src/HOL/Analysis/Borel_Space.thy
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```     1 (*  Title:      HOL/Analysis/Borel_Space.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Armin Heller, TU München
```
```     4 *)
```
```     5
```
```     6 section \<open>Borel spaces\<close>
```
```     7
```
```     8 theory Borel_Space
```
```     9 imports
```
```    10   Measurable Derivative Ordered_Euclidean_Space Extended_Real_Limits
```
```    11 begin
```
```    12
```
```    13 lemma sets_Collect_eventually_sequentially[measurable]:
```
```    14   "(\<And>i. {x\<in>space M. P x i} \<in> sets M) \<Longrightarrow> {x\<in>space M. eventually (P x) sequentially} \<in> sets M"
```
```    15   unfolding eventually_sequentially by simp
```
```    16
```
```    17 lemma topological_basis_trivial: "topological_basis {A. open A}"
```
```    18   by (auto simp: topological_basis_def)
```
```    19
```
```    20 lemma open_prod_generated: "open = generate_topology {A \<times> B | A B. open A \<and> open B}"
```
```    21 proof -
```
```    22   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}))"
```
```    23     by auto
```
```    24   then show ?thesis
```
```    25     by (auto intro: topological_basis_prod topological_basis_trivial topological_basis_imp_subbasis)
```
```    26 qed
```
```    27
```
```    28 definition "mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r \<le> s \<longrightarrow> f r \<le> f s"
```
```    29
```
```    30 lemma mono_onI:
```
```    31   "(\<And>r s. r \<in> A \<Longrightarrow> s \<in> A \<Longrightarrow> r \<le> s \<Longrightarrow> f r \<le> f s) \<Longrightarrow> mono_on f A"
```
```    32   unfolding mono_on_def by simp
```
```    33
```
```    34 lemma mono_onD:
```
```    35   "\<lbrakk>mono_on f A; r \<in> A; s \<in> A; r \<le> s\<rbrakk> \<Longrightarrow> f r \<le> f s"
```
```    36   unfolding mono_on_def by simp
```
```    37
```
```    38 lemma mono_imp_mono_on: "mono f \<Longrightarrow> mono_on f A"
```
```    39   unfolding mono_def mono_on_def by auto
```
```    40
```
```    41 lemma mono_on_subset: "mono_on f A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> mono_on f B"
```
```    42   unfolding mono_on_def by auto
```
```    43
```
```    44 definition "strict_mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r < s \<longrightarrow> f r < f s"
```
```    45
```
```    46 lemma strict_mono_onI:
```
```    47   "(\<And>r s. r \<in> A \<Longrightarrow> s \<in> A \<Longrightarrow> r < s \<Longrightarrow> f r < f s) \<Longrightarrow> strict_mono_on f A"
```
```    48   unfolding strict_mono_on_def by simp
```
```    49
```
```    50 lemma strict_mono_onD:
```
```    51   "\<lbrakk>strict_mono_on f A; r \<in> A; s \<in> A; r < s\<rbrakk> \<Longrightarrow> f r < f s"
```
```    52   unfolding strict_mono_on_def by simp
```
```    53
```
```    54 lemma mono_on_greaterD:
```
```    55   assumes "mono_on g A" "x \<in> A" "y \<in> A" "g x > (g (y::_::linorder) :: _ :: linorder)"
```
```    56   shows "x > y"
```
```    57 proof (rule ccontr)
```
```    58   assume "\<not>x > y"
```
```    59   hence "x \<le> y" by (simp add: not_less)
```
```    60   from assms(1-3) and this have "g x \<le> g y" by (rule mono_onD)
```
```    61   with assms(4) show False by simp
```
```    62 qed
```
```    63
```
```    64 lemma strict_mono_inv:
```
```    65   fixes f :: "('a::linorder) \<Rightarrow> ('b::linorder)"
```
```    66   assumes "strict_mono f" and "surj f" and inv: "\<And>x. g (f x) = x"
```
```    67   shows "strict_mono g"
```
```    68 proof
```
```    69   fix x y :: 'b assume "x < y"
```
```    70   from \<open>surj f\<close> obtain x' y' where [simp]: "x = f x'" "y = f y'" by blast
```
```    71   with \<open>x < y\<close> and \<open>strict_mono f\<close> have "x' < y'" by (simp add: strict_mono_less)
```
```    72   with inv show "g x < g y" by simp
```
```    73 qed
```
```    74
```
```    75 lemma strict_mono_on_imp_inj_on:
```
```    76   assumes "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> (_ :: preorder)) A"
```
```    77   shows "inj_on f A"
```
```    78 proof (rule inj_onI)
```
```    79   fix x y assume "x \<in> A" "y \<in> A" "f x = f y"
```
```    80   thus "x = y"
```
```    81     by (cases x y rule: linorder_cases)
```
```    82        (auto dest: strict_mono_onD[OF assms, of x y] strict_mono_onD[OF assms, of y x])
```
```    83 qed
```
```    84
```
```    85 lemma strict_mono_on_leD:
```
```    86   assumes "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) A" "x \<in> A" "y \<in> A" "x \<le> y"
```
```    87   shows "f x \<le> f y"
```
```    88 proof (insert le_less_linear[of y x], elim disjE)
```
```    89   assume "x < y"
```
```    90   with assms have "f x < f y" by (rule_tac strict_mono_onD[OF assms(1)]) simp_all
```
```    91   thus ?thesis by (rule less_imp_le)
```
```    92 qed (insert assms, simp)
```
```    93
```
```    94 lemma strict_mono_on_eqD:
```
```    95   fixes f :: "(_ :: linorder) \<Rightarrow> (_ :: preorder)"
```
```    96   assumes "strict_mono_on f A" "f x = f y" "x \<in> A" "y \<in> A"
```
```    97   shows "y = x"
```
```    98   using assms by (rule_tac linorder_cases[of x y]) (auto dest: strict_mono_onD)
```
```    99
```
```   100 lemma mono_on_imp_deriv_nonneg:
```
```   101   assumes mono: "mono_on f A" and deriv: "(f has_real_derivative D) (at x)"
```
```   102   assumes "x \<in> interior A"
```
```   103   shows "D \<ge> 0"
```
```   104 proof (rule tendsto_lowerbound)
```
```   105   let ?A' = "(\<lambda>y. y - x) ` interior A"
```
```   106   from deriv show "((\<lambda>h. (f (x + h) - f x) / h) \<longlongrightarrow> D) (at 0)"
```
```   107       by (simp add: field_has_derivative_at has_field_derivative_def)
```
```   108   from mono have mono': "mono_on f (interior A)" by (rule mono_on_subset) (rule interior_subset)
```
```   109
```
```   110   show "eventually (\<lambda>h. (f (x + h) - f x) / h \<ge> 0) (at 0)"
```
```   111   proof (subst eventually_at_topological, intro exI conjI ballI impI)
```
```   112     have "open (interior A)" by simp
```
```   113     hence "open (op + (-x) ` interior A)" by (rule open_translation)
```
```   114     also have "(op + (-x) ` interior A) = ?A'" by auto
```
```   115     finally show "open ?A'" .
```
```   116   next
```
```   117     from \<open>x \<in> interior A\<close> show "0 \<in> ?A'" by auto
```
```   118   next
```
```   119     fix h assume "h \<in> ?A'"
```
```   120     hence "x + h \<in> interior A" by auto
```
```   121     with mono' and \<open>x \<in> interior A\<close> show "(f (x + h) - f x) / h \<ge> 0"
```
```   122       by (cases h rule: linorder_cases[of _ 0])
```
```   123          (simp_all add: divide_nonpos_neg divide_nonneg_pos mono_onD field_simps)
```
```   124   qed
```
```   125 qed simp
```
```   126
```
```   127 lemma strict_mono_on_imp_mono_on:
```
```   128   "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) A \<Longrightarrow> mono_on f A"
```
```   129   by (rule mono_onI, rule strict_mono_on_leD)
```
```   130
```
```   131 lemma mono_on_ctble_discont:
```
```   132   fixes f :: "real \<Rightarrow> real"
```
```   133   fixes A :: "real set"
```
```   134   assumes "mono_on f A"
```
```   135   shows "countable {a\<in>A. \<not> continuous (at a within A) f}"
```
```   136 proof -
```
```   137   have mono: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```   138     using \<open>mono_on f A\<close> by (simp add: mono_on_def)
```
```   139   have "\<forall>a \<in> {a\<in>A. \<not> continuous (at a within A) f}. \<exists>q :: nat \<times> rat.
```
```   140       (fst q = 0 \<and> of_rat (snd q) < f a \<and> (\<forall>x \<in> A. x < a \<longrightarrow> f x < of_rat (snd q))) \<or>
```
```   141       (fst q = 1 \<and> of_rat (snd q) > f a \<and> (\<forall>x \<in> A. x > a \<longrightarrow> f x > of_rat (snd q)))"
```
```   142   proof (clarsimp simp del: One_nat_def)
```
```   143     fix a assume "a \<in> A" assume "\<not> continuous (at a within A) f"
```
```   144     thus "\<exists>q1 q2.
```
```   145             q1 = 0 \<and> real_of_rat q2 < f a \<and> (\<forall>x\<in>A. x < a \<longrightarrow> f x < real_of_rat q2) \<or>
```
```   146             q1 = 1 \<and> f a < real_of_rat q2 \<and> (\<forall>x\<in>A. a < x \<longrightarrow> real_of_rat q2 < f x)"
```
```   147     proof (auto simp add: continuous_within order_tendsto_iff eventually_at)
```
```   148       fix l assume "l < f a"
```
```   149       then obtain q2 where q2: "l < of_rat q2" "of_rat q2 < f a"
```
```   150         using of_rat_dense by blast
```
```   151       assume * [rule_format]: "\<forall>d>0. \<exists>x\<in>A. x \<noteq> a \<and> dist x a < d \<and> \<not> l < f x"
```
```   152       from q2 have "real_of_rat q2 < f a \<and> (\<forall>x\<in>A. x < a \<longrightarrow> f x < real_of_rat q2)"
```
```   153       proof auto
```
```   154         fix x assume "x \<in> A" "x < a"
```
```   155         with q2 *[of "a - x"] show "f x < real_of_rat q2"
```
```   156           apply (auto simp add: dist_real_def not_less)
```
```   157           apply (subgoal_tac "f x \<le> f xa")
```
```   158           by (auto intro: mono)
```
```   159       qed
```
```   160       thus ?thesis by auto
```
```   161     next
```
```   162       fix u assume "u > f a"
```
```   163       then obtain q2 where q2: "f a < of_rat q2" "of_rat q2 < u"
```
```   164         using of_rat_dense by blast
```
```   165       assume *[rule_format]: "\<forall>d>0. \<exists>x\<in>A. x \<noteq> a \<and> dist x a < d \<and> \<not> u > f x"
```
```   166       from q2 have "real_of_rat q2 > f a \<and> (\<forall>x\<in>A. x > a \<longrightarrow> f x > real_of_rat q2)"
```
```   167       proof auto
```
```   168         fix x assume "x \<in> A" "x > a"
```
```   169         with q2 *[of "x - a"] show "f x > real_of_rat q2"
```
```   170           apply (auto simp add: dist_real_def)
```
```   171           apply (subgoal_tac "f x \<ge> f xa")
```
```   172           by (auto intro: mono)
```
```   173       qed
```
```   174       thus ?thesis by auto
```
```   175     qed
```
```   176   qed
```
```   177   hence "\<exists>g :: real \<Rightarrow> nat \<times> rat . \<forall>a \<in> {a\<in>A. \<not> continuous (at a within A) f}.
```
```   178       (fst (g a) = 0 \<and> of_rat (snd (g a)) < f a \<and> (\<forall>x \<in> A. x < a \<longrightarrow> f x < of_rat (snd (g a)))) |
```
```   179       (fst (g a) = 1 \<and> of_rat (snd (g a)) > f a \<and> (\<forall>x \<in> A. x > a \<longrightarrow> f x > of_rat (snd (g a))))"
```
```   180     by (rule bchoice)
```
```   181   then guess g ..
