(*  Title:      HOL/Probability/Borel_Space.thy

     Author:     Johannes Hölzl, TU München

     Author:     Armin Heller, TU München

 *)

 section {*Borel spaces*}

 theory Borel_Space

 imports

   Measurable

   "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"

 begin

 lemma topological_basis_trivial: "topological_basis {A. open A}"

   by (auto simp: topological_basis_def)

 lemma open_prod_generated: "open = generate_topology {A \<times> B | A B. open A \<and> open B}"

 proof -

   have "{A \<times> B :: ('a \<times> 'b) set | A B. open A \<and> open B} = ((\<lambda>(a, b). a \<times> b) ` ({A. open A} \<times> {A. open A}))"

     by auto

   then show ?thesis

     by (auto intro: topological_basis_prod topological_basis_trivial topological_basis_imp_subbasis)  

 qed

 subsection {* Generic Borel spaces *}

 definition borel :: "'a::topological_space measure" where

   "borel = sigma UNIV {S. open S}"

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

 lemma in_borel_measurable:

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

     (\<forall>S \<in> sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)"

   by (auto simp add: measurable_def borel_def)

 lemma in_borel_measurable_borel:

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

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

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

   by (auto simp add: measurable_def borel_def)

 lemma space_borel[simp]: "space borel = UNIV"

   unfolding borel_def by auto

 lemma space_in_borel[measurable]: "UNIV \<in> sets borel"

   unfolding borel_def by auto

 lemma sets_borel: "sets borel = sigma_sets UNIV {S. open S}"

   unfolding borel_def by (rule sets_measure_of) simp

 lemma pred_Collect_borel[measurable (raw)]: "Measurable.pred borel P \<Longrightarrow> {x. P x} \<in> sets borel"

   unfolding borel_def pred_def by auto

 lemma borel_open[measurable (raw generic)]:

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

 proof -

   have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms .

   thus ?thesis unfolding borel_def by auto

 qed

 lemma borel_closed[measurable (raw generic)]:

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

 proof -

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

     using assms unfolding closed_def by (blast intro: borel_open)

   thus ?thesis by simp

 qed

 lemma borel_singleton[measurable]:

   "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets (borel :: 'a::t1_space measure)"

   unfolding insert_def by (rule sets.Un) auto

 lemma borel_comp[measurable]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"

   unfolding Compl_eq_Diff_UNIV by simp

 lemma borel_measurable_vimage:

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

   assumes borel[measurable]: "f \<in> borel_measurable M"

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

   by simp

 lemma borel_measurableI:

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

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

   shows "f \<in> borel_measurable M"

   unfolding borel_def

 proof (rule measurable_measure_of, simp_all)

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

     using assms[of S] by simp

 qed

 lemma borel_measurable_const:

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

   by auto

 lemma borel_measurable_indicator:

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

   shows "indicator A \<in> borel_measurable M"

   unfolding indicator_def [abs_def] using A

   by (auto intro!: measurable_If_set)

 lemma borel_measurable_count_space[measurable (raw)]:

   "f \<in> borel_measurable (count_space S)"

   unfolding measurable_def by auto

 lemma borel_measurable_indicator'[measurable (raw)]:

   assumes [measurable]: "{x\<in>space M. f x \<in> A x} \<in> sets M"

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

   unfolding indicator_def[abs_def]

   by (auto intro!: measurable_If)

 lemma borel_measurable_indicator_iff:

   "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"

     (is "?I \<in> borel_measurable M \<longleftrightarrow> _")

 proof

   assume "?I \<in> borel_measurable M"

   then have "?I -` {1} \<inter> space M \<in> sets M"

     unfolding measurable_def by auto

   also have "?I -` {1} \<inter> space M = A \<inter> space M"

     unfolding indicator_def [abs_def] by auto

   finally show "A \<inter> space M \<in> sets M" .

 next

   assume "A \<inter> space M \<in> sets M"

   moreover have "?I \<in> borel_measurable M \<longleftrightarrow>

     (indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"

     by (intro measurable_cong) (auto simp: indicator_def)

   ultimately show "?I \<in> borel_measurable M" by auto

 qed

 lemma borel_measurable_subalgebra:

   assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"

   shows "f \<in> borel_measurable M"

   using assms unfolding measurable_def by auto

 lemma borel_measurable_restrict_space_iff_ereal:

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

   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"

   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>

     (\<lambda>x. f x * indicator \<Omega> x) \<in> borel_measurable M"

   by (subst measurable_restrict_space_iff)

      (auto simp: indicator_def if_distrib[where f="\<lambda>x. a * x" for a] cong del: if_cong)

 lemma borel_measurable_restrict_space_iff:

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

   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"

   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>

     (\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> borel_measurable M"

   by (subst measurable_restrict_space_iff)

      (auto simp: indicator_def if_distrib[where f="\<lambda>x. x *\<^sub>R a" for a] ac_simps cong del: if_cong)

 lemma cbox_borel[measurable]: "cbox a b \<in> sets borel"

   by (auto intro: borel_closed)

 lemma box_borel[measurable]: "box a b \<in> sets borel"

   by (auto intro: borel_open)

 lemma borel_compact: "compact (A::'a::t2_space set) \<Longrightarrow> A \<in> sets borel"

   by (auto intro: borel_closed dest!: compact_imp_closed)

 lemma second_countable_borel_measurable:

   fixes X :: "'a::second_countable_topology set set"

   assumes eq: "open = generate_topology X"

   shows "borel = sigma UNIV X"

   unfolding borel_def

 proof (intro sigma_eqI sigma_sets_eqI)

   interpret X: sigma_algebra UNIV "sigma_sets UNIV X"

     by (rule sigma_algebra_sigma_sets) simp

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

   then have "generate_topology X S"

     by (auto simp: eq)

   then show "S \<in> sigma_sets UNIV X"

   proof induction

     case (UN K)

     then have K: "\<And>k. k \<in> K \<Longrightarrow> open k"

       unfolding eq by auto

     from ex_countable_basis obtain B :: "'a set set" where

       B:  "\<And>b. b \<in> B \<Longrightarrow> open b" "\<And>X. open X \<Longrightarrow> \<exists>b\<subseteq>B. (\<Union>b) = X" and "countable B"

       by (auto simp: topological_basis_def)

     from B(2)[OF K] obtain m where m: "\<And>k. k \<in> K \<Longrightarrow> m k \<subseteq> B" "\<And>k. k \<in> K \<Longrightarrow> (\<Union>m k) = k"

       by metis

     def U \<equiv> "(\<Union>k\<in>K. m k)"

     with m have "countable U"

       by (intro countable_subset[OF _ `countable B`]) auto

     have "\<Union>U = (\<Union>A\<in>U. A)" by simp

     also have "\<dots> = \<Union>K"

       unfolding U_def UN_simps by (simp add: m)

     finally have "\<Union>U = \<Union>K" .

