src/HOL/Probability/Borel_Space.thy
 author haftmann Fri Jun 19 07:53:35 2015 +0200 (2015-06-19) changeset 60517 f16e4fb20652 parent 60172 423273355b55 child 60771 8558e4a37b48 permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
```     1 (*  Title:      HOL/Probability/Borel_Space.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Armin Heller, TU München
```
```     4 *)
```
```     5
```
```     6 section {*Borel spaces*}
```
```     7
```
```     8 theory Borel_Space
```
```     9 imports
```
```    10   Measurable
```
```    11   "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
```
```    12 begin
```
```    13
```
```    14 lemma topological_basis_trivial: "topological_basis {A. open A}"
```
```    15   by (auto simp: topological_basis_def)
```
```    16
```
```    17 lemma open_prod_generated: "open = generate_topology {A \<times> B | A B. open A \<and> open B}"
```
```    18 proof -
```
```    19   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}))"
```
```    20     by auto
```
```    21   then show ?thesis
```
```    22     by (auto intro: topological_basis_prod topological_basis_trivial topological_basis_imp_subbasis)
```
```    23 qed
```
```    24
```
```    25 subsection {* Generic Borel spaces *}
```
```    26
```
```    27 definition borel :: "'a::topological_space measure" where
```
```    28   "borel = sigma UNIV {S. open S}"
```
```    29
```
```    30 abbreviation "borel_measurable M \<equiv> measurable M borel"
```
```    31
```
```    32 lemma in_borel_measurable:
```
```    33    "f \<in> borel_measurable M \<longleftrightarrow>
```
```    34     (\<forall>S \<in> sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)"
```
```    35   by (auto simp add: measurable_def borel_def)
```
```    36
```
```    37 lemma in_borel_measurable_borel:
```
```    38    "f \<in> borel_measurable M \<longleftrightarrow>
```
```    39     (\<forall>S \<in> sets borel.
```
```    40       f -` S \<inter> space M \<in> sets M)"
```
```    41   by (auto simp add: measurable_def borel_def)
```
```    42
```
```    43 lemma space_borel[simp]: "space borel = UNIV"
```
```    44   unfolding borel_def by auto
```
```    45
```
```    46 lemma space_in_borel[measurable]: "UNIV \<in> sets borel"
```
```    47   unfolding borel_def by auto
```
```    48
```
```    49 lemma sets_borel: "sets borel = sigma_sets UNIV {S. open S}"
```
```    50   unfolding borel_def by (rule sets_measure_of) simp
```
```    51
```
```    52 lemma pred_Collect_borel[measurable (raw)]: "Measurable.pred borel P \<Longrightarrow> {x. P x} \<in> sets borel"
```
```    53   unfolding borel_def pred_def by auto
```
```    54
```
```    55 lemma borel_open[measurable (raw generic)]:
```
```    56   assumes "open A" shows "A \<in> sets borel"
```
```    57 proof -
```
```    58   have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms .
```
```    59   thus ?thesis unfolding borel_def by auto
```
```    60 qed
```
```    61
```
```    62 lemma borel_closed[measurable (raw generic)]:
```
```    63   assumes "closed A" shows "A \<in> sets borel"
```
```    64 proof -
```
```    65   have "space borel - (- A) \<in> sets borel"
```
```    66     using assms unfolding closed_def by (blast intro: borel_open)
```
```    67   thus ?thesis by simp
```
```    68 qed
```
```    69
```
```    70 lemma borel_singleton[measurable]:
```
```    71   "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets (borel :: 'a::t1_space measure)"
```
```    72   unfolding insert_def by (rule sets.Un) auto
```
```    73
```
```    74 lemma borel_comp[measurable]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
```
```    75   unfolding Compl_eq_Diff_UNIV by simp
```
```    76
```
```    77 lemma borel_measurable_vimage:
```
```    78   fixes f :: "'a \<Rightarrow> 'x::t2_space"
```
```    79   assumes borel[measurable]: "f \<in> borel_measurable M"
```
```    80   shows "f -` {x} \<inter> space M \<in> sets M"
```
```    81   by simp
```
```    82
```
```    83 lemma borel_measurableI:
```
```    84   fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space"
```
```    85   assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
```
```    86   shows "f \<in> borel_measurable M"
```
```    87   unfolding borel_def
```
```    88 proof (rule measurable_measure_of, simp_all)
```
```    89   fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"
```
```    90     using assms[of S] by simp
```
```    91 qed
```
```    92
```
```    93 lemma borel_measurable_const:
```
```    94   "(\<lambda>x. c) \<in> borel_measurable M"
```
```    95   by auto
```
```    96
```
```    97 lemma borel_measurable_indicator:
```
```    98   assumes A: "A \<in> sets M"
```
```    99   shows "indicator A \<in> borel_measurable M"
```
```   100   unfolding indicator_def [abs_def] using A
```
```   101   by (auto intro!: measurable_If_set)
```
```   102
```
```   103 lemma borel_measurable_count_space[measurable (raw)]:
```
```   104   "f \<in> borel_measurable (count_space S)"
```
```   105   unfolding measurable_def by auto
```
```   106
```
```   107 lemma borel_measurable_indicator'[measurable (raw)]:
```
```   108   assumes [measurable]: "{x\<in>space M. f x \<in> A x} \<in> sets M"
```
```   109   shows "(\<lambda>x. indicator (A x) (f x)) \<in> borel_measurable M"
```
```   110   unfolding indicator_def[abs_def]
```
```   111   by (auto intro!: measurable_If)
```
```   112
```
```   113 lemma borel_measurable_indicator_iff:
```
```   114   "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
```
```   115     (is "?I \<in> borel_measurable M \<longleftrightarrow> _")
```
```   116 proof
```
```   117   assume "?I \<in> borel_measurable M"
```
```   118   then have "?I -` {1} \<inter> space M \<in> sets M"
```
```   119     unfolding measurable_def by auto
```
```   120   also have "?I -` {1} \<inter> space M = A \<inter> space M"
```
```   121     unfolding indicator_def [abs_def] by auto
```
```   122   finally show "A \<inter> space M \<in> sets M" .
```
```   123 next
```
```   124   assume "A \<inter> space M \<in> sets M"
```
```   125   moreover have "?I \<in> borel_measurable M \<longleftrightarrow>
```
```   126     (indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"
```
```   127     by (intro measurable_cong) (auto simp: indicator_def)
```
```   128   ultimately show "?I \<in> borel_measurable M" by auto
```
```   129 qed
```
```   130
```
```   131 lemma borel_measurable_subalgebra:
```
```   132   assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
```
```   133   shows "f \<in> borel_measurable M"
```
```   134   using assms unfolding measurable_def by auto
```
```   135
```
```   136 lemma borel_measurable_restrict_space_iff_ereal:
```
```   137   fixes f :: "'a \<Rightarrow> ereal"
```
```   138   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
```
```   139   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
```
```   140     (\<lambda>x. f x * indicator \<Omega> x) \<in> borel_measurable M"
```
```   141   by (subst measurable_restrict_space_iff)
```
```   142      (auto simp: indicator_def if_distrib[where f="\<lambda>x. a * x" for a] cong del: if_cong)
```
```   143
```
```   144 lemma borel_measurable_restrict_space_iff:
```
```   145   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   146   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
```
```   147   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
```
```   148     (\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> borel_measurable M"
```
```   149   by (subst measurable_restrict_space_iff)
```
```   150      (auto simp: indicator_def if_distrib[where f="\<lambda>x. x *\<^sub>R a" for a] ac_simps cong del: if_cong)
```
```   151
```
```   152 lemma cbox_borel[measurable]: "cbox a b \<in> sets borel"
```
```   153   by (auto intro: borel_closed)
```
```   154
```
```   155 lemma box_borel[measurable]: "box a b \<in> sets borel"
```
```   156   by (auto intro: borel_open)
```
```   157
```
```   158 lemma borel_compact: "compact (A::'a::t2_space set) \<Longrightarrow> A \<in> sets borel"
```
```   159   by (auto intro: borel_closed dest!: compact_imp_closed)
```
```   160
```
```   161 lemma second_countable_borel_measurable:
```
```   162   fixes X :: "'a::second_countable_topology set set"
```
```   163   assumes eq: "open = generate_topology X"
```
```   164   shows "borel = sigma UNIV X"
```
```   165   unfolding borel_def
```
```   166 proof (intro sigma_eqI sigma_sets_eqI)
```
```   167   interpret X: sigma_algebra UNIV "sigma_sets UNIV X"
```
```   168     by (rule sigma_algebra_sigma_sets) simp
```
```   169
```
```   170   fix S :: "'a set" assume "S \<in> Collect open"
```
```   171   then have "generate_topology X S"
```
```   172     by (auto simp: eq)
```
```   173   then show "S \<in> sigma_sets UNIV X"
```
```   174   proof induction
```
```   175     case (UN K)
```
```   176     then have K: "\<And>k. k \<in> K \<Longrightarrow> open k"
```
```   177       unfolding eq by auto
```
```   178     from ex_countable_basis obtain B :: "'a set set" where
```
```   179       B:  "\<And>b. b \<in> B \<Longrightarrow> open b" "\<And>X. open X \<Longrightarrow> \<exists>b\<subseteq>B. (\<Union>b) = X" and "countable B"
```
```   180       by (auto simp: topological_basis_def)
```
```   181     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"
```
```   182       by metis
```
```   183     def U \<equiv> "(\<Union>k\<in>K. m k)"
```
```   184     with m have "countable U"
```
```   185       by (intro countable_subset[OF _ `countable B`]) auto
```
```   186     have "\<Union>U = (\<Union>A\<in>U. A)" by simp
```
```   187     also have "\<dots> = \<Union>K"
```
```   188       unfolding U_def UN_simps by (simp add: m)
```
```   189     finally have "\<Union>U = \<Union>K" .
