src/HOL/Probability/Borel_Space.thy
author hoelzl
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(*  Title:      HOL/Probability/Borel_Space.thy
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    Author:     Johannes Hölzl, TU München
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    Author:     Armin Heller, TU München
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*)
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header {*Borel spaces*}
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theory Borel_Space
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  imports Sigma_Algebra "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
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begin
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section "Generic Borel spaces"
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definition borel :: "'a::topological_space measure" where
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  "borel = sigma UNIV {S. open S}"
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abbreviation "borel_measurable M \<equiv> measurable M borel"
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lemma in_borel_measurable:
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   "f \<in> borel_measurable M \<longleftrightarrow>
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    (\<forall>S \<in> sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)"
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  by (auto simp add: measurable_def borel_def)
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lemma in_borel_measurable_borel:
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   "f \<in> borel_measurable M \<longleftrightarrow>
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    (\<forall>S \<in> sets borel.
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      f -` S \<inter> space M \<in> sets M)"
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  by (auto simp add: measurable_def borel_def)
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lemma space_borel[simp]: "space borel = UNIV"
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  unfolding borel_def by auto
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lemma space_in_borel[measurable]: "UNIV \<in> sets borel"
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  unfolding borel_def by auto
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lemma pred_Collect_borel[measurable (raw)]: "Sigma_Algebra.pred borel P \<Longrightarrow> {x. P x} \<in> sets borel"
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  unfolding borel_def pred_def by auto
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lemma borel_open[measurable (raw generic)]:
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  assumes "open A" shows "A \<in> sets borel"
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proof -
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  have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms .
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  thus ?thesis unfolding borel_def by auto
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qed
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lemma borel_closed[measurable (raw generic)]:
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  assumes "closed A" shows "A \<in> sets borel"
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proof -
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  have "space borel - (- A) \<in> sets borel"
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    using assms unfolding closed_def by (blast intro: borel_open)
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  thus ?thesis by simp
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qed
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lemma borel_singleton[measurable]:
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  "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets (borel :: 'a::t1_space measure)"
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  unfolding insert_def by (rule Un) auto
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lemma borel_comp[measurable]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
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  unfolding Compl_eq_Diff_UNIV by simp
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lemma borel_measurable_vimage:
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  fixes f :: "'a \<Rightarrow> 'x::t2_space"
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  assumes borel[measurable]: "f \<in> borel_measurable M"
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  shows "f -` {x} \<inter> space M \<in> sets M"
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  by simp
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lemma borel_measurableI:
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  fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space"
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  assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
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  shows "f \<in> borel_measurable M"
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  unfolding borel_def
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proof (rule measurable_measure_of, simp_all)
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  fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"
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    using assms[of S] by simp
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qed
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lemma borel_measurable_const[measurable (raw)]:
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  "(\<lambda>x. c) \<in> borel_measurable M"
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  by auto
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lemma borel_measurable_indicator:
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  assumes A: "A \<in> sets M"
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  shows "indicator A \<in> borel_measurable M"
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  unfolding indicator_def [abs_def] using A
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  by (auto intro!: measurable_If_set)
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lemma borel_measurable_indicator'[measurable]:
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  "{x\<in>space M. x \<in> A} \<in> sets M \<Longrightarrow> indicator A \<in> borel_measurable M"
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  unfolding indicator_def[abs_def]
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  by (auto intro!: measurable_If)
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lemma borel_measurable_indicator_iff:
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  "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
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    (is "?I \<in> borel_measurable M \<longleftrightarrow> _")
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proof
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  assume "?I \<in> borel_measurable M"
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  then have "?I -` {1} \<inter> space M \<in> sets M"
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    unfolding measurable_def by auto
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  also have "?I -` {1} \<inter> space M = A \<inter> space M"
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    unfolding indicator_def [abs_def] by auto
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  finally show "A \<inter> space M \<in> sets M" .
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next
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  assume "A \<inter> space M \<in> sets M"
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  moreover have "?I \<in> borel_measurable M \<longleftrightarrow>
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    (indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"
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    by (intro measurable_cong) (auto simp: indicator_def)
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  ultimately show "?I \<in> borel_measurable M" by auto
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qed
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lemma borel_measurable_subalgebra:
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  assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
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  shows "f \<in> borel_measurable M"
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  using assms unfolding measurable_def by auto
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lemma borel_measurable_continuous_on1:
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  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
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  assumes "continuous_on UNIV f"
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  shows "f \<in> borel_measurable borel"
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  apply(rule borel_measurableI)
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  using continuous_open_preimage[OF assms] unfolding vimage_def by auto
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section "Borel spaces on euclidean spaces"
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lemma borel_measurable_euclidean_component'[measurable]:
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  "(\<lambda>x::'a::euclidean_space. x $$ i) \<in> borel_measurable borel"
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  by (intro continuous_on_euclidean_component continuous_on_id borel_measurable_continuous_on1)
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lemma borel_measurable_euclidean_component:
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  "(f :: 'a \<Rightarrow> 'b::euclidean_space) \<in> borel_measurable M \<Longrightarrow>(\<lambda>x. f x $$ i) \<in> borel_measurable M"
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  by simp
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lemma [measurable]:
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  fixes a b :: "'a\<Colon>ordered_euclidean_space"
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  shows lessThan_borel: "{..< a} \<in> sets borel"
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    and greaterThan_borel: "{a <..} \<in> sets borel"
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    and greaterThanLessThan_borel: "{a<..<b} \<in> sets borel"
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    and atMost_borel: "{..a} \<in> sets borel"
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    and atLeast_borel: "{a..} \<in> sets borel"
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    and atLeastAtMost_borel: "{a..b} \<in> sets borel"
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    and greaterThanAtMost_borel: "{a<..b} \<in> sets borel"
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    and atLeastLessThan_borel: "{a..<b} \<in> sets borel"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   142
  unfolding greaterThanAtMost_def atLeastLessThan_def
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   143
  by (blast intro: borel_open borel_closed)+
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   144
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   145
lemma 
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   146
  shows hafspace_less_borel: "{x::'a::euclidean_space. a < x $$ i} \<in> sets borel"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   147
    and hafspace_greater_borel: "{x::'a::euclidean_space. x $$ i < a} \<in> sets borel"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   148
    and hafspace_less_eq_borel: "{x::'a::euclidean_space. a \<le> x $$ i} \<in> sets borel"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   149
    and hafspace_greater_eq_borel: "{x::'a::euclidean_space. x $$ i \<le> a} \<in> sets borel"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   150
  by simp_all
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   151
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   152
lemma borel_measurable_less[measurable]:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   153
  fixes f :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   154
  assumes f: "f \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   155
  assumes g: "g \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   156
  shows "{w \<in> space M. f w < g w} \<in> sets M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   157
proof -
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   158
  have "{w \<in> space M. f w < g w} = {x \<in> space M. \<exists>r. f x < of_rat r \<and> of_rat r < g x}"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   159
    using Rats_dense_in_real by (auto simp add: Rats_def)
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   160
  with f g show ?thesis
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   161
    by simp
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   162
qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   163
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   164
lemma
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   165
  fixes f :: "'a \<Rightarrow> real"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   166
  assumes f[measurable]: "f \<in> borel_measurable M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   167
  assumes g[measurable]: "g \<in> borel_measurable M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   168
  shows borel_measurable_le[measurable]: "{w \<in> space M. f w \<le> g w} \<in> sets M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   169
    and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   170
    and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   171
  unfolding eq_iff not_less[symmetric] by measurable+
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   172
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   173
subsection "Borel space equals sigma algebras over intervals"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   174
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   175
lemma rational_boxes:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   176
  fixes x :: "'a\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   177
  assumes "0 < e"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   178
  shows "\<exists>a b. (\<forall>i. a $$ i \<in> \<rat>) \<and> (\<forall>i. b $$ i \<in> \<rat>) \<and> x \<in> {a <..< b} \<and> {a <..< b} \<subseteq> ball x e"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   179
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   180
  def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   181
  then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   182
