src/HOL/Analysis/Borel_Space.thy
 author hoelzl Tue Oct 18 12:01:54 2016 +0200 (2016-10-18) changeset 64284 f3b905b2eee2 parent 64283 979cdfdf7a79 child 64287 d85d88722745 permissions -rw-r--r--
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
1 (*  Title:      HOL/Analysis/Borel_Space.thy
2     Author:     Johannes Hölzl, TU München
3     Author:     Armin Heller, TU München
4 *)
6 section \<open>Borel spaces\<close>
8 theory Borel_Space
9 imports
10   Measurable Derivative Ordered_Euclidean_Space Extended_Real_Limits
11 begin
13 lemma sets_Collect_eventually_sequentially[measurable]:
14   "(\<And>i. {x\<in>space M. P x i} \<in> sets M) \<Longrightarrow> {x\<in>space M. eventually (P x) sequentially} \<in> sets M"
15   unfolding eventually_sequentially by simp
17 lemma topological_basis_trivial: "topological_basis {A. open A}"
18   by (auto simp: topological_basis_def)
20 lemma open_prod_generated: "open = generate_topology {A \<times> B | A B. open A \<and> open B}"
21 proof -
22   have "{A \<times> B :: ('a \<times> 'b) set | A B. open A \<and> open B} = ((\<lambda>(a, b). a \<times> b) ` ({A. open A} \<times> {A. open A}))"
23     by auto
24   then show ?thesis
25     by (auto intro: topological_basis_prod topological_basis_trivial topological_basis_imp_subbasis)
26 qed
28 definition "mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r \<le> s \<longrightarrow> f r \<le> f s"
30 lemma mono_onI:
31   "(\<And>r s. r \<in> A \<Longrightarrow> s \<in> A \<Longrightarrow> r \<le> s \<Longrightarrow> f r \<le> f s) \<Longrightarrow> mono_on f A"
32   unfolding mono_on_def by simp
34 lemma mono_onD:
35   "\<lbrakk>mono_on f A; r \<in> A; s \<in> A; r \<le> s\<rbrakk> \<Longrightarrow> f r \<le> f s"
36   unfolding mono_on_def by simp
38 lemma mono_imp_mono_on: "mono f \<Longrightarrow> mono_on f A"
39   unfolding mono_def mono_on_def by auto
41 lemma mono_on_subset: "mono_on f A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> mono_on f B"
42   unfolding mono_on_def by auto
44 definition "strict_mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r < s \<longrightarrow> f r < f s"
46 lemma strict_mono_onI:
47   "(\<And>r s. r \<in> A \<Longrightarrow> s \<in> A \<Longrightarrow> r < s \<Longrightarrow> f r < f s) \<Longrightarrow> strict_mono_on f A"
48   unfolding strict_mono_on_def by simp
50 lemma strict_mono_onD:
51   "\<lbrakk>strict_mono_on f A; r \<in> A; s \<in> A; r < s\<rbrakk> \<Longrightarrow> f r < f s"
52   unfolding strict_mono_on_def by simp
54 lemma mono_on_greaterD:
55   assumes "mono_on g A" "x \<in> A" "y \<in> A" "g x > (g (y::_::linorder) :: _ :: linorder)"
56   shows "x > y"
57 proof (rule ccontr)
58   assume "\<not>x > y"
59   hence "x \<le> y" by (simp add: not_less)
60   from assms(1-3) and this have "g x \<le> g y" by (rule mono_onD)
61   with assms(4) show False by simp
62 qed
64 lemma strict_mono_inv:
65   fixes f :: "('a::linorder) \<Rightarrow> ('b::linorder)"
66   assumes "strict_mono f" and "surj f" and inv: "\<And>x. g (f x) = x"
67   shows "strict_mono g"
68 proof
69   fix x y :: 'b assume "x < y"
70   from \<open>surj f\<close> obtain x' y' where [simp]: "x = f x'" "y = f y'" by blast
71   with \<open>x < y\<close> and \<open>strict_mono f\<close> have "x' < y'" by (simp add: strict_mono_less)
72   with inv show "g x < g y" by simp
73 qed
75 lemma strict_mono_on_imp_inj_on:
76   assumes "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> (_ :: preorder)) A"
77   shows "inj_on f A"
78 proof (rule inj_onI)
79   fix x y assume "x \<in> A" "y \<in> A" "f x = f y"
80   thus "x = y"
81     by (cases x y rule: linorder_cases)
82        (auto dest: strict_mono_onD[OF assms, of x y] strict_mono_onD[OF assms, of y x])
83 qed
85 lemma strict_mono_on_leD:
86   assumes "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) A" "x \<in> A" "y \<in> A" "x \<le> y"
87   shows "f x \<le> f y"
88 proof (insert le_less_linear[of y x], elim disjE)
89   assume "x < y"
90   with assms have "f x < f y" by (rule_tac strict_mono_onD[OF assms(1)]) simp_all
91   thus ?thesis by (rule less_imp_le)
92 qed (insert assms, simp)
94 lemma strict_mono_on_eqD:
95   fixes f :: "(_ :: linorder) \<Rightarrow> (_ :: preorder)"
96   assumes "strict_mono_on f A" "f x = f y" "x \<in> A" "y \<in> A"
97   shows "y = x"
98   using assms by (rule_tac linorder_cases[of x y]) (auto dest: strict_mono_onD)
100 lemma mono_on_imp_deriv_nonneg:
101   assumes mono: "mono_on f A" and deriv: "(f has_real_derivative D) (at x)"
102   assumes "x \<in> interior A"
103   shows "D \<ge> 0"
104 proof (rule tendsto_lowerbound)
105   let ?A' = "(\<lambda>y. y - x) ` interior A"
106   from deriv show "((\<lambda>h. (f (x + h) - f x) / h) \<longlongrightarrow> D) (at 0)"
107       by (simp add: field_has_derivative_at has_field_derivative_def)
108   from mono have mono': "mono_on f (interior A)" by (rule mono_on_subset) (rule interior_subset)
110   show "eventually (\<lambda>h. (f (x + h) - f x) / h \<ge> 0) (at 0)"
111   proof (subst eventually_at_topological, intro exI conjI ballI impI)
112     have "open (interior A)" by simp
113     hence "open (op + (-x) ` interior A)" by (rule open_translation)
114     also have "(op + (-x) ` interior A) = ?A'" by auto
115     finally show "open ?A'" .
116   next
117     from \<open>x \<in> interior A\<close> show "0 \<in> ?A'" by auto
118   next
119     fix h assume "h \<in> ?A'"
120     hence "x + h \<in> interior A" by auto
121     with mono' and \<open>x \<in> interior A\<close> show "(f (x + h) - f x) / h \<ge> 0"
122       by (cases h rule: linorder_cases[of _ 0])
123          (simp_all add: divide_nonpos_neg divide_nonneg_pos mono_onD field_simps)
124   qed
125 qed simp
127 lemma strict_mono_on_imp_mono_on:
128   "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) A \<Longrightarrow> mono_on f A"
129   by (rule mono_onI, rule strict_mono_on_leD)
131 lemma mono_on_ctble_discont:
132   fixes f :: "real \<Rightarrow> real"
133   fixes A :: "real set"
134   assumes "mono_on f A"
135   shows "countable {a\<in>A. \<not> continuous (at a within A) f}"
136 proof -
137   have mono: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
138     using \<open>mono_on f A\<close> by (simp add: mono_on_def)
139   have "\<forall>a \<in> {a\<in>A. \<not> continuous (at a within A) f}. \<exists>q :: nat \<times> rat.
140       (fst q = 0 \<and> of_rat (snd q) < f a \<and> (\<forall>x \<in> A. x < a \<longrightarrow> f x < of_rat (snd q))) \<or>
141       (fst q = 1 \<and> of_rat (snd q) > f a \<and> (\<forall>x \<in> A. x > a \<longrightarrow> f x > of_rat (snd q)))"
142   proof (clarsimp simp del: One_nat_def)
143     fix a assume "a \<in> A" assume "\<not> continuous (at a within A) f"
144     thus "\<exists>q1 q2.
145             q1 = 0 \<and> real_of_rat q2 < f a \<and> (\<forall>x\<in>A. x < a \<longrightarrow> f x < real_of_rat q2) \<or>
146             q1 = 1 \<and> f a < real_of_rat q2 \<and> (\<forall>x\<in>A. a < x \<longrightarrow> real_of_rat q2 < f x)"
147     proof (auto simp add: continuous_within order_tendsto_iff eventually_at)
148       fix l assume "l < f a"
149       then obtain q2 where q2: "l < of_rat q2" "of_rat q2 < f a"
150         using of_rat_dense by blast
151       assume * [rule_format]: "\<forall>d>0. \<exists>x\<in>A. x \<noteq> a \<and> dist x a < d \<and> \<not> l < f x"
152       from q2 have "real_of_rat q2 < f a \<and> (\<forall>x\<in>A. x < a \<longrightarrow> f x < real_of_rat q2)"
153       proof auto
154         fix x assume "x \<in> A" "x < a"
155         with q2 *[of "a - x"] show "f x < real_of_rat q2"
156           apply (auto simp add: dist_real_def not_less)
157           apply (subgoal_tac "f x \<le> f xa")
158           by (auto intro: mono)
159       qed
160       thus ?thesis by auto
161     next
162       fix u assume "u > f a"
163       then obtain q2 where q2: "f a < of_rat q2" "of_rat q2 < u"
164         using of_rat_dense by blast
165       assume *[rule_format]: "\<forall>d>0. \<exists>x\<in>A. x \<noteq> a \<and> dist x a < d \<and> \<not> u > f x"
166       from q2 have "real_of_rat q2 > f a \<and> (\<forall>x\<in>A. x > a \<longrightarrow> f x > real_of_rat q2)"
167       proof auto
168         fix x assume "x \<in> A" "x > a"
169         with q2 *[of "x - a"] show "f x > real_of_rat q2"
170           apply (auto simp add: dist_real_def)
171           apply (subgoal_tac "f x \<ge> f xa")
172           by (auto intro: mono)
173       qed
174       thus ?thesis by auto
175     qed
176   qed
177   hence "\<exists>g :: real \<Rightarrow> nat \<times> rat . \<forall>a \<in> {a\<in>A. \<not> continuous (at a within A) f}.