```
```   190
```
```   191     have "\<forall>b\<in>U. \<exists>k\<in>K. b \<subseteq> k"
```
```   192       using m by (auto simp: U_def)
```
```   193     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"
```
```   194       by metis
```
```   195     then have "(\<Union>b\<in>U. u b) \<subseteq> \<Union>K" "\<Union>U \<subseteq> (\<Union>b\<in>U. u b)"
```
```   196       by auto
```
```   197     then have "\<Union>K = (\<Union>b\<in>U. u b)"
```
```   198       unfolding `\<Union>U = \<Union>K` by auto
```
```   199     also have "\<dots> \<in> sigma_sets UNIV X"
```
```   200       using u UN by (intro X.countable_UN' `countable U`) auto
```
```   201     finally show "\<Union>K \<in> sigma_sets UNIV X" .
```
```   202   qed auto
```
```   203 qed (auto simp: eq intro: generate_topology.Basis)
```
```   204
```
```   205 lemma borel_measurable_continuous_on_restrict:
```
```   206   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
```
```   207   assumes f: "continuous_on A f"
```
```   208   shows "f \<in> borel_measurable (restrict_space borel A)"
```
```   209 proof (rule borel_measurableI)
```
```   210   fix S :: "'b set" assume "open S"
```
```   211   with f obtain T where "f -` S \<inter> A = T \<inter> A" "open T"
```
```   212     by (metis continuous_on_open_invariant)
```
```   213   then show "f -` S \<inter> space (restrict_space borel A) \<in> sets (restrict_space borel A)"
```
```   214     by (force simp add: sets_restrict_space space_restrict_space)
```
```   215 qed
```
```   216
```
```   217 lemma borel_measurable_continuous_on1: "continuous_on UNIV f \<Longrightarrow> f \<in> borel_measurable borel"
```
```   218   by (drule borel_measurable_continuous_on_restrict) simp
```
```   219
```
```   220 lemma borel_measurable_continuous_on_if:
```
```   221   "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> continuous_on (- A) g \<Longrightarrow>
```
```   222     (\<lambda>x. if x \<in> A then f x else g x) \<in> borel_measurable borel"
```
```   223   by (auto simp add: measurable_If_restrict_space_iff Collect_neg_eq
```
```   224            intro!: borel_measurable_continuous_on_restrict)
```
```   225
```
```   226 lemma borel_measurable_continuous_countable_exceptions:
```
```   227   fixes f :: "'a::t1_space \<Rightarrow> 'b::topological_space"
```
```   228   assumes X: "countable X"
```
```   229   assumes "continuous_on (- X) f"
```
```   230   shows "f \<in> borel_measurable borel"
```
```   231 proof (rule measurable_discrete_difference[OF _ X])
```
```   232   have "X \<in> sets borel"
```
```   233     by (rule sets.countable[OF _ X]) auto
```
```   234   then show "(\<lambda>x. if x \<in> X then undefined else f x) \<in> borel_measurable borel"
```
```   235     by (intro borel_measurable_continuous_on_if assms continuous_intros)
```
```   236 qed auto
```
```   237
```
```   238 lemma borel_measurable_continuous_on:
```
```   239   assumes f: "continuous_on UNIV f" and g: "g \<in> borel_measurable M"
```
```   240   shows "(\<lambda>x. f (g x)) \<in> borel_measurable M"
```
```   241   using measurable_comp[OF g borel_measurable_continuous_on1[OF f]] by (simp add: comp_def)
```
```   242
```
```   243 lemma borel_measurable_continuous_on_indicator:
```
```   244   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   245   shows "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable borel"
```
```   246   by (subst borel_measurable_restrict_space_iff[symmetric])
```
```   247      (auto intro: borel_measurable_continuous_on_restrict)
```
```   248
```
```   249 lemma borel_eq_countable_basis:
```
```   250   fixes B::"'a::topological_space set set"
```
```   251   assumes "countable B"
```
```   252   assumes "topological_basis B"
```
```   253   shows "borel = sigma UNIV B"
```
```   254   unfolding borel_def
```
```   255 proof (intro sigma_eqI sigma_sets_eqI, safe)
```
```   256   interpret countable_basis using assms by unfold_locales
```
```   257   fix X::"'a set" assume "open X"
```
```   258   from open_countable_basisE[OF this] guess B' . note B' = this
```
```   259   then show "X \<in> sigma_sets UNIV B"
```
```   260     by (blast intro: sigma_sets_UNION `countable B` countable_subset)
```
```   261 next
```
```   262   fix b assume "b \<in> B"
```
```   263   hence "open b" by (rule topological_basis_open[OF assms(2)])
```
```   264   thus "b \<in> sigma_sets UNIV (Collect open)" by auto
```
```   265 qed simp_all
```
```   266
```
```   267 lemma borel_measurable_Pair[measurable (raw)]:
```
```   268   fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"
```
```   269   assumes f[measurable]: "f \<in> borel_measurable M"
```
```   270   assumes g[measurable]: "g \<in> borel_measurable M"
```
```   271   shows "(\<lambda>x. (f x, g x)) \<in> borel_measurable M"
```
```   272 proof (subst borel_eq_countable_basis)
```
```   273   let ?B = "SOME B::'b set set. countable B \<and> topological_basis B"
```
```   274   let ?C = "SOME B::'c set set. countable B \<and> topological_basis B"
```
```   275   let ?P = "(\<lambda>(b, c). b \<times> c) ` (?B \<times> ?C)"
```
```   276   show "countable ?P" "topological_basis ?P"
```
```   277     by (auto intro!: countable_basis topological_basis_prod is_basis)
```
```   278
```
```   279   show "(\<lambda>x. (f x, g x)) \<in> measurable M (sigma UNIV ?P)"
```
```   280   proof (rule measurable_measure_of)
```
```   281     fix S assume "S \<in> ?P"
```
```   282     then obtain b c where "b \<in> ?B" "c \<in> ?C" and S: "S = b \<times> c" by auto
```
```   283     then have borel: "open b" "open c"
```
```   284       by (auto intro: is_basis topological_basis_open)
```
```   285     have "(\<lambda>x. (f x, g x)) -` S \<inter> space M = (f -` b \<inter> space M) \<inter> (g -` c \<inter> space M)"
```
```   286       unfolding S by auto
```
```   287     also have "\<dots> \<in> sets M"
```
```   288       using borel by simp
```
```   289     finally show "(\<lambda>x. (f x, g x)) -` S \<inter> space M \<in> sets M" .
```
```   290   qed auto
```
```   291 qed
```
```   292
```
```   293 lemma borel_measurable_continuous_Pair:
```
```   294   fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"
```
```   295   assumes [measurable]: "f \<in> borel_measurable M"
```
```   296   assumes [measurable]: "g \<in> borel_measurable M"
```
```   297   assumes H: "continuous_on UNIV (\<lambda>x. H (fst x) (snd x))"
```
```   298   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
```
```   299 proof -
```
```   300   have eq: "(\<lambda>x. H (f x) (g x)) = (\<lambda>x. (\<lambda>x. H (fst x) (snd x)) (f x, g x))" by auto
```
```   301   show ?thesis
```
```   302     unfolding eq by (rule borel_measurable_continuous_on[OF H]) auto
```
```   303 qed
```
```   304
```
```   305 subsection {* Borel spaces on order topologies *}
```
```   306
```
```   307
```
```   308 lemma borel_Iio:
```
```   309   "borel = sigma UNIV (range lessThan :: 'a::{linorder_topology, second_countable_topology} set set)"
```
```   310   unfolding second_countable_borel_measurable[OF open_generated_order]
```
```   311 proof (intro sigma_eqI sigma_sets_eqI)
```
```   312   from countable_dense_setE guess D :: "'a set" . note D = this
```
```   313
```
```   314   interpret L: sigma_algebra UNIV "sigma_sets UNIV (range lessThan)"
```
```   315     by (rule sigma_algebra_sigma_sets) simp
```
```   316
```
```   317   fix A :: "'a set" assume "A \<in> range lessThan \<union> range greaterThan"
```
```   318   then obtain y where "A = {y <..} \<or> A = {..< y}"
```
```   319     by blast
```
```   320   then show "A \<in> sigma_sets UNIV (range lessThan)"
```
```   321   proof
```
```   322     assume A: "A = {y <..}"
```
```   323     show ?thesis
```
```   324     proof cases
```
```   325       assume "\<forall>x>y. \<exists>d. y < d \<and> d < x"
```
```   326       with D(2)[of "{y <..< x}" for x] have "\<forall>x>y. \<exists>d\<in>D. y < d \<and> d < x"
```
```   327         by (auto simp: set_eq_iff)
```
```   328       then have "A = UNIV - (\<Inter>d\<in>{d\<in>D. y < d}. {..< d})"
```
```   329         by (auto simp: A) (metis less_asym)
```
```   330       also have "\<dots> \<in> sigma_sets UNIV (range lessThan)"
```
```   331         using D(1) by (intro L.Diff L.top L.countable_INT'') auto
```
```   332       finally show ?thesis .
```
```   333     next
```
```   334       assume "\<not> (\<forall>x>y. \<exists>d. y < d \<and> d < x)"
```
```   335       then obtain x where "y < x"  "\<And>d. y < d \<Longrightarrow> \<not> d < x"
```
```   336         by auto
```
```   337       then have "A = UNIV - {..< x}"
```
```   338         unfolding A by (auto simp: not_less[symmetric])
```
```   339       also have "\<dots> \<in> sigma_sets UNIV (range lessThan)"
```
```   340         by auto
```
```   341       finally show ?thesis .
