun_eq_iff extensional_def p2e_def e2p_def) 
948 

949 
lemma p2e_e2p[simp]: 

950 
"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

951 
by (auto simp: euclidean_eq_iff[where 'a='a] p2e_def e2p_def) 
49777  952 

953 
interpretation lborel_product: product_sigma_finite "\<lambda>x. lborel::real measure" 

954 
by default 

955 

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

956 
interpretation lborel_space: finite_product_sigma_finite "\<lambda>x. lborel::real measure" "Basis" 
49777  957 
by default auto 
958 

959 
lemma sets_product_borel: 

960 
assumes I: "finite I" 

53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

961 
shows "sets (\<Pi>\<^sub>M i\<in>I. lborel) = sigma_sets (\<Pi>\<^sub>E i\<in>I. UNIV) { \<Pi>\<^sub>E i\<in>I. {..< x i :: real}  x. True}" (is "_ = ?G") 
49777  962 
proof (subst sigma_prod_algebra_sigma_eq[where S="\<lambda>_ i::nat. {..<real i}" and E="\<lambda>_. range lessThan", OF I]) 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

963 
show "sigma_sets (space (Pi\<^sub>M I (\<lambda>i. lborel))) {Pi\<^sub>E I F F. \<forall>i\<in>I. F i \<in> range lessThan} = ?G" 
49777  964 
by (intro arg_cong2[where f=sigma_sets]) (auto simp: space_PiM image_iff bchoice_iff) 
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54257
diff
changeset

965 
qed (auto simp: borel_eq_lessThan eucl_lessThan reals_Archimedean2) 
49777  966 

50003  967 
lemma measurable_e2p[measurable]: 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

968 
"e2p \<in> measurable (borel::'a::ordered_euclidean_space measure) (\<Pi>\<^sub>M (i::'a)\<in>Basis. (lborel :: real measure))" 
49777  969 
proof (rule measurable_sigma_sets[OF sets_product_borel]) 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

970 
fix A :: "('a \<Rightarrow> real) set" assume "A \<in> {\<Pi>\<^sub>E (i::'a)\<in>Basis. {..<x i} x. True} " 
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

971 
then obtain x where "A = (\<Pi>\<^sub>E (i::'a)\<in>Basis. {..<x i})" by auto 
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54257
diff
changeset

972 
then have "e2p ` A = {y :: 'a. eucl_less y (\<Sum>i\<in>Basis. x i *\<^sub>R i)}" 
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54257
diff
changeset

973 
using DIM_positive by (auto simp add: set_eq_iff e2p_def eucl_less_def) 
49777  974 
then show "e2p ` A \<inter> space (borel::'a measure) \<in> sets borel" by simp 
975 
qed (auto simp: e2p_def) 

976 

50003  977 
(* FIXME: conversion in measurable prover *) 
50385  978 
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 
979 
lemma lborelD[measurable (raw)]: "A \<in> sets borel \<Longrightarrow> A \<in> sets lborel" by simp 

50003  980 

981 
lemma measurable_p2e[measurable]: 

53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

982 
"p2e \<in> measurable (\<Pi>\<^sub>M (i::'a)\<in>Basis. (lborel :: real measure)) 
49777  983 
(borel :: 'a::ordered_euclidean_space measure)" 
984 
(is "p2e \<in> measurable ?P _") 

985 
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

986 
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

987 
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

988 
assume "i \<in> Basis" 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

989 
then have "?A = (\<Pi>\<^sub>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

990 
using DIM_positive by (auto simp: space_PiM p2e_def PiE_def split: split_if_asm) 
49777  991 
then show "?A \<in> sets ?P" 
992 
by auto 

993 
qed 

994 

995 
lemma lborel_eq_lborel_space: 

53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

996 
"(lborel :: 'a measure) = distr (\<Pi>\<^sub>M (i::'a::ordered_euclidean_space)\<in>Basis. lborel) borel p2e" 
49777  997 
(is "?B = ?D") 
998 
proof (rule lborel_eqI) 

999 
show "sets ?D = sets borel" by simp 

53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

1000 
let ?P = "(\<Pi>\<^sub>M (i::'a)\<in>Basis. lborel)" 
49777  1001 
fix a b :: 'a 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

1002 
have *: "p2e ` {a .. b} \<inter> space ?P = (\<Pi>\<^sub>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

1003 
by (auto simp: eucl_le[where 'a='a] p2e_def space_PiM PiE_def Pi_iff) 
49777  1004 
have "emeasure ?P (p2e ` {a..b} \<inter> space ?P) = content {a..b}" 
1005 
proof cases 

1006 
assume "{a..b} \<noteq> {}" 

1007 
then have "a \<le> b" 