     have "\<forall>b\<in>U. \<exists>k\<in>K. b \<subseteq> k"

       using m by (auto simp: U_def)

     then obtain u where u: "\<And>b. b \<in> U \<Longrightarrow> u b \<in> K" and "\<And>b. b \<in> U \<Longrightarrow> b \<subseteq> u b"

       by metis

     then have "(\<Union>b\<in>U. u b) \<subseteq> \<Union>K" "\<Union>U \<subseteq> (\<Union>b\<in>U. u b)"

       by auto

     then have "\<Union>K = (\<Union>b\<in>U. u b)"

       unfolding `\<Union>U = \<Union>K` by auto

     also have "\<dots> \<in> sigma_sets UNIV X"

       using u UN by (intro X.countable_UN' `countable U`) auto

     finally show "\<Union>K \<in> sigma_sets UNIV X" .

   qed auto

 qed (auto simp: eq intro: generate_topology.Basis)

 lemma borel_measurable_continuous_on_restrict:

   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"

   assumes f: "continuous_on A f"

   shows "f \<in> borel_measurable (restrict_space borel A)"

 proof (rule borel_measurableI)

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

   with f obtain T where "f -` S \<inter> A = T \<inter> A" "open T"

     by (metis continuous_on_open_invariant)

   then show "f -` S \<inter> space (restrict_space borel A) \<in> sets (restrict_space borel A)"

     by (force simp add: sets_restrict_space space_restrict_space)

 qed

 lemma borel_measurable_continuous_on1: "continuous_on UNIV f \<Longrightarrow> f \<in> borel_measurable borel"

   by (drule borel_measurable_continuous_on_restrict) simp

 lemma borel_measurable_continuous_on_if:

   "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> continuous_on (- A) g \<Longrightarrow>

     (\<lambda>x. if x \<in> A then f x else g x) \<in> borel_measurable borel"

   by (auto simp add: measurable_If_restrict_space_iff Collect_neg_eq

            intro!: borel_measurable_continuous_on_restrict)

 lemma borel_measurable_continuous_countable_exceptions:

   fixes f :: "'a::t1_space \<Rightarrow> 'b::topological_space"

   assumes X: "countable X"

   assumes "continuous_on (- X) f"

   shows "f \<in> borel_measurable borel"

 proof (rule measurable_discrete_difference[OF _ X])

   have "X \<in> sets borel"

     by (rule sets.countable[OF _ X]) auto

   then show "(\<lambda>x. if x \<in> X then undefined else f x) \<in> borel_measurable borel"

     by (intro borel_measurable_continuous_on_if assms continuous_intros)

 qed auto

 lemma borel_measurable_continuous_on:

   assumes f: "continuous_on UNIV f" and g: "g \<in> borel_measurable M"

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

   using measurable_comp[OF g borel_measurable_continuous_on1[OF f]] by (simp add: comp_def)

 lemma borel_measurable_continuous_on_indicator:

   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"

   shows "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable borel"

   by (subst borel_measurable_restrict_space_iff[symmetric])

      (auto intro: borel_measurable_continuous_on_restrict)

 lemma borel_eq_countable_basis:

   fixes B::"'a::topological_space set set"

   assumes "countable B"

   assumes "topological_basis B"

   shows "borel = sigma UNIV B"

   unfolding borel_def

 proof (intro sigma_eqI sigma_sets_eqI, safe)

   interpret countable_basis using assms by unfold_locales

   fix X::"'a set" assume "open X"

   from open_countable_basisE[OF this] guess B' . note B' = this

   then show "X \<in> sigma_sets UNIV B"

     by (blast intro: sigma_sets_UNION `countable B` countable_subset)

 next

   fix b assume "b \<in> B"

   hence "open b" by (rule topological_basis_open[OF assms(2)])

   thus "b \<in> sigma_sets UNIV (Collect open)" by auto

 qed simp_all

 lemma borel_measurable_Pair[measurable (raw)]:

   fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"

   assumes f[measurable]: "f \<in> borel_measurable M"

   assumes g[measurable]: "g \<in> borel_measurable M"

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

 proof (subst borel_eq_countable_basis)

   let ?B = "SOME B::'b set set. countable B \<and> topological_basis B"

   let ?C = "SOME B::'c set set. countable B \<and> topological_basis B"

   let ?P = "(\<lambda>(b, c). b \<times> c) ` (?B \<times> ?C)"

   show "countable ?P" "topological_basis ?P"

     by (auto intro!: countable_basis topological_basis_prod is_basis)

   show "(\<lambda>x. (f x, g x)) \<in> measurable M (sigma UNIV ?P)"

   proof (rule measurable_measure_of)

     fix S assume "S \<in> ?P"

     then obtain b c where "b \<in> ?B" "c \<in> ?C" and S: "S = b \<times> c" by auto

     then have borel: "open b" "open c"

       by (auto intro: is_basis topological_basis_open)

     have "(\<lambda>x. (f x, g x)) -` S \<inter> space M = (f -` b \<inter> space M) \<inter> (g -` c \<inter> space M)"

       unfolding S by auto

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

       using borel by simp

     finally show "(\<lambda>x. (f x, g x)) -` S \<inter> space M \<in> sets M" .

   qed auto

 qed

 lemma borel_measurable_continuous_Pair:

   fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"

   assumes [measurable]: "f \<in> borel_measurable M"

   assumes [measurable]: "g \<in> borel_measurable M"

   assumes H: "continuous_on UNIV (\<lambda>x. H (fst x) (snd x))"

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

 proof -

   have eq: "(\<lambda>x. H (f x) (g x)) = (\<lambda>x. (\<lambda>x. H (fst x) (snd x)) (f x, g x))" by auto

   show ?thesis

     unfolding eq by (rule borel_measurable_continuous_on[OF H]) auto

 qed

 subsection {* Borel spaces on order topologies *}

 lemma borel_Iio:

   "borel = sigma UNIV (range lessThan :: 'a::{linorder_topology, second_countable_topology} set set)"

   unfolding second_countable_borel_measurable[OF open_generated_order]

 proof (intro sigma_eqI sigma_sets_eqI)

   from countable_dense_setE guess D :: "'a set" . note D = this

   interpret L: sigma_algebra UNIV "sigma_sets UNIV (range lessThan)"

     by (rule sigma_algebra_sigma_sets) simp

   fix A :: "'a set" assume "A \<in> range lessThan \<union> range greaterThan"

   then obtain y where "A = {y <..} \<or> A = {..< y}"

     by blast

   then show "A \<in> sigma_sets UNIV (range lessThan)"

   proof

     assume A: "A = {y <..}"

     show ?thesis

     proof cases

       assume "\<forall>x>y. \<exists>d. y < d \<and> d < x"

       with D(2)[of "{y <..< x}" for x] have "\<forall>x>y. \<exists>d\<in>D. y < d \<and> d < x"

         by (auto simp: set_eq_iff)

       then have "A = UNIV - (\<Inter>d\<in>{d\<in>D. y < d}. {..< d})"

         by (auto simp: A) (metis less_asym)

       also have "\<dots> \<in> sigma_sets UNIV (range lessThan)"

         using D(1) by (intro L.Diff L.top L.countable_INT'') auto

       finally show ?thesis .

     next

       assume "\<not> (\<forall>x>y. \<exists>d. y < d \<and> d < x)"

       then obtain x where "y < x"  "\<And>d. y < d \<Longrightarrow> \<not> d < x"

         by auto

       then have "A = UNIV - {..< x}"

         unfolding A by (auto simp: not_less[symmetric])

       also have "\<dots> \<in> sigma_sets UNIV (range lessThan)"

         by auto

       finally show ?thesis .