  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x $$ i \<and> x $$ i - y < e'" (is "\<forall>i. ?th i")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   183
  proof
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   184
    fix i from Rats_dense_in_real[of "x $$ i - e'" "x $$ i"] e
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   185
    show "?th i" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   186
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   187
  from choice[OF this] guess a .. note a = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   188
  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x $$ i < y \<and> y - x $$ i < e'" (is "\<forall>i. ?th i")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   189
  proof
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   190
    fix i from Rats_dense_in_real[of "x $$ i" "x $$ i + e'"] e
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   191
    show "?th i" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   192
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   193
  from choice[OF this] guess b .. note b = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   194
  { fix y :: 'a assume *: "Chi a < y" "y < Chi b"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   195
    have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x $$ i) (y $$ i))\<twosuperior>)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   196
      unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   197
    also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   198
    proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   199
      fix i assume i: "i \<in> {..<DIM('a)}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   200
      have "a i < y$$i \<and> y$$i < b i" using * i eucl_less[where 'a='a] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   201
      moreover have "a i < x$$i" "x$$i - a i < e'" using a by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   202
      moreover have "x$$i < b i" "b i - x$$i < e'" using b by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   203
      ultimately have "\<bar>x$$i - y$$i\<bar> < 2 * e'" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   204
      then have "dist (x $$ i) (y $$ i) < e/sqrt (real (DIM('a)))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   205
        unfolding e'_def by (auto simp: dist_real_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   206
      then have "(dist (x $$ i) (y $$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   207
        by (rule power_strict_mono) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   208
      then show "(dist (x $$ i) (y $$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   209
        by (simp add: power_divide)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   210
    qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   211
    also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   212
    finally have "dist x y < e" . }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   213
  with a b show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   214
    apply (rule_tac exI[of _ "Chi a"])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   215
    apply (rule_tac exI[of _ "Chi b"])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   216
    using eucl_less[where 'a='a] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   217
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   218
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   219
lemma ex_rat_list:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   220
  fixes x :: "'a\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   221
  assumes "\<And> i. x $$ i \<in> \<rat>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   222
  shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x $$ i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   223
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   224
  have "\<forall>i. \<exists>r. x $$ i = of_rat r" using assms unfolding Rats_def by blast
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   225
  from choice[OF this] guess r ..
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   226
  then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   227
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   228
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   229
lemma open_UNION:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   230
  fixes M :: "'a\<Colon>ordered_euclidean_space set"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   231
  assumes "open M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   232
  shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M}
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   233
                   (\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   234
    (is "M = UNION ?idx ?box")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   235
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   236
  fix x assume "x \<in> M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   237
  obtain e where e: "e > 0" "ball x e \<subseteq> M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   238
    using openE[OF assms `x \<in> M`] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   239
  then obtain a b where ab: "x \<in> {a <..< b}" "\<And>i. a $$ i \<in> \<rat>" "\<And>i. b $$ i \<in> \<rat>" "{a <..< b} \<subseteq> ball x e"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   240
    using rational_boxes[OF e(1)] by blast
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   241
  then obtain p q where pq: "length p = DIM ('a)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   242
                            "length q = DIM ('a)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   243
                            "\<forall> i < DIM ('a). of_rat (p ! i) = a $$ i \<and> of_rat (q ! i) = b $$ i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   244
    using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   245
  hence p: "Chi (of_rat \<circ> op ! p) = a"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   246
    using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   247
    unfolding o_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   248
  from pq have q: "Chi (of_rat \<circ> op ! q) = b"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   249
    using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   250
    unfolding o_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   251
  have "x \<in> ?box (p, q)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   252
    using p q ab by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   253
  thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   254
qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   255
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   256
lemma borel_sigma_sets_subset:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   257
  "A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   258
  using sigma_sets_subset[of A borel] by simp
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   259
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   260
lemma borel_eq_sigmaI1:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   261
  fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   262
  assumes borel_eq: "borel = sigma UNIV X"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   263
  assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (range F))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   264
  assumes F: "\<And>i. F i \<in> sets borel"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   265
  shows "borel = sigma UNIV (range F)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   266
  unfolding borel_def
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   267
proof (intro sigma_eqI antisym)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   268
  have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   269
    unfolding borel_def by simp
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   270
  also have "\<dots> = sigma_sets UNIV X"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   271
    unfolding borel_eq by simp
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   272
  also have "\<dots> \<subseteq> sigma_sets UNIV (range F)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   273
    using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   274
  finally show "sigma_sets UNIV {S. open S} \<subseteq> sigma_sets UNIV (range F)" .
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   275
  show "sigma_sets UNIV (range F) \<subseteq> sigma_sets UNIV {S. open S}"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   276
    unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   277
qed auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   278
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   279
lemma borel_eq_sigmaI2:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   280
  fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   281
    and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   282
  assumes borel_eq: "borel = sigma UNIV (range (\<lambda>(i, j). G i j))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   283
  assumes X: "\<And>i j. G i j \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   284
  assumes F: "\<And>i j. F i j \<in> sets borel"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   285
  shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   286
  using assms by (intro borel_eq_sigmaI1[where X="range (\<lambda>(i, j). G i j)" and F="(\<lambda>(i, j). F i j)"]) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   287
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   288
lemma borel_eq_sigmaI3:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   289
  fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   290
  assumes borel_eq: "borel = sigma UNIV X"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   291
  assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   292
  assumes F: "\<And>i j. F i j \<in> sets borel"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   293
  shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   294
  using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   295
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   296
lemma borel_eq_sigmaI4:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   297
  fixes F :: "'i \<Rightarrow> 'a::topological_space set"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   298
    and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   299
  assumes borel_eq: "borel = sigma UNIV (range (\<lambda>(i, j). G i j))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   300
  assumes X: "\<And>i j. G i j \<in> sets (sigma UNIV (range F))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   301
  assumes F: "\<And>i. F i \<in> sets borel"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   302
  shows "borel = sigma UNIV (range F)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   303
  using assms by (intro borel_eq_sigmaI1[where X="range (\<lambda>(i, j). G i j)" and F=F]) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   304
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   305
lemma borel_eq_sigmaI5:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   306
  fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   307
  assumes borel_eq: "borel = sigma UNIV (range G)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   308
  assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   309
  assumes F: "\<And>i j. F i j \<in> sets borel"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   310
  shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   311
  using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   312
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   313
lemma halfspace_gt_in_halfspace:
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   314
  "{x\<Colon>'a. a < x $$ i} \<in> sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a}))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   315
  (is "?set \<in> ?SIGMA")
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   316
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   317
  interpret sigma_algebra UNIV ?SIGMA
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   318
    by (intro sigma_algebra_sigma_sets) simp_all
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   319
  have *: "?set = (\<Union>n. UNIV - {x\<Colon>'a. x $$ i < a + 1 / real (Suc n)})"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   320
  proof (safe, simp_all add: not_less)
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   321
    fix x :: 'a assume "a < x $$ i"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   322
    with reals_Archimedean[of "x $$ i - a"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   323
    obtain n where "a + 1 / real (Suc n) < x $$ i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   324
      by (auto simp: inverse_eq_divide field_simps)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   325
    then show "\<exists>n. a + 1 / real (Suc n) \<le> x $$ i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   326
      by (blast intro: less_imp_le)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   327
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   328
    fix x n
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   329
    have "a < a + 1 / real (Suc n)" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   330
    also assume "\<dots> \<le> x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   331
    finally show "a < x" .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   332
  qed
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   333
  show "?set \<in> ?SIGMA" unfolding *
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   334
    by (auto del: Diff intro!: Diff)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   335
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   336
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   337
lemma borel_eq_halfspace_less:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   338
  "borel = sigma UNIV (range (\<lambda>(a, i). {x::'a::ordered_euclidean_space. x $$ i < a}))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   339
  (is "_ = ?SIGMA")
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   340
proof (rule borel_eq_sigmaI3[OF borel_def])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   341
  fix S :: "'a set" assume "S \<in> {S. open S}"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   342
  then have "open S" by simp
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   343
  from open_UNION[OF this]
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   344
  obtain I where *: "S =
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   345
    (\<Union>(a, b)\<in>I.