```
```   342     qed
```
```   343   qed auto
```
```   344 qed auto
```
```   345
```
```   346 lemma borel_Ioi:
```
```   347   "borel = sigma UNIV (range greaterThan :: 'a::{linorder_topology, second_countable_topology} set set)"
```
```   348   unfolding second_countable_borel_measurable[OF open_generated_order]
```
```   349 proof (intro sigma_eqI sigma_sets_eqI)
```
```   350   from countable_dense_setE guess D :: "'a set" . note D = this
```
```   351
```
```   352   interpret L: sigma_algebra UNIV "sigma_sets UNIV (range greaterThan)"
```
```   353     by (rule sigma_algebra_sigma_sets) simp
```
```   354
```
```   355   fix A :: "'a set" assume "A \<in> range lessThan \<union> range greaterThan"
```
```   356   then obtain y where "A = {y <..} \<or> A = {..< y}"
```
```   357     by blast
```
```   358   then show "A \<in> sigma_sets UNIV (range greaterThan)"
```
```   359   proof
```
```   360     assume A: "A = {..< y}"
```
```   361     show ?thesis
```
```   362     proof cases
```
```   363       assume "\<forall>x<y. \<exists>d. x < d \<and> d < y"
```
```   364       with D(2)[of "{x <..< y}" for x] have "\<forall>x<y. \<exists>d\<in>D. x < d \<and> d < y"
```
```   365         by (auto simp: set_eq_iff)
```
```   366       then have "A = UNIV - (\<Inter>d\<in>{d\<in>D. d < y}. {d <..})"
```
```   367         by (auto simp: A) (metis less_asym)
```
```   368       also have "\<dots> \<in> sigma_sets UNIV (range greaterThan)"
```
```   369         using D(1) by (intro L.Diff L.top L.countable_INT'') auto
```
```   370       finally show ?thesis .
```
```   371     next
```
```   372       assume "\<not> (\<forall>x<y. \<exists>d. x < d \<and> d < y)"
```
```   373       then obtain x where "x < y"  "\<And>d. y > d \<Longrightarrow> x \<ge> d"
```
```   374         by (auto simp: not_less[symmetric])
```
```   375       then have "A = UNIV - {x <..}"
```
```   376         unfolding A Compl_eq_Diff_UNIV[symmetric] by auto
```
```   377       also have "\<dots> \<in> sigma_sets UNIV (range greaterThan)"
```
```   378         by auto
```
```   379       finally show ?thesis .
```
```   380     qed
```
```   381   qed auto
```
```   382 qed auto
```
```   383
```
```   384 lemma borel_measurableI_less:
```
```   385   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
```
```   386   shows "(\<And>y. {x\<in>space M. f x < y} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
```
```   387   unfolding borel_Iio
```
```   388   by (rule measurable_measure_of) (auto simp: Int_def conj_commute)
```
```   389
```
```   390 lemma borel_measurableI_greater:
```
```   391   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
```
```   392   shows "(\<And>y. {x\<in>space M. y < f x} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
```
```   393   unfolding borel_Ioi
```
```   394   by (rule measurable_measure_of) (auto simp: Int_def conj_commute)
```
```   395
```
```   396 lemma borel_measurable_SUP[measurable (raw)]:
```
```   397   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}"
```
```   398   assumes [simp]: "countable I"
```
```   399   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```   400   shows "(\<lambda>x. SUP i:I. F i x) \<in> borel_measurable M"
```
```   401   by (rule borel_measurableI_greater) (simp add: less_SUP_iff)
```
```   402
```
```   403 lemma borel_measurable_INF[measurable (raw)]:
```
```   404   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}"
```
```   405   assumes [simp]: "countable I"
```
```   406   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```   407   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
```
```   408   by (rule borel_measurableI_less) (simp add: INF_less_iff)
```
```   409
```
```   410 lemma borel_measurable_lfp[consumes 1, case_names continuity step]:
```
```   411   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})"
```
```   412   assumes "sup_continuous F"
```
```   413   assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M"
```
```   414   shows "lfp F \<in> borel_measurable M"
```
```   415 proof -
```
```   416   { fix i have "((F ^^ i) bot) \<in> borel_measurable M"
```
```   417       by (induct i) (auto intro!: *) }
```
```   418   then have "(\<lambda>x. SUP i. (F ^^ i) bot x) \<in> borel_measurable M"
```
```   419     by measurable
```
```   420   also have "(\<lambda>x. SUP i. (F ^^ i) bot x) = (SUP i. (F ^^ i) bot)"
```
```   421     by auto
```
```   422   also have "(SUP i. (F ^^ i) bot) = lfp F"
```
```   423     by (rule sup_continuous_lfp[symmetric]) fact
```
```   424   finally show ?thesis .
```
```   425 qed
```
```   426
```
```   427 lemma borel_measurable_gfp[consumes 1, case_names continuity step]:
```
```   428   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})"
```
```   429   assumes "inf_continuous F"
```
```   430   assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M"
```
```   431   shows "gfp F \<in> borel_measurable M"
```
```   432 proof -
```
```   433   { fix i have "((F ^^ i) top) \<in> borel_measurable M"
```
```   434       by (induct i) (auto intro!: * simp: bot_fun_def) }
```
```   435   then have "(\<lambda>x. INF i. (F ^^ i) top x) \<in> borel_measurable M"
```
```   436     by measurable
```
```   437   also have "(\<lambda>x. INF i. (F ^^ i) top x) = (INF i. (F ^^ i) top)"
```
```   438     by auto
```
```   439   also have "\<dots> = gfp F"
```
```   440     by (rule inf_continuous_gfp[symmetric]) fact
```
```   441   finally show ?thesis .
```
```   442 qed
```
```   443
```
```   444 subsection {* Borel spaces on euclidean spaces *}
```
```   445
```
```   446 lemma borel_measurable_inner[measurable (raw)]:
```
```   447   fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_inner}"
```
```   448   assumes "f \<in> borel_measurable M"
```
```   449   assumes "g \<in> borel_measurable M"
```
```   450   shows "(\<lambda>x. f x \<bullet> g x) \<in> borel_measurable M"
```
```   451   using assms
```
```   452   by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```   453
```
```   454 lemma [measurable]:
```
```   455   fixes a b :: "'a\<Colon>linorder_topology"
```
```   456   shows lessThan_borel: "{..< a} \<in> sets borel"
```
```   457     and greaterThan_borel: "{a <..} \<in> sets borel"
```
```   458     and greaterThanLessThan_borel: "{a<..<b} \<in> sets borel"
```
```   459     and atMost_borel: "{..a} \<in> sets borel"
```
```   460     and atLeast_borel: "{a..} \<in> sets borel"
```
```   461     and atLeastAtMost_borel: "{a..b} \<in> sets borel"
```
```   462     and greaterThanAtMost_borel: "{a<..b} \<in> sets borel"
```
```   463     and atLeastLessThan_borel: "{a..<b} \<in> sets borel"
```
```   464   unfolding greaterThanAtMost_def atLeastLessThan_def
```
```   465   by (blast intro: borel_open borel_closed open_lessThan open_greaterThan open_greaterThanLessThan
```
```   466                    closed_atMost closed_atLeast closed_atLeastAtMost)+
```
```   467
```
```   468 notation
```
```   469   eucl_less (infix "<e" 50)
```
```   470
```
```   471 lemma box_oc: "{x. a <e x \<and> x \<le> b} = {x. a <e x} \<inter> {..b}"
```
```   472   and box_co: "{x. a \<le> x \<and> x <e b} = {a..} \<inter> {x. x <e b}"
```
```   473   by auto
```
```   474
```
```   475 lemma eucl_ivals[measurable]:
```
```   476   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   477   shows "{x. x <e a} \<in> sets borel"
```
```   478     and "{x. a <e x} \<in> sets borel"
```
```   479     and "{..a} \<in> sets borel"
```
```   480     and "{a..} \<in> sets borel"
```
```   481     and "{a..b} \<in> sets borel"
```
```   482     and  "{x. a <e x \<and> x \<le> b} \<in> sets borel"
```
```   483     and "{x. a \<le> x \<and>  x <e b} \<in> sets borel"
```
```   484   unfolding box_oc box_co
```
```   485   by (auto intro: borel_open borel_closed)
```
```   486
```
```   487 lemma open_Collect_less:
```
```   488   fixes f g :: "'i::topological_space \<Rightarrow> 'a :: {dense_linorder, linorder_topology}"
```
```   489   assumes "continuous_on UNIV f"
```
```   490   assumes "continuous_on UNIV g"
```
```   491   shows "open {x. f x < g x}"
```
```   492 proof -
```
```   493   have "open (\<Union>y. {x \<in> UNIV. f x \<in> {..< y}} \<inter> {x \<in> UNIV. g x \<in> {y <..}})" (is "open ?X")
```
```   494     by (intro open_UN ballI open_Int continuous_open_preimage assms) auto
```
```   495   also have "?X = {x. f x < g x}"
```
```   496     by (auto intro: dense)
```
```   497   finally show ?thesis .
```
```   498 qed
```
```   499
```
```   500 lemma closed_Collect_le:
```
```   501   fixes f g :: "'i::topological_space \<Rightarrow> 'a :: {dense_linorder, linorder_topology}"
```
```   502   assumes f: "continuous_on UNIV f"
```
```   503   assumes g: "continuous_on UNIV g"
```
```   504   shows "closed {x. f x \<le> g x}"
```
```   505   using open_Collect_less[OF g f] unfolding not_less[symmetric] Collect_neg_eq open_closed .
```
```   506
```
```   507 lemma borel_measurable_less[measurable]:
```
```   508   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, dense_linorder, linorder_topology}"
```
```   509   assumes "f \<in> borel_measurable M"
```
```   510   assumes "g \<in> borel_measurable M"
```
```   511   shows "{w \<in> space M. f w < g w} \<in> sets M"
```
```   512 proof -
```
```   513   have "{w \<in> space M. f w < g w} = (\<lambda>x. (f x, g x)) -` {x. fst x < snd x} \<inter> space M"
```
```   514     by auto
```
```   515   also have "\<dots> \<in> sets M"
```
```   516     by (intro measurable_sets[OF borel_measurable_Pair borel_open, OF assms open_Collect_less]
```
```   517               continuous_intros)
```
```   518   finally show ?thesis .