```
```   182   hence g: "\<And>a x. a \<in> A \<Longrightarrow> \<not> continuous (at a within A) f \<Longrightarrow> x \<in> A \<Longrightarrow>
```
```   183       (fst (g a) = 0 \<and> of_rat (snd (g a)) < f a \<and> (x < a \<longrightarrow> f x < of_rat (snd (g a)))) |
```
```   184       (fst (g a) = 1 \<and> of_rat (snd (g a)) > f a \<and> (x > a \<longrightarrow> f x > of_rat (snd (g a))))"
```
```   185     by auto
```
```   186   have "inj_on g {a\<in>A. \<not> continuous (at a within A) f}"
```
```   187   proof (auto simp add: inj_on_def)
```
```   188     fix w z
```
```   189     assume 1: "w \<in> A" and 2: "\<not> continuous (at w within A) f" and
```
```   190            3: "z \<in> A" and 4: "\<not> continuous (at z within A) f" and
```
```   191            5: "g w = g z"
```
```   192     from g [OF 1 2 3] g [OF 3 4 1] 5
```
```   193     show "w = z" by auto
```
```   194   qed
```
```   195   thus ?thesis
```
```   196     by (rule countableI')
```
```   197 qed
```
```   198
```
```   199 lemma mono_on_ctble_discont_open:
```
```   200   fixes f :: "real \<Rightarrow> real"
```
```   201   fixes A :: "real set"
```
```   202   assumes "open A" "mono_on f A"
```
```   203   shows "countable {a\<in>A. \<not>isCont f a}"
```
```   204 proof -
```
```   205   have "{a\<in>A. \<not>isCont f a} = {a\<in>A. \<not>(continuous (at a within A) f)}"
```
```   206     by (auto simp add: continuous_within_open [OF _ \<open>open A\<close>])
```
```   207   thus ?thesis
```
```   208     apply (elim ssubst)
```
```   209     by (rule mono_on_ctble_discont, rule assms)
```
```   210 qed
```
```   211
```
```   212 lemma mono_ctble_discont:
```
```   213   fixes f :: "real \<Rightarrow> real"
```
```   214   assumes "mono f"
```
```   215   shows "countable {a. \<not> isCont f a}"
```
```   216 using assms mono_on_ctble_discont [of f UNIV] unfolding mono_on_def mono_def by auto
```
```   217
```
```   218 lemma has_real_derivative_imp_continuous_on:
```
```   219   assumes "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
```
```   220   shows "continuous_on A f"
```
```   221   apply (intro differentiable_imp_continuous_on, unfold differentiable_on_def)
```
```   222   apply (intro ballI Deriv.differentiableI)
```
```   223   apply (rule has_field_derivative_subset[OF assms])
```
```   224   apply simp_all
```
```   225   done
```
```   226
```
```   227 lemma closure_contains_Sup:
```
```   228   fixes S :: "real set"
```
```   229   assumes "S \<noteq> {}" "bdd_above S"
```
```   230   shows "Sup S \<in> closure S"
```
```   231 proof-
```
```   232   have "Inf (uminus ` S) \<in> closure (uminus ` S)"
```
```   233       using assms by (intro closure_contains_Inf) auto
```
```   234   also have "Inf (uminus ` S) = -Sup S" by (simp add: Inf_real_def)
```
```   235   also have "closure (uminus ` S) = uminus ` closure S"
```
```   236       by (rule sym, intro closure_injective_linear_image) (auto intro: linearI)
```
```   237   finally show ?thesis by auto
```
```   238 qed
```
```   239
```
```   240 lemma closed_contains_Sup:
```
```   241   fixes S :: "real set"
```
```   242   shows "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> closed S \<Longrightarrow> Sup S \<in> S"
```
```   243   by (subst closure_closed[symmetric], assumption, rule closure_contains_Sup)
```
```   244
```
```   245 lemma deriv_nonneg_imp_mono:
```
```   246   assumes deriv: "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_real_derivative g' x) (at x)"
```
```   247   assumes nonneg: "\<And>x. x \<in> {a..b} \<Longrightarrow> g' x \<ge> 0"
```
```   248   assumes ab: "a \<le> b"
```
```   249   shows "g a \<le> g b"
```
```   250 proof (cases "a < b")
```
```   251   assume "a < b"
```
```   252   from deriv have "\<forall>x. x \<ge> a \<and> x \<le> b \<longrightarrow> (g has_real_derivative g' x) (at x)" by simp
```
```   253   from MVT2[OF \<open>a < b\<close> this] and deriv
```
```   254     obtain \<xi> where \<xi>_ab: "\<xi> > a" "\<xi> < b" and g_ab: "g b - g a = (b - a) * g' \<xi>" by blast
```
```   255   from \<xi>_ab ab nonneg have "(b - a) * g' \<xi> \<ge> 0" by simp
```
```   256   with g_ab show ?thesis by simp
```
```   257 qed (insert ab, simp)
```
```   258
```
```   259 lemma continuous_interval_vimage_Int:
```
```   260   assumes "continuous_on {a::real..b} g" and mono: "\<And>x y. a \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> b \<Longrightarrow> g x \<le> g y"
```
```   261   assumes "a \<le> b" "(c::real) \<le> d" "{c..d} \<subseteq> {g a..g b}"
```
```   262   obtains c' d' where "{a..b} \<inter> g -` {c..d} = {c'..d'}" "c' \<le> d'" "g c' = c" "g d' = d"
```
```   263 proof-
```
```   264   let ?A = "{a..b} \<inter> g -` {c..d}"
```
```   265   from IVT'[of g a c b, OF _ _ \<open>a \<le> b\<close> assms(1)] assms(4,5)
```
```   266   obtain c'' where c'': "c'' \<in> ?A" "g c'' = c" by auto
```
```   267   from IVT'[of g a d b, OF _ _ \<open>a \<le> b\<close> assms(1)] assms(4,5)
```
```   268   obtain d'' where d'': "d'' \<in> ?A" "g d'' = d" by auto
```
```   269   hence [simp]: "?A \<noteq> {}" by blast
```
```   270
```
```   271   define c' where "c' = Inf ?A"
```
```   272   define d' where "d' = Sup ?A"
```
```   273   have "?A \<subseteq> {c'..d'}" unfolding c'_def d'_def
```
```   274     by (intro subsetI) (auto intro: cInf_lower cSup_upper)
```
```   275   moreover from assms have "closed ?A"
```
```   276     using continuous_on_closed_vimage[of "{a..b}" g] by (subst Int_commute) simp
```
```   277   hence c'd'_in_set: "c' \<in> ?A" "d' \<in> ?A" unfolding c'_def d'_def
```
```   278     by ((intro closed_contains_Inf closed_contains_Sup, simp_all)[])+
```
```   279   hence "{c'..d'} \<subseteq> ?A" using assms
```
```   280     by (intro subsetI)
```
```   281        (auto intro!: order_trans[of c "g c'" "g x" for x] order_trans[of "g x" "g d'" d for x]
```
```   282              intro!: mono)
```
```   283   moreover have "c' \<le> d'" using c'd'_in_set(2) unfolding c'_def by (intro cInf_lower) auto
```
```   284   moreover have "g c' \<le> c" "g d' \<ge> d"
```
```   285     apply (insert c'' d'' c'd'_in_set)
```
```   286     apply (subst c''(2)[symmetric])
```
```   287     apply (auto simp: c'_def intro!: mono cInf_lower c'') []
```
```   288     apply (subst d''(2)[symmetric])
```
```   289     apply (auto simp: d'_def intro!: mono cSup_upper d'') []
```
```   290     done
```
```   291   with c'd'_in_set have "g c' = c" "g d' = d" by auto
```
```   292   ultimately show ?thesis using that by blast
```
```   293 qed
```
```   294
```
```   295 subsection \<open>Generic Borel spaces\<close>
```
```   296
```
```   297 definition (in topological_space) borel :: "'a measure" where
```
```   298   "borel = sigma UNIV {S. open S}"
```
```   299
```
```   300 abbreviation "borel_measurable M \<equiv> measurable M borel"
```
```   301
```
```   302 lemma in_borel_measurable:
```
```   303    "f \<in> borel_measurable M \<longleftrightarrow>
```
```   304     (\<forall>S \<in> sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)"
```
```   305   by (auto simp add: measurable_def borel_def)
```
```   306
```
```   307 lemma in_borel_measurable_borel:
```
```   308    "f \<in> borel_measurable M \<longleftrightarrow>
```
```   309     (\<forall>S \<in> sets borel.
```
```   310       f -` S \<inter> space M \<in> sets M)"
```
```   311   by (auto simp add: measurable_def borel_def)
```
```   312
```
```   313 lemma space_borel[simp]: "space borel = UNIV"
```
```   314   unfolding borel_def by auto
```
```   315
```
```   316 lemma space_in_borel[measurable]: "UNIV \<in> sets borel"
```
```   317   unfolding borel_def by auto
```
```   318
```
```   319 lemma sets_borel: "sets borel = sigma_sets UNIV {S. open S}"
```
```   320   unfolding borel_def by (rule sets_measure_of) simp
```
```   321
```
```   322 lemma measurable_sets_borel:
```
```   323     "\<lbrakk>f \<in> measurable borel M; A \<in> sets M\<rbrakk> \<Longrightarrow> f -` A \<in> sets borel"
```
```   324   by (drule (1) measurable_sets) simp
```
```   325
```
```   326 lemma pred_Collect_borel[measurable (raw)]: "Measurable.pred borel P \<Longrightarrow> {x. P x} \<in> sets borel"
```
```   327   unfolding borel_def pred_def by auto
```
```   328
```
```   329 lemma borel_open[measurable (raw generic)]:
```
```   330   assumes "open A" shows "A \<in> sets borel"
```
```   331 proof -
```
```   332   have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms .
```
```   333   thus ?thesis unfolding borel_def by auto
```
```   334 qed
```
```   335
```
```   336 lemma borel_closed[measurable (raw generic)]:
```
```   337   assumes "closed A" shows "A \<in> sets borel"
```
```   338 proof -
```
```   339   have "space borel - (- A) \<in> sets borel"
```
```   340     using assms unfolding closed_def by (blast intro: borel_open)
```
```   341   thus ?thesis by simp
```
```   342 qed
```
```   343
```
```   344 lemma borel_singleton[measurable]:
```
```   345   "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets (borel :: 'a::t1_space measure)"
```
```   346   unfolding insert_def by (rule sets.Un) auto
```
```   347
```
```   348 lemma sets_borel_eq_count_space: "sets (borel :: 'a::{countable, t2_space} measure) = count_space UNIV"
```
```   349 proof -
```
```   350   have "(\<Union>a\<in>A. {a}) \<in> sets borel" for A :: "'a set"
```
```   351     by (intro sets.countable_UN') auto
```
```   352   then show ?thesis
```
```   353     by auto
```
```   354 qed
```
```   355
```
```   356 lemma borel_comp[measurable]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
```
```   357   unfolding Compl_eq_Diff_UNIV by simp
```
```   358
```
```   359 lemma borel_measurable_vimage:
```
```   360   fixes f :: "'a \<Rightarrow> 'x::t2_space"
```
```   361   assumes borel[measurable]: "f \<in> borel_measurable M"
```
```   362   shows "f -` {x} \<inter> space M \<in> sets M"
```
```   363   by simp
```
```   364
```
```   365 lemma borel_measurableI:
```
```   366   fixes f :: "'a \<Rightarrow> 'x::topological_space"
```
```   367   assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
```
```   368   shows "f \<in> borel_measurable M"
```
```   369   unfolding borel_def
```
```   370 proof (rule measurable_measure_of, simp_all)
```
```   371   fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"
```
```   372     using assms[of S] by simp
```
```   373 qed
```
```   374
```
```   375 lemma borel_measurable_const:
```
```   376   "(\<lambda>x. c) \<in> borel_measurable M"
```
```   377   by auto
```
```   378
```
```   379 lemma borel_measurable_indicator:
```
```   380   assumes A: "A \<in> sets M"
```
```   381   shows "indicator A \<in> borel_measurable M"
```
```   382   unfolding indicator_def [abs_def] using A
```
```   383   by (auto intro!: measurable_If_set)
```
```   384
```
```   385 lemma borel_measurable_count_space[measurable (raw)]:
```
```   386   "f \<in> borel_measurable (count_space S)"
```
```   387   unfolding measurable_def by auto
```
```   388
```
```   389 lemma borel_measurable_indicator'[measurable (raw)]:
```
```   390   assumes [measurable]: "{x\<in>space M. f x \<in> A x} \<in> sets M"
```
```   391   shows "(\<lambda>x. indicator (A x) (f x)) \<in> borel_measurable M"
```
```   392   unfolding indicator_def[abs_def]
```
```   393   by (auto intro!: measurable_If)
```
```   394
```
```   395 lemma borel_measurable_indicator_iff:
```
```   396   "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
```
```   397     (is "?I \<in> borel_measurable M \<longleftrightarrow> _")
```
```   398 proof
```
```   399   assume "?I \<in> borel_measurable M"
```
```   400   then have "?I -` {1} \<inter> space M \<in> sets M"
```
```   401     unfolding measurable_def by auto
```
```   402   also have "?I -` {1} \<inter> space M = A \<inter> space M"
```
```   403     unfolding indicator_def [abs_def] by auto
```
```   404   finally show "A \<inter> space M \<in> sets M" .
```
```   405 next
```
```   406   assume "A \<inter> space M \<in> sets M"
```
```   407   moreover have "?I \<in> borel_measurable M \<longleftrightarrow>
```
```   408     (indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"
```
```   409     by (intro measurable_cong) (auto simp: indicator_def)
```
```   410   ultimately show "?I \<in> borel_measurable M" by auto
```
```   411 qed
```
```   412
```
```   413 lemma borel_measurable_subalgebra:
```
```   414   assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
```
```   415   shows "f \<in> borel_measurable M"
```
```   416   using assms unfolding measurable_def by auto
```
```   417
```
```   418 lemma borel_measurable_restrict_space_iff_ereal:
```
```   419   fixes f :: "'a \<Rightarrow> ereal"
```
```   420   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
```
```   421   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
```
```   422     (\<lambda>x. f x * indicator \<Omega> x) \<in> borel_measurable M"
```
```   423   by (subst measurable_restrict_space_iff)
```
```   424      (auto simp: indicator_def if_distrib[where f="\<lambda>x. a * x" for a] cong del: if_weak_cong)
```
```   425
```
```   426 lemma borel_measurable_restrict_space_iff_ennreal:
```
```   427   fixes f :: "'a \<Rightarrow> ennreal"
```
```   428   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
```
```   429   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
```
```   430     (\<lambda>x. f x * indicator \<Omega> x) \<in> borel_measurable M"
```
```   431   by (subst measurable_restrict_space_iff)
```
```   432      (auto simp: indicator_def if_distrib[where f="\<lambda>x. a * x" for a] cong del: if_weak_cong)
```
```   433
```
```   434 lemma borel_measurable_restrict_space_iff:
```
```   435   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   436   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
```
```   437   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
```
```   438     (\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> borel_measurable M"
```
```   439   by (subst measurable_restrict_space_iff)
```
```   440      (auto simp: indicator_def if_distrib[where f="\<lambda>x. x *\<^sub>R a" for a] ac_simps
```
```   441        cong del: if_weak_cong)
```
```   442
```
```   443 lemma cbox_borel[measurable]: "cbox a b \<in> sets borel"
```
```   444   by (auto intro: borel_closed)
```
```   445
```
```   446 lemma box_borel[measurable]: "box a b \<in> sets borel"
```
```   447   by (auto intro: borel_open)
```
```   448
```
```   449 lemma borel_compact: "compact (A::'a::t2_space set) \<Longrightarrow> A \<in> sets borel"
```
```   450   by (auto intro: borel_closed dest!: compact_imp_closed)
```
```   451
```
```   452 lemma borel_sigma_sets_subset:
```
```   453   "A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel"
```
```   454   using sets.sigma_sets_subset[of A borel] by simp
```
```   455
```
```   456 lemma borel_eq_sigmaI1:
```
```   457   fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
```
```   458   assumes borel_eq: "borel = sigma UNIV X"
```
```   459   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (F ` A))"
```
```   460   assumes F: "\<And>i. i \<in> A \<Longrightarrow> F i \<in> sets borel"
```
```   461   shows "borel = sigma UNIV (F ` A)"
```
```   462   unfolding borel_def
```
```   463 proof (intro sigma_eqI antisym)
```
```   464   have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel"
```
```   465     unfolding borel_def by simp
```
```   466   also have "\<dots> = sigma_sets UNIV X"
```
```   467     unfolding borel_eq by simp
```
```   468   also have "\<dots> \<subseteq> sigma_sets UNIV (F`A)"
```
```   469     using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto
```
```   470   finally show "sigma_sets UNIV {S. open S} \<subseteq> sigma_sets UNIV (F`A)" .
```
```   471   show "sigma_sets UNIV (F`A) \<subseteq> sigma_sets UNIV {S. open S}"
```
```   472     unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto
```
```   473 qed auto
```
```   474
```
```   475 lemma borel_eq_sigmaI2:
```
```   476   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set"
```
```   477     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
```
```   478   assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`B)"
```
```   479   assumes X: "\<And>i j. (i, j) \<in> B \<Longrightarrow> G i j \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"
```
```   480   assumes F: "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel"
```
```   481   shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"
```
```   482   using assms
```
```   483   by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` B" and F="(\<lambda>(i, j). F i j)"]) auto
```
```   484
```
```   485 lemma borel_eq_sigmaI3:
```
```   486   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
```
```   487   assumes borel_eq: "borel = sigma UNIV X"
```
```   488   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"
```
```   489   assumes F: "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel"
```
```   490   shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"
```
```   491   using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto
```
```   492
```
```   493 lemma borel_eq_sigmaI4:
```
```   494   fixes F :: "'i \<Rightarrow> 'a::topological_space set"
```
```   495     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
```
```   496   assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`A)"
```
```   497   assumes X: "\<And>i j. (i, j) \<in> A \<Longrightarrow> G i j \<in> sets (sigma UNIV (range F))"
```
```   498   assumes F: "\<And>i. F i \<in> sets borel"
```
```   499   shows "borel = sigma UNIV (range F)"
```
```   500   using assms by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` A" and F=F]) auto
```
```   501
```
```   502 lemma borel_eq_sigmaI5:
```
```   503   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set"
```
```   504   assumes borel_eq: "borel = sigma UNIV (range G)"
```
```   505   assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
```
```   506   assumes F: "\<And>i j. F i j \<in> sets borel"
```
```   507   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
```
```   508   using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto
```
```   509
```
```   510 lemma second_countable_borel_measurable:
```
```   511   fixes X :: "'a::second_countable_topology set set"
```
```   512   assumes eq: "open = generate_topology X"
```
```   513   shows "borel = sigma UNIV X"
```
```   514   unfolding borel_def
```
```   515 proof (intro sigma_eqI sigma_sets_eqI)
```
```   516   interpret X: sigma_algebra UNIV "sigma_sets UNIV X"
```
```   517     by (rule sigma_algebra_sigma_sets) simp
```
```   518
```
```   519   fix S :: "'a set" assume "S \<in> Collect open"
```
```   520   then have "generate_topology X S"
```
```   521     by (auto simp: eq)
```
```   522   then show "S \<in> sigma_sets UNIV X"
```
```   523   proof induction
```
```   524     case (UN K)
```
```   525     then have K: "\<And>k. k \<in> K \<Longrightarrow> open k"
```
```   526       unfolding eq by auto
```
```   527     from ex_countable_basis obtain B :: "'a set set" where
```
```   528       B:  "\<And>b. b \<in> B \<Longrightarrow> open b" "\<And>X. open X \<Longrightarrow> \<exists>b\<subseteq>B. (\<Union>b) = X" and "countable B"
```
```   529       by (auto simp: topological_basis_def)
```
```   530     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"
```
```   531       by metis
```
```   532     define U where "U = (\<Union>k\<in>K. m k)"
```
```   533     with m have "countable U"
```
```   534       by (intro countable_subset[OF _ \<open>countable B\<close>]) auto
```
```   535     have "\<Union>U = (\<Union>A\<in>U. A)" by simp
```
```   536     also have "\<dots> = \<Union>K"
```
```   537       unfolding U_def UN_simps by (simp add: m)
```
```   538     finally have "\<Union>U = \<Union>K" .