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   346
        (\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) $$ i < x $$ i}) \<inter>
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   347
        (\<Inter> i<DIM('a). {x. x $$ i < (Chi (real_of_rat \<circ> op ! b)::'a) $$ i}))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   348
    unfolding greaterThanLessThan_def
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   349
    unfolding eucl_greaterThan_eq_halfspaces[where 'a='a]
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   350
    unfolding eucl_lessThan_eq_halfspaces[where 'a='a]
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   351
    by blast
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   352
  show "S \<in> ?SIGMA"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   353
    unfolding *
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   354
    by (safe intro!: countable_UN Int countable_INT) (auto intro!: halfspace_gt_in_halfspace)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   355
qed auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   356
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   357
lemma borel_eq_halfspace_le:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   358
  "borel = sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x $$ i \<le> a}))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   359
  (is "_ = ?SIGMA")
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   360
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   361
  fix a i
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   362
  have *: "{x::'a. x$$i < a} = (\<Union>n. {x. x$$i \<le> a - 1/real (Suc n)})"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   363
  proof (safe, simp_all)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   364
    fix x::'a assume *: "x$$i < a"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   365
    with reals_Archimedean[of "a - x$$i"]
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   366
    obtain n where "x $$ i < a - 1 / (real (Suc n))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   367
      by (auto simp: field_simps inverse_eq_divide)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   368
    then show "\<exists>n. x $$ i \<le> a - 1 / (real (Suc n))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   369
      by (blast intro: less_imp_le)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   370
  next
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   371
    fix x::'a and n
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   372
    assume "x$$i \<le> a - 1 / real (Suc n)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   373
    also have "\<dots> < a" by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   374
    finally show "x$$i < a" .
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   375
  qed
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   376
  show "{x. x$$i < a} \<in> ?SIGMA" unfolding *
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   377
    by (safe intro!: countable_UN) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   378
qed auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   379
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   380
lemma borel_eq_halfspace_ge:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   381
  "borel = sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x $$ i}))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   382
  (is "_ = ?SIGMA")
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   383
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   384
  fix a i have *: "{x::'a. x$$i < a} = space ?SIGMA - {x::'a. a \<le> x$$i}" by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   385
  show "{x. x$$i < a} \<in> ?SIGMA" unfolding *
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   386
      by (safe intro!: compl_sets) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   387
qed auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   388
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   389
lemma borel_eq_halfspace_greater:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   390
  "borel = sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a < x $$ i}))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   391
  (is "_ = ?SIGMA")
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   392
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   393
  fix a i have *: "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   394
  show "{x. x$$i \<le> a} \<in> ?SIGMA" unfolding *
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   395
    by (safe intro!: compl_sets) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   396
qed auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   397
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   398
lemma borel_eq_atMost:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   399
  "borel = sigma UNIV (range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space}))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   400
  (is "_ = ?SIGMA")
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   401
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   402
  fix a i show "{x. x$$i \<le> a} \<in> ?SIGMA"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   403
  proof cases
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   404
    assume "i < DIM('a)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   405
    then have *: "{x::'a. x$$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   406
    proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   407
      fix x
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   408
      from real_arch_simple[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"] guess k::nat ..
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   409
      then have "\<And>i. i < DIM('a) \<Longrightarrow> x$$i \<le> real k"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   410
        by (subst (asm) Max_le_iff) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   411
      then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia \<le> real k"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   412
        by (auto intro!: exI[of _ k])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   413
    qed
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   414
    show "{x. x$$i \<le> a} \<in> ?SIGMA" unfolding *
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   415
      by (safe intro!: countable_UN) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   416
  qed (auto intro: sigma_sets_top sigma_sets.Empty)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   417
qed auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   418
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   419
lemma borel_eq_greaterThan:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   420
  "borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {a<..}))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   421
  (is "_ = ?SIGMA")
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   422
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   423
  fix a i show "{x. x$$i \<le> a} \<in> ?SIGMA"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   424
  proof cases
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   425
    assume "i < DIM('a)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   426
    have "{x::'a. x$$i \<le> a} = UNIV - {x::'a. a < x$$i}" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   427
    also have *: "{x::'a. a < x$$i} = (\<Union>k::nat. {(\<chi>\<chi> n. if n = i then a else -real k) <..})" using `i <DIM('a)`
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   428
    proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   429
      fix x
44666
8670a39d4420 remove more duplicate lemmas
huffman
parents: 44537
diff changeset
   430
      from reals_Archimedean2[of "Max ((\<lambda>i. -x$$i)`{..<DIM('a)})"]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   431
      guess k::nat .. note k = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   432
      { fix i assume "i < DIM('a)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   433
        then have "-x$$i < real k"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   434
          using k by (subst (asm) Max_less_iff) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   435
        then have "- real k < x$$i" by simp }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   436
      then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x $$ ia"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   437
        by (auto intro!: exI[of _ k])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   438
    qed
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   439
    finally show "{x. x$$i \<le> a} \<in> ?SIGMA"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   440
      apply (simp only:)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   441
      apply (safe intro!: countable_UN Diff)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   442
      apply (auto intro: sigma_sets_top)
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 45288
diff changeset
   443
      done
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   444
  qed (auto intro: sigma_sets_top sigma_sets.Empty)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   445
qed auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   446
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   447
lemma borel_eq_lessThan:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   448
  "borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {..<a}))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   449
  (is "_ = ?SIGMA")
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   450
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   451
  fix a i show "{x. a \<le> x$$i} \<in> ?SIGMA"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   452
  proof cases
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   453
    fix a i assume "i < DIM('a)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   454
    have "{x::'a. a \<le> x$$i} = UNIV - {x::'a. x$$i < a}" by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   455
    also have *: "{x::'a. x$$i < a} = (\<Union>k::nat. {..< (\<chi>\<chi> n. if n = i then a else real k)})" using `i <DIM('a)`
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   456
    proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   457
      fix x
44666
8670a39d4420 remove more duplicate lemmas
huffman
parents: 44537
diff changeset
   458
      from reals_Archimedean2[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   459
      guess k::nat .. note k = this
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   460
      { fix i assume "i < DIM('a)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   461
        then have "x$$i < real k"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   462
          using k by (subst (asm) Max_less_iff) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   463
        then have "x$$i < real k" by simp }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   464
      then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia < real k"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   465
        by (auto intro!: exI[of _ k])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   466
    qed
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   467
    finally show "{x. a \<le> x$$i} \<in> ?SIGMA"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   468
      apply (simp only:)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   469
      apply (safe intro!: countable_UN Diff)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   470
      apply (auto intro: sigma_sets_top)
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 45288
diff changeset
   471
      done
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   472
  qed (auto intro: sigma_sets_top sigma_sets.Empty)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   473
qed auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   474
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   475
lemma borel_eq_atLeastAtMost:
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   476
  "borel = sigma UNIV (range (\<lambda>(a,b). {a..b} \<Colon>'a\<Colon>ordered_euclidean_space set))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   477
  (is "_ = ?SIGMA")
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   478
proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   479
  fix a::'a
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   480
  have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   481
  proof (safe, simp_all add: eucl_le[where 'a='a])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   482
    fix x
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   483
    from real_arch_simple[of "Max ((\<lambda>i. - x$$i)`{..<DIM('a)})"]
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   484
    guess k::nat .. note k = this
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   485
    { fix i assume "i < DIM('a)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   486
      with k have "- x$$i \<le> real k"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   487
        by (subst (asm) Max_le_iff) (auto simp: field_simps)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   488
      then have "- real k \<le> x$$i" by simp }
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   489
    then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x $$ i"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   490
      by (auto intro!: exI[of _ k])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   491
  qed
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   492
  show "{..a} \<in> ?SIGMA" unfolding *
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   493
    by (safe intro!: countable_UN)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   494
       (auto intro!: sigma_sets_top)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   495
qed auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   496
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   497
lemma borel_eq_greaterThanLessThan:
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   498
  "borel = sigma UNIV (range (\<lambda> (a, b). {a <..< b} :: 'a \<Colon> ordered_euclidean_space set))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   499
    (is "_ = ?SIGMA")
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   500
proof (rule borel_eq_sigmaI1[OF borel_def])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   501
  fix M :: "'a set" assume "M \<in> {S. open S}"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   502
  then have "open M" by simp
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   503
  show "M \<in> ?SIGMA"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   504
    apply (subst open_UNION[OF `open M`])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   505
    apply (safe intro!: countable_UN)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   506
    apply auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   507
    done
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   508
qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   509
42862
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   510
lemma borel_eq_atLeastLessThan:
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   511
  "borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA")
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   512
proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   513
  have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   514
  fix x :: real
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   515
  have "{..<x} = (\<Union>i::nat. {-real i ..< x})"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   516
    by (auto simp: move_uminus real_arch_simple)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   517
  then show "{..< x} \<in> ?SIGMA"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   518
    by (auto intro: sigma_sets.intros)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   519
qed auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   520
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   521
lemma borel_measurable_halfspacesI:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   522
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   523
  assumes F: "borel = sigma UNIV (range F)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   524
  and S_eq: "\<And>a i. S a i = f -` F (a,i) \<inter> space M" 
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   525
  and S: "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   526
  shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   527
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   528
  fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   529
  then show "S a i \<in> sets M" unfolding assms
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   530
    by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1))
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   531
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   532
  assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   533
  { fix a i have "S a i \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   534
    proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   535
      assume "i < DIM('c)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   536
      with a show ?thesis unfolding assms(2) by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   537
    next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   538
      assume "\<not> i < DIM('c)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   539
      from S[OF this] show ?thesis .