56188  1008 
by (simp add: 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

1009 
then have "emeasure lborel {a..b} = (\<Prod>x\<in>Basis. emeasure lborel {a \<bullet> x .. b \<bullet> x})" 
56188  1010 
by (auto simp: eucl_le[where 'a='a] content_closed_interval 
49777  1011 
intro!: setprod_ereal[symmetric]) 
1012 
also have "\<dots> = emeasure ?P (p2e ` {a..b} \<inter> space ?P)" 

1013 
unfolding * by (subst lborel_space.measure_times) auto 

1014 
finally show ?thesis by simp 

1015 
qed simp 

1016 
then show "emeasure ?D {a .. b} = content {a .. b}" 

1017 
by (simp add: emeasure_distr measurable_p2e) 

1018 
qed 

1019 

1020 
lemma borel_fubini_positiv_integral: 

1021 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> ereal" 

1022 
assumes f: "f \<in> borel_measurable borel" 

56996  1023 
shows "integral\<^sup>N lborel f = \<integral>\<^sup>+x. f (p2e x) \<partial>(\<Pi>\<^sub>M (i::'a)\<in>Basis. lborel)" 
1024 
by (subst lborel_eq_lborel_space) (simp add: nn_integral_distr measurable_p2e f) 

49777  1025 

1026 
lemma borel_fubini_integrable: 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1027 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> _::{banach, second_countable_topology}" 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

1028 
shows "integrable lborel f \<longleftrightarrow> integrable (\<Pi>\<^sub>M (i::'a)\<in>Basis. lborel) (\<lambda>x. f (p2e x))" 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1029 
unfolding integrable_iff_bounded 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1030 
proof (intro conj_cong[symmetric]) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1031 
show "((\<lambda>x. f (p2e x)) \<in> borel_measurable (Pi\<^sub>M Basis (\<lambda>i. lborel))) = (f \<in> borel_measurable lborel)" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1032 
proof 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1033 
assume "((\<lambda>x. f (p2e x)) \<in> borel_measurable (Pi\<^sub>M Basis (\<lambda>i. lborel)))" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1034 
then have "(\<lambda>x. f (p2e (e2p x))) \<in> borel_measurable borel" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1035 
by measurable 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1036 
then show "f \<in> borel_measurable lborel" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1037 
by simp 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1038 
qed simp 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1039 
qed (simp add: borel_fubini_positiv_integral) 
49777  1040 

1041 
lemma borel_fubini: 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1042 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> _::{banach, second_countable_topology}" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1043 
shows "f \<in> borel_measurable borel \<Longrightarrow> 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1044 
integral\<^sup>L lborel f = \<integral>x. f (p2e x) \<partial>((\<Pi>\<^sub>M (i::'a)\<in>Basis. lborel))" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1045 
by (subst lborel_eq_lborel_space) (simp add: integral_distr) 
47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1046 

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

1047 
lemma integrable_on_borel_integrable: 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1048 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1049 
shows "f \<in> borel_measurable borel \<Longrightarrow> (\<And>x. 0 \<le> f x) \<Longrightarrow> f integrable_on UNIV \<Longrightarrow> integrable lborel f" 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1050 
by (metis borel_measurable_lebesgueI integrable_has_integral_nonneg integrable_on_def 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1051 
lebesgue_integral_eq_borel(1)) 
50418
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1052 

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

1053 
subsection {* Fundamental Theorem of Calculus for the Lebesgue integral *} 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1054 

57138
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1055 
lemma emeasure_bounded_finite: 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1056 
assumes "bounded A" shows "emeasure lborel A < \<infinity>" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1057 
proof  
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1058 
from bounded_subset_cbox[OF `bounded A`] obtain a b where "A \<subseteq> cbox a b" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1059 
by auto 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1060 
then have "emeasure lborel A \<le> emeasure lborel (cbox a b)" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1061 
by (intro emeasure_mono) auto 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1062 
then show ?thesis 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1063 
by auto 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1064 
qed 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1065 

7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1066 
lemma emeasure_compact_finite: "compact A \<Longrightarrow> emeasure lborel A < \<infinity>" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1067 
using emeasure_bounded_finite[of A] by (auto intro: compact_imp_bounded) 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1068 

7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1069 
lemma borel_integrable_compact: 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1070 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> 'b::{banach, second_countable_topology}" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1071 
assumes "compact S" "continuous_on S f" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1072 
shows "integrable lborel (\<lambda>x. indicator S x *\<^sub>R f x)" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1073 
proof cases 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1074 
assume "S \<noteq> {}" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1075 
have "continuous_on S (\<lambda>x. norm (f x))" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1076 
using assms by (intro continuous_intros) 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1077 
from continuous_attains_sup[OF `compact S` `S \<noteq> {}` this] 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1078 
obtain M where M: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> M" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1079 
by auto 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1080 

