src/HOL/Analysis/Borel_Space.thy
 author paulson Wed Sep 28 17:01:01 2016 +0100 (2016-09-28) changeset 63952 354808e9f44b parent 63627 6ddb43c6b711 child 64008 17a20ca86d62 permissions -rw-r--r--
new material connected with HOL Light measure theory, plus more rationalisation
```     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 borel_comp[measurable]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
```
```   349   unfolding Compl_eq_Diff_UNIV by simp
```
```   350
```
```   351 lemma borel_measurable_vimage:
```
```   352   fixes f :: "'a \<Rightarrow> 'x::t2_space"
```
```   353   assumes borel[measurable]: "f \<in> borel_measurable M"
```
```   354   shows "f -` {x} \<inter> space M \<in> sets M"
```
```   355   by simp
```
```   356
```
```   357 lemma borel_measurableI:
```
```   358   fixes f :: "'a \<Rightarrow> 'x::topological_space"
```
```   359   assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
```
```   360   shows "f \<in> borel_measurable M"
```
```   361   unfolding borel_def
```
```   362 proof (rule measurable_measure_of, simp_all)
```
```   363   fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"
```
```   364     using assms[of S] by simp
```
```   365 qed
```
```   366
```
```   367 lemma borel_measurable_const:
```
```   368   "(\<lambda>x. c) \<in> borel_measurable M"
```
```   369   by auto
```
```   370
```
```   371 lemma borel_measurable_indicator:
```
```   372   assumes A: "A \<in> sets M"
```
```   373   shows "indicator A \<in> borel_measurable M"
```
```   374   unfolding indicator_def [abs_def] using A
```
```   375   by (auto intro!: measurable_If_set)
```
```   376
```
```   377 lemma borel_measurable_count_space[measurable (raw)]:
```
```   378   "f \<in> borel_measurable (count_space S)"
```
```   379   unfolding measurable_def by auto
```
```   380
```
```   381 lemma borel_measurable_indicator'[measurable (raw)]:
```
```   382   assumes [measurable]: "{x\<in>space M. f x \<in> A x} \<in> sets M"
```
```   383   shows "(\<lambda>x. indicator (A x) (f x)) \<in> borel_measurable M"
```
```   384   unfolding indicator_def[abs_def]
```
```   385   by (auto intro!: measurable_If)
```
```   386
```
```   387 lemma borel_measurable_indicator_iff:
```
```   388   "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
```
```   389     (is "?I \<in> borel_measurable M \<longleftrightarrow> _")
```
```   390 proof
```
```   391   assume "?I \<in> borel_measurable M"
```
```   392   then have "?I -` {1} \<inter> space M \<in> sets M"
```
```   393     unfolding measurable_def by auto
```
```   394   also have "?I -` {1} \<inter> space M = A \<inter> space M"
```
```   395     unfolding indicator_def [abs_def] by auto
```
```   396   finally show "A \<inter> space M \<in> sets M" .
```
```   397 next
```
```   398   assume "A \<inter> space M \<in> sets M"
```
```   399   moreover have "?I \<in> borel_measurable M \<longleftrightarrow>
```
```   400     (indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"
```
```   401     by (intro measurable_cong) (auto simp: indicator_def)
```
```   402   ultimately show "?I \<in> borel_measurable M" by auto
```
```   403 qed
```
```   404
```
```   405 lemma borel_measurable_subalgebra:
```
```   406   assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
```
```   407   shows "f \<in> borel_measurable M"
```
```   408   using assms unfolding measurable_def by auto
```
```   409
```
```   410 lemma borel_measurable_restrict_space_iff_ereal:
```
```   411   fixes f :: "'a \<Rightarrow> ereal"
```
```   412   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
```
```   413   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
```
```   414     (\<lambda>x. f x * indicator \<Omega> x) \<in> borel_measurable M"
```
```   415   by (subst measurable_restrict_space_iff)
```
```   416      (auto simp: indicator_def if_distrib[where f="\<lambda>x. a * x" for a] cong del: if_weak_cong)
```
```   417
```
```   418 lemma borel_measurable_restrict_space_iff_ennreal:
```
```   419   fixes f :: "'a \<Rightarrow> ennreal"
```
```   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:
```
```   427   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   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. indicator \<Omega> x *\<^sub>R f x) \<in> borel_measurable M"
```
```   431   by (subst measurable_restrict_space_iff)
```
```   432      (auto simp: indicator_def if_distrib[where f="\<lambda>x. x *\<^sub>R a" for a] ac_simps
```
```   433        cong del: if_weak_cong)
```
```   434
```
```   435 lemma cbox_borel[measurable]: "cbox a b \<in> sets borel"
```
```   436   by (auto intro: borel_closed)
```
```   437
```
```   438 lemma box_borel[measurable]: "box a b \<in> sets borel"
```
```   439   by (auto intro: borel_open)
```
```   440
```
```   441 lemma borel_compact: "compact (A::'a::t2_space set) \<Longrightarrow> A \<in> sets borel"
```
```   442   by (auto intro: borel_closed dest!: compact_imp_closed)
```
```   443
```
```   444 lemma borel_sigma_sets_subset:
```
```   445   "A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel"
```
```   446   using sets.sigma_sets_subset[of A borel] by simp
```
```   447
```
```   448 lemma borel_eq_sigmaI1:
```
```   449   fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
```
```   450   assumes borel_eq: "borel = sigma UNIV X"
```
```   451   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (F ` A))"
```
```   452   assumes F: "\<And>i. i \<in> A \<Longrightarrow> F i \<in> sets borel"
```
```   453   shows "borel = sigma UNIV (F ` A)"
```
```   454   unfolding borel_def
```
```   455 proof (intro sigma_eqI antisym)
```
```   456   have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel"
```
```   457     unfolding borel_def by simp
```
```   458   also have "\<dots> = sigma_sets UNIV X"
```
```   459     unfolding borel_eq by simp
```
```   460   also have "\<dots> \<subseteq> sigma_sets UNIV (F`A)"
```
```   461     using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto
```
```   462   finally show "sigma_sets UNIV {S. open S} \<subseteq> sigma_sets UNIV (F`A)" .
```
```   463   show "sigma_sets UNIV (F`A) \<subseteq> sigma_sets UNIV {S. open S}"
```
```   464     unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto
```
```   465 qed auto
```
```   466
```
```   467 lemma borel_eq_sigmaI2:
```
```   468   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set"
```
```   469     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
```
```   470   assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`B)"
```
```   471   assumes X: "\<And>i j. (i, j) \<in> B \<Longrightarrow> G i j \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"
```
```   472   assumes F: "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel"
```
```   473   shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"
```
```   474   using assms
```
```   475   by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` B" and F="(\<lambda>(i, j). F i j)"]) auto
```
```   476
```
```   477 lemma borel_eq_sigmaI3:
```
```   478   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
```
```   479   assumes borel_eq: "borel = sigma UNIV X"
```
```   480   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"
```
```   481   assumes F: "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel"
```
```   482   shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"
```
```   483   using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto
```
```   484
```
```   485 lemma borel_eq_sigmaI4:
```
```   486   fixes F :: "'i \<Rightarrow> 'a::topological_space set"
```
```   487     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
```
```   488   assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`A)"
```
```   489   assumes X: "\<And>i j. (i, j) \<in> A \<Longrightarrow> G i j \<in> sets (sigma UNIV (range F))"
```
```   490   assumes F: "\<And>i. F i \<in> sets borel"
```
```   491   shows "borel = sigma UNIV (range F)"
```
```   492   using assms by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` A" and F=F]) auto
```
```   493
```
```   494 lemma borel_eq_sigmaI5:
```
```   495   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set"
```
```   496   assumes borel_eq: "borel = sigma UNIV (range G)"
```
```   497   assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
```
```   498   assumes F: "\<And>i j. F i j \<in> sets borel"
```
```   499   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
```
```   500   using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto
```
```   501
```
```   502 lemma second_countable_borel_measurable:
```
```   503   fixes X :: "'a::second_countable_topology set set"
```
```   504   assumes eq: "open = generate_topology X"
```
```   505   shows "borel = sigma UNIV X"
```
```   506   unfolding borel_def
```
```   507 proof (intro sigma_eqI sigma_sets_eqI)
```
```   508   interpret X: sigma_algebra UNIV "sigma_sets UNIV X"
```
```   509     by (rule sigma_algebra_sigma_sets) simp
```
```   510
```
```   511   fix S :: "'a set" assume "S \<in> Collect open"
```
```   512   then have "generate_topology X S"
```
```   513     by (auto simp: eq)
```
```   514   then show "S \<in> sigma_sets UNIV X"
```
```   515   proof induction
```
```   516     case (UN K)
```
```   517     then have K: "\<And>k. k \<in> K \<Longrightarrow> open k"
```
```   518       unfolding eq by auto
```
```   519     from ex_countable_basis obtain B :: "'a set set" where
```
```   520       B:  "\<And>b. b \<in> B \<Longrightarrow> open b" "\<And>X. open X \<Longrightarrow> \<exists>b\<subseteq>B. (\<Union>b) = X" and "countable B"
```
```   521       by (auto simp: topological_basis_def)
```
```   522     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"
```
```   523       by metis
```
```   524     define U where "U = (\<Union>k\<in>K. m k)"
```
```   525     with m have "countable U"
```
```   526       by (intro countable_subset[OF _ \<open>countable B\<close>]) auto
```
```   527     have "\<Union>U = (\<Union>A\<in>U. A)" by simp
```
```   528     also have "\<dots> = \<Union>K"
```
```   529       unfolding U_def UN_simps by (simp add: m)
```
```   530     finally have "\<Union>U = \<Union>K" .
```
```   531
```
```   532     have "\<forall>b\<in>U. \<exists>k\<in>K. b \<subseteq> k"
```
```   533       using m by (auto simp: U_def)
```
```   534     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"
```
```   535       by metis
```
```   536     then have "(\<Union>b\<in>U. u b) \<subseteq> \<Union>K" "\<Union>U \<subseteq> (\<Union>b\<in>U. u b)"
```
```   537       by auto
```
```   538     then have "\<Union>K = (\<Union>b\<in>U. u b)"
```
```   539       unfolding \<open>\<Union>U = \<Union>K\<close> by auto
```
```   540     also have "\<dots> \<in> sigma_sets UNIV X"
```
```   541       using u UN by (intro X.countable_UN' \<open>countable U\<close>) auto
```
```   542     finally show "\<Union>K \<in> sigma_sets UNIV X" .