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   540
    qed }
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   541
  then show "f \<in> borel_measurable M"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   542
    by (auto intro!: measurable_measure_of simp: S_eq F)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   543
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   544
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   545
lemma borel_measurable_iff_halfspace_le:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   546
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   547
  shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i \<le> a} \<in> sets M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   548
  by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   549
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   550
lemma borel_measurable_iff_halfspace_less:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   551
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   552
  shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i < a} \<in> sets M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   553
  by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   554
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   555
lemma borel_measurable_iff_halfspace_ge:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   556
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   557
  shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w $$ i} \<in> sets M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   558
  by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   559
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   560
lemma borel_measurable_iff_halfspace_greater:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   561
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   562
  shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w $$ i} \<in> sets M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   563
  by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   564
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   565
lemma borel_measurable_iff_le:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   566
  "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   567
  using borel_measurable_iff_halfspace_le[where 'c=real] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   568
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   569
lemma borel_measurable_iff_less:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   570
  "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   571
  using borel_measurable_iff_halfspace_less[where 'c=real] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   572
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   573
lemma borel_measurable_iff_ge:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   574
  "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   575
  using borel_measurable_iff_halfspace_ge[where 'c=real]
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   576
  by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   577
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   578
lemma borel_measurable_iff_greater:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   579
  "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   580
  using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   581
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   582
lemma borel_measurable_euclidean_space:
39087
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   583
  fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   584
  shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M)"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   585
proof safe
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   586
  fix i assume "f \<in> borel_measurable M"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   587
  then show "(\<lambda>x. f x $$ i) \<in> borel_measurable M"
41025
8b2cd85ecf11 fixed spelling errors
hoelzl
parents: 41023
diff changeset
   588
    by (auto intro: borel_measurable_euclidean_component)
39087
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   589
next
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   590
  assume f: "\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   591
  then show "f \<in> borel_measurable M"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   592
    unfolding borel_measurable_iff_halfspace_le by auto
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   593
qed
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   594
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   595
subsection "Borel measurable operators"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   596
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   597
lemma borel_measurable_continuous_on:
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   598
  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   599
  assumes f: "continuous_on UNIV f" and g: "g \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   600
  shows "(\<lambda>x. f (g x)) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   601
  using measurable_comp[OF g borel_measurable_continuous_on1[OF f]] by (simp add: comp_def)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   602
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   603
lemma borel_measurable_continuous_on_open':
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   604
  fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   605
  assumes cont: "continuous_on A f" "open A"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   606
  shows "(\<lambda>x. if x \<in> A then f x else c) \<in> borel_measurable borel" (is "?f \<in> _")
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   607
proof (rule borel_measurableI)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   608
  fix S :: "'b set" assume "open S"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   609
  then have "open {x\<in>A. f x \<in> S}"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   610
    by (intro continuous_open_preimage[OF cont]) auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   611
  then have *: "{x\<in>A. f x \<in> S} \<in> sets borel" by auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   612
  have "?f -` S \<inter> space borel = 
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   613
    {x\<in>A. f x \<in> S} \<union> (if c \<in> S then space borel - A else {})"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   614
    by (auto split: split_if_asm)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   615
  also have "\<dots> \<in> sets borel"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   616
    using * `open A` by auto
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   617
  finally show "?f -` S \<inter> space borel \<in> sets borel" .
