(* Author: Armin Heller, Johannes Hoelzl, TU Muenchen *)

header {*Borel spaces*}

theory Borel

  imports Sigma_Algebra Positive_Infinite_Real Multivariate_Analysis

begin

section "Generic Borel spaces"

definition "borel_space = sigma (UNIV::'a::topological_space set) open"

abbreviation "borel_measurable M \<equiv> measurable M borel_space"

interpretation borel_space: sigma_algebra borel_space

  using sigma_algebra_sigma by (auto simp: borel_space_def)

lemma in_borel_measurable:

   "f \<in> borel_measurable M \<longleftrightarrow>

    (\<forall>S \<in> sets (sigma UNIV open).

      f -` S \<inter> space M \<in> sets M)"

  by (auto simp add: measurable_def borel_space_def)

lemma in_borel_measurable_borel_space:

   "f \<in> borel_measurable M \<longleftrightarrow>

    (\<forall>S \<in> sets borel_space.

      f -` S \<inter> space M \<in> sets M)"

  by (auto simp add: measurable_def borel_space_def)

lemma space_borel_space[simp]: "space borel_space = UNIV"

  unfolding borel_space_def by auto

lemma borel_space_open[simp]:

  assumes "open A" shows "A \<in> sets borel_space"

proof -

  have "A \<in> open" unfolding mem_def using assms .

  thus ?thesis unfolding borel_space_def sigma_def by (auto intro!: sigma_sets.Basic)

qed

lemma borel_space_closed[simp]:

  assumes "closed A" shows "A \<in> sets borel_space"

proof -

  have "space borel_space - (- A) \<in> sets borel_space"

    using assms unfolding closed_def by (blast intro: borel_space_open)

  thus ?thesis by simp

qed

lemma (in sigma_algebra) borel_measurable_vimage:

  fixes f :: "'a \<Rightarrow> 'x::t2_space"

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

  shows "f -` {x} \<inter> space M \<in> sets M"

proof (cases "x \<in> f ` space M")

  case True then obtain y where "x = f y" by auto

  from closed_sing[of "f y"]

  have "{f y} \<in> sets borel_space" by (rule borel_space_closed)

  with assms show ?thesis

    unfolding in_borel_measurable_borel_space `x = f y` by auto

next

  case False hence "f -` {x} \<inter> space M = {}" by auto

  thus ?thesis by auto

qed

lemma (in sigma_algebra) borel_measurableI:

  fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space"

  assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"

  shows "f \<in> borel_measurable M"

  unfolding borel_space_def

proof (rule measurable_sigma)

  fix S :: "'x set" assume "S \<in> open" thus "f -` S \<inter> space M \<in> sets M"

    using assms[of S] by (simp add: mem_def)

qed simp_all

lemma borel_space_singleton[simp, intro]:

  fixes x :: "'a::t1_space"

  shows "A \<in> sets borel_space \<Longrightarrow> insert x A \<in> sets borel_space"

  proof (rule borel_space.insert_in_sets)

    show "{x} \<in> sets borel_space"

      using closed_sing[of x] by (rule borel_space_closed)

  qed simp

lemma (in sigma_algebra) borel_measurable_const[simp, intro]:

  "(\<lambda>x. c) \<in> borel_measurable M"

  by (auto intro!: measurable_const)

lemma (in sigma_algebra) borel_measurable_indicator[simp, intro!]:

  assumes A: "A \<in> sets M"

  shows "indicator A \<in> borel_measurable M"

  unfolding indicator_def_raw using A

  by (auto intro!: measurable_If_set borel_measurable_const)

lemma borel_measurable_translate:

  assumes "A \<in> sets borel_space" and trans: "\<And>B. open B \<Longrightarrow> f -` B \<in> sets borel_space"

  shows "f -` A \<in> sets borel_space"

proof -

  have "A \<in> sigma_sets UNIV open" using assms

    by (simp add: borel_space_def sigma_def)

  thus ?thesis

  proof (induct rule: sigma_sets.induct)

    case (Basic a) thus ?case using trans[of a] by (simp add: mem_def)

  next

    case (Compl a)

    moreover have "UNIV \<in> sets borel_space"

      by (metis borel_space.top borel_space_def_raw mem_def space_sigma)

    ultimately show ?case

      by (auto simp: vimage_Diff borel_space.Diff)

  qed (auto simp add: vimage_UN)

qed

section "Borel spaces on euclidean spaces"

lemma lessThan_borel[simp, intro]:

  fixes a :: "'a\<Colon>ordered_euclidean_space"

  shows "{..< a} \<in> sets borel_space"

  by (blast intro: borel_space_open)

lemma greaterThan_borel[simp, intro]:

  fixes a :: "'a\<Colon>ordered_euclidean_space"

  shows "{a <..} \<in> sets borel_space"

  by (blast intro: borel_space_open)

lemma greaterThanLessThan_borel[simp, intro]:

  fixes a b :: "'a\<Colon>ordered_euclidean_space"

  shows "{a<..<b} \<in> sets borel_space"

  by (blast intro: borel_space_open)

lemma atMost_borel[simp, intro]:

  fixes a :: "'a\<Colon>ordered_euclidean_space"

  shows "{..a} \<in> sets borel_space"

  by (blast intro: borel_space_closed)

lemma atLeast_borel[simp, intro]:

  fixes a :: "'a\<Colon>ordered_euclidean_space"

  shows "{a..} \<in> sets borel_space"

  by (blast intro: borel_space_closed)

lemma atLeastAtMost_borel[simp, intro]:

  fixes a b :: "'a\<Colon>ordered_euclidean_space"

