src/HOL/Analysis/Lebesgue_Measure.thy
author nipkow
Mon, 17 Oct 2016 11:46:22 +0200
changeset 64267 b9a1486e79be
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child 64272 f76b6dda2e56
permissions -rw-r--r--
setsum -> sum
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Title:      HOL/Analysis/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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    Author:     Jeremy Avigad
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    Author:     Luke Serafin
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*)
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section \<open>Lebesgue measure\<close>
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theory Lebesgue_Measure
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  imports Finite_Product_Measure Bochner_Integration Caratheodory Complete_Measure Summation_Tests Regularity
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begin
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lemma measure_eqI_lessThan:
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  fixes M N :: "real measure"
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  assumes sets: "sets M = sets borel" "sets N = sets borel"
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  assumes fin: "\<And>x. emeasure M {x <..} < \<infinity>"
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  assumes "\<And>x. emeasure M {x <..} = emeasure N {x <..}"
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  shows "M = N"
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proof (rule measure_eqI_generator_eq_countable)
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  let ?LT = "\<lambda>a::real. {a <..}" let ?E = "range ?LT"
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  show "Int_stable ?E"
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    by (auto simp: Int_stable_def lessThan_Int_lessThan)
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  show "?E \<subseteq> Pow UNIV" "sets M = sigma_sets UNIV ?E" "sets N = sigma_sets UNIV ?E"
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    unfolding sets borel_Ioi by auto
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  show "?LT`Rats \<subseteq> ?E" "(\<Union>i\<in>Rats. ?LT i) = UNIV" "\<And>a. a \<in> ?LT`Rats \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
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    using fin by (auto intro: Rats_no_bot_less simp: less_top)
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qed (auto intro: assms countable_rat)
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subsection \<open>Every right continuous and nondecreasing function gives rise to a measure\<close>
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definition interval_measure :: "(real \<Rightarrow> real) \<Rightarrow> real measure" where
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  "interval_measure F = extend_measure UNIV {(a, b). a \<le> b} (\<lambda>(a, b). {a <.. b}) (\<lambda>(a, b). ennreal (F b - F a))"
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lemma emeasure_interval_measure_Ioc:
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  assumes "a \<le> b"
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  assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
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  assumes right_cont_F : "\<And>a. continuous (at_right a) F"
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  shows "emeasure (interval_measure F) {a <.. b} = F b - F a"
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proof (rule extend_measure_caratheodory_pair[OF interval_measure_def \<open>a \<le> b\<close>])
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  show "semiring_of_sets UNIV {{a<..b} |a b :: real. a \<le> b}"
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  proof (unfold_locales, safe)
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    fix a b c d :: real assume *: "a \<le> b" "c \<le> d"
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    then show "\<exists>C\<subseteq>{{a<..b} |a b. a \<le> b}. finite C \<and> disjoint C \<and> {a<..b} - {c<..d} = \<Union>C"
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    proof cases
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      let ?C = "{{a<..b}}"
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      assume "b < c \<or> d \<le> a \<or> d \<le> c"
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      with * have "?C \<subseteq> {{a<..b} |a b. a \<le> b} \<and> finite ?C \<and> disjoint ?C \<and> {a<..b} - {c<..d} = \<Union>?C"
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        by (auto simp add: disjoint_def)
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      thus ?thesis ..
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    next
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      let ?C = "{{a<..c}, {d<..b}}"
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      assume "\<not> (b < c \<or> d \<le> a \<or> d \<le> c)"
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      with * have "?C \<subseteq> {{a<..b} |a b. a \<le> b} \<and> finite ?C \<and> disjoint ?C \<and> {a<..b} - {c<..d} = \<Union>?C"
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        by (auto simp add: disjoint_def Ioc_inj) (metis linear)+
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      thus ?thesis ..
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    qed
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  qed (auto simp: Ioc_inj, metis linear)
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next
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  fix l r :: "nat \<Rightarrow> real" and a b :: real
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  assume l_r[simp]: "\<And>n. l n \<le> r n" and "a \<le> b" and disj: "disjoint_family (\<lambda>n. {l n<..r n})"
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  assume lr_eq_ab: "(\<Union>i. {l i<..r i}) = {a<..b}"
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  have [intro, simp]: "\<And>a b. a \<le> b \<Longrightarrow> F a \<le> F b"
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    by (auto intro!: l_r mono_F)
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  { fix S :: "nat set" assume "finite S"
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    moreover note \<open>a \<le> b\<close>
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    moreover have "\<And>i. i \<in> S \<Longrightarrow> {l i <.. r i} \<subseteq> {a <.. b}"
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      unfolding lr_eq_ab[symmetric] by auto
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    ultimately have "(\<Sum>i\<in>S. F (r i) - F (l i)) \<le> F b - F a"
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    proof (induction S arbitrary: a rule: finite_psubset_induct)
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      case (psubset S)
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      show ?case
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      proof cases
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        assume "\<exists>i\<in>S. l i < r i"
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        with \<open>finite S\<close> have "Min (l ` {i\<in>S. l i < r i}) \<in> l ` {i\<in>S. l i < r i}"
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          by (intro Min_in) auto
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        then obtain m where m: "m \<in> S" "l m < r m" "l m = Min (l ` {i\<in>S. l i < r i})"
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          by fastforce
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        have "(\<Sum>i\<in>S. F (r i) - F (l i)) = (F (r m) - F (l m)) + (\<Sum>i\<in>S - {m}. F (r i) - F (l i))"
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          using m psubset by (intro sum.remove) auto
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        also have "(\<Sum>i\<in>S - {m}. F (r i) - F (l i)) \<le> F b - F (r m)"
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        proof (intro psubset.IH)
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          show "S - {m} \<subset> S"
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            using \<open>m\<in>S\<close> by auto
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          show "r m \<le> b"
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            using psubset.prems(2)[OF \<open>m\<in>S\<close>] \<open>l m < r m\<close> by auto
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        next
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          fix i assume "i \<in> S - {m}"
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          then have i: "i \<in> S" "i \<noteq> m" by auto
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          { assume i': "l i < r i" "l i < r m"
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            with \<open>finite S\<close> i m have "l m \<le> l i"
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              by auto
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    98
            with i' have "{l i <.. r i} \<inter> {l m <.. r m} \<noteq> {}"
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    99
              by auto
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            then have False
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   101
              using disjoint_family_onD[OF disj, of i m] i by auto }
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          then have "l i \<noteq> r i \<Longrightarrow> r m \<le> l i"
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   103
            unfolding not_less[symmetric] using l_r[of i] by auto
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          then show "{l i <.. r i} \<subseteq> {r m <.. b}"
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   105
            using psubset.prems(2)[OF \<open>i\<in>S\<close>] by auto
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        qed
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        also have "F (r m) - F (l m) \<le> F (r m) - F a"
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          using psubset.prems(2)[OF \<open>m \<in> S\<close>] \<open>l m < r m\<close>
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          by (auto simp add: Ioc_subset_iff intro!: mono_F)
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        finally show ?case
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          by (auto intro: add_mono)
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   112
      qed (auto simp add: \<open>a \<le> b\<close> less_le)
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    qed }
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   114
  note claim1 = this
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   115
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   116
  (* second key induction: a lower bound on the measures of any finite collection of Ai's
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   117
     that cover an interval {u..v} *)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   118
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   119
  { fix S u v and l r :: "nat \<Rightarrow> real"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   120
    assume "finite S" "\<And>i. i\<in>S \<Longrightarrow> l i < r i" "{u..v} \<subseteq> (\<Union>i\<in>S. {l i<..< r i})"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   121
    then have "F v - F u \<le> (\<Sum>i\<in>S. F (r i) - F (l i))"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   122
    proof (induction arbitrary: v u rule: finite_psubset_induct)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   123
      case (psubset S)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   124
      show ?case
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   125
      proof cases
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   126
        assume "S = {}" then show ?case
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   127
          using psubset by (simp add: mono_F)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   128
      next
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   129
        assume "S \<noteq> {}"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   130
        then obtain j where "j \<in> S"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   131
          by auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   132
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   133
        let ?R = "r j < u \<or> l j > v \<or> (\<exists>i\<in>S-{j}. l i \<le> l j \<and> r j \<le> r i)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   134
        show ?case
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   135
        proof cases
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   136
          assume "?R"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61762
diff changeset
   137
          with \<open>j \<in> S\<close> psubset.prems have "{u..v} \<subseteq> (\<Union>i\<in>S-{j}. {l i<..< r i})"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   138
            apply (auto simp: subset_eq Ball_def)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   139
            apply (metis Diff_iff less_le_trans leD linear singletonD)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   140
            apply (metis Diff_iff less_le_trans leD linear singletonD)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   141
            apply (metis order_trans less_le_not_le linear)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   142
            done
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61762
diff changeset
   143
          with \<open>j \<in> S\<close> have "F v - F u \<le> (\<Sum>i\<in>S - {j}. F (r i) - F (l i))"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   144
            by (intro psubset) auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   145
          also have "\<dots> \<le> (\<Sum>i\<in>S. F (r i) - F (l i))"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   146
            using psubset.prems
64267
b9a1486e79be setsum -> sum
nipkow
parents: 64008
diff changeset
   147
            by (intro sum_mono2 psubset) (auto intro: less_imp_le)
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   148
          finally show ?thesis .