     qed

   qed auto

 qed auto

 lemma borel_Ioi:

   "borel = sigma UNIV (range greaterThan :: 'a::{linorder_topology, second_countable_topology} set set)"

   unfolding second_countable_borel_measurable[OF open_generated_order]

 proof (intro sigma_eqI sigma_sets_eqI)

   from countable_dense_setE guess D :: "'a set" . note D = this

   interpret L: sigma_algebra UNIV "sigma_sets UNIV (range greaterThan)"

     by (rule sigma_algebra_sigma_sets) simp

   fix A :: "'a set" assume "A \<in> range lessThan \<union> range greaterThan"

   then obtain y where "A = {y <..} \<or> A = {..< y}"

     by blast

   then show "A \<in> sigma_sets UNIV (range greaterThan)"

   proof

     assume A: "A = {..< y}"

     show ?thesis

     proof cases

       assume "\<forall>x<y. \<exists>d. x < d \<and> d < y"

       with D(2)[of "{x <..< y}" for x] have "\<forall>x<y. \<exists>d\<in>D. x < d \<and> d < y"

         by (auto simp: set_eq_iff)

       then have "A = UNIV - (\<Inter>d\<in>{d\<in>D. d < y}. {d <..})"

         by (auto simp: A) (metis less_asym)

       also have "\<dots> \<in> sigma_sets UNIV (range greaterThan)"

         using D(1) by (intro L.Diff L.top L.countable_INT'') auto

       finally show ?thesis .

     next

       assume "\<not> (\<forall>x<y. \<exists>d. x < d \<and> d < y)"

       then obtain x where "x < y"  "\<And>d. y > d \<Longrightarrow> x \<ge> d"

         by (auto simp: not_less[symmetric])

       then have "A = UNIV - {x <..}"

         unfolding A Compl_eq_Diff_UNIV[symmetric] by auto

       also have "\<dots> \<in> sigma_sets UNIV (range greaterThan)"

         by auto

       finally show ?thesis .

     qed

   qed auto

 qed auto

 lemma borel_measurableI_less:

   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"

   shows "(\<And>y. {x\<in>space M. f x < y} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"

   unfolding borel_Iio

   by (rule measurable_measure_of) (auto simp: Int_def conj_commute)

 lemma borel_measurableI_greater:

   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"

   shows "(\<And>y. {x\<in>space M. y < f x} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"

   unfolding borel_Ioi

   by (rule measurable_measure_of) (auto simp: Int_def conj_commute)

 lemma borel_measurable_SUP[measurable (raw)]:

   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}"

   assumes [simp]: "countable I"

   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"

   shows "(\<lambda>x. SUP i:I. F i x) \<in> borel_measurable M"

   by (rule borel_measurableI_greater) (simp add: less_SUP_iff)

 lemma borel_measurable_INF[measurable (raw)]:

   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}"

   assumes [simp]: "countable I"

   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"

   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"

   by (rule borel_measurableI_less) (simp add: INF_less_iff)

 lemma borel_measurable_lfp[consumes 1, case_names continuity step]:

   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})"

   assumes "sup_continuous F"

   assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M"

   shows "lfp F \<in> borel_measurable M"

 proof -

   { fix i have "((F ^^ i) bot) \<in> borel_measurable M"

       by (induct i) (auto intro!: *) }

   then have "(\<lambda>x. SUP i. (F ^^ i) bot x) \<in> borel_measurable M"

     by measurable

   also have "(\<lambda>x. SUP i. (F ^^ i) bot x) = (SUP i. (F ^^ i) bot)"

     by auto

   also have "(SUP i. (F ^^ i) bot) = lfp F"

     by (rule sup_continuous_lfp[symmetric]) fact

   finally show ?thesis .

 qed

 lemma borel_measurable_gfp[consumes 1, case_names continuity step]:

   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})"

   assumes "inf_continuous F"

   assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M"

   shows "gfp F \<in> borel_measurable M"

 proof -

   { fix i have "((F ^^ i) top) \<in> borel_measurable M"

       by (induct i) (auto intro!: * simp: bot_fun_def) }

   then have "(\<lambda>x. INF i. (F ^^ i) top x) \<in> borel_measurable M"

     by measurable

   also have "(\<lambda>x. INF i. (F ^^ i) top x) = (INF i. (F ^^ i) top)"

     by auto

   also have "\<dots> = gfp F"

     by (rule inf_continuous_gfp[symmetric]) fact

   finally show ?thesis .

 qed

 subsection {* Borel spaces on euclidean spaces *}

 lemma borel_measurable_inner[measurable (raw)]:

   fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_inner}"

   assumes "f \<in> borel_measurable M"

   assumes "g \<in> borel_measurable M"

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

   using assms

   by (rule borel_measurable_continuous_Pair) (intro continuous_intros)

 lemma [measurable]:

   fixes a b :: "'a::linorder_topology"

   shows lessThan_borel: "{..< a} \<in> sets borel"

     and greaterThan_borel: "{a <..} \<in> sets borel"

     and greaterThanLessThan_borel: "{a<..<b} \<in> sets borel"

     and atMost_borel: "{..a} \<in> sets borel"

     and atLeast_borel: "{a..} \<in> sets borel"

     and atLeastAtMost_borel: "{a..b} \<in> sets borel"

     and greaterThanAtMost_borel: "{a<..b} \<in> sets borel"

     and atLeastLessThan_borel: "{a..<b} \<in> sets borel"

   unfolding greaterThanAtMost_def atLeastLessThan_def

   by (blast intro: borel_open borel_closed open_lessThan open_greaterThan open_greaterThanLessThan

                    closed_atMost closed_atLeast closed_atLeastAtMost)+

 notation

   eucl_less (infix "<e" 50)

 lemma box_oc: "{x. a <e x \<and> x \<le> b} = {x. a <e x} \<inter> {..b}"

   and box_co: "{x. a \<le> x \<and> x <e b} = {a..} \<inter> {x. x <e b}"

   by auto

 lemma eucl_ivals[measurable]:

   fixes a b :: "'a::ordered_euclidean_space"

   shows "{x. x <e a} \<in> sets borel"

     and "{x. a <e x} \<in> sets borel"

     and "{..a} \<in> sets borel"

     and "{a..} \<in> sets borel"

     and "{a..b} \<in> sets borel"

     and  "{x. a <e x \<and> x \<le> b} \<in> sets borel"

     and "{x. a \<le> x \<and>  x <e b} \<in> sets borel"

   unfolding box_oc box_co

   by (auto intro: borel_open borel_closed)

 lemma open_Collect_less:

   fixes f g :: "'i::topological_space \<Rightarrow> 'a :: {dense_linorder, linorder_topology}"

   assumes "continuous_on UNIV f"

   assumes "continuous_on UNIV g"

   shows "open {x. f x < g x}"

 proof -

   have "open (\<Union>y. {x \<in> UNIV. f x \<in> {..< y}} \<inter> {x \<in> UNIV. g x \<in> {y <..}})" (is "open ?X")

     by (intro open_UN ballI open_Int continuous_open_preimage assms) auto

   also have "?X = {x. f x < g x}"

     by (auto intro: dense)

   finally show ?thesis .