178       (fst (g a) = 0 \<and> of_rat (snd (g a)) < f a \<and> (\<forall>x \<in> A. x < a \<longrightarrow> f x < of_rat (snd (g a)))) |
179       (fst (g a) = 1 \<and> of_rat (snd (g a)) > f a \<and> (\<forall>x \<in> A. x > a \<longrightarrow> f x > of_rat (snd (g a))))"
180     by (rule bchoice)
181   then guess g ..
182   hence g: "\<And>a x. a \<in> A \<Longrightarrow> \<not> continuous (at a within A) f \<Longrightarrow> x \<in> A \<Longrightarrow>
183       (fst (g a) = 0 \<and> of_rat (snd (g a)) < f a \<and> (x < a \<longrightarrow> f x < of_rat (snd (g a)))) |
184       (fst (g a) = 1 \<and> of_rat (snd (g a)) > f a \<and> (x > a \<longrightarrow> f x > of_rat (snd (g a))))"
185     by auto
186   have "inj_on g {a\<in>A. \<not> continuous (at a within A) f}"
187   proof (auto simp add: inj_on_def)
188     fix w z
189     assume 1: "w \<in> A" and 2: "\<not> continuous (at w within A) f" and
190            3: "z \<in> A" and 4: "\<not> continuous (at z within A) f" and
191            5: "g w = g z"
192     from g [OF 1 2 3] g [OF 3 4 1] 5
193     show "w = z" by auto
194   qed
195   thus ?thesis
196     by (rule countableI')
197 qed
199 lemma mono_on_ctble_discont_open:
200   fixes f :: "real \<Rightarrow> real"
201   fixes A :: "real set"
202   assumes "open A" "mono_on f A"
203   shows "countable {a\<in>A. \<not>isCont f a}"
204 proof -
205   have "{a\<in>A. \<not>isCont f a} = {a\<in>A. \<not>(continuous (at a within A) f)}"
206     by (auto simp add: continuous_within_open [OF _ \<open>open A\<close>])
207   thus ?thesis
208     apply (elim ssubst)
209     by (rule mono_on_ctble_discont, rule assms)
210 qed
212 lemma mono_ctble_discont:
213   fixes f :: "real \<Rightarrow> real"
214   assumes "mono f"
215   shows "countable {a. \<not> isCont f a}"
216 using assms mono_on_ctble_discont [of f UNIV] unfolding mono_on_def mono_def by auto
218 lemma has_real_derivative_imp_continuous_on:
219   assumes "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
220   shows "continuous_on A f"
221   apply (intro differentiable_imp_continuous_on, unfold differentiable_on_def)
222   apply (intro ballI Deriv.differentiableI)
223   apply (rule has_field_derivative_subset[OF assms])
224   apply simp_all
225   done
227 lemma closure_contains_Sup:
228   fixes S :: "real set"
229   assumes "S \<noteq> {}" "bdd_above S"
230   shows "Sup S \<in> closure S"
231 proof-
232   have "Inf (uminus ` S) \<in> closure (uminus ` S)"
233       using assms by (intro closure_contains_Inf) auto
234   also have "Inf (uminus ` S) = -Sup S" by (simp add: Inf_real_def)
235   also have "closure (uminus ` S) = uminus ` closure S"
236       by (rule sym, intro closure_injective_linear_image) (auto intro: linearI)
237   finally show ?thesis by auto
238 qed
240 lemma closed_contains_Sup:
241   fixes S :: "real set"
242   shows "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> closed S \<Longrightarrow> Sup S \<in> S"
243   by (subst closure_closed[symmetric], assumption, rule closure_contains_Sup)
245 lemma deriv_nonneg_imp_mono:
246   assumes deriv: "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_real_derivative g' x) (at x)"
247   assumes nonneg: "\<And>x. x \<in> {a..b} \<Longrightarrow> g' x \<ge> 0"
248   assumes ab: "a \<le> b"
249   shows "g a \<le> g b"
250 proof (cases "a < b")
251   assume "a < b"
252   from deriv have "\<forall>x. x \<ge> a \<and> x \<le> b \<longrightarrow> (g has_real_derivative g' x) (at x)" by simp
253   from MVT2[OF \<open>a < b\<close> this] and deriv
254     obtain \<xi> where \<xi>_ab: "\<xi> > a" "\<xi> < b" and g_ab: "g b - g a = (b - a) * g' \<xi>" by blast
255   from \<xi>_ab ab nonneg have "(b - a) * g' \<xi> \<ge> 0" by simp
256   with g_ab show ?thesis by simp
257 qed (insert ab, simp)
259 lemma continuous_interval_vimage_Int:
260   assumes "continuous_on {a::real..b} g" and mono: "\<And>x y. a \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> b \<Longrightarrow> g x \<le> g y"
261   assumes "a \<le> b" "(c::real) \<le> d" "{c..d} \<subseteq> {g a..g b}"
262   obtains c' d' where "{a..b} \<inter> g -` {c..d} = {c'..d'}" "c' \<le> d'" "g c' = c" "g d' = d"
263 proof-
264   let ?A = "{a..b} \<inter> g -` {c..d}"
265   from IVT'[of g a c b, OF _ _ \<open>a \<le> b\<close> assms(1)] assms(4,5)
266   obtain c'' where c'': "c'' \<in> ?A" "g c'' = c" by auto
267   from IVT'[of g a d b, OF _ _ \<open>a \<le> b\<close> assms(1)] assms(4,5)
268   obtain d'' where d'': "d'' \<in> ?A" "g d'' = d" by auto
269   hence [simp]: "?A \<noteq> {}" by blast
271   define c' where "c' = Inf ?A"
272   define d' where "d' = Sup ?A"
273   have "?A \<subseteq> {c'..d'}" unfolding c'_def d'_def
274     by (intro subsetI) (auto intro: cInf_lower cSup_upper)
275   moreover from assms have "closed ?A"
276     using continuous_on_closed_vimage[of "{a..b}" g] by (subst Int_commute) simp
277   hence c'd'_in_set: "c' \<in> ?A" "d' \<in> ?A" unfolding c'_def d'_def
278     by ((intro closed_contains_Inf closed_contains_Sup, simp_all)[])+
279   hence "{c'..d'} \<subseteq> ?A" using assms
280     by (intro subsetI)
281        (auto intro!: order_trans[of c "g c'" "g x" for x] order_trans[of "g x" "g d'" d for x]
282              intro!: mono)
283   moreover have "c' \<le> d'" using c'd'_in_set(2) unfolding c'_def by (intro cInf_lower) auto
284   moreover have "g c' \<le> c" "g d' \<ge> d"
285     apply (insert c'' d'' c'd'_in_set)
286     apply (subst c''(2)[symmetric])
287     apply (auto simp: c'_def intro!: mono cInf_lower c'') []
288     apply (subst d''(2)[symmetric])
289     apply (auto simp: d'_def intro!: mono cSup_upper d'') []
290     done
291   with c'd'_in_set have "g c' = c" "g d' = d" by auto
292   ultimately show ?thesis using that by blast
293 qed
295 subsection \<open>Generic Borel spaces\<close>
297 definition (in topological_space) borel :: "'a measure" where
298   "borel = sigma UNIV {S. open S}"
300 abbreviation "borel_measurable M \<equiv> measurable M borel"
302 lemma in_borel_measurable:
303    "f \<in> borel_measurable M \<longleftrightarrow>
304     (\<forall>S \<in> sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)"
305   by (auto simp add: measurable_def borel_def)
307 lemma in_borel_measurable_borel:
308    "f \<in> borel_measurable M \<longleftrightarrow>
309     (\<forall>S \<in> sets borel.
310       f -` S \<inter> space M \<in> sets M)"
311   by (auto simp add: measurable_def borel_def)
313 lemma space_borel[simp]: "space borel = UNIV"
314   unfolding borel_def by auto
316 lemma space_in_borel[measurable]: "UNIV \<in> sets borel"
317   unfolding borel_def by auto
319 lemma sets_borel: "sets borel = sigma_sets UNIV {S. open S}"
320   unfolding borel_def by (rule sets_measure_of) simp
322 lemma measurable_sets_borel:
323     "\<lbrakk>f \<in> measurable borel M; A \<in> sets M\<rbrakk> \<Longrightarrow> f -` A \<in> sets borel"
324   by (drule (1) measurable_sets) simp
326 lemma pred_Collect_borel[measurable (raw)]: "Measurable.pred borel P \<Longrightarrow> {x. P x} \<in> sets borel"
327   unfolding borel_def pred_def by auto
329 lemma borel_open[measurable (raw generic)]:
330   assumes "open A" shows "A \<in> sets borel"
331 proof -
332   have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms .
333   thus ?thesis unfolding borel_def by auto
334 qed
336 lemma borel_closed[measurable (raw generic)]:
337   assumes "closed A" shows "A \<in> sets borel"
338 proof -
339   have "space borel - (- A) \<in> sets borel"
340     using assms unfolding closed_def by (blast intro: borel_open)
341   thus ?thesis by simp
342 qed
344 lemma borel_singleton[measurable]:
345   "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets (borel :: 'a::t1_space measure)"
346   unfolding insert_def by (rule sets.Un) auto
348 lemma borel_comp[measurable]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
349   unfolding Compl_eq_Diff_UNIV by simp
351 lemma borel_measurable_vimage:
352   fixes f :: "'a \<Rightarrow> 'x::t2_space"
353   assumes borel[measurable]: "f \<in> borel_measurable M"
354   shows "f -` {x} \<inter> space M \<in> sets M"
355   by simp
357 lemma borel_measurableI:
358   fixes f :: "'a \<Rightarrow> 'x::topological_space"
359   assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
360   shows "f \<in> borel_measurable M"
361   unfolding borel_def
362 proof (rule measurable_measure_of, simp_all)
363   fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"
364     using assms[of S] by simp
365 qed
367 lemma borel_measurable_const:
368   "(\<lambda>x. c) \<in> borel_measurable M"
369   by auto
371 lemma borel_measurable_indicator:
372   assumes A: "A \<in> sets M"
373   shows "indicator A \<in> borel_measurable M"
374   unfolding indicator_def [abs_def] using A
375   by (auto intro!: measurable_If_set)
377 lemma borel_measurable_count_space[measurable (raw)]:
378   "f \<in> borel_measurable (count_space S)"
379   unfolding measurable_def by auto
381 lemma borel_measurable_indicator'[measurable (raw)]:
382   assumes [measurable]: "{x\<in>space M. f x \<in> A x} \<in> sets M"
383   shows "(\<lambda>x. indicator (A x) (f x)) \<in> borel_measurable M"
384   unfolding indicator_def[abs_def]
385   by (auto intro!: measurable_If)
387 lemma borel_measurable_indicator_iff:
388   "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
389     (is "?I \<in> borel_measurable M \<longleftrightarrow> _")
390 proof
391   assume "?I \<in> borel_measurable M"
392   then have "?I -` {1} \<inter> space M \<in> sets M"
393     unfolding measurable_def by auto
394   also have "?I -` {1} \<inter> space M = A \<inter> space M"
395     unfolding indicator_def [abs_def] by auto
396   finally show "A \<inter> space M \<in> sets M" .
397 next
398   assume "A \<inter> space M \<in> sets M"
399   moreover have "?I \<in> borel_measurable M \<longleftrightarrow>
400     (indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"
401     by (intro measurable_cong) (auto simp: indicator_def)
402   ultimately show "?I \<in> borel_measurable M" by auto
403 qed
405 lemma borel_measurable_subalgebra:
406   assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
407   shows "f \<in> borel_measurable M"
408   using assms unfolding measurable_def by auto
410 lemma borel_measurable_restrict_space_iff_ereal:
411   fixes f :: "'a \<Rightarrow> ereal"
412   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
413   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
414     (\<lambda>x. f x * indicator \<Omega> x) \<in> borel_measurable M"
415   by (subst measurable_restrict_space_iff)
416      (auto simp: indicator_def if_distrib[where f="\<lambda>x. a * x" for a] cong del: if_weak_cong)
418 lemma borel_measurable_restrict_space_iff_ennreal:
419   fixes f :: "'a \<Rightarrow> ennreal"
420   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
421   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
422     (\<lambda>x. f x * indicator \<Omega> x) \<in> borel_measurable M"
423   by (subst measurable_restrict_space_iff)
424      (auto simp: indicator_def if_distrib[where f="\<lambda>x. a * x" for a] cong del: if_weak_cong)
426 lemma borel_measurable_restrict_space_iff:
427   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
428   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
429   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
430     (\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> borel_measurable M"
431   by (subst measurable_restrict_space_iff)
432      (auto simp: indicator_def if_distrib[where f="\<lambda>x. x *\<^sub>R a" for a] ac_simps
433        cong del: if_weak_cong)
435 lemma cbox_borel[measurable]: "cbox a b \<in> sets borel"
436   by (auto intro: borel_closed)
438 lemma box_borel[measurable]: "box a b \<in> sets borel"
439   by (auto intro: borel_open)
441 lemma borel_compact: "compact (A::'a::t2_space set) \<Longrightarrow> A \<in> sets borel"
442   by (auto intro: borel_closed dest!: compact_imp_closed)
444 lemma borel_sigma_sets_subset:
445   "A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel"
446   using sets.sigma_sets_subset[of A borel] by simp
448 lemma borel_eq_sigmaI1:
449   fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
450   assumes borel_eq: "borel = sigma UNIV X"
451   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (F ` A))"
452   assumes F: "\<And>i. i \<in> A \<Longrightarrow> F i \<in> sets borel"
453   shows "borel = sigma UNIV (F ` A)"
454   unfolding borel_def
455 proof (intro sigma_eqI antisym)
456   have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel"
457     unfolding borel_def by simp
458   also have "\<dots> = sigma_sets UNIV X"
459     unfolding borel_eq by simp
460   also have "\<dots> \<subseteq> sigma_sets UNIV (F`A)"
461     using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto
462   finally show "sigma_sets UNIV {S. open S} \<subseteq> sigma_sets UNIV (F`A)" .
463   show "sigma_sets UNIV (F`A) \<subseteq> sigma_sets UNIV {S. open S}"
464     unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto
465 qed auto
467 lemma borel_eq_sigmaI2:
468   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set"
469     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
470   assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`B)"
471   assumes X: "\<And>i j. (i, j) \<in> B \<Longrightarrow> G i j \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"
472   assumes F: "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel"
473   shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"
474   using assms
475   by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` B" and F="(\<lambda>(i, j). F i j)"]) auto
477 lemma borel_eq_sigmaI3:
478   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
479   assumes borel_eq: "borel = sigma UNIV X"
480   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"
481   assumes F: "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel"
482   shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"
483   using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto
485 lemma borel_eq_sigmaI4:
486   fixes F :: "'i \<Rightarrow> 'a::topological_space set"
487     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
488   assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`A)"
489   assumes X: "\<And>i j. (i, j) \<in> A \<Longrightarrow> G i j \<in> sets (sigma UNIV (range F))"
490   assumes F: "\<And>i. F i \<in> sets borel"
491   shows "borel = sigma UNIV (range F)"
492   using assms by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` A" and F=F]) auto
494 lemma borel_eq_sigmaI5:
495   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set"
496   assumes borel_eq: "borel = sigma UNIV (range G)"
497   assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
498   assumes F: "\<And>i j. F i j \<in> sets borel"
499   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
500   using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto
502 lemma second_countable_borel_measurable:
503   fixes X :: "'a::second_countable_topology set set"
504   assumes eq: "open = generate_topology X"
505   shows "borel = sigma UNIV X"
506   unfolding borel_def
507 proof (intro sigma_eqI sigma_sets_eqI)
508   interpret X: sigma_algebra UNIV "sigma_sets UNIV X"
509     by (rule sigma_algebra_sigma_sets) simp
511   fix S :: "'a set" assume "S \<in> Collect open"
512   then have "generate_topology X S"
513     by (auto simp: eq)
514   then show "S \<in> sigma_sets UNIV X"
515   proof induction
516     case (UN K)
517     then have K: "\<And>k. k \<in> K \<Longrightarrow> open k"
518       unfolding eq by auto
519     from ex_countable_basis obtain B :: "'a set set" where
520       B:  "\<And>b. b \<in> B \<Longrightarrow> open b" "\<And>X. open X \<Longrightarrow> \<exists>b\<subseteq>B. (\<Union>b) = X" and "countable B"
521       by (auto simp: topological_basis_def)
522     from B(2)[OF K] obtain m where m: "\<And>k. k \<in> K \<Longrightarrow> m k \<subseteq> B" "\<And>k. k \<in> K \<Longrightarrow> (\<Union>m k) = k"
523       by metis
524     define U where "U = (\<Union>k\<in>K. m k)"
525     with m have "countable U"
526       by (intro countable_subset[OF _ \<open>countable B\<close>]) auto
527     have "\<Union>U = (\<Union>A\<in>U. A)" by simp
528     also have "\<dots> = \<Union>K"
529       unfolding U_def UN_simps by (simp add: m)
530     finally have "\<Union>U = \<Union>K" .
532     have "\<forall>b\<in>U. \<exists>k\<in>K. b \<subseteq> k"
533       using m by (auto simp: U_def)
534     then obtain u where u: "\<And>b. b \<in> U \<Longrightarrow> u b \<in> K" and "\<And>b. b \<in> U \<Longrightarrow> b \<subseteq> u b"
535       by metis
536     then have "(\<Union>b\<in>U. u b) \<subseteq> \<Union>K" "\<Union>U \<subseteq> (\<Union>b\<in>U. u b)"
537       by auto
538     then have "\<Union>K = (\<Union>b\<in>U. u b)"
539       unfolding \<open>\<Union>U = \<Union>K\<close> by auto
540     also have "\<dots> \<in> sigma_sets UNIV X"
541       using u UN by (intro X.countable_UN' \<open>countable U\<close>) auto
542     finally show "\<Union>K \<in> sigma_sets UNIV X" .