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   149
        next
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   150
          assume "\<not> ?R"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   151
          then have j: "u \<le> r j" "l j \<le> v" "\<And>i. i \<in> S - {j} \<Longrightarrow> r i < r j \<or> l i > l j"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   152
            by (auto simp: not_less)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   153
          let ?S1 = "{i \<in> S. l i < l j}"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   154
          let ?S2 = "{i \<in> S. r i > r j}"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 38656
diff changeset
   155
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   156
          have "(\<Sum>i\<in>S. F (r i) - F (l i)) \<ge> (\<Sum>i\<in>?S1 \<union> ?S2 \<union> {j}. F (r i) - F (l i))"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61762
diff changeset
   157
            using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
64267
b9a1486e79be setsum -> sum
nipkow
parents: 64008
diff changeset
   158
            by (intro sum_mono2) (auto intro: less_imp_le)
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   159
          also have "(\<Sum>i\<in>?S1 \<union> ?S2 \<union> {j}. F (r i) - F (l i)) =
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   160
            (\<Sum>i\<in>?S1. F (r i) - F (l i)) + (\<Sum>i\<in>?S2 . F (r i) - F (l i)) + (F (r j) - F (l j))"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   161
            using psubset(1) psubset.prems(1) j
64267
b9a1486e79be setsum -> sum
nipkow
parents: 64008
diff changeset
   162
            apply (subst sum.union_disjoint)
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   163
            apply simp_all
64267
b9a1486e79be setsum -> sum
nipkow
parents: 64008
diff changeset
   164
            apply (subst sum.union_disjoint)
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   165
            apply auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   166
            apply (metis less_le_not_le)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   167
            done
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   168
          also (xtrans) have "(\<Sum>i\<in>?S1. F (r i) - F (l i)) \<ge> F (l j) - F u"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61762
diff changeset
   169
            using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   170
            apply (intro psubset.IH psubset)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   171
            apply (auto simp: subset_eq Ball_def)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   172
            apply (metis less_le_trans not_le)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   173
            done
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   174
          also (xtrans) have "(\<Sum>i\<in>?S2. F (r i) - F (l i)) \<ge> F v - F (r j)"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61762
diff changeset
   175
            using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   176
            apply (intro psubset.IH psubset)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   177
            apply (auto simp: subset_eq Ball_def)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   178
            apply (metis le_less_trans not_le)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   179
            done
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   180
          finally (xtrans) show ?case
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   181
            by (auto simp: add_mono)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   182
        qed
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   183
      qed
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   184
    qed }
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   185
  note claim2 = this
49777
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   186
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   187
  (* now prove the inequality going the other way *)
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   188
  have "ennreal (F b - F a) \<le> (\<Sum>i. ennreal (F (r i) - F (l i)))"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   189
  proof (rule ennreal_le_epsilon)
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   190
    fix epsilon :: real assume egt0: "epsilon > 0"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   191
    have "\<forall>i. \<exists>d>0. F (r i + d) < F (r i) + epsilon / 2^(i+2)"
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   192
    proof
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   193
      fix i
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   194
      note right_cont_F [of "r i"]
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   195
      thus "\<exists>d>0. F (r i + d) < F (r i) + epsilon / 2^(i+2)"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   196
        apply -
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   197
        apply (subst (asm) continuous_at_right_real_increasing)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   198
        apply (rule mono_F, assumption)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   199
        apply (drule_tac x = "epsilon / 2 ^ (i + 2)" in spec)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   200
        apply (erule impE)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   201
        using egt0 by (auto simp add: field_simps)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   202
    qed
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   203
    then obtain delta where
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   204
        deltai_gt0: "\<And>i. delta i > 0" and
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   205
        deltai_prop: "\<And>i. F (r i + delta i) < F (r i) + epsilon / 2^(i+2)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   206
      by metis
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   207
    have "\<exists>a' > a. F a' - F a < epsilon / 2"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   208
      apply (insert right_cont_F [of a])
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   209
      apply (subst (asm) continuous_at_right_real_increasing)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   210
      using mono_F apply force
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   211
      apply (drule_tac x = "epsilon / 2" in spec)
59554
4044f53326c9 inlined rules to free user-space from technical names
haftmann
parents: 59425
diff changeset
   212
      using egt0 unfolding mult.commute [of 2] by force
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   213
    then obtain a' where a'lea [arith]: "a' > a" and
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   214
      a_prop: "F a' - F a < epsilon / 2"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   215
      by auto
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62975
diff changeset
   216
    define S' where "S' = {i. l i < r i}"
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   217
    obtain S :: "nat set" where
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   218
      "S \<subseteq> S'" and finS: "finite S" and
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   219
      Sprop: "{a'..b} \<subseteq> (\<Union>i \<in> S. {l i<..<r i + delta i})"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   220
    proof (rule compactE_image)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   221
      show "compact {a'..b}"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   222
        by (rule compact_Icc)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   223
      show "\<forall>i \<in> S'. open ({l i<..<r i + delta i})" by auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   224
      have "{a'..b} \<subseteq> {a <.. b}"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   225
        by auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   226
      also have "{a <.. b} = (\<Union>i\<in>S'. {l i<..r i})"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   227
        unfolding lr_eq_ab[symmetric] by (fastforce simp add: S'_def intro: less_le_trans)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   228
      also have "\<dots> \<subseteq> (\<Union>i \<in> S'. {l i<..<r i + delta i})"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   229
        apply (intro UN_mono)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   230
        apply (auto simp: S'_def)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   231
        apply (cut_tac i=i in deltai_gt0)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   232
        apply simp
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   233
        done
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   234
      finally show "{a'..b} \<subseteq> (\<Union>i \<in> S'. {l i<..<r i + delta i})" .
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   235
    qed
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   236
    with S'_def have Sprop2: "\<And>i. i \<in> S \<Longrightarrow> l i < r i" by auto
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   237
    from finS have "\<exists>n. \<forall>i \<in> S. i \<le> n"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   238
      by (subst finite_nat_set_iff_bounded_le [symmetric])
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   239
    then obtain n where Sbound [rule_format]: "\<forall>i \<in> S. i \<le> n" ..
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   240
    have "F b - F a' \<le> (\<Sum>i\<in>S. F (r i + delta i) - F (l i))"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   241
      apply (rule claim2 [rule_format])
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   242
      using finS Sprop apply auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   243
      apply (frule Sprop2)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   244
      apply (subgoal_tac "delta i > 0")
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   245
      apply arith
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   246
      by (rule deltai_gt0)
61954
1d43f86f48be more symbols;
wenzelm
parents: 61945
diff changeset
   247
    also have "... \<le> (\<Sum>i \<in> S. F(r i) - F(l i) + epsilon / 2^(i+2))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 64008
diff changeset
   248
      apply (rule sum_mono)
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   249
      apply simp
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   250
      apply (rule order_trans)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   251
      apply (rule less_imp_le)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   252
      apply (rule deltai_prop)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   253
      by auto
61954
1d43f86f48be more symbols;
wenzelm
parents: 61945
diff changeset
   254
    also have "... = (\<Sum>i \<in> S. F(r i) - F(l i)) +
1d43f86f48be more symbols;
wenzelm
parents: 61945
diff changeset
   255
        (epsilon / 4) * (\<Sum>i \<in> S. (1 / 2)^i)" (is "_ = ?t + _")
64267
b9a1486e79be setsum -> sum
nipkow
parents: 64008
diff changeset
   256
      by (subst sum.distrib) (simp add: field_simps sum_distrib_left)
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   257
    also have "... \<le> ?t + (epsilon / 4) * (\<Sum> i < Suc n. (1 / 2)^i)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   258
      apply (rule add_left_mono)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   259
      apply (rule mult_left_mono)
64267
b9a1486e79be setsum -> sum
nipkow
parents: 64008
diff changeset
   260
      apply (rule sum_mono2)
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   261
      using egt0 apply auto
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   262
      by (frule Sbound, auto)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   263
    also have "... \<le> ?t + (epsilon / 2)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   264
      apply (rule add_left_mono)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   265
      apply (subst geometric_sum)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   266
      apply auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   267
      apply (rule mult_left_mono)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   268
      using egt0 apply auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   269
      done
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   270
    finally have aux2: "F b - F a' \<le> (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon / 2"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   271
      by simp
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50418
diff changeset
   272
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   273
    have "F b - F a = (F b - F a') + (F a' - F a)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   274
      by auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   275
    also have "... \<le> (F b - F a') + epsilon / 2"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   276
      using a_prop by (intro add_left_mono) simp
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   277
    also have "... \<le> (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon / 2 + epsilon / 2"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   278
      apply (intro add_right_mono)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   279
      apply (rule aux2)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   280
      done
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   281
    also have "... = (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   282
      by auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   283
    also have "... \<le> (\<Sum>i\<le>n. F (r i) - F (l i)) + epsilon"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 64008
diff changeset
   284
      using finS Sbound Sprop by (auto intro!: add_right_mono sum_mono3)
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   285
    finally have "ennreal (F b - F a) \<le> (\<Sum>i\<le>n. ennreal (F (r i) - F (l i))) + epsilon"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 64008
diff changeset
   286
      using egt0 by (simp add: ennreal_plus[symmetric] sum_nonneg del: ennreal_plus)
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   287
    then show "ennreal (F b - F a) \<le> (\<Sum>i. ennreal (F (r i) - F (l i))) + (epsilon :: real)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 64008
diff changeset
   288
      by (rule order_trans) (auto intro!: add_mono sum_le_suminf simp del: sum_ennreal)
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   289
  qed
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   290
  moreover have "(\<Sum>i. ennreal (F (r i) - F (l i))) \<le> ennreal (F b - F a)"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   291
    using \<open>a \<le> b\<close> by (auto intro!: suminf_le_const ennreal_le_iff[THEN iffD2] claim1)
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   292
  ultimately show "(\<Sum>n. ennreal (F (r n) - F (l n))) = ennreal (F b - F a)"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   293
    by (rule antisym[rotated])
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61610
diff changeset
   294