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   618
qed
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   619
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   620
lemma borel_measurable_continuous_on_open:
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   621
  fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   622
  assumes cont: "continuous_on A f" "open A"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   623
  assumes g: "g \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   624
  shows "(\<lambda>x. if g x \<in> A then f (g x) else c) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   625
  using measurable_comp[OF g borel_measurable_continuous_on_open'[OF cont], of c]
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   626
  by (simp add: comp_def)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   627
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   628
lemma borel_measurable_uminus[measurable (raw)]:
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   629
  fixes g :: "'a \<Rightarrow> real"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   630
  assumes g: "g \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   631
  shows "(\<lambda>x. - g x) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   632
  by (rule borel_measurable_continuous_on[OF _ g]) (auto intro: continuous_on_minus continuous_on_id)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   633
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   634
lemma euclidean_component_prod:
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   635
  fixes x :: "'a :: euclidean_space \<times> 'b :: euclidean_space"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   636
  shows "x $$ i = (if i < DIM('a) then fst x $$ i else snd x $$ (i - DIM('a)))"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   637
  unfolding euclidean_component_def basis_prod_def inner_prod_def by auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   638
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   639
lemma borel_measurable_Pair[measurable (raw)]:
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   640
  fixes f :: "'a \<Rightarrow> 'b::ordered_euclidean_space" and g :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   641
  assumes f: "f \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   642
  assumes g: "g \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   643
  shows "(\<lambda>x. (f x, g x)) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   644
proof (intro borel_measurable_iff_halfspace_le[THEN iffD2] allI impI)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   645
  fix i and a :: real assume i: "i < DIM('b \<times> 'c)"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   646
  have [simp]: "\<And>P A B C. {w. (P \<longrightarrow> A w \<and> B w) \<and> (\<not> P \<longrightarrow> A w \<and> C w)} = 
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   647
    {w. A w \<and> (P \<longrightarrow> B w) \<and> (\<not> P \<longrightarrow> C w)}" by auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   648
  from i f g show "{w \<in> space M. (f w, g w) $$ i \<le> a} \<in> sets M"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   649
    by (auto simp: euclidean_component_prod)
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   650
qed
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   651
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   652
lemma continuous_on_fst: "continuous_on UNIV fst"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   653
proof -
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   654
  have [simp]: "range fst = UNIV" by (auto simp: image_iff)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   655
  show ?thesis
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   656
    using closed_vimage_fst
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   657
    by (auto simp: continuous_on_closed closed_closedin vimage_def)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   658
qed
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   659
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   660
lemma continuous_on_snd: "continuous_on UNIV snd"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   661
proof -
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   662
  have [simp]: "range snd = UNIV" by (auto simp: image_iff)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   663
  show ?thesis
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   664
    using closed_vimage_snd
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   665
    by (auto simp: continuous_on_closed closed_closedin vimage_def)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   666
qed
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   667
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   668
lemma borel_measurable_continuous_Pair:
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   669
  fixes f :: "'a \<Rightarrow> 'b::ordered_euclidean_space" and g :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   670
  assumes [measurable]: "f \<in> borel_measurable M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   671
  assumes [measurable]: "g \<in> borel_measurable M"
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   672
  assumes H: "continuous_on UNIV (\<lambda>x. H (fst x) (snd x))"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   673
  shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   674
proof -
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   675
  have eq: "(\<lambda>x. H (f x) (g x)) = (\<lambda>x. (\<lambda>x. H (fst x) (snd x)) (f x, g x))" by auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   676
  show ?thesis
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   677
    unfolding eq by (rule borel_measurable_continuous_on[OF H]) auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   678
qed
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   679
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   680
lemma borel_measurable_add[measurable (raw)]:
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   681
  fixes f g :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   682
  assumes f: "f \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   683
  assumes g: "g \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   684
  shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   685
  using f g
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   686
  by (rule borel_measurable_continuous_Pair)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   687
     (auto intro: continuous_on_fst continuous_on_snd continuous_on_add)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   688
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   689
lemma borel_measurable_setsum[measurable (raw)]:
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   690
  fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   691
  assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   692
  shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   693
proof cases
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   694
  assume "finite S"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   695
  thus ?thesis using assms by induct auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   696
qed simp
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   697
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   698
lemma borel_measurable_diff[measurable (raw)]:
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   699
  fixes f :: "'a \<Rightarrow> real"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   700
  assumes f: "f \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   701
  assumes g: "g \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   702
  shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   703
  unfolding diff_minus using assms by simp
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   704
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   705
lemma borel_measurable_times[measurable (raw)]:
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   706
  fixes f :: "'a \<Rightarrow> real"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   707
  assumes f: "f \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   708
  assumes g: "g \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   709
  shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   710
  using f g
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   711
  by (rule borel_measurable_continuous_Pair)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   712
     (auto intro: continuous_on_fst continuous_on_snd continuous_on_mult)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   713
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   714
lemma continuous_on_dist:
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   715
  fixes f :: "'a :: t2_space \<Rightarrow> 'b :: metric_space"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   716
  shows "continuous_on A f \<Longrightarrow> continuous_on A g \<Longrightarrow> continuous_on A (\<lambda>x. dist (f x) (g x))"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   717
  unfolding continuous_on_eq_continuous_within by (auto simp: continuous_dist)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   718
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   719
lemma borel_measurable_dist[measurable (raw)]:
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   720
  fixes g f :: "'a \<Rightarrow> 'b::ordered_euclidean_space"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   721
  assumes f: "f \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   722
  assumes g: "g \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   723
  shows "(\<lambda>x. dist (f x) (g x)) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   724
  using f g
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   725
  by (rule borel_measurable_continuous_Pair)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   726
     (intro continuous_on_dist continuous_on_fst continuous_on_snd)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   727
  
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   728
lemma borel_measurable_scaleR[measurable (raw)]:
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   729
  fixes g :: "'a \<Rightarrow> 'b::ordered_euclidean_space"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   730
  assumes f: "f \<in> borel_measurable M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   731
  assumes g: "g \<in> borel_measurable M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   732
  shows "(\<lambda>x. f x *\<^sub>R g x) \<in> borel_measurable M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   733
  by (rule borel_measurable_continuous_Pair[OF f g])
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   734
     (auto intro!: continuous_on_scaleR continuous_on_fst continuous_on_snd)
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   735
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   736
lemma affine_borel_measurable_vector:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   737
  fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   738
  assumes "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   739
  shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   740
proof (rule borel_measurableI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   741
  fix S :: "'x set" assume "open S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   742
  show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   743
  proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   744
    assume "b \<noteq> 0"
44537
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
   745
    with `open S` have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
   746
      by (auto intro!: open_affinity simp: scaleR_add_right)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   747
    hence "?S \<in> sets borel" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   748
    moreover
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   749
    from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   750
      apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   751
    ultimately show ?thesis using assms unfolding in_borel_measurable_borel
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   752
      by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   753
  qed simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   754
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   755
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   756
lemma borel_measurable_const_scaleR[measurable (raw)]:
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   757
  "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. b *\<^sub>R f x ::'a::real_normed_vector) \<in> borel_measurable M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   758
  using affine_borel_measurable_vector[of f M 0 b] by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   759
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   760
lemma borel_measurable_const_add[measurable (raw)]:
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   761
  "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. a + f x ::'a::real_normed_vector) \<in> borel_measurable M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   762
  using affine_borel_measurable_vector[of f M a 1] by simp
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   763
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   764
lemma borel_measurable_setprod[measurable (raw)]:
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   765
  fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   766
  assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   767
  shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   768
proof cases
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   769
  assume "finite S"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   770
  thus ?thesis using assms by induct auto
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   771
qed simp
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   772
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   773
lemma borel_measurable_inverse[measurable (raw)]:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   774
  fixes f :: "'a \<Rightarrow> real"
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   775
  assumes f: "f \<in> borel_measurable M"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   776
  shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   777
proof -
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   778
  have "(\<lambda>x::real. if x \<in> UNIV - {0} then inverse x else 0) \<in> borel_measurable borel"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   779
    by (intro borel_measurable_continuous_on_open' continuous_on_inverse continuous_on_id) auto
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   780
  also have "(\<lambda>x::real. if x \<in> UNIV - {0} then inverse x else 0) = inverse" by (intro ext) auto
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   781
  finally show ?thesis using f by simp
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   782
qed
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   783
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   784
lemma borel_measurable_divide[measurable (raw)]:
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   785
  "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. f x / g x::real) \<in> borel_measurable M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   786
  by (simp add: field_divide_inverse)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   787
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   788
lemma borel_measurable_max[measurable (raw)]:
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   789
  "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. max (g x) (f x) :: real) \<in> borel_measurable M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   790
  by (simp add: max_def)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   791
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   792
lemma borel_measurable_min[measurable (raw)]:
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   793
  "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. min (g x) (f x) :: real) \<in> borel_measurable M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   794
  by (simp add: min_def)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   795
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   796
lemma borel_measurable_abs[measurable (raw)]:
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   797
  "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   798
  unfolding abs_real_def by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   799
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   800
lemma borel_measurable_nth[measurable (raw)]:
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   801
  "(\<lambda>x::real^'n. x $ i) \<in> borel_measurable borel"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   802
  by (simp add: nth_conv_component)
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   803
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   804
lemma convex_measurable:
42990
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
   805
  fixes a b :: real
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
   806
  assumes X: "X \<in> borel_measurable M" "X ` space M \<subseteq> { a <..< b}"
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
   807
  assumes q: "convex_on { a <..< b} q"
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   808
  shows "(\<lambda>x. q (X x)) \<in> borel_measurable M"
42990
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
   809
proof -
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   810
  have "(\<lambda>x. if X x \<in> {a <..< b} then q (X x) else 0) \<in> borel_measurable M" (is "?qX")
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   811
  proof (rule borel_measurable_continuous_on_open[OF _ _ X(1)])
42990
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
   812
    show "open {a<..<b}" by auto
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
   813
    from this q show "continuous_on {a<..<b} q"
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
   814
      by (rule convex_on_continuous)
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
   815
  qed
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   816
  also have "?qX \<longleftrightarrow> (\<lambda>x. q (X x)) \<in> borel_measurable M"
42990
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
   817
    using X by (intro measurable_cong) auto
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   818
  finally show ?thesis .