543   qed auto
544 qed (auto simp: eq intro: generate_topology.Basis)
546 lemma borel_eq_closed: "borel = sigma UNIV (Collect closed)"
547   unfolding borel_def
548 proof (intro sigma_eqI sigma_sets_eqI, safe)
549   fix x :: "'a set" assume "open x"
550   hence "x = UNIV - (UNIV - x)" by auto
551   also have "\<dots> \<in> sigma_sets UNIV (Collect closed)"
552     by (force intro: sigma_sets.Compl simp: \<open>open x\<close>)
553   finally show "x \<in> sigma_sets UNIV (Collect closed)" by simp
554 next
555   fix x :: "'a set" assume "closed x"
556   hence "x = UNIV - (UNIV - x)" by auto
557   also have "\<dots> \<in> sigma_sets UNIV (Collect open)"
558     by (force intro: sigma_sets.Compl simp: \<open>closed x\<close>)
559   finally show "x \<in> sigma_sets UNIV (Collect open)" by simp
560 qed simp_all
562 lemma borel_eq_countable_basis:
563   fixes B::"'a::topological_space set set"
564   assumes "countable B"
565   assumes "topological_basis B"
566   shows "borel = sigma UNIV B"
567   unfolding borel_def
568 proof (intro sigma_eqI sigma_sets_eqI, safe)
569   interpret countable_basis using assms by unfold_locales
570   fix X::"'a set" assume "open X"
571   from open_countable_basisE[OF this] guess B' . note B' = this
572   then show "X \<in> sigma_sets UNIV B"
573     by (blast intro: sigma_sets_UNION \<open>countable B\<close> countable_subset)
574 next
575   fix b assume "b \<in> B"
576   hence "open b" by (rule topological_basis_open[OF assms(2)])
577   thus "b \<in> sigma_sets UNIV (Collect open)" by auto
578 qed simp_all
580 lemma borel_measurable_continuous_on_restrict:
581   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
582   assumes f: "continuous_on A f"
583   shows "f \<in> borel_measurable (restrict_space borel A)"
584 proof (rule borel_measurableI)
585   fix S :: "'b set" assume "open S"
586   with f obtain T where "f -` S \<inter> A = T \<inter> A" "open T"
587     by (metis continuous_on_open_invariant)
588   then show "f -` S \<inter> space (restrict_space borel A) \<in> sets (restrict_space borel A)"
589     by (force simp add: sets_restrict_space space_restrict_space)
590 qed
592 lemma borel_measurable_continuous_on1: "continuous_on UNIV f \<Longrightarrow> f \<in> borel_measurable borel"
593   by (drule borel_measurable_continuous_on_restrict) simp
595 lemma borel_measurable_continuous_on_if:
596   "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> continuous_on (- A) g \<Longrightarrow>
597     (\<lambda>x. if x \<in> A then f x else g x) \<in> borel_measurable borel"
598   by (auto simp add: measurable_If_restrict_space_iff Collect_neg_eq
599            intro!: borel_measurable_continuous_on_restrict)
601 lemma borel_measurable_continuous_countable_exceptions:
602   fixes f :: "'a::t1_space \<Rightarrow> 'b::topological_space"
603   assumes X: "countable X"
604   assumes "continuous_on (- X) f"
605   shows "f \<in> borel_measurable borel"
606 proof (rule measurable_discrete_difference[OF _ X])
607   have "X \<in> sets borel"
608     by (rule sets.countable[OF _ X]) auto
609   then show "(\<lambda>x. if x \<in> X then undefined else f x) \<in> borel_measurable borel"
610     by (intro borel_measurable_continuous_on_if assms continuous_intros)
611 qed auto
613 lemma borel_measurable_continuous_on:
614   assumes f: "continuous_on UNIV f" and g: "g \<in> borel_measurable M"
615   shows "(\<lambda>x. f (g x)) \<in> borel_measurable M"
616   using measurable_comp[OF g borel_measurable_continuous_on1[OF f]] by (simp add: comp_def)
618 lemma borel_measurable_continuous_on_indicator:
619   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
620   shows "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable borel"
621   by (subst borel_measurable_restrict_space_iff[symmetric])
622      (auto intro: borel_measurable_continuous_on_restrict)
624 lemma borel_measurable_Pair[measurable (raw)]:
625   fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"
626   assumes f[measurable]: "f \<in> borel_measurable M"
627   assumes g[measurable]: "g \<in> borel_measurable M"
628   shows "(\<lambda>x. (f x, g x)) \<in> borel_measurable M"
629 proof (subst borel_eq_countable_basis)
630   let ?B = "SOME B::'b set set. countable B \<and> topological_basis B"
631   let ?C = "SOME B::'c set set. countable B \<and> topological_basis B"
632   let ?P = "(\<lambda>(b, c). b \<times> c) ` (?B \<times> ?C)"
633   show "countable ?P" "topological_basis ?P"
634     by (auto intro!: countable_basis topological_basis_prod is_basis)
636   show "(\<lambda>x. (f x, g x)) \<in> measurable M (sigma UNIV ?P)"
637   proof (rule measurable_measure_of)
638     fix S assume "S \<in> ?P"
639     then obtain b c where "b \<in> ?B" "c \<in> ?C" and S: "S = b \<times> c" by auto
640     then have borel: "open b" "open c"
641       by (auto intro: is_basis topological_basis_open)
642     have "(\<lambda>x. (f x, g x)) -` S \<inter> space M = (f -` b \<inter> space M) \<inter> (g -` c \<inter> space M)"
643       unfolding S by auto
644     also have "\<dots> \<in> sets M"
645       using borel by simp
646     finally show "(\<lambda>x. (f x, g x)) -` S \<inter> space M \<in> sets M" .
647   qed auto
648 qed
650 lemma borel_measurable_continuous_Pair:
651   fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"
652   assumes [measurable]: "f \<in> borel_measurable M"
653   assumes [measurable]: "g \<in> borel_measurable M"
654   assumes H: "continuous_on UNIV (\<lambda>x. H (fst x) (snd x))"
655   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
656 proof -
657   have eq: "(\<lambda>x. H (f x) (g x)) = (\<lambda>x. (\<lambda>x. H (fst x) (snd x)) (f x, g x))" by auto
658   show ?thesis
659     unfolding eq by (rule borel_measurable_continuous_on[OF H]) auto
660 qed
662 subsection \<open>Borel spaces on order topologies\<close>
664 lemma [measurable]:
665   fixes a b :: "'a::linorder_topology"
666   shows lessThan_borel: "{..< a} \<in> sets borel"
667     and greaterThan_borel: "{a <..} \<in> sets borel"
668     and greaterThanLessThan_borel: "{a<..<b} \<in> sets borel"
669     and atMost_borel: "{..a} \<in> sets borel"
670     and atLeast_borel: "{a..} \<in> sets borel"
671     and atLeastAtMost_borel: "{a..b} \<in> sets borel"
672     and greaterThanAtMost_borel: "{a<..b} \<in> sets borel"
673     and atLeastLessThan_borel: "{a..<b} \<in> sets borel"
674   unfolding greaterThanAtMost_def atLeastLessThan_def
675   by (blast intro: borel_open borel_closed open_lessThan open_greaterThan open_greaterThanLessThan
676                    closed_atMost closed_atLeast closed_atLeastAtMost)+
678 lemma borel_Iio:
679   "borel = sigma UNIV (range lessThan :: 'a::{linorder_topology, second_countable_topology} set set)"
680   unfolding second_countable_borel_measurable[OF open_generated_order]
681 proof (intro sigma_eqI sigma_sets_eqI)
682   from countable_dense_setE guess D :: "'a set" . note D = this
684   interpret L: sigma_algebra UNIV "sigma_sets UNIV (range lessThan)"
685     by (rule sigma_algebra_sigma_sets) simp
687   fix A :: "'a set" assume "A \<in> range lessThan \<union> range greaterThan"
688   then obtain y where "A = {y <..} \<or> A = {..< y}"
689     by blast
690   then show "A \<in> sigma_sets UNIV (range lessThan)"
691   proof
692     assume A: "A = {y <..}"
693     show ?thesis
694     proof cases
695       assume "\<forall>x>y. \<exists>d. y < d \<and> d < x"
696       with D(2)[of "{y <..< x}" for x] have "\<forall>x>y. \<exists>d\<in>D. y < d \<and> d < x"
697         by (auto simp: set_eq_iff)
698       then have "A = UNIV - (\<Inter>d\<in>{d\<in>D. y < d}. {..< d})"
699         by (auto simp: A) (metis less_asym)
700       also have "\<dots> \<in> sigma_sets UNIV (range lessThan)"
701         using D(1) by (intro L.Diff L.top L.countable_INT'') auto
702       finally show ?thesis .
703     next
704       assume "\<not> (\<forall>x>y. \<exists>d. y < d \<and> d < x)"
705       then obtain x where "y < x"  "\<And>d. y < d \<Longrightarrow> \<not> d < x"
706         by auto
707       then have "A = UNIV - {..< x}"
708         unfolding A by (auto simp: not_less[symmetric])
709       also have "\<dots> \<in> sigma_sets UNIV (range lessThan)"
710         by auto
711       finally show ?thesis .
712     qed
713   qed auto
714 qed auto
716 lemma borel_Ioi:
717   "borel = sigma UNIV (range greaterThan :: 'a::{linorder_topology, second_countable_topology} set set)"
718   unfolding second_countable_borel_measurable[OF open_generated_order]
719 proof (intro sigma_eqI sigma_sets_eqI)
720   from countable_dense_setE guess D :: "'a set" . note D = this
722   interpret L: sigma_algebra UNIV "sigma_sets UNIV (range greaterThan)"
723     by (rule sigma_algebra_sigma_sets) simp
725   fix A :: "'a set" assume "A \<in> range lessThan \<union> range greaterThan"
726   then obtain y where "A = {y <..} \<or> A = {..< y}"
727     by blast
728   then show "A \<in> sigma_sets UNIV (range greaterThan)"
729   proof
730     assume A: "A = {..< y}"
731     show ?thesis
732     proof cases
733       assume "\<forall>x<y. \<exists>d. x < d \<and> d < y"
734       with D(2)[of "{x <..< y}" for x] have "\<forall>x<y. \<exists>d\<in>D. x < d \<and> d < y"
735         by (auto simp: set_eq_iff)
736       then have "A = UNIV - (\<Inter>d\<in>{d\<in>D. d < y}. {d <..})"
737         by (auto simp: A) (metis less_asym)
738       also have "\<dots> \<in> sigma_sets UNIV (range greaterThan)"
739         using D(1) by (intro L.Diff L.top L.countable_INT'') auto
740       finally show ?thesis .