qed (auto simp: Ioc_inj mono_F)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents:
diff changeset
   295
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   296
lemma measure_interval_measure_Ioc:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   297
  assumes "a \<le> b"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   298
  assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   299
  assumes right_cont_F : "\<And>a. continuous (at_right a) F"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   300
  shows "measure (interval_measure F) {a <.. b} = F b - F a"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   301
  unfolding measure_def
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   302
  apply (subst emeasure_interval_measure_Ioc)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   303
  apply fact+
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   304
  apply (simp add: assms)
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   305
  done
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   306
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   307
lemma emeasure_interval_measure_Ioc_eq:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   308
  "(\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y) \<Longrightarrow> (\<And>a. continuous (at_right a) F) \<Longrightarrow>
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   309
    emeasure (interval_measure F) {a <.. b} = (if a \<le> b then F b - F a else 0)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   310
  using emeasure_interval_measure_Ioc[of a b F] by auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   311
59048
7dc8ac6f0895 add congruence solver to measurability prover
hoelzl
parents: 59000
diff changeset
   312
lemma sets_interval_measure [simp, measurable_cong]: "sets (interval_measure F) = sets borel"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   313
  apply (simp add: sets_extend_measure interval_measure_def borel_sigma_sets_Ioc)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   314
  apply (rule sigma_sets_eqI)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   315
  apply auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   316
  apply (case_tac "a \<le> ba")
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   317
  apply (auto intro: sigma_sets.Empty)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   318
  done
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   319
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   320
lemma space_interval_measure [simp]: "space (interval_measure F) = UNIV"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   321
  by (simp add: interval_measure_def space_extend_measure)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   322
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   323
lemma emeasure_interval_measure_Icc:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   324
  assumes "a \<le> b"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   325
  assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   326
  assumes cont_F : "continuous_on UNIV F"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   327
  shows "emeasure (interval_measure F) {a .. b} = F b - F a"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   328
proof (rule tendsto_unique)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   329
  { fix a b :: real assume "a \<le> b" then have "emeasure (interval_measure F) {a <.. b} = F b - F a"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   330
      using cont_F
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   331
      by (subst emeasure_interval_measure_Ioc)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   332
         (auto intro: mono_F continuous_within_subset simp: continuous_on_eq_continuous_within) }
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   333
  note * = this
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents:
diff changeset
   334
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   335
  let ?F = "interval_measure F"
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   336
  show "((\<lambda>a. F b - F a) \<longlongrightarrow> emeasure ?F {a..b}) (at_left a)"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   337
  proof (rule tendsto_at_left_sequentially)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   338
    show "a - 1 < a" by simp
61969
e01015e49041 more symbols;
wenzelm
parents: 61954
diff changeset
   339
    fix X assume "\<And>n. X n < a" "incseq X" "X \<longlonglongrightarrow> a"
e01015e49041 more symbols;
wenzelm
parents: 61954
diff changeset
   340
    with \<open>a \<le> b\<close> have "(\<lambda>n. emeasure ?F {X n<..b}) \<longlonglongrightarrow> emeasure ?F (\<Inter>n. {X n <..b})"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   341
      apply (intro Lim_emeasure_decseq)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   342
      apply (auto simp: decseq_def incseq_def emeasure_interval_measure_Ioc *)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   343
      apply force
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   344
      apply (subst (asm ) *)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   345
      apply (auto intro: less_le_trans less_imp_le)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   346
      done
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   347
    also have "(\<Inter>n. {X n <..b}) = {a..b}"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61762
diff changeset
   348
      using \<open>\<And>n. X n < a\<close>
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   349
      apply auto
61969
e01015e49041 more symbols;
wenzelm
parents: 61954
diff changeset
   350
      apply (rule LIMSEQ_le_const2[OF \<open>X \<longlonglongrightarrow> a\<close>])
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   351
      apply (auto intro: less_imp_le)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   352
      apply (auto intro: less_le_trans)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   353
      done
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   354
    also have "(\<lambda>n. emeasure ?F {X n<..b}) = (\<lambda>n. F b - F (X n))"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61762
diff changeset
   355
      using \<open>\<And>n. X n < a\<close> \<open>a \<le> b\<close> by (subst *) (auto intro: less_imp_le less_le_trans)
61969
e01015e49041 more symbols;
wenzelm
parents: 61954
diff changeset
   356
    finally show "(\<lambda>n. F b - F (X n)) \<longlonglongrightarrow> emeasure ?F {a..b}" .
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   357
  qed
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   358
  show "((\<lambda>a. ennreal (F b - F a)) \<longlongrightarrow> F b - F a) (at_left a)"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   359
    by (rule continuous_on_tendsto_compose[where g="\<lambda>x. x" and s=UNIV])
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   360
       (auto simp: continuous_on_ennreal continuous_on_diff cont_F continuous_on_const)
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   361
qed (rule trivial_limit_at_left_real)
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   362
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   363
lemma sigma_finite_interval_measure:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   364
  assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   365
  assumes right_cont_F : "\<And>a. continuous (at_right a) F"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   366
  shows "sigma_finite_measure (interval_measure F)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   367
  apply unfold_locales
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   368
  apply (intro exI[of _ "(\<lambda>(a, b). {a <.. b}) ` (\<rat> \<times> \<rat>)"])
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   369
  apply (auto intro!: Rats_no_top_le Rats_no_bot_less countable_rat simp: emeasure_interval_measure_Ioc_eq[OF assms])
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   370
  done
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   371
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61762
diff changeset
   372
subsection \<open>Lebesgue-Borel measure\<close>
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   373
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   374
definition lborel :: "('a :: euclidean_space) measure" where
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   375
  "lborel = distr (\<Pi>\<^sub>M b\<in>Basis. interval_measure (\<lambda>x. x)) borel (\<lambda>f. \<Sum>b\<in>Basis. f b *\<^sub>R b)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   376
63958
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   377
abbreviation lebesgue :: "'a::euclidean_space measure"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   378
  where "lebesgue \<equiv> completion lborel"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   379
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   380
abbreviation lebesgue_on :: "'a set \<Rightarrow> 'a::euclidean_space measure"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   381
  where "lebesgue_on \<Omega> \<equiv> restrict_space (completion lborel) \<Omega>"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   382
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   383
lemma
59048
7dc8ac6f0895 add congruence solver to measurability prover
hoelzl
parents: 59000
diff changeset
   384
  shows sets_lborel[simp, measurable_cong]: "sets lborel = sets borel"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   385
    and space_lborel[simp]: "space lborel = space borel"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   386
    and measurable_lborel1[simp]: "measurable M lborel = measurable M borel"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   387
    and measurable_lborel2[simp]: "measurable lborel M = measurable borel M"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   388
  by (simp_all add: lborel_def)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   389
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   390
context
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   391
begin
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   392
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   393
interpretation sigma_finite_measure "interval_measure (\<lambda>x. x)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   394
  by (rule sigma_finite_interval_measure) auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   395
interpretation finite_product_sigma_finite "\<lambda>_. interval_measure (\<lambda>x. x)" Basis
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   396
  proof qed simp
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   397
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   398
lemma lborel_eq_real: "lborel = interval_measure (\<lambda>x. x)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   399
  unfolding lborel_def Basis_real_def
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   400
  using distr_id[of "interval_measure (\<lambda>x. x)"]
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   401
  by (subst distr_component[symmetric])
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   402
     (simp_all add: distr_distr comp_def del: distr_id cong: distr_cong)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   403
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   404
lemma lborel_eq: "lborel = distr (\<Pi>\<^sub>M b\<in>Basis. lborel) borel (\<lambda>f. \<Sum>b\<in>Basis. f b *\<^sub>R b)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   405
  by (subst lborel_def) (simp add: lborel_eq_real)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   406
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   407
lemma nn_integral_lborel_setprod:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   408
  assumes [measurable]: "\<And>b. b \<in> Basis \<Longrightarrow> f b \<in> borel_measurable borel"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   409
  assumes nn[simp]: "\<And>b x. b \<in> Basis \<Longrightarrow> 0 \<le> f b x"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   410
  shows "(\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. f b (x \<bullet> b)) \<partial>lborel) = (\<Prod>b\<in>Basis. (\<integral>\<^sup>+x. f b x \<partial>lborel))"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   411
  by (simp add: lborel_def nn_integral_distr product_nn_integral_setprod
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   412
                product_nn_integral_singleton)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   413
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   414
lemma emeasure_lborel_Icc[simp]:
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   415
  fixes l u :: real
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   416
  assumes [simp]: "l \<le> u"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   417
  shows "emeasure lborel {l .. u} = u - l"
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
   418
proof -
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   419
  have "((\<lambda>f. f 1) -` {l..u} \<inter> space (Pi\<^sub>M {1} (\<lambda>b. interval_measure (\<lambda>x. x)))) = {1::real} \<rightarrow>\<^sub>E {l..u}"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   420
    by (auto simp: space_PiM)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   421
  then show ?thesis
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   422
    by (simp add: lborel_def emeasure_distr emeasure_PiM emeasure_interval_measure_Icc continuous_on_id)
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 50003
diff changeset
   423
qed
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 50003
diff changeset
   424
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   425
lemma emeasure_lborel_Icc_eq: "emeasure lborel {l .. u} = ennreal (if l \<le> u then u - l else 0)"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   426
  by simp
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   427
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   428
lemma emeasure_lborel_cbox[simp]:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   429
  assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   430
  shows "emeasure lborel (cbox l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
41654
32fe42892983 Gauge measure removed
hoelzl
parents: 41546
diff changeset
   431
proof -
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   432
  have "(\<lambda>x. \<Prod>b\<in>Basis. indicator {l\<bullet>b .. u\<bullet>b} (x \<bullet> b) :: ennreal) = indicator (cbox l u)"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   433
    by (auto simp: fun_eq_iff cbox_def split: split_indicator)
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   434
  then have "emeasure lborel (cbox l u) = (\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. indicator {l\<bullet>b .. u\<bullet>b} (x \<bullet> b)) \<partial>lborel)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   435
    by simp
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   436
  also have "\<dots> = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   437
    by (subst nn_integral_lborel_setprod) (simp_all add: setprod_ennreal inner_diff_left)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   438
  finally show ?thesis .