 qed

 lemma closed_Collect_le:

   fixes f g :: "'i::topological_space \<Rightarrow> 'a :: {dense_linorder, linorder_topology}"

   assumes f: "continuous_on UNIV f"

   assumes g: "continuous_on UNIV g"

   shows "closed {x. f x \<le> g x}"

   using open_Collect_less[OF g f] unfolding not_less[symmetric] Collect_neg_eq open_closed .

 lemma borel_measurable_less[measurable]:

   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, dense_linorder, linorder_topology}"

   assumes "f \<in> borel_measurable M"

   assumes "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} = (\<lambda>x. (f x, g x)) -` {x. fst x < snd x} \<inter> space M"

     by auto

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

     by (intro measurable_sets[OF borel_measurable_Pair borel_open, OF assms open_Collect_less]

               continuous_intros)

   finally show ?thesis .

 qed

 lemma

   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, dense_linorder, linorder_topology}"

   assumes f[measurable]: "f \<in> borel_measurable M"

   assumes g[measurable]: "g \<in> borel_measurable M"

   shows borel_measurable_le[measurable]: "{w \<in> space M. f w \<le> g w} \<in> sets M"

     and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M"

     and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M"

   unfolding eq_iff not_less[symmetric]

   by measurable

 lemma 

   fixes i :: "'a::{second_countable_topology, real_inner}"

   shows hafspace_less_borel: "{x. a < x \<bullet> i} \<in> sets borel"

     and hafspace_greater_borel: "{x. x \<bullet> i < a} \<in> sets borel"

     and hafspace_less_eq_borel: "{x. a \<le> x \<bullet> i} \<in> sets borel"

     and hafspace_greater_eq_borel: "{x. x \<bullet> i \<le> a} \<in> sets borel"

   by simp_all

 subsection "Borel space equals sigma algebras over intervals"

 lemma borel_sigma_sets_subset:

   "A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel"

   using sets.sigma_sets_subset[of A borel] by simp

 lemma borel_eq_sigmaI1:

   fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"

   assumes borel_eq: "borel = sigma UNIV X"

   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (F ` A))"

   assumes F: "\<And>i. i \<in> A \<Longrightarrow> F i \<in> sets borel"

   shows "borel = sigma UNIV (F ` A)"

   unfolding borel_def

 proof (intro sigma_eqI antisym)

   have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel"

     unfolding borel_def by simp

   also have "\<dots> = sigma_sets UNIV X"

     unfolding borel_eq by simp

   also have "\<dots> \<subseteq> sigma_sets UNIV (F`A)"

     using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto

   finally show "sigma_sets UNIV {S. open S} \<subseteq> sigma_sets UNIV (F`A)" .

   show "sigma_sets UNIV (F`A) \<subseteq> sigma_sets UNIV {S. open S}"

     unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto

 qed auto

 lemma borel_eq_sigmaI2:

   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set"

     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"

   assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`B)"

   assumes X: "\<And>i j. (i, j) \<in> B \<Longrightarrow> G i j \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"

   assumes F: "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel"

   shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"

   using assms

   by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` B" and F="(\<lambda>(i, j). F i j)"]) auto

 lemma borel_eq_sigmaI3:

   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"

   assumes borel_eq: "borel = sigma UNIV X"

   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"

   assumes F: "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel"

   shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"

   using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto

 lemma borel_eq_sigmaI4:

   fixes F :: "'i \<Rightarrow> 'a::topological_space set"

     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"

   assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`A)"

   assumes X: "\<And>i j. (i, j) \<in> A \<Longrightarrow> G i j \<in> sets (sigma UNIV (range F))"

   assumes F: "\<And>i. F i \<in> sets borel"

   shows "borel = sigma UNIV (range F)"

   using assms by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` A" and F=F]) auto

 lemma borel_eq_sigmaI5:

   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set"

   assumes borel_eq: "borel = sigma UNIV (range G)"

   assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"

   assumes F: "\<And>i j. F i j \<in> sets borel"

   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"

   using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto

 lemma borel_eq_box:

   "borel = sigma UNIV (range (\<lambda> (a, b). box a b :: 'a :: euclidean_space set))"

     (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI1[OF borel_def])

   fix M :: "'a set" assume "M \<in> {S. open S}"

   then have "open M" by simp

   show "M \<in> ?SIGMA"

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

     apply (safe intro!: sets.countable_UN' countable_PiE countable_Collect)

     apply (auto intro: countable_rat)

     done

 qed (auto simp: box_def)

 lemma halfspace_gt_in_halfspace:

   assumes i: "i \<in> A"

   shows "{x::'a. a < x \<bullet> i} \<in> 

     sigma_sets UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> A))"

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

 proof -

   interpret sigma_algebra UNIV ?SIGMA

     by (intro sigma_algebra_sigma_sets) simp_all

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

   proof (safe, simp_all add: not_less del: real_of_nat_Suc)

     fix x :: 'a assume "a < x \<bullet> i"

     with reals_Archimedean[of "x \<bullet> i - a"]

     obtain n where "a + 1 / real (Suc n) < x \<bullet> i"

       by (auto simp: field_simps)

     then show "\<exists>n. a + 1 / real (Suc n) \<le> x \<bullet> 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> ?SIGMA" unfolding *

     by (auto intro!: Diff sigma_sets_Inter i)

 qed

 lemma borel_eq_halfspace_less:

   "borel = sigma UNIV ((\<lambda>(a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> Basis))"

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI2[OF borel_eq_box])

   fix a b :: 'a

   have "box a b = {x\<in>space ?SIGMA. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"

     by (auto simp: box_def)

   also have "\<dots> \<in> sets ?SIGMA"

     by (intro sets.sets_Collect_conj sets.sets_Collect_finite_All sets.sets_Collect_const)

        (auto intro!: halfspace_gt_in_halfspace countable_PiE countable_rat)

   finally show "box a b \<in> sets ?SIGMA" .

 qed auto

 lemma borel_eq_halfspace_le:

   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i \<le> a}) ` (UNIV \<times> Basis))"

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])

   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"

   then have i: "i \<in> Basis" by auto

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

   proof (safe, simp_all del: real_of_nat_Suc)

     fix x::'a assume *: "x\<bullet>i < a"

     with reals_Archimedean[of "a - x\<bullet>i"]

     obtain n where "x \<bullet> i < a - 1 / (real (Suc n))"

       by (auto simp: field_simps)

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

       by (blast intro: less_imp_le)

   next

     fix x::'a and n

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

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

     finally show "x\<bullet>i < a" .

   qed

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

     by (intro sets.countable_UN) (auto intro: i)

 qed auto

 lemma borel_eq_halfspace_ge:

   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. a \<le> x \<bullet> i}) ` (UNIV \<times> Basis))"

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])

   fix a :: real and i :: 'a assume i: "(a, i) \<in> UNIV \<times> Basis"

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

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

     using i by (intro sets.compl_sets) auto

 qed auto

 lemma borel_eq_halfspace_greater:

   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. a < x \<bullet> i}) ` (UNIV \<times> Basis))"

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])

   fix a :: real and i :: 'a assume "(a, i) \<in> (UNIV \<times> Basis)"

   then have i: "i \<in> Basis" by auto

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

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

     by (intro sets.compl_sets) (auto intro: i)

 qed auto

 lemma borel_eq_atMost:

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

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])