  shows "{a..b} \<in> sets borel_space"

  by (blast intro: borel_space_closed)

lemma greaterThanAtMost_borel[simp, intro]:

  fixes a b :: "'a\<Colon>ordered_euclidean_space"

  shows "{a<..b} \<in> sets borel_space"

  unfolding greaterThanAtMost_def by blast

lemma atLeastLessThan_borel[simp, intro]:

  fixes a b :: "'a\<Colon>ordered_euclidean_space"

  shows "{a..<b} \<in> sets borel_space"

  unfolding atLeastLessThan_def by blast

lemma hafspace_less_borel[simp, intro]:

  fixes a :: real

  shows "{x::'a::euclidean_space. a < x $$ i} \<in> sets borel_space"

  by (auto intro!: borel_space_open open_halfspace_component_gt)

lemma hafspace_greater_borel[simp, intro]:

  fixes a :: real

  shows "{x::'a::euclidean_space. x $$ i < a} \<in> sets borel_space"

  by (auto intro!: borel_space_open open_halfspace_component_lt)

lemma hafspace_less_eq_borel[simp, intro]:

  fixes a :: real

  shows "{x::'a::euclidean_space. a \<le> x $$ i} \<in> sets borel_space"

  by (auto intro!: borel_space_closed closed_halfspace_component_ge)

lemma hafspace_greater_eq_borel[simp, intro]:

  fixes a :: real

  shows "{x::'a::euclidean_space. x $$ i \<le> a} \<in> sets borel_space"

  by (auto intro!: borel_space_closed closed_halfspace_component_le)

lemma (in sigma_algebra) borel_measurable_less[simp, intro]:

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

  assumes f: "f \<in> borel_measurable M"

  assumes g: "g \<in> borel_measurable M"

  shows "{w \<in> space M. f w < g w} \<in> sets M"

proof -

  have "{w \<in> space M. f w < g w} =

        (\<Union>r. (f -` {..< of_rat r} \<inter> space M) \<inter> (g -` {of_rat r <..} \<inter> space M))"

    using Rats_dense_in_real by (auto simp add: Rats_def)

  then show ?thesis using f g

    by simp (blast intro: measurable_sets)

qed

lemma (in sigma_algebra) borel_measurable_le[simp, intro]:

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

  assumes f: "f \<in> borel_measurable M"

  assumes g: "g \<in> borel_measurable M"

  shows "{w \<in> space M. f w \<le> g w} \<in> sets M"

proof -

  have "{w \<in> space M. f w \<le> g w} = space M - {w \<in> space M. g w < f w}"

    by auto

  thus ?thesis using f g

    by simp blast

qed

lemma (in sigma_algebra) borel_measurable_eq[simp, intro]:

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

  assumes f: "f \<in> borel_measurable M"

  assumes g: "g \<in> borel_measurable M"

  shows "{w \<in> space M. f w = g w} \<in> sets M"

proof -

  have "{w \<in> space M. f w = g w} =

        {w \<in> space M. f w \<le> g w} \<inter> {w \<in> space M. g w \<le> f w}"

    by auto

  thus ?thesis using f g by auto

qed

lemma (in sigma_algebra) borel_measurable_neq[simp, intro]:

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

  assumes f: "f \<in> borel_measurable M"

  assumes g: "g \<in> borel_measurable M"

  shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"

proof -

  have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}"

    by auto

  thus ?thesis using f g by auto

qed

subsection "Borel space equals sigma algebras over intervals"

lemma rational_boxes:

  fixes x :: "'a\<Colon>ordered_euclidean_space"

  assumes "0 < e"

  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"

proof -

  def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"

  then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos)

  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x $$ i \<and> x $$ i - y < e'" (is "\<forall>i. ?th i")

  proof

    fix i from Rats_dense_in_real[of "x $$ i - e'" "x $$ i"] e

    show "?th i" by auto

  qed

  from choice[OF this] guess a .. note a = this

  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x $$ i < y \<and> y - x $$ i < e'" (is "\<forall>i. ?th i")

  proof

    fix i from Rats_dense_in_real[of "x $$ i" "x $$ i + e'"] e

    show "?th i" by auto

  qed

  from choice[OF this] guess b .. note b = this

  { fix y :: 'a assume *: "Chi a < y" "y < Chi b"

    have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x $$ i) (y $$ i))\<twosuperior>)"

      unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

    also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))"

    proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

      fix i assume i: "i \<in> {..<DIM('a)}"

      have "a i < y$$i \<and> y$$i < b i" using * i eucl_less[where 'a='a] by auto

      moreover have "a i < x$$i" "x$$i - a i < e'" using a by auto

      moreover have "x$$i < b i" "b i - x$$i < e'" using b by auto

      ultimately have "\<bar>x$$i - y$$i\<bar> < 2 * e'" by auto

      then have "dist (x $$ i) (y $$ i) < e/sqrt (real (DIM('a)))"

        unfolding e'_def by (auto simp: dist_real_def)

      then have "(dist (x $$ i) (y $$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"

        by (rule power_strict_mono) auto

      then show "(dist (x $$ i) (y $$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)"

        by (simp add: power_divide)

    qed auto

    also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive)

    finally have "dist x y < e" . }

  with a b show ?thesis

    apply (rule_tac exI[of _ "Chi a"])

    apply (rule_tac exI[of _ "Chi b"])

    using eucl_less[where 'a='a] by auto

qed

lemma ex_rat_list:

  fixes x :: "'a\<Colon>ordered_euclidean_space"

  assumes "\<And> i. x $$ i \<in> \<rat>"

  shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x $$ i)"

proof -

  have "\<forall>i. \<exists>r. x $$ i = of_rat r" using assms unfolding Rats_def by blast

  from choice[OF this] guess r ..