```
```   543   qed auto
```
```   544 qed (auto simp: eq intro: generate_topology.Basis)
```
```   545
```
```   546 lemma borel_eq_closed: "borel = sigma UNIV (Collect closed)"
```
```   547   unfolding borel_def
```
```   548 proof (intro sigma_eqI sigma_sets_eqI, safe)
```
```   549   fix x :: "'a set" assume "open x"
```
```   550   hence "x = UNIV - (UNIV - x)" by auto
```
```   551   also have "\<dots> \<in> sigma_sets UNIV (Collect closed)"
```
```   552     by (force intro: sigma_sets.Compl simp: \<open>open x\<close>)
```
```   553   finally show "x \<in> sigma_sets UNIV (Collect closed)" by simp
```
```   554 next
```
```   555   fix x :: "'a set" assume "closed x"
```
```   556   hence "x = UNIV - (UNIV - x)" by auto
```
```   557   also have "\<dots> \<in> sigma_sets UNIV (Collect open)"
```
```   558     by (force intro: sigma_sets.Compl simp: \<open>closed x\<close>)
```
```   559   finally show "x \<in> sigma_sets UNIV (Collect open)" by simp
```
```   560 qed simp_all
```
```   561
```
```   562 lemma borel_eq_countable_basis:
```
```   563   fixes B::"'a::topological_space set set"
```
```   564   assumes "countable B"
```
```   565   assumes "topological_basis B"
```
```   566   shows "borel = sigma UNIV B"
```
```   567   unfolding borel_def
```
```   568 proof (intro sigma_eqI sigma_sets_eqI, safe)
```
```   569   interpret countable_basis using assms by unfold_locales
```
```   570   fix X::"'a set" assume "open X"
```
```   571   from open_countable_basisE[OF this] guess B' . note B' = this
```
```   572   then show "X \<in> sigma_sets UNIV B"
```
```   573     by (blast intro: sigma_sets_UNION \<open>countable B\<close> countable_subset)
```
```   574 next
```
```   575   fix b assume "b \<in> B"
```
```   576   hence "open b" by (rule topological_basis_open[OF assms(2)])
```
```   577   thus "b \<in> sigma_sets UNIV (Collect open)" by auto
```
```   578 qed simp_all
```
```   579
```
```   580 lemma borel_measurable_continuous_on_restrict:
```
```   581   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
```
```   582   assumes f: "continuous_on A f"
```
```   583   shows "f \<in> borel_measurable (restrict_space borel A)"
```
```   584 proof (rule borel_measurableI)
```
```   585   fix S :: "'b set" assume "open S"
```
```   586   with f obtain T where "f -` S \<inter> A = T \<inter> A" "open T"
```
```   587     by (metis continuous_on_open_invariant)
```
```   588   then show "f -` S \<inter> space (restrict_space borel A) \<in> sets (restrict_space borel A)"
```
```   589     by (force simp add: sets_restrict_space space_restrict_space)
```
```   590 qed
```
```   591
```
```   592 lemma borel_measurable_continuous_on1: "continuous_on UNIV f \<Longrightarrow> f \<in> borel_measurable borel"
```
```   593   by (drule borel_measurable_continuous_on_restrict) simp
```
```   594
```
```   595 lemma borel_measurable_continuous_on_if:
```
```   596   "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> continuous_on (- A) g \<Longrightarrow>
```
```   597     (\<lambda>x. if x \<in> A then f x else g x) \<in> borel_measurable borel"
```
```   598   by (auto simp add: measurable_If_restrict_space_iff Collect_neg_eq
```
```   599            intro!: borel_measurable_continuous_on_restrict)
```
```   600
```
```   601 lemma borel_measurable_continuous_countable_exceptions:
```
```   602   fixes f :: "'a::t1_space \<Rightarrow> 'b::topological_space"
```
```   603   assumes X: "countable X"
```
```   604   assumes "continuous_on (- X) f"
```
```   605   shows "f \<in> borel_measurable borel"
```
```   606 proof (rule measurable_discrete_difference[OF _ X])
```
```   607   have "X \<in> sets borel"
```
```   608     by (rule sets.countable[OF _ X]) auto
```
```   609   then show "(\<lambda>x. if x \<in> X then undefined else f x) \<in> borel_measurable borel"
```
```   610     by (intro borel_measurable_continuous_on_if assms continuous_intros)
```
```   611 qed auto
```
```   612
```
```   613 lemma borel_measurable_continuous_on:
```
```   614   assumes f: "continuous_on UNIV f" and g: "g \<in> borel_measurable M"
```
```   615   shows "(\<lambda>x. f (g x)) \<in> borel_measurable M"
```
```   616   using measurable_comp[OF g borel_measurable_continuous_on1[OF f]] by (simp add: comp_def)
```
```   617
```
```   618 lemma borel_measurable_continuous_on_indicator:
```
```   619   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   620   shows "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable borel"
```
```   621   by (subst borel_measurable_restrict_space_iff[symmetric])
```
```   622      (auto intro: borel_measurable_continuous_on_restrict)
```
```   623
```
```   624 lemma borel_measurable_Pair[measurable (raw)]:
```
```   625   fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"
```
```   626   assumes f[measurable]: "f \<in> borel_measurable M"
```
```   627   assumes g[measurable]: "g \<in> borel_measurable M"
```
```   628   shows "(\<lambda>x. (f x, g x)) \<in> borel_measurable M"
```
```   629 proof (subst borel_eq_countable_basis)
```
```   630   let ?B = "SOME B::'b set set. countable B \<and> topological_basis B"
```
```   631   let ?C = "SOME B::'c set set. countable B \<and> topological_basis B"
```
```   632   let ?P = "(\<lambda>(b, c). b \<times> c) ` (?B \<times> ?C)"
```
```   633   show "countable ?P" "topological_basis ?P"
```
```   634     by (auto intro!: countable_basis topological_basis_prod is_basis)
```
```   635
```
```   636   show "(\<lambda>x. (f x, g x)) \<in> measurable M (sigma UNIV ?P)"
```
```   637   proof (rule measurable_measure_of)
```
```   638     fix S assume "S \<in> ?P"
```
```   639     then obtain b c where "b \<in> ?B" "c \<in> ?C" and S: "S = b \<times> c" by auto
```
```   640     then have borel: "open b" "open c"
```
```   641       by (auto intro: is_basis topological_basis_open)
```
```   642     have "(\<lambda>x. (f x, g x)) -` S \<inter> space M = (f -` b \<inter> space M) \<inter> (g -` c \<inter> space M)"
```
```   643       unfolding S by auto
```
```   644     also have "\<dots> \<in> sets M"
```
```   645       using borel by simp
```
```   646     finally show "(\<lambda>x. (f x, g x)) -` S \<inter> space M \<in> sets M" .
```
```   647   qed auto
```
```   648 qed
```
```   649
```
```   650 lemma borel_measurable_continuous_Pair:
```
```   651   fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"
```
```   652   assumes [measurable]: "f \<in> borel_measurable M"
```
```   653   assumes [measurable]: "g \<in> borel_measurable M"
```
```   654   assumes H: "continuous_on UNIV (\<lambda>x. H (fst x) (snd x))"
```
```   655   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
```
```   656 proof -
```
```   657   have eq: "(\<lambda>x. H (f x) (g x)) = (\<lambda>x. (\<lambda>x. H (fst x) (snd x)) (f x, g x))" by auto
```
```   658   show ?thesis
```
```   659     unfolding eq by (rule borel_measurable_continuous_on[OF H]) auto
```
```   660 qed
```
```   661
```
```   662 subsection \<open>Borel spaces on order topologies\<close>
```
```   663
```
```   664 lemma [measurable]:
```
```   665   fixes a b :: "'a::linorder_topology"
```
```   666   shows lessThan_borel: "{..< a} \<in> sets borel"
```
```   667     and greaterThan_borel: "{a <..} \<in> sets borel"
```
```   668     and greaterThanLessThan_borel: "{a<..<b} \<in> sets borel"
```
```   669     and atMost_borel: "{..a} \<in> sets borel"
```
```   670     and atLeast_borel: "{a..} \<in> sets borel"
```
```   671     and atLeastAtMost_borel: "{a..b} \<in> sets borel"
```
```   672     and greaterThanAtMost_borel: "{a<..b} \<in> sets borel"
```
```   673     and atLeastLessThan_borel: "{a..<b} \<in> sets borel"
```
```   674   unfolding greaterThanAtMost_def atLeastLessThan_def
```
```   675   by (blast intro: borel_open borel_closed open_lessThan open_greaterThan open_greaterThanLessThan
```
```   676                    closed_atMost closed_atLeast closed_atLeastAtMost)+
```
```   677
```
```   678 lemma borel_Iio:
```
```   679   "borel = sigma UNIV (range lessThan :: 'a::{linorder_topology, second_countable_topology} set set)"
```
```   680   unfolding second_countable_borel_measurable[OF open_generated_order]
```
```   681 proof (intro sigma_eqI sigma_sets_eqI)
```
```   682   from countable_dense_setE guess D :: "'a set" . note D = this
```
```   683
```
```   684   interpret L: sigma_algebra UNIV "sigma_sets UNIV (range lessThan)"
```
```   685     by (rule sigma_algebra_sigma_sets) simp
```
```   686
```
```   687   fix A :: "'a set" assume "A \<in> range lessThan \<union> range greaterThan"
```
```   688   then obtain y where "A = {y <..} \<or> A = {..< y}"
```
```   689     by blast
```
```   690   then show "A \<in> sigma_sets UNIV (range lessThan)"
```
```   691   proof
```
```   692     assume A: "A = {y <..}"
```
```   693     show ?thesis
```
```   694     proof cases
```
```   695       assume "\<forall>x>y. \<exists>d. y < d \<and> d < x"
```
```   696       with D(2)[of "{y <..< x}" for x] have "\<forall>x>y. \<exists>d\<in>D. y < d \<and> d < x"
```
```   697         by (auto simp: set_eq_iff)
```
```   698       then have "A = UNIV - (\<Inter>d\<in>{d\<in>D. y < d}. {..< d})"
```
```   699         by (auto simp: A) (metis less_asym)
```
```   700       also have "\<dots> \<in> sigma_sets UNIV (range lessThan)"
```
```   701         using D(1) by (intro L.Diff L.top L.countable_INT'') auto
```
```   702       finally show ?thesis .
```
```   703     next
```
```   704       assume "\<not> (\<forall>x>y. \<exists>d. y < d \<and> d < x)"
```
```   705       then obtain x where "y < x"  "\<And>d. y < d \<Longrightarrow> \<not> d < x"
```
```   706         by auto
```
```   707       then have "A = UNIV - {..< x}"
```
```   708         unfolding A by (auto simp: not_less[symmetric])
```
```   709       also have "\<dots> \<in> sigma_sets UNIV (range lessThan)"
```
```   710         by auto
```
```   711       finally show ?thesis .