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents:
diff changeset
   439
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents:
diff changeset
   440
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   441
lemma AE_lborel_singleton: "AE x in lborel::'a::euclidean_space measure. x \<noteq> c"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   442
  using SOME_Basis AE_discrete_difference [of "{c}" lborel] emeasure_lborel_cbox [of c c]
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   443
  by (auto simp add: cbox_sing setprod_constant power_0_left)
47757
5e6fe71e2390 equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents: 47694
diff changeset
   444
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   445
lemma emeasure_lborel_Ioo[simp]:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   446
  assumes [simp]: "l \<le> u"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   447
  shows "emeasure lborel {l <..< u} = ennreal (u - l)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 38656
diff changeset
   448
proof -
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   449
  have "emeasure lborel {l <..< u} = emeasure lborel {l .. u}"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   450
    using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   451
  then show ?thesis
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   452
    by simp
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   453
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents:
diff changeset
   454
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   455
lemma emeasure_lborel_Ioc[simp]:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   456
  assumes [simp]: "l \<le> u"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   457
  shows "emeasure lborel {l <.. u} = ennreal (u - l)"
41654
32fe42892983 Gauge measure removed
hoelzl
parents: 41546
diff changeset
   458
proof -
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   459
  have "emeasure lborel {l <.. u} = emeasure lborel {l .. u}"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   460
    using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   461
  then show ?thesis
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   462
    by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents:
diff changeset
   463
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents:
diff changeset
   464
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   465
lemma emeasure_lborel_Ico[simp]:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   466
  assumes [simp]: "l \<le> u"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   467
  shows "emeasure lborel {l ..< u} = ennreal (u - l)"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   468
proof -
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   469
  have "emeasure lborel {l ..< u} = emeasure lborel {l .. u}"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   470
    using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   471
  then show ?thesis
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   472
    by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents:
diff changeset
   473
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents:
diff changeset
   474
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   475
lemma emeasure_lborel_box[simp]:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   476
  assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   477
  shows "emeasure lborel (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   478
proof -
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   479
  have "(\<lambda>x. \<Prod>b\<in>Basis. indicator {l\<bullet>b <..< u\<bullet>b} (x \<bullet> b) :: ennreal) = indicator (box l u)"
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   480
    by (auto simp: fun_eq_iff box_def split: split_indicator)
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   481
  then have "emeasure lborel (box l u) = (\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. indicator {l\<bullet>b <..< u\<bullet>b} (x \<bullet> b)) \<partial>lborel)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   482
    by simp
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   483
  also have "\<dots> = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   484
    by (subst nn_integral_lborel_setprod) (simp_all add: setprod_ennreal inner_diff_left)
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   485
  finally show ?thesis .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 38656
diff changeset
   486
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents:
diff changeset
   487
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   488
lemma emeasure_lborel_cbox_eq:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   489
  "emeasure lborel (cbox l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   490
  using box_eq_empty(2)[THEN iffD2, of u l] by (auto simp: not_le)
41654
32fe42892983 Gauge measure removed
hoelzl
parents: 41546
diff changeset
   491
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   492
lemma emeasure_lborel_box_eq:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   493
  "emeasure lborel (box l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   494
  using box_eq_empty(1)[THEN iffD2, of u l] by (auto simp: not_le dest!: less_imp_le) force
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 38656
diff changeset
   495
63886
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   496
lemma emeasure_lborel_singleton[simp]: "emeasure lborel {x} = 0"
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   497
  using emeasure_lborel_cbox[of x x] nonempty_Basis
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   498
  by (auto simp del: emeasure_lborel_cbox nonempty_Basis simp add: cbox_sing setprod_constant)
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   499
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 38656
diff changeset
   500
lemma
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   501
  fixes l u :: real
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   502
  assumes [simp]: "l \<le> u"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   503
  shows measure_lborel_Icc[simp]: "measure lborel {l .. u} = u - l"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   504
    and measure_lborel_Ico[simp]: "measure lborel {l ..< u} = u - l"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   505
    and measure_lborel_Ioc[simp]: "measure lborel {l <.. u} = u - l"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   506
    and measure_lborel_Ioo[simp]: "measure lborel {l <..< u} = u - l"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   507
  by (simp_all add: measure_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 38656
diff changeset
   508
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   509
lemma
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   510
  assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   511
  shows measure_lborel_box[simp]: "measure lborel (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   512
    and measure_lborel_cbox[simp]: "measure lborel (cbox l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   513
  by (simp_all add: measure_def inner_diff_left setprod_nonneg)
41654
32fe42892983 Gauge measure removed
hoelzl
parents: 41546
diff changeset
   514
63886
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   515
lemma measure_lborel_cbox_eq:
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   516
  "measure lborel (cbox l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   517
  using box_eq_empty(2)[THEN iffD2, of u l] by (auto simp: not_le)
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   518
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   519
lemma measure_lborel_box_eq:
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   520
  "measure lborel (box l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   521
  using box_eq_empty(1)[THEN iffD2, of u l] by (auto simp: not_le dest!: less_imp_le) force
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   522
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   523
lemma measure_lborel_singleton[simp]: "measure lborel {x} = 0"
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   524
  by (simp add: measure_def)
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   525
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   526
lemma sigma_finite_lborel: "sigma_finite_measure lborel"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   527
proof
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   528
  show "\<exists>A::'a set set. countable A \<and> A \<subseteq> sets lborel \<and> \<Union>A = space lborel \<and> (\<forall>a\<in>A. emeasure lborel a \<noteq> \<infinity>)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   529
    by (intro exI[of _ "range (\<lambda>n::nat. box (- real n *\<^sub>R One) (real n *\<^sub>R One))"])
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   530
       (auto simp: emeasure_lborel_cbox_eq UN_box_eq_UNIV)
49777
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   531
qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 38656
diff changeset
   532
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   533
end
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
   534
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   535
lemma emeasure_lborel_UNIV: "emeasure lborel (UNIV::'a::euclidean_space set) = \<infinity>"
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   536
proof -
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   537
  { fix n::nat
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   538
    let ?Ba = "Basis :: 'a set"
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   539
    have "real n \<le> (2::real) ^ card ?Ba * real n"
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   540
      by (simp add: mult_le_cancel_right1)
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   541
    also
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   542
    have "... \<le> (2::real) ^ card ?Ba * real (Suc n) ^ card ?Ba"
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   543
      apply (rule mult_left_mono)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61284
diff changeset
   544
      apply (metis DIM_positive One_nat_def less_eq_Suc_le less_imp_le of_nat_le_iff of_nat_power self_le_power zero_less_Suc)
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   545
      apply (simp add: DIM_positive)
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   546
      done
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   547
    finally have "real n \<le> (2::real) ^ card ?Ba * real (Suc n) ^ card ?Ba" .