7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1081 
show ?thesis 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1082 
proof (rule integrable_bound) 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1083 
show "integrable lborel (\<lambda>x. indicator S x * M)" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1084 
using assms by (auto intro!: emeasure_compact_finite borel_compact integrable_mult_left) 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1085 
show "(\<lambda>x. indicator S x *\<^sub>R f x) \<in> borel_measurable lborel" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1086 
using assms by (auto intro!: borel_measurable_continuous_on_indicator borel_compact) 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1087 
show "AE x in lborel. norm (indicator S x *\<^sub>R f x) \<le> norm (indicator S x * M)" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1088 
by (auto split: split_indicator simp: abs_real_def dest!: M) 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1089 
qed 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1090 
qed simp 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1091 

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

1092 
lemma borel_integrable_atLeastAtMost: 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1093 
fixes f :: "real \<Rightarrow> real" 
50418
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1094 
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

1095 
shows "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)" (is "integrable _ ?f") 
57138
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1096 
proof  
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1097 
have "integrable lborel (\<lambda>x. indicator {a .. b} x *\<^sub>R f x)" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1098 
proof (rule borel_integrable_compact) 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1099 
from f show "continuous_on {a..b} f" 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1100 
by (auto intro: continuous_at_imp_continuous_on) 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1101 
qed simp 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1102 
then show ?thesis 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1103 
by (auto simp: mult_commute) 
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset

1104 
qed 
50418
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1105 

56181
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56020
diff
changeset

1106 
lemma has_field_derivative_subset: 
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56020
diff
changeset

1107 
"(f has_field_derivative y) (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_field_derivative y) (at x within t)" 
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56020
diff
changeset

1108 
unfolding has_field_derivative_def by (rule has_derivative_subset) 
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56020
diff
changeset

1109 

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

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

1111 
fixes a b :: real 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1112 
assumes "a \<le> b" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1113 
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

1114 
and f: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x" 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

1115 
shows "integral\<^sup>L lborel (\<lambda>x. f x * indicator {a .. b} x) = F b  F a" 
50418
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1116 
proof  
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1117 
let ?f = "\<lambda>x. f x * indicator {a .. b} x" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1118 
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

1119 
using borel_integrable_atLeastAtMost[OF f] 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1120 
by (rule has_integral_lebesgue_integral) 
50418
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1121 
moreover 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1122 
have "(f has_integral F b  F a) {a .. b}" 
56181
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56020
diff
changeset

1123 
by (intro fundamental_theorem_of_calculus) 
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56020
diff
changeset

1124 
(auto simp: has_field_derivative_iff_has_vector_derivative[symmetric] 
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56020
diff
changeset

1125 
intro: has_field_derivative_subset[OF F] assms(1)) 
50418
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1126 
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

1127 
by (subst has_integral_eq_eq[where g=f]) auto 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1128 
then have "(?f has_integral F b  F a) UNIV" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1129 
by (intro has_integral_on_superset[where t=UNIV and s="{a..b}"]) auto 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset

1130 
ultimately show "integral\<^sup>L lborel ?f = F b  F a" 
50418
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1131 
by (rule has_integral_unique) 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1132 
qed 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1133 

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

1134 
text {* 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1135 

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

1136 
For the positive integral we replace continuity with Borelmeasurability. 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1137 

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

1138 
*} 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1139 

56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1140 
lemma 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1141 
fixes f :: "real \<Rightarrow> real" 
50418
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1142 
assumes f_borel: "f \<in> borel_measurable borel" 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1143 
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" 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1144 
shows integral_FTC_Icc_nonneg: "(\<integral>x. f x * indicator {a .. b} x \<partial>lborel) = F b  F a" (is ?eq) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1145 
and integrable_FTC_Icc_nonneg: "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)" (is ?int) 
50418
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

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

1147 
have i: "(f has_integral F b  F a) {a..b}" 
56181
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56020
diff
changeset

1148 
by (intro fundamental_theorem_of_calculus) 
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56020
diff
changeset

1149 
(auto simp: has_field_derivative_iff_has_vector_derivative[symmetric] 
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56020
diff
changeset

1150 
intro: has_field_derivative_subset[OF f(1)] `a \<le> b`) 
50418
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1151 
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

1152 
by (rule has_integral_eq[OF _ i]) auto 
bd68cf816dd3
fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents:
50385
diff
changeset

1153 
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

1154 
by (rule has_integral_on_superset[OF _ _ i]) auto 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1155 
from i f f_borel show ?eq 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1156 
by (intro integral_has_integral_nonneg) (auto split: split_indicator) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1157 
from i f f_borel show ?int 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1158 
by (intro integrable_has_integral_nonneg) (auto split: split_indicator) 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1159 
qed 
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents:
56218
diff
changeset

1160 

56996  1161 
lemma nn_integral_FTC_atLeastAtMost: 
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from realvalued function to functions on realnormed vector spaces
hoelzl
parents: 