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
   819
qed
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
   820
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   821
lemma borel_measurable_ln[measurable (raw)]:
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   822
  assumes f: "f \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   823
  shows "(\<lambda>x. ln (f x)) \<in> borel_measurable M"
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
   824
proof -
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
   825
  { fix x :: real assume x: "x \<le> 0"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
   826
    { fix x::real assume "x \<le> 0" then have "\<And>u. exp u = x \<longleftrightarrow> False" by auto }
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   827
    from this[of x] x this[of 0] have "ln 0 = ln x"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   828
      by (auto simp: ln_def) }
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   829
  note ln_imp = this
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   830
  have "(\<lambda>x. if f x \<in> {0<..} then ln (f x) else ln 0) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   831
  proof (rule borel_measurable_continuous_on_open[OF _ _ f])
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   832
    show "continuous_on {0<..} ln"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   833
      by (auto intro!: continuous_at_imp_continuous_on DERIV_ln DERIV_isCont
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
   834
               simp: continuous_isCont[symmetric])
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
   835
    show "open ({0<..}::real set)" by auto
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
   836
  qed
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   837
  also have "(\<lambda>x. if x \<in> {0<..} then ln x else ln 0) = ln"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   838
    by (simp add: fun_eq_iff not_less ln_imp)
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
   839
  finally show ?thesis .
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
   840
qed
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
   841
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   842
lemma borel_measurable_log[measurable (raw)]:
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   843
  "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. log (g x) (f x)) \<in> borel_measurable M"
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   844
  unfolding log_def by auto
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
   845
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   846
lemma measurable_count_space_eq2_countable:
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   847
  fixes f :: "'a => 'c::countable"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   848
  shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   849
proof -
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   850
  { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   851
    then have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   852
      by auto
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   853
    moreover assume "\<And>a. a\<in>A \<Longrightarrow> f -` {a} \<inter> space M \<in> sets M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   854
    ultimately have "f -` X \<inter> space M \<in> sets M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   855
      using `X \<subseteq> A` by (simp add: subset_eq del: UN_simps) }
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   856
  then show ?thesis
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   857
    unfolding measurable_def by auto
47761
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47694
diff changeset
   858
qed
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47694
diff changeset
   859
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   860
lemma measurable_real_floor[measurable]:
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   861
  "(floor :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
47761
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47694
diff changeset
   862
proof -
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   863
  have "\<And>a x. \<lfloor>x\<rfloor> = a \<longleftrightarrow> (real a \<le> x \<and> x < real (a + 1))"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   864
    by (auto intro: floor_eq2)
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   865
  then show ?thesis
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   866
    by (auto simp: vimage_def measurable_count_space_eq2_countable)
47761
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47694
diff changeset
   867
qed
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47694
diff changeset
   868
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   869
lemma measurable_real_natfloor[measurable]:
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   870
  "(natfloor :: real \<Rightarrow> nat) \<in> measurable borel (count_space UNIV)"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   871
  by (simp add: natfloor_def[abs_def])
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   872
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   873
lemma measurable_real_ceiling[measurable]:
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   874
  "(ceiling :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   875
  unfolding ceiling_def[abs_def] by simp
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   876
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   877
lemma borel_measurable_real_floor: "(\<lambda>x::real. real \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   878
  by simp
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   879
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   880
lemma borel_measurable_real_natfloor:
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   881
  "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. real (natfloor (f x))) \<in> borel_measurable M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   882
  by simp
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   883
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   884
subsection "Borel space on the extended reals"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   885
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   886
lemma borel_measurable_ereal[measurable (raw)]:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
   887
  assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   888
  using continuous_on_ereal f by (rule borel_measurable_continuous_on)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   889
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   890
lemma borel_measurable_real_of_ereal[measurable (raw)]:
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   891
  fixes f :: "'a \<Rightarrow> ereal" 
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   892
  assumes f: "f \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   893
  shows "(\<lambda>x. real (f x)) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   894
proof -
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   895
  have "(\<lambda>x. if f x \<in> UNIV - { \<infinity>, - \<infinity> } then real (f x) else 0) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   896
    using continuous_on_real
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   897
    by (rule borel_measurable_continuous_on_open[OF _ _ f]) auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   898
  also have "(\<lambda>x. if f x \<in> UNIV - { \<infinity>, - \<infinity> } then real (f x) else 0) = (\<lambda>x. real (f x))"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   899
    by auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   900
  finally show ?thesis .
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   901
qed
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   902
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   903
lemma borel_measurable_ereal_cases:
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   904
  fixes f :: "'a \<Rightarrow> ereal" 
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   905
  assumes f: "f \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   906
  assumes H: "(\<lambda>x. H (ereal (real (f x)))) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   907
  shows "(\<lambda>x. H (f x)) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   908
proof -
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   909
  let ?F = "\<lambda>x. if f x = \<infinity> then H \<infinity> else if f x = - \<infinity> then H (-\<infinity>) else H (ereal (real (f x)))"
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   910
  { fix x have "H (f x) = ?F x" by (cases "f x") auto }
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   911
  with f H show ?thesis by simp
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   912
qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   913
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   914
lemma
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   915
  fixes f :: "'a \<Rightarrow> ereal" assumes f[measurable]: "f \<in> borel_measurable M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   916
  shows borel_measurable_ereal_abs[measurable(raw)]: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   917
    and borel_measurable_ereal_inverse[measurable(raw)]: "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   918
    and borel_measurable_uminus_ereal[measurable(raw)]: "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M"
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   919
  by (auto simp del: abs_real_of_ereal simp: borel_measurable_ereal_cases[OF f] measurable_If)
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   920
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   921
lemma borel_measurable_uminus_eq_ereal[simp]:
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   922
  "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   923
proof
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   924
  assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   925
qed auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   926
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   927
lemma set_Collect_ereal2:
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   928
  fixes f g :: "'a \<Rightarrow> ereal" 
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   929
  assumes f: "f \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   930
  assumes g: "g \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   931
  assumes H: "{x \<in> space M. H (ereal (real (f x))) (ereal (real (g x)))} \<in> sets M"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   932
    "{x \<in> space borel. H (-\<infinity>) (ereal x)} \<in> sets borel"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   933
    "{x \<in> space borel. H (\<infinity>) (ereal x)} \<in> sets borel"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   934
    "{x \<in> space borel. H (ereal x) (-\<infinity>)} \<in> sets borel"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   935
    "{x \<in> space borel. H (ereal x) (\<infinity>)} \<in> sets borel"
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   936
  shows "{x \<in> space M. H (f x) (g x)} \<in> sets M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   937
proof -
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   938
  let ?G = "\<lambda>y x. if g x = \<infinity> then H y \<infinity> else if g x = -\<infinity> then H y (-\<infinity>) else H y (ereal (real (g x)))"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   939
  let ?F = "\<lambda>x. if f x = \<infinity> then ?G \<infinity> x else if f x = -\<infinity> then ?G (-\<infinity>) x else ?G (ereal (real (f x))) x"
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   940
  { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   941
  note * = this
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   942
  from assms show ?thesis
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   943
    by (subst *) (simp del: space_borel split del: split_if)
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   944
qed
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   945
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   946
lemma [measurable]:
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   947
  fixes f g :: "'a \<Rightarrow> ereal"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   948
  assumes f: "f \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
   949
  assumes g: "g \<in> borel_measurable M"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   950
  shows borel_measurable_ereal_le: "{x \<in> space M. f x \<le> g x} \<in> sets M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   951
    and borel_measurable_ereal_less: "{x \<in> space M. f x < g x} \<in> sets M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   952
    and borel_measurable_ereal_eq: "{w \<in> space M. f w = g w} \<in> sets M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   953
  using f g by (simp_all add: set_Collect_ereal2)
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   954
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   955
lemma borel_measurable_ereal_neq:
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   956
  "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> {w \<in> space M. f w \<noteq> (g w :: ereal)} \<in> sets M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   957
  by simp
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   958
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   959
lemma borel_measurable_ereal_iff:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
   960
  shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   961
proof
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
   962
  assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
   963
  from borel_measurable_real_of_ereal[OF this]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   964
  show "f \<in> borel_measurable M" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   965
qed auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   966
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   967
lemma borel_measurable_ereal_iff_real:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   968
  fixes f :: "'a \<Rightarrow> ereal"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   969
  shows "f \<in> borel_measurable M \<longleftrightarrow>
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   970
    ((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   971
proof safe
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   972
  assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   973
  have "f -` {\<infinity>} \<inter> space M = {x\<in>space M. f x = \<infinity>}" "f -` {-\<infinity>} \<inter> space M = {x\<in>space M. f x = -\<infinity>}" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   974
  with * have **: "{x\<in>space M. f x = \<infinity>} \<in> sets M" "{x\<in>space M. f x = -\<infinity>} \<in> sets M" by simp_all
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 45288
diff changeset
   975
  let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real (f x))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   976
  have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
   977
  also have "?f = f" by (auto simp: fun_eq_iff ereal_real)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   978
  finally show "f \<in> borel_measurable M" .