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   548
  } note [intro!] = this
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   549
  show ?thesis
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   550
    unfolding UN_box_eq_UNIV[symmetric]
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   551
    apply (subst SUP_emeasure_incseq[symmetric])
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   552
    apply (auto simp: incseq_def subset_box inner_add_left setprod_constant
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   553
      simp del: Sup_eq_top_iff SUP_eq_top_iff
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   554
      intro!: ennreal_SUP_eq_top)
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   555
    done
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59554
diff changeset
   556
qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 38656
diff changeset
   557
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   558
lemma emeasure_lborel_countable:
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   559
  fixes A :: "'a::euclidean_space set"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   560
  assumes "countable A"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   561
  shows "emeasure lborel A = 0"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   562
proof -
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   563
  have "A \<subseteq> (\<Union>i. {from_nat_into A i})" using from_nat_into_surj assms by force
63262
e497387de7af remove smt call in Lebesge_Measure
hoelzl
parents: 63040
diff changeset
   564
  then have "emeasure lborel A \<le> emeasure lborel (\<Union>i. {from_nat_into A i})"
e497387de7af remove smt call in Lebesge_Measure
hoelzl
parents: 63040
diff changeset
   565
    by (intro emeasure_mono) auto
e497387de7af remove smt call in Lebesge_Measure
hoelzl
parents: 63040
diff changeset
   566
  also have "emeasure lborel (\<Union>i. {from_nat_into A i}) = 0"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   567
    by (rule emeasure_UN_eq_0) auto
63262
e497387de7af remove smt call in Lebesge_Measure
hoelzl
parents: 63040
diff changeset
   568
  finally show ?thesis
e497387de7af remove smt call in Lebesge_Measure
hoelzl
parents: 63040
diff changeset
   569
    by (auto simp add: )
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 38656
diff changeset
   570
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 38656
diff changeset
   571
59425
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   572
lemma countable_imp_null_set_lborel: "countable A \<Longrightarrow> A \<in> null_sets lborel"
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   573
  by (simp add: null_sets_def emeasure_lborel_countable sets.countable)
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   574
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   575
lemma finite_imp_null_set_lborel: "finite A \<Longrightarrow> A \<in> null_sets lborel"
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   576
  by (intro countable_imp_null_set_lborel countable_finite)
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   577
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   578
lemma lborel_neq_count_space[simp]: "lborel \<noteq> count_space (A::('a::ordered_euclidean_space) set)"
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   579
proof
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   580
  assume asm: "lborel = count_space A"
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   581
  have "space lborel = UNIV" by simp
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   582
  hence [simp]: "A = UNIV" by (subst (asm) asm) (simp only: space_count_space)
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   583
  have "emeasure lborel {undefined::'a} = 1"
59425
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   584
      by (subst asm, subst emeasure_count_space_finite) auto
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   585
  moreover have "emeasure lborel {undefined} \<noteq> 1" by simp
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   586
  ultimately show False by contradiction
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   587
qed
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   588
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61762
diff changeset
   589
subsection \<open>Affine transformation on the Lebesgue-Borel\<close>
49777
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   590
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   591
lemma lborel_eqI:
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   592
  fixes M :: "'a::euclidean_space measure"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   593
  assumes emeasure_eq: "\<And>l u. (\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b) \<Longrightarrow> emeasure M (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
49777
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   594
  assumes sets_eq: "sets M = sets borel"
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   595
  shows "lborel = M"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   596
proof (rule measure_eqI_generator_eq)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   597
  let ?E = "range (\<lambda>(a, b). box a b::'a set)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   598
  show "Int_stable ?E"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   599
    by (auto simp: Int_stable_def box_Int_box)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   600
49777
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   601
  show "?E \<subseteq> Pow UNIV" "sets lborel = sigma_sets UNIV ?E" "sets M = sigma_sets UNIV ?E"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   602
    by (simp_all add: borel_eq_box sets_eq)
49777
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   603
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   604
  let ?A = "\<lambda>n::nat. box (- (real n *\<^sub>R One)) (real n *\<^sub>R One) :: 'a set"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   605
  show "range ?A \<subseteq> ?E" "(\<Union>i. ?A i) = UNIV"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   606
    unfolding UN_box_eq_UNIV by auto
49777
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   607
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   608
  { fix i show "emeasure lborel (?A i) \<noteq> \<infinity>" by auto }
49777
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   609
  { fix X assume "X \<in> ?E" then show "emeasure lborel X = emeasure M X"
63886
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   610
      apply (auto simp: emeasure_eq emeasure_lborel_box_eq)
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   611
      apply (subst box_eq_empty(1)[THEN iffD2])
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   612
      apply (auto intro: less_imp_le simp: not_le)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   613
      done }
49777
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   614
qed
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   615
63886
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   616
lemma lborel_affine_euclidean:
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   617
  fixes c :: "'a::euclidean_space \<Rightarrow> real" and t
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   618
  defines "T x \<equiv> t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j)"
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   619
  assumes c: "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0"
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   620
  shows "lborel = density (distr lborel borel T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))" (is "_ = ?D")
49777
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   621
proof (rule lborel_eqI)
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   622
  let ?B = "Basis :: 'a set"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   623
  fix l u assume le: "\<And>b. b \<in> ?B \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
63886
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   624
  have [measurable]: "T \<in> borel \<rightarrow>\<^sub>M borel"
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   625
    by (simp add: T_def[abs_def])
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   626
  have eq: "T -` box l u = box
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   627
    (\<Sum>j\<in>Basis. (((if 0 < c j then l - t else u - t) \<bullet> j) / c j) *\<^sub>R j)
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   628
    (\<Sum>j\<in>Basis. (((if 0 < c j then u - t else l - t) \<bullet> j) / c j) *\<^sub>R j)"
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   629
    using c by (auto simp: box_def T_def field_simps inner_simps divide_less_eq)
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   630
  with le c show "emeasure ?D (box l u) = (\<Prod>b\<in>?B. (u - l) \<bullet> b)"
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   631
    by (auto simp: emeasure_density emeasure_distr nn_integral_multc emeasure_lborel_box_eq inner_simps
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   632
                   field_simps divide_simps ennreal_mult'[symmetric] setprod_nonneg setprod.distrib[symmetric]
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   633
             intro!: setprod.cong)
49777
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   634
qed simp
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   635
63886
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   636
lemma lborel_affine:
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   637
  fixes t :: "'a::euclidean_space"
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   638
  shows "c \<noteq> 0 \<Longrightarrow> lborel = density (distr lborel borel (\<lambda>x. t + c *\<^sub>R x)) (\<lambda>_. \<bar>c\<bar>^DIM('a))"
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   639
  using lborel_affine_euclidean[where c="\<lambda>_::'a. c" and t=t]
64267
b9a1486e79be setsum -> sum
nipkow
parents: 64008
diff changeset
   640
  unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric] euclidean_representation setprod_constant by simp
63886
685fb01256af move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents: 63627
diff changeset
   641
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   642
lemma lborel_real_affine:
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   643
  "c \<noteq> 0 \<Longrightarrow> lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. ennreal (abs c))"
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   644
  using lborel_affine[of c t] by simp
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   645
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   646
lemma AE_borel_affine:
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   647
  fixes P :: "real \<Rightarrow> bool"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   648
  shows "c \<noteq> 0 \<Longrightarrow> Measurable.pred borel P \<Longrightarrow> AE x in lborel. P x \<Longrightarrow> AE x in lborel. P (t + c * x)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   649
  by (subst lborel_real_affine[where t="- t / c" and c="1 / c"])
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   650
     (simp_all add: AE_density AE_distr_iff field_simps)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   651
56996
891e992e510f renamed positive_integral to nn_integral
hoelzl
parents: 56993
diff changeset
   652
lemma nn_integral_real_affine:
56993
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   653
  fixes c :: real assumes [measurable]: "f \<in> borel_measurable borel" and c: "c \<noteq> 0"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   654
  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 real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   655
  by (subst lborel_real_affine[OF c, of t])
56996
891e992e510f renamed positive_integral to nn_integral
hoelzl
parents: 56993
diff changeset
   656
     (simp add: nn_integral_density nn_integral_distr nn_integral_cmult)
56993
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   657
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   658
lemma lborel_integrable_real_affine:
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   659
  fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
56993
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   660
  assumes f: "integrable lborel f"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   661
  shows "c \<noteq> 0 \<Longrightarrow> integrable lborel (\<lambda>x. f (t + c * x))"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   662
  using f f[THEN borel_measurable_integrable] unfolding integrable_iff_bounded
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   663
  by (subst (asm) nn_integral_real_affine[where c=c and t=t]) (auto simp: ennreal_mult_less_top)
56993
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   664
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   665
lemma lborel_integrable_real_affine_iff:
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   666
  fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   667
  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 real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   668
  using
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   669
    lborel_integrable_real_affine[of f c t]
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   670
    lborel_integrable_real_affine[of "\<lambda>x. f (t + c * x)" "1/c" "-t/c"]
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   671
  by (auto simp add: field_simps)
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   672
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   673
lemma lborel_integral_real_affine:
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   674
  fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}" and c :: real
57166
5cfcc616d485 use 0 as integral-value for non-integrable functions, simplify a couple of rewrite rules
hoelzl
parents: 57138
diff changeset
   675
  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 integral-value for non-integrable functions, simplify a couple of rewrite rules
hoelzl
parents: 57138
diff changeset
   676
proof cases
5cfcc616d485 use 0 as integral-value for non-integrable functions, simplify a couple of rewrite rules
hoelzl
parents: 57138
diff changeset
   677
  assume f[measurable]: "integrable lborel f" then show ?thesis
5cfcc616d485 use 0 as integral-value for non-integrable functions, simplify a couple of rewrite rules
hoelzl
parents: 57138
diff changeset
   678
    using c f f[THEN borel_measurable_integrable] f[THEN lborel_integrable_real_affine, of c t]
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   679
    by (subst lborel_real_affine[OF c, of t])
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   680
       (simp add: integral_density integral_distr)
57166
5cfcc616d485 use 0 as integral-value for non-integrable functions, simplify a couple of rewrite rules
hoelzl
parents: 57138
diff changeset
   681
next
5cfcc616d485 use 0 as integral-value for non-integrable functions, simplify a couple of rewrite rules
hoelzl
parents: 57138
diff changeset
   682
  assume "\<not> integrable lborel f" with c show ?thesis
5cfcc616d485 use 0 as integral-value for non-integrable functions, simplify a couple of rewrite rules
hoelzl
parents: 57138
diff changeset
   683
    by (simp add: lborel_integrable_real_affine_iff not_integrable_integral_eq)
5cfcc616d485 use 0 as integral-value for non-integrable functions, simplify a couple of rewrite rules
hoelzl
parents: 57138
diff changeset
   684
qed
56993
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   685
63958
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   686
lemma
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   687
  fixes c :: "'a::euclidean_space \<Rightarrow> real" and t
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   688
  assumes c: "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   689
  defines "T == (\<lambda>x. t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j))"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   690
  shows lebesgue_affine_euclidean: "lebesgue = density (distr lebesgue lebesgue T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))" (is "_ = ?D")
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   691
    and lebesgue_affine_measurable: "T \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   692
proof -
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   693
  have T_borel[measurable]: "T \<in> borel \<rightarrow>\<^sub>M borel"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   694
    by (auto simp: T_def[abs_def])
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   695
  { fix A :: "'a set" assume A: "A \<in> sets borel"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   696
    then have "emeasure lborel A = 0 \<longleftrightarrow> emeasure (density (distr lborel borel T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))) A = 0"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   697
      unfolding T_def using c by (subst lborel_affine_euclidean[symmetric]) auto
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   698
    also have "\<dots> \<longleftrightarrow> emeasure (distr lebesgue lborel T) A = 0"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   699
      using A c by (simp add: distr_completion emeasure_density nn_integral_cmult setprod_nonneg cong: distr_cong)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   700
    finally have "emeasure lborel A = 0 \<longleftrightarrow> emeasure (distr lebesgue lborel T) A = 0" . }
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   701
  then have eq: "null_sets lborel = null_sets (distr lebesgue lborel T)"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   702
    by (auto simp: null_sets_def)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   703
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   704
  show "T \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   705
    by (rule completion.measurable_completion2) (auto simp: eq measurable_completion)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   706
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   707
  have "lebesgue = completion (density (distr lborel borel T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>)))"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   708
    using c by (subst lborel_affine_euclidean[of c t]) (simp_all add: T_def[abs_def])
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   709
  also have "\<dots> = density (completion (distr lebesgue lborel T)) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   710
    using c by (auto intro!: always_eventually setprod_pos completion_density_eq simp: distr_completion cong: distr_cong)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   711
  also have "\<dots> = density (distr lebesgue lebesgue T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   712
    by (subst completion.completion_distr_eq) (auto simp: eq measurable_completion)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   713
  finally show "lebesgue = density (distr lebesgue lebesgue T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))" .