```
```   539
```
```   540     have "\<forall>b\<in>U. \<exists>k\<in>K. b \<subseteq> k"
```
```   541       using m by (auto simp: U_def)
```
```   542     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"
```
```   543       by metis
```
```   544     then have "(\<Union>b\<in>U. u b) \<subseteq> \<Union>K" "\<Union>U \<subseteq> (\<Union>b\<in>U. u b)"
```
```   545       by auto
```
```   546     then have "\<Union>K = (\<Union>b\<in>U. u b)"
```
```   547       unfolding \<open>\<Union>U = \<Union>K\<close> by auto
```
```   548     also have "\<dots> \<in> sigma_sets UNIV X"
```
```   549       using u UN by (intro X.countable_UN' \<open>countable U\<close>) auto
```
```   550     finally show "\<Union>K \<in> sigma_sets UNIV X" .
```
```   551   qed auto
```
```   552 qed (auto simp: eq intro: generate_topology.Basis)
```
```   553
```
```   554 lemma borel_eq_closed: "borel = sigma UNIV (Collect closed)"
```
```   555   unfolding borel_def
```
```   556 proof (intro sigma_eqI sigma_sets_eqI, safe)
```
```   557   fix x :: "'a set" assume "open x"
```
```   558   hence "x = UNIV - (UNIV - x)" by auto
```
```   559   also have "\<dots> \<in> sigma_sets UNIV (Collect closed)"
```
```   560     by (force intro: sigma_sets.Compl simp: \<open>open x\<close>)
```
```   561   finally show "x \<in> sigma_sets UNIV (Collect closed)" by simp
```
```   562 next
```
```   563   fix x :: "'a set" assume "closed x"
```
```   564   hence "x = UNIV - (UNIV - x)" by auto
```
```   565   also have "\<dots> \<in> sigma_sets UNIV (Collect open)"
```
```   566     by (force intro: sigma_sets.Compl simp: \<open>closed x\<close>)
```
```   567   finally show "x \<in> sigma_sets UNIV (Collect open)" by simp
```
```   568 qed simp_all
```
```   569
```
```   570 lemma borel_eq_countable_basis:
```
```   571   fixes B::"'a::topological_space set set"
```
```   572   assumes "countable B"
```
```   573   assumes "topological_basis B"
```
```   574   shows "borel = sigma UNIV B"
```
```   575   unfolding borel_def
```
```   576 proof (intro sigma_eqI sigma_sets_eqI, safe)
```
```   577   interpret countable_basis using assms by unfold_locales
```
```   578   fix X::"'a set" assume "open X"
```
```   579   from open_countable_basisE[OF this] guess B' . note B' = this
```
```   580   then show "X \<in> sigma_sets UNIV B"
```
```   581     by (blast intro: sigma_sets_UNION \<open>countable B\<close> countable_subset)
```
```   582 next
```
```   583   fix b assume "b \<in> B"
```
```   584   hence "open b" by (rule topological_basis_open[OF assms(2)])
```
```   585   thus "b \<in> sigma_sets UNIV (Collect open)" by auto
```
```   586 qed simp_all
```
```   587
```
```   588 lemma borel_measurable_continuous_on_restrict:
```
```   589   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
```
```   590   assumes f: "continuous_on A f"
```
```   591   shows "f \<in> borel_measurable (restrict_space borel A)"
```
```   592 proof (rule borel_measurableI)
```
```   593   fix S :: "'b set" assume "open S"
```
```   594   with f obtain T where "f -` S \<inter> A = T \<inter> A" "open T"
```
```   595     by (metis continuous_on_open_invariant)
```
```   596   then show "f -` S \<inter> space (restrict_space borel A) \<in> sets (restrict_space borel A)"
```
```   597     by (force simp add: sets_restrict_space space_restrict_space)
```
```   598 qed
```
```   599
```
```   600 lemma borel_measurable_continuous_on1: "continuous_on UNIV f \<Longrightarrow> f \<in> borel_measurable borel"
```
```   601   by (drule borel_measurable_continuous_on_restrict) simp
```
```   602
```
```   603 lemma borel_measurable_continuous_on_if:
```
```   604   "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> continuous_on (- A) g \<Longrightarrow>
```
```   605     (\<lambda>x. if x \<in> A then f x else g x) \<in> borel_measurable borel"
```
```   606   by (auto simp add: measurable_If_restrict_space_iff Collect_neg_eq
```
```   607            intro!: borel_measurable_continuous_on_restrict)
```
```   608
```
```   609 lemma borel_measurable_continuous_countable_exceptions:
```
```   610   fixes f :: "'a::t1_space \<Rightarrow> 'b::topological_space"
```
```   611   assumes X: "countable X"
```
```   612   assumes "continuous_on (- X) f"
```
```   613   shows "f \<in> borel_measurable borel"
```
```   614 proof (rule measurable_discrete_difference[OF _ X])
```
```   615   have "X \<in> sets borel"
```
```   616     by (rule sets.countable[OF _ X]) auto
```
```   617   then show "(\<lambda>x. if x \<in> X then undefined else f x) \<in> borel_measurable borel"
```
```   618     by (intro borel_measurable_continuous_on_if assms continuous_intros)
```
```   619 qed auto
```
```   620
```
```   621 lemma borel_measurable_continuous_on:
```
```   622   assumes f: "continuous_on UNIV f" and g: "g \<in> borel_measurable M"
```
```   623   shows "(\<lambda>x. f (g x)) \<in> borel_measurable M"
```
```   624   using measurable_comp[OF g borel_measurable_continuous_on1[OF f]] by (simp add: comp_def)
```
```   625
```
```   626 lemma borel_measurable_continuous_on_indicator:
```
```   627   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   628   shows "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable borel"
```
```   629   by (subst borel_measurable_restrict_space_iff[symmetric])
```
```   630      (auto intro: borel_measurable_continuous_on_restrict)
```
```   631
```
```   632 lemma borel_measurable_Pair[measurable (raw)]:
```
```   633   fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"
```
```   634   assumes f[measurable]: "f \<in> borel_measurable M"
```
```   635   assumes g[measurable]: "g \<in> borel_measurable M"
```
```   636   shows "(\<lambda>x. (f x, g x)) \<in> borel_measurable M"
```
```   637 proof (subst borel_eq_countable_basis)
```
```   638   let ?B = "SOME B::'b set set. countable B \<and> topological_basis B"
```
```   639   let ?C = "SOME B::'c set set. countable B \<and> topological_basis B"
```
```   640   let ?P = "(\<lambda>(b, c). b \<times> c) ` (?B \<times> ?C)"
```
```   641   show "countable ?P" "topological_basis ?P"
```
```   642     by (auto intro!: countable_basis topological_basis_prod is_basis)
```
```   643
```
```   644   show "(\<lambda>x. (f x, g x)) \<in> measurable M (sigma UNIV ?P)"
```
```   645   proof (rule measurable_measure_of)
```
```   646     fix S assume "S \<in> ?P"
```
```   647     then obtain b c where "b \<in> ?B" "c \<in> ?C" and S: "S = b \<times> c" by auto
```
```   648     then have borel: "open b" "open c"
```
```   649       by (auto intro: is_basis topological_basis_open)
```
```   650     have "(\<lambda>x. (f x, g x)) -` S \<inter> space M = (f -` b \<inter> space M) \<inter> (g -` c \<inter> space M)"
```
```   651       unfolding S by auto
```
```   652     also have "\<dots> \<in> sets M"
```
```   653       using borel by simp
```
```   654     finally show "(\<lambda>x. (f x, g x)) -` S \<inter> space M \<in> sets M" .
```
```   655   qed auto
```
```   656 qed
```
```   657
```
```   658 lemma borel_measurable_continuous_Pair:
```
```   659   fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"
```
```   660   assumes [measurable]: "f \<in> borel_measurable M"
```
```   661   assumes [measurable]: "g \<in> borel_measurable M"
```
```   662   assumes H: "continuous_on UNIV (\<lambda>x. H (fst x) (snd x))"
```
```   663   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
```
```   664 proof -
```
```   665   have eq: "(\<lambda>x. H (f x) (g x)) = (\<lambda>x. (\<lambda>x. H (fst x) (snd x)) (f x, g x))" by auto
```
```   666   show ?thesis
```
```   667     unfolding eq by (rule borel_measurable_continuous_on[OF H]) auto
```
```   668 qed
```
```   669
```
```   670 subsection \<open>Borel spaces on order topologies\<close>
```
```   671
```
```   672 lemma [measurable]:
```
```   673   fixes a b :: "'a::linorder_topology"
```
```   674   shows lessThan_borel: "{..< a} \<in> sets borel"
```
```   675     and greaterThan_borel: "{a <..} \<in> sets borel"
```
```   676     and greaterThanLessThan_borel: "{a<..<b} \<in> sets borel"
```
```   677     and atMost_borel: "{..a} \<in> sets borel"
```
```   678     and atLeast_borel: "{a..} \<in> sets borel"
```
```   679     and atLeastAtMost_borel: "{a..b} \<in> sets borel"
```
```   680     and greaterThanAtMost_borel: "{a<..b} \<in> sets borel"
```
```   681     and atLeastLessThan_borel: "{a..<b} \<in> sets borel"
```
```   682   unfolding greaterThanAtMost_def atLeastLessThan_def
```
```   683   by (blast intro: borel_open borel_closed open_lessThan open_greaterThan open_greaterThanLessThan
```
```   684                    closed_atMost closed_atLeast closed_atLeastAtMost)+
```
```   685
```
```   686 lemma borel_Iio:
```
```   687   "borel = sigma UNIV (range lessThan :: 'a::{linorder_topology, second_countable_topology} set set)"
```
```   688   unfolding second_countable_borel_measurable[OF open_generated_order]
```
```   689 proof (intro sigma_eqI sigma_sets_eqI)
```
```   690   from countable_dense_setE guess D :: "'a set" . note D = this
```
```   691
```
```   692   interpret L: sigma_algebra UNIV "sigma_sets UNIV (range lessThan)"
```
```   693     by (rule sigma_algebra_sigma_sets) simp
```
```   694
```
```   695   fix A :: "'a set" assume "A \<in> range lessThan \<union> range greaterThan"
```
```   696   then obtain y where "A = {y <..} \<or> A = {..< y}"
```
```   697     by blast
```
```   698   then show "A \<in> sigma_sets UNIV (range lessThan)"
```
```   699   proof
```
```   700     assume A: "A = {y <..}"
```
```   701     show ?thesis
```
```   702     proof cases
```
```   703       assume "\<forall>x>y. \<exists>d. y < d \<and> d < x"
```
```   704       with D(2)[of "{y <..< x}" for x] have "\<forall>x>y. \<exists>d\<in>D. y < d \<and> d < x"
```
```   705         by (auto simp: set_eq_iff)
```
```   706       then have "A = UNIV - (\<Inter>d\<in>{d\<in>D. y < d}. {..< d})"
```
```   707         by (auto simp: A) (metis less_asym)
```
```   708       also have "\<dots> \<in> sigma_sets UNIV (range lessThan)"
```
```   709         using D(1) by (intro L.Diff L.top L.countable_INT'') auto
```
```   710       finally show ?thesis .
```
```   711     next
```
```   712       assume "\<not> (\<forall>x>y. \<exists>d. y < d \<and> d < x)"
```
```   713       then obtain x where "y < x"  "\<And>d. y < d \<Longrightarrow> \<not> d < x"
```
```   714         by auto
```
```   715       then have "A = UNIV - {..< x}"
```
```   716         unfolding A by (auto simp: not_less[symmetric])
```
```   717       also have "\<dots> \<in> sigma_sets UNIV (range lessThan)"
```
```   718         by auto
```
```   719       finally show ?thesis .
```
```   720     qed
```
```   721   qed auto
```
```   722 qed auto
```
```   723
```
```   724 lemma borel_Ioi:
```
```   725   "borel = sigma UNIV (range greaterThan :: 'a::{linorder_topology, second_countable_topology} set set)"
```
```   726   unfolding second_countable_borel_measurable[OF open_generated_order]
```
```   727 proof (intro sigma_eqI sigma_sets_eqI)
```
```   728   from countable_dense_setE guess D :: "'a set" . note D = this
```
```   729
```
```   730   interpret L: sigma_algebra UNIV "sigma_sets UNIV (range greaterThan)"
```
```   731     by (rule sigma_algebra_sigma_sets) simp
```
```   732
```
```   733   fix A :: "'a set" assume "A \<in> range lessThan \<union> range greaterThan"
```
```   734   then obtain y where "A = {y <..} \<or> A = {..< y}"
```
```   735     by blast
```
```   736   then show "A \<in> sigma_sets UNIV (range greaterThan)"
```
```   737   proof
```
```   738     assume A: "A = {..< y}"
```
```   739     show ?thesis
```
```   740     proof cases
```
```   741       assume "\<forall>x<y. \<exists>d. x < d \<and> d < y"
```
```   742       with D(2)[of "{x <..< y}" for x] have "\<forall>x<y. \<exists>d\<in>D. x < d \<and> d < y"
```
```   743         by (auto simp: set_eq_iff)
```
```   744       then have "A = UNIV - (\<Inter>d\<in>{d\<in>D. d < y}. {d <..})"
```
```   745         by (auto simp: A) (metis less_asym)
```
```   746       also have "\<dots> \<in> sigma_sets UNIV (range greaterThan)"
```
```   747         using D(1) by (intro L.Diff L.top L.countable_INT'') auto
```
```   748       finally show ?thesis .
```
```   749     next
```
```   750       assume "\<not> (\<forall>x<y. \<exists>d. x < d \<and> d < y)"
```
```   751       then obtain x where "x < y"  "\<And>d. y > d \<Longrightarrow> x \<ge> d"
```
```   752         by (auto simp: not_less[symmetric])
```
```   753       then have "A = UNIV - {x <..}"
```
```   754         unfolding A Compl_eq_Diff_UNIV[symmetric] by auto
```
```   755       also have "\<dots> \<in> sigma_sets UNIV (range greaterThan)"
```
```   756         by auto
```
```   757       finally show ?thesis .
```
```   758     qed
```
```   759   qed auto
```
```   760 qed auto
```
```   761
```
```   762 lemma borel_measurableI_less:
```
```   763   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
```
```   764   shows "(\<And>y. {x\<in>space M. f x < y} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
```
```   765   unfolding borel_Iio
```
```   766   by (rule measurable_measure_of) (auto simp: Int_def conj_commute)
```
```   767
```
```   768 lemma borel_measurableI_greater:
```
```   769   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
```
```   770   shows "(\<And>y. {x\<in>space M. y < f x} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
```
```   771   unfolding borel_Ioi
```
```   772   by (rule measurable_measure_of) (auto simp: Int_def conj_commute)
```
```   773
```
```   774 lemma borel_measurableI_le:
```
```   775   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
```
```   776   shows "(\<And>y. {x\<in>space M. f x \<le> y} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
```
```   777   by (rule borel_measurableI_greater) (auto simp: not_le[symmetric])
```
```   778
```
```   779 lemma borel_measurableI_ge:
```
```   780   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
```
```   781   shows "(\<And>y. {x\<in>space M. y \<le> f x} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
```
```   782   by (rule borel_measurableI_less) (auto simp: not_le[symmetric])
```
```   783
```
```   784 lemma borel_measurable_less[measurable]:
```
```   785   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
```
```   786   assumes "f \<in> borel_measurable M"
```
```   787   assumes "g \<in> borel_measurable M"
```
```   788   shows "{w \<in> space M. f w < g w} \<in> sets M"
```
```   789 proof -
```
```   790   have "{w \<in> space M. f w < g w} = (\<lambda>x. (f x, g x)) -` {x. fst x < snd x} \<inter> space M"
```
```   791     by auto
```
```   792   also have "\<dots> \<in> sets M"
```
```   793     by (intro measurable_sets[OF borel_measurable_Pair borel_open, OF assms open_Collect_less]
```
```   794               continuous_intros)
```
```   795   finally show ?thesis .