741     next
742       assume "\<not> (\<forall>x<y. \<exists>d. x < d \<and> d < y)"
743       then obtain x where "x < y"  "\<And>d. y > d \<Longrightarrow> x \<ge> d"
744         by (auto simp: not_less[symmetric])
745       then have "A = UNIV - {x <..}"
746         unfolding A Compl_eq_Diff_UNIV[symmetric] by auto
747       also have "\<dots> \<in> sigma_sets UNIV (range greaterThan)"
748         by auto
749       finally show ?thesis .
750     qed
751   qed auto
752 qed auto
754 lemma borel_measurableI_less:
755   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
756   shows "(\<And>y. {x\<in>space M. f x < y} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
757   unfolding borel_Iio
758   by (rule measurable_measure_of) (auto simp: Int_def conj_commute)
760 lemma borel_measurableI_greater:
761   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
762   shows "(\<And>y. {x\<in>space M. y < f x} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
763   unfolding borel_Ioi
764   by (rule measurable_measure_of) (auto simp: Int_def conj_commute)
766 lemma borel_measurableI_le:
767   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
768   shows "(\<And>y. {x\<in>space M. f x \<le> y} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
769   by (rule borel_measurableI_greater) (auto simp: not_le[symmetric])
771 lemma borel_measurableI_ge:
772   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
773   shows "(\<And>y. {x\<in>space M. y \<le> f x} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
774   by (rule borel_measurableI_less) (auto simp: not_le[symmetric])
776 lemma borel_measurable_less[measurable]:
777   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
778   assumes "f \<in> borel_measurable M"
779   assumes "g \<in> borel_measurable M"
780   shows "{w \<in> space M. f w < g w} \<in> sets M"
781 proof -
782   have "{w \<in> space M. f w < g w} = (\<lambda>x. (f x, g x)) -` {x. fst x < snd x} \<inter> space M"
783     by auto
784   also have "\<dots> \<in> sets M"
785     by (intro measurable_sets[OF borel_measurable_Pair borel_open, OF assms open_Collect_less]
786               continuous_intros)
787   finally show ?thesis .
788 qed
790 lemma
791   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
792   assumes f[measurable]: "f \<in> borel_measurable M"
793   assumes g[measurable]: "g \<in> borel_measurable M"
794   shows borel_measurable_le[measurable]: "{w \<in> space M. f w \<le> g w} \<in> sets M"
795     and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M"
796     and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
797   unfolding eq_iff not_less[symmetric]
798   by measurable
800 lemma borel_measurable_SUP[measurable (raw)]:
801   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}"
802   assumes [simp]: "countable I"
803   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
804   shows "(\<lambda>x. SUP i:I. F i x) \<in> borel_measurable M"
805   by (rule borel_measurableI_greater) (simp add: less_SUP_iff)
807 lemma borel_measurable_INF[measurable (raw)]:
808   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}"
809   assumes [simp]: "countable I"
810   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
811   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
812   by (rule borel_measurableI_less) (simp add: INF_less_iff)
814 lemma borel_measurable_cSUP[measurable (raw)]:
815   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology, second_countable_topology}"
816   assumes [simp]: "countable I"
817   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
818   assumes bdd: "\<And>x. x \<in> space M \<Longrightarrow> bdd_above ((\<lambda>i. F i x) ` I)"
819   shows "(\<lambda>x. SUP i:I. F i x) \<in> borel_measurable M"
820 proof cases
821   assume "I = {}" then show ?thesis
822     unfolding \<open>I = {}\<close> image_empty by simp
823 next
824   assume "I \<noteq> {}"
825   show ?thesis
826   proof (rule borel_measurableI_le)
827     fix y
828     have "{x \<in> space M. \<forall>i\<in>I. F i x \<le> y} \<in> sets M"
829       by measurable
830     also have "{x \<in> space M. \<forall>i\<in>I. F i x \<le> y} = {x \<in> space M. (SUP i:I. F i x) \<le> y}"
831       by (simp add: cSUP_le_iff \<open>I \<noteq> {}\<close> bdd cong: conj_cong)
832     finally show "{x \<in> space M. (SUP i:I. F i x) \<le>  y} \<in> sets M"  .
833   qed
834 qed
836 lemma borel_measurable_cINF[measurable (raw)]:
837   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology, second_countable_topology}"
838   assumes [simp]: "countable I"
839   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
840   assumes bdd: "\<And>x. x \<in> space M \<Longrightarrow> bdd_below ((\<lambda>i. F i x) ` I)"
841   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
842 proof cases
843   assume "I = {}" then show ?thesis
844     unfolding \<open>I = {}\<close> image_empty by simp
845 next
846   assume "I \<noteq> {}"
847   show ?thesis
848   proof (rule borel_measurableI_ge)
849     fix y
850     have "{x \<in> space M. \<forall>i\<in>I. y \<le> F i x} \<in> sets M"
851       by measurable
852     also have "{x \<in> space M. \<forall>i\<in>I. y \<le> F i x} = {x \<in> space M. y \<le> (INF i:I. F i x)}"
853       by (simp add: le_cINF_iff \<open>I \<noteq> {}\<close> bdd cong: conj_cong)
854     finally show "{x \<in> space M. y \<le> (INF i:I. F i x)} \<in> sets M"  .
855   qed
856 qed
858 lemma borel_measurable_lfp[consumes 1, case_names continuity step]:
859   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})"
860   assumes "sup_continuous F"
861   assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M"
862   shows "lfp F \<in> borel_measurable M"
863 proof -
864   { fix i have "((F ^^ i) bot) \<in> borel_measurable M"
865       by (induct i) (auto intro!: *) }
866   then have "(\<lambda>x. SUP i. (F ^^ i) bot x) \<in> borel_measurable M"
867     by measurable
868   also have "(\<lambda>x. SUP i. (F ^^ i) bot x) = (SUP i. (F ^^ i) bot)"
869     by auto
870   also have "(SUP i. (F ^^ i) bot) = lfp F"
871     by (rule sup_continuous_lfp[symmetric]) fact
872   finally show ?thesis .
873 qed
875 lemma borel_measurable_gfp[consumes 1, case_names continuity step]:
876   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})"
877   assumes "inf_continuous F"
878   assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M"
879   shows "gfp F \<in> borel_measurable M"
880 proof -
881   { fix i have "((F ^^ i) top) \<in> borel_measurable M"
882       by (induct i) (auto intro!: * simp: bot_fun_def) }
883   then have "(\<lambda>x. INF i. (F ^^ i) top x) \<in> borel_measurable M"
884     by measurable
885   also have "(\<lambda>x. INF i. (F ^^ i) top x) = (INF i. (F ^^ i) top)"
886     by auto
887   also have "\<dots> = gfp F"
888     by (rule inf_continuous_gfp[symmetric]) fact
889   finally show ?thesis .
890 qed
892 lemma borel_measurable_max[measurable (raw)]:
893   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. max (g x) (f x) :: 'b::{second_countable_topology, linorder_topology}) \<in> borel_measurable M"
894   by (rule borel_measurableI_less) simp
896 lemma borel_measurable_min[measurable (raw)]:
897   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. min (g x) (f x) :: 'b::{second_countable_topology, linorder_topology}) \<in> borel_measurable M"
898   by (rule borel_measurableI_greater) simp
900 lemma borel_measurable_Min[measurable (raw)]:
901   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M) \<Longrightarrow> (\<lambda>x. Min ((\<lambda>i. f i x)`I) :: 'b::{second_countable_topology, linorder_topology}) \<in> borel_measurable M"
902 proof (induct I rule: finite_induct)
903   case (insert i I) then show ?case
904     by (cases "I = {}") auto
905 qed auto
907 lemma borel_measurable_Max[measurable (raw)]:
908   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M) \<Longrightarrow> (\<lambda>x. Max ((\<lambda>i. f i x)`I) :: 'b::{second_countable_topology, linorder_topology}) \<in> borel_measurable M"
909 proof (induct I rule: finite_induct)
910   case (insert i I) then show ?case
911     by (cases "I = {}") auto
912 qed auto
914 lemma borel_measurable_sup[measurable (raw)]:
915   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. sup (g x) (f x) :: 'b::{lattice, second_countable_topology, linorder_topology}) \<in> borel_measurable M"
916   unfolding sup_max by measurable
918 lemma borel_measurable_inf[measurable (raw)]:
919   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. inf (g x) (f x) :: 'b::{lattice, second_countable_topology, linorder_topology}) \<in> borel_measurable M"
920   unfolding inf_min by measurable
922 lemma [measurable (raw)]:
923   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
924   assumes "\<And>i. f i \<in> borel_measurable M"
925   shows borel_measurable_liminf: "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
926     and borel_measurable_limsup: "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
927   unfolding liminf_SUP_INF limsup_INF_SUP using assms by auto
929 lemma measurable_convergent[measurable (raw)]:
930   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