```
```   712     qed
```
```   713   qed auto
```
```   714 qed auto
```
```   715
```
```   716 lemma borel_Ioi:
```
```   717   "borel = sigma UNIV (range greaterThan :: 'a::{linorder_topology, second_countable_topology} set set)"
```
```   718   unfolding second_countable_borel_measurable[OF open_generated_order]
```
```   719 proof (intro sigma_eqI sigma_sets_eqI)
```
```   720   from countable_dense_setE guess D :: "'a set" . note D = this
```
```   721
```
```   722   interpret L: sigma_algebra UNIV "sigma_sets UNIV (range greaterThan)"
```
```   723     by (rule sigma_algebra_sigma_sets) simp
```
```   724
```
```   725   fix A :: "'a set" assume "A \<in> range lessThan \<union> range greaterThan"
```
```   726   then obtain y where "A = {y <..} \<or> A = {..< y}"
```
```   727     by blast
```
```   728   then show "A \<in> sigma_sets UNIV (range greaterThan)"
```
```   729   proof
```
```   730     assume A: "A = {..< y}"
```
```   731     show ?thesis
```
```   732     proof cases
```
```   733       assume "\<forall>x<y. \<exists>d. x < d \<and> d < y"
```
```   734       with D(2)[of "{x <..< y}" for x] have "\<forall>x<y. \<exists>d\<in>D. x < d \<and> d < y"
```
```   735         by (auto simp: set_eq_iff)
```
```   736       then have "A = UNIV - (\<Inter>d\<in>{d\<in>D. d < y}. {d <..})"
```
```   737         by (auto simp: A) (metis less_asym)
```
```   738       also have "\<dots> \<in> sigma_sets UNIV (range greaterThan)"
```
```   739         using D(1) by (intro L.Diff L.top L.countable_INT'') auto
```
```   740       finally show ?thesis .
```
```   741     next
```
```   742       assume "\<not> (\<forall>x<y. \<exists>d. x < d \<and> d < y)"
```
```   743       then obtain x where "x < y"  "\<And>d. y > d \<Longrightarrow> x \<ge> d"
```
```   744         by (auto simp: not_less[symmetric])
```
```   745       then have "A = UNIV - {x <..}"
```
```   746         unfolding A Compl_eq_Diff_UNIV[symmetric] by auto
```
```   747       also have "\<dots> \<in> sigma_sets UNIV (range greaterThan)"
```
```   748         by auto
```
```   749       finally show ?thesis .
```
```   750     qed
```
```   751   qed auto
```
```   752 qed auto
```
```   753
```
```   754 lemma borel_measurableI_less:
```
```   755   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
```
```   756   shows "(\<And>y. {x\<in>space M. f x < y} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
```
```   757   unfolding borel_Iio
```
```   758   by (rule measurable_measure_of) (auto simp: Int_def conj_commute)
```
```   759
```
```   760 lemma borel_measurableI_greater:
```
```   761   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
```
```   762   shows "(\<And>y. {x\<in>space M. y < f x} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
```
```   763   unfolding borel_Ioi
```
```   764   by (rule measurable_measure_of) (auto simp: Int_def conj_commute)
```
```   765
```
```   766 lemma borel_measurableI_le:
```
```   767   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
```
```   768   shows "(\<And>y. {x\<in>space M. f x \<le> y} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
```
```   769   by (rule borel_measurableI_greater) (auto simp: not_le[symmetric])
```
```   770
```
```   771 lemma borel_measurableI_ge:
```
```   772   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
```
```   773   shows "(\<And>y. {x\<in>space M. y \<le> f x} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
```
```   774   by (rule borel_measurableI_less) (auto simp: not_le[symmetric])
```
```   775
```
```   776 lemma borel_measurable_less[measurable]:
```
```   777   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
```
```   778   assumes "f \<in> borel_measurable M"
```
```   779   assumes "g \<in> borel_measurable M"
```
```   780   shows "{w \<in> space M. f w < g w} \<in> sets M"
```
```   781 proof -
```
```   782   have "{w \<in> space M. f w < g w} = (\<lambda>x. (f x, g x)) -` {x. fst x < snd x} \<inter> space M"
```
```   783     by auto
```
```   784   also have "\<dots> \<in> sets M"
```
```   785     by (intro measurable_sets[OF borel_measurable_Pair borel_open, OF assms open_Collect_less]
```
```   786               continuous_intros)
```
```   787   finally show ?thesis .
```
```   788 qed
```
```   789
```
```   790 lemma
```
```   791   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
```
```   792   assumes f[measurable]: "f \<in> borel_measurable M"
```
```   793   assumes g[measurable]: "g \<in> borel_measurable M"
```
```   794   shows borel_measurable_le[measurable]: "{w \<in> space M. f w \<le> g w} \<in> sets M"
```
```   795     and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M"
```
```   796     and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
```
```   797   unfolding eq_iff not_less[symmetric]
```
```   798   by measurable
```
```   799
```
```   800 lemma borel_measurable_SUP[measurable (raw)]:
```
```   801   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}"
```
```   802   assumes [simp]: "countable I"
```
```   803   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```   804   shows "(\<lambda>x. SUP i:I. F i x) \<in> borel_measurable M"
```
```   805   by (rule borel_measurableI_greater) (simp add: less_SUP_iff)
```
```   806
```
```   807 lemma borel_measurable_INF[measurable (raw)]:
```
```   808   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}"
```
```   809   assumes [simp]: "countable I"
```
```   810   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```   811   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
```
```   812   by (rule borel_measurableI_less) (simp add: INF_less_iff)
```
```   813
```
```   814 lemma borel_measurable_cSUP[measurable (raw)]:
```
```   815   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology, second_countable_topology}"
```
```   816   assumes [simp]: "countable I"
```
```   817   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```   818   assumes bdd: "\<And>x. x \<in> space M \<Longrightarrow> bdd_above ((\<lambda>i. F i x) ` I)"
```
```   819   shows "(\<lambda>x. SUP i:I. F i x) \<in> borel_measurable M"
```
```   820 proof cases
```
```   821   assume "I = {}" then show ?thesis
```
```   822     unfolding \<open>I = {}\<close> image_empty by simp
```
```   823 next
```
```   824   assume "I \<noteq> {}"
```
```   825   show ?thesis
```
```   826   proof (rule borel_measurableI_le)
```
```   827     fix y
```
```   828     have "{x \<in> space M. \<forall>i\<in>I. F i x \<le> y} \<in> sets M"
```
```   829       by measurable
```
```   830     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}"
```
```   831       by (simp add: cSUP_le_iff \<open>I \<noteq> {}\<close> bdd cong: conj_cong)
```
```   832     finally show "{x \<in> space M. (SUP i:I. F i x) \<le>  y} \<in> sets M"  .
```
```   833   qed
```
```   834 qed
```
```   835
```
```   836 lemma borel_measurable_cINF[measurable (raw)]:
```
```   837   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology, second_countable_topology}"
```
```   838   assumes [simp]: "countable I"
```
```   839   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```   840   assumes bdd: "\<And>x. x \<in> space M \<Longrightarrow> bdd_below ((\<lambda>i. F i x) ` I)"
```
```   841   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
```
```   842 proof cases
```
```   843   assume "I = {}" then show ?thesis
```
```   844     unfolding \<open>I = {}\<close> image_empty by simp
```
```   845 next
```
```   846   assume "I \<noteq> {}"
```
```   847   show ?thesis
```
```   848   proof (rule borel_measurableI_ge)
```
```   849     fix y
```
```   850     have "{x \<in> space M. \<forall>i\<in>I. y \<le> F i x} \<in> sets M"
```
```   851       by measurable
```
```   852     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)}"
```
```   853       by (simp add: le_cINF_iff \<open>I \<noteq> {}\<close> bdd cong: conj_cong)
```
```   854     finally show "{x \<in> space M. y \<le> (INF i:I. F i x)} \<in> sets M"  .
```
```   855   qed
```
```   856 qed
```
```   857
```
```   858 lemma borel_measurable_lfp[consumes 1, case_names continuity step]:
```
```   859   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})"
```
```   860   assumes "sup_continuous F"
```
```   861   assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M"
```
```   862   shows "lfp F \<in> borel_measurable M"
```
```   863 proof -
```
```   864   { fix i have "((F ^^ i) bot) \<in> borel_measurable M"
```
```   865       by (induct i) (auto intro!: *) }
```
```   866   then have "(\<lambda>x. SUP i. (F ^^ i) bot x) \<in> borel_measurable M"
```
```   867     by measurable
```
```   868   also have "(\<lambda>x. SUP i. (F ^^ i) bot x) = (SUP i. (F ^^ i) bot)"
```
```   869     by auto
```
```   870   also have "(SUP i. (F ^^ i) bot) = lfp F"
```
```   871     by (rule sup_continuous_lfp[symmetric]) fact
```
```   872   finally show ?thesis .
```
```   873 qed
```
```   874
```
```   875 lemma borel_measurable_gfp[consumes 1, case_names continuity step]:
```
```   876   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})"
```
```   877   assumes "inf_continuous F"
```
```   878   assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M"
```
```   879   shows "gfp F \<in> borel_measurable M"
```
```   880 proof -
```
```   881   { fix i have "((F ^^ i) top) \<in> borel_measurable M"
```
```   882       by (induct i) (auto intro!: * simp: bot_fun_def) }
```
```   883   then have "(\<lambda>x. INF i. (F ^^ i) top x) \<in> borel_measurable M"
```
```   884     by measurable
```
```   885   also have "(\<lambda>x. INF i. (F ^^ i) top x) = (INF i. (F ^^ i) top)"
```
```   886     by auto
```
```   887   also have "\<dots> = gfp F"
```
```   888     by (rule inf_continuous_gfp[symmetric]) fact
```
```   889   finally show ?thesis .