```
```   796 qed
```
```   797
```
```   798 lemma
```
```   799   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
```
```   800   assumes f[measurable]: "f \<in> borel_measurable M"
```
```   801   assumes g[measurable]: "g \<in> borel_measurable M"
```
```   802   shows borel_measurable_le[measurable]: "{w \<in> space M. f w \<le> g w} \<in> sets M"
```
```   803     and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M"
```
```   804     and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
```
```   805   unfolding eq_iff not_less[symmetric]
```
```   806   by measurable
```
```   807
```
```   808 lemma borel_measurable_SUP[measurable (raw)]:
```
```   809   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}"
```
```   810   assumes [simp]: "countable I"
```
```   811   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```   812   shows "(\<lambda>x. SUP i:I. F i x) \<in> borel_measurable M"
```
```   813   by (rule borel_measurableI_greater) (simp add: less_SUP_iff)
```
```   814
```
```   815 lemma borel_measurable_INF[measurable (raw)]:
```
```   816   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}"
```
```   817   assumes [simp]: "countable I"
```
```   818   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```   819   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
```
```   820   by (rule borel_measurableI_less) (simp add: INF_less_iff)
```
```   821
```
```   822 lemma borel_measurable_cSUP[measurable (raw)]:
```
```   823   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology, second_countable_topology}"
```
```   824   assumes [simp]: "countable I"
```
```   825   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```   826   assumes bdd: "\<And>x. x \<in> space M \<Longrightarrow> bdd_above ((\<lambda>i. F i x) ` I)"
```
```   827   shows "(\<lambda>x. SUP i:I. F i x) \<in> borel_measurable M"
```
```   828 proof cases
```
```   829   assume "I = {}" then show ?thesis
```
```   830     unfolding \<open>I = {}\<close> image_empty by simp
```
```   831 next
```
```   832   assume "I \<noteq> {}"
```
```   833   show ?thesis
```
```   834   proof (rule borel_measurableI_le)
```
```   835     fix y
```
```   836     have "{x \<in> space M. \<forall>i\<in>I. F i x \<le> y} \<in> sets M"
```
```   837       by measurable
```
```   838     also have "{x \<in> space M. \<forall>i\<in>I. F i x \<le> y} = {x \<in> space M. (SUP i:I. F i x) \<le> y}"
```
```   839       by (simp add: cSUP_le_iff \<open>I \<noteq> {}\<close> bdd cong: conj_cong)
```
```   840     finally show "{x \<in> space M. (SUP i:I. F i x) \<le>  y} \<in> sets M"  .
```
```   841   qed
```
```   842 qed
```
```   843
```
```   844 lemma borel_measurable_cINF[measurable (raw)]:
```
```   845   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology, second_countable_topology}"
```
```   846   assumes [simp]: "countable I"
```
```   847   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```   848   assumes bdd: "\<And>x. x \<in> space M \<Longrightarrow> bdd_below ((\<lambda>i. F i x) ` I)"
```
```   849   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
```
```   850 proof cases
```
```   851   assume "I = {}" then show ?thesis
```
```   852     unfolding \<open>I = {}\<close> image_empty by simp
```
```   853 next
```
```   854   assume "I \<noteq> {}"
```
```   855   show ?thesis
```
```   856   proof (rule borel_measurableI_ge)
```
```   857     fix y
```
```   858     have "{x \<in> space M. \<forall>i\<in>I. y \<le> F i x} \<in> sets M"
```
```   859       by measurable
```
```   860     also have "{x \<in> space M. \<forall>i\<in>I. y \<le> F i x} = {x \<in> space M. y \<le> (INF i:I. F i x)}"
```
```   861       by (simp add: le_cINF_iff \<open>I \<noteq> {}\<close> bdd cong: conj_cong)
```
```   862     finally show "{x \<in> space M. y \<le> (INF i:I. F i x)} \<in> sets M"  .
```
```   863   qed
```
```   864 qed
```
```   865
```
```   866 lemma borel_measurable_lfp[consumes 1, case_names continuity step]:
```
```   867   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})"
```
```   868   assumes "sup_continuous F"
```
```   869   assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M"
```
```   870   shows "lfp F \<in> borel_measurable M"
```
```   871 proof -
```
```   872   { fix i have "((F ^^ i) bot) \<in> borel_measurable M"
```
```   873       by (induct i) (auto intro!: *) }
```
```   874   then have "(\<lambda>x. SUP i. (F ^^ i) bot x) \<in> borel_measurable M"
```
```   875     by measurable
```
```   876   also have "(\<lambda>x. SUP i. (F ^^ i) bot x) = (SUP i. (F ^^ i) bot)"
```
```   877     by auto
```
```   878   also have "(SUP i. (F ^^ i) bot) = lfp F"
```
```   879     by (rule sup_continuous_lfp[symmetric]) fact
```
```   880   finally show ?thesis .
```
```   881 qed
```
```   882
```
```   883 lemma borel_measurable_gfp[consumes 1, case_names continuity step]:
```
```   884   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})"
```
```   885   assumes "inf_continuous F"
```
```   886   assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M"
```
```   887   shows "gfp F \<in> borel_measurable M"
```
```   888 proof -
```
```   889   { fix i have "((F ^^ i) top) \<in> borel_measurable M"
```
```   890       by (induct i) (auto intro!: * simp: bot_fun_def) }
```
```   891   then have "(\<lambda>x. INF i. (F ^^ i) top x) \<in> borel_measurable M"
```
```   892     by measurable
```
```   893   also have "(\<lambda>x. INF i. (F ^^ i) top x) = (INF i. (F ^^ i) top)"
```
```   894     by auto
```
```   895   also have "\<dots> = gfp F"
```
```   896     by (rule inf_continuous_gfp[symmetric]) fact
```
```   897   finally show ?thesis .
```
```   898 qed
```
```   899
```
```   900 lemma borel_measurable_max[measurable (raw)]:
```
```   901   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. max (g x) (f x) :: 'b::{second_countable_topology, linorder_topology}) \<in> borel_measurable M"
```
```   902   by (rule borel_measurableI_less) simp
```
```   903
```
```   904 lemma borel_measurable_min[measurable (raw)]:
```
```   905   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. min (g x) (f x) :: 'b::{second_countable_topology, linorder_topology}) \<in> borel_measurable M"
```
```   906   by (rule borel_measurableI_greater) simp
```
```   907
```
```   908 lemma borel_measurable_Min[measurable (raw)]:
```
```   909   "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, linorder_topology}) \<in> borel_measurable M"
```
```   910 proof (induct I rule: finite_induct)
```
```   911   case (insert i I) then show ?case
```
```   912     by (cases "I = {}") auto
```
```   913 qed auto
```
```   914
```
```   915 lemma borel_measurable_Max[measurable (raw)]:
```
```   916   "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, linorder_topology}) \<in> borel_measurable M"
```
```   917 proof (induct I rule: finite_induct)
```
```   918   case (insert i I) then show ?case
```
```   919     by (cases "I = {}") auto
```
```   920 qed auto
```
```   921
```
```   922 lemma borel_measurable_sup[measurable (raw)]:
```
```   923   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. sup (g x) (f x) :: 'b::{lattice, second_countable_topology, linorder_topology}) \<in> borel_measurable M"
```
```   924   unfolding sup_max by measurable
```
```   925
```
```   926 lemma borel_measurable_inf[measurable (raw)]:
```
```   927   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. inf (g x) (f x) :: 'b::{lattice, second_countable_topology, linorder_topology}) \<in> borel_measurable M"
```
```   928   unfolding inf_min by measurable
```
```   929
```
```   930 lemma [measurable (raw)]:
```
```   931   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   932   assumes "\<And>i. f i \<in> borel_measurable M"
```
```   933   shows borel_measurable_liminf: "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```   934     and borel_measurable_limsup: "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```   935   unfolding liminf_SUP_INF limsup_INF_SUP using assms by auto
```
```   936
```
```   937 lemma measurable_convergent[measurable (raw)]:
```
```   938   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   939   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```   940   shows "Measurable.pred M (\<lambda>x. convergent (\<lambda>i. f i x))"
```
```   941   unfolding convergent_ereal by measurable
```
```   942
```
```   943 lemma sets_Collect_convergent[measurable]:
```
```   944   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   945   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```   946   shows "{x\<in>space M. convergent (\<lambda>i. f i x)} \<in> sets M"
```
```   947   by measurable
```
```   948
```
```   949 lemma borel_measurable_lim[measurable (raw)]:
```
```   950   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   951   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```   952   shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```   953 proof -
```
```   954   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))"
```
```   955     by (simp add: lim_def convergent_def convergent_limsup_cl)
```
```   956   then show ?thesis
```
```   957     by simp
```
```   958 qed
```
```   959
```
```   960 lemma borel_measurable_LIMSEQ_order:
```
```   961   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   962   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
```
```   963   and u: "\<And>i. u i \<in> borel_measurable M"
```
```   964   shows "u' \<in> borel_measurable M"
```
```   965 proof -
```
```   966   have "\<And>x. x \<in> space M \<Longrightarrow> u' x = liminf (\<lambda>n. u n x)"
```
```   967     using u' by (simp add: lim_imp_Liminf[symmetric])
```
```   968   with u show ?thesis by (simp cong: measurable_cong)
```
```   969 qed
```
```   970
```
```   971 subsection \<open>Borel spaces on topological monoids\<close>
```
```   972
```
```   973 lemma borel_measurable_add[measurable (raw)]:
```
```   974   fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, topological_monoid_add}"
```
```   975   assumes f: "f \<in> borel_measurable M"
```
```   976   assumes g: "g \<in> borel_measurable M"
```
```   977   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```   978   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```   979
```
```   980 lemma borel_measurable_sum[measurable (raw)]:
```
```   981   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, topological_comm_monoid_add}"
```
```   982   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   983   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```   984 proof cases
```
```   985   assume "finite S"
```
```   986   thus ?thesis using assms by induct auto
```
```   987 qed simp
```
```   988
```
```   989 lemma borel_measurable_suminf_order[measurable (raw)]:
```
```   990   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology, topological_comm_monoid_add}"
```
```   991   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```   992   shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```   993   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
```
```   994
```
```   995 subsection \<open>Borel spaces on Euclidean spaces\<close>
```
```   996
```
```   997 lemma borel_measurable_inner[measurable (raw)]:
```
```   998   fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_inner}"
```
```   999   assumes "f \<in> borel_measurable M"
```
```  1000   assumes "g \<in> borel_measurable M"
```
```  1001   shows "(\<lambda>x. f x \<bullet> g x) \<in> borel_measurable M"
```
```  1002   using assms
```
```  1003   by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```  1004
```
```  1005 notation
```
```  1006   eucl_less (infix "<e" 50)
```
```  1007
```
```  1008 lemma box_oc: "{x. a <e x \<and> x \<le> b} = {x. a <e x} \<inter> {..b}"
```
```  1009   and box_co: "{x. a \<le> x \<and> x <e b} = {a..} \<inter> {x. x <e b}"
```
```  1010   by auto
```
```  1011
```
```  1012 lemma eucl_ivals[measurable]:
```
```  1013   fixes a b :: "'a::ordered_euclidean_space"
```
```  1014   shows "{x. x <e a} \<in> sets borel"
```
```  1015     and "{x. a <e x} \<in> sets borel"
```
```  1016     and "{..a} \<in> sets borel"
```
```  1017     and "{a..} \<in> sets borel"
```
```  1018     and "{a..b} \<in> sets borel"
```
```  1019     and  "{x. a <e x \<and> x \<le> b} \<in> sets borel"
```
```  1020     and "{x. a \<le> x \<and>  x <e b} \<in> sets borel"
```
```  1021   unfolding box_oc box_co
```
```  1022   by (auto intro: borel_open borel_closed)
```
```  1023
```
```  1024 lemma
```
```  1025   fixes i :: "'a::{second_countable_topology, real_inner}"
```
```  1026   shows hafspace_less_borel: "{x. a < x \<bullet> i} \<in> sets borel"
```
```  1027     and hafspace_greater_borel: "{x. x \<bullet> i < a} \<in> sets borel"
```
```  1028     and hafspace_less_eq_borel: "{x. a \<le> x \<bullet> i} \<in> sets borel"
```
```  1029     and hafspace_greater_eq_borel: "{x. x \<bullet> i \<le> a} \<in> sets borel"
```
```  1030   by simp_all
```
```  1031
```
```  1032 lemma borel_eq_box:
```
```  1033   "borel = sigma UNIV (range (\<lambda> (a, b). box a b :: 'a :: euclidean_space set))"
```
```  1034     (is "_ = ?SIGMA")
```
```  1035 proof (rule borel_eq_sigmaI1[OF borel_def])
```
```  1036   fix M :: "'a set" assume "M \<in> {S. open S}"
```
```  1037   then have "open M" by simp
```
```  1038   show "M \<in> ?SIGMA"
```
```  1039     apply (subst open_UNION_box[OF \<open>open M\<close>])
```
```  1040     apply (safe intro!: sets.countable_UN' countable_PiE countable_Collect)
```
```  1041     apply (auto intro: countable_rat)
```
```  1042     done
```
```  1043 qed (auto simp: box_def)
```
```  1044
```
```  1045 lemma halfspace_gt_in_halfspace:
```
```  1046   assumes i: "i \<in> A"
```
```  1047   shows "{x::'a. a < x \<bullet> i} \<in>
```
```  1048     sigma_sets UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> A))"
```
```  1049   (is "?set \<in> ?SIGMA")
```
```  1050 proof -
```
```  1051   interpret sigma_algebra UNIV ?SIGMA
```
```  1052     by (intro sigma_algebra_sigma_sets) simp_all
```
```  1053   have *: "?set = (\<Union>n. UNIV - {x::'a. x \<bullet> i < a + 1 / real (Suc n)})"
```
```  1054   proof (safe, simp_all add: not_less del: of_nat_Suc)
```
```  1055     fix x :: 'a assume "a < x \<bullet> i"
```
```  1056     with reals_Archimedean[of "x \<bullet> i - a"]
```
```  1057     obtain n where "a + 1 / real (Suc n) < x \<bullet> i"
```
```  1058       by (auto simp: field_simps)
```
```  1059     then show "\<exists>n. a + 1 / real (Suc n) \<le> x \<bullet> i"
```
```  1060       by (blast intro: less_imp_le)
```
```  1061   next
```
```  1062     fix x n
```
```  1063     have "a < a + 1 / real (Suc n)" by auto
```
```  1064     also assume "\<dots> \<le> x"
```
```  1065     finally show "a < x" .
```
```  1066   qed
```
```  1067   show "?set \<in> ?SIGMA" unfolding *
```
```  1068     by (auto intro!: Diff sigma_sets_Inter i)
```
```  1069 qed
```
```  1070
```
```  1071 lemma borel_eq_halfspace_less:
```
```  1072   "borel = sigma UNIV ((\<lambda>(a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> Basis))"
```
```  1073   (is "_ = ?SIGMA")
```
```  1074 proof (rule borel_eq_sigmaI2[OF borel_eq_box])
```
```  1075   fix a b :: 'a
```
```  1076   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}"
```
```  1077     by (auto simp: box_def)
```
```  1078   also have "\<dots> \<in> sets ?SIGMA"
```
```  1079     by (intro sets.sets_Collect_conj sets.sets_Collect_finite_All sets.sets_Collect_const)
```
```  1080        (auto intro!: halfspace_gt_in_halfspace countable_PiE countable_rat)
```
```  1081   finally show "box a b \<in> sets ?SIGMA" .
```
```  1082 qed auto
```
```  1083
```
```  1084 lemma borel_eq_halfspace_le:
```
```  1085   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i \<le> a}) ` (UNIV \<times> Basis))"
```
```  1086   (is "_ = ?SIGMA")
```
```  1087 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
```
```  1088   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```  1089   then have i: "i \<in> Basis" by auto
```
```  1090   have *: "{x::'a. x\<bullet>i < a} = (\<Union>n. {x. x\<bullet>i \<le> a - 1/real (Suc n)})"
```
```  1091   proof (safe, simp_all del: of_nat_Suc)
```
```  1092     fix x::'a assume *: "x\<bullet>i < a"
```
```  1093     with reals_Archimedean[of "a - x\<bullet>i"]
```
```  1094     obtain n where "x \<bullet> i < a - 1 / (real (Suc n))"
```
```  1095       by (auto simp: field_simps)
```
```  1096     then show "\<exists>n. x \<bullet> i \<le> a - 1 / (real (Suc n))"
```
```  1097       by (blast intro: less_imp_le)
```
```  1098   next
```
```  1099     fix x::'a and n
```
```  1100     assume "x\<bullet>i \<le> a - 1 / real (Suc n)"
```
```  1101     also have "\<dots> < a" by auto
```
```  1102     finally show "x\<bullet>i < a" .
