src/HOL/Analysis/Borel_Space.thy
 author Angeliki KoutsoukouArgyraki Tue Aug 28 13:28:39 2018 +0100 (13 months ago) changeset 68833 fde093888c16 parent 68635 8094b853a92f child 69022 e2858770997a permissions -rw-r--r--
tagged 21 theories in the Analysis library for the manual
```     1 (*  Title:      HOL/Analysis/Borel_Space.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Armin Heller, TU München
```
```     4 *)
```
```     5
```
```     6 section%important \<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%unimportant 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%unimportant topological_basis_trivial: "topological_basis {A. open A}"
```
```    18   by (auto simp: topological_basis_def)
```
```    19
```
```    20 lemma%important open_prod_generated: "open = generate_topology {A \<times> B | A B. open A \<and> open B}"
```
```    21 proof%unimportant -
```
```    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%important "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%unimportant 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%unimportant 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%unimportant mono_imp_mono_on: "mono f \<Longrightarrow> mono_on f A"
```
```    39   unfolding mono_def mono_on_def by auto
```
```    40
```
```    41 lemma%unimportant 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%important "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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%important 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%unimportant (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 ((+) (-x) ` interior A)" by (rule open_translation)
```
```   114     also have "((+) (-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%unimportant 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%important 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%unimportant -
```
```   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%important 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%unimportant -
```
```   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%important mono_ctble_discont:
```
```   213   fixes f :: "real \<Rightarrow> real"
```
```   214   assumes "mono f"
```
```   215   shows "countable {a. \<not> isCont f a}"
```
```   216 using%unimportant assms mono_on_ctble_discont [of f UNIV] unfolding mono_on_def mono_def by auto
```
```   217
```
```   218 lemma%important 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%important closure_contains_Sup:
```
```   228   fixes S :: "real set"
```
```   229   assumes "S \<noteq> {}" "bdd_above S"
```
```   230   shows "Sup S \<in> closure S"
```
```   231 proof%unimportant -
```
```   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%unimportant 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%unimportant closed_subset_contains_Sup:
```
```   246   fixes A C :: "real set"
```
```   247   shows "closed C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> A \<noteq> {} \<Longrightarrow> bdd_above A \<Longrightarrow> Sup A \<in> C"
```
```   248   by (metis closure_contains_Sup closure_minimal subset_eq)
```
```   249
```
```   250 lemma%important deriv_nonneg_imp_mono:
```
```   251   assumes deriv: "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_real_derivative g' x) (at x)"
```
```   252   assumes nonneg: "\<And>x. x \<in> {a..b} \<Longrightarrow> g' x \<ge> 0"
```
```   253   assumes ab: "a \<le> b"
```
```   254   shows "g a \<le> g b"
```
```   255 proof%unimportant (cases "a < b")
```
```   256   assume "a < b"
```
```   257   from deriv have "\<And>x. \<lbrakk>x \<ge> a; x \<le> b\<rbrakk> \<Longrightarrow> (g has_real_derivative g' x) (at x)" by simp
```
```   258   with MVT2[OF \<open>a < b\<close>] and deriv
```
```   259     obtain \<xi> where \<xi>_ab: "\<xi> > a" "\<xi> < b" and g_ab: "g b - g a = (b - a) * g' \<xi>" by blast
```
```   260   from \<xi>_ab ab nonneg have "(b - a) * g' \<xi> \<ge> 0" by simp
```
```   261   with g_ab show ?thesis by simp
```
```   262 qed (insert ab, simp)
```
```   263
```
```   264 lemma%important continuous_interval_vimage_Int:
```
```   265   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"
```
```   266   assumes "a \<le> b" "(c::real) \<le> d" "{c..d} \<subseteq> {g a..g b}"
```
```   267   obtains c' d' where "{a..b} \<inter> g -` {c..d} = {c'..d'}" "c' \<le> d'" "g c' = c" "g d' = d"
```
```   268 proof%unimportant-
```
```   269   let ?A = "{a..b} \<inter> g -` {c..d}"
```
```   270   from IVT'[of g a c b, OF _ _ \<open>a \<le> b\<close> assms(1)] assms(4,5)
```
```   271   obtain c'' where c'': "c'' \<in> ?A" "g c'' = c" by auto
```
```   272   from IVT'[of g a d b, OF _ _ \<open>a \<le> b\<close> assms(1)] assms(4,5)
```
```   273   obtain d'' where d'': "d'' \<in> ?A" "g d'' = d" by auto
```
```   274   hence [simp]: "?A \<noteq> {}" by blast
```
```   275
```
```   276   define c' where "c' = Inf ?A"
```
```   277   define d' where "d' = Sup ?A"
```
```   278   have "?A \<subseteq> {c'..d'}" unfolding c'_def d'_def
```
```   279     by (intro subsetI) (auto intro: cInf_lower cSup_upper)
```
```   280   moreover from assms have "closed ?A"
```
```   281     using continuous_on_closed_vimage[of "{a..b}" g] by (subst Int_commute) simp
```
```   282   hence c'd'_in_set: "c' \<in> ?A" "d' \<in> ?A" unfolding c'_def d'_def
```
```   283     by ((intro closed_contains_Inf closed_contains_Sup, simp_all)[])+
```
```   284   hence "{c'..d'} \<subseteq> ?A" using assms
```
```   285     by (intro subsetI)
```
```   286        (auto intro!: order_trans[of c "g c'" "g x" for x] order_trans[of "g x" "g d'" d for x]
```
```   287              intro!: mono)
```
```   288   moreover have "c' \<le> d'" using c'd'_in_set(2) unfolding c'_def by (intro cInf_lower) auto
```
```   289   moreover have "g c' \<le> c" "g d' \<ge> d"
```
```   290     apply (insert c'' d'' c'd'_in_set)
```
```   291     apply (subst c''(2)[symmetric])
```
```   292     apply (auto simp: c'_def intro!: mono cInf_lower c'') []
```
```   293     apply (subst d''(2)[symmetric])
```
```   294     apply (auto simp: d'_def intro!: mono cSup_upper d'') []
```
```   295     done
```
```   296   with c'd'_in_set have "g c' = c" "g d' = d" by auto
```
```   297   ultimately show ?thesis using that by blast
```
```   298 qed
```
```   299
```
```   300 subsection%important \<open>Generic Borel spaces\<close>
```
```   301
```
```   302 definition%important (in topological_space) borel :: "'a measure" where
```
```   303   "borel = sigma UNIV {S. open S}"
```
```   304
```
```   305 abbreviation "borel_measurable M \<equiv> measurable M borel"
```
```   306
```
```   307 lemma%important in_borel_measurable:
```
```   308    "f \<in> borel_measurable M \<longleftrightarrow>
```
```   309     (\<forall>S \<in> sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)"
```
```   310   by%unimportant (auto simp add: measurable_def borel_def)
```
```   311
```
```   312 lemma%important in_borel_measurable_borel:
```
```   313    "f \<in> borel_measurable M \<longleftrightarrow>
```
```   314     (\<forall>S \<in> sets borel.
```
```   315       f -` S \<inter> space M \<in> sets M)"
```
```   316   by%unimportant (auto simp add: measurable_def borel_def)
```
```   317
```
```   318 lemma%unimportant space_borel[simp]: "space borel = UNIV"
```
```   319   unfolding borel_def by auto
```
```   320
```
```   321 lemma%unimportant space_in_borel[measurable]: "UNIV \<in> sets borel"
```
```   322   unfolding borel_def by auto
```
```   323
```
```   324 lemma%unimportant sets_borel: "sets borel = sigma_sets UNIV {S. open S}"
```
```   325   unfolding borel_def by (rule sets_measure_of) simp
```
```   326
```
```   327 lemma%unimportant measurable_sets_borel:
```
```   328     "\<lbrakk>f \<in> measurable borel M; A \<in> sets M\<rbrakk> \<Longrightarrow> f -` A \<in> sets borel"
```
```   329   by (drule (1) measurable_sets) simp
```
```   330
```
```   331 lemma%unimportant pred_Collect_borel[measurable (raw)]: "Measurable.pred borel P \<Longrightarrow> {x. P x} \<in> sets borel"
```
```   332   unfolding borel_def pred_def by auto
```
```   333
```
```   334 lemma%unimportant borel_open[measurable (raw generic)]:
```
```   335   assumes "open A" shows "A \<in> sets borel"
```
```   336 proof -
```
```   337   have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms .
```
```   338   thus ?thesis unfolding borel_def by auto
```
```   339 qed
```
```   340
```
```   341 lemma%unimportant borel_closed[measurable (raw generic)]:
```
```   342   assumes "closed A" shows "A \<in> sets borel"
```
```   343 proof -
```
```   344   have "space borel - (- A) \<in> sets borel"
```
```   345     using assms unfolding closed_def by (blast intro: borel_open)
```
```   346   thus ?thesis by simp
```
```   347 qed
```
```   348
```
```   349 lemma%unimportant borel_singleton[measurable]:
```
```   350   "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets (borel :: 'a::t1_space measure)"
```
```   351   unfolding insert_def by (rule sets.Un) auto
```
```   352
```
```   353 lemma%unimportant sets_borel_eq_count_space: "sets (borel :: 'a::{countable, t2_space} measure) = count_space UNIV"
```
```   354 proof -
```
```   355   have "(\<Union>a\<in>A. {a}) \<in> sets borel" for A :: "'a set"
```
```   356     by (intro sets.countable_UN') auto
```
```   357   then show ?thesis
```
```   358     by auto
```
```   359 qed
```
```   360
```
```   361 lemma%unimportant borel_comp[measurable]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
```
```   362   unfolding Compl_eq_Diff_UNIV by simp
```
```   363
```
```   364 lemma%unimportant borel_measurable_vimage:
```
```   365   fixes f :: "'a \<Rightarrow> 'x::t2_space"
```
```   366   assumes borel[measurable]: "f \<in> borel_measurable M"
```
```   367   shows "f -` {x} \<inter> space M \<in> sets M"
```
```   368   by simp
```
```   369
```
```   370 lemma%unimportant borel_measurableI:
```
```   371   fixes f :: "'a \<Rightarrow> 'x::topological_space"
```
```   372   assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
```
```   373   shows "f \<in> borel_measurable M"
```
```   374   unfolding borel_def
```
```   375 proof (rule measurable_measure_of, simp_all)
```
```   376   fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"
```
```   377     using assms[of S] by simp
```
```   378 qed
```
```   379
```
```   380 lemma%unimportant borel_measurable_const:
```
```   381   "(\<lambda>x. c) \<in> borel_measurable M"
```
```   382   by auto
```
```   383
```
```   384 lemma%unimportant borel_measurable_indicator:
```
```   385   assumes A: "A \<in> sets M"
```
```   386   shows "indicator A \<in> borel_measurable M"
```
```   387   unfolding indicator_def [abs_def] using A
```
```   388   by (auto intro!: measurable_If_set)
```
```   389
```
```   390 lemma%unimportant borel_measurable_count_space[measurable (raw)]:
```
```   391   "f \<in> borel_measurable (count_space S)"
```
```   392   unfolding measurable_def by auto
```
```   393
```
```   394 lemma%unimportant borel_measurable_indicator'[measurable (raw)]:
```
```   395   assumes [measurable]: "{x\<in>space M. f x \<in> A x} \<in> sets M"
```
```   396   shows "(\<lambda>x. indicator (A x) (f x)) \<in> borel_measurable M"
```
```   397   unfolding indicator_def[abs_def]
```
```   398   by (auto intro!: measurable_If)
```
```   399
```
```   400 lemma%important borel_measurable_indicator_iff:
```
```   401   "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
```
```   402     (is "?I \<in> borel_measurable M \<longleftrightarrow> _")
```
```   403 proof%unimportant
```
```   404   assume "?I \<in> borel_measurable M"
```
```   405   then have "?I -` {1} \<inter> space M \<in> sets M"
```
```   406     unfolding measurable_def by auto
```
```   407   also have "?I -` {1} \<inter> space M = A \<inter> space M"
```
```   408     unfolding indicator_def [abs_def] by auto
```
```   409   finally show "A \<inter> space M \<in> sets M" .
```
```   410 next
```
```   411   assume "A \<inter> space M \<in> sets M"
```
```   412   moreover have "?I \<in> borel_measurable M \<longleftrightarrow>
```
```   413     (indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"
```
```   414     by (intro measurable_cong) (auto simp: indicator_def)
```
```   415   ultimately show "?I \<in> borel_measurable M" by auto
```
```   416 qed
```
```   417
```
```   418 lemma%unimportant borel_measurable_subalgebra:
```
```   419   assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
```
```   420   shows "f \<in> borel_measurable M"
```
```   421   using assms unfolding measurable_def by auto
```
```   422
```
```   423 lemma%unimportant borel_measurable_restrict_space_iff_ereal:
```
```   424   fixes f :: "'a \<Rightarrow> ereal"
```
```   425   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
```
```   426   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
```
```   427     (\<lambda>x. f x * indicator \<Omega> x) \<in> borel_measurable M"
```
```   428   by (subst measurable_restrict_space_iff)
```
```   429      (auto simp: indicator_def if_distrib[where f="\<lambda>x. a * x" for a] cong del: if_weak_cong)
```
```   430
```
```   431 lemma%unimportant borel_measurable_restrict_space_iff_ennreal:
```
```   432   fixes f :: "'a \<Rightarrow> ennreal"
```
```   433   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
```
```   434   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
```
```   435     (\<lambda>x. f x * indicator \<Omega> x) \<in> borel_measurable M"
```
```   436   by (subst measurable_restrict_space_iff)
```
```   437      (auto simp: indicator_def if_distrib[where f="\<lambda>x. a * x" for a] cong del: if_weak_cong)
```
```   438
```
```   439 lemma%unimportant borel_measurable_restrict_space_iff:
```
```   440   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   441   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
```
```   442   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
```
```   443     (\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> borel_measurable M"
```
```   444   by (subst measurable_restrict_space_iff)
```
```   445      (auto simp: indicator_def if_distrib[where f="\<lambda>x. x *\<^sub>R a" for a] ac_simps
```
```   446        cong del: if_weak_cong)
```
```   447
```
```   448 lemma%unimportant cbox_borel[measurable]: "cbox a b \<in> sets borel"
```
```   449   by (auto intro: borel_closed)
```
```   450
```
```   451 lemma%unimportant box_borel[measurable]: "box a b \<in> sets borel"
```
```   452   by (auto intro: borel_open)
```
```   453
```
```   454 lemma%unimportant borel_compact: "compact (A::'a::t2_space set) \<Longrightarrow> A \<in> sets borel"
```
```   455   by (auto intro: borel_closed dest!: compact_imp_closed)
```
```   456
```
```   457 lemma%unimportant borel_sigma_sets_subset:
```
```   458   "A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel"
```
```   459   using sets.sigma_sets_subset[of A borel] by simp
```
```   460
```
```   461 lemma%important borel_eq_sigmaI1:
```
```   462   fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
```
```   463   assumes borel_eq: "borel = sigma UNIV X"
```
```   464   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (F ` A))"
```
```   465   assumes F: "\<And>i. i \<in> A \<Longrightarrow> F i \<in> sets borel"
```
```   466   shows "borel = sigma UNIV (F ` A)"
```
```   467   unfolding borel_def
```
```   468 proof%unimportant (intro sigma_eqI antisym)
```
```   469   have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel"
```
```   470     unfolding borel_def by simp
```
```   471   also have "\<dots> = sigma_sets UNIV X"
```
```   472     unfolding borel_eq by simp
```
```   473   also have "\<dots> \<subseteq> sigma_sets UNIV (F`A)"
```
```   474     using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto
```
```   475   finally show "sigma_sets UNIV {S. open S} \<subseteq> sigma_sets UNIV (F`A)" .