```
```   519 qed
```
```   520
```
```   521 lemma
```
```   522   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, dense_linorder, linorder_topology}"
```
```   523   assumes f[measurable]: "f \<in> borel_measurable M"
```
```   524   assumes g[measurable]: "g \<in> borel_measurable M"
```
```   525   shows borel_measurable_le[measurable]: "{w \<in> space M. f w \<le> g w} \<in> sets M"
```
```   526     and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M"
```
```   527     and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
```
```   528   unfolding eq_iff not_less[symmetric]
```
```   529   by measurable
```
```   530
```
```   531 lemma
```
```   532   fixes i :: "'a::{second_countable_topology, real_inner}"
```
```   533   shows hafspace_less_borel: "{x. a < x \<bullet> i} \<in> sets borel"
```
```   534     and hafspace_greater_borel: "{x. x \<bullet> i < a} \<in> sets borel"
```
```   535     and hafspace_less_eq_borel: "{x. a \<le> x \<bullet> i} \<in> sets borel"
```
```   536     and hafspace_greater_eq_borel: "{x. x \<bullet> i \<le> a} \<in> sets borel"
```
```   537   by simp_all
```
```   538
```
```   539 subsection "Borel space equals sigma algebras over intervals"
```
```   540
```
```   541 lemma borel_sigma_sets_subset:
```
```   542   "A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel"
```
```   543   using sets.sigma_sets_subset[of A borel] by simp
```
```   544
```
```   545 lemma borel_eq_sigmaI1:
```
```   546   fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
```
```   547   assumes borel_eq: "borel = sigma UNIV X"
```
```   548   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (F ` A))"
```
```   549   assumes F: "\<And>i. i \<in> A \<Longrightarrow> F i \<in> sets borel"
```
```   550   shows "borel = sigma UNIV (F ` A)"
```
```   551   unfolding borel_def
```
```   552 proof (intro sigma_eqI antisym)
```
```   553   have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel"
```
```   554     unfolding borel_def by simp
```
```   555   also have "\<dots> = sigma_sets UNIV X"
```
```   556     unfolding borel_eq by simp
```
```   557   also have "\<dots> \<subseteq> sigma_sets UNIV (F`A)"
```
```   558     using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto
```
```   559   finally show "sigma_sets UNIV {S. open S} \<subseteq> sigma_sets UNIV (F`A)" .
```
```   560   show "sigma_sets UNIV (F`A) \<subseteq> sigma_sets UNIV {S. open S}"
```
```   561     unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto
```
```   562 qed auto
```
```   563
```
```   564 lemma borel_eq_sigmaI2:
```
```   565   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set"
```
```   566     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
```
```   567   assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`B)"
```
```   568   assumes X: "\<And>i j. (i, j) \<in> B \<Longrightarrow> G i j \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"
```
```   569   assumes F: "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel"
```
```   570   shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"
```
```   571   using assms
```
```   572   by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` B" and F="(\<lambda>(i, j). F i j)"]) auto
```
```   573
```
```   574 lemma borel_eq_sigmaI3:
```
```   575   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
```
```   576   assumes borel_eq: "borel = sigma UNIV X"
```
```   577   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"
```
```   578   assumes F: "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel"
```
```   579   shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"
```
```   580   using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto
```
```   581
```
```   582 lemma borel_eq_sigmaI4:
```
```   583   fixes F :: "'i \<Rightarrow> 'a::topological_space set"
```
```   584     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
```
```   585   assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`A)"
```
```   586   assumes X: "\<And>i j. (i, j) \<in> A \<Longrightarrow> G i j \<in> sets (sigma UNIV (range F))"
```
```   587   assumes F: "\<And>i. F i \<in> sets borel"
```
```   588   shows "borel = sigma UNIV (range F)"
```
```   589   using assms by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` A" and F=F]) auto
```
```   590
```
```   591 lemma borel_eq_sigmaI5:
```
```   592   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set"
```
```   593   assumes borel_eq: "borel = sigma UNIV (range G)"
```
```   594   assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
```
```   595   assumes F: "\<And>i j. F i j \<in> sets borel"
```
```   596   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
```
```   597   using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto
```
```   598
```
```   599 lemma borel_eq_box:
```
```   600   "borel = sigma UNIV (range (\<lambda> (a, b). box a b :: 'a \<Colon> euclidean_space set))"
```
```   601     (is "_ = ?SIGMA")
```
```   602 proof (rule borel_eq_sigmaI1[OF borel_def])
```
```   603   fix M :: "'a set" assume "M \<in> {S. open S}"
```
```   604   then have "open M" by simp
```
```   605   show "M \<in> ?SIGMA"
```
```   606     apply (subst open_UNION_box[OF `open M`])
```
```   607     apply (safe intro!: sets.countable_UN' countable_PiE countable_Collect)
```
```   608     apply (auto intro: countable_rat)
```
```   609     done
```
```   610 qed (auto simp: box_def)
```
```   611
```
```   612 lemma halfspace_gt_in_halfspace:
```
```   613   assumes i: "i \<in> A"
```
```   614   shows "{x\<Colon>'a. a < x \<bullet> i} \<in>
```
```   615     sigma_sets UNIV ((\<lambda> (a, i). {x\<Colon>'a\<Colon>euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> A))"
```
```   616   (is "?set \<in> ?SIGMA")
```
```   617 proof -
```
```   618   interpret sigma_algebra UNIV ?SIGMA
```
```   619     by (intro sigma_algebra_sigma_sets) simp_all
```
```   620   have *: "?set = (\<Union>n. UNIV - {x\<Colon>'a. x \<bullet> i < a + 1 / real (Suc n)})"
```
```   621   proof (safe, simp_all add: not_less)
```
```   622     fix x :: 'a assume "a < x \<bullet> i"
```
```   623     with reals_Archimedean[of "x \<bullet> i - a"]
```
```   624     obtain n where "a + 1 / real (Suc n) < x \<bullet> i"
```
```   625       by (auto simp: field_simps)
```
```   626     then show "\<exists>n. a + 1 / real (Suc n) \<le> x \<bullet> i"
```
```   627       by (blast intro: less_imp_le)
```
```   628   next
```
```   629     fix x n
```
```   630     have "a < a + 1 / real (Suc n)" by auto
```
```   631     also assume "\<dots> \<le> x"
```
```   632     finally show "a < x" .
```
```   633   qed
```
```   634   show "?set \<in> ?SIGMA" unfolding *
```
```   635     by (auto del: Diff intro!: Diff i)
```
```   636 qed
```
```   637
```
```   638 lemma borel_eq_halfspace_less:
```
```   639   "borel = sigma UNIV ((\<lambda>(a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> Basis))"
```
```   640   (is "_ = ?SIGMA")
```
```   641 proof (rule borel_eq_sigmaI2[OF borel_eq_box])
```
```   642   fix a b :: 'a
```
```   643   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}"
```
```   644     by (auto simp: box_def)
```
```   645   also have "\<dots> \<in> sets ?SIGMA"
```
```   646     by (intro sets.sets_Collect_conj sets.sets_Collect_finite_All sets.sets_Collect_const)
```
```   647        (auto intro!: halfspace_gt_in_halfspace countable_PiE countable_rat)
```
```   648   finally show "box a b \<in> sets ?SIGMA" .
```
```   649 qed auto
```
```   650
```
```   651 lemma borel_eq_halfspace_le:
```
```   652   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i \<le> a}) ` (UNIV \<times> Basis))"
```
```   653   (is "_ = ?SIGMA")
```
```   654 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
```
```   655   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```   656   then have i: "i \<in> Basis" by auto
```
```   657   have *: "{x::'a. x\<bullet>i < a} = (\<Union>n. {x. x\<bullet>i \<le> a - 1/real (Suc n)})"
```
```   658   proof (safe, simp_all)
```
```   659     fix x::'a assume *: "x\<bullet>i < a"
```
```   660     with reals_Archimedean[of "a - x\<bullet>i"]
```
```   661     obtain n where "x \<bullet> i < a - 1 / (real (Suc n))"
```
```   662       by (auto simp: field_simps)
```
```   663     then show "\<exists>n. x \<bullet> i \<le> a - 1 / (real (Suc n))"
```
```   664       by (blast intro: less_imp_le)
```
```   665   next
```
```   666     fix x::'a and n
```
```   667     assume "x\<bullet>i \<le> a - 1 / real (Suc n)"
```
```   668     also have "\<dots> < a" by auto
```
```   669     finally show "x\<bullet>i < a" .
```
```   670   qed
```
```   671   show "{x. x\<bullet>i < a} \<in> ?SIGMA" unfolding *
```
```   672     by (intro sets.countable_UN) (auto intro: i)
```
```   673 qed auto
```
```   674
```
```   675 lemma borel_eq_halfspace_ge:
```
```   676   "borel = sigma UNIV ((\<lambda> (a, i). {x\<Colon>'a\<Colon>euclidean_space. a \<le> x \<bullet> i}) ` (UNIV \<times> Basis))"
```
```   677   (is "_ = ?SIGMA")
```
```   678 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
```
```   679   fix a :: real and i :: 'a assume i: "(a, i) \<in> UNIV \<times> Basis"
```
```   680   have *: "{x::'a. x\<bullet>i < a} = space ?SIGMA - {x::'a. a \<le> x\<bullet>i}" by auto
```
```   681   show "{x. x\<bullet>i < a} \<in> ?SIGMA" unfolding *
```
```   682     using i by (intro sets.compl_sets) auto
```
```   683 qed auto
```
```   684
```
```   685 lemma borel_eq_halfspace_greater:
```
```   686   "borel = sigma UNIV ((\<lambda> (a, i). {x\<Colon>'a\<Colon>euclidean_space. a < x \<bullet> i}) ` (UNIV \<times> Basis))"
```
```   687   (is "_ = ?SIGMA")
```
```   688 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])
```
```   689   fix a :: real and i :: 'a assume "(a, i) \<in> (UNIV \<times> Basis)"
```
```   690   then have i: "i \<in> Basis" by auto
```
```   691   have *: "{x::'a. x\<bullet>i \<le> a} = space ?SIGMA - {x::'a. a < x\<bullet>i}" by auto
```
```   692   show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA" unfolding *
```
```   693     by (intro sets.compl_sets) (auto intro: i)
```
```   694 qed auto
```
```   695
```
```   696 lemma borel_eq_atMost:
```
```   697   "borel = sigma UNIV (range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space}))"
```
```   698   (is "_ = ?SIGMA")
```
```   699 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
```
```   700   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```   701   then have "i \<in> Basis" by auto
```
```   702   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)})"
```
```   703   proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm)
```
```   704     fix x :: 'a
```
```   705     from real_arch_simple[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"] guess k::nat ..