```
```  1103   qed
```
```  1104   show "{x. x\<bullet>i < a} \<in> ?SIGMA" unfolding *
```
```  1105     by (intro sets.countable_UN) (auto intro: i)
```
```  1106 qed auto
```
```  1107
```
```  1108 lemma borel_eq_halfspace_ge:
```
```  1109   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. a \<le> x \<bullet> i}) ` (UNIV \<times> Basis))"
```
```  1110   (is "_ = ?SIGMA")
```
```  1111 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
```
```  1112   fix a :: real and i :: 'a assume i: "(a, i) \<in> UNIV \<times> Basis"
```
```  1113   have *: "{x::'a. x\<bullet>i < a} = space ?SIGMA - {x::'a. a \<le> x\<bullet>i}" by auto
```
```  1114   show "{x. x\<bullet>i < a} \<in> ?SIGMA" unfolding *
```
```  1115     using i by (intro sets.compl_sets) auto
```
```  1116 qed auto
```
```  1117
```
```  1118 lemma borel_eq_halfspace_greater:
```
```  1119   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. a < x \<bullet> i}) ` (UNIV \<times> Basis))"
```
```  1120   (is "_ = ?SIGMA")
```
```  1121 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])
```
```  1122   fix a :: real and i :: 'a assume "(a, i) \<in> (UNIV \<times> Basis)"
```
```  1123   then have i: "i \<in> Basis" by auto
```
```  1124   have *: "{x::'a. x\<bullet>i \<le> a} = space ?SIGMA - {x::'a. a < x\<bullet>i}" by auto
```
```  1125   show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA" unfolding *
```
```  1126     by (intro sets.compl_sets) (auto intro: i)
```
```  1127 qed auto
```
```  1128
```
```  1129 lemma borel_eq_atMost:
```
```  1130   "borel = sigma UNIV (range (\<lambda>a. {..a::'a::ordered_euclidean_space}))"
```
```  1131   (is "_ = ?SIGMA")
```
```  1132 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
```
```  1133   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```  1134   then have "i \<in> Basis" by auto
```
```  1135   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)})"
```
```  1136   proof (safe, simp_all add: eucl_le[where 'a='a] split: if_split_asm)
```
```  1137     fix x :: 'a
```
```  1138     from real_arch_simple[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"] guess k::nat ..
```
```  1139     then have "\<And>i. i \<in> Basis \<Longrightarrow> x\<bullet>i \<le> real k"
```
```  1140       by (subst (asm) Max_le_iff) auto
```
```  1141     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> x \<bullet> ia \<le> real k"
```
```  1142       by (auto intro!: exI[of _ k])
```
```  1143   qed
```
```  1144   show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA" unfolding *
```
```  1145     by (intro sets.countable_UN) auto
```
```  1146 qed auto
```
```  1147
```
```  1148 lemma borel_eq_greaterThan:
```
```  1149   "borel = sigma UNIV (range (\<lambda>a::'a::ordered_euclidean_space. {x. a <e x}))"
```
```  1150   (is "_ = ?SIGMA")
```
```  1151 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
```
```  1152   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```  1153   then have i: "i \<in> Basis" by auto
```
```  1154   have "{x::'a. x\<bullet>i \<le> a} = UNIV - {x::'a. a < x\<bullet>i}" by auto
```
```  1155   also have *: "{x::'a. a < x\<bullet>i} =
```
```  1156       (\<Union>k::nat. {x. (\<Sum>n\<in>Basis. (if n = i then a else -real k) *\<^sub>R n) <e x})" using i
```
```  1157   proof (safe, simp_all add: eucl_less_def split: if_split_asm)
```
```  1158     fix x :: 'a
```
```  1159     from reals_Archimedean2[of "Max ((\<lambda>i. -x\<bullet>i)`Basis)"]
```
```  1160     guess k::nat .. note k = this
```
```  1161     { fix i :: 'a assume "i \<in> Basis"
```
```  1162       then have "-x\<bullet>i < real k"
```
```  1163         using k by (subst (asm) Max_less_iff) auto
```
```  1164       then have "- real k < x\<bullet>i" by simp }
```
```  1165     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> -real k < x \<bullet> ia"
```
```  1166       by (auto intro!: exI[of _ k])
```
```  1167   qed
```
```  1168   finally show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA"
```
```  1169     apply (simp only:)
```
```  1170     apply (intro sets.countable_UN sets.Diff)
```
```  1171     apply (auto intro: sigma_sets_top)
```
```  1172     done
```
```  1173 qed auto
```
```  1174
```
```  1175 lemma borel_eq_lessThan:
```
```  1176   "borel = sigma UNIV (range (\<lambda>a::'a::ordered_euclidean_space. {x. x <e a}))"
```
```  1177   (is "_ = ?SIGMA")
```
```  1178 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge])
```
```  1179   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```  1180   then have i: "i \<in> Basis" by auto
```
```  1181   have "{x::'a. a \<le> x\<bullet>i} = UNIV - {x::'a. x\<bullet>i < a}" by auto
```
```  1182   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 \<open>i\<in> Basis\<close>
```
```  1183   proof (safe, simp_all add: eucl_less_def split: if_split_asm)
```
```  1184     fix x :: 'a
```
```  1185     from reals_Archimedean2[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"]
```
```  1186     guess k::nat .. note k = this
```
```  1187     { fix i :: 'a assume "i \<in> Basis"
```
```  1188       then have "x\<bullet>i < real k"
```
```  1189         using k by (subst (asm) Max_less_iff) auto
```
```  1190       then have "x\<bullet>i < real k" by simp }
```
```  1191     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> x \<bullet> ia < real k"
```
```  1192       by (auto intro!: exI[of _ k])
```
```  1193   qed
```
```  1194   finally show "{x. a \<le> x\<bullet>i} \<in> ?SIGMA"
```
```  1195     apply (simp only:)
```
```  1196     apply (intro sets.countable_UN sets.Diff)
```
```  1197     apply (auto intro: sigma_sets_top )
```
```  1198     done
```
```  1199 qed auto
```
```  1200
```
```  1201 lemma borel_eq_atLeastAtMost:
```
```  1202   "borel = sigma UNIV (range (\<lambda>(a,b). {a..b} ::'a::ordered_euclidean_space set))"
```
```  1203   (is "_ = ?SIGMA")
```
```  1204 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
```
```  1205   fix a::'a
```
```  1206   have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
```
```  1207   proof (safe, simp_all add: eucl_le[where 'a='a])
```
```  1208     fix x :: 'a
```
```  1209     from real_arch_simple[of "Max ((\<lambda>i. - x\<bullet>i)`Basis)"]
```
```  1210     guess k::nat .. note k = this
```
```  1211     { fix i :: 'a assume "i \<in> Basis"
```
```  1212       with k have "- x\<bullet>i \<le> real k"
```
```  1213         by (subst (asm) Max_le_iff) (auto simp: field_simps)
```
```  1214       then have "- real k \<le> x\<bullet>i" by simp }
```
```  1215     then show "\<exists>n::nat. \<forall>i\<in>Basis. - real n \<le> x \<bullet> i"
```
```  1216       by (auto intro!: exI[of _ k])
```
```  1217   qed
```
```  1218   show "{..a} \<in> ?SIGMA" unfolding *
```
```  1219     by (intro sets.countable_UN)
```
```  1220        (auto intro!: sigma_sets_top)
```
```  1221 qed auto
```
```  1222
```
```  1223 lemma borel_set_induct[consumes 1, case_names empty interval compl union]:
```
```  1224   assumes "A \<in> sets borel"
```
```  1225   assumes empty: "P {}" and int: "\<And>a b. a \<le> b \<Longrightarrow> P {a..b}" and compl: "\<And>A. A \<in> sets borel \<Longrightarrow> P A \<Longrightarrow> P (-A)" and
```
```  1226           un: "\<And>f. disjoint_family f \<Longrightarrow> (\<And>i. f i \<in> sets borel) \<Longrightarrow>  (\<And>i. P (f i)) \<Longrightarrow> P (\<Union>i::nat. f i)"
```
```  1227   shows "P (A::real set)"
```
```  1228 proof-
```
```  1229   let ?G = "range (\<lambda>(a,b). {a..b::real})"
```
```  1230   have "Int_stable ?G" "?G \<subseteq> Pow UNIV" "A \<in> sigma_sets UNIV ?G"
```
```  1231       using assms(1) by (auto simp add: borel_eq_atLeastAtMost Int_stable_def)
```
```  1232   thus ?thesis
```
```  1233   proof (induction rule: sigma_sets_induct_disjoint)
```
```  1234     case (union f)
```
```  1235       from union.hyps(2) have "\<And>i. f i \<in> sets borel" by (auto simp: borel_eq_atLeastAtMost)
```
```  1236       with union show ?case by (auto intro: un)
```
```  1237   next
```
```  1238     case (basic A)
```
```  1239     then obtain a b where "A = {a .. b}" by auto
```
```  1240     then show ?case
```
```  1241       by (cases "a \<le> b") (auto intro: int empty)
```
```  1242   qed (auto intro: empty compl simp: Compl_eq_Diff_UNIV[symmetric] borel_eq_atLeastAtMost)
```
```  1243 qed
```
```  1244
```
```  1245 lemma borel_sigma_sets_Ioc: "borel = sigma UNIV (range (\<lambda>(a, b). {a <.. b::real}))"
```
```  1246 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
```
```  1247   fix i :: real
```
```  1248   have "{..i} = (\<Union>j::nat. {-j <.. i})"
```
```  1249     by (auto simp: minus_less_iff reals_Archimedean2)
```
```  1250   also have "\<dots> \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))"
```
```  1251     by (intro sets.countable_nat_UN) auto
```
```  1252   finally show "{..i} \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))" .
```
```  1253 qed simp
```
```  1254
```
```  1255 lemma eucl_lessThan: "{x::real. x <e a} = lessThan a"
```
```  1256   by (simp add: eucl_less_def lessThan_def)
```
```  1257
```
```  1258 lemma borel_eq_atLeastLessThan:
```
```  1259   "borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA")
```
```  1260 proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan])
```
```  1261   have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto
```
```  1262   fix x :: real
```
```  1263   have "{..<x} = (\<Union>i::nat. {-real i ..< x})"
```
```  1264     by (auto simp: move_uminus real_arch_simple)
```
```  1265   then show "{y. y <e x} \<in> ?SIGMA"
```
```  1266     by (auto intro: sigma_sets.intros(2-) simp: eucl_lessThan)
```
```  1267 qed auto
```
```  1268
```
```  1269 lemma borel_measurable_halfspacesI:
```
```  1270   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1271   assumes F: "borel = sigma UNIV (F ` (UNIV \<times> Basis))"
```
```  1272   and S_eq: "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
```
```  1273   shows "f \<in> borel_measurable M = (\<forall>i\<in>Basis. \<forall>a::real. S a i \<in> sets M)"
```
```  1274 proof safe
```
```  1275   fix a :: real and i :: 'b assume i: "i \<in> Basis" and f: "f \<in> borel_measurable M"
```
```  1276   then show "S a i \<in> sets M" unfolding assms
```
```  1277     by (auto intro!: measurable_sets simp: assms(1))
```
```  1278 next
```
```  1279   assume a: "\<forall>i\<in>Basis. \<forall>a. S a i \<in> sets M"
```
```  1280   then show "f \<in> borel_measurable M"
```
```  1281     by (auto intro!: measurable_measure_of simp: S_eq F)
```
```  1282 qed
```
```  1283
```
```  1284 lemma borel_measurable_iff_halfspace_le:
```
```  1285   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1286   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)"
```
```  1287   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
```
```  1288
```
```  1289 lemma borel_measurable_iff_halfspace_less:
```
```  1290   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1291   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)"
```
```  1292   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
```
```  1293
```
```  1294 lemma borel_measurable_iff_halfspace_ge:
```
```  1295   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1296   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)"
```
```  1297   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
```
```  1298
```
```  1299 lemma borel_measurable_iff_halfspace_greater:
```
```  1300   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1301   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)"
```
```  1302   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto
```
```  1303
```
```  1304 lemma borel_measurable_iff_le:
```
```  1305   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
```
```  1306   using borel_measurable_iff_halfspace_le[where 'c=real] by simp
```
```  1307
```
```  1308 lemma borel_measurable_iff_less:
```
```  1309   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
```
```  1310   using borel_measurable_iff_halfspace_less[where 'c=real] by simp
```
```  1311
```
```  1312 lemma borel_measurable_iff_ge:
```
```  1313   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
```
```  1314   using borel_measurable_iff_halfspace_ge[where 'c=real]
```
```  1315   by simp
```
```  1316
```
```  1317 lemma borel_measurable_iff_greater:
```
```  1318   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
```
```  1319   using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
```
```  1320
```
```  1321 lemma borel_measurable_euclidean_space:
```
```  1322   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1323   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M)"
```
```  1324 proof safe
```
```  1325   assume f: "\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M"
```
```  1326   then show "f \<in> borel_measurable M"
```
```  1327     by (subst borel_measurable_iff_halfspace_le) auto
```
```  1328 qed auto
```
```  1329
```
```  1330 subsection "Borel measurable operators"
```
```  1331
```
```  1332 lemma borel_measurable_norm[measurable]: "norm \<in> borel_measurable borel"
```
```  1333   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1334
```
```  1335 lemma borel_measurable_sgn [measurable]: "(sgn::'a::real_normed_vector \<Rightarrow> 'a) \<in> borel_measurable borel"
```
```  1336   by (rule borel_measurable_continuous_countable_exceptions[where X="{0}"])
```
```  1337      (auto intro!: continuous_on_sgn continuous_on_id)
```
```  1338
```
```  1339 lemma borel_measurable_uminus[measurable (raw)]:
```
```  1340   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```  1341   assumes g: "g \<in> borel_measurable M"
```
```  1342   shows "(\<lambda>x. - g x) \<in> borel_measurable M"
```
```  1343   by (rule borel_measurable_continuous_on[OF _ g]) (intro continuous_intros)
```
```  1344
```
```  1345 lemma borel_measurable_diff[measurable (raw)]:
```
```  1346   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```  1347   assumes f: "f \<in> borel_measurable M"
```
```  1348   assumes g: "g \<in> borel_measurable M"
```
```  1349   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```  1350   using borel_measurable_add [of f M "- g"] assms by (simp add: fun_Compl_def)