931   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
932   shows "Measurable.pred M (\<lambda>x. convergent (\<lambda>i. f i x))"
933   unfolding convergent_ereal by measurable
935 lemma sets_Collect_convergent[measurable]:
936   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
937   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
938   shows "{x\<in>space M. convergent (\<lambda>i. f i x)} \<in> sets M"
939   by measurable
941 lemma borel_measurable_lim[measurable (raw)]:
942   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
943   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
944   shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
945 proof -
946   have "\<And>x. lim (\<lambda>i. f i x) = (if convergent (\<lambda>i. f i x) then limsup (\<lambda>i. f i x) else (THE i. False))"
947     by (simp add: lim_def convergent_def convergent_limsup_cl)
948   then show ?thesis
949     by simp
950 qed
952 lemma borel_measurable_LIMSEQ_order:
953   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
954   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
955   and u: "\<And>i. u i \<in> borel_measurable M"
956   shows "u' \<in> borel_measurable M"
957 proof -
958   have "\<And>x. x \<in> space M \<Longrightarrow> u' x = liminf (\<lambda>n. u n x)"
959     using u' by (simp add: lim_imp_Liminf[symmetric])
960   with u show ?thesis by (simp cong: measurable_cong)
961 qed
963 subsection \<open>Borel spaces on topological monoids\<close>
965 lemma borel_measurable_add[measurable (raw)]:
966   fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, topological_monoid_add}"
967   assumes f: "f \<in> borel_measurable M"
968   assumes g: "g \<in> borel_measurable M"
969   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
970   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
972 lemma borel_measurable_sum[measurable (raw)]:
973   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, topological_comm_monoid_add}"
974   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
975   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
976 proof cases
977   assume "finite S"
978   thus ?thesis using assms by induct auto
979 qed simp
981 lemma borel_measurable_suminf_order[measurable (raw)]:
982   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology, topological_comm_monoid_add}"
983   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
984   shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M"
985   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
987 subsection \<open>Borel spaces on Euclidean spaces\<close>
989 lemma borel_measurable_inner[measurable (raw)]:
990   fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_inner}"
991   assumes "f \<in> borel_measurable M"
992   assumes "g \<in> borel_measurable M"
993   shows "(\<lambda>x. f x \<bullet> g x) \<in> borel_measurable M"
994   using assms
995   by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
997 notation
998   eucl_less (infix "<e" 50)
1000 lemma box_oc: "{x. a <e x \<and> x \<le> b} = {x. a <e x} \<inter> {..b}"
1001   and box_co: "{x. a \<le> x \<and> x <e b} = {a..} \<inter> {x. x <e b}"
1002   by auto
1004 lemma eucl_ivals[measurable]:
1005   fixes a b :: "'a::ordered_euclidean_space"
1006   shows "{x. x <e a} \<in> sets borel"
1007     and "{x. a <e x} \<in> sets borel"
1008     and "{..a} \<in> sets borel"
1009     and "{a..} \<in> sets borel"
1010     and "{a..b} \<in> sets borel"
1011     and  "{x. a <e x \<and> x \<le> b} \<in> sets borel"
1012     and "{x. a \<le> x \<and>  x <e b} \<in> sets borel"
1013   unfolding box_oc box_co
1014   by (auto intro: borel_open borel_closed)
1016 lemma
1017   fixes i :: "'a::{second_countable_topology, real_inner}"
1018   shows hafspace_less_borel: "{x. a < x \<bullet> i} \<in> sets borel"
1019     and hafspace_greater_borel: "{x. x \<bullet> i < a} \<in> sets borel"
1020     and hafspace_less_eq_borel: "{x. a \<le> x \<bullet> i} \<in> sets borel"
1021     and hafspace_greater_eq_borel: "{x. x \<bullet> i \<le> a} \<in> sets borel"
1022   by simp_all
1024 lemma borel_eq_box:
1025   "borel = sigma UNIV (range (\<lambda> (a, b). box a b :: 'a :: euclidean_space set))"
1026     (is "_ = ?SIGMA")
1027 proof (rule borel_eq_sigmaI1[OF borel_def])
1028   fix M :: "'a set" assume "M \<in> {S. open S}"
1029   then have "open M" by simp
1030   show "M \<in> ?SIGMA"
1031     apply (subst open_UNION_box[OF \<open>open M\<close>])
1032     apply (safe intro!: sets.countable_UN' countable_PiE countable_Collect)
1033     apply (auto intro: countable_rat)
1034     done
1035 qed (auto simp: box_def)
1037 lemma halfspace_gt_in_halfspace:
1038   assumes i: "i \<in> A"
1039   shows "{x::'a. a < x \<bullet> i} \<in>
1040     sigma_sets UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> A))"
1041   (is "?set \<in> ?SIGMA")
1042 proof -
1043   interpret sigma_algebra UNIV ?SIGMA
1044     by (intro sigma_algebra_sigma_sets) simp_all
1045   have *: "?set = (\<Union>n. UNIV - {x::'a. x \<bullet> i < a + 1 / real (Suc n)})"
1046   proof (safe, simp_all add: not_less del: of_nat_Suc)
1047     fix x :: 'a assume "a < x \<bullet> i"
1048     with reals_Archimedean[of "x \<bullet> i - a"]
1049     obtain n where "a + 1 / real (Suc n) < x \<bullet> i"
1050       by (auto simp: field_simps)
1051     then show "\<exists>n. a + 1 / real (Suc n) \<le> x \<bullet> i"
1052       by (blast intro: less_imp_le)
1053   next
1054     fix x n
1055     have "a < a + 1 / real (Suc n)" by auto
1056     also assume "\<dots> \<le> x"
1057     finally show "a < x" .
1058   qed
1059   show "?set \<in> ?SIGMA" unfolding *
1060     by (auto intro!: Diff sigma_sets_Inter i)
1061 qed
1063 lemma borel_eq_halfspace_less:
1064   "borel = sigma UNIV ((\<lambda>(a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> Basis))"
1065   (is "_ = ?SIGMA")
1066 proof (rule borel_eq_sigmaI2[OF borel_eq_box])
1067   fix a b :: 'a
1068   have "box a b = {x\<in>space ?SIGMA. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"
1069     by (auto simp: box_def)
1070   also have "\<dots> \<in> sets ?SIGMA"
1071     by (intro sets.sets_Collect_conj sets.sets_Collect_finite_All sets.sets_Collect_const)
1072        (auto intro!: halfspace_gt_in_halfspace countable_PiE countable_rat)
1073   finally show "box a b \<in> sets ?SIGMA" .
1074 qed auto
1076 lemma borel_eq_halfspace_le:
1077   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i \<le> a}) ` (UNIV \<times> Basis))"
1078   (is "_ = ?SIGMA")
1079 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
1080   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
1081   then have i: "i \<in> Basis" by auto
1082   have *: "{x::'a. x\<bullet>i < a} = (\<Union>n. {x. x\<bullet>i \<le> a - 1/real (Suc n)})"
1083   proof (safe, simp_all del: of_nat_Suc)
1084     fix x::'a assume *: "x\<bullet>i < a"
1085     with reals_Archimedean[of "a - x\<bullet>i"]
1086     obtain n where "x \<bullet> i < a - 1 / (real (Suc n))"
1087       by (auto simp: field_simps)
1088     then show "\<exists>n. x \<bullet> i \<le> a - 1 / (real (Suc n))"
1089       by (blast intro: less_imp_le)
1090   next
1091     fix x::'a and n
1092     assume "x\<bullet>i \<le> a - 1 / real (Suc n)"
1093     also have "\<dots> < a" by auto
1094     finally show "x\<bullet>i < a" .
1095   qed
1096   show "{x. x\<bullet>i < a} \<in> ?SIGMA" unfolding *
1097     by (intro sets.countable_UN) (auto intro: i)
1098 qed auto
1100 lemma borel_eq_halfspace_ge:
1101   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. a \<le> x \<bullet> i}) ` (UNIV \<times> Basis))"
1102   (is "_ = ?SIGMA")
1103 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
1104   fix a :: real and i :: 'a assume i: "(a, i) \<in> UNIV \<times> Basis"
1105   have *: "{x::'a. x\<bullet>i < a} = space ?SIGMA - {x::'a. a \<le> x\<bullet>i}" by auto
1106   show "{x. x\<bullet>i < a} \<in> ?SIGMA" unfolding *
1107     using i by (intro sets.compl_sets) auto
1108 qed auto
1110 lemma borel_eq_halfspace_greater:
1111   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. a < x \<bullet> i}) ` (UNIV \<times> Basis))"
1112   (is "_ = ?SIGMA")
1113 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])
1114   fix a :: real and i :: 'a assume "(a, i) \<in> (UNIV \<times> Basis)"
1115   then have i: "i \<in> Basis" by auto
1116   have *: "{x::'a. x\<bullet>i \<le> a} = space ?SIGMA - {x::'a. a < x\<bullet>i}" by auto
1117   show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA" unfolding *
1118     by (intro sets.compl_sets) (auto intro: i)
1119 qed auto
1121 lemma borel_eq_atMost:
1122   "borel = sigma UNIV (range (\<lambda>a. {..a::'a::ordered_euclidean_space}))"
1123   (is "_ = ?SIGMA")
1124 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
1125   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
1126   then have "i \<in> Basis" by auto
1127   then have *: "{x::'a. x\<bullet>i \<le> a} = (\<Union>k::nat. {.. (\<Sum>n\<in>Basis. (if n = i then a else real k)*\<^sub>R n)})"
1128   proof (safe, simp_all add: eucl_le[where 'a='a] split: if_split_asm)
1129     fix x :: 'a
1130     from real_arch_simple[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"] guess k::nat ..