```
```   890 qed
```
```   891
```
```   892 lemma borel_measurable_max[measurable (raw)]:
```
```   893   "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"
```
```   894   by (rule borel_measurableI_less) simp
```
```   895
```
```   896 lemma borel_measurable_min[measurable (raw)]:
```
```   897   "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"
```
```   898   by (rule borel_measurableI_greater) simp
```
```   899
```
```   900 lemma borel_measurable_Min[measurable (raw)]:
```
```   901   "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"
```
```   902 proof (induct I rule: finite_induct)
```
```   903   case (insert i I) then show ?case
```
```   904     by (cases "I = {}") auto
```
```   905 qed auto
```
```   906
```
```   907 lemma borel_measurable_Max[measurable (raw)]:
```
```   908   "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"
```
```   909 proof (induct I rule: finite_induct)
```
```   910   case (insert i I) then show ?case
```
```   911     by (cases "I = {}") auto
```
```   912 qed auto
```
```   913
```
```   914 lemma borel_measurable_sup[measurable (raw)]:
```
```   915   "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"
```
```   916   unfolding sup_max by measurable
```
```   917
```
```   918 lemma borel_measurable_inf[measurable (raw)]:
```
```   919   "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"
```
```   920   unfolding inf_min by measurable
```
```   921
```
```   922 lemma [measurable (raw)]:
```
```   923   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   924   assumes "\<And>i. f i \<in> borel_measurable M"
```
```   925   shows borel_measurable_liminf: "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```   926     and borel_measurable_limsup: "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```   927   unfolding liminf_SUP_INF limsup_INF_SUP using assms by auto
```
```   928
```
```   929 lemma measurable_convergent[measurable (raw)]:
```
```   930   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   931   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```   932   shows "Measurable.pred M (\<lambda>x. convergent (\<lambda>i. f i x))"
```
```   933   unfolding convergent_ereal by measurable
```
```   934
```
```   935 lemma sets_Collect_convergent[measurable]:
```
```   936   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   937   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```   938   shows "{x\<in>space M. convergent (\<lambda>i. f i x)} \<in> sets M"
```
```   939   by measurable
```
```   940
```
```   941 lemma borel_measurable_lim[measurable (raw)]:
```
```   942   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   943   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```   944   shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```   945 proof -
```
```   946   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))"
```
```   947     by (simp add: lim_def convergent_def convergent_limsup_cl)
```
```   948   then show ?thesis
```
```   949     by simp
```
```   950 qed
```
```   951
```
```   952 lemma borel_measurable_LIMSEQ_order:
```
```   953   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   954   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
```
```   955   and u: "\<And>i. u i \<in> borel_measurable M"
```
```   956   shows "u' \<in> borel_measurable M"
```
```   957 proof -
```
```   958   have "\<And>x. x \<in> space M \<Longrightarrow> u' x = liminf (\<lambda>n. u n x)"
```
```   959     using u' by (simp add: lim_imp_Liminf[symmetric])
```
```   960   with u show ?thesis by (simp cong: measurable_cong)
```
```   961 qed
```
```   962
```
```   963 subsection \<open>Borel spaces on topological monoids\<close>
```
```   964
```
```   965 lemma borel_measurable_add[measurable (raw)]:
```
```   966   fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, topological_monoid_add}"
```
```   967   assumes f: "f \<in> borel_measurable M"
```
```   968   assumes g: "g \<in> borel_measurable M"
```
```   969   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```   970   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```   971
```
```   972 lemma borel_measurable_setsum[measurable (raw)]:
```
```   973   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, topological_comm_monoid_add}"
```
```   974   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   975   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```   976 proof cases
```
```   977   assume "finite S"
```
```   978   thus ?thesis using assms by induct auto
```
```   979 qed simp
```
```   980
```
```   981 lemma borel_measurable_suminf_order[measurable (raw)]:
```
```   982   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology, topological_comm_monoid_add}"
```
```   983   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```   984   shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```   985   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
```
```   986
```
```   987 subsection \<open>Borel spaces on Euclidean spaces\<close>
```
```   988
```
```   989 lemma borel_measurable_inner[measurable (raw)]:
```
```   990   fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_inner}"
```
```   991   assumes "f \<in> borel_measurable M"
```
```   992   assumes "g \<in> borel_measurable M"
```
```   993   shows "(\<lambda>x. f x \<bullet> g x) \<in> borel_measurable M"
```
```   994   using assms
```
```   995   by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```   996
```
```   997 notation
```
```   998   eucl_less (infix "<e" 50)
```
```   999
```
```  1000 lemma box_oc: "{x. a <e x \<and> x \<le> b} = {x. a <e x} \<inter> {..b}"
```
```  1001   and box_co: "{x. a \<le> x \<and> x <e b} = {a..} \<inter> {x. x <e b}"
```
```  1002   by auto
```
```  1003
```
```  1004 lemma eucl_ivals[measurable]:
```
```  1005   fixes a b :: "'a::ordered_euclidean_space"
```
```  1006   shows "{x. x <e a} \<in> sets borel"
```
```  1007     and "{x. a <e x} \<in> sets borel"
```
```  1008     and "{..a} \<in> sets borel"
```
```  1009     and "{a..} \<in> sets borel"
```
```  1010     and "{a..b} \<in> sets borel"
```
```  1011     and  "{x. a <e x \<and> x \<le> b} \<in> sets borel"
```
```  1012     and "{x. a \<le> x \<and>  x <e b} \<in> sets borel"
```
```  1013   unfolding box_oc box_co
```
```  1014   by (auto intro: borel_open borel_closed)
```
```  1015
```
```  1016 lemma
```
```  1017   fixes i :: "'a::{second_countable_topology, real_inner}"
```
```  1018   shows hafspace_less_borel: "{x. a < x \<bullet> i} \<in> sets borel"
```
```  1019     and hafspace_greater_borel: "{x. x \<bullet> i < a} \<in> sets borel"
```
```  1020     and hafspace_less_eq_borel: "{x. a \<le> x \<bullet> i} \<in> sets borel"
```
```  1021     and hafspace_greater_eq_borel: "{x. x \<bullet> i \<le> a} \<in> sets borel"
```
```  1022   by simp_all
```
```  1023
```
```  1024 lemma borel_eq_box:
```
```  1025   "borel = sigma UNIV (range (\<lambda> (a, b). box a b :: 'a :: euclidean_space set))"
```
```  1026     (is "_ = ?SIGMA")
```
```  1027 proof (rule borel_eq_sigmaI1[OF borel_def])
```
```  1028   fix M :: "'a set" assume "M \<in> {S. open S}"
```
```  1029   then have "open M" by simp
```
```  1030   show "M \<in> ?SIGMA"
```
```  1031     apply (subst open_UNION_box[OF \<open>open M\<close>])
```
```  1032     apply (safe intro!: sets.countable_UN' countable_PiE countable_Collect)
```
```  1033     apply (auto intro: countable_rat)
```
```  1034     done
```
```  1035 qed (auto simp: box_def)
```
```  1036
```
```  1037 lemma halfspace_gt_in_halfspace:
```
```  1038   assumes i: "i \<in> A"
```
```  1039   shows "{x::'a. a < x \<bullet> i} \<in>
```
```  1040     sigma_sets UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> A))"
```
```  1041   (is "?set \<in> ?SIGMA")
```
```  1042 proof -
```
```  1043   interpret sigma_algebra UNIV ?SIGMA
```
```  1044     by (intro sigma_algebra_sigma_sets) simp_all
```
```  1045   have *: "?set = (\<Union>n. UNIV - {x::'a. x \<bullet> i < a + 1 / real (Suc n)})"
```
```  1046   proof (safe, simp_all add: not_less del: of_nat_Suc)
```
```  1047     fix x :: 'a assume "a < x \<bullet> i"
```
```  1048     with reals_Archimedean[of "x \<bullet> i - a"]
```
```  1049     obtain n where "a + 1 / real (Suc n) < x \<bullet> i"
```
```  1050       by (auto simp: field_simps)
```
```  1051     then show "\<exists>n. a + 1 / real (Suc n) \<le> x \<bullet> i"
```
```  1052       by (blast intro: less_imp_le)
```
```  1053   next
```
```  1054     fix x n
```
```  1055     have "a < a + 1 / real (Suc n)" by auto
```
```  1056     also assume "\<dots> \<le> x"
```
```  1057     finally show "a < x" .
```
```  1058   qed
```
```  1059   show "?set \<in> ?SIGMA" unfolding *
```
```  1060     by (auto intro!: Diff sigma_sets_Inter i)
```
```  1061 qed
```
```  1062
```
```  1063 lemma borel_eq_halfspace_less:
```
```  1064   "borel = sigma UNIV ((\<lambda>(a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> Basis))"
```
```  1065   (is "_ = ?SIGMA")
```
```  1066 proof (rule borel_eq_sigmaI2[OF borel_eq_box])
```
```  1067   fix a b :: 'a
```
```  1068   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}"
```
```  1069     by (auto simp: box_def)
```
```  1070   also have "\<dots> \<in> sets ?SIGMA"
```
```  1071     by (intro sets.sets_Collect_conj sets.sets_Collect_finite_All sets.sets_Collect_const)
```
```  1072        (auto intro!: halfspace_gt_in_halfspace countable_PiE countable_rat)
```
```  1073   finally show "box a b \<in> sets ?SIGMA" .
```
```  1074 qed auto
```
```  1075
```
```  1076 lemma borel_eq_halfspace_le:
```
```  1077   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i \<le> a}) ` (UNIV \<times> Basis))"
```
```  1078   (is "_ = ?SIGMA")
```
```  1079 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
```
```  1080   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```  1081   then have i: "i \<in> Basis" by auto
```
```  1082   have *: "{x::'a. x\<bullet>i < a} = (\<Union>n. {x. x\<bullet>i \<le> a - 1/real (Suc n)})"
```
```  1083   proof (safe, simp_all del: of_nat_Suc)
```
```  1084     fix x::'a assume *: "x\<bullet>i < a"
```
```  1085     with reals_Archimedean[of "a - x\<bullet>i"]
```
```  1086     obtain n where "x \<bullet> i < a - 1 / (real (Suc n))"
```
```  1087       by (auto simp: field_simps)
```
```  1088     then show "\<exists>n. x \<bullet> i \<le> a - 1 / (real (Suc n))"
```
```  1089       by (blast intro: less_imp_le)
```
```  1090   next
```
```  1091     fix x::'a and n
```
```  1092     assume "x\<bullet>i \<le> a - 1 / real (Suc n)"
```
```  1093     also have "\<dots> < a" by auto
```
```  1094     finally show "x\<bullet>i < a" .
```
```  1095   qed
```
```  1096   show "{x. x\<bullet>i < a} \<in> ?SIGMA" unfolding *
```
```  1097     by (intro sets.countable_UN) (auto intro: i)
```
```  1098 qed auto
```
```  1099
```
```  1100 lemma borel_eq_halfspace_ge:
```
```  1101   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. a \<le> x \<bullet> i}) ` (UNIV \<times> Basis))"
```
```  1102   (is "_ = ?SIGMA")
```
```  1103 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
```
```  1104   fix a :: real and i :: 'a assume i: "(a, i) \<in> UNIV \<times> Basis"
```
```  1105   have *: "{x::'a. x\<bullet>i < a} = space ?SIGMA - {x::'a. a \<le> x\<bullet>i}" by auto
```
```  1106   show "{x. x\<bullet>i < a} \<in> ?SIGMA" unfolding *
```
```  1107     using i by (intro sets.compl_sets) auto
```
```  1108 qed auto
```
```  1109
```
```  1110 lemma borel_eq_halfspace_greater:
```
```  1111   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. a < x \<bullet> i}) ` (UNIV \<times> Basis))"
```
```  1112   (is "_ = ?SIGMA")
```
```  1113 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])
```
```  1114   fix a :: real and i :: 'a assume "(a, i) \<in> (UNIV \<times> Basis)"
```
```  1115   then have i: "i \<in> Basis" by auto
```
```  1116   have *: "{x::'a. x\<bullet>i \<le> a} = space ?SIGMA - {x::'a. a < x\<bullet>i}" by auto
```
```  1117   show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA" unfolding *
```
```  1118     by (intro sets.compl_sets) (auto intro: i)
```
```  1119 qed auto
```
```  1120
```
```  1121 lemma borel_eq_atMost:
```
```  1122   "borel = sigma UNIV (range (\<lambda>a. {..a::'a::ordered_euclidean_space}))"
```
```  1123   (is "_ = ?SIGMA")
```
```  1124 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
```
```  1125   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```  1126   then have "i \<in> Basis" by auto
```
```  1127   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)})"
```
```  1128   proof (safe, simp_all add: eucl_le[where 'a='a] split: if_split_asm)
```
```  1129     fix x :: 'a
```
```  1130     from real_arch_simple[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"] guess k::nat ..