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   714
qed
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   715
63959
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   716
lemma lebesgue_measurable_scaling[measurable]: "op *\<^sub>R x \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   717
proof cases
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   718
  assume "x = 0"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   719
  then have "op *\<^sub>R x = (\<lambda>x. 0::'a)"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   720
    by (auto simp: fun_eq_iff)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   721
  then show ?thesis by auto
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   722
next
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   723
  assume "x \<noteq> 0" then show ?thesis
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   724
    using lebesgue_affine_measurable[of "\<lambda>_. x" 0]
64267
b9a1486e79be setsum -> sum
nipkow
parents: 64008
diff changeset
   725
    unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric] euclidean_representation
63959
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   726
    by (auto simp add: ac_simps)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   727
qed
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   728
63958
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   729
lemma
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   730
  fixes m :: real and \<delta> :: "'a::euclidean_space"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   731
  defines "T r d x \<equiv> r *\<^sub>R x + d"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   732
  shows emeasure_lebesgue_affine: "emeasure lebesgue (T m \<delta> ` S) = \<bar>m\<bar> ^ DIM('a) * emeasure lebesgue S" (is ?e)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   733
    and measure_lebesgue_affine: "measure lebesgue (T m \<delta> ` S) = \<bar>m\<bar> ^ DIM('a) * measure lebesgue S" (is ?m)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   734
proof -
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   735
  show ?e
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   736
  proof cases
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   737
    assume "m = 0" then show ?thesis
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   738
      by (simp add: image_constant_conv T_def[abs_def])
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   739
  next
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   740
    let ?T = "T m \<delta>" and ?T' = "T (1 / m) (- ((1/m) *\<^sub>R \<delta>))"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   741
    assume "m \<noteq> 0"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   742
    then have s_comp_s: "?T' \<circ> ?T = id" "?T \<circ> ?T' = id"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   743
      by (auto simp: T_def[abs_def] fun_eq_iff scaleR_add_right scaleR_diff_right)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   744
    then have "inv ?T' = ?T" "bij ?T'"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   745
      by (auto intro: inv_unique_comp o_bij)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   746
    then have eq: "T m \<delta> ` S = T (1 / m) ((-1/m) *\<^sub>R \<delta>) -` S \<inter> space lebesgue"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   747
      using bij_vimage_eq_inv_image[OF \<open>bij ?T'\<close>, of S] by auto
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   748
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   749
    have trans_eq_T: "(\<lambda>x. \<delta> + (\<Sum>j\<in>Basis. (m * (x \<bullet> j)) *\<^sub>R j)) = T m \<delta>" for m \<delta>
64267
b9a1486e79be setsum -> sum
nipkow
parents: 64008
diff changeset
   750
      unfolding T_def[abs_def] scaleR_scaleR[symmetric] scaleR_sum_right[symmetric]
63958
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   751
      by (auto simp add: euclidean_representation ac_simps)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   752
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   753
    have T[measurable]: "T r d \<in> lebesgue \<rightarrow>\<^sub>M lebesgue" for r d
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   754
      using lebesgue_affine_measurable[of "\<lambda>_. r" d]
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   755
      by (cases "r = 0") (auto simp: trans_eq_T T_def[abs_def])
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   756
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   757
    show ?thesis
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   758
    proof cases
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   759
      assume "S \<in> sets lebesgue" with \<open>m \<noteq> 0\<close> show ?thesis
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   760
        unfolding eq
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   761
        apply (subst lebesgue_affine_euclidean[of "\<lambda>_. m" \<delta>])
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   762
        apply (simp_all add: emeasure_density trans_eq_T nn_integral_cmult emeasure_distr
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   763
                        del: space_completion emeasure_completion)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   764
        apply (simp add: vimage_comp s_comp_s setprod_constant)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   765
        done
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   766
    next
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   767
      assume "S \<notin> sets lebesgue"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   768
      moreover have "?T ` S \<notin> sets lebesgue"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   769
      proof
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   770
        assume "?T ` S \<in> sets lebesgue"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   771
        then have "?T -` (?T ` S) \<inter> space lebesgue \<in> sets lebesgue"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   772
          by (rule measurable_sets[OF T])
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   773
        also have "?T -` (?T ` S) \<inter> space lebesgue = S"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   774
          by (simp add: vimage_comp s_comp_s eq)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   775
        finally show False using \<open>S \<notin> sets lebesgue\<close> by auto
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   776
      qed
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   777
      ultimately show ?thesis
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   778
        by (simp add: emeasure_notin_sets)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   779
    qed
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   780
  qed
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   781
  show ?m
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   782
    unfolding measure_def \<open>?e\<close> by (simp add: enn2real_mult setprod_nonneg)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   783
qed
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   784
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   785
lemma divideR_right:
56993
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   786
  fixes x y :: "'a::real_normed_vector"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   787
  shows "r \<noteq> 0 \<Longrightarrow> y = x /\<^sub>R r \<longleftrightarrow> r *\<^sub>R y = x"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   788
  using scaleR_cancel_left[of r y "x /\<^sub>R r"] by simp
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   789
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   790
lemma lborel_has_bochner_integral_real_affine_iff:
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   791
  fixes x :: "'a :: {banach, second_countable_topology}"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   792
  shows "c \<noteq> 0 \<Longrightarrow>
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   793
    has_bochner_integral lborel f x \<longleftrightarrow>
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   794
    has_bochner_integral lborel (\<lambda>x. f (t + c * x)) (x /\<^sub>R \<bar>c\<bar>)"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   795
  unfolding has_bochner_integral_iff lborel_integrable_real_affine_iff
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   796
  by (simp_all add: lborel_integral_real_affine[symmetric] divideR_right cong: conj_cong)
49777
6ac97ab9b295 tuned Lebesgue measure proofs
hoelzl
parents: 47757
diff changeset
   797
59425
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   798
lemma lborel_distr_uminus: "distr lborel borel uminus = (lborel :: real measure)"
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   799
  by (subst lborel_real_affine[of "-1" 0])
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   800
     (auto simp: density_1 one_ennreal_def[symmetric])
59425
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   801
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   802
lemma lborel_distr_mult:
59425
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   803
  assumes "(c::real) \<noteq> 0"
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   804
  shows "distr lborel borel (op * c) = density lborel (\<lambda>_. inverse \<bar>c\<bar>)"
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   805
proof-
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   806
  have "distr lborel borel (op * c) = distr lborel lborel (op * c)" by (simp cong: distr_cong)
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   807
  also from assms have "... = density lborel (\<lambda>_. inverse \<bar>c\<bar>)"
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   808
    by (subst lborel_real_affine[of "inverse c" 0]) (auto simp: o_def distr_density_distr)
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   809
  finally show ?thesis .
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   810
qed
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   811
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   812
lemma lborel_distr_mult':
59425
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   813
  assumes "(c::real) \<noteq> 0"
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61808
diff changeset
   814
  shows "lborel = density (distr lborel borel (op * c)) (\<lambda>_. \<bar>c\<bar>)"
59425
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   815
proof-
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   816
  have "lborel = density lborel (\<lambda>_. 1)" by (rule density_1[symmetric])
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   817
  also from assms have "(\<lambda>_. 1 :: ennreal) = (\<lambda>_. inverse \<bar>c\<bar> * \<bar>c\<bar>)" by (intro ext) simp
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61808
diff changeset
   818
  also have "density lborel ... = density (density lborel (\<lambda>_. inverse \<bar>c\<bar>)) (\<lambda>_. \<bar>c\<bar>)"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   819
    by (subst density_density_eq) (auto simp: ennreal_mult)
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61808
diff changeset
   820
  also from assms have "density lborel (\<lambda>_. inverse \<bar>c\<bar>) = distr lborel borel (op * c)"
59425
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   821
    by (rule lborel_distr_mult[symmetric])
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   822
  finally show ?thesis .