1131     then have "\<And>i. i \<in> Basis \<Longrightarrow> x\<bullet>i \<le> real k"
1132       by (subst (asm) Max_le_iff) auto
1133     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> x \<bullet> ia \<le> real k"
1134       by (auto intro!: exI[of _ k])
1135   qed
1136   show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA" unfolding *
1137     by (intro sets.countable_UN) auto
1138 qed auto
1140 lemma borel_eq_greaterThan:
1141   "borel = sigma UNIV (range (\<lambda>a::'a::ordered_euclidean_space. {x. a <e x}))"
1142   (is "_ = ?SIGMA")
1143 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
1144   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
1145   then have i: "i \<in> Basis" by auto
1146   have "{x::'a. x\<bullet>i \<le> a} = UNIV - {x::'a. a < x\<bullet>i}" by auto
1147   also have *: "{x::'a. a < x\<bullet>i} =
1148       (\<Union>k::nat. {x. (\<Sum>n\<in>Basis. (if n = i then a else -real k) *\<^sub>R n) <e x})" using i
1149   proof (safe, simp_all add: eucl_less_def split: if_split_asm)
1150     fix x :: 'a
1151     from reals_Archimedean2[of "Max ((\<lambda>i. -x\<bullet>i)`Basis)"]
1152     guess k::nat .. note k = this
1153     { fix i :: 'a assume "i \<in> Basis"
1154       then have "-x\<bullet>i < real k"
1155         using k by (subst (asm) Max_less_iff) auto
1156       then have "- real k < x\<bullet>i" by simp }
1157     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> -real k < x \<bullet> ia"
1158       by (auto intro!: exI[of _ k])
1159   qed
1160   finally show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA"
1161     apply (simp only:)
1162     apply (intro sets.countable_UN sets.Diff)
1163     apply (auto intro: sigma_sets_top)
1164     done
1165 qed auto
1167 lemma borel_eq_lessThan:
1168   "borel = sigma UNIV (range (\<lambda>a::'a::ordered_euclidean_space. {x. x <e a}))"
1169   (is "_ = ?SIGMA")
1170 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge])
1171   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
1172   then have i: "i \<in> Basis" by auto
1173   have "{x::'a. a \<le> x\<bullet>i} = UNIV - {x::'a. x\<bullet>i < a}" by auto
1174   also have *: "{x::'a. x\<bullet>i < a} = (\<Union>k::nat. {x. x <e (\<Sum>n\<in>Basis. (if n = i then a else real k) *\<^sub>R n)})" using \<open>i\<in> Basis\<close>
1175   proof (safe, simp_all add: eucl_less_def split: if_split_asm)
1176     fix x :: 'a
1177     from reals_Archimedean2[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"]
1178     guess k::nat .. note k = this
1179     { fix i :: 'a assume "i \<in> Basis"
1180       then have "x\<bullet>i < real k"
1181         using k by (subst (asm) Max_less_iff) auto
1182       then have "x\<bullet>i < real k" by simp }
1183     then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> x \<bullet> ia < real k"
1184       by (auto intro!: exI[of _ k])
1185   qed
1186   finally show "{x. a \<le> x\<bullet>i} \<in> ?SIGMA"
1187     apply (simp only:)
1188     apply (intro sets.countable_UN sets.Diff)
1189     apply (auto intro: sigma_sets_top )
1190     done
1191 qed auto
1193 lemma borel_eq_atLeastAtMost:
1194   "borel = sigma UNIV (range (\<lambda>(a,b). {a..b} ::'a::ordered_euclidean_space set))"
1195   (is "_ = ?SIGMA")
1196 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
1197   fix a::'a
1198   have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
1199   proof (safe, simp_all add: eucl_le[where 'a='a])
1200     fix x :: 'a
1201     from real_arch_simple[of "Max ((\<lambda>i. - x\<bullet>i)`Basis)"]
1202     guess k::nat .. note k = this
1203     { fix i :: 'a assume "i \<in> Basis"
1204       with k have "- x\<bullet>i \<le> real k"
1205         by (subst (asm) Max_le_iff) (auto simp: field_simps)
1206       then have "- real k \<le> x\<bullet>i" by simp }
1207     then show "\<exists>n::nat. \<forall>i\<in>Basis. - real n \<le> x \<bullet> i"
1208       by (auto intro!: exI[of _ k])
1209   qed
1210   show "{..a} \<in> ?SIGMA" unfolding *
1211     by (intro sets.countable_UN)
1212        (auto intro!: sigma_sets_top)
1213 qed auto
1215 lemma borel_set_induct[consumes 1, case_names empty interval compl union]:
1216   assumes "A \<in> sets borel"
1217   assumes empty: "P {}" and int: "\<And>a b. a \<le> b \<Longrightarrow> P {a..b}" and compl: "\<And>A. A \<in> sets borel \<Longrightarrow> P A \<Longrightarrow> P (-A)" and
1218           un: "\<And>f. disjoint_family f \<Longrightarrow> (\<And>i. f i \<in> sets borel) \<Longrightarrow>  (\<And>i. P (f i)) \<Longrightarrow> P (\<Union>i::nat. f i)"
1219   shows "P (A::real set)"
1220 proof-
1221   let ?G = "range (\<lambda>(a,b). {a..b::real})"
1222   have "Int_stable ?G" "?G \<subseteq> Pow UNIV" "A \<in> sigma_sets UNIV ?G"
1223       using assms(1) by (auto simp add: borel_eq_atLeastAtMost Int_stable_def)
1224   thus ?thesis
1225   proof (induction rule: sigma_sets_induct_disjoint)
1226     case (union f)
1227       from union.hyps(2) have "\<And>i. f i \<in> sets borel" by (auto simp: borel_eq_atLeastAtMost)
1228       with union show ?case by (auto intro: un)
1229   next
1230     case (basic A)
1231     then obtain a b where "A = {a .. b}" by auto
1232     then show ?case
1233       by (cases "a \<le> b") (auto intro: int empty)
1234   qed (auto intro: empty compl simp: Compl_eq_Diff_UNIV[symmetric] borel_eq_atLeastAtMost)
1235 qed
1237 lemma borel_sigma_sets_Ioc: "borel = sigma UNIV (range (\<lambda>(a, b). {a <.. b::real}))"
1238 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
1239   fix i :: real
1240   have "{..i} = (\<Union>j::nat. {-j <.. i})"
1241     by (auto simp: minus_less_iff reals_Archimedean2)
1242   also have "\<dots> \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))"
1243     by (intro sets.countable_nat_UN) auto
1244   finally show "{..i} \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))" .
1245 qed simp
1247 lemma eucl_lessThan: "{x::real. x <e a} = lessThan a"
1248   by (simp add: eucl_less_def lessThan_def)
1250 lemma borel_eq_atLeastLessThan:
1251   "borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA")
1252 proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan])
1253   have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto
1254   fix x :: real
1255   have "{..<x} = (\<Union>i::nat. {-real i ..< x})"
1256     by (auto simp: move_uminus real_arch_simple)
1257   then show "{y. y <e x} \<in> ?SIGMA"
1258     by (auto intro: sigma_sets.intros(2-) simp: eucl_lessThan)
1259 qed auto
1261 lemma borel_measurable_halfspacesI:
1262   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
1263   assumes F: "borel = sigma UNIV (F ` (UNIV \<times> Basis))"
1264   and S_eq: "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
1265   shows "f \<in> borel_measurable M = (\<forall>i\<in>Basis. \<forall>a::real. S a i \<in> sets M)"
1266 proof safe
1267   fix a :: real and i :: 'b assume i: "i \<in> Basis" and f: "f \<in> borel_measurable M"
1268   then show "S a i \<in> sets M" unfolding assms
1269     by (auto intro!: measurable_sets simp: assms(1))
1270 next
1271   assume a: "\<forall>i\<in>Basis. \<forall>a. S a i \<in> sets M"
1272   then show "f \<in> borel_measurable M"
1273     by (auto intro!: measurable_measure_of simp: S_eq F)
1274 qed
1276 lemma borel_measurable_iff_halfspace_le:
1277   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
1278   shows "f \<in> borel_measurable M = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. f w \<bullet> i \<le> a} \<in> sets M)"
1279   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
1281 lemma borel_measurable_iff_halfspace_less:
1282   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
1283   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. f w \<bullet> i < a} \<in> sets M)"
1284   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
1286 lemma borel_measurable_iff_halfspace_ge:
1287   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
1288   shows "f \<in> borel_measurable M = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. a \<le> f w \<bullet> i} \<in> sets M)"
1289   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
1291 lemma borel_measurable_iff_halfspace_greater:
1292   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
1293   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. a < f w \<bullet> i} \<in> sets M)"
1294   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto
1296 lemma borel_measurable_iff_le:
1297   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
1298   using borel_measurable_iff_halfspace_le[where 'c=real] by simp
1300 lemma borel_measurable_iff_less:
1301   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
1302   using borel_measurable_iff_halfspace_less[where 'c=real] by simp
1304 lemma borel_measurable_iff_ge:
1305   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
1306   using borel_measurable_iff_halfspace_ge[where 'c=real]
1307   by simp
1309 lemma borel_measurable_iff_greater:
1310   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
1311   using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
1313 lemma borel_measurable_euclidean_space:
1314   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
1315   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M)"
1316 proof safe
1317   assume f: "\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M"
1318   then show "f \<in> borel_measurable M"
1319     by (subst borel_measurable_iff_halfspace_le) auto
1320 qed auto
1322 subsection "Borel measurable operators"
1324 lemma borel_measurable_norm[measurable]: "norm \<in> borel_measurable borel"
1325   by (intro borel_measurable_continuous_on1 continuous_intros)
1327 lemma borel_measurable_sgn [measurable]: "(sgn::'a::real_normed_vector \<Rightarrow> 'a) \<in> borel_measurable borel"
1328   by (rule borel_measurable_continuous_countable_exceptions[where X="{0}"])
1329      (auto intro!: continuous_on_sgn continuous_on_id)
1331 lemma borel_measurable_uminus[measurable (raw)]:
1332   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
1333   assumes g: "g \<in> borel_measurable M"
1334   shows "(\<lambda>x. - g x) \<in> borel_measurable M"
1335   by (rule borel_measurable_continuous_on[OF _ g]) (intro continuous_intros)
1337 lemma borel_measurable_diff[measurable (raw)]:
1338   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
1339   assumes f: "f \<in> borel_measurable M"
1340   assumes g: "g \<in> borel_measurable M"