```
```  1131     then have "\<And>i. i \<in> Basis \<Longrightarrow> x\<bullet>i \<le> real k"
```
```  1132       by (subst (asm) Max_le_iff) auto
```
```  1133     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> x \<bullet> ia \<le> real k"
```
```  1134       by (auto intro!: exI[of _ k])
```
```  1135   qed
```
```  1136   show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA" unfolding *
```
```  1137     by (intro sets.countable_UN) auto
```
```  1138 qed auto
```
```  1139
```
```  1140 lemma borel_eq_greaterThan:
```
```  1141   "borel = sigma UNIV (range (\<lambda>a::'a::ordered_euclidean_space. {x. a <e x}))"
```
```  1142   (is "_ = ?SIGMA")
```
```  1143 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
```
```  1144   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```  1145   then have i: "i \<in> Basis" by auto
```
```  1146   have "{x::'a. x\<bullet>i \<le> a} = UNIV - {x::'a. a < x\<bullet>i}" by auto
```
```  1147   also have *: "{x::'a. a < x\<bullet>i} =
```
```  1148       (\<Union>k::nat. {x. (\<Sum>n\<in>Basis. (if n = i then a else -real k) *\<^sub>R n) <e x})" using i
```
```  1149   proof (safe, simp_all add: eucl_less_def split: if_split_asm)
```
```  1150     fix x :: 'a
```
```  1151     from reals_Archimedean2[of "Max ((\<lambda>i. -x\<bullet>i)`Basis)"]
```
```  1152     guess k::nat .. note k = this
```
```  1153     { fix i :: 'a assume "i \<in> Basis"
```
```  1154       then have "-x\<bullet>i < real k"
```
```  1155         using k by (subst (asm) Max_less_iff) auto
```
```  1156       then have "- real k < x\<bullet>i" by simp }
```
```  1157     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> -real k < x \<bullet> ia"
```
```  1158       by (auto intro!: exI[of _ k])
```
```  1159   qed
```
```  1160   finally show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA"
```
```  1161     apply (simp only:)
```
```  1162     apply (intro sets.countable_UN sets.Diff)
```
```  1163     apply (auto intro: sigma_sets_top)
```
```  1164     done
```
```  1165 qed auto
```
```  1166
```
```  1167 lemma borel_eq_lessThan:
```
```  1168   "borel = sigma UNIV (range (\<lambda>a::'a::ordered_euclidean_space. {x. x <e a}))"
```
```  1169   (is "_ = ?SIGMA")
```
```  1170 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge])
```
```  1171   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```  1172   then have i: "i \<in> Basis" by auto
```
```  1173   have "{x::'a. a \<le> x\<bullet>i} = UNIV - {x::'a. x\<bullet>i < a}" by auto
```
```  1174   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>
```
```  1175   proof (safe, simp_all add: eucl_less_def split: if_split_asm)
```
```  1176     fix x :: 'a
```
```  1177     from reals_Archimedean2[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"]
```
```  1178     guess k::nat .. note k = this
```
```  1179     { fix i :: 'a assume "i \<in> Basis"
```
```  1180       then have "x\<bullet>i < real k"
```
```  1181         using k by (subst (asm) Max_less_iff) auto
```
```  1182       then have "x\<bullet>i < real k" by simp }
```
```  1183     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> x \<bullet> ia < real k"
```
```  1184       by (auto intro!: exI[of _ k])
```
```  1185   qed
```
```  1186   finally show "{x. a \<le> x\<bullet>i} \<in> ?SIGMA"
```
```  1187     apply (simp only:)
```
```  1188     apply (intro sets.countable_UN sets.Diff)
```
```  1189     apply (auto intro: sigma_sets_top )
```
```  1190     done
```
```  1191 qed auto
```
```  1192
```
```  1193 lemma borel_eq_atLeastAtMost:
```
```  1194   "borel = sigma UNIV (range (\<lambda>(a,b). {a..b} ::'a::ordered_euclidean_space set))"
```
```  1195   (is "_ = ?SIGMA")
```
```  1196 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
```
```  1197   fix a::'a
```
```  1198   have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
```
```  1199   proof (safe, simp_all add: eucl_le[where 'a='a])
```
```  1200     fix x :: 'a
```
```  1201     from real_arch_simple[of "Max ((\<lambda>i. - x\<bullet>i)`Basis)"]
```
```  1202     guess k::nat .. note k = this
```
```  1203     { fix i :: 'a assume "i \<in> Basis"
```
```  1204       with k have "- x\<bullet>i \<le> real k"
```
```  1205         by (subst (asm) Max_le_iff) (auto simp: field_simps)
```
```  1206       then have "- real k \<le> x\<bullet>i" by simp }
```
```  1207     then show "\<exists>n::nat. \<forall>i\<in>Basis. - real n \<le> x \<bullet> i"
```
```  1208       by (auto intro!: exI[of _ k])
```
```  1209   qed
```
```  1210   show "{..a} \<in> ?SIGMA" unfolding *
```
```  1211     by (intro sets.countable_UN)
```
```  1212        (auto intro!: sigma_sets_top)
```
```  1213 qed auto
```
```  1214
```
```  1215 lemma borel_set_induct[consumes 1, case_names empty interval compl union]:
```
```  1216   assumes "A \<in> sets borel"
```
```  1217   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
```
```  1218           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)"
```
```  1219   shows "P (A::real set)"
```
```  1220 proof-
```
```  1221   let ?G = "range (\<lambda>(a,b). {a..b::real})"
```
```  1222   have "Int_stable ?G" "?G \<subseteq> Pow UNIV" "A \<in> sigma_sets UNIV ?G"
```
```  1223       using assms(1) by (auto simp add: borel_eq_atLeastAtMost Int_stable_def)
```
```  1224   thus ?thesis
```
```  1225   proof (induction rule: sigma_sets_induct_disjoint)
```
```  1226     case (union f)
```
```  1227       from union.hyps(2) have "\<And>i. f i \<in> sets borel" by (auto simp: borel_eq_atLeastAtMost)
```
```  1228       with union show ?case by (auto intro: un)
```
```  1229   next
```
```  1230     case (basic A)
```
```  1231     then obtain a b where "A = {a .. b}" by auto
```
```  1232     then show ?case
```
```  1233       by (cases "a \<le> b") (auto intro: int empty)
```
```  1234   qed (auto intro: empty compl simp: Compl_eq_Diff_UNIV[symmetric] borel_eq_atLeastAtMost)
```
```  1235 qed
```
```  1236
```
```  1237 lemma borel_sigma_sets_Ioc: "borel = sigma UNIV (range (\<lambda>(a, b). {a <.. b::real}))"
```
```  1238 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
```
```  1239   fix i :: real
```
```  1240   have "{..i} = (\<Union>j::nat. {-j <.. i})"
```
```  1241     by (auto simp: minus_less_iff reals_Archimedean2)
```
```  1242   also have "\<dots> \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))"
```
```  1243     by (intro sets.countable_nat_UN) auto
```
```  1244   finally show "{..i} \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))" .
```
```  1245 qed simp
```
```  1246
```
```  1247 lemma eucl_lessThan: "{x::real. x <e a} = lessThan a"
```
```  1248   by (simp add: eucl_less_def lessThan_def)
```
```  1249
```
```  1250 lemma borel_eq_atLeastLessThan:
```
```  1251   "borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA")
```
```  1252 proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan])
```
```  1253   have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto
```
```  1254   fix x :: real
```
```  1255   have "{..<x} = (\<Union>i::nat. {-real i ..< x})"
```
```  1256     by (auto simp: move_uminus real_arch_simple)
```
```  1257   then show "{y. y <e x} \<in> ?SIGMA"
```
```  1258     by (auto intro: sigma_sets.intros(2-) simp: eucl_lessThan)
```
```  1259 qed auto
```
```  1260
```
```  1261 lemma borel_measurable_halfspacesI:
```
```  1262   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1263   assumes F: "borel = sigma UNIV (F ` (UNIV \<times> Basis))"
```
```  1264   and S_eq: "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
```
```  1265   shows "f \<in> borel_measurable M = (\<forall>i\<in>Basis. \<forall>a::real. S a i \<in> sets M)"
```
```  1266 proof safe
```
```  1267   fix a :: real and i :: 'b assume i: "i \<in> Basis" and f: "f \<in> borel_measurable M"
```
```  1268   then show "S a i \<in> sets M" unfolding assms
```
```  1269     by (auto intro!: measurable_sets simp: assms(1))
```
```  1270 next
```
```  1271   assume a: "\<forall>i\<in>Basis. \<forall>a. S a i \<in> sets M"
```
```  1272   then show "f \<in> borel_measurable M"
```
```  1273     by (auto intro!: measurable_measure_of simp: S_eq F)
```
```  1274 qed
```
```  1275
```
```  1276 lemma borel_measurable_iff_halfspace_le:
```
```  1277   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1278   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)"
```
```  1279   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
```
```  1280
```
```  1281 lemma borel_measurable_iff_halfspace_less:
```
```  1282   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1283   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)"
```
```  1284   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
```
```  1285
```
```  1286 lemma borel_measurable_iff_halfspace_ge:
```
```  1287   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1288   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)"
```
```  1289   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
```
```  1290
```
```  1291 lemma borel_measurable_iff_halfspace_greater:
```
```  1292   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1293   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)"
```
```  1294   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto
```
```  1295
```
```  1296 lemma borel_measurable_iff_le:
```
```  1297   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
```
```  1298   using borel_measurable_iff_halfspace_le[where 'c=real] by simp
```
```  1299
```
```  1300 lemma borel_measurable_iff_less:
```
```  1301   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
```
```  1302   using borel_measurable_iff_halfspace_less[where 'c=real] by simp
```
```  1303
```
```  1304 lemma borel_measurable_iff_ge:
```
```  1305   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
```
```  1306   using borel_measurable_iff_halfspace_ge[where 'c=real]
```
```  1307   by simp
```
```  1308
```
```  1309 lemma borel_measurable_iff_greater:
```
```  1310   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
```
```  1311   using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
```
```  1312
```
```  1313 lemma borel_measurable_euclidean_space:
```
```  1314   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1315   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M)"
```
```  1316 proof safe
```
```  1317   assume f: "\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M"
```
```  1318   then show "f \<in> borel_measurable M"
```
```  1319     by (subst borel_measurable_iff_halfspace_le) auto
```
```  1320 qed auto
```
```  1321
```
```  1322 subsection "Borel measurable operators"
```
```  1323
```
```  1324 lemma borel_measurable_norm[measurable]: "norm \<in> borel_measurable borel"
```
```  1325   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1326
```
```  1327 lemma borel_measurable_sgn [measurable]: "(sgn::'a::real_normed_vector \<Rightarrow> 'a) \<in> borel_measurable borel"
```
```  1328   by (rule borel_measurable_continuous_countable_exceptions[where X="{0}"])
```
```  1329      (auto intro!: continuous_on_sgn continuous_on_id)
```
```  1330
```
```  1331 lemma borel_measurable_uminus[measurable (raw)]:
```
```  1332   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```  1333   assumes g: "g \<in> borel_measurable M"
```
```  1334   shows "(\<lambda>x. - g x) \<in> borel_measurable M"
```
```  1335   by (rule borel_measurable_continuous_on[OF _ g]) (intro continuous_intros)
```
```  1336
```
```  1337 lemma borel_measurable_diff[measurable (raw)]:
```
```  1338   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```  1339   assumes f: "f \<in> borel_measurable M"
```
```  1340   assumes g: "g \<in> borel_measurable M"
```
```  1341   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```  1342   using borel_measurable_add [of f M "- g"] assms by (simp add: fun_Compl_def)
```
```  1343
```
```  1344 lemma borel_measurable_times[measurable (raw)]:
```
```  1345   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_algebra}"
```
```  1346   assumes f: "f \<in> borel_measurable M"
```
```  1347   assumes g: "g \<in> borel_measurable M"
```
```  1348   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1349   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```  1350
```
```  1351 lemma borel_measurable_setprod[measurable (raw)]:
```
```  1352   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, real_normed_field}"
```
```  1353   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1354   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1355 proof cases
```
```  1356   assume "finite S"
```
```  1357   thus ?thesis using assms by induct auto
```
```  1358 qed simp
```
```  1359
```
```  1360 lemma borel_measurable_dist[measurable (raw)]:
```
```  1361   fixes g f :: "'a \<Rightarrow> 'b::{second_countable_topology, metric_space}"
```
```  1362   assumes f: "f \<in> borel_measurable M"