```
```   706     then have "\<And>i. i \<in> Basis \<Longrightarrow> x\<bullet>i \<le> real k"
```
```   707       by (subst (asm) Max_le_iff) auto
```
```   708     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> x \<bullet> ia \<le> real k"
```
```   709       by (auto intro!: exI[of _ k])
```
```   710   qed
```
```   711   show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA" unfolding *
```
```   712     by (intro sets.countable_UN) auto
```
```   713 qed auto
```
```   714
```
```   715 lemma borel_eq_greaterThan:
```
```   716   "borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {x. a <e x}))"
```
```   717   (is "_ = ?SIGMA")
```
```   718 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
```
```   719   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```   720   then have i: "i \<in> Basis" by auto
```
```   721   have "{x::'a. x\<bullet>i \<le> a} = UNIV - {x::'a. a < x\<bullet>i}" by auto
```
```   722   also have *: "{x::'a. a < x\<bullet>i} =
```
```   723       (\<Union>k::nat. {x. (\<Sum>n\<in>Basis. (if n = i then a else -real k) *\<^sub>R n) <e x})" using i
```
```   724   proof (safe, simp_all add: eucl_less_def split: split_if_asm)
```
```   725     fix x :: 'a
```
```   726     from reals_Archimedean2[of "Max ((\<lambda>i. -x\<bullet>i)`Basis)"]
```
```   727     guess k::nat .. note k = this
```
```   728     { fix i :: 'a assume "i \<in> Basis"
```
```   729       then have "-x\<bullet>i < real k"
```
```   730         using k by (subst (asm) Max_less_iff) auto
```
```   731       then have "- real k < x\<bullet>i" by simp }
```
```   732     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> -real k < x \<bullet> ia"
```
```   733       by (auto intro!: exI[of _ k])
```
```   734   qed
```
```   735   finally show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA"
```
```   736     apply (simp only:)
```
```   737     apply (intro sets.countable_UN sets.Diff)
```
```   738     apply (auto intro: sigma_sets_top)
```
```   739     done
```
```   740 qed auto
```
```   741
```
```   742 lemma borel_eq_lessThan:
```
```   743   "borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {x. x <e a}))"
```
```   744   (is "_ = ?SIGMA")
```
```   745 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge])
```
```   746   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```   747   then have i: "i \<in> Basis" by auto
```
```   748   have "{x::'a. a \<le> x\<bullet>i} = UNIV - {x::'a. x\<bullet>i < a}" by auto
```
```   749   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`
```
```   750   proof (safe, simp_all add: eucl_less_def split: split_if_asm)
```
```   751     fix x :: 'a
```
```   752     from reals_Archimedean2[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"]
```
```   753     guess k::nat .. note k = this
```
```   754     { fix i :: 'a assume "i \<in> Basis"
```
```   755       then have "x\<bullet>i < real k"
```
```   756         using k by (subst (asm) Max_less_iff) auto
```
```   757       then have "x\<bullet>i < real k" by simp }
```
```   758     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> x \<bullet> ia < real k"
```
```   759       by (auto intro!: exI[of _ k])
```
```   760   qed
```
```   761   finally show "{x. a \<le> x\<bullet>i} \<in> ?SIGMA"
```
```   762     apply (simp only:)
```
```   763     apply (intro sets.countable_UN sets.Diff)
```
```   764     apply (auto intro: sigma_sets_top )
```
```   765     done
```
```   766 qed auto
```
```   767
```
```   768 lemma borel_eq_atLeastAtMost:
```
```   769   "borel = sigma UNIV (range (\<lambda>(a,b). {a..b} \<Colon>'a\<Colon>ordered_euclidean_space set))"
```
```   770   (is "_ = ?SIGMA")
```
```   771 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
```
```   772   fix a::'a
```
```   773   have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
```
```   774   proof (safe, simp_all add: eucl_le[where 'a='a])
```
```   775     fix x :: 'a
```
```   776     from real_arch_simple[of "Max ((\<lambda>i. - x\<bullet>i)`Basis)"]
```
```   777     guess k::nat .. note k = this
```
```   778     { fix i :: 'a assume "i \<in> Basis"
```
```   779       with k have "- x\<bullet>i \<le> real k"
```
```   780         by (subst (asm) Max_le_iff) (auto simp: field_simps)
```
```   781       then have "- real k \<le> x\<bullet>i" by simp }
```
```   782     then show "\<exists>n::nat. \<forall>i\<in>Basis. - real n \<le> x \<bullet> i"
```
```   783       by (auto intro!: exI[of _ k])
```
```   784   qed
```
```   785   show "{..a} \<in> ?SIGMA" unfolding *
```
```   786     by (intro sets.countable_UN)
```
```   787        (auto intro!: sigma_sets_top)
```
```   788 qed auto
```
```   789
```
```   790 lemma borel_sigma_sets_Ioc: "borel = sigma UNIV (range (\<lambda>(a, b). {a <.. b::real}))"
```
```   791 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
```
```   792   fix i :: real
```
```   793   have "{..i} = (\<Union>j::nat. {-j <.. i})"
```
```   794     by (auto simp: minus_less_iff reals_Archimedean2)
```
```   795   also have "\<dots> \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))"
```
```   796     by (intro sets.countable_nat_UN) auto
```
```   797   finally show "{..i} \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))" .
```
```   798 qed simp
```
```   799
```
```   800 lemma eucl_lessThan: "{x::real. x <e a} = lessThan a"
```
```   801   by (simp add: eucl_less_def lessThan_def)
```
```   802
```
```   803 lemma borel_eq_atLeastLessThan:
```
```   804   "borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA")
```
```   805 proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan])
```
```   806   have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto
```
```   807   fix x :: real
```
```   808   have "{..<x} = (\<Union>i::nat. {-real i ..< x})"
```
```   809     by (auto simp: move_uminus real_arch_simple)
```
```   810   then show "{y. y <e x} \<in> ?SIGMA"
```
```   811     by (auto intro: sigma_sets.intros(2-) simp: eucl_lessThan)
```
```   812 qed auto
```
```   813
```
```   814 lemma borel_eq_closed: "borel = sigma UNIV (Collect closed)"
```
```   815   unfolding borel_def
```
```   816 proof (intro sigma_eqI sigma_sets_eqI, safe)
```
```   817   fix x :: "'a set" assume "open x"
```
```   818   hence "x = UNIV - (UNIV - x)" by auto
```
```   819   also have "\<dots> \<in> sigma_sets UNIV (Collect closed)"
```
```   820     by (force intro: sigma_sets.Compl simp: `open x`)
```
```   821   finally show "x \<in> sigma_sets UNIV (Collect closed)" by simp
```
```   822 next
```
```   823   fix x :: "'a set" assume "closed x"
```
```   824   hence "x = UNIV - (UNIV - x)" by auto
```
```   825   also have "\<dots> \<in> sigma_sets UNIV (Collect open)"
```
```   826     by (force intro: sigma_sets.Compl simp: `closed x`)
```
```   827   finally show "x \<in> sigma_sets UNIV (Collect open)" by simp
```
```   828 qed simp_all
```
```   829
```
```   830 lemma borel_measurable_halfspacesI:
```
```   831   fixes f :: "'a \<Rightarrow> 'c\<Colon>euclidean_space"
```
```   832   assumes F: "borel = sigma UNIV (F ` (UNIV \<times> Basis))"
```
```   833   and S_eq: "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
```
```   834   shows "f \<in> borel_measurable M = (\<forall>i\<in>Basis. \<forall>a::real. S a i \<in> sets M)"
```
```   835 proof safe
```
```   836   fix a :: real and i :: 'b assume i: "i \<in> Basis" and f: "f \<in> borel_measurable M"
```
```   837   then show "S a i \<in> sets M" unfolding assms
```
```   838     by (auto intro!: measurable_sets simp: assms(1))
```
```   839 next
```
```   840   assume a: "\<forall>i\<in>Basis. \<forall>a. S a i \<in> sets M"
```
```   841   then show "f \<in> borel_measurable M"
```
```   842     by (auto intro!: measurable_measure_of simp: S_eq F)
```
```   843 qed
```
```   844
```
```   845 lemma borel_measurable_iff_halfspace_le:
```
```   846   fixes f :: "'a \<Rightarrow> 'c\<Colon>euclidean_space"
```
```   847   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)"
```
```   848   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
```
```   849
```
```   850 lemma borel_measurable_iff_halfspace_less:
```
```   851   fixes f :: "'a \<Rightarrow> 'c\<Colon>euclidean_space"
```
```   852   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)"
```
```   853   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
```
```   854
```
```   855 lemma borel_measurable_iff_halfspace_ge:
```
```   856   fixes f :: "'a \<Rightarrow> 'c\<Colon>euclidean_space"
```
```   857   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)"
```
```   858   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
```
```   859
```
```   860 lemma borel_measurable_iff_halfspace_greater:
```
```   861   fixes f :: "'a \<Rightarrow> 'c\<Colon>euclidean_space"
```
```   862   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)"
```
```   863   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto
```
```   864
```
```   865 lemma borel_measurable_iff_le:
```
```   866   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
```
```   867   using borel_measurable_iff_halfspace_le[where 'c=real] by simp
```
```   868
```
```   869 lemma borel_measurable_iff_less:
```
```   870   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
```
```   871   using borel_measurable_iff_halfspace_less[where 'c=real] by simp
```
```   872
```
```   873 lemma borel_measurable_iff_ge:
```
```   874   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
```
```   875   using borel_measurable_iff_halfspace_ge[where 'c=real]
```
```   876   by simp
```
```   877
```
```   878 lemma borel_measurable_iff_greater:
```
```   879   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
```
```   880   using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
```
```   881
```
```   882 lemma borel_measurable_euclidean_space:
```
```   883   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```   884   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M)"
```
```   885 proof safe
```
```   886   assume f: "\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M"
```
```   887   then show "f \<in> borel_measurable M"
```
```   888     by (subst borel_measurable_iff_halfspace_le) auto
```
```   889 qed auto
```
```   890
```
```   891 subsection "Borel measurable operators"
```
```   892
```
```   893 lemma borel_measurable_norm[measurable]: "norm \<in> borel_measurable borel"
```
```   894   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```   895
```
```   896 lemma borel_measurable_sgn [measurable]: "(sgn::'a::real_normed_vector \<Rightarrow> 'a) \<in> borel_measurable borel"
```