```
```   476   show "sigma_sets UNIV (F`A) \<subseteq> sigma_sets UNIV {S. open S}"
```
```   477     unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto
```
```   478 qed auto
```
```   479
```
```   480 lemma%unimportant borel_eq_sigmaI2:
```
```   481   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set"
```
```   482     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
```
```   483   assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`B)"
```
```   484   assumes X: "\<And>i j. (i, j) \<in> B \<Longrightarrow> G i j \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"
```
```   485   assumes F: "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel"
```
```   486   shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"
```
```   487   using assms
```
```   488   by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` B" and F="(\<lambda>(i, j). F i j)"]) auto
```
```   489
```
```   490 lemma%unimportant borel_eq_sigmaI3:
```
```   491   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
```
```   492   assumes borel_eq: "borel = sigma UNIV X"
```
```   493   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"
```
```   494   assumes F: "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel"
```
```   495   shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"
```
```   496   using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto
```
```   497
```
```   498 lemma%unimportant borel_eq_sigmaI4:
```
```   499   fixes F :: "'i \<Rightarrow> 'a::topological_space set"
```
```   500     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
```
```   501   assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`A)"
```
```   502   assumes X: "\<And>i j. (i, j) \<in> A \<Longrightarrow> G i j \<in> sets (sigma UNIV (range F))"
```
```   503   assumes F: "\<And>i. F i \<in> sets borel"
```
```   504   shows "borel = sigma UNIV (range F)"
```
```   505   using assms by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` A" and F=F]) auto
```
```   506
```
```   507 lemma%unimportant borel_eq_sigmaI5:
```
```   508   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set"
```
```   509   assumes borel_eq: "borel = sigma UNIV (range G)"
```
```   510   assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
```
```   511   assumes F: "\<And>i j. F i j \<in> sets borel"
```
```   512   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
```
```   513   using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto
```
```   514
```
```   515 lemma%important second_countable_borel_measurable:
```
```   516   fixes X :: "'a::second_countable_topology set set"
```
```   517   assumes eq: "open = generate_topology X"
```
```   518   shows "borel = sigma UNIV X"
```
```   519   unfolding borel_def
```
```   520 proof%unimportant (intro sigma_eqI sigma_sets_eqI)
```
```   521   interpret X: sigma_algebra UNIV "sigma_sets UNIV X"
```
```   522     by (rule sigma_algebra_sigma_sets) simp
```
```   523
```
```   524   fix S :: "'a set" assume "S \<in> Collect open"
```
```   525   then have "generate_topology X S"
```
```   526     by (auto simp: eq)
```
```   527   then show "S \<in> sigma_sets UNIV X"
```
```   528   proof induction
```
```   529     case (UN K)
```
```   530     then have K: "\<And>k. k \<in> K \<Longrightarrow> open k"
```
```   531       unfolding eq by auto
```
```   532     from ex_countable_basis obtain B :: "'a set set" where
```
```   533       B:  "\<And>b. b \<in> B \<Longrightarrow> open b" "\<And>X. open X \<Longrightarrow> \<exists>b\<subseteq>B. (\<Union>b) = X" and "countable B"
```
```   534       by (auto simp: topological_basis_def)
```
```   535     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"
```
```   536       by metis
```
```   537     define U where "U = (\<Union>k\<in>K. m k)"
```
```   538     with m have "countable U"
```
```   539       by (intro countable_subset[OF _ \<open>countable B\<close>]) auto
```
```   540     have "\<Union>U = (\<Union>A\<in>U. A)" by simp
```
```   541     also have "\<dots> = \<Union>K"
```
```   542       unfolding U_def UN_simps by (simp add: m)
```
```   543     finally have "\<Union>U = \<Union>K" .
```
```   544
```
```   545     have "\<forall>b\<in>U. \<exists>k\<in>K. b \<subseteq> k"
```
```   546       using m by (auto simp: U_def)
```
```   547     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"
```
```   548       by metis
```
```   549     then have "(\<Union>b\<in>U. u b) \<subseteq> \<Union>K" "\<Union>U \<subseteq> (\<Union>b\<in>U. u b)"
```
```   550       by auto
```
```   551     then have "\<Union>K = (\<Union>b\<in>U. u b)"
```
```   552       unfolding \<open>\<Union>U = \<Union>K\<close> by auto
```
```   553     also have "\<dots> \<in> sigma_sets UNIV X"
```
```   554       using u UN by (intro X.countable_UN' \<open>countable U\<close>) auto
```
```   555     finally show "\<Union>K \<in> sigma_sets UNIV X" .
```
```   556   qed auto
```
```   557 qed (auto simp: eq intro: generate_topology.Basis)
```
```   558
```
```   559 lemma%unimportant borel_eq_closed: "borel = sigma UNIV (Collect closed)"
```
```   560   unfolding borel_def
```
```   561 proof (intro sigma_eqI sigma_sets_eqI, safe)
```
```   562   fix x :: "'a set" assume "open x"
```
```   563   hence "x = UNIV - (UNIV - x)" by auto
```
```   564   also have "\<dots> \<in> sigma_sets UNIV (Collect closed)"
```
```   565     by (force intro: sigma_sets.Compl simp: \<open>open x\<close>)
```
```   566   finally show "x \<in> sigma_sets UNIV (Collect closed)" by simp
```
```   567 next
```
```   568   fix x :: "'a set" assume "closed x"
```
```   569   hence "x = UNIV - (UNIV - x)" by auto
```
```   570   also have "\<dots> \<in> sigma_sets UNIV (Collect open)"
```
```   571     by (force intro: sigma_sets.Compl simp: \<open>closed x\<close>)
```
```   572   finally show "x \<in> sigma_sets UNIV (Collect open)" by simp
```
```   573 qed simp_all
```
```   574
```
```   575 lemma%important borel_eq_countable_basis:
```
```   576   fixes B::"'a::topological_space set set"
```
```   577   assumes "countable B"
```
```   578   assumes "topological_basis B"
```
```   579   shows "borel = sigma UNIV B"
```
```   580   unfolding borel_def
```
```   581 proof%unimportant (intro sigma_eqI sigma_sets_eqI, safe)
```
```   582   interpret countable_basis using assms by unfold_locales
```
```   583   fix X::"'a set" assume "open X"
```
```   584   from open_countable_basisE[OF this] guess B' . note B' = this
```
```   585   then show "X \<in> sigma_sets UNIV B"
```
```   586     by (blast intro: sigma_sets_UNION \<open>countable B\<close> countable_subset)
```
```   587 next
```
```   588   fix b assume "b \<in> B"
```
```   589   hence "open b" by (rule topological_basis_open[OF assms(2)])
```
```   590   thus "b \<in> sigma_sets UNIV (Collect open)" by auto
```
```   591 qed simp_all
```
```   592
```
```   593 lemma%unimportant borel_measurable_continuous_on_restrict:
```
```   594   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
```
```   595   assumes f: "continuous_on A f"
```
```   596   shows "f \<in> borel_measurable (restrict_space borel A)"
```
```   597 proof (rule borel_measurableI)
```
```   598   fix S :: "'b set" assume "open S"
```
```   599   with f obtain T where "f -` S \<inter> A = T \<inter> A" "open T"
```
```   600     by (metis continuous_on_open_invariant)
```
```   601   then show "f -` S \<inter> space (restrict_space borel A) \<in> sets (restrict_space borel A)"
```
```   602     by (force simp add: sets_restrict_space space_restrict_space)
```
```   603 qed
```
```   604
```
```   605 lemma%unimportant borel_measurable_continuous_on1: "continuous_on UNIV f \<Longrightarrow> f \<in> borel_measurable borel"
```
```   606   by (drule borel_measurable_continuous_on_restrict) simp
```
```   607
```
```   608 lemma%unimportant borel_measurable_continuous_on_if:
```
```   609   "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> continuous_on (- A) g \<Longrightarrow>
```
```   610     (\<lambda>x. if x \<in> A then f x else g x) \<in> borel_measurable borel"
```
```   611   by (auto simp add: measurable_If_restrict_space_iff Collect_neg_eq
```
```   612            intro!: borel_measurable_continuous_on_restrict)
```
```   613
```
```   614 lemma%unimportant borel_measurable_continuous_countable_exceptions:
```
```   615   fixes f :: "'a::t1_space \<Rightarrow> 'b::topological_space"
```
```   616   assumes X: "countable X"
```
```   617   assumes "continuous_on (- X) f"
```
```   618   shows "f \<in> borel_measurable borel"
```
```   619 proof (rule measurable_discrete_difference[OF _ X])
```
```   620   have "X \<in> sets borel"
```
```   621     by (rule sets.countable[OF _ X]) auto
```
```   622   then show "(\<lambda>x. if x \<in> X then undefined else f x) \<in> borel_measurable borel"
```
```   623     by (intro borel_measurable_continuous_on_if assms continuous_intros)
```
```   624 qed auto
```
```   625
```
```   626 lemma%unimportant borel_measurable_continuous_on:
```
```   627   assumes f: "continuous_on UNIV f" and g: "g \<in> borel_measurable M"
```
```   628   shows "(\<lambda>x. f (g x)) \<in> borel_measurable M"
```
```   629   using measurable_comp[OF g borel_measurable_continuous_on1[OF f]] by (simp add: comp_def)
```
```   630
```
```   631 lemma%unimportant borel_measurable_continuous_on_indicator:
```
```   632   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   633   shows "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable borel"
```
```   634   by (subst borel_measurable_restrict_space_iff[symmetric])
```
```   635      (auto intro: borel_measurable_continuous_on_restrict)
```
```   636
```
```   637 lemma%important borel_measurable_Pair[measurable (raw)]:
```
```   638   fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"
```
```   639   assumes f[measurable]: "f \<in> borel_measurable M"
```
```   640   assumes g[measurable]: "g \<in> borel_measurable M"
```
```   641   shows "(\<lambda>x. (f x, g x)) \<in> borel_measurable M"
```
```   642 proof%unimportant (subst borel_eq_countable_basis)
```
```   643   let ?B = "SOME B::'b set set. countable B \<and> topological_basis B"
```
```   644   let ?C = "SOME B::'c set set. countable B \<and> topological_basis B"
```
```   645   let ?P = "(\<lambda>(b, c). b \<times> c) ` (?B \<times> ?C)"
```
```   646   show "countable ?P" "topological_basis ?P"
```
```   647     by (auto intro!: countable_basis topological_basis_prod is_basis)
```
```   648
```
```   649   show "(\<lambda>x. (f x, g x)) \<in> measurable M (sigma UNIV ?P)"
```
```   650   proof (rule measurable_measure_of)
```
```   651     fix S assume "S \<in> ?P"
```
```   652     then obtain b c where "b \<in> ?B" "c \<in> ?C" and S: "S = b \<times> c" by auto
```
```   653     then have borel: "open b" "open c"
```
```   654       by (auto intro: is_basis topological_basis_open)
```
```   655     have "(\<lambda>x. (f x, g x)) -` S \<inter> space M = (f -` b \<inter> space M) \<inter> (g -` c \<inter> space M)"
```
```   656       unfolding S by auto
```
```   657     also have "\<dots> \<in> sets M"
```
```   658       using borel by simp
```
```   659     finally show "(\<lambda>x. (f x, g x)) -` S \<inter> space M \<in> sets M" .
```
```   660   qed auto
```
```   661 qed
```
```   662
```
```   663 lemma%important borel_measurable_continuous_Pair:
```
```   664   fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"
```
```   665   assumes [measurable]: "f \<in> borel_measurable M"
```
```   666   assumes [measurable]: "g \<in> borel_measurable M"
```
```   667   assumes H: "continuous_on UNIV (\<lambda>x. H (fst x) (snd x))"
```
```   668   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
```
```   669 proof%unimportant -
```
```   670   have eq: "(\<lambda>x. H (f x) (g x)) = (\<lambda>x. (\<lambda>x. H (fst x) (snd x)) (f x, g x))" by auto
```
```   671   show ?thesis
```
```   672     unfolding eq by (rule borel_measurable_continuous_on[OF H]) auto
```
```   673 qed
```
```   674
```
```   675 subsection%important \<open>Borel spaces on order topologies\<close>
```
```   676
```
```   677 lemma%unimportant [measurable]:
```
```   678   fixes a b :: "'a::linorder_topology"
```
```   679   shows lessThan_borel: "{..< a} \<in> sets borel"
```
```   680     and greaterThan_borel: "{a <..} \<in> sets borel"
```
```   681     and greaterThanLessThan_borel: "{a<..<b} \<in> sets borel"
```
```   682     and atMost_borel: "{..a} \<in> sets borel"
```
```   683     and atLeast_borel: "{a..} \<in> sets borel"
```
```   684     and atLeastAtMost_borel: "{a..b} \<in> sets borel"
```
```   685     and greaterThanAtMost_borel: "{a<..b} \<in> sets borel"
```
```   686     and atLeastLessThan_borel: "{a..<b} \<in> sets borel"
```
```   687   unfolding greaterThanAtMost_def atLeastLessThan_def
```
```   688   by (blast intro: borel_open borel_closed open_lessThan open_greaterThan open_greaterThanLessThan
```
```   689                    closed_atMost closed_atLeast closed_atLeastAtMost)+
```
```   690
```
```   691 lemma%unimportant borel_Iio:
```
```   692   "borel = sigma UNIV (range lessThan :: 'a::{linorder_topology, second_countable_topology} set set)"
```
```   693   unfolding second_countable_borel_measurable[OF open_generated_order]
```
```   694 proof (intro sigma_eqI sigma_sets_eqI)
```
```   695   from countable_dense_setE guess D :: "'a set" . note D = this
```
```   696
```
```   697   interpret L: sigma_algebra UNIV "sigma_sets UNIV (range lessThan)"
```
```   698     by (rule sigma_algebra_sigma_sets) simp
```
```   699
```
```   700   fix A :: "'a set" assume "A \<in> range lessThan \<union> range greaterThan"
```
```   701   then obtain y where "A = {y <..} \<or> A = {..< y}"
```
```   702     by blast
```
```   703   then show "A \<in> sigma_sets UNIV (range lessThan)"
```
```   704   proof
```
```   705     assume A: "A = {y <..}"
```
```   706     show ?thesis
```
```   707     proof cases
```
```   708       assume "\<forall>x>y. \<exists>d. y < d \<and> d < x"
```
```   709       with D(2)[of "{y <..< x}" for x] have "\<forall>x>y. \<exists>d\<in>D. y < d \<and> d < x"
```
```   710         by (auto simp: set_eq_iff)
```
```   711       then have "A = UNIV - (\<Inter>d\<in>{d\<in>D. y < d}. {..< d})"
```
```   712         by (auto simp: A) (metis less_asym)
```
```   713       also have "\<dots> \<in> sigma_sets UNIV (range lessThan)"
```
```   714         using D(1) by (intro L.Diff L.top L.countable_INT'') auto
```
```   715       finally show ?thesis .