```
```  1363   assumes g: "g \<in> borel_measurable M"
```
```  1364   shows "(\<lambda>x. dist (f x) (g x)) \<in> borel_measurable M"
```
```  1365   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```  1366
```
```  1367 lemma borel_measurable_scaleR[measurable (raw)]:
```
```  1368   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```  1369   assumes f: "f \<in> borel_measurable M"
```
```  1370   assumes g: "g \<in> borel_measurable M"
```
```  1371   shows "(\<lambda>x. f x *\<^sub>R g x) \<in> borel_measurable M"
```
```  1372   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```  1373
```
```  1374 lemma affine_borel_measurable_vector:
```
```  1375   fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
```
```  1376   assumes "f \<in> borel_measurable M"
```
```  1377   shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
```
```  1378 proof (rule borel_measurableI)
```
```  1379   fix S :: "'x set" assume "open S"
```
```  1380   show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
```
```  1381   proof cases
```
```  1382     assume "b \<noteq> 0"
```
```  1383     with \<open>open S\<close> have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")
```
```  1384       using open_affinity [of S "inverse b" "- a /\<^sub>R b"]
```
```  1385       by (auto simp: algebra_simps)
```
```  1386     hence "?S \<in> sets borel" by auto
```
```  1387     moreover
```
```  1388     from \<open>b \<noteq> 0\<close> have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
```
```  1389       apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
```
```  1390     ultimately show ?thesis using assms unfolding in_borel_measurable_borel
```
```  1391       by auto
```
```  1392   qed simp
```
```  1393 qed
```
```  1394
```
```  1395 lemma borel_measurable_const_scaleR[measurable (raw)]:
```
```  1396   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. b *\<^sub>R f x ::'a::real_normed_vector) \<in> borel_measurable M"
```
```  1397   using affine_borel_measurable_vector[of f M 0 b] by simp
```
```  1398
```
```  1399 lemma borel_measurable_const_add[measurable (raw)]:
```
```  1400   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. a + f x ::'a::real_normed_vector) \<in> borel_measurable M"
```
```  1401   using affine_borel_measurable_vector[of f M a 1] by simp
```
```  1402
```
```  1403 lemma borel_measurable_inverse[measurable (raw)]:
```
```  1404   fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
```
```  1405   assumes f: "f \<in> borel_measurable M"
```
```  1406   shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
```
```  1407   apply (rule measurable_compose[OF f])
```
```  1408   apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])
```
```  1409   apply (auto intro!: continuous_on_inverse continuous_on_id)
```
```  1410   done
```
```  1411
```
```  1412 lemma borel_measurable_divide[measurable (raw)]:
```
```  1413   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>
```
```  1414     (\<lambda>x. f x / g x::'b::{second_countable_topology, real_normed_div_algebra}) \<in> borel_measurable M"
```
```  1415   by (simp add: divide_inverse)
```
```  1416
```
```  1417 lemma borel_measurable_abs[measurable (raw)]:
```
```  1418   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
```
```  1419   unfolding abs_real_def by simp
```
```  1420
```
```  1421 lemma borel_measurable_nth[measurable (raw)]:
```
```  1422   "(\<lambda>x::real^'n. x \$ i) \<in> borel_measurable borel"
```
```  1423   by (simp add: cart_eq_inner_axis)
```
```  1424
```
```  1425 lemma convex_measurable:
```
```  1426   fixes A :: "'a :: euclidean_space set"
```
```  1427   shows "X \<in> borel_measurable M \<Longrightarrow> X ` space M \<subseteq> A \<Longrightarrow> open A \<Longrightarrow> convex_on A q \<Longrightarrow>
```
```  1428     (\<lambda>x. q (X x)) \<in> borel_measurable M"
```
```  1429   by (rule measurable_compose[where f=X and N="restrict_space borel A"])
```
```  1430      (auto intro!: borel_measurable_continuous_on_restrict convex_on_continuous measurable_restrict_space2)
```
```  1431
```
```  1432 lemma borel_measurable_ln[measurable (raw)]:
```
```  1433   assumes f: "f \<in> borel_measurable M"
```
```  1434   shows "(\<lambda>x. ln (f x :: real)) \<in> borel_measurable M"
```
```  1435   apply (rule measurable_compose[OF f])
```
```  1436   apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])
```
```  1437   apply (auto intro!: continuous_on_ln continuous_on_id)
```
```  1438   done
```
```  1439
```
```  1440 lemma borel_measurable_log[measurable (raw)]:
```
```  1441   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. log (g x) (f x)) \<in> borel_measurable M"
```
```  1442   unfolding log_def by auto
```
```  1443
```
```  1444 lemma borel_measurable_exp[measurable]:
```
```  1445   "(exp::'a::{real_normed_field,banach}\<Rightarrow>'a) \<in> borel_measurable borel"
```
```  1446   by (intro borel_measurable_continuous_on1 continuous_at_imp_continuous_on ballI isCont_exp)
```
```  1447
```
```  1448 lemma measurable_real_floor[measurable]:
```
```  1449   "(floor :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
```
```  1450 proof -
```
```  1451   have "\<And>a x. \<lfloor>x\<rfloor> = a \<longleftrightarrow> (real_of_int a \<le> x \<and> x < real_of_int (a + 1))"
```
```  1452     by (auto intro: floor_eq2)
```
```  1453   then show ?thesis
```
```  1454     by (auto simp: vimage_def measurable_count_space_eq2_countable)
```
```  1455 qed
```
```  1456
```
```  1457 lemma measurable_real_ceiling[measurable]:
```
```  1458   "(ceiling :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
```
```  1459   unfolding ceiling_def[abs_def] by simp
```
```  1460
```
```  1461 lemma borel_measurable_real_floor: "(\<lambda>x::real. real_of_int \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
```
```  1462   by simp
```
```  1463
```
```  1464 lemma borel_measurable_root [measurable]: "root n \<in> borel_measurable borel"
```
```  1465   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1466
```
```  1467 lemma borel_measurable_sqrt [measurable]: "sqrt \<in> borel_measurable borel"
```
```  1468   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1469
```
```  1470 lemma borel_measurable_power [measurable (raw)]:
```
```  1471   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```  1472   assumes f: "f \<in> borel_measurable M"
```
```  1473   shows "(\<lambda>x. (f x) ^ n) \<in> borel_measurable M"
```
```  1474   by (intro borel_measurable_continuous_on [OF _ f] continuous_intros)
```
```  1475
```
```  1476 lemma borel_measurable_Re [measurable]: "Re \<in> borel_measurable borel"
```
```  1477   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1478
```
```  1479 lemma borel_measurable_Im [measurable]: "Im \<in> borel_measurable borel"
```
```  1480   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1481
```
```  1482 lemma borel_measurable_of_real [measurable]: "(of_real :: _ \<Rightarrow> (_::real_normed_algebra)) \<in> borel_measurable borel"
```
```  1483   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1484
```
```  1485 lemma borel_measurable_sin [measurable]: "(sin :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"
```
```  1486   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1487
```
```  1488 lemma borel_measurable_cos [measurable]: "(cos :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"
```
```  1489   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1490
```
```  1491 lemma borel_measurable_arctan [measurable]: "arctan \<in> borel_measurable borel"
```
```  1492   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1493
```
```  1494 lemma borel_measurable_complex_iff:
```
```  1495   "f \<in> borel_measurable M \<longleftrightarrow>
```
```  1496     (\<lambda>x. Re (f x)) \<in> borel_measurable M \<and> (\<lambda>x. Im (f x)) \<in> borel_measurable M"
```
```  1497   apply auto
```
```  1498   apply (subst fun_complex_eq)
```
```  1499   apply (intro borel_measurable_add)
```
```  1500   apply auto
```
```  1501   done
```
```  1502
```
```  1503 subsection "Borel space on the extended reals"
```
```  1504
```
```  1505 lemma borel_measurable_ereal[measurable (raw)]:
```
```  1506   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
```
```  1507   using continuous_on_ereal f by (rule borel_measurable_continuous_on) (rule continuous_on_id)
```
```  1508
```
```  1509 lemma borel_measurable_real_of_ereal[measurable (raw)]:
```
```  1510   fixes f :: "'a \<Rightarrow> ereal"
```
```  1511   assumes f: "f \<in> borel_measurable M"
```
```  1512   shows "(\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M"
```
```  1513   apply (rule measurable_compose[OF f])
```
```  1514   apply (rule borel_measurable_continuous_countable_exceptions[of "{\<infinity>, -\<infinity> }"])
```
```  1515   apply (auto intro: continuous_on_real simp: Compl_eq_Diff_UNIV)
```
```  1516   done
```
```  1517
```
```  1518 lemma borel_measurable_ereal_cases:
```
```  1519   fixes f :: "'a \<Rightarrow> ereal"
```
```  1520   assumes f: "f \<in> borel_measurable M"
```
```  1521   assumes H: "(\<lambda>x. H (ereal (real_of_ereal (f x)))) \<in> borel_measurable M"
```
```  1522   shows "(\<lambda>x. H (f x)) \<in> borel_measurable M"
```
```  1523 proof -
```
```  1524   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)))"
```
```  1525   { fix x have "H (f x) = ?F x" by (cases "f x") auto }
```
```  1526   with f H show ?thesis by simp
```
```  1527 qed
```
```  1528
```
```  1529 lemma
```
```  1530   fixes f :: "'a \<Rightarrow> ereal" assumes f[measurable]: "f \<in> borel_measurable M"
```
```  1531   shows borel_measurable_ereal_abs[measurable(raw)]: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
```
```  1532     and borel_measurable_ereal_inverse[measurable(raw)]: "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M"
```
```  1533     and borel_measurable_uminus_ereal[measurable(raw)]: "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M"
```
```  1534   by (auto simp del: abs_real_of_ereal simp: borel_measurable_ereal_cases[OF f] measurable_If)
```
```  1535
```
```  1536 lemma borel_measurable_uminus_eq_ereal[simp]:
```
```  1537   "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
```
```  1538 proof
```
```  1539   assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp
```
```  1540 qed auto
```
```  1541
```
```  1542 lemma set_Collect_ereal2:
```
```  1543   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1544   assumes f: "f \<in> borel_measurable M"
```
```  1545   assumes g: "g \<in> borel_measurable M"
```
```  1546   assumes H: "{x \<in> space M. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))} \<in> sets M"
```
```  1547     "{x \<in> space borel. H (-\<infinity>) (ereal x)} \<in> sets borel"
```
```  1548     "{x \<in> space borel. H (\<infinity>) (ereal x)} \<in> sets borel"
```
```  1549     "{x \<in> space borel. H (ereal x) (-\<infinity>)} \<in> sets borel"
```
```  1550     "{x \<in> space borel. H (ereal x) (\<infinity>)} \<in> sets borel"
```
```  1551   shows "{x \<in> space M. H (f x) (g x)} \<in> sets M"
```
```  1552 proof -
```
```  1553   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)))"
```
```  1554   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"
```
```  1555   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
```
```  1556   note * = this
```
```  1557   from assms show ?thesis
```
```  1558     by (subst *) (simp del: space_borel split del: if_split)
```
```  1559 qed
```
```  1560
```
```  1561 lemma borel_measurable_ereal_iff:
```
```  1562   shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
```
```  1563 proof
```
```  1564   assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
```
```  1565   from borel_measurable_real_of_ereal[OF this]
```
```  1566   show "f \<in> borel_measurable M" by auto
```
```  1567 qed auto
```
```  1568
```
```  1569 lemma borel_measurable_erealD[measurable_dest]:
```
```  1570   "(\<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"
```
```  1571   unfolding borel_measurable_ereal_iff by simp
```
```  1572
```
```  1573 lemma borel_measurable_ereal_iff_real:
```
```  1574   fixes f :: "'a \<Rightarrow> ereal"
```
```  1575   shows "f \<in> borel_measurable M \<longleftrightarrow>
```
```  1576     ((\<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)"
```
```  1577 proof safe
```
```  1578   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"
```
```  1579   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
```
```  1580   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
```
```  1581   let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real_of_ereal (f x))"
```
```  1582   have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
```
```  1583   also have "?f = f" by (auto simp: fun_eq_iff ereal_real)
```
```  1584   finally show "f \<in> borel_measurable M" .