```   897   by (rule borel_measurable_continuous_countable_exceptions[where X="{0}"])
```
```   898      (auto intro!: continuous_on_sgn continuous_on_id)
```
```   899
```
```   900 lemma borel_measurable_uminus[measurable (raw)]:
```
```   901   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```   902   assumes g: "g \<in> borel_measurable M"
```
```   903   shows "(\<lambda>x. - g x) \<in> borel_measurable M"
```
```   904   by (rule borel_measurable_continuous_on[OF _ g]) (intro continuous_intros)
```
```   905
```
```   906 lemma borel_measurable_add[measurable (raw)]:
```
```   907   fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```   908   assumes f: "f \<in> borel_measurable M"
```
```   909   assumes g: "g \<in> borel_measurable M"
```
```   910   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```   911   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```   912
```
```   913 lemma borel_measurable_setsum[measurable (raw)]:
```
```   914   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```   915   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   916   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```   917 proof cases
```
```   918   assume "finite S"
```
```   919   thus ?thesis using assms by induct auto
```
```   920 qed simp
```
```   921
```
```   922 lemma borel_measurable_diff[measurable (raw)]:
```
```   923   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```   924   assumes f: "f \<in> borel_measurable M"
```
```   925   assumes g: "g \<in> borel_measurable M"
```
```   926   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```   927   using borel_measurable_add [of f M "- g"] assms by (simp add: fun_Compl_def)
```
```   928
```
```   929 lemma borel_measurable_times[measurable (raw)]:
```
```   930   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_algebra}"
```
```   931   assumes f: "f \<in> borel_measurable M"
```
```   932   assumes g: "g \<in> borel_measurable M"
```
```   933   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```   934   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```   935
```
```   936 lemma borel_measurable_setprod[measurable (raw)]:
```
```   937   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, real_normed_field}"
```
```   938   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   939   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```   940 proof cases
```
```   941   assume "finite S"
```
```   942   thus ?thesis using assms by induct auto
```
```   943 qed simp
```
```   944
```
```   945 lemma borel_measurable_dist[measurable (raw)]:
```
```   946   fixes g f :: "'a \<Rightarrow> 'b::{second_countable_topology, metric_space}"
```
```   947   assumes f: "f \<in> borel_measurable M"
```
```   948   assumes g: "g \<in> borel_measurable M"
```
```   949   shows "(\<lambda>x. dist (f x) (g x)) \<in> borel_measurable M"
```
```   950   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```   951
```
```   952 lemma borel_measurable_scaleR[measurable (raw)]:
```
```   953   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```   954   assumes f: "f \<in> borel_measurable M"
```
```   955   assumes g: "g \<in> borel_measurable M"
```
```   956   shows "(\<lambda>x. f x *\<^sub>R g x) \<in> borel_measurable M"
```
```   957   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```   958
```
```   959 lemma affine_borel_measurable_vector:
```
```   960   fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
```
```   961   assumes "f \<in> borel_measurable M"
```
```   962   shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
```
```   963 proof (rule borel_measurableI)
```
```   964   fix S :: "'x set" assume "open S"
```
```   965   show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
```
```   966   proof cases
```
```   967     assume "b \<noteq> 0"
```
```   968     with `open S` have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")
```
```   969       using open_affinity [of S "inverse b" "- a /\<^sub>R b"]
```
```   970       by (auto simp: algebra_simps)
```
```   971     hence "?S \<in> sets borel" by auto
```
```   972     moreover
```
```   973     from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
```
```   974       apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
```
```   975     ultimately show ?thesis using assms unfolding in_borel_measurable_borel
```
```   976       by auto
```
```   977   qed simp
```
```   978 qed
```
```   979
```
```   980 lemma borel_measurable_const_scaleR[measurable (raw)]:
```
```   981   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. b *\<^sub>R f x ::'a::real_normed_vector) \<in> borel_measurable M"
```
```   982   using affine_borel_measurable_vector[of f M 0 b] by simp
```
```   983
```
```   984 lemma borel_measurable_const_add[measurable (raw)]:
```
```   985   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. a + f x ::'a::real_normed_vector) \<in> borel_measurable M"
```
```   986   using affine_borel_measurable_vector[of f M a 1] by simp
```
```   987
```
```   988 lemma borel_measurable_inverse[measurable (raw)]:
```
```   989   fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
```
```   990   assumes f: "f \<in> borel_measurable M"
```
```   991   shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
```
```   992   apply (rule measurable_compose[OF f])
```
```   993   apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])
```
```   994   apply (auto intro!: continuous_on_inverse continuous_on_id)
```
```   995   done
```
```   996
```
```   997 lemma borel_measurable_divide[measurable (raw)]:
```
```   998   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>
```
```   999     (\<lambda>x. f x / g x::'b::{second_countable_topology, real_normed_div_algebra}) \<in> borel_measurable M"
```
```  1000   by (simp add: divide_inverse)
```
```  1001
```
```  1002 lemma borel_measurable_max[measurable (raw)]:
```
```  1003   "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"
```
```  1004   by (simp add: max_def)
```
```  1005
```
```  1006 lemma borel_measurable_min[measurable (raw)]:
```
```  1007   "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"
```
```  1008   by (simp add: min_def)
```
```  1009
```
```  1010 lemma borel_measurable_Min[measurable (raw)]:
```
```  1011   "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"
```
```  1012 proof (induct I rule: finite_induct)
```
```  1013   case (insert i I) then show ?case
```
```  1014     by (cases "I = {}") auto
```
```  1015 qed auto
```
```  1016
```
```  1017 lemma borel_measurable_Max[measurable (raw)]:
```
```  1018   "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"
```
```  1019 proof (induct I rule: finite_induct)
```
```  1020   case (insert i I) then show ?case
```
```  1021     by (cases "I = {}") auto
```
```  1022 qed auto
```
```  1023
```
```  1024 lemma borel_measurable_abs[measurable (raw)]:
```
```  1025   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
```
```  1026   unfolding abs_real_def by simp
```
```  1027
```
```  1028 lemma borel_measurable_nth[measurable (raw)]:
```
```  1029   "(\<lambda>x::real^'n. x \$ i) \<in> borel_measurable borel"
```
```  1030   by (simp add: cart_eq_inner_axis)
```
```  1031
```
```  1032 lemma convex_measurable:
```
```  1033   fixes A :: "'a :: euclidean_space set"
```
```  1034   shows "X \<in> borel_measurable M \<Longrightarrow> X ` space M \<subseteq> A \<Longrightarrow> open A \<Longrightarrow> convex_on A q \<Longrightarrow>
```
```  1035     (\<lambda>x. q (X x)) \<in> borel_measurable M"
```
```  1036   by (rule measurable_compose[where f=X and N="restrict_space borel A"])
```
```  1037      (auto intro!: borel_measurable_continuous_on_restrict convex_on_continuous measurable_restrict_space2)
```
```  1038
```
```  1039 lemma borel_measurable_ln[measurable (raw)]:
```
```  1040   assumes f: "f \<in> borel_measurable M"
```
```  1041   shows "(\<lambda>x. ln (f x :: real)) \<in> borel_measurable M"
```
```  1042   apply (rule measurable_compose[OF f])
```
```  1043   apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])
```
```  1044   apply (auto intro!: continuous_on_ln continuous_on_id)
```
```  1045   done
```
```  1046
```
```  1047 lemma borel_measurable_log[measurable (raw)]:
```
```  1048   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. log (g x) (f x)) \<in> borel_measurable M"
```
```  1049   unfolding log_def by auto
```
```  1050
```
```  1051 lemma borel_measurable_exp[measurable]:
```
```  1052   "(exp::'a::{real_normed_field,banach}\<Rightarrow>'a) \<in> borel_measurable borel"
```
```  1053   by (intro borel_measurable_continuous_on1 continuous_at_imp_continuous_on ballI isCont_exp)
```
```  1054
```
```  1055 lemma measurable_real_floor[measurable]:
```
```  1056   "(floor :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
```
```  1057 proof -
```
```  1058   have "\<And>a x. \<lfloor>x\<rfloor> = a \<longleftrightarrow> (real a \<le> x \<and> x < real (a + 1))"
```
```  1059     by (auto intro: floor_eq2)
```
```  1060   then show ?thesis
```
```  1061     by (auto simp: vimage_def measurable_count_space_eq2_countable)
```
```  1062 qed
```
```  1063
```
```  1064 lemma measurable_real_ceiling[measurable]:
```
```  1065   "(ceiling :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
```
```  1066   unfolding ceiling_def[abs_def] by simp
```
```  1067
```
```  1068 lemma borel_measurable_real_floor: "(\<lambda>x::real. real \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
```
```  1069   by simp
```
```  1070
```
```  1071 lemma borel_measurable_root [measurable]: "root n \<in> borel_measurable borel"
```
```  1072   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1073
```
```  1074 lemma borel_measurable_sqrt [measurable]: "sqrt \<in> borel_measurable borel"
```
```  1075   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1076
```
```  1077 lemma borel_measurable_power [measurable (raw)]:
```
```  1078   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```  1079   assumes f: "f \<in> borel_measurable M"
```
```  1080   shows "(\<lambda>x. (f x) ^ n) \<in> borel_measurable M"
```
```  1081   by (intro borel_measurable_continuous_on [OF _ f] continuous_intros)
```
```  1082
```
```  1083 lemma borel_measurable_Re [measurable]: "Re \<in> borel_measurable borel"
```
```  1084   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1085
```
```  1086 lemma borel_measurable_Im [measurable]: "Im \<in> borel_measurable borel"
```
```  1087   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1088
```
```  1089 lemma borel_measurable_of_real [measurable]: "(of_real :: _ \<Rightarrow> (_::real_normed_algebra)) \<in> borel_measurable borel"