  then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"])

qed

lemma open_UNION:

  fixes M :: "'a\<Colon>ordered_euclidean_space set"

  assumes "open M"

  shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M}

                   (\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})"

    (is "M = UNION ?idx ?box")

proof safe

  fix x assume "x \<in> M"

  obtain e where e: "e > 0" "ball x e \<subseteq> M"

    using openE[OF assms `x \<in> M`] by auto

  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"

    using rational_boxes[OF e(1)] by blast

  then obtain p q where pq: "length p = DIM ('a)"

                            "length q = DIM ('a)"

                            "\<forall> i < DIM ('a). of_rat (p ! i) = a $$ i \<and> of_rat (q ! i) = b $$ i"

    using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast

  hence p: "Chi (of_rat \<circ> op ! p) = a"

    using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a]

    unfolding o_def by auto

  from pq have q: "Chi (of_rat \<circ> op ! q) = b"

    using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b]

    unfolding o_def by auto

  have "x \<in> ?box (p, q)"

    using p q ab by auto

  thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto

qed auto

lemma halfspace_span_open:

  "sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})))

    \<subseteq> sets borel_space"

  by (auto intro!: borel_space.sigma_sets_subset[simplified] borel_space_open

                   open_halfspace_component_lt simp: sets_sigma)

lemma halfspace_lt_in_halfspace:

  "{x\<Colon>'a. x $$ i < a} \<in> sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})))"

  unfolding sets_sigma by (rule sigma_sets.Basic) auto

lemma halfspace_gt_in_halfspace:

  "{x\<Colon>'a. a < x $$ i} \<in> sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})))"

    (is "?set \<in> sets ?SIGMA")

proof -

  interpret sigma_algebra ?SIGMA by (rule sigma_algebra_sigma) simp

  have *: "?set = (\<Union>n. space ?SIGMA - {x\<Colon>'a. x $$ i < a + 1 / real (Suc n)})"

  proof (safe, simp_all add: not_less)

    fix x assume "a < x $$ i"

    with reals_Archimedean[of "x $$ i - a"]

    obtain n where "a + 1 / real (Suc n) < x $$ i"

      by (auto simp: inverse_eq_divide field_simps)

    then show "\<exists>n. a + 1 / real (Suc n) \<le> x $$ i"

      by (blast intro: less_imp_le)

  next

    fix x n

    have "a < a + 1 / real (Suc n)" by auto

    also assume "\<dots> \<le> x"

    finally show "a < x" .

  qed

  show "?set \<in> sets ?SIGMA" unfolding *

    by (safe intro!: countable_UN Diff halfspace_lt_in_halfspace)

qed

lemma (in sigma_algebra) sets_sigma_subset:

  assumes "A = space M"

  assumes "B \<subseteq> sets M"

  shows "sets (sigma A B) \<subseteq> sets M"

  by (unfold assms sets_sigma, rule sigma_sets_subset, rule assms)

lemma open_span_halfspace:

  "sets borel_space \<subseteq> sets (sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x $$ i < a})))"

    (is "_ \<subseteq> sets ?SIGMA")

proof (unfold borel_space_def, rule sigma_algebra.sets_sigma_subset, safe)

  show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) simp

  then interpret sigma_algebra ?SIGMA .

  fix S :: "'a set" assume "S \<in> open" then have "open S" unfolding mem_def .

  from open_UNION[OF this]

  obtain I where *: "S =

    (\<Union>(a, b)\<in>I.

        (\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) $$ i < x $$ i}) \<inter>

        (\<Inter> i<DIM('a). {x. x $$ i < (Chi (real_of_rat \<circ> op ! b)::'a) $$ i}))"

    unfolding greaterThanLessThan_def

    unfolding eucl_greaterThan_eq_halfspaces[where 'a='a]

    unfolding eucl_lessThan_eq_halfspaces[where 'a='a]

    by blast

  show "S \<in> sets ?SIGMA"

    unfolding *

    by (auto intro!: countable_UN Int countable_INT halfspace_lt_in_halfspace halfspace_gt_in_halfspace)

qed auto

lemma halfspace_span_halfspace_le:

  "sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a}))) \<subseteq>

   sets (sigma UNIV (range (\<lambda> (a, i). {x. x $$ i \<le> a})))"

  (is "_ \<subseteq> sets ?SIGMA")

proof (rule sigma_algebra.sets_sigma_subset, safe)

  show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto

  then interpret sigma_algebra ?SIGMA .

  fix a i

  have *: "{x::'a. x$$i < a} = (\<Union>n. {x. x$$i \<le> a - 1/real (Suc n)})"

  proof (safe, simp_all)

    fix x::'a assume *: "x$$i < a"

    with reals_Archimedean[of "a - x$$i"]

    obtain n where "x $$ i < a - 1 / (real (Suc n))"

      by (auto simp: field_simps inverse_eq_divide)

    then show "\<exists>n. x $$ i \<le> a - 1 / (real (Suc n))"

      by (blast intro: less_imp_le)

  next

    fix x::'a and n

    assume "x$$i \<le> a - 1 / real (Suc n)"

    also have "\<dots> < a" by auto

    finally show "x$$i < a" .

  qed

  show "{x. x$$i < a} \<in> sets ?SIGMA" unfolding *

    by (safe intro!: countable_UN)

       (auto simp: sets_sigma intro!: sigma_sets.Basic)

qed auto

lemma halfspace_span_halfspace_ge:

  "sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a}))) \<subseteq> 

   sets (sigma UNIV (range (\<lambda> (a, i). {x. a \<le> x $$ i})))"

  (is "_ \<subseteq> sets ?SIGMA")

proof (rule sigma_algebra.sets_sigma_subset, safe)

  show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto

  then interpret sigma_algebra ?SIGMA .

  fix a i have *: "{x::'a. x$$i < a} = space ?SIGMA - {x::'a. a \<le> x$$i}" by auto

  show "{x. x$$i < a} \<in> sets ?SIGMA" unfolding *

    by (safe intro!: Diff)

       (auto simp: sets_sigma intro!: sigma_sets.Basic)

qed auto

lemma halfspace_le_span_halfspace_gt:

  "sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a}))) \<subseteq> 

   sets (sigma UNIV (range (\<lambda> (a, i). {x. a < x $$ i})))"

  (is "_ \<subseteq> sets ?SIGMA")

proof (rule sigma_algebra.sets_sigma_subset, safe)

  show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto

  then interpret sigma_algebra ?SIGMA .

  fix a i have *: "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto

  show "{x. x$$i \<le> a} \<in> sets ?SIGMA" unfolding *

    by (safe intro!: Diff)

       (auto simp: sets_sigma intro!: sigma_sets.Basic)

qed auto

lemma halfspace_le_span_atMost:

  "sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a}))) \<subseteq>

   sets (sigma UNIV (range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})))"

  (is "_ \<subseteq> sets ?SIGMA")

proof (rule sigma_algebra.sets_sigma_subset, safe)

  show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto

  then interpret sigma_algebra ?SIGMA .

  fix a i

  show "{x. x$$i \<le> a} \<in> sets ?SIGMA"

  proof cases

    assume "i < DIM('a)"

    then have *: "{x::'a. x$$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})"

    proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm)

      fix x

      from real_arch_simple[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"] guess k::nat ..

      then have "\<And>i. i < DIM('a) \<Longrightarrow> x$$i \<le> real k"

        by (subst (asm) Max_le_iff) auto

      then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia \<le> real k"

        by (auto intro!: exI[of _ k])

    qed

    show "{x. x$$i \<le> a} \<in> sets ?SIGMA" unfolding *

      by (safe intro!: countable_UN)

         (auto simp: sets_sigma intro!: sigma_sets.Basic)

  next

    assume "\<not> i < DIM('a)"

    then show "{x. x$$i \<le> a} \<in> sets ?SIGMA"

      using top by auto

  qed

qed auto

lemma halfspace_le_span_greaterThan:

  "sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a}))) \<subseteq>

   sets (sigma UNIV (range (\<lambda>a. {a<..})))"

  (is "_ \<subseteq> sets ?SIGMA")

proof (rule sigma_algebra.sets_sigma_subset, safe)

  show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto

  then interpret sigma_algebra ?SIGMA .

  fix a i

  show "{x. x$$i \<le> a} \<in> sets ?SIGMA"

  proof cases

    assume "i < DIM('a)"

    have "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto

    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)`

    proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)

      fix x

      from real_arch_lt[of "Max ((\<lambda>i. -x$$i)`{..<DIM('a)})"]

      guess k::nat .. note k = this

      { fix i assume "i < DIM('a)"

        then have "-x$$i < real k"

          using k by (subst (asm) Max_less_iff) auto

        then have "- real k < x$$i" by simp }

      then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x $$ ia"

        by (auto intro!: exI[of _ k])

    qed

    finally show "{x. x$$i \<le> a} \<in> sets ?SIGMA"

      apply (simp only:)

      apply (safe intro!: countable_UN Diff)

      by (auto simp: sets_sigma intro!: sigma_sets.Basic)

  next

    assume "\<not> i < DIM('a)"

    then show "{x. x$$i \<le> a} \<in> sets ?SIGMA"

      using top by auto

  qed

qed auto

lemma atMost_span_atLeastAtMost:

  "sets (sigma UNIV (range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space}))) \<subseteq>

   sets (sigma UNIV (range (\<lambda>(a,b). {a..b})))"

  (is "_ \<subseteq> sets ?SIGMA")

proof (rule sigma_algebra.sets_sigma_subset, safe)

  show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto

  then interpret sigma_algebra ?SIGMA .

  fix a::'a

  have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"

  proof (safe, simp_all add: eucl_le[where 'a='a])

    fix x

    from real_arch_simple[of "Max ((\<lambda>i. - x$$i)`{..<DIM('a)})"]

    guess k::nat .. note k = this

    { fix i assume "i < DIM('a)"

      with k have "- x$$i \<le> real k"

        by (subst (asm) Max_le_iff) (auto simp: field_simps)

      then have "- real k \<le> x$$i" by simp }

    then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x $$ i"

      by (auto intro!: exI[of _ k])

  qed

  show "{..a} \<in> sets ?SIGMA" unfolding *

    by (safe intro!: countable_UN)

       (auto simp: sets_sigma intro!: sigma_sets.Basic)

qed auto

lemma borel_space_eq_greaterThanLessThan:

  "sets borel_space = sets (sigma UNIV (range (\<lambda> (a, b). {a <..< (b :: 'a \<Colon> ordered_euclidean_space)})))"

    (is "_ = sets ?SIGMA")

proof (rule antisym)

  show "sets ?SIGMA \<subseteq> sets borel_space"

    by (rule borel_space.sets_sigma_subset) auto

  show "sets borel_space \<subseteq> sets ?SIGMA"

    unfolding borel_space_def

  proof (rule sigma_algebra.sets_sigma_subset, safe)

    show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto

    then interpret sigma_algebra ?SIGMA .