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   979
qed simp_all
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
   980
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   981
lemma borel_measurable_eq_atMost_ereal:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   982
  fixes f :: "'a \<Rightarrow> ereal"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   983
  shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   984
proof (intro iffI allI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   985
  assume pos[rule_format]: "\<forall>a. f -` {..a} \<inter> space M \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   986
  show "f \<in> borel_measurable M"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
   987
    unfolding borel_measurable_ereal_iff_real borel_measurable_iff_le
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   988
  proof (intro conjI allI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   989
    fix a :: real
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
   990
    { fix x :: ereal assume *: "\<forall>i::nat. real i < x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   991
      have "x = \<infinity>"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
   992
      proof (rule ereal_top)
44666
8670a39d4420 remove more duplicate lemmas
huffman
parents: 44537
diff changeset
   993
        fix B from reals_Archimedean2[of B] guess n ..
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
   994
        then have "ereal B < real n" by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   995
        with * show "B \<le> x" by (metis less_trans less_imp_le)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   996
      qed }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   997
    then have "f -` {\<infinity>} \<inter> space M = space M - (\<Union>i::nat. f -` {.. real i} \<inter> space M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   998
      by (auto simp: not_le)
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
   999
    then show "f -` {\<infinity>} \<inter> space M \<in> sets M" using pos
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1000
      by (auto simp del: UN_simps)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1001
    moreover
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1002
    have "{-\<infinity>::ereal} = {..-\<infinity>}" by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1003
    then show "f -` {-\<infinity>} \<inter> space M \<in> sets M" using pos by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1004
    moreover have "{x\<in>space M. f x \<le> ereal a} \<in> sets M"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1005
      using pos[of "ereal a"] by (simp add: vimage_def Int_def conj_commute)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1006
    moreover have "{w \<in> space M. real (f w) \<le> a} =
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1007
      (if a < 0 then {w \<in> space M. f w \<le> ereal a} - f -` {-\<infinity>} \<inter> space M
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1008
      else {w \<in> space M. f w \<le> ereal a} \<union> (f -` {\<infinity>} \<inter> space M) \<union> (f -` {-\<infinity>} \<inter> space M))" (is "?l = ?r")
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1009
      proof (intro set_eqI) fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases "f x") auto qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1010
    ultimately show "{w \<in> space M. real (f w) \<le> a} \<in> sets M" by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents: 35347
diff changeset
  1011
  qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1012
qed (simp add: measurable_sets)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents: 35347
diff changeset
  1013
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1014
lemma borel_measurable_eq_atLeast_ereal:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1015
  "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1016
proof
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1017
  assume pos: "\<forall>a. f -` {a..} \<inter> space M \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1018
  moreover have "\<And>a. (\<lambda>x. - f x) -` {..a} = f -` {-a ..}"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1019
    by (auto simp: ereal_uminus_le_reorder)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1020
  ultimately have "(\<lambda>x. - f x) \<in> borel_measurable M"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1021
    unfolding borel_measurable_eq_atMost_ereal by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1022
  then show "f \<in> borel_measurable M" by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1023
qed (simp add: measurable_sets)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents: 35347
diff changeset
  1024
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1025
lemma greater_eq_le_measurable:
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1026
  fixes f :: "'a \<Rightarrow> 'c::linorder"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1027
  shows "f -` {..< a} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {a ..} \<inter> space M \<in> sets M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1028
proof
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1029
  assume "f -` {a ..} \<inter> space M \<in> sets M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1030
  moreover have "f -` {..< a} \<inter> space M = space M - f -` {a ..} \<inter> space M" by auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1031
  ultimately show "f -` {..< a} \<inter> space M \<in> sets M" by auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1032
next
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1033
  assume "f -` {..< a} \<inter> space M \<in> sets M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1034
  moreover have "f -` {a ..} \<inter> space M = space M - f -` {..< a} \<inter> space M" by auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1035
  ultimately show "f -` {a ..} \<inter> space M \<in> sets M" by auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1036
qed
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1037
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1038
lemma borel_measurable_ereal_iff_less:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1039
  "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1040
  unfolding borel_measurable_eq_atLeast_ereal greater_eq_le_measurable ..
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1041
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1042
lemma less_eq_ge_measurable:
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1043
  fixes f :: "'a \<Rightarrow> 'c::linorder"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1044
  shows "f -` {a <..} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {..a} \<inter> space M \<in> sets M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1045
proof
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1046
  assume "f -` {a <..} \<inter> space M \<in> sets M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1047
  moreover have "f -` {..a} \<inter> space M = space M - f -` {a <..} \<inter> space M" by auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1048
  ultimately show "f -` {..a} \<inter> space M \<in> sets M" by auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1049
next
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1050
  assume "f -` {..a} \<inter> space M \<in> sets M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1051
  moreover have "f -` {a <..} \<inter> space M = space M - f -` {..a} \<inter> space M" by auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1052
  ultimately show "f -` {a <..} \<inter> space M \<in> sets M" by auto
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1053
qed
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1054
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1055
lemma borel_measurable_ereal_iff_ge:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1056
  "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1057
  unfolding borel_measurable_eq_atMost_ereal less_eq_ge_measurable ..