```
```   716     next
```
```   717       assume "\<not> (\<forall>x>y. \<exists>d. y < d \<and> d < x)"
```
```   718       then obtain x where "y < x"  "\<And>d. y < d \<Longrightarrow> \<not> d < x"
```
```   719         by auto
```
```   720       then have "A = UNIV - {..< x}"
```
```   721         unfolding A by (auto simp: not_less[symmetric])
```
```   722       also have "\<dots> \<in> sigma_sets UNIV (range lessThan)"
```
```   723         by auto
```
```   724       finally show ?thesis .
```
```   725     qed
```
```   726   qed auto
```
```   727 qed auto
```
```   728
```
```   729 lemma%unimportant borel_Ioi:
```
```   730   "borel = sigma UNIV (range greaterThan :: 'a::{linorder_topology, second_countable_topology} set set)"
```
```   731   unfolding second_countable_borel_measurable[OF open_generated_order]
```
```   732 proof (intro sigma_eqI sigma_sets_eqI)
```
```   733   from countable_dense_setE guess D :: "'a set" . note D = this
```
```   734
```
```   735   interpret L: sigma_algebra UNIV "sigma_sets UNIV (range greaterThan)"
```
```   736     by (rule sigma_algebra_sigma_sets) simp
```
```   737
```
```   738   fix A :: "'a set" assume "A \<in> range lessThan \<union> range greaterThan"
```
```   739   then obtain y where "A = {y <..} \<or> A = {..< y}"
```
```   740     by blast
```
```   741   then show "A \<in> sigma_sets UNIV (range greaterThan)"
```
```   742   proof
```
```   743     assume A: "A = {..< y}"
```
```   744     show ?thesis
```
```   745     proof cases
```
```   746       assume "\<forall>x<y. \<exists>d. x < d \<and> d < y"
```
```   747       with D(2)[of "{x <..< y}" for x] have "\<forall>x<y. \<exists>d\<in>D. x < d \<and> d < y"
```
```   748         by (auto simp: set_eq_iff)
```
```   749       then have "A = UNIV - (\<Inter>d\<in>{d\<in>D. d < y}. {d <..})"
```
```   750         by (auto simp: A) (metis less_asym)
```
```   751       also have "\<dots> \<in> sigma_sets UNIV (range greaterThan)"
```
```   752         using D(1) by (intro L.Diff L.top L.countable_INT'') auto
```
```   753       finally show ?thesis .
```
```   754     next
```
```   755       assume "\<not> (\<forall>x<y. \<exists>d. x < d \<and> d < y)"
```
```   756       then obtain x where "x < y"  "\<And>d. y > d \<Longrightarrow> x \<ge> d"
```
```   757         by (auto simp: not_less[symmetric])
```
```   758       then have "A = UNIV - {x <..}"
```
```   759         unfolding A Compl_eq_Diff_UNIV[symmetric] by auto
```
```   760       also have "\<dots> \<in> sigma_sets UNIV (range greaterThan)"
```
```   761         by auto
```
```   762       finally show ?thesis .
```
```   763     qed
```
```   764   qed auto
```
```   765 qed auto
```
```   766
```
```   767 lemma%unimportant borel_measurableI_less:
```
```   768   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
```
```   769   shows "(\<And>y. {x\<in>space M. f x < y} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
```
```   770   unfolding borel_Iio
```
```   771   by (rule measurable_measure_of) (auto simp: Int_def conj_commute)
```
```   772
```
```   773 lemma%important borel_measurableI_greater:
```
```   774   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
```
```   775   shows "(\<And>y. {x\<in>space M. y < f x} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
```
```   776   unfolding borel_Ioi
```
```   777   by%unimportant (rule measurable_measure_of) (auto simp: Int_def conj_commute)
```
```   778
```
```   779 lemma%unimportant borel_measurableI_le:
```
```   780   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
```
```   781   shows "(\<And>y. {x\<in>space M. f x \<le> y} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
```
```   782   by (rule borel_measurableI_greater) (auto simp: not_le[symmetric])
```
```   783
```
```   784 lemma%unimportant borel_measurableI_ge:
```
```   785   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
```
```   786   shows "(\<And>y. {x\<in>space M. y \<le> f x} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
```
```   787   by (rule borel_measurableI_less) (auto simp: not_le[symmetric])
```
```   788
```
```   789 lemma%unimportant borel_measurable_less[measurable]:
```
```   790   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
```
```   791   assumes "f \<in> borel_measurable M"
```
```   792   assumes "g \<in> borel_measurable M"
```
```   793   shows "{w \<in> space M. f w < g w} \<in> sets M"
```
```   794 proof -
```
```   795   have "{w \<in> space M. f w < g w} = (\<lambda>x. (f x, g x)) -` {x. fst x < snd x} \<inter> space M"
```
```   796     by auto
```
```   797   also have "\<dots> \<in> sets M"
```
```   798     by (intro measurable_sets[OF borel_measurable_Pair borel_open, OF assms open_Collect_less]
```
```   799               continuous_intros)
```
```   800   finally show ?thesis .
```
```   801 qed
```
```   802
```
```   803 lemma%important
```
```   804   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
```
```   805   assumes f[measurable]: "f \<in> borel_measurable M"
```
```   806   assumes g[measurable]: "g \<in> borel_measurable M"
```
```   807   shows borel_measurable_le[measurable]: "{w \<in> space M. f w \<le> g w} \<in> sets M"
```
```   808     and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M"
```
```   809     and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
```
```   810   unfolding eq_iff not_less[symmetric]
```
```   811   by%unimportant measurable
```
```   812
```
```   813 lemma%important borel_measurable_SUP[measurable (raw)]:
```
```   814   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}"
```
```   815   assumes [simp]: "countable I"
```
```   816   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```   817   shows "(\<lambda>x. SUP i:I. F i x) \<in> borel_measurable M"
```
```   818   by%unimportant (rule borel_measurableI_greater) (simp add: less_SUP_iff)
```
```   819
```
```   820 lemma%unimportant borel_measurable_INF[measurable (raw)]:
```
```   821   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}"
```
```   822   assumes [simp]: "countable I"
```
```   823   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```   824   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
```
```   825   by (rule borel_measurableI_less) (simp add: INF_less_iff)
```
```   826
```
```   827 lemma%unimportant borel_measurable_cSUP[measurable (raw)]:
```
```   828   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology, second_countable_topology}"
```
```   829   assumes [simp]: "countable I"
```
```   830   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```   831   assumes bdd: "\<And>x. x \<in> space M \<Longrightarrow> bdd_above ((\<lambda>i. F i x) ` I)"
```
```   832   shows "(\<lambda>x. SUP i:I. F i x) \<in> borel_measurable M"
```
```   833 proof cases
```
```   834   assume "I = {}" then show ?thesis
```
```   835     unfolding \<open>I = {}\<close> image_empty by simp
```
```   836 next
```
```   837   assume "I \<noteq> {}"
```
```   838   show ?thesis
```
```   839   proof (rule borel_measurableI_le)
```
```   840     fix y
```
```   841     have "{x \<in> space M. \<forall>i\<in>I. F i x \<le> y} \<in> sets M"
```
```   842       by measurable
```
```   843     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}"
```
```   844       by (simp add: cSUP_le_iff \<open>I \<noteq> {}\<close> bdd cong: conj_cong)
```
```   845     finally show "{x \<in> space M. (SUP i:I. F i x) \<le>  y} \<in> sets M"  .
```
```   846   qed
```
```   847 qed
```
```   848
```
```   849 lemma%important borel_measurable_cINF[measurable (raw)]:
```
```   850   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology, second_countable_topology}"
```
```   851   assumes [simp]: "countable I"
```
```   852   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```   853   assumes bdd: "\<And>x. x \<in> space M \<Longrightarrow> bdd_below ((\<lambda>i. F i x) ` I)"
```
```   854   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
```
```   855 proof%unimportant cases
```
```   856   assume "I = {}" then show ?thesis
```
```   857     unfolding \<open>I = {}\<close> image_empty by simp
```
```   858 next
```
```   859   assume "I \<noteq> {}"
```
```   860   show ?thesis
```
```   861   proof (rule borel_measurableI_ge)
```
```   862     fix y
```
```   863     have "{x \<in> space M. \<forall>i\<in>I. y \<le> F i x} \<in> sets M"
```
```   864       by measurable
```
```   865     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)}"
```
```   866       by (simp add: le_cINF_iff \<open>I \<noteq> {}\<close> bdd cong: conj_cong)
```
```   867     finally show "{x \<in> space M. y \<le> (INF i:I. F i x)} \<in> sets M"  .
```
```   868   qed
```
```   869 qed
```
```   870
```
```   871 lemma%unimportant borel_measurable_lfp[consumes 1, case_names continuity step]:
```
```   872   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})"
```
```   873   assumes "sup_continuous F"
```
```   874   assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M"
```
```   875   shows "lfp F \<in> borel_measurable M"
```
```   876 proof -
```
```   877   { fix i have "((F ^^ i) bot) \<in> borel_measurable M"
```
```   878       by (induct i) (auto intro!: *) }
```
```   879   then have "(\<lambda>x. SUP i. (F ^^ i) bot x) \<in> borel_measurable M"
```
```   880     by measurable
```
```   881   also have "(\<lambda>x. SUP i. (F ^^ i) bot x) = (SUP i. (F ^^ i) bot)"
```
```   882     by auto
```
```   883   also have "(SUP i. (F ^^ i) bot) = lfp F"
```
```   884     by (rule sup_continuous_lfp[symmetric]) fact
```
```   885   finally show ?thesis .
```
```   886 qed
```
```   887
```
```   888 lemma%unimportant borel_measurable_gfp[consumes 1, case_names continuity step]:
```
```   889   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})"
```
```   890   assumes "inf_continuous F"
```
```   891   assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M"
```
```   892   shows "gfp F \<in> borel_measurable M"
```
```   893 proof -
```
```   894   { fix i have "((F ^^ i) top) \<in> borel_measurable M"
```
```   895       by (induct i) (auto intro!: * simp: bot_fun_def) }
```
```   896   then have "(\<lambda>x. INF i. (F ^^ i) top x) \<in> borel_measurable M"
```
```   897     by measurable
```
```   898   also have "(\<lambda>x. INF i. (F ^^ i) top x) = (INF i. (F ^^ i) top)"
```
```   899     by auto
```
```   900   also have "\<dots> = gfp F"
```
```   901     by (rule inf_continuous_gfp[symmetric]) fact
```
```   902   finally show ?thesis .