```
```  1351
```
```  1352 lemma borel_measurable_times[measurable (raw)]:
```
```  1353   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_algebra}"
```
```  1354   assumes f: "f \<in> borel_measurable M"
```
```  1355   assumes g: "g \<in> borel_measurable M"
```
```  1356   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1357   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```  1358
```
```  1359 lemma borel_measurable_prod[measurable (raw)]:
```
```  1360   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, real_normed_field}"
```
```  1361   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1362   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1363 proof cases
```
```  1364   assume "finite S"
```
```  1365   thus ?thesis using assms by induct auto
```
```  1366 qed simp
```
```  1367
```
```  1368 lemma borel_measurable_dist[measurable (raw)]:
```
```  1369   fixes g f :: "'a \<Rightarrow> 'b::{second_countable_topology, metric_space}"
```
```  1370   assumes f: "f \<in> borel_measurable M"
```
```  1371   assumes g: "g \<in> borel_measurable M"
```
```  1372   shows "(\<lambda>x. dist (f x) (g x)) \<in> borel_measurable M"
```
```  1373   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```  1374
```
```  1375 lemma borel_measurable_scaleR[measurable (raw)]:
```
```  1376   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```  1377   assumes f: "f \<in> borel_measurable M"
```
```  1378   assumes g: "g \<in> borel_measurable M"
```
```  1379   shows "(\<lambda>x. f x *\<^sub>R g x) \<in> borel_measurable M"
```
```  1380   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```  1381
```
```  1382 lemma affine_borel_measurable_vector:
```
```  1383   fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
```
```  1384   assumes "f \<in> borel_measurable M"
```
```  1385   shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
```
```  1386 proof (rule borel_measurableI)
```
```  1387   fix S :: "'x set" assume "open S"
```
```  1388   show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
```
```  1389   proof cases
```
```  1390     assume "b \<noteq> 0"
```
```  1391     with \<open>open S\<close> have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")
```
```  1392       using open_affinity [of S "inverse b" "- a /\<^sub>R b"]
```
```  1393       by (auto simp: algebra_simps)
```
```  1394     hence "?S \<in> sets borel" by auto
```
```  1395     moreover
```
```  1396     from \<open>b \<noteq> 0\<close> have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
```
```  1397       apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
```
```  1398     ultimately show ?thesis using assms unfolding in_borel_measurable_borel
```
```  1399       by auto
```
```  1400   qed simp
```
```  1401 qed
```
```  1402
```
```  1403 lemma borel_measurable_const_scaleR[measurable (raw)]:
```
```  1404   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. b *\<^sub>R f x ::'a::real_normed_vector) \<in> borel_measurable M"
```
```  1405   using affine_borel_measurable_vector[of f M 0 b] by simp
```
```  1406
```
```  1407 lemma borel_measurable_const_add[measurable (raw)]:
```
```  1408   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. a + f x ::'a::real_normed_vector) \<in> borel_measurable M"
```
```  1409   using affine_borel_measurable_vector[of f M a 1] by simp
```
```  1410
```
```  1411 lemma borel_measurable_inverse[measurable (raw)]:
```
```  1412   fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
```
```  1413   assumes f: "f \<in> borel_measurable M"
```
```  1414   shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
```
```  1415   apply (rule measurable_compose[OF f])
```
```  1416   apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])
```
```  1417   apply (auto intro!: continuous_on_inverse continuous_on_id)
```
```  1418   done
```
```  1419
```
```  1420 lemma borel_measurable_divide[measurable (raw)]:
```
```  1421   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>
```
```  1422     (\<lambda>x. f x / g x::'b::{second_countable_topology, real_normed_div_algebra}) \<in> borel_measurable M"
```
```  1423   by (simp add: divide_inverse)
```
```  1424
```
```  1425 lemma borel_measurable_abs[measurable (raw)]:
```
```  1426   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
```
```  1427   unfolding abs_real_def by simp
```
```  1428
```
```  1429 lemma borel_measurable_nth[measurable (raw)]:
```
```  1430   "(\<lambda>x::real^'n. x \$ i) \<in> borel_measurable borel"
```
```  1431   by (simp add: cart_eq_inner_axis)
```
```  1432
```
```  1433 lemma convex_measurable:
```
```  1434   fixes A :: "'a :: euclidean_space set"
```
```  1435   shows "X \<in> borel_measurable M \<Longrightarrow> X ` space M \<subseteq> A \<Longrightarrow> open A \<Longrightarrow> convex_on A q \<Longrightarrow>
```
```  1436     (\<lambda>x. q (X x)) \<in> borel_measurable M"
```
```  1437   by (rule measurable_compose[where f=X and N="restrict_space borel A"])
```
```  1438      (auto intro!: borel_measurable_continuous_on_restrict convex_on_continuous measurable_restrict_space2)
```
```  1439
```
```  1440 lemma borel_measurable_ln[measurable (raw)]:
```
```  1441   assumes f: "f \<in> borel_measurable M"
```
```  1442   shows "(\<lambda>x. ln (f x :: real)) \<in> borel_measurable M"
```
```  1443   apply (rule measurable_compose[OF f])
```
```  1444   apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])
```
```  1445   apply (auto intro!: continuous_on_ln continuous_on_id)
```
```  1446   done
```
```  1447
```
```  1448 lemma borel_measurable_log[measurable (raw)]:
```
```  1449   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. log (g x) (f x)) \<in> borel_measurable M"
```
```  1450   unfolding log_def by auto
```
```  1451
```
```  1452 lemma borel_measurable_exp[measurable]:
```
```  1453   "(exp::'a::{real_normed_field,banach}\<Rightarrow>'a) \<in> borel_measurable borel"
```
```  1454   by (intro borel_measurable_continuous_on1 continuous_at_imp_continuous_on ballI isCont_exp)
```
```  1455
```
```  1456 lemma measurable_real_floor[measurable]:
```
```  1457   "(floor :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
```
```  1458 proof -
```
```  1459   have "\<And>a x. \<lfloor>x\<rfloor> = a \<longleftrightarrow> (real_of_int a \<le> x \<and> x < real_of_int (a + 1))"
```
```  1460     by (auto intro: floor_eq2)
```
```  1461   then show ?thesis
```
```  1462     by (auto simp: vimage_def measurable_count_space_eq2_countable)
```
```  1463 qed
```
```  1464
```
```  1465 lemma measurable_real_ceiling[measurable]:
```
```  1466   "(ceiling :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
```
```  1467   unfolding ceiling_def[abs_def] by simp
```
```  1468
```
```  1469 lemma borel_measurable_real_floor: "(\<lambda>x::real. real_of_int \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
```
```  1470   by simp
```
```  1471
```
```  1472 lemma borel_measurable_root [measurable]: "root n \<in> borel_measurable borel"
```
```  1473   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1474
```
```  1475 lemma borel_measurable_sqrt [measurable]: "sqrt \<in> borel_measurable borel"
```
```  1476   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1477
```
```  1478 lemma borel_measurable_power [measurable (raw)]:
```
```  1479   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```  1480   assumes f: "f \<in> borel_measurable M"
```
```  1481   shows "(\<lambda>x. (f x) ^ n) \<in> borel_measurable M"
```
```  1482   by (intro borel_measurable_continuous_on [OF _ f] continuous_intros)
```
```  1483
```
```  1484 lemma borel_measurable_Re [measurable]: "Re \<in> borel_measurable borel"
```
```  1485   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1486
```
```  1487 lemma borel_measurable_Im [measurable]: "Im \<in> borel_measurable borel"
```
```  1488   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1489
```
```  1490 lemma borel_measurable_of_real [measurable]: "(of_real :: _ \<Rightarrow> (_::real_normed_algebra)) \<in> borel_measurable borel"
```
```  1491   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1492
```
```  1493 lemma borel_measurable_sin [measurable]: "(sin :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"
```
```  1494   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1495
```
```  1496 lemma borel_measurable_cos [measurable]: "(cos :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"
```
```  1497   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1498
```
```  1499 lemma borel_measurable_arctan [measurable]: "arctan \<in> borel_measurable borel"
```
```  1500   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1501
```
```  1502 lemma borel_measurable_complex_iff:
```
```  1503   "f \<in> borel_measurable M \<longleftrightarrow>
```
```  1504     (\<lambda>x. Re (f x)) \<in> borel_measurable M \<and> (\<lambda>x. Im (f x)) \<in> borel_measurable M"
```
```  1505   apply auto
```
```  1506   apply (subst fun_complex_eq)
```
```  1507   apply (intro borel_measurable_add)
```
```  1508   apply auto
```
```  1509   done
```
```  1510
```
```  1511 lemma measurable_of_bool[measurable]: "of_bool \<in> count_space UNIV \<rightarrow>\<^sub>M borel"
```
```  1512   by simp
```
```  1513
```
```  1514 subsection "Borel space on the extended reals"
```
```  1515
```
```  1516 lemma borel_measurable_ereal[measurable (raw)]:
```
```  1517   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
```
```  1518   using continuous_on_ereal f by (rule borel_measurable_continuous_on) (rule continuous_on_id)
```
```  1519
```
```  1520 lemma borel_measurable_real_of_ereal[measurable (raw)]:
```
```  1521   fixes f :: "'a \<Rightarrow> ereal"
```
```  1522   assumes f: "f \<in> borel_measurable M"
```
```  1523   shows "(\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M"
```
```  1524   apply (rule measurable_compose[OF f])
```
```  1525   apply (rule borel_measurable_continuous_countable_exceptions[of "{\<infinity>, -\<infinity> }"])
```
```  1526   apply (auto intro: continuous_on_real simp: Compl_eq_Diff_UNIV)
```
```  1527   done
```
```  1528
```
```  1529 lemma borel_measurable_ereal_cases:
```
```  1530   fixes f :: "'a \<Rightarrow> ereal"
```
```  1531   assumes f: "f \<in> borel_measurable M"
```
```  1532   assumes H: "(\<lambda>x. H (ereal (real_of_ereal (f x)))) \<in> borel_measurable M"
```
```  1533   shows "(\<lambda>x. H (f x)) \<in> borel_measurable M"
```
```  1534 proof -
```
```  1535   let ?F = "\<lambda>x. if f x = \<infinity> then H \<infinity> else if f x = - \<infinity> then H (-\<infinity>) else H (ereal (real_of_ereal (f x)))"
```
```  1536   { fix x have "H (f x) = ?F x" by (cases "f x") auto }
```
```  1537   with f H show ?thesis by simp
```
```  1538 qed
```
```  1539
```
```  1540 lemma
```
```  1541   fixes f :: "'a \<Rightarrow> ereal" assumes f[measurable]: "f \<in> borel_measurable M"
```
```  1542   shows borel_measurable_ereal_abs[measurable(raw)]: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
```
```  1543     and borel_measurable_ereal_inverse[measurable(raw)]: "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M"
```
```  1544     and borel_measurable_uminus_ereal[measurable(raw)]: "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M"
```
```  1545   by (auto simp del: abs_real_of_ereal simp: borel_measurable_ereal_cases[OF f] measurable_If)
```
```  1546
```
```  1547 lemma borel_measurable_uminus_eq_ereal[simp]:
```
```  1548   "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
```
```  1549 proof
```
```  1550   assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp
```
```  1551 qed auto
```
```  1552
```
```  1553 lemma set_Collect_ereal2:
```
```  1554   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1555   assumes f: "f \<in> borel_measurable M"
```
```  1556   assumes g: "g \<in> borel_measurable M"
```
```  1557   assumes H: "{x \<in> space M. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))} \<in> sets M"
```
```  1558     "{x \<in> space borel. H (-\<infinity>) (ereal x)} \<in> sets borel"
```
```  1559     "{x \<in> space borel. H (\<infinity>) (ereal x)} \<in> sets borel"
```
```  1560     "{x \<in> space borel. H (ereal x) (-\<infinity>)} \<in> sets borel"
```
```  1561     "{x \<in> space borel. H (ereal x) (\<infinity>)} \<in> sets borel"
```
```  1562   shows "{x \<in> space M. H (f x) (g x)} \<in> sets M"
```
```  1563 proof -
```
```  1564   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_of_ereal (g x)))"
```
```  1565   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_of_ereal (f x))) x"
```
```  1566   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
```
```  1567   note * = this
```
```  1568   from assms show ?thesis
```
```  1569     by (subst *) (simp del: space_borel split del: if_split)
```
```  1570 qed
```
```  1571
```
```  1572 lemma borel_measurable_ereal_iff:
```
```  1573   shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
```
```  1574 proof
```
```  1575   assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
```
```  1576   from borel_measurable_real_of_ereal[OF this]
```
```  1577   show "f \<in> borel_measurable M" by auto
```
```  1578 qed auto
```
```  1579
```
```  1580 lemma borel_measurable_erealD[measurable_dest]:
```
```  1581   "(\<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"
```
```  1582   unfolding borel_measurable_ereal_iff by simp
```
```  1583
```
```  1584 lemma borel_measurable_ereal_iff_real:
```
```  1585   fixes f :: "'a \<Rightarrow> ereal"
```
```  1586   shows "f \<in> borel_measurable M \<longleftrightarrow>
```
```  1587     ((\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)"
```
```  1588 proof safe
```
```  1589   assume *: "(\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M"
```
```  1590   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
```
```  1591   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
```
```  1592   let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real_of_ereal (f x))"
```
```  1593   have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
```
```  1594   also have "?f = f" by (auto simp: fun_eq_iff ereal_real)
```
```  1595   finally show "f \<in> borel_measurable M" .