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1058
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1059
lemma borel_measurable_ereal2:
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1060
  fixes f g :: "'a \<Rightarrow> ereal" 
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1061
  assumes f: "f \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1062
  assumes g: "g \<in> borel_measurable M"
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1063
  assumes H: "(\<lambda>x. H (ereal (real (f x))) (ereal (real (g x)))) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1064
    "(\<lambda>x. H (-\<infinity>) (ereal (real (g x)))) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1065
    "(\<lambda>x. H (\<infinity>) (ereal (real (g x)))) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1066
    "(\<lambda>x. H (ereal (real (f x))) (-\<infinity>)) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1067
    "(\<lambda>x. H (ereal (real (f x))) (\<infinity>)) \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1068
  shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1069
proof -
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1070
  let ?G = "\<lambda>y x. if g x = \<infinity> then H y \<infinity> else if g x = - \<infinity> then H y (-\<infinity>) else H y (ereal (real (g x)))"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1071
  let ?F = "\<lambda>x. if f x = \<infinity> then ?G \<infinity> x else if f x = - \<infinity> then ?G (-\<infinity>) x else ?G (ereal (real (f x))) x"
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1072
  { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1073
  note * = this
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1074
  from assms show ?thesis unfolding * by simp
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1075
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1076
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1077
lemma
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1078
  fixes f :: "'a \<Rightarrow> ereal" assumes f: "f \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1079
  shows borel_measurable_ereal_eq_const: "{x\<in>space M. f x = c} \<in> sets M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1080
    and borel_measurable_ereal_neq_const: "{x\<in>space M. f x \<noteq> c} \<in> sets M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1081
  using f by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1082
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1083
lemma [measurable(raw)]:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1084
  fixes f :: "'a \<Rightarrow> ereal"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1085
  assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1086
  shows borel_measurable_ereal_add: "(\<lambda>x. f x + g x) \<in> borel_measurable M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1087
    and borel_measurable_ereal_times: "(\<lambda>x. f x * g x) \<in> borel_measurable M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1088
    and borel_measurable_ereal_min: "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1089
    and borel_measurable_ereal_max: "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1090
  by (simp_all add: borel_measurable_ereal2 min_def max_def)
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1091
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1092
lemma [measurable(raw)]:
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1093
  fixes f g :: "'a \<Rightarrow> ereal"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1094
  assumes "f \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1095
  assumes "g \<in> borel_measurable M"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1096
  shows borel_measurable_ereal_diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1097
    and borel_measurable_ereal_divide: "(\<lambda>x. f x / g x) \<in> borel_measurable M"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1098
  using assms by (simp_all add: minus_ereal_def divide_ereal_def)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1099
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1100
lemma borel_measurable_ereal_setsum[measurable (raw)]:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1101
  fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
41096
843c40bbc379 integral over setprod
hoelzl
parents: 41083
diff changeset
  1102
  assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
843c40bbc379 integral over setprod
hoelzl
parents: 41083
diff changeset
  1103
  shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
843c40bbc379 integral over setprod
hoelzl
parents: 41083
diff changeset
  1104
proof cases
843c40bbc379 integral over setprod
hoelzl
parents: 41083
diff changeset
  1105
  assume "finite S"
843c40bbc379 integral over setprod
hoelzl
parents: 41083
diff changeset
  1106
  thus ?thesis using assms
843c40bbc379 integral over setprod
hoelzl
parents: 41083
diff changeset
  1107
    by induct auto
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1108
qed simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1109
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1110
lemma borel_measurable_ereal_setprod[measurable (raw)]:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1111
  fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1112
  assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
41096
843c40bbc379 integral over setprod
hoelzl
parents: 41083
diff changeset
  1113
  shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1114
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1115
  assume "finite S"
41096
843c40bbc379 integral over setprod
hoelzl
parents: 41083
diff changeset
  1116
  thus ?thesis using assms by induct auto
843c40bbc379 integral over setprod
hoelzl
parents: 41083
diff changeset
  1117
qed simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1118
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1119
lemma borel_measurable_SUP[measurable (raw)]:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1120
  fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> ereal"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1121
  assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
41097
a1abfa4e2b44 use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents: 41096
diff changeset
  1122
  shows "(\<lambda>x. SUP i : A. f i x) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M")
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1123
  unfolding borel_measurable_ereal_iff_ge
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1124
proof
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1125
  fix a
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1126
  have "?sup -` {a<..} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46731
diff changeset
  1127
    by (auto simp: less_SUP_iff)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1128
  then show "?sup -` {a<..} \<inter> space M \<in> sets M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1129
    using assms by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1130
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1131
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1132
lemma borel_measurable_INF[measurable (raw)]:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1133
  fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> ereal"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1134
  assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
41097
a1abfa4e2b44 use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents: 41096
diff changeset
  1135
  shows "(\<lambda>x. INF i : A. f i x) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M")
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1136
  unfolding borel_measurable_ereal_iff_less
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1137
proof
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1138
  fix a
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1139
  have "?inf -` {..<a} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46731
diff changeset
  1140
    by (auto simp: INF_less_iff)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1141
  then show "?inf -` {..<a} \<inter> space M \<in> sets M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1142
    using assms by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1143
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1144
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1145
lemma [measurable (raw)]:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1146
  fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1147
  assumes "\<And>i. f i \<in> borel_measurable M"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1148
  shows borel_measurable_liminf: "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1149
    and borel_measurable_limsup: "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1150
  unfolding liminf_SUPR_INFI limsup_INFI_SUPR using assms by auto
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  1151
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1152
lemma sets_Collect_eventually_sequientially[measurable]:
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1153
  "(\<And>i. {x\<in>space M. P x i} \<in> sets M) \<Longrightarrow> {x\<in>space M. eventually (P x) sequentially} \<in> sets M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1154
  unfolding eventually_sequentially by simp
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1155
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1156
lemma convergent_ereal:
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1157
  fixes X :: "nat \<Rightarrow> ereal"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1158
  shows "convergent X \<longleftrightarrow> limsup X = liminf X"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1159
  using ereal_Liminf_eq_Limsup_iff[of sequentially]
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1160
  by (auto simp: convergent_def)
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1161
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1162
lemma convergent_ereal_limsup:
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1163
  fixes X :: "nat \<Rightarrow> ereal"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1164
  shows "convergent X \<Longrightarrow> limsup X = lim X"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1165
  by (auto simp: convergent_def limI lim_imp_Limsup)
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1166
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1167
lemma sets_Collect_ereal_convergent[measurable]: 
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1168
  fixes f :: "nat \<Rightarrow> 'a => ereal"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1169
  assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1170
  shows "{x\<in>space M. convergent (\<lambda>i. f i x)} \<in> sets M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1171
  unfolding convergent_ereal by auto
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1172
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1173
lemma borel_measurable_extreal_lim[measurable (raw)]:
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1174
  fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1175
  assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1176
  shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1177
proof -
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1178
  have "\<And>x. lim (\<lambda>i. f i x) = (if convergent (\<lambda>i. f i x) then limsup (\<lambda>i. f i x) else (THE i. False))"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1179
    using convergent_ereal_limsup by (simp add: lim_def convergent_def)
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1180
  then show ?thesis
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1181
    by simp
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1182
qed
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1183
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1184
lemma borel_measurable_ereal_LIMSEQ:
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1185
  fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1186
  assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1187
  and u: "\<And>i. u i \<in> borel_measurable M"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1188
  shows "u' \<in> borel_measurable M"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1189
proof -
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1190
  have "\<And>x. x \<in> space M \<Longrightarrow> u' x = liminf (\<lambda>n. u n x)"
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1191
    using u' by (simp add: lim_imp_Liminf[symmetric])
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1192
  with u show ?thesis by (simp cong: measurable_cong)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1193
qed
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1194
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1195
lemma borel_measurable_extreal_suminf[measurable (raw)]:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1196
  fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1197
  assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1198
  shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1199
  unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
  1200
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
  1201
section "LIMSEQ is borel measurable"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
  1202
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1203
lemma borel_measurable_LIMSEQ:
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
  1204
  fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
  1205
  assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
  1206
  and u: "\<And>i. u i \<in> borel_measurable M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
  1207
  shows "u' \<in> borel_measurable M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
  1208
proof -
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1209
  have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)"
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 45288
diff changeset
  1210
    using u' by (simp add: lim_imp_Liminf)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1211
  moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
  1212
    by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1213
  ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff)
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
  1214
qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
  1215
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1216
lemma sets_Collect_Cauchy[measurable]: 
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1217
  fixes f :: "nat \<Rightarrow> 'a => real"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1218
  assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1219
  shows "{x\<in>space M. Cauchy (\<lambda>i. f i x)} \<in> sets M"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1220
  unfolding Cauchy_iff2 using f by auto
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1221
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1222
lemma borel_measurable_lim[measurable (raw)]:
49774
dfa8ddb874ce use continuity to show Borel-measurability
hoelzl
parents: 47761
diff changeset
  1223
  fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1224
  assumes f[measurable]: "\<