    fix M :: "'a set" assume "M \<in> open"

    then have "open M" by (simp add: mem_def)

    show "M \<in> sets ?SIGMA"

      apply (subst open_UNION[OF `open M`])

      apply (safe intro!: countable_UN)

      by (auto simp add: sigma_def intro!: sigma_sets.Basic)

  qed auto

qed

lemma borel_space_eq_atMost:

  "sets borel_space = sets (sigma UNIV (range (\<lambda> a. {.. a::'a\<Colon>ordered_euclidean_space})))"

    (is "_ = sets ?SIGMA")

proof (rule antisym)

  show "sets borel_space \<subseteq> sets ?SIGMA"

    using halfspace_le_span_atMost halfspace_span_halfspace_le open_span_halfspace

    by auto

  show "sets ?SIGMA \<subseteq> sets borel_space"

    by (rule borel_space.sets_sigma_subset) auto

qed

lemma borel_space_eq_atLeastAtMost:

  "sets borel_space = sets (sigma UNIV (range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space, b). {a .. b})))"

   (is "_ = sets ?SIGMA")

proof (rule antisym)

  show "sets borel_space \<subseteq> sets ?SIGMA"

    using atMost_span_atLeastAtMost halfspace_le_span_atMost

      halfspace_span_halfspace_le open_span_halfspace

    by auto

  show "sets ?SIGMA \<subseteq> sets borel_space"

    by (rule borel_space.sets_sigma_subset) auto

qed

lemma borel_space_eq_greaterThan:

  "sets borel_space = sets (sigma UNIV (range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {a <..})))"

   (is "_ = sets ?SIGMA")

proof (rule antisym)

  show "sets borel_space \<subseteq> sets ?SIGMA"

    using halfspace_le_span_greaterThan

      halfspace_span_halfspace_le open_span_halfspace

    by auto

  show "sets ?SIGMA \<subseteq> sets borel_space"

    by (rule borel_space.sets_sigma_subset) auto

qed

lemma borel_space_eq_halfspace_le:

  "sets borel_space = sets (sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x$$i \<le> a})))"

   (is "_ = sets ?SIGMA")

proof (rule antisym)

  show "sets borel_space \<subseteq> sets ?SIGMA"

    using open_span_halfspace halfspace_span_halfspace_le by auto

  show "sets ?SIGMA \<subseteq> sets borel_space"

    by (rule borel_space.sets_sigma_subset) auto

qed

lemma borel_space_eq_halfspace_less:

  "sets borel_space = sets (sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x$$i < a})))"

   (is "_ = sets ?SIGMA")

proof (rule antisym)

  show "sets borel_space \<subseteq> sets ?SIGMA"

    using open_span_halfspace .

  show "sets ?SIGMA \<subseteq> sets borel_space"

    by (rule borel_space.sets_sigma_subset) auto

qed

lemma borel_space_eq_halfspace_gt:

  "sets borel_space = sets (sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a < x$$i})))"

   (is "_ = sets ?SIGMA")

proof (rule antisym)

  show "sets borel_space \<subseteq> sets ?SIGMA"

    using halfspace_le_span_halfspace_gt open_span_halfspace halfspace_span_halfspace_le by auto

  show "sets ?SIGMA \<subseteq> sets borel_space"

    by (rule borel_space.sets_sigma_subset) auto

qed

lemma borel_space_eq_halfspace_ge:

  "sets borel_space = sets (sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a \<le> x$$i})))"

   (is "_ = sets ?SIGMA")

proof (rule antisym)

  show "sets borel_space \<subseteq> sets ?SIGMA"

    using halfspace_span_halfspace_ge open_span_halfspace by auto

  show "sets ?SIGMA \<subseteq> sets borel_space"

    by (rule borel_space.sets_sigma_subset) auto

qed

lemma (in sigma_algebra) borel_measurable_halfspacesI:

  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"

  assumes "sets borel_space = sets (sigma UNIV (range F))"

  and "\<And>a i. S a i = f -` F (a,i) \<inter> space M"

  and "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M"

  shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)"

proof safe

  fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M"

  then show "S a i \<in> sets M" unfolding assms

    by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1) sigma_def)

next

  assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M"

  { fix a i have "S a i \<in> sets M"

    proof cases

      assume "i < DIM('c)"

      with a show ?thesis unfolding assms(2) by simp

    next

      assume "\<not> i < DIM('c)"

      from assms(3)[OF this] show ?thesis .

    qed }

  then have "f \<in> measurable M (sigma UNIV (range F))"

    by (auto intro!: measurable_sigma simp: assms(2))

  then show "f \<in> borel_measurable M" unfolding measurable_def

    unfolding assms(1) by simp

qed

lemma (in sigma_algebra) borel_measurable_iff_halfspace_le:

  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"

  shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i \<le> a} \<in> sets M)"

  by (rule borel_measurable_halfspacesI[OF borel_space_eq_halfspace_le]) auto

lemma (in sigma_algebra) borel_measurable_iff_halfspace_less:

  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"

  shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i < a} \<in> sets M)"

  by (rule borel_measurable_halfspacesI[OF borel_space_eq_halfspace_less]) auto

lemma (in sigma_algebra) borel_measurable_iff_halfspace_ge:

  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"

  shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w $$ i} \<in> sets M)"

  by (rule borel_measurable_halfspacesI[OF borel_space_eq_halfspace_ge]) auto

lemma (in sigma_algebra) borel_measurable_iff_halfspace_greater:

  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"

  shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w $$ i} \<in> sets M)"

  by (rule borel_measurable_halfspacesI[OF borel_space_eq_halfspace_gt]) auto

lemma (in sigma_algebra) borel_measurable_iff_le:

  "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"

  using borel_measurable_iff_halfspace_le[where 'c=real] by simp

lemma (in sigma_algebra) borel_measurable_iff_less:

  "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"

  using borel_measurable_iff_halfspace_less[where 'c=real] by simp

lemma (in sigma_algebra) borel_measurable_iff_ge:

  "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"

  using borel_measurable_iff_halfspace_ge[where 'c=real] by simp

lemma (in sigma_algebra) borel_measurable_iff_greater:

  "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"

  using borel_measurable_iff_halfspace_greater[where 'c=real] by simp

subsection "Borel measurable operators"

lemma (in sigma_algebra) affine_borel_measurable_vector:

  fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"

  assumes "f \<in> borel_measurable M"

  shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"

proof (rule borel_measurableI)

  fix S :: "'x set" assume "open S"

  show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"

  proof cases

    assume "b \<noteq> 0"

    with `open S` have "((\<lambda>x. (- a + x) /\<^sub>R b) ` S) \<in> open" (is "?S \<in> open")

      by (auto intro!: open_affinity simp: scaleR.add_right mem_def)

    hence "?S \<in> sets borel_space"

      unfolding borel_space_def by (auto simp: sigma_def intro!: sigma_sets.Basic)

    moreover

    from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"

      apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)

    ultimately show ?thesis using assms unfolding in_borel_measurable_borel_space

      by auto

  qed simp

qed

lemma (in sigma_algebra) affine_borel_measurable:

  fixes g :: "'a \<Rightarrow> real"

  assumes g: "g \<in> borel_measurable M"

  shows "(\<lambda>x. a + (g x) * b) \<in> borel_measurable M"

  using affine_borel_measurable_vector[OF assms] by (simp add: mult_commute)

lemma (in sigma_algebra) borel_measurable_add[simp, intro]:

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

  assumes f: "f \<in> borel_measurable M"

  assumes g: "g \<in> borel_measurable M"

  shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"

proof -

  have 1: "\<And>a. {w\<in>space M. a \<le> f w + g w} = {w \<in> space M. a + g w * -1 \<le> f w}"

    by auto

  have "\<And>a. (\<lambda>w. a + (g w) * -1) \<in> borel_measurable M"

    by (rule affine_borel_measurable [OF g])

  then have "\<And>a. {w \<in> space M. (\<lambda>w. a + (g w) * -1)(w) \<le> f w} \<in> sets M" using f

    by auto

  then have "\<And>a. {w \<in> space M. a \<le> f w + g w} \<in> sets M"

    by (simp add: 1)

  then show ?thesis

    by (simp add: borel_measurable_iff_ge)

qed

lemma (in sigma_algebra) borel_measurable_square:

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

  assumes f: "f \<in> borel_measurable M"

  shows "(\<lambda>x. (f x)^2) \<in> borel_measurable M"

proof -

  {

    fix a

    have "{w \<in> space M. (f w)\<twosuperior> \<le> a} \<in> sets M"

    proof (cases rule: linorder_cases [of a 0])

      case less

      hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = {}"

        by auto (metis less order_le_less_trans power2_less_0)

      also have "... \<in> sets M"

        by (rule empty_sets)

      finally show ?thesis .

    next

      case equal

      hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =

             {w \<in> space M. f w \<le> 0} \<inter> {w \<in> space M. 0 \<le> f w}"

        by auto

      also have "... \<in> sets M"

        apply (insert f)

        apply (rule Int)

        apply (simp add: borel_measurable_iff_le)

        apply (simp add: borel_measurable_iff_ge)

        done

      finally show ?thesis .

    next

      case greater

      have "\<forall>x. (f x ^ 2 \<le> sqrt a ^ 2) = (- sqrt a  \<le> f x & f x \<le> sqrt a)"

        by (metis abs_le_interval_iff abs_of_pos greater real_sqrt_abs

                  real_sqrt_le_iff real_sqrt_power)

      hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =

             {w \<in> space M. -(sqrt a) \<le> f w} \<inter> {w \<in> space M. f w \<le> sqrt a}"

        using greater by auto

      also have "... \<in> sets M"

        apply (insert f)

        apply (rule Int)

        apply (simp add: borel_measurable_iff_ge)

        apply (simp add: borel_measurable_iff_le)

        done

      finally show ?thesis .

    qed

  }

  thus ?thesis by (auto simp add: borel_measurable_iff_le)

qed

lemma times_eq_sum_squares:

   fixes x::real

   shows"x*y = ((x+y)^2)/4 - ((x-y)^ 2)/4"

by (simp add: power2_eq_square ring_distribs diff_divide_distrib [symmetric])

lemma (in sigma_algebra) borel_measurable_uminus[simp, intro]:

  fixes g :: "'a \<Rightarrow> real"

  assumes g: "g \<in> borel_measurable M"

  shows "(\<lambda>x. - g x) \<in> borel_measurable M"

proof -

  have "(\<lambda>x. - g x) = (\<lambda>x. 0 + (g x) * -1)"

    by simp

  also have "... \<in> borel_measurable M"

    by (fast intro: affine_borel_measurable g)

  finally show ?thesis .

qed

lemma (in sigma_algebra) borel_measurable_times[simp, intro]:

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

  assumes f: "f \<in> borel_measurable M"

  assumes g: "g \<in> borel_measurable M"

  shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"

proof -

  have 1: "(\<lambda>x. 0 + (f x + g x)\<twosuperior> * inverse 4) \<in> borel_measurable M"

    using assms by (fast intro: affine_borel_measurable borel_measurable_square)

  have "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) =

        (\<lambda>x. 0 + ((f x + -g x) ^ 2 * inverse -4))"

    by (simp add: minus_divide_right)

  also have "... \<in> borel_measurable M"

    using f g by (fast intro: affine_borel_measurable borel_measurable_square f g)

  finally have 2: "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) \<in> borel_measurable M" .

  show ?thesis

    apply (simp add: times_eq_sum_squares diff_minus)

    using 1 2 by simp

qed

lemma (in sigma_algebra) borel_measurable_diff[simp, intro]:

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

  assumes f: "f \<in> borel_measurable M"

  assumes g: "g \<in> borel_measurable M"

  shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"

  unfolding diff_minus using assms by fast

lemma (in sigma_algebra) borel_measurable_setsum[simp, intro]:

  fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"

  assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"

  shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"

proof cases

  assume "finite S"

  thus ?thesis using assms by induct auto

qed simp

lemma (in sigma_algebra) borel_measurable_inverse[simp, intro]:

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

  assumes "f \<in> borel_measurable M"

  shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"

  unfolding borel_measurable_iff_ge unfolding inverse_eq_divide

proof safe

  fix a :: real

  have *: "{w \<in> space M. a \<le> 1 / f w} =

      ({w \<in> space M. 0 < f w} \<inter> {w \<in> space M. a * f w \<le> 1}) \<union>

      ({w \<in> space M. f w < 0} \<inter> {w \<in> space M. 1 \<le> a * f w}) \<union>

      ({w \<in> space M. f w = 0} \<inter> {w \<in> space M. a \<le> 0})" by (auto simp: le_divide_eq)

  show "{w \<in> space M. a \<le> 1 / f w} \<in> sets M" using assms unfolding *

    by (auto intro!: Int Un)

qed

lemma (in sigma_algebra) borel_measurable_divide[simp, intro]:

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

  assumes "f \<in> borel_measurable M"

  and "g \<in> borel_measurable M"

  shows "(\<lambda>x. f x / g x) \<in> borel_measurable M"

  unfolding field_divide_inverse

  by (rule borel_measurable_inverse borel_measurable_times assms)+

lemma (in sigma_algebra) borel_measurable_max[intro, simp]:

  fixes f g :: "'a \<Rightarrow> real"

  assumes "f \<in> borel_measurable M"

  assumes "g \<in> borel_measurable M"

  shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"

  unfolding borel_measurable_iff_le

proof safe

  fix a

  have "{x \<in> space M. max (g x) (f x) \<le> a} =

    {x \<in> space M. g x \<le> a} \<inter> {x \<in> space M. f x \<le> a}" by auto

  thus "{x \<in> space M. max (g x) (f x) \<le> a} \<in> sets M"

    using assms unfolding borel_measurable_iff_le

    by (auto intro!: Int)

qed

lemma (in sigma_algebra) borel_measurable_min[intro, simp]:

  fixes f g :: "'a \<Rightarrow> real"

  assumes "f \<in> borel_measurable M"

  assumes "g \<in> borel_measurable M"

  shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"

  unfolding borel_measurable_iff_ge

proof safe

  fix a

  have "{x \<in> space M. a \<le> min (g x) (f x)} =

    {x \<in> space M. a \<le> g x} \<inter> {x \<in> space M. a \<le> f x}" by auto

  thus "{x \<in> space M. a \<le> min (g x) (f x)} \<in> sets M"

    using assms unfolding borel_measurable_iff_ge

    by (auto intro!: Int)

qed

lemma (in sigma_algebra) borel_measurable_abs[simp, intro]:

  assumes "f \<in> borel_measurable M"

  shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"

proof -

  have *: "\<And>x. \<bar>f x\<bar> = max 0 (f x) + max 0 (- f x)" by (simp add: max_def)

  show ?thesis unfolding * using assms by auto

qed

section "Borel space over the real line with infinity"

lemma borel_space_Real_measurable:

  "A \<in> sets borel_space \<Longrightarrow> Real -` A \<in> sets borel_space"

proof (rule borel_measurable_translate)

  fix B :: "pinfreal set" assume "open B"

  then obtain T x where T: "open T" "Real ` (T \<inter> {0..}) = B - {\<omega>}" and

    x: "\<omega> \<in> B \<Longrightarrow> 0 \<le> x" "\<omega> \<in> B \<Longrightarrow> {Real x <..} \<subseteq> B"

    unfolding open_pinfreal_def by blast

  have "Real -` B = Real -` (B - {\<omega>})" by auto

  also have "\<dots> = Real -` (Real ` (T \<inter> {0..}))" using T by simp

  also have "\<dots> = (if 0 \<in> T then T \<union> {.. 0} else T \<inter> {0..})"

    apply (auto simp add: Real_eq_Real image_iff)

    apply (rule_tac x="max 0 x" in bexI)

    by (auto simp: max_def)

  finally show "Real -` B \<in> sets borel_space"

    using `open T` by auto

qed simp

lemma borel_space_real_measurable:

  "A \<in> sets borel_space \<Longrightarrow> (real -` A :: pinfreal set) \<in> sets borel_space"

proof (rule borel_measurable_translate)

  fix B :: "real set" assume "open B"