```
```  1585 qed simp_all
```
```  1586
```
```  1587 lemma borel_measurable_ereal_iff_Iio:
```
```  1588   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
```
```  1589   by (auto simp: borel_Iio measurable_iff_measure_of)
```
```  1590
```
```  1591 lemma borel_measurable_ereal_iff_Ioi:
```
```  1592   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
```
```  1593   by (auto simp: borel_Ioi measurable_iff_measure_of)
```
```  1594
```
```  1595 lemma vimage_sets_compl_iff:
```
```  1596   "f -` A \<inter> space M \<in> sets M \<longleftrightarrow> f -` (- A) \<inter> space M \<in> sets M"
```
```  1597 proof -
```
```  1598   { fix A assume "f -` A \<inter> space M \<in> sets M"
```
```  1599     moreover have "f -` (- A) \<inter> space M = space M - f -` A \<inter> space M" by auto
```
```  1600     ultimately have "f -` (- A) \<inter> space M \<in> sets M" by auto }
```
```  1601   from this[of A] this[of "-A"] show ?thesis
```
```  1602     by (metis double_complement)
```
```  1603 qed
```
```  1604
```
```  1605 lemma borel_measurable_iff_Iic_ereal:
```
```  1606   "(f::'a\<Rightarrow>ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
```
```  1607   unfolding borel_measurable_ereal_iff_Ioi vimage_sets_compl_iff[where A="{a <..}" for a] by simp
```
```  1608
```
```  1609 lemma borel_measurable_iff_Ici_ereal:
```
```  1610   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
```
```  1611   unfolding borel_measurable_ereal_iff_Iio vimage_sets_compl_iff[where A="{..< a}" for a] by simp
```
```  1612
```
```  1613 lemma borel_measurable_ereal2:
```
```  1614   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1615   assumes f: "f \<in> borel_measurable M"
```
```  1616   assumes g: "g \<in> borel_measurable M"
```
```  1617   assumes H: "(\<lambda>x. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
```
```  1618     "(\<lambda>x. H (-\<infinity>) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
```
```  1619     "(\<lambda>x. H (\<infinity>) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
```
```  1620     "(\<lambda>x. H (ereal (real_of_ereal (f x))) (-\<infinity>)) \<in> borel_measurable M"
```
```  1621     "(\<lambda>x. H (ereal (real_of_ereal (f x))) (\<infinity>)) \<in> borel_measurable M"
```
```  1622   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
```
```  1623 proof -
```
```  1624   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)))"
```
```  1625   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"
```
```  1626   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
```
```  1627   note * = this
```
```  1628   from assms show ?thesis unfolding * by simp
```
```  1629 qed
```
```  1630
```
```  1631 lemma [measurable(raw)]:
```
```  1632   fixes f :: "'a \<Rightarrow> ereal"
```
```  1633   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1634   shows borel_measurable_ereal_add: "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```  1635     and borel_measurable_ereal_times: "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1636   by (simp_all add: borel_measurable_ereal2)
```
```  1637
```
```  1638 lemma [measurable(raw)]:
```
```  1639   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1640   assumes "f \<in> borel_measurable M"
```
```  1641   assumes "g \<in> borel_measurable M"
```
```  1642   shows borel_measurable_ereal_diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```  1643     and borel_measurable_ereal_divide: "(\<lambda>x. f x / g x) \<in> borel_measurable M"
```
```  1644   using assms by (simp_all add: minus_ereal_def divide_ereal_def)
```
```  1645
```
```  1646 lemma borel_measurable_ereal_setsum[measurable (raw)]:
```
```  1647   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1648   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1649   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1650   using assms by (induction S rule: infinite_finite_induct) auto
```
```  1651
```
```  1652 lemma borel_measurable_ereal_setprod[measurable (raw)]:
```
```  1653   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1654   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1655   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1656   using assms by (induction S rule: infinite_finite_induct) auto
```
```  1657
```
```  1658 lemma borel_measurable_extreal_suminf[measurable (raw)]:
```
```  1659   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1660   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1661   shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
```
```  1662   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
```
```  1663
```
```  1664 subsection "Borel space on the extended non-negative reals"
```
```  1665
```
```  1666 text \<open> @{type ennreal} is a topological monoid, so no rules for plus are required, also all order
```
```  1667   statements are usually done on type classes. \<close>
```
```  1668
```
```  1669 lemma measurable_enn2ereal[measurable]: "enn2ereal \<in> borel \<rightarrow>\<^sub>M borel"
```
```  1670   by (intro borel_measurable_continuous_on1 continuous_on_enn2ereal)
```
```  1671
```
```  1672 lemma measurable_e2ennreal[measurable]: "e2ennreal \<in> borel \<rightarrow>\<^sub>M borel"
```
```  1673   by (intro borel_measurable_continuous_on1 continuous_on_e2ennreal)
```
```  1674
```
```  1675 lemma borel_measurable_enn2real[measurable (raw)]:
```
```  1676   "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. enn2real (f x)) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1677   unfolding enn2real_def[abs_def] by measurable
```
```  1678
```
```  1679 definition [simp]: "is_borel f M \<longleftrightarrow> f \<in> borel_measurable M"
```
```  1680
```
```  1681 lemma is_borel_transfer[transfer_rule]: "rel_fun (rel_fun op = pcr_ennreal) op = is_borel is_borel"
```
```  1682   unfolding is_borel_def[abs_def]
```
```  1683 proof (safe intro!: rel_funI ext dest!: rel_fun_eq_pcr_ennreal[THEN iffD1])
```
```  1684   fix f and M :: "'a measure"
```
```  1685   show "f \<in> borel_measurable M" if f: "enn2ereal \<circ> f \<in> borel_measurable M"
```
```  1686     using measurable_compose[OF f measurable_e2ennreal] by simp
```
```  1687 qed simp
```
```  1688
```
```  1689 context
```
```  1690   includes ennreal.lifting
```
```  1691 begin
```
```  1692
```
```  1693 lemma measurable_ennreal[measurable]: "ennreal \<in> borel \<rightarrow>\<^sub>M borel"
```
```  1694   unfolding is_borel_def[symmetric]
```
```  1695   by transfer simp
```
```  1696
```
```  1697 lemma borel_measurable_ennreal_iff[simp]:
```
```  1698   assumes [simp]: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
```
```  1699   shows "(\<lambda>x. ennreal (f x)) \<in> M \<rightarrow>\<^sub>M borel \<longleftrightarrow> f \<in> M \<rightarrow>\<^sub>M borel"
```
```  1700 proof safe
```
```  1701   assume "(\<lambda>x. ennreal (f x)) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1702   then have "(\<lambda>x. enn2real (ennreal (f x))) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1703     by measurable
```
```  1704   then show "f \<in> M \<rightarrow>\<^sub>M borel"
```
```  1705     by (rule measurable_cong[THEN iffD1, rotated]) auto
```
```  1706 qed measurable
```
```  1707
```
```  1708 lemma borel_measurable_times_ennreal[measurable (raw)]:
```
```  1709   fixes f g :: "'a \<Rightarrow> ennreal"
```
```  1710   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"
```
```  1711   unfolding is_borel_def[symmetric] by transfer simp
```
```  1712
```
```  1713 lemma borel_measurable_inverse_ennreal[measurable (raw)]:
```
```  1714   fixes f :: "'a \<Rightarrow> ennreal"
```
```  1715   shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. inverse (f x)) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1716   unfolding is_borel_def[symmetric] by transfer simp
```
```  1717
```
```  1718 lemma borel_measurable_divide_ennreal[measurable (raw)]:
```
```  1719   fixes f :: "'a \<Rightarrow> ennreal"
```
```  1720   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"
```
```  1721   unfolding divide_ennreal_def by simp
```
```  1722
```
```  1723 lemma borel_measurable_minus_ennreal[measurable (raw)]:
```
```  1724   fixes f :: "'a \<Rightarrow> ennreal"
```
```  1725   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"
```
```  1726   unfolding is_borel_def[symmetric] by transfer simp
```
```  1727
```
```  1728 lemma borel_measurable_setprod_ennreal[measurable (raw)]:
```
```  1729   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ennreal"
```
```  1730   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1731   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1732   using assms by (induction S rule: infinite_finite_induct) auto
```
```  1733
```
```  1734 end
```
```  1735
```
```  1736 hide_const (open) is_borel
```
```  1737
```
```  1738 subsection \<open>LIMSEQ is borel measurable\<close>
```
```  1739
```
```  1740 lemma borel_measurable_LIMSEQ_real:
```
```  1741   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
```
```  1742   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
```
```  1743   and u: "\<And>i. u i \<in> borel_measurable M"
```
```  1744   shows "u' \<in> borel_measurable M"
```
```  1745 proof -
```
```  1746   have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)"
```
```  1747     using u' by (simp add: lim_imp_Liminf)
```
```  1748   moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M"
```
```  1749     by auto
```
```  1750   ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff)
```
```  1751 qed
```
```  1752
```
```  1753 lemma borel_measurable_LIMSEQ_metric:
```
```  1754   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b :: metric_space"
```
```  1755   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1756   assumes lim: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. f i x) \<longlonglongrightarrow> g x"
```
```  1757   shows "g \<in> borel_measurable M"
```
```  1758   unfolding borel_eq_closed
```
```  1759 proof (safe intro!: measurable_measure_of)
```
```  1760   fix A :: "'b set" assume "closed A"
```
```  1761
```
```  1762   have [measurable]: "(\<lambda>x. infdist (g x) A) \<in> borel_measurable M"
```
```  1763   proof (rule borel_measurable_LIMSEQ_real)
```
```  1764     show "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. infdist (f i x) A) \<longlonglongrightarrow> infdist (g x) A"
```
```  1765       by (intro tendsto_infdist lim)
```
```  1766     show "\<And>i. (\<lambda>x. infdist (f i x) A) \<in> borel_measurable M"
```
```  1767       by (intro borel_measurable_continuous_on[where f="\<lambda>x. infdist x A"]
```
```  1768         continuous_at_imp_continuous_on ballI continuous_infdist continuous_ident) auto
```
```  1769   qed
```
```  1770
```
```  1771   show "g -` A \<inter> space M \<in> sets M"
```
```  1772   proof cases
```
```  1773     assume "A \<noteq> {}"
```
```  1774     then have "\<And>x. infdist x A = 0 \<longleftrightarrow> x \<in> A"
```
```  1775       using \<open>closed A\<close> by (simp add: in_closed_iff_infdist_zero)
```
```  1776     then have "g -` A \<inter> space M = {x\<in>space M. infdist (g x) A = 0}"
```
```  1777       by auto
```
```  1778     also have "\<dots> \<in> sets M"
```
```  1779       by measurable
```
```  1780     finally show ?thesis .