```
```  1090   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1091
```
```  1092 lemma borel_measurable_sin [measurable]: "(sin :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"
```
```  1093   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1094
```
```  1095 lemma borel_measurable_cos [measurable]: "(cos :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"
```
```  1096   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1097
```
```  1098 lemma borel_measurable_arctan [measurable]: "arctan \<in> borel_measurable borel"
```
```  1099   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1100
```
```  1101 lemma borel_measurable_complex_iff:
```
```  1102   "f \<in> borel_measurable M \<longleftrightarrow>
```
```  1103     (\<lambda>x. Re (f x)) \<in> borel_measurable M \<and> (\<lambda>x. Im (f x)) \<in> borel_measurable M"
```
```  1104   apply auto
```
```  1105   apply (subst fun_complex_eq)
```
```  1106   apply (intro borel_measurable_add)
```
```  1107   apply auto
```
```  1108   done
```
```  1109
```
```  1110 subsection "Borel space on the extended reals"
```
```  1111
```
```  1112 lemma borel_measurable_ereal[measurable (raw)]:
```
```  1113   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
```
```  1114   using continuous_on_ereal f by (rule borel_measurable_continuous_on)
```
```  1115
```
```  1116 lemma borel_measurable_real_of_ereal[measurable (raw)]:
```
```  1117   fixes f :: "'a \<Rightarrow> ereal"
```
```  1118   assumes f: "f \<in> borel_measurable M"
```
```  1119   shows "(\<lambda>x. real (f x)) \<in> borel_measurable M"
```
```  1120   apply (rule measurable_compose[OF f])
```
```  1121   apply (rule borel_measurable_continuous_countable_exceptions[of "{\<infinity>, -\<infinity> }"])
```
```  1122   apply (auto intro: continuous_on_real simp: Compl_eq_Diff_UNIV)
```
```  1123   done
```
```  1124
```
```  1125 lemma borel_measurable_ereal_cases:
```
```  1126   fixes f :: "'a \<Rightarrow> ereal"
```
```  1127   assumes f: "f \<in> borel_measurable M"
```
```  1128   assumes H: "(\<lambda>x. H (ereal (real (f x)))) \<in> borel_measurable M"
```
```  1129   shows "(\<lambda>x. H (f x)) \<in> borel_measurable M"
```
```  1130 proof -
```
```  1131   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)))"
```
```  1132   { fix x have "H (f x) = ?F x" by (cases "f x") auto }
```
```  1133   with f H show ?thesis by simp
```
```  1134 qed
```
```  1135
```
```  1136 lemma
```
```  1137   fixes f :: "'a \<Rightarrow> ereal" assumes f[measurable]: "f \<in> borel_measurable M"
```
```  1138   shows borel_measurable_ereal_abs[measurable(raw)]: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
```
```  1139     and borel_measurable_ereal_inverse[measurable(raw)]: "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M"
```
```  1140     and borel_measurable_uminus_ereal[measurable(raw)]: "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M"
```
```  1141   by (auto simp del: abs_real_of_ereal simp: borel_measurable_ereal_cases[OF f] measurable_If)
```
```  1142
```
```  1143 lemma borel_measurable_uminus_eq_ereal[simp]:
```
```  1144   "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
```
```  1145 proof
```
```  1146   assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp
```
```  1147 qed auto
```
```  1148
```
```  1149 lemma set_Collect_ereal2:
```
```  1150   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1151   assumes f: "f \<in> borel_measurable M"
```
```  1152   assumes g: "g \<in> borel_measurable M"
```
```  1153   assumes H: "{x \<in> space M. H (ereal (real (f x))) (ereal (real (g x)))} \<in> sets M"
```
```  1154     "{x \<in> space borel. H (-\<infinity>) (ereal x)} \<in> sets borel"
```
```  1155     "{x \<in> space borel. H (\<infinity>) (ereal x)} \<in> sets borel"
```
```  1156     "{x \<in> space borel. H (ereal x) (-\<infinity>)} \<in> sets borel"
```
```  1157     "{x \<in> space borel. H (ereal x) (\<infinity>)} \<in> sets borel"
```
```  1158   shows "{x \<in> space M. H (f x) (g x)} \<in> sets M"
```
```  1159 proof -
```
```  1160   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)))"
```
```  1161   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"
```
```  1162   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
```
```  1163   note * = this
```
```  1164   from assms show ?thesis
```
```  1165     by (subst *) (simp del: space_borel split del: split_if)
```
```  1166 qed
```
```  1167
```
```  1168 lemma borel_measurable_ereal_iff:
```
```  1169   shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
```
```  1170 proof
```
```  1171   assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
```
```  1172   from borel_measurable_real_of_ereal[OF this]
```
```  1173   show "f \<in> borel_measurable M" by auto
```
```  1174 qed auto
```
```  1175
```
```  1176 lemma borel_measurable_erealD[measurable_dest]:
```
```  1177   "(\<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"
```
```  1178   unfolding borel_measurable_ereal_iff by simp
```
```  1179
```
```  1180 lemma borel_measurable_ereal_iff_real:
```
```  1181   fixes f :: "'a \<Rightarrow> ereal"
```
```  1182   shows "f \<in> borel_measurable M \<longleftrightarrow>
```
```  1183     ((\<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)"
```
```  1184 proof safe
```
```  1185   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"
```
```  1186   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
```
```  1187   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
```
```  1188   let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real (f x))"
```
```  1189   have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
```
```  1190   also have "?f = f" by (auto simp: fun_eq_iff ereal_real)
```
```  1191   finally show "f \<in> borel_measurable M" .
```
```  1192 qed simp_all
```
```  1193
```
```  1194 lemma borel_measurable_ereal_iff_Iio:
```
```  1195   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
```
```  1196   by (auto simp: borel_Iio measurable_iff_measure_of)
```
```  1197
```
```  1198 lemma borel_measurable_ereal_iff_Ioi:
```
```  1199   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
```
```  1200   by (auto simp: borel_Ioi measurable_iff_measure_of)
```
```  1201
```
```  1202 lemma vimage_sets_compl_iff:
```
```  1203   "f -` A \<inter> space M \<in> sets M \<longleftrightarrow> f -` (- A) \<inter> space M \<in> sets M"
```
```  1204 proof -
```
```  1205   { fix A assume "f -` A \<inter> space M \<in> sets M"
```
```  1206     moreover have "f -` (- A) \<inter> space M = space M - f -` A \<inter> space M" by auto
```
```  1207     ultimately have "f -` (- A) \<inter> space M \<in> sets M" by auto }
```
```  1208   from this[of A] this[of "-A"] show ?thesis
```
```  1209     by (metis double_complement)
```
```  1210 qed
```
```  1211
```
```  1212 lemma borel_measurable_iff_Iic_ereal:
```
```  1213   "(f::'a\<Rightarrow>ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
```
```  1214   unfolding borel_measurable_ereal_iff_Ioi vimage_sets_compl_iff[where A="{a <..}" for a] by simp
```
```  1215
```
```  1216 lemma borel_measurable_iff_Ici_ereal:
```
```  1217   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
```
```  1218   unfolding borel_measurable_ereal_iff_Iio vimage_sets_compl_iff[where A="{..< a}" for a] by simp
```
```  1219
```
```  1220 lemma borel_measurable_ereal2:
```
```  1221   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1222   assumes f: "f \<in> borel_measurable M"
```
```  1223   assumes g: "g \<in> borel_measurable M"
```
```  1224   assumes H: "(\<lambda>x. H (ereal (real (f x))) (ereal (real (g x)))) \<in> borel_measurable M"
```
```  1225     "(\<lambda>x. H (-\<infinity>) (ereal (real (g x)))) \<in> borel_measurable M"
```
```  1226     "(\<lambda>x. H (\<infinity>) (ereal (real (g x)))) \<in> borel_measurable M"
```
```  1227     "(\<lambda>x. H (ereal (real (f x))) (-\<infinity>)) \<in> borel_measurable M"
```
```  1228     "(\<lambda>x. H (ereal (real (f x))) (\<infinity>)) \<in> borel_measurable M"
```
```  1229   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
```
```  1230 proof -
```
```  1231   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)))"
```
```  1232   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"
```
```  1233   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
```
```  1234   note * = this
```
```  1235   from assms show ?thesis unfolding * by simp
```
```  1236 qed
```
```  1237
```
```  1238 lemma
```
```  1239   fixes f :: "'a \<Rightarrow> ereal" assumes f: "f \<in> borel_measurable M"
```
```  1240   shows borel_measurable_ereal_eq_const: "{x\<in>space M. f x = c} \<in> sets M"
```
```  1241     and borel_measurable_ereal_neq_const: "{x\<in>space M. f x \<noteq> c} \<in> sets M"
```
```  1242   using f by auto
```
```  1243
```
```  1244 lemma [measurable(raw)]:
```
```  1245   fixes f :: "'a \<Rightarrow> ereal"
```
```  1246   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1247   shows borel_measurable_ereal_add: "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```  1248     and borel_measurable_ereal_times: "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1249     and borel_measurable_ereal_min: "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
```
```  1250     and borel_measurable_ereal_max: "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
```
```  1251   by (simp_all add: borel_measurable_ereal2 min_def max_def)
```
```  1252
```
```  1253 lemma [measurable(raw)]:
```
```  1254   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1255   assumes "f \<in> borel_measurable M"
```
```  1256   assumes "g \<in> borel_measurable M"
```
```  1257   shows borel_measurable_ereal_diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```  1258     and borel_measurable_ereal_divide: "(\<lambda>x. f x / g x) \<in> borel_measurable M"
```
```  1259   using assms by (simp_all add: minus_ereal_def divide_ereal_def)
```
```  1260
```
```  1261 lemma borel_measurable_ereal_setsum[measurable (raw)]:
```
```  1262   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1263   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1264   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1265   using assms by (induction S rule: infinite_finite_induct) auto
```
```  1266
```
```  1267 lemma borel_measurable_ereal_setprod[measurable (raw)]:
```
```  1268   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1269   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1270   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1271   using assms by (induction S rule: infinite_finite_induct) auto
```
```  1272
```