```
```   903 qed
```
```   904
```
```   905 lemma%unimportant borel_measurable_max[measurable (raw)]:
```
```   906   "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"
```
```   907   by (rule borel_measurableI_less) simp
```
```   908
```
```   909 lemma%unimportant borel_measurable_min[measurable (raw)]:
```
```   910   "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"
```
```   911   by (rule borel_measurableI_greater) simp
```
```   912
```
```   913 lemma%unimportant borel_measurable_Min[measurable (raw)]:
```
```   914   "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"
```
```   915 proof (induct I rule: finite_induct)
```
```   916   case (insert i I) then show ?case
```
```   917     by (cases "I = {}") auto
```
```   918 qed auto
```
```   919
```
```   920 lemma%unimportant borel_measurable_Max[measurable (raw)]:
```
```   921   "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"
```
```   922 proof (induct I rule: finite_induct)
```
```   923   case (insert i I) then show ?case
```
```   924     by (cases "I = {}") auto
```
```   925 qed auto
```
```   926
```
```   927 lemma%unimportant borel_measurable_sup[measurable (raw)]:
```
```   928   "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"
```
```   929   unfolding sup_max by measurable
```
```   930
```
```   931 lemma%unimportant borel_measurable_inf[measurable (raw)]:
```
```   932   "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"
```
```   933   unfolding inf_min by measurable
```
```   934
```
```   935 lemma%unimportant [measurable (raw)]:
```
```   936   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   937   assumes "\<And>i. f i \<in> borel_measurable M"
```
```   938   shows borel_measurable_liminf: "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```   939     and borel_measurable_limsup: "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```   940   unfolding liminf_SUP_INF limsup_INF_SUP using assms by auto
```
```   941
```
```   942 lemma%unimportant measurable_convergent[measurable (raw)]:
```
```   943   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   944   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```   945   shows "Measurable.pred M (\<lambda>x. convergent (\<lambda>i. f i x))"
```
```   946   unfolding convergent_ereal by measurable
```
```   947
```
```   948 lemma%unimportant sets_Collect_convergent[measurable]:
```
```   949   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   950   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```   951   shows "{x\<in>space M. convergent (\<lambda>i. f i x)} \<in> sets M"
```
```   952   by measurable
```
```   953
```
```   954 lemma%unimportant borel_measurable_lim[measurable (raw)]:
```
```   955   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   956   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```   957   shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```   958 proof -
```
```   959   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))"
```
```   960     by (simp add: lim_def convergent_def convergent_limsup_cl)
```
```   961   then show ?thesis
```
```   962     by simp
```
```   963 qed
```
```   964
```
```   965 lemma%unimportant borel_measurable_LIMSEQ_order:
```
```   966   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
```
```   967   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
```
```   968   and u: "\<And>i. u i \<in> borel_measurable M"
```
```   969   shows "u' \<in> borel_measurable M"
```
```   970 proof -
```
```   971   have "\<And>x. x \<in> space M \<Longrightarrow> u' x = liminf (\<lambda>n. u n x)"
```
```   972     using u' by (simp add: lim_imp_Liminf[symmetric])
```
```   973   with u show ?thesis by (simp cong: measurable_cong)
```
```   974 qed
```
```   975
```
```   976 subsection%important \<open>Borel spaces on topological monoids\<close>
```
```   977
```
```   978 lemma%unimportant borel_measurable_add[measurable (raw)]:
```
```   979   fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, topological_monoid_add}"
```
```   980   assumes f: "f \<in> borel_measurable M"
```
```   981   assumes g: "g \<in> borel_measurable M"
```
```   982   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```   983   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```   984
```
```   985 lemma%unimportant borel_measurable_sum[measurable (raw)]:
```
```   986   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, topological_comm_monoid_add}"
```
```   987   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   988   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```   989 proof cases
```
```   990   assume "finite S"
```
```   991   thus ?thesis using assms by induct auto
```
```   992 qed simp
```
```   993
```
```   994 lemma%important borel_measurable_suminf_order[measurable (raw)]:
```
```   995   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology, topological_comm_monoid_add}"
```
```   996   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```   997   shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```   998   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
```
```   999
```
```  1000 subsection%important \<open>Borel spaces on Euclidean spaces\<close>
```
```  1001
```
```  1002 lemma%important borel_measurable_inner[measurable (raw)]:
```
```  1003   fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_inner}"
```
```  1004   assumes "f \<in> borel_measurable M"
```
```  1005   assumes "g \<in> borel_measurable M"
```
```  1006   shows "(\<lambda>x. f x \<bullet> g x) \<in> borel_measurable M"
```
```  1007   using assms
```
```  1008   by%unimportant (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```  1009
```
```  1010 notation
```
```  1011   eucl_less (infix "<e" 50)
```
```  1012
```
```  1013 lemma%important box_oc: "{x. a <e x \<and> x \<le> b} = {x. a <e x} \<inter> {..b}"
```
```  1014   and box_co: "{x. a \<le> x \<and> x <e b} = {a..} \<inter> {x. x <e b}"
```
```  1015   by auto
```
```  1016
```
```  1017 lemma%important eucl_ivals[measurable]:
```
```  1018   fixes a b :: "'a::ordered_euclidean_space"
```
```  1019   shows "{x. x <e a} \<in> sets borel"
```
```  1020     and "{x. a <e x} \<in> sets borel"
```
```  1021     and "{..a} \<in> sets borel"
```
```  1022     and "{a..} \<in> sets borel"
```
```  1023     and "{a..b} \<in> sets borel"
```
```  1024     and  "{x. a <e x \<and> x \<le> b} \<in> sets borel"
```
```  1025     and "{x. a \<le> x \<and>  x <e b} \<in> sets borel"
```
```  1026   unfolding box_oc box_co
```
```  1027   by (auto intro: borel_open borel_closed)
```
```  1028
```
```  1029 lemma%unimportant (*FIX ME this has no name *)
```
```  1030   fixes i :: "'a::{second_countable_topology, real_inner}"
```
```  1031   shows hafspace_less_borel: "{x. a < x \<bullet> i} \<in> sets borel"
```
```  1032     and hafspace_greater_borel: "{x. x \<bullet> i < a} \<in> sets borel"
```
```  1033     and hafspace_less_eq_borel: "{x. a \<le> x \<bullet> i} \<in> sets borel"
```
```  1034     and hafspace_greater_eq_borel: "{x. x \<bullet> i \<le> a} \<in> sets borel"
```
```  1035   by simp_all
```
```  1036
```
```  1037 lemma%unimportant borel_eq_box:
```
```  1038   "borel = sigma UNIV (range (\<lambda> (a, b). box a b :: 'a :: euclidean_space set))"
```
```  1039     (is "_ = ?SIGMA")
```
```  1040 proof (rule borel_eq_sigmaI1[OF borel_def])
```
```  1041   fix M :: "'a set" assume "M \<in> {S. open S}"
```
```  1042   then have "open M" by simp
```
```  1043   show "M \<in> ?SIGMA"
```
```  1044     apply (subst open_UNION_box[OF \<open>open M\<close>])
```
```  1045     apply (safe intro!: sets.countable_UN' countable_PiE countable_Collect)
```
```  1046     apply (auto intro: countable_rat)
```
```  1047     done
```
```  1048 qed (auto simp: box_def)
```
```  1049
```
```  1050 lemma%unimportant halfspace_gt_in_halfspace:
```
```  1051   assumes i: "i \<in> A"
```
```  1052   shows "{x::'a. a < x \<bullet> i} \<in>
```
```  1053     sigma_sets UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> A))"
```
```  1054   (is "?set \<in> ?SIGMA")
```
```  1055 proof -
```
```  1056   interpret sigma_algebra UNIV ?SIGMA
```
```  1057     by (intro sigma_algebra_sigma_sets) simp_all
```
```  1058   have *: "?set = (\<Union>n. UNIV - {x::'a. x \<bullet> i < a + 1 / real (Suc n)})"
```
```  1059   proof (safe, simp_all add: not_less del: of_nat_Suc)
```
```  1060     fix x :: 'a assume "a < x \<bullet> i"
```
```  1061     with reals_Archimedean[of "x \<bullet> i - a"]
```
```  1062     obtain n where "a + 1 / real (Suc n) < x \<bullet> i"
```
```  1063       by (auto simp: field_simps)
```
```  1064     then show "\<exists>n. a + 1 / real (Suc n) \<le> x \<bullet> i"
```
```  1065       by (blast intro: less_imp_le)
```
```  1066   next
```
```  1067     fix x n
```
```  1068     have "a < a + 1 / real (Suc n)" by auto
```
```  1069     also assume "\<dots> \<le> x"
```
```  1070     finally show "a < x" .
```
```  1071   qed
```
```  1072   show "?set \<in> ?SIGMA" unfolding *
```
```  1073     by (auto intro!: Diff sigma_sets_Inter i)
```
```  1074 qed
```
```  1075
```
```  1076 lemma%unimportant borel_eq_halfspace_less:
```
```  1077   "borel = sigma UNIV ((\<lambda>(a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> Basis))"
```
```  1078   (is "_ = ?SIGMA")
```
```  1079 proof (rule borel_eq_sigmaI2[OF borel_eq_box])
```
```  1080   fix a b :: 'a
```
```  1081   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}"
```
```  1082     by (auto simp: box_def)
```
```  1083   also have "\<dots> \<in> sets ?SIGMA"
```
```  1084     by (intro sets.sets_Collect_conj sets.sets_Collect_finite_All sets.sets_Collect_const)
```
```  1085        (auto intro!: halfspace_gt_in_halfspace countable_PiE countable_rat)
```
```  1086   finally show "box a b \<in> sets ?SIGMA" .
```
```  1087 qed auto
```
```  1088
```
```  1089 lemma%unimportant borel_eq_halfspace_le:
```
```  1090   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i \<le> a}) ` (UNIV \<times> Basis))"
```
```  1091   (is "_ = ?SIGMA")
```
```  1092 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
```
```  1093   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```  1094   then have i: "i \<in> Basis" by auto
```
```  1095   have *: "{x::'a. x\<bullet>i < a} = (\<Union>n. {x. x\<bullet>i \<le> a - 1/real (Suc n)})"
```
```  1096   proof (safe, simp_all del: of_nat_Suc)
```
```  1097     fix x::'a assume *: "x\<bullet>i < a"
```
```  1098     with reals_Archimedean[of "a - x\<bullet>i"]
```
```  1099     obtain n where "x \<bullet> i < a - 1 / (real (Suc n))"
```
```  1100       by (auto simp: field_simps)
```
```  1101     then show "\<exists>n. x \<bullet> i \<le> a - 1 / (real (Suc n))"
```
```  1102       by (blast intro: less_imp_le)
```
```  1103   next
```
```  1104     fix x::'a and n
```
```  1105     assume "x\<bullet>i \<le> a - 1 / real (Suc n)"
```
```  1106     also have "\<dots> < a" by auto
```
```  1107     finally show "x\<bullet>i < a" .
```
```  1108   qed
```
```  1109   show "{x. x\<bullet>i < a} \<in> ?SIGMA" unfolding *
```
```  1110     by (intro sets.countable_UN) (auto intro: i)
```
```  1111 qed auto
```
```  1112
```
```  1113 lemma%unimportant borel_eq_halfspace_ge:
```
```  1114   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. a \<le> x \<bullet> i}) ` (UNIV \<times> Basis))"
```
```  1115   (is "_ = ?SIGMA")
```
```  1116 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
```
```  1117   fix a :: real and i :: 'a assume i: "(a, i) \<in> UNIV \<times> Basis"
```
```  1118   have *: "{x::'a. x\<bullet>i < a} = space ?SIGMA - {x::'a. a \<le> x\<bullet>i}" by auto
```
```  1119   show "{x. x\<bullet>i < a} \<in> ?SIGMA" unfolding *
```
```  1120     using i by (intro sets.compl_sets) auto
```
```  1121 qed auto
```
```  1122
```
```  1123 lemma%important borel_eq_halfspace_greater:
```
```  1124   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. a < x \<bullet> i}) ` (UNIV \<times> Basis))"
```
```  1125   (is "_ = ?SIGMA")
```
```  1126 proof%unimportant (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])
```
```  1127   fix a :: real and i :: 'a assume "(a, i) \<in> (UNIV \<times> Basis)"
```
```  1128   then have i: "i \<in> Basis" by auto
```
```  1129   have *: "{x::'a. x\<bullet>i \<le> a} = space ?SIGMA - {x::'a. a < x\<bullet>i}" by auto
```
```  1130   show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA" unfolding *
```
```  1131     by (intro sets.compl_sets) (auto intro: i)
```
```  1132 qed auto
```
```  1133
```
```  1134 lemma%unimportant borel_eq_atMost:
```
```  1135   "borel = sigma UNIV (range (\<lambda>a. {..a::'a::ordered_euclidean_space}))"
```
```  1136   (is "_ = ?SIGMA")
```
```  1137 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
```
```  1138   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```  1139   then have "i \<in> Basis" by auto
```
```  1140   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)})"
```
```  1141   proof (safe, simp_all add: eucl_le[where 'a='a] split: if_split_asm)
```
```  1142     fix x :: 'a
```
```  1143     from real_arch_simple[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"] guess k::nat ..
```
```  1144     then have "\<And>i. i \<in> Basis \<Longrightarrow> x\<bullet>i \<le> real k"
```
```  1145       by (subst (asm) Max_le_iff) auto
```
```  1146     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> x \<bullet> ia \<le> real k"
```
```  1147       by (auto intro!: exI[of _ k])
```
```  1148   qed
```
```  1149   show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA" unfolding *
```
```  1150     by (intro sets.countable_UN) auto
```
```  1151 qed auto
```
```  1152
```
```  1153 lemma%unimportant borel_eq_greaterThan:
```
```  1154   "borel = sigma UNIV (range (\<lambda>a::'a::ordered_euclidean_space. {x. a <e x}))"
```
```  1155   (is "_ = ?SIGMA")
```
```  1156 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
```
```  1157   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```  1158   then have i: "i \<in> Basis" by auto
```
```  1159   have "{x::'a. x\<bullet>i \<le> a} = UNIV - {x::'a. a < x\<bullet>i}" by auto
```
```  1160   also have *: "{x::'a. a < x\<bullet>i} =
```
```  1161       (\<Union>k::nat. {x. (\<Sum>n\<in>Basis. (if n = i then a else -real k) *\<^sub>R n) <e x})" using i
```
```  1162   proof (safe, simp_all add: eucl_less_def split: if_split_asm)
```
```  1163     fix x :: 'a
```
```  1164     from reals_Archimedean2[of "Max ((\<lambda>i. -x\<bullet>i)`Basis)"]
```
```  1165     guess k::nat .. note k = this
```
```  1166     { fix i :: 'a assume "i \<in> Basis"
```
```  1167       then have "-x\<bullet>i < real k"
```
```  1168         using k by (subst (asm) Max_less_iff) auto
```
```  1169       then have "- real k < x\<bullet>i" by simp }
```
```  1170     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> -real k < x \<bullet> ia"
```
```  1171       by (auto intro!: exI[of _ k])
```
```  1172   qed
```
```  1173   finally show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA"
```
```  1174     apply (simp only:)
```
```  1175     apply (intro sets.countable_UN sets.Diff)
```
```  1176     apply (auto intro: sigma_sets_top)
```
```  1177     done
```
```  1178 qed auto
```
```  1179
```
```  1180 lemma%unimportant borel_eq_lessThan:
```
```  1181   "borel = sigma UNIV (range (\<lambda>a::'a::ordered_euclidean_space. {x. x <e a}))"
```
```  1182   (is "_ = ?SIGMA")
```
```  1183 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge])
```
```  1184   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
```
```  1185   then have i: "i \<in> Basis" by auto
```
```  1186   have "{x::'a. a \<le> x\<bullet>i} = UNIV - {x::'a. x\<bullet>i < a}" by auto
```
```  1187   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>
```
```  1188   proof (safe, simp_all add: eucl_less_def split: if_split_asm)
```
```  1189     fix x :: 'a
```
```  1190     from reals_Archimedean2[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"]
```
```  1191     guess k::nat .. note k = this
```
```  1192     { fix i :: 'a assume "i \<in> Basis"
```
```  1193       then have "x\<bullet>i < real k"
```
```  1194         using k by (subst (asm) Max_less_iff) auto
```
```  1195       then have "x\<bullet>i < real k" by simp }
```
```  1196     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> x \<bullet> ia < real k"
```
```  1197       by (auto intro!: exI[of _ k])
```
```  1198   qed
```
```  1199   finally show "{x. a \<le> x\<bullet>i} \<in> ?SIGMA"
```
```  1200     apply (simp only:)
```
```  1201     apply (intro sets.countable_UN sets.Diff)
```
```  1202     apply (auto intro: sigma_sets_top )
```
```  1203     done
```
```  1204 qed auto
```
```  1205
```
```  1206 lemma%important borel_eq_atLeastAtMost:
```
```  1207   "borel = sigma UNIV (range (\<lambda>(a,b). {a..b} ::'a::ordered_euclidean_space set))"
```
```  1208   (is "_ = ?SIGMA")
```
```  1209 proof%unimportant (rule borel_eq_sigmaI5[OF borel_eq_atMost])
```
```  1210   fix a::'a
```
```  1211   have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
```
```  1212   proof (safe, simp_all add: eucl_le[where 'a='a])
```
```  1213     fix x :: 'a
```
```  1214     from real_arch_simple[of "Max ((\<lambda>i. - x\<bullet>i)`Basis)"]
```
```  1215     guess k::nat .. note k = this
```
```  1216     { fix i :: 'a assume "i \<in> Basis"
```
```  1217       with k have "- x\<bullet>i \<le> real k"
```
```  1218         by (subst (asm) Max_le_iff) (auto simp: field_simps)
```
```  1219       then have "- real k \<le> x\<bullet>i" by simp }
```
```  1220     then show "\<exists>n::nat. \<forall>i\<in>Basis. - real n \<le> x \<bullet> i"
```
```  1221       by (auto intro!: exI[of _ k])
```
```  1222   qed
```
```  1223   show "{..a} \<in> ?SIGMA" unfolding *
```
```  1224     by (intro sets.countable_UN)
```
```  1225        (auto intro!: sigma_sets_top)
```
```  1226 qed auto
```
```  1227
```
```  1228 lemma%important borel_set_induct[consumes 1, case_names empty interval compl union]:
```
```  1229   assumes "A \<in> sets borel"
```
```  1230   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
```
```  1231           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)"
```
```  1232   shows "P (A::real set)"
```
```  1233 proof%unimportant -
```
```  1234   let ?G = "range (\<lambda>(a,b). {a..b::real})"
```
```  1235   have "Int_stable ?G" "?G \<subseteq> Pow UNIV" "A \<in> sigma_sets UNIV ?G"
```
```  1236       using assms(1) by (auto simp add: borel_eq_atLeastAtMost Int_stable_def)
```
```  1237   thus ?thesis
```
```  1238   proof (induction rule: sigma_sets_induct_disjoint)
```
```  1239     case (union f)
```
```  1240       from union.hyps(2) have "\<And>i. f i \<in> sets borel" by (auto simp: borel_eq_atLeastAtMost)
```
```  1241       with union show ?case by (auto intro: un)
```
```  1242   next
```
```  1243     case (basic A)
```
```  1244     then obtain a b where "A = {a .. b}" by auto
```
```  1245     then show ?case
```
```  1246       by (cases "a \<le> b") (auto intro: int empty)
```
```  1247   qed (auto intro: empty compl simp: Compl_eq_Diff_UNIV[symmetric] borel_eq_atLeastAtMost)
```
```  1248 qed
```
```  1249
```
```  1250 lemma%unimportant borel_sigma_sets_Ioc: "borel = sigma UNIV (range (\<lambda>(a, b). {a <.. b::real}))"
```
```  1251 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
```
```  1252   fix i :: real
```
```  1253   have "{..i} = (\<Union>j::nat. {-j <.. i})"
```
```  1254     by (auto simp: minus_less_iff reals_Archimedean2)
```
```  1255   also have "\<dots> \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))"
```
```  1256     by (intro sets.countable_nat_UN) auto
```
```  1257   finally show "{..i} \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))" .