```
```  1596 qed simp_all
```
```  1597
```
```  1598 lemma borel_measurable_ereal_iff_Iio:
```
```  1599   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
```
```  1600   by (auto simp: borel_Iio measurable_iff_measure_of)
```
```  1601
```
```  1602 lemma borel_measurable_ereal_iff_Ioi:
```
```  1603   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
```
```  1604   by (auto simp: borel_Ioi measurable_iff_measure_of)
```
```  1605
```
```  1606 lemma vimage_sets_compl_iff:
```
```  1607   "f -` A \<inter> space M \<in> sets M \<longleftrightarrow> f -` (- A) \<inter> space M \<in> sets M"
```
```  1608 proof -
```
```  1609   { fix A assume "f -` A \<inter> space M \<in> sets M"
```
```  1610     moreover have "f -` (- A) \<inter> space M = space M - f -` A \<inter> space M" by auto
```
```  1611     ultimately have "f -` (- A) \<inter> space M \<in> sets M" by auto }
```
```  1612   from this[of A] this[of "-A"] show ?thesis
```
```  1613     by (metis double_complement)
```
```  1614 qed
```
```  1615
```
```  1616 lemma borel_measurable_iff_Iic_ereal:
```
```  1617   "(f::'a\<Rightarrow>ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
```
```  1618   unfolding borel_measurable_ereal_iff_Ioi vimage_sets_compl_iff[where A="{a <..}" for a] by simp
```
```  1619
```
```  1620 lemma borel_measurable_iff_Ici_ereal:
```
```  1621   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
```
```  1622   unfolding borel_measurable_ereal_iff_Iio vimage_sets_compl_iff[where A="{..< a}" for a] by simp
```
```  1623
```
```  1624 lemma borel_measurable_ereal2:
```
```  1625   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1626   assumes f: "f \<in> borel_measurable M"
```
```  1627   assumes g: "g \<in> borel_measurable M"
```
```  1628   assumes H: "(\<lambda>x. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
```
```  1629     "(\<lambda>x. H (-\<infinity>) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
```
```  1630     "(\<lambda>x. H (\<infinity>) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
```
```  1631     "(\<lambda>x. H (ereal (real_of_ereal (f x))) (-\<infinity>)) \<in> borel_measurable M"
```
```  1632     "(\<lambda>x. H (ereal (real_of_ereal (f x))) (\<infinity>)) \<in> borel_measurable M"
```
```  1633   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
```
```  1634 proof -
```
```  1635   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_of_ereal (g x)))"
```
```  1636   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_of_ereal (f x))) x"
```
```  1637   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
```
```  1638   note * = this
```
```  1639   from assms show ?thesis unfolding * by simp
```
```  1640 qed
```
```  1641
```
```  1642 lemma [measurable(raw)]:
```
```  1643   fixes f :: "'a \<Rightarrow> ereal"
```
```  1644   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1645   shows borel_measurable_ereal_add: "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```  1646     and borel_measurable_ereal_times: "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1647   by (simp_all add: borel_measurable_ereal2)
```
```  1648
```
```  1649 lemma [measurable(raw)]:
```
```  1650   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1651   assumes "f \<in> borel_measurable M"
```
```  1652   assumes "g \<in> borel_measurable M"
```
```  1653   shows borel_measurable_ereal_diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```  1654     and borel_measurable_ereal_divide: "(\<lambda>x. f x / g x) \<in> borel_measurable M"
```
```  1655   using assms by (simp_all add: minus_ereal_def divide_ereal_def)
```
```  1656
```
```  1657 lemma borel_measurable_ereal_sum[measurable (raw)]:
```
```  1658   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1659   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1660   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1661   using assms by (induction S rule: infinite_finite_induct) auto
```
```  1662
```
```  1663 lemma borel_measurable_ereal_prod[measurable (raw)]:
```
```  1664   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1665   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1666   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1667   using assms by (induction S rule: infinite_finite_induct) auto
```
```  1668
```
```  1669 lemma borel_measurable_extreal_suminf[measurable (raw)]:
```
```  1670   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1671   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1672   shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
```
```  1673   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
```
```  1674
```
```  1675 subsection "Borel space on the extended non-negative reals"
```
```  1676
```
```  1677 text \<open> @{type ennreal} is a topological monoid, so no rules for plus are required, also all order
```
```  1678   statements are usually done on type classes. \<close>
```
```  1679
```
```  1680 lemma measurable_enn2ereal[measurable]: "enn2ereal \<in> borel \<rightarrow>\<^sub>M borel"
```
```  1681   by (intro borel_measurable_continuous_on1 continuous_on_enn2ereal)
```
```  1682
```
```  1683 lemma measurable_e2ennreal[measurable]: "e2ennreal \<in> borel \<rightarrow>\<^sub>M borel"
```
```  1684   by (intro borel_measurable_continuous_on1 continuous_on_e2ennreal)
```
```  1685
```
```  1686 lemma borel_measurable_enn2real[measurable (raw)]:
```
```  1687   "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. enn2real (f x)) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1688   unfolding enn2real_def[abs_def] by measurable
```
```  1689
```
```  1690 definition [simp]: "is_borel f M \<longleftrightarrow> f \<in> borel_measurable M"
```
```  1691
```
```  1692 lemma is_borel_transfer[transfer_rule]: "rel_fun (rel_fun op = pcr_ennreal) op = is_borel is_borel"
```
```  1693   unfolding is_borel_def[abs_def]
```
```  1694 proof (safe intro!: rel_funI ext dest!: rel_fun_eq_pcr_ennreal[THEN iffD1])
```
```  1695   fix f and M :: "'a measure"
```
```  1696   show "f \<in> borel_measurable M" if f: "enn2ereal \<circ> f \<in> borel_measurable M"
```
```  1697     using measurable_compose[OF f measurable_e2ennreal] by simp
```
```  1698 qed simp
```
```  1699
```
```  1700 context
```
```  1701   includes ennreal.lifting
```
```  1702 begin
```
```  1703
```
```  1704 lemma measurable_ennreal[measurable]: "ennreal \<in> borel \<rightarrow>\<^sub>M borel"
```
```  1705   unfolding is_borel_def[symmetric]
```
```  1706   by transfer simp
```
```  1707
```
```  1708 lemma borel_measurable_ennreal_iff[simp]:
```
```  1709   assumes [simp]: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
```
```  1710   shows "(\<lambda>x. ennreal (f x)) \<in> M \<rightarrow>\<^sub>M borel \<longleftrightarrow> f \<in> M \<rightarrow>\<^sub>M borel"
```
```  1711 proof safe
```
```  1712   assume "(\<lambda>x. ennreal (f x)) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1713   then have "(\<lambda>x. enn2real (ennreal (f x))) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1714     by measurable
```
```  1715   then show "f \<in> M \<rightarrow>\<^sub>M borel"
```
```  1716     by (rule measurable_cong[THEN iffD1, rotated]) auto
```
```  1717 qed measurable
```
```  1718
```
```  1719 lemma borel_measurable_times_ennreal[measurable (raw)]:
```
```  1720   fixes f g :: "'a \<Rightarrow> ennreal"
```
```  1721   shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> g \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. f x * g x) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1722   unfolding is_borel_def[symmetric] by transfer simp
```
```  1723
```
```  1724 lemma borel_measurable_inverse_ennreal[measurable (raw)]:
```
```  1725   fixes f :: "'a \<Rightarrow> ennreal"
```
```  1726   shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. inverse (f x)) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1727   unfolding is_borel_def[symmetric] by transfer simp
```
```  1728
```
```  1729 lemma borel_measurable_divide_ennreal[measurable (raw)]:
```
```  1730   fixes f :: "'a \<Rightarrow> ennreal"
```
```  1731   shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> g \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. f x / g x) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1732   unfolding divide_ennreal_def by simp
```
```  1733
```
```  1734 lemma borel_measurable_minus_ennreal[measurable (raw)]:
```
```  1735   fixes f :: "'a \<Rightarrow> ennreal"
```
```  1736   shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> g \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. f x - g x) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1737   unfolding is_borel_def[symmetric] by transfer simp
```
```  1738
```
```  1739 lemma borel_measurable_prod_ennreal[measurable (raw)]:
```
```  1740   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ennreal"
```
```  1741   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1742   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1743   using assms by (induction S rule: infinite_finite_induct) auto
```
```  1744
```
```  1745 end
```
```  1746
```
```  1747 hide_const (open) is_borel
```
```  1748
```
```  1749 subsection \<open>LIMSEQ is borel measurable\<close>
```
```  1750
```
```  1751 lemma borel_measurable_LIMSEQ_real:
```
```  1752   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
```
```  1753   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
```
```  1754   and u: "\<And>i. u i \<in> borel_measurable M"
```
```  1755   shows "u' \<in> borel_measurable M"
```
```  1756 proof -
```
```  1757   have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)"
```
```  1758     using u' by (simp add: lim_imp_Liminf)
```
```  1759   moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M"
```
```  1760     by auto
```
```  1761   ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff)
```
```  1762 qed
```
```  1763
```
```  1764 lemma borel_measurable_LIMSEQ_metric:
```
```  1765   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b :: metric_space"
```
```  1766   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1767   assumes lim: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. f i x) \<longlonglongrightarrow> g x"
```
```  1768   shows "g \<in> borel_measurable M"
```
```  1769   unfolding borel_eq_closed
```
```  1770 proof (safe intro!: measurable_measure_of)
```
```  1771   fix A :: "'b set" assume "closed A"
```
```  1772
```
```  1773   have [measurable]: "(\<lambda>x. infdist (g x) A) \<in> borel_measurable M"
```
```  1774   proof (rule borel_measurable_LIMSEQ_real)
```
```  1775     show "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. infdist (f i x) A) \<longlonglongrightarrow> infdist (g x) A"
```
```  1776       by (intro tendsto_infdist lim)
```
```  1777     show "\<And>i. (\<lambda>x. infdist (f i x) A) \<in> borel_measurable M"
```
```  1778       by (intro borel_measurable_continuous_on[where f="\<lambda>x. infdist x A"]
```
```  1779         continuous_at_imp_continuous_on ballI continuous_infdist continuous_ident) auto
```
```  1780   qed
```
```  1781
```
```  1782   show "g -` A \<inter> space M \<in> sets M"
```
```  1783   proof cases
```
```  1784     assume "A \<noteq> {}"
```
```  1785     then have "\<And>x. infdist x A = 0 \<longleftrightarrow> x \<in> A"
```
```  1786       using \<open>closed A\<close> by (simp add: in_closed_iff_infdist_zero)
```
```  1787     then have "g -` A \<inter> space M = {x\<in>space M. infdist (g x) A = 0}"
```
```  1788       by auto
```
```  1789     also have "\<dots> \<in> sets M"
```
```  1790       by measurable
```
```  1791     finally show ?thesis .
```
```  1792   qed simp
```
```  1793 qed auto
```
```  1794
```
```  1795 lemma sets_Collect_Cauchy[measurable]:
```
```  1796   fixes f :: "nat \<Rightarrow> 'a => 'b::{metric_space, second_countable_topology}"
```
```  1797   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1798   shows "{x\<in>space M. Cauchy (\<lambda>i. f i x)} \<in> sets M"
```
```  1799   unfolding metric_Cauchy_iff2 using f by auto
```
```  1800
```
```  1801 lemma borel_measurable_lim_metric[measurable (raw)]:
```
```  1802   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```  1803   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1804   shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1805 proof -
```
```  1806   define u' where "u' x = lim (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)" for x
```
```  1807   then have *: "\<And>x. lim (\<lambda>i. f i x) = (if Cauchy (\<lambda>i. f i x) then u' x else (THE x. False))"
```
```  1808     by (auto simp: lim_def convergent_eq_Cauchy[symmetric])
```
```  1809   have "u' \<in> borel_measurable M"
```
```  1810   proof (rule borel_measurable_LIMSEQ_metric)
```
```  1811     fix x
```
```  1812     have "convergent (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)"
```
```  1813       by (cases "Cauchy (\<lambda>i. f i x)")
```
```  1814          (auto simp add: convergent_eq_Cauchy[symmetric] convergent_def)
```
```  1815     then show "(\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0) \<longlonglongrightarrow> u' x"
```
```  1816       unfolding u'_def
```
```  1817       by (rule convergent_LIMSEQ_iff[THEN iffD1])
```
```  1818   qed measurable
```
```  1819   then show ?thesis
```
```  1820     unfolding * by measurable
```
```  1821 qed
```
```  1822
```
```  1823 lemma borel_measurable_suminf[measurable (raw)]:
```
```  1824   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```  1825   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1826   shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1827   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
```
```  1828
```
```  1829 lemma Collect_closed_imp_pred_borel: "closed {x. P x} \<Longrightarrow> Measurable.pred borel P"
```
```  1830   by (simp add: pred_def)
```
```  1831
```
```  1832 (* Proof by Jeremy Avigad and Luke Serafin *)
```
```  1833 lemma isCont_borel_pred[measurable]:
```
```  1834   fixes f :: "'b::metric_space \<Rightarrow> 'a::metric_space"
```
```  1835   shows "Measurable.pred borel (isCont f)"
```
```  1836 proof (subst measurable_cong)
```
```  1837   let ?I = "\<lambda>j. inverse(real (Suc j))"
```
```  1838   show "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)" for x
```
```  1839     unfolding continuous_at_eps_delta
```
```  1840   proof safe
```
```  1841     fix i assume "\<forall>e>0. \<exists>d>0. \<forall>y. dist y x < d \<longrightarrow> dist (f y) (f x) < e"
```
```  1842     moreover have "0 < ?I i / 2"
```
```  1843       by simp
```
```  1844     ultimately obtain d where d: "0 < d" "\<And>y. dist x y < d \<Longrightarrow> dist (f y) (f x) < ?I i / 2"
```
```  1845       by (metis dist_commute)
```
```  1846     then obtain j where j: "?I j < d"
```
```  1847       by (metis reals_Archimedean)
```
```  1848
```
```  1849     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"
```
```  1850     proof (safe intro!: exI[where x=j])
```
```  1851       fix y z assume *: "dist x y < ?I j" "dist x z < ?I j"
```
```  1852       have "dist (f y) (f z) \<le> dist (f y) (f x) + dist (f z) (f x)"
```
```  1853         by (rule dist_triangle2)
```
```  1854       also have "\<dots> < ?I i / 2 + ?I i / 2"
```
```  1855         by (intro add_strict_mono d less_trans[OF _ j] *)
```
```  1856       also have "\<dots> \<le> ?I i"
```
```  1857         by (simp add: field_simps of_nat_Suc)
```
```  1858       finally show "dist (f y) (f z) \<le> ?I i"
```
```  1859         by simp
```
```  1860     qed
```
```  1861   next
```
```  1862     fix e::real assume "0 < e"
```
```  1863     then obtain n where n: "?I n < e"
```
```  1864       by (metis reals_Archimedean)
```
```  1865     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"
```
```  1866     from this[THEN spec, of "Suc n"]
```
```  1867     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)"
```
```  1868       by auto
```
```  1869
```
```  1870     show "\<exists>d>0. \<forall>y. dist y x < d \<longrightarrow> dist (f y) (f x) < e"
```
```  1871     proof (safe intro!: exI[of _ "?I j"])
```
```  1872       fix y assume "dist y x < ?I j"
```
```  1873       then have "dist (f y) (f x) \<le> ?I (Suc n)"
```
```  1874         by (intro j) (auto simp: dist_commute)
```
```  1875       also have "?I (Suc n) < ?I n"
```
```  1876         by simp
```
```  1877       also note n
```
```  1878       finally show "dist (f y) (f x) < e" .
```
```  1879     qed simp
```
```  1880   qed
```
```  1881 qed (intro pred_intros_countable closed_Collect_all closed_Collect_le open_Collect_less
```
```  1882            Collect_closed_imp_pred_borel closed_Collect_imp open_Collect_conj continuous_intros)
```
```  1883
```
```  1884 lemma isCont_borel:
```
```  1885   fixes f :: "'b::metric_space \<Rightarrow> 'a::metric_space"
```
```  1886   shows "{x. isCont f x} \<in> sets borel"
```
```  1887   by simp
```
```  1888
```
```  1889 lemma is_real_interval:
```
```  1890   assumes S: "is_interval S"
```
```  1891   shows "\<exists>a b::real. S = {} \<or> S = UNIV \<or> S = {..<b} \<or> S = {..b} \<or> S = {a<..} \<or> S = {a..} \<or>
```
```  1892     S = {a<..<b} \<or> S = {a<..b} \<or> S = {a..<b} \<or> S = {a..b}"
```
```  1893   using S unfolding is_interval_1 by (blast intro: interval_cases)
```
```  1894
```
```  1895 lemma real_interval_borel_measurable:
```
```  1896   assumes "is_interval (S::real set)"
```
```  1897   shows "S \<in> sets borel"
```
```  1898 proof -
```
```  1899   from assms is_real_interval have "\<exists>a b::real. S = {} \<or> S = UNIV \<or> S = {..<b} \<or> S = {..b} \<or>
```
```  1900     S = {a<..} \<or> S = {a..} \<or> S = {a<..<b} \<or> S = {a<..b} \<or> S = {a..<b} \<or> S = {a..b}" by auto
```
```  1901   then guess a ..
```
```  1902   then guess b ..