```  1273 lemma [measurable (raw)]:
```
```  1274   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1275   assumes "\<And>i. f i \<in> borel_measurable M"
```
```  1276   shows borel_measurable_liminf: "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1277     and borel_measurable_limsup: "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1278   unfolding liminf_SUP_INF limsup_INF_SUP using assms by auto
```
```  1279
```
```  1280 lemma sets_Collect_eventually_sequentially[measurable]:
```
```  1281   "(\<And>i. {x\<in>space M. P x i} \<in> sets M) \<Longrightarrow> {x\<in>space M. eventually (P x) sequentially} \<in> sets M"
```
```  1282   unfolding eventually_sequentially by simp
```
```  1283
```
```  1284 lemma sets_Collect_ereal_convergent[measurable]:
```
```  1285   fixes f :: "nat \<Rightarrow> 'a => ereal"
```
```  1286   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1287   shows "{x\<in>space M. convergent (\<lambda>i. f i x)} \<in> sets M"
```
```  1288   unfolding convergent_ereal by auto
```
```  1289
```
```  1290 lemma borel_measurable_extreal_lim[measurable (raw)]:
```
```  1291   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1292   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1293   shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1294 proof -
```
```  1295   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))"
```
```  1296     by (simp add: lim_def convergent_def convergent_limsup_cl)
```
```  1297   then show ?thesis
```
```  1298     by simp
```
```  1299 qed
```
```  1300
```
```  1301 lemma borel_measurable_ereal_LIMSEQ:
```
```  1302   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1303   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
```
```  1304   and u: "\<And>i. u i \<in> borel_measurable M"
```
```  1305   shows "u' \<in> borel_measurable M"
```
```  1306 proof -
```
```  1307   have "\<And>x. x \<in> space M \<Longrightarrow> u' x = liminf (\<lambda>n. u n x)"
```
```  1308     using u' by (simp add: lim_imp_Liminf[symmetric])
```
```  1309   with u show ?thesis by (simp cong: measurable_cong)
```
```  1310 qed
```
```  1311
```
```  1312 lemma borel_measurable_extreal_suminf[measurable (raw)]:
```
```  1313   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1314   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1315   shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
```
```  1316   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
```
```  1317
```
```  1318 subsection {* LIMSEQ is borel measurable *}
```
```  1319
```
```  1320 lemma borel_measurable_LIMSEQ:
```
```  1321   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
```
```  1322   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
```
```  1323   and u: "\<And>i. u i \<in> borel_measurable M"
```
```  1324   shows "u' \<in> borel_measurable M"
```
```  1325 proof -
```
```  1326   have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)"
```
```  1327     using u' by (simp add: lim_imp_Liminf)
```
```  1328   moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M"
```
```  1329     by auto
```
```  1330   ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff)
```
```  1331 qed
```
```  1332
```
```  1333 lemma borel_measurable_LIMSEQ_metric:
```
```  1334   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b :: metric_space"
```
```  1335   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1336   assumes lim: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. f i x) ----> g x"
```
```  1337   shows "g \<in> borel_measurable M"
```
```  1338   unfolding borel_eq_closed
```
```  1339 proof (safe intro!: measurable_measure_of)
```
```  1340   fix A :: "'b set" assume "closed A"
```
```  1341
```
```  1342   have [measurable]: "(\<lambda>x. infdist (g x) A) \<in> borel_measurable M"
```
```  1343   proof (rule borel_measurable_LIMSEQ)
```
```  1344     show "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. infdist (f i x) A) ----> infdist (g x) A"
```
```  1345       by (intro tendsto_infdist lim)
```
```  1346     show "\<And>i. (\<lambda>x. infdist (f i x) A) \<in> borel_measurable M"
```
```  1347       by (intro borel_measurable_continuous_on[where f="\<lambda>x. infdist x A"]
```
```  1348         continuous_at_imp_continuous_on ballI continuous_infdist continuous_ident) auto
```
```  1349   qed
```
```  1350
```
```  1351   show "g -` A \<inter> space M \<in> sets M"
```
```  1352   proof cases
```
```  1353     assume "A \<noteq> {}"
```
```  1354     then have "\<And>x. infdist x A = 0 \<longleftrightarrow> x \<in> A"
```
```  1355       using `closed A` by (simp add: in_closed_iff_infdist_zero)
```
```  1356     then have "g -` A \<inter> space M = {x\<in>space M. infdist (g x) A = 0}"
```
```  1357       by auto
```
```  1358     also have "\<dots> \<in> sets M"
```
```  1359       by measurable
```
```  1360     finally show ?thesis .
```
```  1361   qed simp
```
```  1362 qed auto
```
```  1363
```
```  1364 lemma sets_Collect_Cauchy[measurable]:
```
```  1365   fixes f :: "nat \<Rightarrow> 'a => 'b::{metric_space, second_countable_topology}"
```
```  1366   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1367   shows "{x\<in>space M. Cauchy (\<lambda>i. f i x)} \<in> sets M"
```
```  1368   unfolding metric_Cauchy_iff2 using f by auto
```
```  1369
```
```  1370 lemma borel_measurable_lim[measurable (raw)]:
```
```  1371   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```  1372   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1373   shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1374 proof -
```
```  1375   def u' \<equiv> "\<lambda>x. lim (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)"
```
```  1376   then have *: "\<And>x. lim (\<lambda>i. f i x) = (if Cauchy (\<lambda>i. f i x) then u' x else (THE x. False))"
```
```  1377     by (auto simp: lim_def convergent_eq_cauchy[symmetric])
```
```  1378   have "u' \<in> borel_measurable M"
```
```  1379   proof (rule borel_measurable_LIMSEQ_metric)
```
```  1380     fix x
```
```  1381     have "convergent (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)"
```
```  1382       by (cases "Cauchy (\<lambda>i. f i x)")
```
```  1383          (auto simp add: convergent_eq_cauchy[symmetric] convergent_def)
```
```  1384     then show "(\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0) ----> u' x"
```
```  1385       unfolding u'_def
```
```  1386       by (rule convergent_LIMSEQ_iff[THEN iffD1])
```
```  1387   qed measurable
```
```  1388   then show ?thesis
```
```  1389     unfolding * by measurable
```
```  1390 qed
```
```  1391
```
```  1392 lemma borel_measurable_suminf[measurable (raw)]:
```
```  1393   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```  1394   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1395   shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1396   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
```
```  1397
```
```  1398 lemma borel_measurable_sup[measurable (raw)]:
```
```  1399   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>
```
```  1400     (\<lambda>x. sup (f x) (g x)::ereal) \<in> borel_measurable M"
```
```  1401   by simp
```
```  1402
```
```  1403 (* Proof by Jeremy Avigad and Luke Serafin *)
```
```  1404 lemma isCont_borel:
```
```  1405   fixes f :: "'b::metric_space \<Rightarrow> 'a::metric_space"
```
```  1406   shows "{x. isCont f x} \<in> sets borel"
```
```  1407 proof -
```
```  1408   let ?I = "\<lambda>j. inverse(real (Suc j))"
```
```  1409
```
```  1410   { fix x
```
```  1411     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)"
```
```  1412       unfolding continuous_at_eps_delta
```
```  1413     proof safe
```
```  1414       fix i assume "\<forall>e>0. \<exists>d>0. \<forall>y. dist y x < d \<longrightarrow> dist (f y) (f x) < e"
```
```  1415       moreover have "0 < ?I i / 2"
```
```  1416         by simp
```
```  1417       ultimately obtain d where d: "0 < d" "\<And>y. dist x y < d \<Longrightarrow> dist (f y) (f x) < ?I i / 2"
```
```  1418         by (metis dist_commute)
```
```  1419       then obtain j where j: "?I j < d"
```
```  1420         by (metis reals_Archimedean)
```
```  1421
```
```  1422       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"
```
```  1423       proof (safe intro!: exI[where x=j])
```
```  1424         fix y z assume *: "dist x y < ?I j" "dist x z < ?I j"
```
```  1425         have "dist (f y) (f z) \<le> dist (f y) (f x) + dist (f z) (f x)"
```
```  1426           by (rule dist_triangle2)
```
```  1427         also have "\<dots> < ?I i / 2 + ?I i / 2"
```
```  1428           by (intro add_strict_mono d less_trans[OF _ j] *)
```
```  1429         also have "\<dots> \<le> ?I i"
```
```  1430           by (simp add: field_simps real_of_nat_Suc)
```
```  1431         finally show "dist (f y) (f z) \<le> ?I i"
```
```  1432           by simp
```
```  1433       qed
```
```  1434     next
```
```  1435       fix e::real assume "0 < e"
```
```  1436       then obtain n where n: "?I n < e"
```
```  1437         by (metis reals_Archimedean)
```
```  1438       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"
```
```  1439       from this[THEN spec, of "Suc n"]
```
```  1440       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)"
```
```  1441         by auto
```
```  1442
```
```  1443       show "\<exists>d>0. \<forall>y. dist y x < d \<longrightarrow> dist (f y) (f x) < e"
```
```  1444       proof (safe intro!: exI[of _ "?I j"])
```
```  1445         fix y assume "dist y x < ?I j"
```
```  1446         then have "dist (f y) (f x) \<le> ?I (Suc n)"
```
```  1447           by (intro j) (auto simp: dist_commute)
```
```  1448         also have "?I (Suc n) < ?I n"
```
```  1449           by simp
```
```  1450         also note n
```
```  1451         finally show "dist (f y) (f x) < e" .
```
```  1452       qed simp
```
```  1453     qed }
```
```  1454   note * = this
```
```  1455
```
```  1456   have **: "\<And>e y. open {x. dist x y < e}"
```
```  1457     using open_ball by (simp_all add: ball_def dist_commute)
```
```  1458
```
```  1459   have "{x\<in>space borel. isCont f x} \<in> sets borel"
```
```  1460     unfolding *
```
```  1461     apply (intro sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex)
```
```  1462     apply (simp add: Collect_all_eq)
```
```  1463     apply (intro borel_closed closed_INT ballI closed_Collect_imp open_Collect_conj **)
```
```  1464     apply auto
```
```  1465     done
```
```  1466   then show ?thesis
```
```  1467     by simp
```
```  1468 qed
```
```  1469
```
```  1470 no_notation
```
```  1471   eucl_less (infix "<e" 50)
```
```  1472
```
```  1473 end
```