```
```  1258 qed simp
```
```  1259
```
```  1260 lemma%unimportant eucl_lessThan: "{x::real. x <e a} = lessThan a"
```
```  1261   by (simp add: eucl_less_def lessThan_def)
```
```  1262
```
```  1263 lemma%unimportant borel_eq_atLeastLessThan:
```
```  1264   "borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA")
```
```  1265 proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan])
```
```  1266   have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto
```
```  1267   fix x :: real
```
```  1268   have "{..<x} = (\<Union>i::nat. {-real i ..< x})"
```
```  1269     by (auto simp: move_uminus real_arch_simple)
```
```  1270   then show "{y. y <e x} \<in> ?SIGMA"
```
```  1271     by (auto intro: sigma_sets.intros(2-) simp: eucl_lessThan)
```
```  1272 qed auto
```
```  1273
```
```  1274 lemma%unimportant borel_measurable_halfspacesI:
```
```  1275   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1276   assumes F: "borel = sigma UNIV (F ` (UNIV \<times> Basis))"
```
```  1277   and S_eq: "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
```
```  1278   shows "f \<in> borel_measurable M = (\<forall>i\<in>Basis. \<forall>a::real. S a i \<in> sets M)"
```
```  1279 proof safe
```
```  1280   fix a :: real and i :: 'b assume i: "i \<in> Basis" and f: "f \<in> borel_measurable M"
```
```  1281   then show "S a i \<in> sets M" unfolding assms
```
```  1282     by (auto intro!: measurable_sets simp: assms(1))
```
```  1283 next
```
```  1284   assume a: "\<forall>i\<in>Basis. \<forall>a. S a i \<in> sets M"
```
```  1285   then show "f \<in> borel_measurable M"
```
```  1286     by (auto intro!: measurable_measure_of simp: S_eq F)
```
```  1287 qed
```
```  1288
```
```  1289 lemma%unimportant borel_measurable_iff_halfspace_le:
```
```  1290   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1291   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)"
```
```  1292   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
```
```  1293
```
```  1294 lemma%unimportant borel_measurable_iff_halfspace_less:
```
```  1295   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1296   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)"
```
```  1297   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
```
```  1298
```
```  1299 lemma%unimportant borel_measurable_iff_halfspace_ge:
```
```  1300   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1301   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)"
```
```  1302   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
```
```  1303
```
```  1304 lemma%unimportant borel_measurable_iff_halfspace_greater:
```
```  1305   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1306   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)"
```
```  1307   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto
```
```  1308
```
```  1309 lemma%unimportant borel_measurable_iff_le:
```
```  1310   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
```
```  1311   using borel_measurable_iff_halfspace_le[where 'c=real] by simp
```
```  1312
```
```  1313 lemma%unimportant borel_measurable_iff_less:
```
```  1314   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
```
```  1315   using borel_measurable_iff_halfspace_less[where 'c=real] by simp
```
```  1316
```
```  1317 lemma%unimportant borel_measurable_iff_ge:
```
```  1318   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
```
```  1319   using borel_measurable_iff_halfspace_ge[where 'c=real]
```
```  1320   by simp
```
```  1321
```
```  1322 lemma%unimportant borel_measurable_iff_greater:
```
```  1323   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
```
```  1324   using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
```
```  1325
```
```  1326 lemma%important borel_measurable_euclidean_space:
```
```  1327   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
```
```  1328   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M)"
```
```  1329 proof%unimportant safe
```
```  1330   assume f: "\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M"
```
```  1331   then show "f \<in> borel_measurable M"
```
```  1332     by (subst borel_measurable_iff_halfspace_le) auto
```
```  1333 qed auto
```
```  1334
```
```  1335 subsection%important "Borel measurable operators"
```
```  1336
```
```  1337 lemma%important borel_measurable_norm[measurable]: "norm \<in> borel_measurable borel"
```
```  1338   by%unimportant (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1339
```
```  1340 lemma%unimportant borel_measurable_sgn [measurable]: "(sgn::'a::real_normed_vector \<Rightarrow> 'a) \<in> borel_measurable borel"
```
```  1341   by (rule borel_measurable_continuous_countable_exceptions[where X="{0}"])
```
```  1342      (auto intro!: continuous_on_sgn continuous_on_id)
```
```  1343
```
```  1344 lemma%important borel_measurable_uminus[measurable (raw)]:
```
```  1345   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```  1346   assumes g: "g \<in> borel_measurable M"
```
```  1347   shows "(\<lambda>x. - g x) \<in> borel_measurable M"
```
```  1348   by%unimportant (rule borel_measurable_continuous_on[OF _ g]) (intro continuous_intros)
```
```  1349
```
```  1350 lemma%important borel_measurable_diff[measurable (raw)]:
```
```  1351   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```  1352   assumes f: "f \<in> borel_measurable M"
```
```  1353   assumes g: "g \<in> borel_measurable M"
```
```  1354   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```  1355   using%unimportant borel_measurable_add [of f M "- g"] assms by (simp add: fun_Compl_def)
```
```  1356
```
```  1357 lemma%unimportant borel_measurable_times[measurable (raw)]:
```
```  1358   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_algebra}"
```
```  1359   assumes f: "f \<in> borel_measurable M"
```
```  1360   assumes g: "g \<in> borel_measurable M"
```
```  1361   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1362   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```  1363
```
```  1364 lemma%important borel_measurable_prod[measurable (raw)]:
```
```  1365   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, real_normed_field}"
```
```  1366   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1367   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1368 proof%unimportant cases
```
```  1369   assume "finite S"
```
```  1370   thus ?thesis using assms by induct auto
```
```  1371 qed simp
```
```  1372
```
```  1373 lemma%important borel_measurable_dist[measurable (raw)]:
```
```  1374   fixes g f :: "'a \<Rightarrow> 'b::{second_countable_topology, metric_space}"
```
```  1375   assumes f: "f \<in> borel_measurable M"
```
```  1376   assumes g: "g \<in> borel_measurable M"
```
```  1377   shows "(\<lambda>x. dist (f x) (g x)) \<in> borel_measurable M"
```
```  1378   using%unimportant f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```  1379
```
```  1380 lemma%unimportant borel_measurable_scaleR[measurable (raw)]:
```
```  1381   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```  1382   assumes f: "f \<in> borel_measurable M"
```
```  1383   assumes g: "g \<in> borel_measurable M"
```
```  1384   shows "(\<lambda>x. f x *\<^sub>R g x) \<in> borel_measurable M"
```
```  1385   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
```
```  1386
```
```  1387 lemma%unimportant borel_measurable_uminus_eq [simp]:
```
```  1388   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
```
```  1389   shows "(\<lambda>x. - f x) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
```
```  1390 proof
```
```  1391   assume ?l from borel_measurable_uminus[OF this] show ?r by simp
```
```  1392 qed auto
```
```  1393
```
```  1394 lemma%unimportant affine_borel_measurable_vector:
```
```  1395   fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
```
```  1396   assumes "f \<in> borel_measurable M"
```
```  1397   shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
```
```  1398 proof (rule borel_measurableI)
```
```  1399   fix S :: "'x set" assume "open S"
```
```  1400   show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
```
```  1401   proof cases
```
```  1402     assume "b \<noteq> 0"
```
```  1403     with \<open>open S\<close> have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")
```
```  1404       using open_affinity [of S "inverse b" "- a /\<^sub>R b"]
```
```  1405       by (auto simp: algebra_simps)
```
```  1406     hence "?S \<in> sets borel" by auto
```
```  1407     moreover
```
```  1408     from \<open>b \<noteq> 0\<close> have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
```
```  1409       apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
```
```  1410     ultimately show ?thesis using assms unfolding in_borel_measurable_borel
```
```  1411       by auto
```
```  1412   qed simp
```
```  1413 qed
```
```  1414
```
```  1415 lemma%unimportant borel_measurable_const_scaleR[measurable (raw)]:
```
```  1416   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. b *\<^sub>R f x ::'a::real_normed_vector) \<in> borel_measurable M"
```
```  1417   using affine_borel_measurable_vector[of f M 0 b] by simp
```
```  1418
```
```  1419 lemma%unimportant borel_measurable_const_add[measurable (raw)]:
```
```  1420   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. a + f x ::'a::real_normed_vector) \<in> borel_measurable M"
```
```  1421   using affine_borel_measurable_vector[of f M a 1] by simp
```
```  1422
```
```  1423 lemma%unimportant borel_measurable_inverse[measurable (raw)]:
```
```  1424   fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
```
```  1425   assumes f: "f \<in> borel_measurable M"
```
```  1426   shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
```
```  1427   apply (rule measurable_compose[OF f])
```
```  1428   apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])
```
```  1429   apply (auto intro!: continuous_on_inverse continuous_on_id)
```
```  1430   done
```
```  1431
```
```  1432 lemma%unimportant borel_measurable_divide[measurable (raw)]:
```
```  1433   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>
```
```  1434     (\<lambda>x. f x / g x::'b::{second_countable_topology, real_normed_div_algebra}) \<in> borel_measurable M"
```
```  1435   by (simp add: divide_inverse)
```
```  1436
```
```  1437 lemma%unimportant borel_measurable_abs[measurable (raw)]:
```
```  1438   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
```
```  1439   unfolding abs_real_def by simp
```
```  1440
```
```  1441 lemma%unimportant borel_measurable_nth[measurable (raw)]:
```
```  1442   "(\<lambda>x::real^'n. x \$ i) \<in> borel_measurable borel"
```
```  1443   by (simp add: cart_eq_inner_axis)
```
```  1444
```
```  1445 lemma%important convex_measurable:
```
```  1446   fixes A :: "'a :: euclidean_space set"
```
```  1447   shows "X \<in> borel_measurable M \<Longrightarrow> X ` space M \<subseteq> A \<Longrightarrow> open A \<Longrightarrow> convex_on A q \<Longrightarrow>
```
```  1448     (\<lambda>x. q (X x)) \<in> borel_measurable M"
```
```  1449   by%unimportant (rule measurable_compose[where f=X and N="restrict_space borel A"])
```
```  1450      (auto intro!: borel_measurable_continuous_on_restrict convex_on_continuous measurable_restrict_space2)
```
```  1451
```
```  1452 lemma%unimportant borel_measurable_ln[measurable (raw)]:
```
```  1453   assumes f: "f \<in> borel_measurable M"
```
```  1454   shows "(\<lambda>x. ln (f x :: real)) \<in> borel_measurable M"
```
```  1455   apply (rule measurable_compose[OF f])
```
```  1456   apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])
```
```  1457   apply (auto intro!: continuous_on_ln continuous_on_id)
```
```  1458   done
```
```  1459
```
```  1460 lemma%unimportant borel_measurable_log[measurable (raw)]:
```
```  1461   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. log (g x) (f x)) \<in> borel_measurable M"
```
```  1462   unfolding log_def by auto
```
```  1463
```
```  1464 lemma%unimportant borel_measurable_exp[measurable]:
```
```  1465   "(exp::'a::{real_normed_field,banach}\<Rightarrow>'a) \<in> borel_measurable borel"
```
```  1466   by (intro borel_measurable_continuous_on1 continuous_at_imp_continuous_on ballI isCont_exp)
```
```  1467
```
```  1468 lemma%unimportant measurable_real_floor[measurable]:
```
```  1469   "(floor :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
```
```  1470 proof -
```
```  1471   have "\<And>a x. \<lfloor>x\<rfloor> = a \<longleftrightarrow> (real_of_int a \<le> x \<and> x < real_of_int (a + 1))"
```
```  1472     by (auto intro: floor_eq2)
```
```  1473   then show ?thesis
```
```  1474     by (auto simp: vimage_def measurable_count_space_eq2_countable)
```
```  1475 qed
```
```  1476
```
```  1477 lemma%unimportant measurable_real_ceiling[measurable]:
```
```  1478   "(ceiling :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
```
```  1479   unfolding ceiling_def[abs_def] by simp
```
```  1480
```
```  1481 lemma%unimportant borel_measurable_real_floor: "(\<lambda>x::real. real_of_int \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
```
```  1482   by simp
```
```  1483
```
```  1484 lemma%unimportant borel_measurable_root [measurable]: "root n \<in> borel_measurable borel"
```
```  1485   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1486
```
```  1487 lemma%unimportant borel_measurable_sqrt [measurable]: "sqrt \<in> borel_measurable borel"
```
```  1488   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1489
```
```  1490 lemma%unimportant borel_measurable_power [measurable (raw)]:
```
```  1491   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```  1492   assumes f: "f \<in> borel_measurable M"
```
```  1493   shows "(\<lambda>x. (f x) ^ n) \<in> borel_measurable M"
```
```  1494   by (intro borel_measurable_continuous_on [OF _ f] continuous_intros)
```
```  1495
```
```  1496 lemma%unimportant borel_measurable_Re [measurable]: "Re \<in> borel_measurable borel"
```
```  1497   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1498
```
```  1499 lemma%unimportant borel_measurable_Im [measurable]: "Im \<in> borel_measurable borel"
```
```  1500   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1501
```
```  1502 lemma%unimportant borel_measurable_of_real [measurable]: "(of_real :: _ \<Rightarrow> (_::real_normed_algebra)) \<in> borel_measurable borel"
```
```  1503   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1504
```
```  1505 lemma%unimportant borel_measurable_sin [measurable]: "(sin :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"
```
```  1506   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1507
```
```  1508 lemma%unimportant borel_measurable_cos [measurable]: "(cos :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"
```
```  1509   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1510
```
```  1511 lemma%unimportant borel_measurable_arctan [measurable]: "arctan \<in> borel_measurable borel"
```
```  1512   by (intro borel_measurable_continuous_on1 continuous_intros)
```
```  1513
```
```  1514 lemma%important borel_measurable_complex_iff:
```
```  1515   "f \<in> borel_measurable M \<longleftrightarrow>
```
```  1516     (\<lambda>x. Re (f x)) \<in> borel_measurable M \<and> (\<lambda>x. Im (f x)) \<in> borel_measurable M"
```
```  1517   apply auto
```
```  1518   apply (subst fun_complex_eq)
```
```  1519   apply (intro borel_measurable_add)
```
```  1520   apply auto
```
```  1521   done
```
```  1522
```
```  1523 lemma%important powr_real_measurable [measurable]:
```
```  1524   assumes "f \<in> measurable M borel" "g \<in> measurable M borel"
```
```  1525   shows   "(\<lambda>x. f x powr g x :: real) \<in> measurable M borel"
```
```  1526   using%unimportant assms by (simp_all add: powr_def)
```
```  1527
```
```  1528 lemma%unimportant measurable_of_bool[measurable]: "of_bool \<in> count_space UNIV \<rightarrow>\<^sub>M borel"
```
```  1529   by simp
```
```  1530
```
```  1531 subsection%important "Borel space on the extended reals"
```
```  1532
```
```  1533 lemma%unimportant borel_measurable_ereal[measurable (raw)]:
```
```  1534   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
```
```  1535   using continuous_on_ereal f by (rule borel_measurable_continuous_on) (rule continuous_on_id)
```
```  1536
```