  { fix x have "0 < real x \<longleftrightarrow> (\<exists>r>0. x = Real r)" by (cases x) auto }

  note Ex_less_real = this

  have *: "real -` B = (if 0 \<in> B then real -` (B \<inter> {0 <..}) \<union> {0, \<omega>} else real -` (B \<inter> {0 <..}))"

    by (force simp: Ex_less_real)

  have "open (real -` (B \<inter> {0 <..}) :: pinfreal set)"

    unfolding open_pinfreal_def using `open B`

    by (auto intro!: open_Int exI[of _ "B \<inter> {0 <..}"] simp: image_iff Ex_less_real)

  then show "(real -` B :: pinfreal set) \<in> sets borel_space" unfolding * by auto

qed simp

lemma (in sigma_algebra) borel_measurable_Real[intro, simp]:

  assumes "f \<in> borel_measurable M"

  shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M"

  unfolding in_borel_measurable_borel_space

proof safe

  fix S :: "pinfreal set" assume "S \<in> sets borel_space"

  from borel_space_Real_measurable[OF this]

  have "(Real \<circ> f) -` S \<inter> space M \<in> sets M"

    using assms

    unfolding vimage_compose in_borel_measurable_borel_space

    by auto

  thus "(\<lambda>x. Real (f x)) -` S \<inter> space M \<in> sets M" by (simp add: comp_def)

qed

lemma (in sigma_algebra) borel_measurable_real[intro, simp]:

  fixes f :: "'a \<Rightarrow> pinfreal"

  assumes "f \<in> borel_measurable M"

  shows "(\<lambda>x. real (f x)) \<in> borel_measurable M"

  unfolding in_borel_measurable_borel_space

proof safe

  fix S :: "real set" assume "S \<in> sets borel_space"

  from borel_space_real_measurable[OF this]

  have "(real \<circ> f) -` S \<inter> space M \<in> sets M"

    using assms

    unfolding vimage_compose in_borel_measurable_borel_space

    by auto

  thus "(\<lambda>x. real (f x)) -` S \<inter> space M \<in> sets M" by (simp add: comp_def)

qed

lemma (in sigma_algebra) borel_measurable_Real_eq:

  assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"

  shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"

proof

  have [simp]: "(\<lambda>x. Real (f x)) -` {\<omega>} \<inter> space M = {}"

    by auto

  assume "(\<lambda>x. Real (f x)) \<in> borel_measurable M"

  hence "(\<lambda>x. real (Real (f x))) \<in> borel_measurable M"

    by (rule borel_measurable_real)

  moreover have "\<And>x. x \<in> space M \<Longrightarrow> real (Real (f x)) = f x"

    using assms by auto

  ultimately show "f \<in> borel_measurable M"

    by (simp cong: measurable_cong)

qed auto

lemma (in sigma_algebra) borel_measurable_pinfreal_eq_real:

  "f \<in> borel_measurable M \<longleftrightarrow>

    ((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<omega>} \<inter> space M \<in> sets M)"

proof safe

  assume "f \<in> borel_measurable M"

  then show "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<omega>} \<inter> space M \<in> sets M"

    by (auto intro: borel_measurable_vimage borel_measurable_real)

next

  assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<omega>} \<inter> space M \<in> sets M"

  have "f -` {\<omega>} \<inter> space M = {x\<in>space M. f x = \<omega>}" by auto

  with * have **: "{x\<in>space M. f x = \<omega>} \<in> sets M" by simp

  have f: "f = (\<lambda>x. if f x = \<omega> then \<omega> else Real (real (f x)))"

    by (simp add: expand_fun_eq Real_real)

  show "f \<in> borel_measurable M"

    apply (subst f)

    apply (rule measurable_If)

    using * ** by auto

qed

lemma (in sigma_algebra) less_eq_ge_measurable:

  fixes f :: "'a \<Rightarrow> 'c::linorder"

  shows "{x\<in>space M. a < f x} \<in> sets M \<longleftrightarrow> {x\<in>space M. f x \<le> a} \<in> sets M"

proof

  assume "{x\<in>space M. f x \<le> a} \<in> sets M"

  moreover have "{x\<in>space M. a < f x} = space M - {x\<in>space M. f x \<le> a}" by auto

  ultimately show "{x\<in>space M. a < f x} \<in> sets M" by auto

next

  assume "{x\<in>space M. a < f x} \<in> sets M"

  moreover have "{x\<in>space M. f x \<le> a} = space M - {x\<in>space M. a < f x}" by auto

  ultimately show "{x\<in>space M. f x \<le> a} \<in> sets M" by auto

qed

lemma (in sigma_algebra) greater_eq_le_measurable:

  fixes f :: "'a \<Rightarrow> 'c::linorder"

  shows "{x\<in>space M. f x < a} \<in> sets M \<longleftrightarrow> {x\<in>space M. a \<le> f x} \<in> sets M"

proof

  assume "{x\<in>space M. a \<le> f x} \<in> sets M"

  moreover have "{x\<in>space M. f x < a} = space M - {x\<in>space M. a \<le> f x}" by auto

  ultimately show "{x\<in>space M. f x < a} \<in> sets M" by auto

next

  assume "{x\<in>space M. f x < a} \<in> sets M"

  moreover have "{x\<in>space M. a \<le> f x} = space M - {x\<in>space M. f x < a}" by auto

  ultimately show "{x\<in>space M. a \<le> f x} \<in> sets M" by auto

qed

lemma (in sigma_algebra) less_eq_le_pinfreal_measurable:

  fixes f :: "'a \<Rightarrow> pinfreal"

  shows "(\<forall>a. {x\<in>space M. a < f x} \<in> sets M) \<longleftrightarrow> (\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M)"