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   823
qed
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   824
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   825
lemma lborel_distr_plus: "distr lborel borel (op + c) = (lborel :: real measure)"
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   826
  by (subst lborel_real_affine[of 1 c]) (auto simp: density_1 one_ennreal_def[symmetric])
59425
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   827
61605
1bf7b186542e qualifier is mandatory by default;
wenzelm
parents: 61284
diff changeset
   828
interpretation lborel: sigma_finite_measure lborel
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   829
  by (rule sigma_finite_lborel)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   830
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   831
interpretation lborel_pair: pair_sigma_finite lborel lborel ..
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   832
59425
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   833
lemma lborel_prod:
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   834
  "lborel \<Otimes>\<^sub>M lborel = (lborel :: ('a::euclidean_space \<times> 'b::euclidean_space) measure)"
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   835
proof (rule lborel_eqI[symmetric], clarify)
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   836
  fix la ua :: 'a and lb ub :: 'b
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   837
  assume lu: "\<And>a b. (a, b) \<in> Basis \<Longrightarrow> (la, lb) \<bullet> (a, b) \<le> (ua, ub) \<bullet> (a, b)"
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   838
  have [simp]:
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   839
    "\<And>b. b \<in> Basis \<Longrightarrow> la \<bullet> b \<le> ua \<bullet> b"
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   840
    "\<And>b. b \<in> Basis \<Longrightarrow> lb \<bullet> b \<le> ub \<bullet> b"
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   841
    "inj_on (\<lambda>u. (u, 0)) Basis" "inj_on (\<lambda>u. (0, u)) Basis"
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   842
    "(\<lambda>u. (u, 0)) ` Basis \<inter> (\<lambda>u. (0, u)) ` Basis = {}"
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   843
    "box (la, lb) (ua, ub) = box la ua \<times> box lb ub"
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   844
    using lu[of _ 0] lu[of 0] by (auto intro!: inj_onI simp add: Basis_prod_def ball_Un box_def)
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   845
  show "emeasure (lborel \<Otimes>\<^sub>M lborel) (box (la, lb) (ua, ub)) =
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   846
      ennreal (setprod (op \<bullet> ((ua, ub) - (la, lb))) Basis)"
59425
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   847
    by (simp add: lborel.emeasure_pair_measure_Times Basis_prod_def setprod.union_disjoint
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   848
                  setprod.reindex ennreal_mult inner_diff_left setprod_nonneg)
59425
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   849
qed (simp add: borel_prod[symmetric])
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 59357
diff changeset
   850
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   851
(* FIXME: conversion in measurable prover *)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   852
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
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   853
lemma lborelD[measurable (raw)]: "A \<in> sets borel \<Longrightarrow> A \<in> sets lborel" by simp
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   854
57138
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   855
lemma emeasure_bounded_finite:
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   856
  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
   857
proof -
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61762
diff changeset
   858
  from bounded_subset_cbox[OF \<open>bounded A\<close>] obtain a b where "A \<subseteq> cbox a b"
57138
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   859
    by auto
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   860
  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
   861
    by (intro emeasure_mono) auto
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   862
  then show ?thesis
62975
1d066f6ab25d Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents: 62390
diff changeset
   863
    by (auto simp: emeasure_lborel_cbox_eq setprod_nonneg less_top[symmetric] top_unique split: if_split_asm)
57138
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   864
qed
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   865
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   866
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
   867
  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
   868
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   869
lemma borel_integrable_compact:
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57275
diff changeset
   870
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{banach, second_countable_topology}"
57138
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   871
  assumes "compact S" "continuous_on S f"
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   872
  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
   873
proof cases
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   874
  assume "S \<noteq> {}"
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   875
  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
   876
    using assms by (intro continuous_intros)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61762
diff changeset
   877
  from continuous_attains_sup[OF \<open>compact S\<close> \<open>S \<noteq> {}\<close> this]
57138
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   878
  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
   879
    by auto
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   880
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   881
  show ?thesis
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   882
  proof (rule integrable_bound)
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   883
    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
   884
      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
   885
    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
   886
      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
   887
    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
   888
      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
   889
  qed
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   890
qed simp
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   891
50418
bd68cf816dd3 fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents: 50385
diff changeset
   892
lemma borel_integrable_atLeastAtMost:
56993
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56218
diff changeset
   893
  fixes f :: "real \<Rightarrow> real"
50418
bd68cf816dd3 fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents: 50385
diff changeset
   894
  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
   895
  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
   896
proof -
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   897
  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
   898
  proof (rule borel_integrable_compact)
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   899
    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
   900
      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
   901
  qed simp
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   902
  then show ?thesis
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57447
diff changeset
   903
    by (auto simp: mult.commute)
57138
7b3146180291 generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents: 57137
diff changeset
   904
qed
50418
bd68cf816dd3 fundamental theorem of calculus for the Lebesgue integral
hoelzl
parents: 50385
diff changeset
   905
63958
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   906
abbreviation lmeasurable :: "'a::euclidean_space set set"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   907
where
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   908
  "lmeasurable \<equiv> fmeasurable lebesgue"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   909
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   910
lemma lmeasurable_iff_integrable:
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   911
  "S \<in> lmeasurable \<longleftrightarrow> integrable lebesgue (indicator S :: 'a::euclidean_space \<Rightarrow> real)"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   912
  by (auto simp: fmeasurable_def integrable_iff_bounded borel_measurable_indicator_iff ennreal_indicator)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   913
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   914
lemma lmeasurable_cbox [iff]: "cbox a b \<in> lmeasurable"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   915
  and lmeasurable_box [iff]: "box a b \<in> lmeasurable"
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   916
  by (auto simp: fmeasurable_def emeasure_lborel_box_eq emeasure_lborel_cbox_eq)
02de4a58e210 HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
hoelzl
parents: 63918
diff changeset
   917
63959
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   918
lemma lmeasurable_compact: "compact S \<Longrightarrow> S \<in> lmeasurable"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   919
  using emeasure_compact_finite[of S] by (intro fmeasurableI) (auto simp: borel_compact)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   920
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   921
lemma lmeasurable_open: "bounded S \<Longrightarrow> open S \<Longrightarrow> S \<in> lmeasurable"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   922
  using emeasure_bounded_finite[of S] by (intro fmeasurableI) (auto simp: borel_open)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   923
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   924
lemma lmeasurable_ball: "ball a r \<in> lmeasurable"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   925
  by (simp add: lmeasurable_open)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   926
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   927
lemma lmeasurable_interior: "bounded S \<Longrightarrow> interior S \<in> lmeasurable"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   928
  by (simp add: bounded_interior lmeasurable_open)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   929
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   930
lemma null_sets_cbox_Diff_box: "cbox a b - box a b \<in> null_sets lborel"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   931
proof -
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   932
  have "emeasure lborel (cbox a b - box a b) = 0"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   933
    by (subst emeasure_Diff) (auto simp: emeasure_lborel_cbox_eq emeasure_lborel_box_eq box_subset_cbox)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   934
  then have "cbox a b - box a b \<in> null_sets lborel"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   935
    by (auto simp: null_sets_def)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   936
  then show ?thesis
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   937
    by (auto dest!: AE_not_in)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   938
qed
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   939
subsection\<open> A nice lemma for negligibility proofs.\<close>
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   940
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   941
lemma summable_iff_suminf_neq_top: "(\<And>n. f n \<ge> 0) \<Longrightarrow> \<not> summable f \<Longrightarrow> (\<Sum>i. ennreal (f i)) = top"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   942
  by (metis summable_suminf_not_top)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   943
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   944
proposition starlike_negligible_bounded_gmeasurable:
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   945
  fixes S :: "'a :: euclidean_space set"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   946
  assumes S: "S \<in> sets lebesgue" and "bounded S"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   947
      and eq1: "\<And>c x. \<lbrakk>(c *\<^sub>R x) \<in> S; 0 \<le> c; x \<in> S\<rbrakk> \<Longrightarrow> c = 1"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   948
    shows "S \<in> null_sets lebesgue"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   949
proof -
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   950
  obtain M where "0 < M" "S \<subseteq> ball 0 M"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   951
    using \<open>bounded S\<close> by (auto dest: bounded_subset_ballD)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   952
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   953
  let ?f = "\<lambda>n. root DIM('a) (Suc n)"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   954
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   955
  have vimage_eq_image: "op *\<^sub>R (?f n) -` S = op *\<^sub>R (1 / ?f n) ` S" for n
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   956
    apply safe
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   957
    subgoal for x by (rule image_eqI[of _ _ "?f n *\<^sub>R x"]) auto
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   958
    subgoal by auto
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   959
    done
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   960
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   961
  have eq: "(1 / ?f n) ^ DIM('a) = 1 / Suc n" for n
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   962
    by (simp add: field_simps)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   963
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   964
  { fix n x assume x: "root DIM('a) (1 + real n) *\<^sub>R x \<in> S"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   965