```  1537 lemma%unimportant borel_measurable_real_of_ereal[measurable (raw)]:
```
```  1538   fixes f :: "'a \<Rightarrow> ereal"
```
```  1539   assumes f: "f \<in> borel_measurable M"
```
```  1540   shows "(\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M"
```
```  1541   apply (rule measurable_compose[OF f])
```
```  1542   apply (rule borel_measurable_continuous_countable_exceptions[of "{\<infinity>, -\<infinity> }"])
```
```  1543   apply (auto intro: continuous_on_real simp: Compl_eq_Diff_UNIV)
```
```  1544   done
```
```  1545
```
```  1546 lemma%unimportant borel_measurable_ereal_cases:
```
```  1547   fixes f :: "'a \<Rightarrow> ereal"
```
```  1548   assumes f: "f \<in> borel_measurable M"
```
```  1549   assumes H: "(\<lambda>x. H (ereal (real_of_ereal (f x)))) \<in> borel_measurable M"
```
```  1550   shows "(\<lambda>x. H (f x)) \<in> borel_measurable M"
```
```  1551 proof -
```
```  1552   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)))"
```
```  1553   { fix x have "H (f x) = ?F x" by (cases "f x") auto }
```
```  1554   with f H show ?thesis by simp
```
```  1555 qed
```
```  1556
```
```  1557 lemma%unimportant (*FIX ME needs a name *)
```
```  1558   fixes f :: "'a \<Rightarrow> ereal" assumes f[measurable]: "f \<in> borel_measurable M"
```
```  1559   shows borel_measurable_ereal_abs[measurable(raw)]: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
```
```  1560     and borel_measurable_ereal_inverse[measurable(raw)]: "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M"
```
```  1561     and borel_measurable_uminus_ereal[measurable(raw)]: "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M"
```
```  1562   by (auto simp del: abs_real_of_ereal simp: borel_measurable_ereal_cases[OF f] measurable_If)
```
```  1563
```
```  1564 lemma%unimportant borel_measurable_uminus_eq_ereal[simp]:
```
```  1565   "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
```
```  1566 proof
```
```  1567   assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp
```
```  1568 qed auto
```
```  1569
```
```  1570 lemma%important set_Collect_ereal2:
```
```  1571   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1572   assumes f: "f \<in> borel_measurable M"
```
```  1573   assumes g: "g \<in> borel_measurable M"
```
```  1574   assumes H: "{x \<in> space M. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))} \<in> sets M"
```
```  1575     "{x \<in> space borel. H (-\<infinity>) (ereal x)} \<in> sets borel"
```
```  1576     "{x \<in> space borel. H (\<infinity>) (ereal x)} \<in> sets borel"
```
```  1577     "{x \<in> space borel. H (ereal x) (-\<infinity>)} \<in> sets borel"
```
```  1578     "{x \<in> space borel. H (ereal x) (\<infinity>)} \<in> sets borel"
```
```  1579   shows "{x \<in> space M. H (f x) (g x)} \<in> sets M"
```
```  1580 proof%unimportant -
```
```  1581   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)))"
```
```  1582   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"
```
```  1583   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
```
```  1584   note * = this
```
```  1585   from assms show ?thesis
```
```  1586     by (subst *) (simp del: space_borel split del: if_split)
```
```  1587 qed
```
```  1588
```
```  1589 lemma%unimportant borel_measurable_ereal_iff:
```
```  1590   shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
```
```  1591 proof
```
```  1592   assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
```
```  1593   from borel_measurable_real_of_ereal[OF this]
```
```  1594   show "f \<in> borel_measurable M" by auto
```
```  1595 qed auto
```
```  1596
```
```  1597 lemma%unimportant borel_measurable_erealD[measurable_dest]:
```
```  1598   "(\<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"
```
```  1599   unfolding borel_measurable_ereal_iff by simp
```
```  1600
```
```  1601 lemma%important borel_measurable_ereal_iff_real:
```
```  1602   fixes f :: "'a \<Rightarrow> ereal"
```
```  1603   shows "f \<in> borel_measurable M \<longleftrightarrow>
```
```  1604     ((\<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)"
```
```  1605 proof%unimportant safe
```
```  1606   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"
```
```  1607   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
```
```  1608   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
```
```  1609   let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real_of_ereal (f x))"
```
```  1610   have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
```
```  1611   also have "?f = f" by (auto simp: fun_eq_iff ereal_real)
```
```  1612   finally show "f \<in> borel_measurable M" .
```
```  1613 qed simp_all
```
```  1614
```
```  1615 lemma%unimportant borel_measurable_ereal_iff_Iio:
```
```  1616   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
```
```  1617   by (auto simp: borel_Iio measurable_iff_measure_of)
```
```  1618
```
```  1619 lemma%unimportant borel_measurable_ereal_iff_Ioi:
```
```  1620   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
```
```  1621   by (auto simp: borel_Ioi measurable_iff_measure_of)
```
```  1622
```
```  1623 lemma%unimportant vimage_sets_compl_iff:
```
```  1624   "f -` A \<inter> space M \<in> sets M \<longleftrightarrow> f -` (- A) \<inter> space M \<in> sets M"
```
```  1625 proof -
```
```  1626   { fix A assume "f -` A \<inter> space M \<in> sets M"
```
```  1627     moreover have "f -` (- A) \<inter> space M = space M - f -` A \<inter> space M" by auto
```
```  1628     ultimately have "f -` (- A) \<inter> space M \<in> sets M" by auto }
```
```  1629   from this[of A] this[of "-A"] show ?thesis
```
```  1630     by (metis double_complement)
```
```  1631 qed
```
```  1632
```
```  1633 lemma%unimportant borel_measurable_iff_Iic_ereal:
```
```  1634   "(f::'a\<Rightarrow>ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
```
```  1635   unfolding borel_measurable_ereal_iff_Ioi vimage_sets_compl_iff[where A="{a <..}" for a] by simp
```
```  1636
```
```  1637 lemma%unimportant borel_measurable_iff_Ici_ereal:
```
```  1638   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
```
```  1639   unfolding borel_measurable_ereal_iff_Iio vimage_sets_compl_iff[where A="{..< a}" for a] by simp
```
```  1640
```
```  1641 lemma%important borel_measurable_ereal2:
```
```  1642   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1643   assumes f: "f \<in> borel_measurable M"
```
```  1644   assumes g: "g \<in> borel_measurable M"
```
```  1645   assumes H: "(\<lambda>x. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
```
```  1646     "(\<lambda>x. H (-\<infinity>) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
```
```  1647     "(\<lambda>x. H (\<infinity>) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
```
```  1648     "(\<lambda>x. H (ereal (real_of_ereal (f x))) (-\<infinity>)) \<in> borel_measurable M"
```
```  1649     "(\<lambda>x. H (ereal (real_of_ereal (f x))) (\<infinity>)) \<in> borel_measurable M"
```
```  1650   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
```
```  1651 proof%unimportant -
```
```  1652   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)))"
```
```  1653   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"
```
```  1654   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
```
```  1655   note * = this
```
```  1656   from assms show ?thesis unfolding * by simp
```
```  1657 qed
```
```  1658
```
```  1659 lemma%unimportant [measurable(raw)]:
```
```  1660   fixes f :: "'a \<Rightarrow> ereal"
```
```  1661   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1662   shows borel_measurable_ereal_add: "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```  1663     and borel_measurable_ereal_times: "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1664   by (simp_all add: borel_measurable_ereal2)
```
```  1665
```
```  1666 lemma%unimportant [measurable(raw)]:
```
```  1667   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1668   assumes "f \<in> borel_measurable M"
```
```  1669   assumes "g \<in> borel_measurable M"
```
```  1670   shows borel_measurable_ereal_diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```  1671     and borel_measurable_ereal_divide: "(\<lambda>x. f x / g x) \<in> borel_measurable M"
```
```  1672   using assms by (simp_all add: minus_ereal_def divide_ereal_def)
```
```  1673
```
```  1674 lemma%unimportant borel_measurable_ereal_sum[measurable (raw)]:
```
```  1675   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1676   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1677   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1678   using assms by (induction S rule: infinite_finite_induct) auto
```
```  1679
```
```  1680 lemma%unimportant borel_measurable_ereal_prod[measurable (raw)]:
```
```  1681   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1682   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1683   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1684   using assms by (induction S rule: infinite_finite_induct) auto
```
```  1685
```
```  1686 lemma%unimportant borel_measurable_extreal_suminf[measurable (raw)]:
```
```  1687   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1688   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1689   shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
```
```  1690   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
```
```  1691
```
```  1692 subsection%important "Borel space on the extended non-negative reals"
```
```  1693
```
```  1694 text \<open> @{type ennreal} is a topological monoid, so no rules for plus are required, also all order
```
```  1695   statements are usually done on type classes. \<close>
```
```  1696
```
```  1697 lemma%unimportant measurable_enn2ereal[measurable]: "enn2ereal \<in> borel \<rightarrow>\<^sub>M borel"
```
```  1698   by (intro borel_measurable_continuous_on1 continuous_on_enn2ereal)
```
```  1699
```
```  1700 lemma%unimportant measurable_e2ennreal[measurable]: "e2ennreal \<in> borel \<rightarrow>\<^sub>M borel"
```
```  1701   by (intro borel_measurable_continuous_on1 continuous_on_e2ennreal)
```
```  1702
```
```  1703 lemma%unimportant borel_measurable_enn2real[measurable (raw)]:
```
```  1704   "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. enn2real (f x)) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1705   unfolding enn2real_def[abs_def] by measurable
```
```  1706
```
```  1707 definition%important [simp]: "is_borel f M \<longleftrightarrow> f \<in> borel_measurable M"
```
```  1708
```
```  1709 lemma%unimportant is_borel_transfer[transfer_rule]: "rel_fun (rel_fun (=) pcr_ennreal) (=) is_borel is_borel"
```
```  1710   unfolding is_borel_def[abs_def]
```
```  1711 proof (safe intro!: rel_funI ext dest!: rel_fun_eq_pcr_ennreal[THEN iffD1])
```
```  1712   fix f and M :: "'a measure"
```
```  1713   show "f \<in> borel_measurable M" if f: "enn2ereal \<circ> f \<in> borel_measurable M"
```
```  1714     using measurable_compose[OF f measurable_e2ennreal] by simp
```
```  1715 qed simp
```
```  1716
```
```  1717 context
```
```  1718   includes ennreal.lifting
```
```  1719 begin
```
```  1720
```
```  1721 lemma%unimportant measurable_ennreal[measurable]: "ennreal \<in> borel \<rightarrow>\<^sub>M borel"
```
```  1722   unfolding is_borel_def[symmetric]
```
```  1723   by transfer simp
```
```  1724
```
```  1725 lemma%important borel_measurable_ennreal_iff[simp]:
```
```  1726   assumes [simp]: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
```
```  1727   shows "(\<lambda>x. ennreal (f x)) \<in> M \<rightarrow>\<^sub>M borel \<longleftrightarrow> f \<in> M \<rightarrow>\<^sub>M borel"
```
```  1728 proof%unimportant safe
```
```  1729   assume "(\<lambda>x. ennreal (f x)) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1730   then have "(\<lambda>x. enn2real (ennreal (f x))) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1731     by measurable
```
```  1732   then show "f \<in> M \<rightarrow>\<^sub>M borel"
```
```  1733     by (rule measurable_cong[THEN iffD1, rotated]) auto
```
```  1734 qed measurable
```
```  1735
```
```  1736 lemma%unimportant borel_measurable_times_ennreal[measurable (raw)]:
```
```  1737   fixes f g :: "'a \<Rightarrow> ennreal"
```
```  1738   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"
```
```  1739   unfolding is_borel_def[symmetric] by transfer simp
```
```  1740
```
```  1741 lemma%unimportant borel_measurable_inverse_ennreal[measurable (raw)]:
```
```  1742   fixes f :: "'a \<Rightarrow> ennreal"
```
```  1743   shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. inverse (f x)) \<in> M \<rightarrow>\<^sub>M borel"
```
```  1744   unfolding is_borel_def[symmetric] by transfer simp
```
```  1745
```
```  1746 lemma%unimportant borel_measurable_divide_ennreal[measurable (raw)]:
```
```  1747   fixes f :: "'a \<Rightarrow> ennreal"
```
```  1748   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"
```
```  1749   unfolding divide_ennreal_def by simp
```
```  1750
```
```  1751 lemma%unimportant borel_measurable_minus_ennreal[measurable (raw)]:
```
```  1752   fixes f :: "'a \<Rightarrow> ennreal"
```
```  1753   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"
```
```  1754   unfolding is_borel_def[symmetric] by transfer simp
```
```  1755
```
```  1756 lemma%important borel_measurable_prod_ennreal[measurable (raw)]:
```
```  1757   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ennreal"
```
```  1758   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1759   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1760   using%unimportant assms by (induction S rule: infinite_finite_induct) auto
```
```  1761
```
```  1762 end
```
```  1763
```
```  1764 hide_const (open) is_borel
```
```  1765
```
```  1766 subsection%important \<open>LIMSEQ is borel measurable\<close>
```
```  1767
```
```  1768 lemma%important borel_measurable_LIMSEQ_real:
```
```  1769   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
```
```  1770   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
```
```  1771   and u: "\<And>i. u i \<in> borel_measurable M"
```
```  1772   shows "u' \<in> borel_measurable M"
```
```  1773 proof%unimportant -
```
```  1774   have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)"
```
```  1775     using u' by (simp add: lim_imp_Liminf)
```
```  1776   moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M"
```
```  1777     by auto
```
```  1778   ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff)
```
```  1779 qed
```
```  1780
```
```  1781 lemma%important borel_measurable_LIMSEQ_metric:
```
```  1782   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b :: metric_space"
```
```  1783   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1784   assumes lim: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. f i x) \<longlonglongrightarrow> g x"
```
```  1785   shows "g \<in> borel_measurable M"
```
```  1786   unfolding borel_eq_closed
```
```  1787 proof%unimportant (safe intro!: measurable_measure_of)
```
```  1788   fix A :: "'b set" assume "closed A"
```
```  1789
```
```  1790   have [measurable]: "(\<lambda>x. infdist (g x) A) \<in> borel_measurable M"
```
```  1791   proof (rule borel_measurable_LIMSEQ_real)
```
```  1792     show "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. infdist (f i x) A) \<longlonglongrightarrow> infdist (g x) A"
```
```  1793       by (intro tendsto_infdist lim)
```
```  1794     show "\<And>i. (\<lambda>x. infdist (f i x) A) \<in> borel_measurable M"
```
```  1795       by (intro borel_measurable_continuous_on[where f="\<lambda>x. infdist x A"]
```
```  1796         continuous_at_imp_continuous_on ballI continuous_infdist continuous_ident) auto
```
```  1797   qed
```
```  1798
```
```  1799   show "g -` A \<inter> space M \<in> sets M"
```
```  1800   proof cases
```
```  1801     assume "A \<noteq> {}"
```
```  1802     then have "\<And>x. infdist x A = 0 \<longleftrightarrow> x \<in> A"
```
```  1803       using \<open>closed A\<close> by (simp add: in_closed_iff_infdist_zero)
```
```  1804     then have "g -` A \<inter> space M = {x\<in>space M. infdist (g x) A = 0}"
```
```  1805       by auto
```
```  1806     also have "\<dots> \<in> sets M"
```
```  1807       by measurable
```
```  1808     finally show ?thesis .