```
```  1903   thus ?thesis
```
```  1904     by auto
```
```  1905 qed
```
```  1906
```
```  1907 text \<open>The next lemmas hold in any second countable linorder (including ennreal or ereal for instance),
```
```  1908 but in the current state they are restricted to reals.\<close>
```
```  1909
```
```  1910 lemma borel_measurable_mono_on_fnc:
```
```  1911   fixes f :: "real \<Rightarrow> real" and A :: "real set"
```
```  1912   assumes "mono_on f A"
```
```  1913   shows "f \<in> borel_measurable (restrict_space borel A)"
```
```  1914   apply (rule measurable_restrict_countable[OF mono_on_ctble_discont[OF assms]])
```
```  1915   apply (auto intro!: image_eqI[where x="{x}" for x] simp: sets_restrict_space)
```
```  1916   apply (auto simp add: sets_restrict_restrict_space continuous_on_eq_continuous_within
```
```  1917               cong: measurable_cong_sets
```
```  1918               intro!: borel_measurable_continuous_on_restrict intro: continuous_within_subset)
```
```  1919   done
```
```  1920
```
```  1921 lemma borel_measurable_piecewise_mono:
```
```  1922   fixes f::"real \<Rightarrow> real" and C::"real set set"
```
```  1923   assumes "countable C" "\<And>c. c \<in> C \<Longrightarrow> c \<in> sets borel" "\<And>c. c \<in> C \<Longrightarrow> mono_on f c" "(\<Union>C) = UNIV"
```
```  1924   shows "f \<in> borel_measurable borel"
```
```  1925 by (rule measurable_piecewise_restrict[of C], auto intro: borel_measurable_mono_on_fnc simp: assms)
```
```  1926
```
```  1927 lemma borel_measurable_mono:
```
```  1928   fixes f :: "real \<Rightarrow> real"
```
```  1929   shows "mono f \<Longrightarrow> f \<in> borel_measurable borel"
```
```  1930   using borel_measurable_mono_on_fnc[of f UNIV] by (simp add: mono_def mono_on_def)
```
```  1931
```
```  1932 lemma measurable_bdd_below_real[measurable (raw)]:
```
```  1933   fixes F :: "'a \<Rightarrow> 'i \<Rightarrow> real"
```
```  1934   assumes [simp]: "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> M \<rightarrow>\<^sub>M borel"
```
```  1935   shows "Measurable.pred M (\<lambda>x. bdd_below ((\<lambda>i. F i x)`I))"
```
```  1936 proof (subst measurable_cong)
```
```  1937   show "bdd_below ((\<lambda>i. F i x)`I) \<longleftrightarrow> (\<exists>q\<in>\<int>. \<forall>i\<in>I. q \<le> F i x)" for x
```
```  1938     by (auto simp: bdd_below_def intro!: bexI[of _ "of_int (floor _)"] intro: order_trans of_int_floor_le)
```
```  1939   show "Measurable.pred M (\<lambda>w. \<exists>q\<in>\<int>. \<forall>i\<in>I. q \<le> F i w)"
```
```  1940     using countable_int by measurable
```
```  1941 qed
```
```  1942
```
```  1943 lemma borel_measurable_cINF_real[measurable (raw)]:
```
```  1944   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> real"
```
```  1945   assumes [simp]: "countable I"
```
```  1946   assumes F[measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```  1947   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
```
```  1948 proof (rule measurable_piecewise_restrict)
```
```  1949   let ?\<Omega> = "{x\<in>space M. bdd_below ((\<lambda>i. F i x)`I)}"
```
```  1950   show "countable {?\<Omega>, - ?\<Omega>}" "space M \<subseteq> \<Union>{?\<Omega>, - ?\<Omega>}" "\<And>X. X \<in> {?\<Omega>, - ?\<Omega>} \<Longrightarrow> X \<inter> space M \<in> sets M"
```
```  1951     by auto
```
```  1952   fix X assume "X \<in> {?\<Omega>, - ?\<Omega>}" then show "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable (restrict_space M X)"
```
```  1953   proof safe
```
```  1954     show "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable (restrict_space M ?\<Omega>)"
```
```  1955       by (intro borel_measurable_cINF measurable_restrict_space1 F)
```
```  1956          (auto simp: space_restrict_space)
```
```  1957     show "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable (restrict_space M (-?\<Omega>))"
```
```  1958     proof (subst measurable_cong)
```
```  1959       fix x assume "x \<in> space (restrict_space M (-?\<Omega>))"
```
```  1960       then have "\<not> (\<forall>i\<in>I. - F i x \<le> y)" for y
```
```  1961         by (auto simp: space_restrict_space bdd_above_def bdd_above_uminus[symmetric])
```
```  1962       then show "(INF i:I. F i x) = - (THE x. False)"
```
```  1963         by (auto simp: space_restrict_space Inf_real_def Sup_real_def Least_def simp del: Set.ball_simps(10))
```
```  1964     qed simp
```
```  1965   qed
```
```  1966 qed
```
```  1967
```
```  1968 lemma borel_Ici: "borel = sigma UNIV (range (\<lambda>x::real. {x ..}))"
```
```  1969 proof (safe intro!: borel_eq_sigmaI1[OF borel_Iio])
```
```  1970   fix x :: real
```
```  1971   have eq: "{..<x} = space (sigma UNIV (range atLeast)) - {x ..}"
```
```  1972     by auto
```
```  1973   show "{..<x} \<in> sets (sigma UNIV (range atLeast))"
```
```  1974     unfolding eq by (intro sets.compl_sets) auto
```
```  1975 qed auto
```
```  1976
```
```  1977 lemma borel_measurable_pred_less[measurable (raw)]:
```
```  1978   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
```
```  1979   shows "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> Measurable.pred M (\<lambda>w. f w < g w)"
```
```  1980   unfolding Measurable.pred_def by (rule borel_measurable_less)
```
```  1981
```
```  1982 no_notation
```
```  1983   eucl_less (infix "<e" 50)
```
```  1984
```
```  1985 lemma borel_measurable_Max2[measurable (raw)]:
```
```  1986   fixes f::"_ \<Rightarrow> _ \<Rightarrow> 'a::{second_countable_topology, dense_linorder, linorder_topology}"
```
```  1987   assumes "finite I"
```
```  1988     and [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1989   shows "(\<lambda>x. Max{f i x |i. i \<in> I}) \<in> borel_measurable M"
```
```  1990 by (simp add: borel_measurable_Max[OF assms(1), where ?f=f and ?M=M] Setcompr_eq_image)
```
```  1991
```
```  1992 lemma measurable_compose_n [measurable (raw)]:
```
```  1993   assumes "T \<in> measurable M M"
```
```  1994   shows "(T^^n) \<in> measurable M M"
```
```  1995 by (induction n, auto simp add: measurable_compose[OF _ assms])
```
```  1996
```
```  1997 lemma measurable_real_imp_nat:
```
```  1998   fixes f::"'a \<Rightarrow> nat"
```
```  1999   assumes [measurable]: "(\<lambda>x. real(f x)) \<in> borel_measurable M"
```
```  2000   shows "f \<in> measurable M (count_space UNIV)"
```
```  2001 proof -
```
```  2002   let ?g = "(\<lambda>x. real(f x))"
```
```  2003   have "\<And>(n::nat). ?g-`({real n}) \<inter> space M = f-`{n} \<inter> space M" by auto
```
```  2004   moreover have "\<And>(n::nat). ?g-`({real n}) \<inter> space M \<in> sets M" using assms by measurable
```
```  2005   ultimately have "\<And>(n::nat). f-`{n} \<inter> space M \<in> sets M" by simp
```
```  2006   then show ?thesis using measurable_count_space_eq2_countable by blast
```
```  2007 qed
```
```  2008
```
```  2009 lemma measurable_equality_set [measurable]:
```
```  2010   fixes f g::"_\<Rightarrow> 'a::{second_countable_topology, t2_space}"
```
```  2011   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  2012   shows "{x \<in> space M. f x = g x} \<in> sets M"
```
```  2013
```
```  2014 proof -
```
```  2015   define A where "A = {x \<in> space M. f x = g x}"
```
```  2016   define B where "B = {y. \<exists>x::'a. y = (x,x)}"
```
```  2017   have "A = (\<lambda>x. (f x, g x))-`B \<inter> space M" unfolding A_def B_def by auto
```
```  2018   moreover have "(\<lambda>x. (f x, g x)) \<in> borel_measurable M" by simp
```
```  2019   moreover have "B \<in> sets borel" unfolding B_def by (simp add: closed_diagonal)
```
```  2020   ultimately have "A \<in> sets M" by simp
```
```  2021   then show ?thesis unfolding A_def by simp
```
```  2022 qed
```
```  2023
```
```  2024 lemma measurable_inequality_set [measurable]:
```
```  2025   fixes f g::"_ \<Rightarrow> 'a::{second_countable_topology, linorder_topology}"
```
```  2026   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  2027   shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
```
```  2028         "{x \<in> space M. f x < g x} \<in> sets M"
```
```  2029         "{x \<in> space M. f x \<ge> g x} \<in> sets M"
```
```  2030         "{x \<in> space M. f x > g x} \<in> sets M"
```
```  2031 proof -
```
```  2032   define F where "F = (\<lambda>x. (f x, g x))"
```
```  2033   have * [measurable]: "F \<in> borel_measurable M" unfolding F_def by simp
```
```  2034
```
```  2035   have "{x \<in> space M. f x \<le> g x} = F-`{(x, y) | x y. x \<le> y} \<inter> space M" unfolding F_def by auto
```
```  2036   moreover have "{(x, y) | x y. x \<le> (y::'a)} \<in> sets borel" using closed_subdiagonal borel_closed by blast
```
```  2037   ultimately show "{x \<in> space M. f x \<le> g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
```
```  2038
```
```  2039   have "{x \<in> space M. f x < g x} = F-`{(x, y) | x y. x < y} \<inter> space M" unfolding F_def by auto
```
```  2040   moreover have "{(x, y) | x y. x < (y::'a)} \<in> sets borel" using open_subdiagonal borel_open by blast
```
```  2041   ultimately show "{x \<in> space M. f x < g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
```
```  2042
```
```  2043   have "{x \<in> space M. f x \<ge> g x} = F-`{(x, y) | x y. x \<ge> y} \<inter> space M" unfolding F_def by auto
```
```  2044   moreover have "{(x, y) | x y. x \<ge> (y::'a)} \<in> sets borel" using closed_superdiagonal borel_closed by blast
```
```  2045   ultimately show "{x \<in> space M. f x \<ge> g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
```
```  2046
```
```  2047   have "{x \<in> space M. f x > g x} = F-`{(x, y) | x y. x > y} \<inter> space M" unfolding F_def by auto
```
```  2048   moreover have "{(x, y) | x y. x > (y::'a)} \<in> sets borel" using open_superdiagonal borel_open by blast
```
```  2049   ultimately show "{x \<in> space M. f x > g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
```
```  2050 qed
```
```  2051
```
```  2052 lemma measurable_limit [measurable]:
```
```  2053   fixes f::"nat \<Rightarrow> 'a \<Rightarrow> 'b::first_countable_topology"
```
```  2054   assumes [measurable]: "\<And>n::nat. f n \<in> borel_measurable M"
```
```  2055   shows "Measurable.pred M (\<lambda>x. (\<lambda>n. f n x) \<longlonglongrightarrow> c)"
```
```  2056 proof -
```
```  2057   obtain A :: "nat \<Rightarrow> 'b set" where A:
```
```  2058     "\<And>i. open (A i)"
```
```  2059     "\<And>i. c \<in> A i"
```
```  2060     "\<And>S. open S \<Longrightarrow> c \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
```
```  2061   by (rule countable_basis_at_decseq) blast
```
```  2062
```
```  2063   have [measurable]: "\<And>N i. (f N)-`(A i) \<inter> space M \<in> sets M" using A(1) by auto
```
```  2064   then have mes: "(\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. (f N)-`(A i) \<inter> space M) \<in> sets M" by blast
```
```  2065
```
```  2066   have "(u \<longlonglongrightarrow> c) \<longleftrightarrow> (\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)" for u::"nat \<Rightarrow> 'b"
```
```  2067   proof
```
```  2068     assume "u \<longlonglongrightarrow> c"
```
```  2069     then have "eventually (\<lambda>n. u n \<in> A i) sequentially" for i using A(1)[of i] A(2)[of i]
```
```  2070       by (simp add: topological_tendstoD)
```
```  2071     then show "(\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)" by auto
```
```  2072   next
```
```  2073     assume H: "(\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)"
```
```  2074     show "(u \<longlonglongrightarrow> c)"
```
```  2075     proof (rule topological_tendstoI)
```
```  2076       fix S assume "open S" "c \<in> S"
```
```  2077       with A(3)[OF this] obtain i where "A i \<subseteq> S"
```
```  2078         using eventually_False_sequentially eventually_mono by blast
```
```  2079       moreover have "eventually (\<lambda>n. u n \<in> A i) sequentially" using H by simp
```
```  2080       ultimately show "\<forall>\<^sub>F n in sequentially. u n \<in> S"
```
```  2081         by (simp add: eventually_mono subset_eq)
```
```  2082     qed
```
```  2083   qed
```
```  2084   then have "{x. (\<lambda>n. f n x) \<longlonglongrightarrow> c} = (\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. (f N)-`(A i))"
```
```  2085     by (auto simp add: atLeast_def eventually_at_top_linorder)
```
```  2086   then have "{x \<in> space M. (\<lambda>n. f n x) \<longlonglongrightarrow> c} = (\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. (f N)-`(A i) \<inter> space M)"
```
```  2087     by auto
```
```  2088   then have "{x \<in> space M. (\<lambda>n. f n x) \<longlonglongrightarrow> c} \<in> sets M" using mes by simp
```
```  2089   then show ?thesis by auto
```
```  2090 qed
```
```  2091
```
```  2092 lemma measurable_limit2 [measurable]:
```
```  2093   fixes u::"nat \<Rightarrow> 'a \<Rightarrow> real"
```
```  2094   assumes [measurable]: "\<And>n. u n \<in> borel_measurable M" "v \<in> borel_measurable M"
```
```  2095   shows "Measurable.pred M (\<lambda>x. (\<lambda>n. u n x) \<longlonglongrightarrow> v x)"
```
```  2096 proof -
```
```  2097   define w where "w = (\<lambda>n x. u n x - v x)"
```
```  2098   have [measurable]: "w n \<in> borel_measurable M" for n unfolding w_def by auto
```
```  2099   have "((\<lambda>n. u n x) \<longlonglongrightarrow> v x) \<longleftrightarrow> ((\<lambda>n. w n x) \<longlonglongrightarrow> 0)" for x
```
```  2100     unfolding w_def using Lim_null by auto
```
```  2101   then show ?thesis using measurable_limit by auto
```
```  2102 qed
```
```  2103
```
```  2104 lemma measurable_P_restriction [measurable (raw)]:
```
```  2105   assumes [measurable]: "Measurable.pred M P" "A \<in> sets M"
```
```  2106   shows "{x \<in> A. P x} \<in> sets M"
```
```  2107 proof -
```
```  2108   have "A \<subseteq> space M" using sets.sets_into_space[OF assms(2)].
```
```  2109   then have "{x \<in> A. P x} = A \<inter> {x \<in> space M. P x}" by blast
```
```  2110   then show ?thesis by auto
```
```  2111 qed
```
```  2112
```
```  2113 lemma measurable_sum_nat [measurable (raw)]:
```
```  2114   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> nat"
```
```  2115   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> measurable M (count_space UNIV)"
```
```  2116   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> measurable M (count_space UNIV)"
```
```  2117 proof cases
```
```  2118   assume "finite S"
```
```  2119   then show ?thesis using assms by induct auto
```
```  2120 qed simp
```
```  2121
```
```  2122
```
```  2123 lemma measurable_abs_powr [measurable]:
```
```  2124   fixes p::real
```
```  2125   assumes [measurable]: "f \<in> borel_measurable M"
```
```  2126   shows "(\<lambda>x. \<bar>f x\<bar> powr p) \<in> borel_measurable M"
```
```  2127 unfolding powr_def by auto
```
```  2128
```
```  2129 text \<open>The next one is a variation around \verb+measurable_restrict_space+.\<close>
```
```  2130
```
```  2131 lemma measurable_restrict_space3:
```
```  2132   assumes "f \<in> measurable M N" and
```
```  2133           "f \<in> A \<rightarrow> B"
```
```  2134   shows "f \<in> measurable (restrict_space M A) (restrict_space N B)"
```
```  2135 proof -
```
```  2136   have "f \<in> measurable (restrict_space M A) N" using assms(1) measurable_restrict_space1 by auto
```
```  2137   then show ?thesis by (metis Int_iff funcsetI funcset_mem
```
```  2138       measurable_restrict_space2[of f, of "restrict_space M A", of B, of N] assms(2) space_restrict_space)
```
```  2139 qed
```
```  2140
```
```  2141 text \<open>The next one is a variation around \verb+measurable_piecewise_restrict+.\<close>
```
```  2142
```
```  2143 lemma measurable_piecewise_restrict2:
```
```  2144   assumes [measurable]: "\<And>n. A n \<in> sets M"
```
```  2145       and "space M = (\<Union>(n::nat). A n)"
```
```  2146           "\<And>n. \<exists>h \<in> measurable M N. (\<forall>x \<in> A n. f x = h x)"
```
```  2147   shows "f \<in> measurable M N"
```
```  2148 proof (rule measurableI)
```
```  2149   fix B assume [measurable]: "B \<in> sets N"
```
```  2150   {
```
```  2151     fix n::nat
```
```  2152     obtain h where [measurable]: "h \<in> measurable M N" and "\<forall>x \<in> A n. f x = h x" using assms(3) by blast
```
```  2153     then have *: "f-`B \<inter> A n = h-`B \<inter> A n" by auto
```
```  2154     have "h-`B \<inter> A n = h-`B \<inter> space M \<inter> A n" using assms(2) sets.sets_into_space by auto
```
```  2155     then have "h-`B \<inter> A n \<in> sets M" by simp
```
```  2156     then have "f-`B \<inter> A n \<in> sets M" using * by simp
```
```  2157   }
```
```  2158   then have "(\<Union>n. f-`B \<inter> A n) \<in> sets M" by measurable
```
```  2159   moreover have "f-`B \<inter> space M = (\<Union>n. f-`B \<inter> A n)" using assms(2) by blast
```
```  2160   ultimately show "f-`B \<inter> space M \<in> sets M" by simp
```
```  2161 next
```
```  2162   fix x assume "x \<in> space M"
```
```  2163   then obtain n where "x \<in> A n" using assms(2) by blast
```
```  2164   obtain h where [measurable]: "h \<in> measurable M N" and "\<forall>x \<in> A n. f x = h x" using assms(3) by blast
```
```  2165   then have "f x = h x" using \<open>x \<in> A n\<close> by blast
```
```  2166   moreover have "h x \<in> space N" by (metis measurable_space \<open>x \<in> space M\<close> \<open>h \<in> measurable M N\<close>)
```
```  2167   ultimately show "f x \<in> space N" by simp
```
```  2168 qed
```
```  2169
```
```  2170 end
```