   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"

   then have "i \<in> Basis" by auto

   then have *: "{x::'a. x\<bullet>i \<le> a} = (\<Union>k::nat. {.. (\<Sum>n\<in>Basis. (if n = i then a else real k)*\<^sub>R n)})"

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

     fix x :: 'a

     from real_arch_simple[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"] guess k::nat ..

     then have "\<And>i. i \<in> Basis \<Longrightarrow> x\<bullet>i \<le> real k"

       by (subst (asm) Max_le_iff) auto

     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> x \<bullet> ia \<le> real k"

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

   qed

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

     by (intro sets.countable_UN) auto

 qed auto

 lemma borel_eq_greaterThan:

   "borel = sigma UNIV (range (\<lambda>a::'a::ordered_euclidean_space. {x. a <e x}))"

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])

   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"

   then have i: "i \<in> Basis" by auto

   have "{x::'a. x\<bullet>i \<le> a} = UNIV - {x::'a. a < x\<bullet>i}" by auto

   also have *: "{x::'a. a < x\<bullet>i} =

       (\<Union>k::nat. {x. (\<Sum>n\<in>Basis. (if n = i then a else -real k) *\<^sub>R n) <e x})" using i

   proof (safe, simp_all add: eucl_less_def split: split_if_asm)

     fix x :: 'a

     from reals_Archimedean2[of "Max ((\<lambda>i. -x\<bullet>i)`Basis)"]

     guess k::nat .. note k = this

     { fix i :: 'a assume "i \<in> Basis"

       then have "-x\<bullet>i < real k"

         using k by (subst (asm) Max_less_iff) auto

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

     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> -real k < x \<bullet> ia"

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

   qed

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

     apply (simp only:)

     apply (intro sets.countable_UN sets.Diff)

     apply (auto intro: sigma_sets_top)

     done

 qed auto

 lemma borel_eq_lessThan:

   "borel = sigma UNIV (range (\<lambda>a::'a::ordered_euclidean_space. {x. x <e a}))"

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge])

   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"

   then have i: "i \<in> Basis" by auto

   have "{x::'a. a \<le> x\<bullet>i} = UNIV - {x::'a. x\<bullet>i < a}" by auto

   also have *: "{x::'a. x\<bullet>i < a} = (\<Union>k::nat. {x. x <e (\<Sum>n\<in>Basis. (if n = i then a else real k) *\<^sub>R n)})" using `i\<in> Basis`

   proof (safe, simp_all add: eucl_less_def split: split_if_asm)

     fix x :: 'a

     from reals_Archimedean2[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"]

     guess k::nat .. note k = this

     { fix i :: 'a assume "i \<in> Basis"

       then have "x\<bullet>i < real k"

         using k by (subst (asm) Max_less_iff) auto

       then have "x\<bullet>i < real k" by simp }

     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> x \<bullet> ia < real k"

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

   qed

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

     apply (simp only:)

     apply (intro sets.countable_UN sets.Diff)

     apply (auto intro: sigma_sets_top )

     done

 qed auto

 lemma borel_eq_atLeastAtMost:

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

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])

   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 :: 'a

     from real_arch_simple[of "Max ((\<lambda>i. - x\<bullet>i)`Basis)"]

     guess k::nat .. note k = this

     { fix i :: 'a assume "i \<in> Basis"

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

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

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

     then show "\<exists>n::nat. \<forall>i\<in>Basis. - real n \<le> x \<bullet> i"

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

   qed

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

     by (intro sets.countable_UN)

        (auto intro!: sigma_sets_top)

 qed auto

 lemma borel_sigma_sets_Ioc: "borel = sigma UNIV (range (\<lambda>(a, b). {a <.. b::real}))"

 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])

   fix i :: real

   have "{..i} = (\<Union>j::nat. {-j <.. i})"

     by (auto simp: minus_less_iff reals_Archimedean2)

   also have "\<dots> \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))"

     by (intro sets.countable_nat_UN) auto 

   finally show "{..i} \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))" .

 qed simp

 lemma eucl_lessThan: "{x::real. x <e a} = lessThan a"

   by (simp add: eucl_less_def lessThan_def)

 lemma borel_eq_atLeastLessThan:

   "borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan])

   have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto

   fix x :: real

   have "{..<x} = (\<Union>i::nat. {-real i ..< x})"

     by (auto simp: move_uminus real_arch_simple)

   then show "{y. y <e x} \<in> ?SIGMA"

     by (auto intro: sigma_sets.intros(2-) simp: eucl_lessThan)

 qed auto

 lemma borel_eq_closed: "borel = sigma UNIV (Collect closed)"

   unfolding borel_def

 proof (intro sigma_eqI sigma_sets_eqI, safe)

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

   hence "x = UNIV - (UNIV - x)" by auto

   also have "\<dots> \<in> sigma_sets UNIV (Collect closed)"

     by (force intro: sigma_sets.Compl simp: `open x`)

   finally show "x \<in> sigma_sets UNIV (Collect closed)" by simp

 next

   fix x :: "'a set" assume "closed x"

   hence "x = UNIV - (UNIV - x)" by auto

   also have "\<dots> \<in> sigma_sets UNIV (Collect open)"

     by (force intro: sigma_sets.Compl simp: `closed x`)

   finally show "x \<in> sigma_sets UNIV (Collect open)" by simp

 qed simp_all

 lemma borel_measurable_halfspacesI:

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

   assumes F: "borel = sigma UNIV (F ` (UNIV \<times> Basis))"

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

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

 proof safe

   fix a :: real and i :: 'b assume i: "i \<in> Basis" and f: "f \<in> borel_measurable M"

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

     by (auto intro!: measurable_sets simp: assms(1))

 next

   assume a: "\<forall>i\<in>Basis. \<forall>a. S a i \<in> sets M"

   then show "f \<in> borel_measurable M"

     by (auto intro!: measurable_measure_of simp: S_eq F)

 qed

 lemma borel_measurable_iff_halfspace_le:

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

   shows "f \<in> borel_measurable M = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. f w \<bullet> i \<le> a} \<in> sets M)"

   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto

 lemma borel_measurable_iff_halfspace_less:

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

   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. f w \<bullet> i < a} \<in> sets M)"

   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto

 lemma borel_measurable_iff_halfspace_ge:

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

   shows "f \<in> borel_measurable M = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. a \<le> f w \<bullet> i} \<in> sets M)"

   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto

 lemma borel_measurable_iff_halfspace_greater:

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

   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. a < f w \<bullet> i} \<in> sets M)"

   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto

 lemma 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 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 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 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

 lemma borel_measurable_euclidean_space:

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

   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M)"

 proof safe

   assume f: "\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M"

   then show "f \<in> borel_measurable M"

     by (subst borel_measurable_iff_halfspace_le) auto

 qed auto

 subsection "Borel measurable operators"

 lemma borel_measurable_norm[measurable]: "norm \<in> borel_measurable borel"

   by (intro borel_measurable_continuous_on1 continuous_intros)

 lemma borel_measurable_sgn [measurable]: "(sgn::'a::real_normed_vector \<Rightarrow> 'a) \<in> borel_measurable borel"

   by (rule borel_measurable_continuous_countable_exceptions[where X="{0}"])

      (auto intro!: continuous_on_sgn continuous_on_id)

 lemma borel_measurable_uminus[measurable (raw)]:

   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"

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

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

   by (rule borel_measurable_continuous_on[OF _ g]) (intro continuous_intros)

 lemma borel_measurable_add[measurable (raw)]:

   fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"

   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"

   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)

 lemma borel_measurable_setsum[measurable (raw)]:

   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"

   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 borel_measurable_diff[measurable (raw)]:

   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"

   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"

   using borel_measurable_add [of f M "- g"] assms by (simp add: fun_Compl_def)

 lemma borel_measurable_times[measurable (raw)]:

   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_algebra}"

   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"

   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)

 lemma borel_measurable_setprod[measurable (raw)]:

   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, real_normed_field}"

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

   shows "(\<lambda>x. \<Prod>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 borel_measurable_dist[measurable (raw)]:

   fixes g f :: "'a \<Rightarrow> 'b::{second_countable_topology, metric_space}"

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

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

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

   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)

 lemma borel_measurable_scaleR[measurable (raw)]:

   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"

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

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

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

   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)

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

       using open_affinity [of S "inverse b" "- a /\<^sub>R b"]

       by (auto simp: algebra_simps)

     hence "?S \<in> sets borel" by auto

     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

       by auto

   qed simp

 qed

 lemma borel_measurable_const_scaleR[measurable (raw)]:

   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. b *\<^sub>R f x ::'a::real_normed_vector) \<in> borel_measurable M"

   using affine_borel_measurable_vector[of f M 0 b] by simp

 lemma borel_measurable_const_add[measurable (raw)]:

   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. a + f x ::'a::real_normed_vector) \<in> borel_measurable M"

   using affine_borel_measurable_vector[of f M a 1] by simp

 lemma borel_measurable_inverse[measurable (raw)]:

   fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"

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

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

   apply (rule measurable_compose[OF f])

   apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])

   apply (auto intro!: continuous_on_inverse continuous_on_id)

   done

 lemma borel_measurable_divide[measurable (raw)]:

   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>

     (\<lambda>x. f x / g x::'b::{second_countable_topology, real_normed_div_algebra}) \<in> borel_measurable M"

   by (simp add: divide_inverse)

 lemma borel_measurable_max[measurable (raw)]:

   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. max (g x) (f x) :: 'b::{second_countable_topology, dense_linorder, linorder_topology}) \<in> borel_measurable M"

   by (simp add: max_def)

 lemma borel_measurable_min[measurable (raw)]:

   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. min (g x) (f x) :: 'b::{second_countable_topology, dense_linorder, linorder_topology}) \<in> borel_measurable M"

   by (simp add: min_def)

 lemma borel_measurable_Min[measurable (raw)]:

   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M) \<Longrightarrow> (\<lambda>x. Min ((\<lambda>i. f i x)`I) :: 'b::{second_countable_topology, dense_linorder, linorder_topology}) \<in> borel_measurable M"

 proof (induct I rule: finite_induct)

   case (insert i I) then show ?case

     by (cases "I = {}") auto

 qed auto

 lemma borel_measurable_Max[measurable (raw)]:

   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M) \<Longrightarrow> (\<lambda>x. Max ((\<lambda>i. f i x)`I) :: 'b::{second_countable_topology, dense_linorder, linorder_topology}) \<in> borel_measurable M"

 proof (induct I rule: finite_induct)

   case (insert i I) then show ?case

     by (cases "I = {}") auto

 qed auto

 lemma borel_measurable_abs[measurable (raw)]:

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

   unfolding abs_real_def by simp

 lemma borel_measurable_nth[measurable (raw)]:

   "(\<lambda>x::real^'n. x $ i) \<in> borel_measurable borel"

   by (simp add: cart_eq_inner_axis)

 lemma convex_measurable:

   fixes A :: "'a :: euclidean_space set"

   shows "X \<in> borel_measurable M \<Longrightarrow> X ` space M \<subseteq> A \<Longrightarrow> open A \<Longrightarrow> convex_on A q \<Longrightarrow> 

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

   by (rule measurable_compose[where f=X and N="restrict_space borel A"])

      (auto intro!: borel_measurable_continuous_on_restrict convex_on_continuous measurable_restrict_space2)

 lemma borel_measurable_ln[measurable (raw)]:

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

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

   apply (rule measurable_compose[OF f])

   apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])

   apply (auto intro!: continuous_on_ln continuous_on_id)

   done

 lemma borel_measurable_log[measurable (raw)]:

   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. log (g x) (f x)) \<in> borel_measurable M"

   unfolding log_def by auto

 lemma borel_measurable_exp[measurable]:

   "(exp::'a::{real_normed_field,banach}\<Rightarrow>'a) \<in> borel_measurable borel"

   by (intro borel_measurable_continuous_on1 continuous_at_imp_continuous_on ballI isCont_exp)

 lemma measurable_real_floor[measurable]:

   "(floor :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"

 proof -

   have "\<And>a x. \<lfloor>x\<rfloor> = a \<longleftrightarrow> (real a \<le> x \<and> x < real (a + 1))"

     by (auto intro: floor_eq2)

   then show ?thesis

     by (auto simp: vimage_def measurable_count_space_eq2_countable)

 qed

 lemma measurable_real_ceiling[measurable]:

   "(ceiling :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"

   unfolding ceiling_def[abs_def] by simp

 lemma borel_measurable_real_floor: "(\<lambda>x::real. real \<lfloor>x\<rfloor>) \<in> borel_measurable borel"

   by simp

 lemma borel_measurable_root [measurable]: "root n \<in> borel_measurable borel"

   by (intro borel_measurable_continuous_on1 continuous_intros)

 lemma borel_measurable_sqrt [measurable]: "sqrt \<in> borel_measurable borel"

   by (intro borel_measurable_continuous_on1 continuous_intros)

 lemma borel_measurable_power [measurable (raw)]:

   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"

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

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

   by (intro borel_measurable_continuous_on [OF _ f] continuous_intros)

 lemma borel_measurable_Re [measurable]: "Re \<in> borel_measurable borel"

   by (intro borel_measurable_continuous_on1 continuous_intros)

 lemma borel_measurable_Im [measurable]: "Im \<in> borel_measurable borel"

   by (intro borel_measurable_continuous_on1 continuous_intros)

 lemma borel_measurable_of_real [measurable]: "(of_real :: _ \<Rightarrow> (_::real_normed_algebra)) \<in> borel_measurable borel"

   by (intro borel_measurable_continuous_on1 continuous_intros)

 lemma borel_measurable_sin [measurable]: "(sin :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"

   by (intro borel_measurable_continuous_on1 continuous_intros)

 lemma borel_measurable_cos [measurable]: "(cos :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"

   by (intro borel_measurable_continuous_on1 continuous_intros)

 lemma borel_measurable_arctan [measurable]: "arctan \<in> borel_measurable borel"

   by (intro borel_measurable_continuous_on1 continuous_intros)

 lemma borel_measurable_complex_iff:

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

     (\<lambda>x. Re (f x)) \<in> borel_measurable M \<and> (\<lambda>x. Im (f x)) \<in> borel_measurable M"

   apply auto

   apply (subst fun_complex_eq)

   apply (intro borel_measurable_add)

   apply auto

   done

 subsection "Borel space on the extended reals"

 lemma borel_measurable_ereal[measurable (raw)]:

   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"

   using continuous_on_ereal f by (rule borel_measurable_continuous_on) (rule continuous_on_id)

 lemma borel_measurable_real_of_ereal[measurable (raw)]:

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

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

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

   apply (rule measurable_compose[OF f])

   apply (rule borel_measurable_continuous_countable_exceptions[of "{\<infinity>, -\<infinity> }"])

   apply (auto intro: continuous_on_real simp: Compl_eq_Diff_UNIV)

   done

 lemma borel_measurable_ereal_cases:

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

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

   assumes H: "(\<lambda>x. H (ereal (real (f x)))) \<in> borel_measurable M"

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

 proof -

   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)))"

   { fix x have "H (f x) = ?F x" by (cases "f x") auto }

   with f H show ?thesis by simp

 qed

 lemma

   fixes f :: "'a \<Rightarrow> ereal" assumes f[measurable]: "f \<in> borel_measurable M"

   shows borel_measurable_ereal_abs[measurable(raw)]: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"

     and borel_measurable_ereal_inverse[measurable(raw)]: "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M"

     and borel_measurable_uminus_ereal[measurable(raw)]: "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M"

   by (auto simp del: abs_real_of_ereal simp: borel_measurable_ereal_cases[OF f] measurable_If)

 lemma borel_measurable_uminus_eq_ereal[simp]:

   "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")

 proof

   assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp

 qed auto

 lemma set_Collect_ereal2:

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

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

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

   assumes H: "{x \<in> space M. H (ereal (real (f x))) (ereal (real (g x)))} \<in> sets M"

     "{x \<in> space borel. H (-\<infinity>) (ereal x)} \<in> sets borel"

     "{x \<in> space borel. H (\<infinity>) (ereal x)} \<in> sets borel"

     "{x \<in> space borel. H (ereal x) (-\<infinity>)} \<in> sets borel"

     "{x \<in> space borel. H (ereal x) (\<infinity>)} \<in> sets borel"

   shows "{x \<in> space M. H (f x) (g x)} \<in> sets M"

 proof -

   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)))"

   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"

   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }

   note * = this

   from assms show ?thesis

     by (subst *) (simp del: space_borel split del: split_if)

 qed

 lemma borel_measurable_ereal_iff:

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

 proof

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

   from borel_measurable_real_of_ereal[OF this]

   show "f \<in> borel_measurable M" by auto

 qed auto

 lemma borel_measurable_erealD[measurable_dest]:

   "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<Longrightarrow> g \<in> measurable N M \<Longrightarrow> (\<lambda>x. f (g x)) \<in> borel_measurable N"

   unfolding borel_measurable_ereal_iff by simp

 lemma borel_measurable_ereal_iff_real:

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

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

     ((\<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)"

 proof safe

   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"

   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

   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

   let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real (f x))"

   have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto

   also have "?f = f" by (auto simp: fun_eq_iff ereal_real)

   finally show "f \<in> borel_measurable M" .

 qed simp_all

 lemma borel_measurable_ereal_iff_Iio:

   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"

   by (auto simp: borel_Iio measurable_iff_measure_of)

 lemma borel_measurable_ereal_iff_Ioi:

   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"

   by (auto simp: borel_Ioi measurable_iff_measure_of)

 lemma vimage_sets_compl_iff:

   "f -` A \<inter> space M \<in> sets M \<longleftrightarrow> f -` (- A) \<inter> space M \<in> sets M"

 proof -

   { fix A assume "f -` A \<inter> space M \<in> sets M"

     moreover have "f -` (- A) \<inter> space M = space M - f -` A \<inter> space M" by auto

     ultimately have "f -` (- A) \<inter> space M \<in> sets M" by auto }

   from this[of A] this[of "-A"] show ?thesis

     by (metis double_complement)

 qed

 lemma borel_measurable_iff_Iic_ereal:

   "(f::'a\<Rightarrow>ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"

   unfolding borel_measurable_ereal_iff_Ioi vimage_sets_compl_iff[where A="{a <..}" for a] by simp

 lemma borel_measurable_iff_Ici_ereal:

   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"

   unfolding borel_measurable_ereal_iff_Iio vimage_sets_compl_iff[where A="{..< a}" for a] by simp

 lemma borel_measurable_ereal2:

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

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

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

   assumes H: "(\<lambda>x. H (ereal (real (f x))) (ereal (real (g x)))) \<in> borel_measurable M"

     "(\<lambda>x. H (-\<infinity>) (ereal (real (g x)))) \<in> borel_measurable M"

     "(\<lambda>x. H (\<infinity>) (ereal (real (g x)))) \<in> borel_measurable M"

     "(\<lambda>x. H (ereal (real (f x))) (-\<infinity>)) \<in> borel_measurable M"

     "(\<lambda>x. H (ereal (real (f x))) (\<infinity>)) \<in> borel_measurable M"

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

 proof -

   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)))"

   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"

   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }

   note * = this

   from assms show ?thesis unfolding * by simp

 qed

 lemma

   fixes f :: "'a \<Rightarrow> ereal" assumes f: "f \<in> borel_measurable M"

   shows borel_measurable_ereal_eq_const: "{x\<in>space M. f x = c} \<in> sets M"

     and borel_measurable_ereal_neq_const: "{x\<in>space M. f x \<noteq> c} \<in> sets M"

   using f by auto

 lemma [measurable(raw)]:

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

   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"

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

     and borel_measurable_ereal_times: "(\<lambda>x. f x * g x) \<in> borel_measurable M"

     and borel_measurable_ereal_min: "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"

     and borel_measurable_ereal_max: "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"

   by (simp_all add: borel_measurable_ereal2 min_def max_def)

 lemma [measurable(raw)]:

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

   assumes "f \<in> borel_measurable M"

   assumes "g \<in> borel_measurable M"

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

     and borel_measurable_ereal_divide: "(\<lambda>x. f x / g x) \<in> borel_measurable M"

   using assms by (simp_all add: minus_ereal_def divide_ereal_def)

 lemma borel_measurable_ereal_setsum[measurable (raw)]:

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

   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"

   using assms by (induction S rule: infinite_finite_induct) auto

 lemma borel_measurable_ereal_setprod[measurable (raw)]:

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

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

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

   using assms by (induction S rule: infinite_finite_induct) auto

 lemma [measurable (raw)]:

   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"

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

   shows borel_measurable_liminf: "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"

     and borel_measurable_limsup: "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"

   unfolding liminf_SUP_INF limsup_INF_SUP using assms by auto

 lemma sets_Collect_eventually_sequentially[measurable]:

   "(\<And>i. {x\<in>space M. P x i} \<in> sets M) \<Longrightarrow> {x\<in>space M. eventually (P x) sequentially} \<in> sets M"

   unfolding eventually_sequentially by simp

 lemma sets_Collect_ereal_convergent[measurable]: 

   fixes f :: "nat \<Rightarrow> 'a => ereal"

   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"

   shows "{x\<in>space M. convergent (\<lambda>i. f i x)} \<in> sets M"

   unfolding convergent_ereal by auto

 lemma borel_measurable_extreal_lim[measurable (raw)]:

   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"

   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"

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

 proof -

   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))"

     by (simp add: lim_def convergent_def convergent_limsup_cl)

   then show ?thesis

     by simp

 qed

 lemma borel_measurable_ereal_LIMSEQ:

   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"

   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"

   and u: "\<And>i. u i \<in> borel_measurable M"

   shows "u' \<in> borel_measurable M"

 proof -

   have "\<And>x. x \<in> space M \<Longrightarrow> u' x = liminf (\<lambda>n. u n x)"

     using u' by (simp add: lim_imp_Liminf[symmetric])

   with u show ?thesis by (simp cong: measurable_cong)

 qed

 lemma borel_measurable_extreal_suminf[measurable (raw)]:

   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"

   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"

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

   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp

 subsection {* LIMSEQ is borel measurable *}

 lemma borel_measurable_LIMSEQ:

   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"

   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"

   and u: "\<And>i. u i \<in> borel_measurable M"

   shows "u' \<in> borel_measurable M"

 proof -

   have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)"

     using u' by (simp add: lim_imp_Liminf)

   moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M"

     by auto

   ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff)

 qed

 lemma borel_measurable_LIMSEQ_metric:

   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b :: metric_space"

   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"

   assumes lim: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. f i x) ----> g x"

   shows "g \<in> borel_measurable M"

   unfolding borel_eq_closed

 proof (safe intro!: measurable_measure_of)

   fix A :: "'b set" assume "closed A" 

   have [measurable]: "(\<lambda>x. infdist (g x) A) \<in> borel_measurable M"

   proof (rule borel_measurable_LIMSEQ)

     show "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. infdist (f i x) A) ----> infdist (g x) A"

       by (intro tendsto_infdist lim)

     show "\<And>i. (\<lambda>x. infdist (f i x) A) \<in> borel_measurable M"

       by (intro borel_measurable_continuous_on[where f="\<lambda>x. infdist x A"]

         continuous_at_imp_continuous_on ballI continuous_infdist continuous_ident) auto

   qed

   show "g -` A \<inter> space M \<in> sets M"

   proof cases

     assume "A \<noteq> {}"

     then have "\<And>x. infdist x A = 0 \<longleftrightarrow> x \<in> A"

       using `closed A` by (simp add: in_closed_iff_infdist_zero)

     then have "g -` A \<inter> space M = {x\<in>space M. infdist (g x) A = 0}"

       by auto

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

       by measurable

     finally show ?thesis .

   qed simp

 qed auto

 lemma sets_Collect_Cauchy[measurable]: 

   fixes f :: "nat \<Rightarrow> 'a => 'b::{metric_space, second_countable_topology}"

   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"

   shows "{x\<in>space M. Cauchy (\<lambda>i. f i x)} \<in> sets M"

   unfolding metric_Cauchy_iff2 using f by auto

 lemma borel_measurable_lim[measurable (raw)]:

   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"

   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"

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

 proof -

   def u' \<equiv> "\<lambda>x. lim (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)"

   then have *: "\<And>x. lim (\<lambda>i. f i x) = (if Cauchy (\<lambda>i. f i x) then u' x else (THE x. False))"

     by (auto simp: lim_def convergent_eq_cauchy[symmetric])

   have "u' \<in> borel_measurable M"

   proof (rule borel_measurable_LIMSEQ_metric)

     fix x

     have "convergent (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)"

       by (cases "Cauchy (\<lambda>i. f i x)")

          (auto simp add: convergent_eq_cauchy[symmetric] convergent_def)

     then show "(\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0) ----> u' x"

       unfolding u'_def 

       by (rule convergent_LIMSEQ_iff[THEN iffD1])

   qed measurable

   then show ?thesis

     unfolding * by measurable

 qed

 lemma borel_measurable_suminf[measurable (raw)]:

   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"

   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"

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

   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp

 lemma borel_measurable_sup[measurable (raw)]:

   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>

     (\<lambda>x. sup (f x) (g x)::ereal) \<in> borel_measurable M"

   by simp

 (* Proof by Jeremy Avigad and Luke Serafin *)

 lemma isCont_borel:

   fixes f :: "'b::metric_space \<Rightarrow> 'a::metric_space"

   shows "{x. isCont f x} \<in> sets borel"

 proof -

   let ?I = "\<lambda>j. inverse(real (Suc j))"

   { fix x

     have "isCont f x = (\<forall>i. \<exists>j. \<forall>y z. dist x y < ?I j \<and> dist x z < ?I j \<longrightarrow> dist (f y) (f z) \<le> ?I i)"

       unfolding continuous_at_eps_delta

     proof safe

       fix i assume "\<forall>e>0. \<exists>d>0. \<forall>y. dist y x < d \<longrightarrow> dist (f y) (f x) < e"

       moreover have "0 < ?I i / 2"

         by simp

       ultimately obtain d where d: "0 < d" "\<And>y. dist x y < d \<Longrightarrow> dist (f y) (f x) < ?I i / 2"

         by (metis dist_commute)

       then obtain j where j: "?I j < d"

         by (metis reals_Archimedean)

       show "\<exists>j. \<forall>y z. dist x y < ?I j \<and> dist x z < ?I j \<longrightarrow> dist (f y) (f z) \<le> ?I i"

       proof (safe intro!: exI[where x=j])

         fix y z assume *: "dist x y < ?I j" "dist x z < ?I j"

         have "dist (f y) (f z) \<le> dist (f y) (f x) + dist (f z) (f x)"

           by (rule dist_triangle2)

         also have "\<dots> < ?I i / 2 + ?I i / 2"

           by (intro add_strict_mono d less_trans[OF _ j] *)

         also have "\<dots> \<le> ?I i"

           by (simp add: field_simps real_of_nat_Suc)

         finally show "dist (f y) (f z) \<le> ?I i"

           by simp

       qed

     next

       fix e::real assume "0 < e"

       then obtain n where n: "?I n < e"

         by (metis reals_Archimedean)

       assume "\<forall>i. \<exists>j. \<forall>y z. dist x y < ?I j \<and> dist x z < ?I j \<longrightarrow> dist (f y) (f z) \<le> ?I i"

       from this[THEN spec, of "Suc n"]

       obtain j where j: "\<And>y z. dist x y < ?I j \<Longrightarrow> dist x z < ?I j \<Longrightarrow> dist (f y) (f z) \<le> ?I (Suc n)"

         by auto

       show "\<exists>d>0. \<forall>y. dist y x < d \<longrightarrow> dist (f y) (f x) < e"

       proof (safe intro!: exI[of _ "?I j"])

         fix y assume "dist y x < ?I j"

         then have "dist (f y) (f x) \<le> ?I (Suc n)"

           by (intro j) (auto simp: dist_commute)

         also have "?I (Suc n) < ?I n"

           by simp

         also note n

         finally show "dist (f y) (f x) < e" .

       qed simp

     qed }

   note * = this

   have **: "\<And>e y. open {x. dist x y < e}"

     using open_ball by (simp_all add: ball_def dist_commute)

   have "{x\<in>space borel. isCont f x} \<in> sets borel"

     unfolding *

     apply (intro sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex)

     apply (simp add: Collect_all_eq)

     apply (intro borel_closed closed_INT ballI closed_Collect_imp open_Collect_conj **)

     apply auto

     done

   then show ?thesis

     by simp

 qed

 no_notation

   eucl_less (infix "<e" 50)

 end