```
```  1809   qed simp
```
```  1810 qed auto
```
```  1811
```
```  1812 lemma%important sets_Collect_Cauchy[measurable]:
```
```  1813   fixes f :: "nat \<Rightarrow> 'a => 'b::{metric_space, second_countable_topology}"
```
```  1814   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1815   shows "{x\<in>space M. Cauchy (\<lambda>i. f i x)} \<in> sets M"
```
```  1816   unfolding metric_Cauchy_iff2 using f by auto
```
```  1817
```
```  1818 lemma%unimportant borel_measurable_lim_metric[measurable (raw)]:
```
```  1819   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```  1820   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1821   shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1822 proof -
```
```  1823   define u' where "u' x = lim (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)" for x
```
```  1824   then have *: "\<And>x. lim (\<lambda>i. f i x) = (if Cauchy (\<lambda>i. f i x) then u' x else (THE x. False))"
```
```  1825     by (auto simp: lim_def convergent_eq_Cauchy[symmetric])
```
```  1826   have "u' \<in> borel_measurable M"
```
```  1827   proof (rule borel_measurable_LIMSEQ_metric)
```
```  1828     fix x
```
```  1829     have "convergent (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)"
```
```  1830       by (cases "Cauchy (\<lambda>i. f i x)")
```
```  1831          (auto simp add: convergent_eq_Cauchy[symmetric] convergent_def)
```
```  1832     then show "(\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0) \<longlonglongrightarrow> u' x"
```
```  1833       unfolding u'_def
```
```  1834       by (rule convergent_LIMSEQ_iff[THEN iffD1])
```
```  1835   qed measurable
```
```  1836   then show ?thesis
```
```  1837     unfolding * by measurable
```
```  1838 qed
```
```  1839
```
```  1840 lemma%unimportant borel_measurable_suminf[measurable (raw)]:
```
```  1841   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```  1842   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  1843   shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1844   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
```
```  1845
```
```  1846 lemma%unimportant Collect_closed_imp_pred_borel: "closed {x. P x} \<Longrightarrow> Measurable.pred borel P"
```
```  1847   by (simp add: pred_def)
```
```  1848
```
```  1849 (* Proof by Jeremy Avigad and Luke Serafin *)
```
```  1850 lemma%unimportant isCont_borel_pred[measurable]:
```
```  1851   fixes f :: "'b::metric_space \<Rightarrow> 'a::metric_space"
```
```  1852   shows "Measurable.pred borel (isCont f)"
```
```  1853 proof (subst measurable_cong)
```
```  1854   let ?I = "\<lambda>j. inverse(real (Suc j))"
```
```  1855   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
```
```  1856     unfolding continuous_at_eps_delta
```
```  1857   proof safe
```
```  1858     fix i assume "\<forall>e>0. \<exists>d>0. \<forall>y. dist y x < d \<longrightarrow> dist (f y) (f x) < e"
```
```  1859     moreover have "0 < ?I i / 2"
```
```  1860       by simp
```
```  1861     ultimately obtain d where d: "0 < d" "\<And>y. dist x y < d \<Longrightarrow> dist (f y) (f x) < ?I i / 2"
```
```  1862       by (metis dist_commute)
```
```  1863     then obtain j where j: "?I j < d"
```
```  1864       by (metis reals_Archimedean)
```
```  1865
```
```  1866     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"
```
```  1867     proof (safe intro!: exI[where x=j])
```
```  1868       fix y z assume *: "dist x y < ?I j" "dist x z < ?I j"
```
```  1869       have "dist (f y) (f z) \<le> dist (f y) (f x) + dist (f z) (f x)"
```
```  1870         by (rule dist_triangle2)
```
```  1871       also have "\<dots> < ?I i / 2 + ?I i / 2"
```
```  1872         by (intro add_strict_mono d less_trans[OF _ j] *)
```
```  1873       also have "\<dots> \<le> ?I i"
```
```  1874         by (simp add: field_simps of_nat_Suc)
```
```  1875       finally show "dist (f y) (f z) \<le> ?I i"
```
```  1876         by simp
```
```  1877     qed
```
```  1878   next
```
```  1879     fix e::real assume "0 < e"
```
```  1880     then obtain n where n: "?I n < e"
```
```  1881       by (metis reals_Archimedean)
```
```  1882     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"
```
```  1883     from this[THEN spec, of "Suc n"]
```
```  1884     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)"
```
```  1885       by auto
```
```  1886
```
```  1887     show "\<exists>d>0. \<forall>y. dist y x < d \<longrightarrow> dist (f y) (f x) < e"
```
```  1888     proof (safe intro!: exI[of _ "?I j"])
```
```  1889       fix y assume "dist y x < ?I j"
```
```  1890       then have "dist (f y) (f x) \<le> ?I (Suc n)"
```
```  1891         by (intro j) (auto simp: dist_commute)
```
```  1892       also have "?I (Suc n) < ?I n"
```
```  1893         by simp
```
```  1894       also note n
```
```  1895       finally show "dist (f y) (f x) < e" .
```
```  1896     qed simp
```
```  1897   qed
```
```  1898 qed (intro pred_intros_countable closed_Collect_all closed_Collect_le open_Collect_less
```
```  1899            Collect_closed_imp_pred_borel closed_Collect_imp open_Collect_conj continuous_intros)
```
```  1900
```
```  1901 lemma%unimportant isCont_borel:
```
```  1902   fixes f :: "'b::metric_space \<Rightarrow> 'a::metric_space"
```
```  1903   shows "{x. isCont f x} \<in> sets borel"
```
```  1904   by simp
```
```  1905
```
```  1906 lemma%important is_real_interval:
```
```  1907   assumes S: "is_interval S"
```
```  1908   shows "\<exists>a b::real. S = {} \<or> S = UNIV \<or> S = {..<b} \<or> S = {..b} \<or> S = {a<..} \<or> S = {a..} \<or>
```
```  1909     S = {a<..<b} \<or> S = {a<..b} \<or> S = {a..<b} \<or> S = {a..b}"
```
```  1910   using S unfolding is_interval_1 by (blast intro: interval_cases)
```
```  1911
```
```  1912 lemma%important real_interval_borel_measurable:
```
```  1913   assumes "is_interval (S::real set)"
```
```  1914   shows "S \<in> sets borel"
```
```  1915 proof%unimportant -
```
```  1916   from assms is_real_interval have "\<exists>a b::real. S = {} \<or> S = UNIV \<or> S = {..<b} \<or> S = {..b} \<or>
```
```  1917     S = {a<..} \<or> S = {a..} \<or> S = {a<..<b} \<or> S = {a<..b} \<or> S = {a..<b} \<or> S = {a..b}" by auto
```
```  1918   then guess a ..
```
```  1919   then guess b ..
```
```  1920   thus ?thesis
```
```  1921     by auto
```
```  1922 qed
```
```  1923
```
```  1924 text \<open>The next lemmas hold in any second countable linorder (including ennreal or ereal for instance),
```
```  1925 but in the current state they are restricted to reals.\<close>
```
```  1926
```
```  1927 lemma%important borel_measurable_mono_on_fnc:
```
```  1928   fixes f :: "real \<Rightarrow> real" and A :: "real set"
```
```  1929   assumes "mono_on f A"
```
```  1930   shows "f \<in> borel_measurable (restrict_space borel A)"
```
```  1931   apply (rule measurable_restrict_countable[OF mono_on_ctble_discont[OF assms]])
```
```  1932   apply (auto intro!: image_eqI[where x="{x}" for x] simp: sets_restrict_space)
```
```  1933   apply (auto simp add: sets_restrict_restrict_space continuous_on_eq_continuous_within
```
```  1934               cong: measurable_cong_sets
```
```  1935               intro!: borel_measurable_continuous_on_restrict intro: continuous_within_subset)
```
```  1936   done
```
```  1937
```
```  1938 lemma%unimportant borel_measurable_piecewise_mono:
```
```  1939   fixes f::"real \<Rightarrow> real" and C::"real set set"
```
```  1940   assumes "countable C" "\<And>c. c \<in> C \<Longrightarrow> c \<in> sets borel" "\<And>c. c \<in> C \<Longrightarrow> mono_on f c" "(\<Union>C) = UNIV"
```
```  1941   shows "f \<in> borel_measurable borel"
```
```  1942   by (rule measurable_piecewise_restrict[of C], auto intro: borel_measurable_mono_on_fnc simp: assms)
```
```  1943
```
```  1944 lemma%unimportant borel_measurable_mono:
```
```  1945   fixes f :: "real \<Rightarrow> real"
```
```  1946   shows "mono f \<Longrightarrow> f \<in> borel_measurable borel"
```
```  1947   using borel_measurable_mono_on_fnc[of f UNIV] by (simp add: mono_def mono_on_def)
```
```  1948
```
```  1949 lemma%unimportant measurable_bdd_below_real[measurable (raw)]:
```
```  1950   fixes F :: "'a \<Rightarrow> 'i \<Rightarrow> real"
```
```  1951   assumes [simp]: "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> M \<rightarrow>\<^sub>M borel"
```
```  1952   shows "Measurable.pred M (\<lambda>x. bdd_below ((\<lambda>i. F i x)`I))"
```
```  1953 proof (subst measurable_cong)
```
```  1954   show "bdd_below ((\<lambda>i. F i x)`I) \<longleftrightarrow> (\<exists>q\<in>\<int>. \<forall>i\<in>I. q \<le> F i x)" for x
```
```  1955     by (auto simp: bdd_below_def intro!: bexI[of _ "of_int (floor _)"] intro: order_trans of_int_floor_le)
```
```  1956   show "Measurable.pred M (\<lambda>w. \<exists>q\<in>\<int>. \<forall>i\<in>I. q \<le> F i w)"
```
```  1957     using countable_int by measurable
```
```  1958 qed
```
```  1959
```
```  1960 lemma%important borel_measurable_cINF_real[measurable (raw)]:
```
```  1961   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> real"
```
```  1962   assumes [simp]: "countable I"
```
```  1963   assumes F[measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
```
```  1964   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
```
```  1965 proof%unimportant (rule measurable_piecewise_restrict)
```
```  1966   let ?\<Omega> = "{x\<in>space M. bdd_below ((\<lambda>i. F i x)`I)}"
```
```  1967   show "countable {?\<Omega>, - ?\<Omega>}" "space M \<subseteq> \<Union>{?\<Omega>, - ?\<Omega>}" "\<And>X. X \<in> {?\<Omega>, - ?\<Omega>} \<Longrightarrow> X \<inter> space M \<in> sets M"
```
```  1968     by auto
```
```  1969   fix X assume "X \<in> {?\<Omega>, - ?\<Omega>}" then show "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable (restrict_space M X)"
```
```  1970   proof safe
```
```  1971     show "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable (restrict_space M ?\<Omega>)"
```
```  1972       by (intro borel_measurable_cINF measurable_restrict_space1 F)
```
```  1973          (auto simp: space_restrict_space)
```
```  1974     show "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable (restrict_space M (-?\<Omega>))"
```
```  1975     proof (subst measurable_cong)
```
```  1976       fix x assume "x \<in> space (restrict_space M (-?\<Omega>))"
```
```  1977       then have "\<not> (\<forall>i\<in>I. - F i x \<le> y)" for y
```
```  1978         by (auto simp: space_restrict_space bdd_above_def bdd_above_uminus[symmetric])
```
```  1979       then show "(INF i:I. F i x) = - (THE x. False)"
```
```  1980         by (auto simp: space_restrict_space Inf_real_def Sup_real_def Least_def simp del: Set.ball_simps(10))
```
```  1981     qed simp
```
```  1982   qed
```
```  1983 qed
```
```  1984
```
```  1985 lemma%unimportant borel_Ici: "borel = sigma UNIV (range (\<lambda>x::real. {x ..}))"
```
```  1986 proof (safe intro!: borel_eq_sigmaI1[OF borel_Iio])
```
```  1987   fix x :: real
```
```  1988   have eq: "{..<x} = space (sigma UNIV (range atLeast)) - {x ..}"
```
```  1989     by auto
```
```  1990   show "{..<x} \<in> sets (sigma UNIV (range atLeast))"
```
```  1991     unfolding eq by (intro sets.compl_sets) auto
```
```  1992 qed auto
```
```  1993
```
```  1994 lemma%unimportant borel_measurable_pred_less[measurable (raw)]:
```
```  1995   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
```
```  1996   shows "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> Measurable.pred M (\<lambda>w. f w < g w)"
```
```  1997   unfolding Measurable.pred_def by (rule borel_measurable_less)
```
```  1998
```
```  1999 no_notation
```
```  2000   eucl_less (infix "<e" 50)
```
```  2001
```
```  2002 lemma%important borel_measurable_Max2[measurable (raw)]:
```
```  2003   fixes f::"_ \<Rightarrow> _ \<Rightarrow> 'a::{second_countable_topology, dense_linorder, linorder_topology}"
```
```  2004   assumes "finite I"
```
```  2005     and [measurable]: "\<And>i. f i \<in> borel_measurable M"