```
```  1781   qed simp
```
```  1782 qed auto
```
```  1783
```
```  1784 lemma sets_Collect_Cauchy[measurable]:
```
```  1785   fixes f :: "nat \<Rightarrow> 'a => 'b::{metric_space, second_countable_topology}"
```
```  1786   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1787   shows "{x\<in>space M. Cauchy (\<lambda>i. f i x)} \<in> sets M"
```
```  1788   unfolding metric_Cauchy_iff2 using f by auto
```
```  1789
```
```  1790 lemma borel_measurable_lim_metric[measurable (raw)]:
```
```  1791   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```  1792   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1793   shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1794 proof -
```
```  1795   define u' where "u' x = lim (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)" for x
```
```  1796   then have *: "\<And>x. lim (\<lambda>i. f i x) = (if Cauchy (\<lambda>i. f i x) then u' x else (THE x. False))"
```
```  1797     by (auto simp: lim_def convergent_eq_cauchy[symmetric])
```
```  1798   have "u' \<in> borel_measurable M"
```
```  1799   proof (rule borel_measurable_LIMSEQ_metric)
```
```  1800     fix x
```
```  1801     have "convergent (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)"
```
```  1802       by (cases "Cauchy (\<lambda>i. f i x)")
```
```  1803          (auto simp add: convergent_eq_cauchy[symmetric] convergent_def)
```
```  1804     then show "(\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0) \<longlonglongrightarrow> u' x"
```
```  1805       unfolding u'_def
```
```  1806       by (rule convergent_LIMSEQ_iff[THEN iffD1])
```
```  1807   qed measurable
```
```  1808   then show ?thesis
```
```  1809     unfolding * by measurable
```
```  1810 qed
```
```  1811
```
```  1812 lemma borel_measurable_suminf[measurable (raw)]:
```
```  1813   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```  1814   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1815   shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1816   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
```
```  1817
```
```  1818 lemma Collect_closed_imp_pred_borel: "closed {x. P x} \<Longrightarrow> Measurable.pred borel P"
```
```  1819   by (simp add: pred_def)
```
```  1820
```
```  1821 (* Proof by Jeremy Avigad and Luke Serafin *)
```
```  1822 lemma isCont_borel_pred[measurable]:
```
```  1823   fixes f :: "'b::metric_space \<Rightarrow> 'a::metric_space"
```
```  1824   shows "Measurable.pred borel (isCont f)"
```
```  1825 proof (subst measurable_cong)
```
```  1826   let ?I = "\<lambda>j. inverse(real (Suc j))"
```
```  1827   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
```
```  1828     unfolding continuous_at_eps_delta
```
```  1829   proof safe
```
```  1830     fix i assume "\<forall>e>0. \<exists>d>0. \<forall>y. dist y x < d \<longrightarrow> dist (f y) (f x) < e"
```
```  1831     moreover have "0 < ?I i / 2"
```
```  1832       by simp
```
```  1833     ultimately obtain d where d: "0 < d" "\<And>y. dist x y < d \<Longrightarrow> dist (f y) (f x) < ?I i / 2"
```
```  1834       by (metis dist_commute)
```
```  1835     then obtain j where j: "?I j < d"
```
```  1836       by (metis reals_Archimedean)
```
```  1837
```
```  1838     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"
```
```  1839     proof (safe intro!: exI[where x=j])
```
```  1840       fix y z assume *: "dist x y < ?I j" "dist x z < ?I j"
```
```  1841       have "dist (f y) (f z) \<le> dist (f y) (f x) + dist (f z) (f x)"
```
```  1842         by (rule dist_triangle2)
```
```  1843       also have "\<dots> < ?I i / 2 + ?I i / 2"
```
```  1844         by (intro add_strict_mono d less_trans[OF _ j] *)
```
```  1845       also have "\<dots> \<le> ?I i"
```
```  1846         by (simp add: field_simps of_nat_Suc)
```
```  1847       finally show "dist (f y) (f z) \<le> ?I i"
```
```  1848         by simp
```
```  1849     qed
```
```  1850   next
```
```  1851     fix e::real assume "0 < e"
```
```  1852     then obtain n where n: "?I n < e"
```
```  1853       by (metis reals_Archimedean)
```
```  1854     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"
```
```  1855     from this[THEN spec, of "Suc n"]
```
```  1856     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)"
```
```  1857       by auto
```
```  1858
```
```  1859     show "\<exists>d>0. \<forall>y. dist y x < d \<longrightarrow> dist (f y) (f x) < e"
```
```  1860     proof (safe intro!: exI[of _ "?I j"])
```
```  1861       fix y assume "dist y x < ?I j"
```
```  1862       then have "dist (f y) (f x) \<le> ?I (Suc n)"
```
```  1863         by (intro j) (auto simp: dist_commute)
```
```  1864       also have "?I (Suc n) < ?I n"
```
```  1865         by simp
```
```  1866       also note n
```
```  1867       finally show "dist (f y) (f x) < e" .
```
```  1868     qed simp
```
```  1869   qed
```
```  1870 qed (intro pred_intros_countable closed_Collect_all closed_Collect_le open_Collect_less
```
```  1871            Collect_closed_imp_pred_borel closed_Collect_imp open_Collect_conj continuous_intros)
```
```  1872
```
```  1873 lemma isCont_borel:
```
```  1874   fixes f :: "'b::metric_space \<Rightarrow> 'a::metric_space"
```
```  1875   shows "{x. isCont f x} \<in> sets borel"
```
```  1876   by simp
```
```  1877
```
```  1878 lemma is_real_interval:
```
```  1879   assumes S: "is_interval S"
```
```  1880   shows "\<exists>a b::real. S = {} \<or> S = UNIV \<or> S = {..<b} \<or> S = {..b} \<or> S = {a<..} \<or> S = {a..} \<or>
```
```  1881     S = {a<..<b} \<or> S = {a<..b} \<or> S = {a..<b} \<or> S = {a..b}"
```
```  1882   using S unfolding is_interval_1 by (blast intro: interval_cases)
```
```  1883
```
```  1884 lemma real_interval_borel_measurable:
```
```  1885   assumes "is_interval (S::real set)"
```
```  1886   shows "S \<in> sets borel"
```
```  1887 proof -
```
```  1888   from assms is_real_interval have "\<exists>a b::real. S = {} \<or> S = UNIV \<or> S = {..<b} \<or> S = {..b} \<or>
```
```  1889     S = {a<..} \<or> S = {a..} \<or> S = {a<..<b} \<or> S = {a<..b} \<or> S = {a..<b} \<or> S = {a..b}" by auto
```
```  1890   then guess a ..
```
```  1891   then guess b ..
```
```  1892   thus ?thesis
```
```  1893     by auto
```
```  1894 qed
```
```  1895
```
```  1896 lemma borel_measurable_mono_on_fnc:
```
```  1897   fixes f :: "real \<Rightarrow> real" and A :: "real set"
```
```  1898   assumes "mono_on f A"
```
```  1899   shows "f \<in> borel_measurable (restrict_space borel A)"
```
```  1900   apply (rule measurable_restrict_countable[OF mono_on_ctble_discont[OF assms]])
```
```  1901   apply (auto intro!: image_eqI[where x="{x}" for x] simp: sets_restrict_space)
```
```  1902   apply (auto simp add: sets_restrict_restrict_space continuous_on_eq_continuous_within
```
```  1903               cong: measurable_cong_sets
```
```  1904               intro!: borel_measurable_continuous_on_restrict intro: continuous_within_subset)
```
```  1905   done
```
```  1906
```
```  1907 lemma borel_measurable_mono:
```
```  1908   fixes f :: "real \<Rightarrow> real"
```
```  1909   shows "mono f \<Longrightarrow> f \<in> borel_measurable borel"
```
```  1910   using borel_measurable_mono_on_fnc[of f UNIV] by (simp add: mono_def mono_on_def)
```
```  1911
```
```  1912 no_notation
```
```  1913   eucl_less (infix "<e" 50)
```
```  1914
```
```  1915 end
```