    have "1 * norm x \<le> root DIM('a) (1 + real n) * norm x"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   966
      by (rule mult_mono) auto
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   967
    also have "\<dots> < M"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   968
      using x \<open>S \<subseteq> ball 0 M\<close> by auto
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   969
    finally have "norm x < M" by simp }
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   970
  note less_M = this
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   971
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   972
  have "(\<Sum>n. ennreal (1 / Suc n)) = top"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   973
    using not_summable_harmonic[where 'a=real] summable_Suc_iff[where f="\<lambda>n. 1 / (real n)"]
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   974
    by (intro summable_iff_suminf_neq_top) (auto simp add: inverse_eq_divide)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   975
  then have "top * emeasure lebesgue S = (\<Sum>n. (1 / ?f n)^DIM('a) * emeasure lebesgue S)"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   976
    unfolding ennreal_suminf_multc eq by simp
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   977
  also have "\<dots> = (\<Sum>n. emeasure lebesgue (op *\<^sub>R (?f n) -` S))"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   978
    unfolding vimage_eq_image using emeasure_lebesgue_affine[of "1 / ?f n" 0 S for n] by simp
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   979
  also have "\<dots> = emeasure lebesgue (\<Union>n. op *\<^sub>R (?f n) -` S)"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   980
  proof (intro suminf_emeasure)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   981
    show "disjoint_family (\<lambda>n. op *\<^sub>R (?f n) -` S)"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   982
      unfolding disjoint_family_on_def
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   983
    proof safe
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   984
      fix m n :: nat and x assume "m \<noteq> n" "?f m *\<^sub>R x \<in> S" "?f n *\<^sub>R x \<in> S"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   985
      with eq1[of "?f m / ?f n" "?f n *\<^sub>R x"] show "x \<in> {}"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   986
        by auto
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   987
    qed
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   988
    have "op *\<^sub>R (?f i) -` S \<in> sets lebesgue" for i
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   989
      using measurable_sets[OF lebesgue_measurable_scaling[of "?f i"] S] by auto
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   990
    then show "range (\<lambda>i. op *\<^sub>R (?f i) -` S) \<subseteq> sets lebesgue"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   991
      by auto
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   992
  qed
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   993
  also have "\<dots> \<le> emeasure lebesgue (ball 0 M :: 'a set)"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   994
    using less_M by (intro emeasure_mono) auto
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   995
  also have "\<dots> < top"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   996
    using lmeasurable_ball by (auto simp: fmeasurable_def)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   997
  finally have "emeasure lebesgue S = 0"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   998
    by (simp add: ennreal_top_mult split: if_split_asm)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
   999
  then show "S \<in> null_sets lebesgue"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
  1000
    unfolding null_sets_def using \<open>S \<in> sets lebesgue\<close> by auto
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
  1001
qed
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
  1002
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
  1003
corollary starlike_negligible_compact:
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
  1004
  "compact S \<Longrightarrow> (\<And>c x. \<lbrakk>(c *\<^sub>R x) \<in> S; 0 \<le> c; x \<in> S\<rbrakk> \<Longrightarrow> c = 1) \<Longrightarrow> S \<in> null_sets lebesgue"
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
  1005
  using starlike_negligible_bounded_gmeasurable[of S] by (auto simp: compact_eq_bounded_closed)
f77dca1abf1b HOL-Analysis: prove that a starlike set is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63958
diff changeset
  1006
63968
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1007
lemma outer_regular_lborel:
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1008
  assumes B: "B \<in> fmeasurable lborel" "0 < (e::real)"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1009
  shows "\<exists>U. open U \<and> B \<subseteq> U \<and> emeasure lborel U \<le> emeasure lborel B + e"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1010
proof -
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1011
  let ?\<mu> = "emeasure lborel"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1012
  let ?B = "\<lambda>n::nat. ball 0 n :: 'a set"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1013
  have B[measurable]: "B \<in> sets borel"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1014
    using B by auto
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1015
  let ?e = "\<lambda>n. e*((1/2)^Suc n)"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1016
  have "\<forall>n. \<exists>U. open U \<and> ?B n \<inter> B \<subseteq> U \<and> ?\<mu> (U - B) < ?e n"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1017
  proof
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1018
    fix n :: nat
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1019
    let ?A = "density lborel (indicator (?B n))"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1020
    have emeasure_A: "X \<in> sets borel \<Longrightarrow> emeasure ?A X = ?\<mu> (?B n \<inter> X)" for X
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1021
      by (auto simp add: emeasure_density borel_measurable_indicator indicator_inter_arith[symmetric])
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1022
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1023
    have finite_A: "emeasure ?A (space ?A) \<noteq> \<infinity>"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1024
      using emeasure_bounded_finite[of "?B n"] by (auto simp add: emeasure_A)
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1025
    interpret A: finite_measure ?A
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1026
      by rule fact
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1027
    have "emeasure ?A B + ?e n > (INF U:{U. B \<subseteq> U \<and> open U}. emeasure ?A U)"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1028
      using \<open>0<e\<close> by (auto simp: outer_regular[OF _ finite_A B, symmetric])
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1029
    then obtain U where U: "B \<subseteq> U" "open U" "?\<mu> (?B n \<inter> B) + ?e n > ?\<mu> (?B n \<inter> U)"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1030
      unfolding INF_less_iff by (auto simp: emeasure_A)
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1031
    moreover
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1032
    { have "?\<mu> ((?B n \<inter> U) - B) = ?\<mu> ((?B n \<inter> U) - (?B n \<inter> B))"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1033
        using U by (intro arg_cong[where f="?\<mu>"]) auto
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1034
      also have "\<dots> = ?\<mu> (?B n \<inter> U) - ?\<mu> (?B n \<inter> B)"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1035
        using U A.emeasure_finite[of B]
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1036
        by (intro emeasure_Diff) (auto simp del: A.emeasure_finite simp: emeasure_A)
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1037
      also have "\<dots> < ?e n"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1038
        using U(1,2,3) A.emeasure_finite[of B]
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1039
        by (subst minus_less_iff_ennreal)
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1040
          (auto simp del: A.emeasure_finite simp: emeasure_A less_top ac_simps intro!: emeasure_mono)
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1041
      finally have "?\<mu> ((?B n \<inter> U) - B) < ?e n" . }
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1042
    ultimately show "\<exists>U. open U \<and> ?B n \<inter> B \<subseteq> U \<and> ?\<mu> (U - B) < ?e n"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1043
      by (intro exI[of _ "?B n \<inter> U"]) auto
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1044
  qed
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1045
  then obtain U
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1046
    where U: "\<And>n. open (U n)" "\<And>n. ?B n \<inter> B \<subseteq> U n" "\<And>n. ?\<mu> (U n - B) < ?e n"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1047
    by metis
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1048
  then show ?thesis
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1049
  proof (intro exI conjI)
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1050
    { fix x assume "x \<in> B"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1051
      moreover
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1052
      have "\<exists>n. norm x < real n"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1053
        by (simp add: reals_Archimedean2)
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1054
      then guess n ..
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1055
      ultimately have "x \<in> (\<Union>n. U n)"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1056
        using U(2)[of n] by auto }
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1057
    note * = this
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1058
    then show "open (\<Union>n. U n)" "B \<subseteq> (\<Union>n. U n)"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1059
      using U(1,2) by auto
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1060
    have "?\<mu> (\<Union>n. U n) = ?\<mu> (B \<union> (\<Union>n. U n - B))"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1061
      using * U(2) by (intro arg_cong[where ?f="?\<mu>"]) auto
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1062
    also have "\<dots> = ?\<mu> B + ?\<mu> (\<Union>n. U n - B)"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1063
      using U(1) by (intro plus_emeasure[symmetric]) auto
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1064
    also have "\<dots> \<le> ?\<mu> B + (\<Sum>n. ?\<mu> (U n - B))"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1065
      using U(1) by (intro add_mono emeasure_subadditive_countably) auto
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1066
    also have "\<dots> \<le> ?\<mu> B + (\<Sum>n. ennreal (?e n))"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1067
      using U(3) by (intro add_mono suminf_le) (auto intro: less_imp_le)
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1068
    also have "(\<Sum>n. ennreal (?e n)) = ennreal (e * 1)"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1069
      using \<open>0<e\<close> by (intro suminf_ennreal_eq sums_mult power_half_series) auto
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1070
    finally show "emeasure lborel (\<Union>n. U n) \<le> emeasure lborel B + ennreal e"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1071
      by simp
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1072
  qed
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1073
qed
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1074
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1075
lemma lmeasurable_outer_open:
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1076
  assumes S: "S \<in> lmeasurable" and "0 < e"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1077
  obtains T where "open T" "S \<subseteq> T" "T \<in> lmeasurable" "measure lebesgue T \<le> measure lebesgue S + e"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1078
proof -
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1079
  obtain S' where S': "S \<subseteq> S'" "S' \<in> sets borel" "emeasure lborel S' = emeasure lebesgue S"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1080
    using completion_upper[of S lborel] S by auto
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1081
  then have f_S': "S' \<in> fmeasurable lborel"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1082
    using S by (auto simp: fmeasurable_def)
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1083
  from outer_regular_lborel[OF this \<open>0<e\<close>] guess U .. note U = this
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1084
  show thesis
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1085
  proof (rule that)
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1086
    show "open U" "S \<subseteq> U" "U \<in> lmeasurable"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1087
      using f_S' U S' by (auto simp: fmeasurable_def less_top[symmetric] top_unique)
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1088
    then have "U \<in> fmeasurable lborel"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1089
      by (auto simp: fmeasurable_def)
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1090
    with S U \<open>0<e\<close> show "measure lebesgue U \<le> measure lebesgue S + e"
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1091
      unfolding S'(3) by (simp add: emeasure_eq_measure2 ennreal_plus[symmetric] del: ennreal_plus)
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1092
  qed
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1093
qed
4359400adfe7 HOL-Analysis: the image of a negligible set under a Lipschitz continuous function is negligible (based on HOL Light proof ported by L. C. Paulson)
hoelzl
parents: 63959
diff changeset
  1094
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents:
diff changeset
  1095
end