```
```  2006   shows "(\<lambda>x. Max{f i x |i. i \<in> I}) \<in> borel_measurable M"
```
```  2007 by%unimportant (simp add: borel_measurable_Max[OF assms(1), where ?f=f and ?M=M] Setcompr_eq_image)
```
```  2008
```
```  2009 lemma%unimportant measurable_compose_n [measurable (raw)]:
```
```  2010   assumes "T \<in> measurable M M"
```
```  2011   shows "(T^^n) \<in> measurable M M"
```
```  2012 by (induction n, auto simp add: measurable_compose[OF _ assms])
```
```  2013
```
```  2014 lemma%unimportant measurable_real_imp_nat:
```
```  2015   fixes f::"'a \<Rightarrow> nat"
```
```  2016   assumes [measurable]: "(\<lambda>x. real(f x)) \<in> borel_measurable M"
```
```  2017   shows "f \<in> measurable M (count_space UNIV)"
```
```  2018 proof -
```
```  2019   let ?g = "(\<lambda>x. real(f x))"
```
```  2020   have "\<And>(n::nat). ?g-`({real n}) \<inter> space M = f-`{n} \<inter> space M" by auto
```
```  2021   moreover have "\<And>(n::nat). ?g-`({real n}) \<inter> space M \<in> sets M" using assms by measurable
```
```  2022   ultimately have "\<And>(n::nat). f-`{n} \<inter> space M \<in> sets M" by simp
```
```  2023   then show ?thesis using measurable_count_space_eq2_countable by blast
```
```  2024 qed
```
```  2025
```
```  2026 lemma%unimportant measurable_equality_set [measurable]:
```
```  2027   fixes f g::"_\<Rightarrow> 'a::{second_countable_topology, t2_space}"
```
```  2028   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  2029   shows "{x \<in> space M. f x = g x} \<in> sets M"
```
```  2030
```
```  2031 proof -
```
```  2032   define A where "A = {x \<in> space M. f x = g x}"
```
```  2033   define B where "B = {y. \<exists>x::'a. y = (x,x)}"
```
```  2034   have "A = (\<lambda>x. (f x, g x))-`B \<inter> space M" unfolding A_def B_def by auto
```
```  2035   moreover have "(\<lambda>x. (f x, g x)) \<in> borel_measurable M" by simp
```
```  2036   moreover have "B \<in> sets borel" unfolding B_def by (simp add: closed_diagonal)
```
```  2037   ultimately have "A \<in> sets M" by simp
```
```  2038   then show ?thesis unfolding A_def by simp
```
```  2039 qed
```
```  2040
```
```  2041 lemma%unimportant measurable_inequality_set [measurable]:
```
```  2042   fixes f g::"_ \<Rightarrow> 'a::{second_countable_topology, linorder_topology}"
```
```  2043   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  2044   shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
```
```  2045         "{x \<in> space M. f x < g x} \<in> sets M"
```
```  2046         "{x \<in> space M. f x \<ge> g x} \<in> sets M"
```
```  2047         "{x \<in> space M. f x > g x} \<in> sets M"
```
```  2048 proof -
```
```  2049   define F where "F = (\<lambda>x. (f x, g x))"
```
```  2050   have * [measurable]: "F \<in> borel_measurable M" unfolding F_def by simp
```
```  2051
```
```  2052   have "{x \<in> space M. f x \<le> g x} = F-`{(x, y) | x y. x \<le> y} \<inter> space M" unfolding F_def by auto
```
```  2053   moreover have "{(x, y) | x y. x \<le> (y::'a)} \<in> sets borel" using closed_subdiagonal borel_closed by blast
```
```  2054   ultimately show "{x \<in> space M. f x \<le> g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
```
```  2055
```
```  2056   have "{x \<in> space M. f x < g x} = F-`{(x, y) | x y. x < y} \<inter> space M" unfolding F_def by auto
```
```  2057   moreover have "{(x, y) | x y. x < (y::'a)} \<in> sets borel" using open_subdiagonal borel_open by blast
```
```  2058   ultimately show "{x \<in> space M. f x < g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
```
```  2059
```
```  2060   have "{x \<in> space M. f x \<ge> g x} = F-`{(x, y) | x y. x \<ge> y} \<inter> space M" unfolding F_def by auto
```
```  2061   moreover have "{(x, y) | x y. x \<ge> (y::'a)} \<in> sets borel" using closed_superdiagonal borel_closed by blast
```
```  2062   ultimately show "{x \<in> space M. f x \<ge> g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
```
```  2063
```
```  2064   have "{x \<in> space M. f x > g x} = F-`{(x, y) | x y. x > y} \<inter> space M" unfolding F_def by auto
```
```  2065   moreover have "{(x, y) | x y. x > (y::'a)} \<in> sets borel" using open_superdiagonal borel_open by blast
```
```  2066   ultimately show "{x \<in> space M. f x > g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
```
```  2067 qed
```
```  2068
```
```  2069 lemma%unimportant measurable_limit [measurable]:
```
```  2070   fixes f::"nat \<Rightarrow> 'a \<Rightarrow> 'b::first_countable_topology"
```
```  2071   assumes [measurable]: "\<And>n::nat. f n \<in> borel_measurable M"
```
```  2072   shows "Measurable.pred M (\<lambda>x. (\<lambda>n. f n x) \<longlonglongrightarrow> c)"
```
```  2073 proof -
```
```  2074   obtain A :: "nat \<Rightarrow> 'b set" where A:
```
```  2075     "\<And>i. open (A i)"
```
```  2076     "\<And>i. c \<in> A i"
```
```  2077     "\<And>S. open S \<Longrightarrow> c \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
```
```  2078   by (rule countable_basis_at_decseq) blast
```
```  2079
```
```  2080   have [measurable]: "\<And>N i. (f N)-`(A i) \<inter> space M \<in> sets M" using A(1) by auto
```
```  2081   then have mes: "(\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. (f N)-`(A i) \<inter> space M) \<in> sets M" by blast
```
```  2082
```
```  2083   have "(u \<longlonglongrightarrow> c) \<longleftrightarrow> (\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)" for u::"nat \<Rightarrow> 'b"
```
```  2084   proof
```
```  2085     assume "u \<longlonglongrightarrow> c"
```
```  2086     then have "eventually (\<lambda>n. u n \<in> A i) sequentially" for i using A(1)[of i] A(2)[of i]
```
```  2087       by (simp add: topological_tendstoD)
```
```  2088     then show "(\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)" by auto
```
```  2089   next
```
```  2090     assume H: "(\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)"
```
```  2091     show "(u \<longlonglongrightarrow> c)"
```
```  2092     proof (rule topological_tendstoI)
```
```  2093       fix S assume "open S" "c \<in> S"
```
```  2094       with A(3)[OF this] obtain i where "A i \<subseteq> S"
```
```  2095         using eventually_False_sequentially eventually_mono by blast
```
```  2096       moreover have "eventually (\<lambda>n. u n \<in> A i) sequentially" using H by simp
```
```  2097       ultimately show "\<forall>\<^sub>F n in sequentially. u n \<in> S"
```
```  2098         by (simp add: eventually_mono subset_eq)
```
```  2099     qed
```
```  2100   qed
```
```  2101   then have "{x. (\<lambda>n. f n x) \<longlonglongrightarrow> c} = (\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. (f N)-`(A i))"
```
```  2102     by (auto simp add: atLeast_def eventually_at_top_linorder)
```
```  2103   then have "{x \<in> space M. (\<lambda>n. f n x) \<longlonglongrightarrow> c} = (\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. (f N)-`(A i) \<inter> space M)"
```
```  2104     by auto
```
```  2105   then have "{x \<in> space M. (\<lambda>n. f n x) \<longlonglongrightarrow> c} \<in> sets M" using mes by simp
```
```  2106   then show ?thesis by auto
```
```  2107 qed
```
```  2108
```
```  2109 lemma%important measurable_limit2 [measurable]:
```
```  2110   fixes u::"nat \<Rightarrow> 'a \<Rightarrow> real"
```
```  2111   assumes [measurable]: "\<And>n. u n \<in> borel_measurable M" "v \<in> borel_measurable M"
```
```  2112   shows "Measurable.pred M (\<lambda>x. (\<lambda>n. u n x) \<longlonglongrightarrow> v x)"
```
```  2113 proof%unimportant -
```
```  2114   define w where "w = (\<lambda>n x. u n x - v x)"
```
```  2115   have [measurable]: "w n \<in> borel_measurable M" for n unfolding w_def by auto
```
```  2116   have "((\<lambda>n. u n x) \<longlonglongrightarrow> v x) \<longleftrightarrow> ((\<lambda>n. w n x) \<longlonglongrightarrow> 0)" for x
```
```  2117     unfolding w_def using Lim_null by auto
```
```  2118   then show ?thesis using measurable_limit by auto
```
```  2119 qed
```
```  2120
```
```  2121 lemma%unimportant measurable_P_restriction [measurable (raw)]:
```
```  2122   assumes [measurable]: "Measurable.pred M P" "A \<in> sets M"
```
```  2123   shows "{x \<in> A. P x} \<in> sets M"
```
```  2124 proof -
```
```  2125   have "A \<subseteq> space M" using sets.sets_into_space[OF assms(2)].
```
```  2126   then have "{x \<in> A. P x} = A \<inter> {x \<in> space M. P x}" by blast
```
```  2127   then show ?thesis by auto
```
```  2128 qed
```
```  2129
```
```  2130 lemma%unimportant measurable_sum_nat [measurable (raw)]:
```
```  2131   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> nat"
```
```  2132   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> measurable M (count_space UNIV)"
```
```  2133   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> measurable M (count_space UNIV)"
```
```  2134 proof cases
```
```  2135   assume "finite S"
```
```  2136   then show ?thesis using assms by induct auto
```
```  2137 qed simp
```
```  2138
```
```  2139
```
```  2140 lemma%unimportant measurable_abs_powr [measurable]:
```
```  2141   fixes p::real
```
```  2142   assumes [measurable]: "f \<in> borel_measurable M"
```
```  2143   shows "(\<lambda>x. \<bar>f x\<bar> powr p) \<in> borel_measurable M"
```
```  2144 unfolding powr_def by auto
```
```  2145
```
```  2146 text \<open>The next one is a variation around \verb+measurable_restrict_space+.\<close>
```
```  2147
```
```  2148 lemma%unimportant measurable_restrict_space3:
```
```  2149   assumes "f \<in> measurable M N" and
```
```  2150           "f \<in> A \<rightarrow> B"
```
```  2151   shows "f \<in> measurable (restrict_space M A) (restrict_space N B)"
```
```  2152 proof -
```
```  2153   have "f \<in> measurable (restrict_space M A) N" using assms(1) measurable_restrict_space1 by auto
```
```  2154   then show ?thesis by (metis Int_iff funcsetI funcset_mem
```
```  2155       measurable_restrict_space2[of f, of "restrict_space M A", of B, of N] assms(2) space_restrict_space)
```
```  2156 qed
```
```  2157
```
```  2158 text \<open>The next one is a variation around \verb+measurable_piecewise_restrict+.\<close>
```
```  2159
```
```  2160 lemma%important measurable_piecewise_restrict2:
```
```  2161   assumes [measurable]: "\<And>n. A n \<in> sets M"
```
```  2162       and "space M = (\<Union>(n::nat). A n)"
```
```  2163           "\<And>n. \<exists>h \<in> measurable M N. (\<forall>x \<in> A n. f x = h x)"
```
```  2164   shows "f \<in> measurable M N"
```
```  2165 proof%unimportant (rule measurableI)
```
```  2166   fix B assume [measurable]: "B \<in> sets N"
```
```  2167   {
```
```  2168     fix n::nat
```
```  2169     obtain h where [measurable]: "h \<in> measurable M N" and "\<forall>x \<in> A n. f x = h x" using assms(3) by blast
```
```  2170     then have *: "f-`B \<inter> A n = h-`B \<inter> A n" by auto
```
```  2171     have "h-`B \<inter> A n = h-`B \<inter> space M \<inter> A n" using assms(2) sets.sets_into_space by auto
```
```  2172     then have "h-`B \<inter> A n \<in> sets M" by simp
```
```  2173     then have "f-`B \<inter> A n \<in> sets M" using * by simp
```
```  2174   }
```
```  2175   then have "(\<Union>n. f-`B \<inter> A n) \<in> sets M" by measurable
```
```  2176   moreover have "f-`B \<inter> space M = (\<Union>n. f-`B \<inter> A n)" using assms(2) by blast
```
```  2177   ultimately show "f-`B \<inter> space M \<in> sets M" by simp
```
```  2178 next
```
```  2179   fix x assume "x \<in> space M"
```
```  2180   then obtain n where "x \<in> A n" using assms(2) by blast
```
```  2181   obtain h where [measurable]: "h \<in> measurable M N" and "\<forall>x \<in> A n. f x = h x" using assms(3) by blast
```
```  2182   then have "f x = h x" using \<open>x \<in> A n\<close> by blast
```
```  2183   moreover have "h x \<in> space N" by (metis measurable_space \<open>x \<in> space M\<close> \<open>h \<in> measurable M N\<close>)
```
```  2184   ultimately show "f x \<in> space N" by simp
```
```  2185 qed
```
```  2186
```
```  2187 end
```