1341   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
1342   using borel_measurable_add [of f M "- g"] assms by (simp add: fun_Compl_def)
1344 lemma borel_measurable_times[measurable (raw)]:
1345   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_algebra}"
1346   assumes f: "f \<in> borel_measurable M"
1347   assumes g: "g \<in> borel_measurable M"
1348   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
1349   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
1351 lemma borel_measurable_prod[measurable (raw)]:
1352   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, real_normed_field}"
1353   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
1354   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
1355 proof cases
1356   assume "finite S"
1357   thus ?thesis using assms by induct auto
1358 qed simp
1360 lemma borel_measurable_dist[measurable (raw)]:
1361   fixes g f :: "'a \<Rightarrow> 'b::{second_countable_topology, metric_space}"
1362   assumes f: "f \<in> borel_measurable M"
1363   assumes g: "g \<in> borel_measurable M"
1364   shows "(\<lambda>x. dist (f x) (g x)) \<in> borel_measurable M"
1365   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
1367 lemma borel_measurable_scaleR[measurable (raw)]:
1368   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
1369   assumes f: "f \<in> borel_measurable M"
1370   assumes g: "g \<in> borel_measurable M"
1371   shows "(\<lambda>x. f x *\<^sub>R g x) \<in> borel_measurable M"
1372   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
1374 lemma affine_borel_measurable_vector:
1375   fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
1376   assumes "f \<in> borel_measurable M"
1377   shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
1378 proof (rule borel_measurableI)
1379   fix S :: "'x set" assume "open S"
1380   show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
1381   proof cases
1382     assume "b \<noteq> 0"
1383     with \<open>open S\<close> have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")
1384       using open_affinity [of S "inverse b" "- a /\<^sub>R b"]
1385       by (auto simp: algebra_simps)
1386     hence "?S \<in> sets borel" by auto
1387     moreover
1388     from \<open>b \<noteq> 0\<close> have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
1389       apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
1390     ultimately show ?thesis using assms unfolding in_borel_measurable_borel
1391       by auto
1392   qed simp
1393 qed
1395 lemma borel_measurable_const_scaleR[measurable (raw)]:
1396   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. b *\<^sub>R f x ::'a::real_normed_vector) \<in> borel_measurable M"
1397   using affine_borel_measurable_vector[of f M 0 b] by simp
1399 lemma borel_measurable_const_add[measurable (raw)]:
1400   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. a + f x ::'a::real_normed_vector) \<in> borel_measurable M"
1401   using affine_borel_measurable_vector[of f M a 1] by simp
1403 lemma borel_measurable_inverse[measurable (raw)]:
1404   fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
1405   assumes f: "f \<in> borel_measurable M"
1406   shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
1407   apply (rule measurable_compose[OF f])
1408   apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])
1409   apply (auto intro!: continuous_on_inverse continuous_on_id)
1410   done
1412 lemma borel_measurable_divide[measurable (raw)]:
1413   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>
1414     (\<lambda>x. f x / g x::'b::{second_countable_topology, real_normed_div_algebra}) \<in> borel_measurable M"
1415   by (simp add: divide_inverse)
1417 lemma borel_measurable_abs[measurable (raw)]:
1418   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
1419   unfolding abs_real_def by simp
1421 lemma borel_measurable_nth[measurable (raw)]:
1422   "(\<lambda>x::real^'n. x \$ i) \<in> borel_measurable borel"
1423   by (simp add: cart_eq_inner_axis)
1425 lemma convex_measurable:
1426   fixes A :: "'a :: euclidean_space set"
1427   shows "X \<in> borel_measurable M \<Longrightarrow> X ` space M \<subseteq> A \<Longrightarrow> open A \<Longrightarrow> convex_on A q \<Longrightarrow>
1428     (\<lambda>x. q (X x)) \<in> borel_measurable M"
1429   by (rule measurable_compose[where f=X and N="restrict_space borel A"])
1430      (auto intro!: borel_measurable_continuous_on_restrict convex_on_continuous measurable_restrict_space2)
1432 lemma borel_measurable_ln[measurable (raw)]:
1433   assumes f: "f \<in> borel_measurable M"
1434   shows "(\<lambda>x. ln (f x :: real)) \<in> borel_measurable M"
1435   apply (rule measurable_compose[OF f])
1436   apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])
1437   apply (auto intro!: continuous_on_ln continuous_on_id)
1438   done
1440 lemma borel_measurable_log[measurable (raw)]:
1441   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. log (g x) (f x)) \<in> borel_measurable M"
1442   unfolding log_def by auto
1444 lemma borel_measurable_exp[measurable]:
1445   "(exp::'a::{real_normed_field,banach}\<Rightarrow>'a) \<in> borel_measurable borel"
1446   by (intro borel_measurable_continuous_on1 continuous_at_imp_continuous_on ballI isCont_exp)
1448 lemma measurable_real_floor[measurable]:
1449   "(floor :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
1450 proof -
1451   have "\<And>a x. \<lfloor>x\<rfloor> = a \<longleftrightarrow> (real_of_int a \<le> x \<and> x < real_of_int (a + 1))"
1452     by (auto intro: floor_eq2)
1453   then show ?thesis
1454     by (auto simp: vimage_def measurable_count_space_eq2_countable)
1455 qed
1457 lemma measurable_real_ceiling[measurable]:
1458   "(ceiling :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
1459   unfolding ceiling_def[abs_def] by simp
1461 lemma borel_measurable_real_floor: "(\<lambda>x::real. real_of_int \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
1462   by simp
1464 lemma borel_measurable_root [measurable]: "root n \<in> borel_measurable borel"
1465   by (intro borel_measurable_continuous_on1 continuous_intros)
1467 lemma borel_measurable_sqrt [measurable]: "sqrt \<in> borel_measurable borel"
1468   by (intro borel_measurable_continuous_on1 continuous_intros)
1470 lemma borel_measurable_power [measurable (raw)]:
1471   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
1472   assumes f: "f \<in> borel_measurable M"
1473   shows "(\<lambda>x. (f x) ^ n) \<in> borel_measurable M"
1474   by (intro borel_measurable_continuous_on [OF _ f] continuous_intros)
1476 lemma borel_measurable_Re [measurable]: "Re \<in> borel_measurable borel"
1477   by (intro borel_measurable_continuous_on1 continuous_intros)
1479 lemma borel_measurable_Im [measurable]: "Im \<in> borel_measurable borel"
1480   by (intro borel_measurable_continuous_on1 continuous_intros)
1482 lemma borel_measurable_of_real [measurable]: "(of_real :: _ \<Rightarrow> (_::real_normed_algebra)) \<in> borel_measurable borel"
1483   by (intro borel_measurable_continuous_on1 continuous_intros)
1485 lemma borel_measurable_sin [measurable]: "(sin :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"
1486   by (intro borel_measurable_continuous_on1 continuous_intros)
1488 lemma borel_measurable_cos [measurable]: "(cos :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"
1489   by (intro borel_measurable_continuous_on1 continuous_intros)
1491 lemma borel_measurable_arctan [measurable]: "arctan \<in> borel_measurable borel"
1492   by (intro borel_measurable_continuous_on1 continuous_intros)
1494 lemma borel_measurable_complex_iff:
1495   "f \<in> borel_measurable M \<longleftrightarrow>
1496     (\<lambda>x. Re (f x)) \<in> borel_measurable M \<and> (\<lambda>x. Im (f x)) \<in> borel_measurable M"
1497   apply auto
1498   apply (subst fun_complex_eq)
1499   apply (intro borel_measurable_add)
1500   apply auto
1501   done
1503 lemma measurable_of_bool[measurable]: "of_bool \<in> count_space UNIV \<rightarrow>\<^sub>M borel"
1504   by simp
1506 subsection "Borel space on the extended reals"
1508 lemma borel_measurable_ereal[measurable (raw)]:
1509   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
1510   using continuous_on_ereal f by (rule borel_measurable_continuous_on) (rule continuous_on_id)
1512 lemma borel_measurable_real_of_ereal[measurable (raw)]:
1513   fixes f :: "'a \<Rightarrow> ereal"
1514   assumes f: "f \<in> borel_measurable M"
1515   shows "(\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M"
1516   apply (rule measurable_compose[OF f])
1517   apply (rule borel_measurable_continuous_countable_exceptions[of "{\<infinity>, -\<infinity> }"])
1518   apply (auto intro: continuous_on_real simp: Compl_eq_Diff_UNIV)
1519   done
1521 lemma borel_measurable_ereal_cases:
1522   fixes f :: "'a \<Rightarrow> ereal"
1523   assumes f: "f \<in> borel_measurable M"
1524   assumes H: "(\<lambda>x. H (ereal (real_of_ereal (f x)))) \<in> borel_measurable M"
1525   shows "(\<lambda>x. H (f x)) \<in> borel_measurable M"
1526 proof -
1527   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)))"
1528   { fix x have "H (f x) = ?F x" by (cases "f x") auto }
1529   with f H show ?thesis by simp
1530 qed
1532 lemma
1533   fixes f :: "'a \<Rightarrow> ereal" assumes f[measurable]: "f \<in> borel_measurable M"
1534   shows borel_measurable_ereal_abs[measurable(raw)]: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
1535     and borel_measurable_ereal_inverse[measurable(raw)]: "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M"
1536     and borel_measurable_uminus_ereal[measurable(raw)]: "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M"
1537   by (auto simp del: abs_real_of_ereal simp: borel_measurable_ereal_cases[OF f] measurable_If)
1539 lemma borel_measurable_uminus_eq_ereal[simp]:
1540   "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
1541 proof
1542   assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp
1543 qed auto
1545 lemma set_Collect_ereal2:
1546   fixes f g :: "'a \<Rightarrow> ereal"
1547   assumes f: "f \<in> borel_measurable M"
1548   assumes g: "g \<in> borel_measurable M"
1549   assumes H: "{x \<in> space M. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))} \<in> sets M"
1550     "{x \<in> space borel. H (-\<infinity>) (ereal x)} \<in> sets borel"
1551     "{x \<in> space borel. H (\<infinity>) (ereal x)} \<in> sets borel"
1552     "{x \<in> space borel. H (ereal x) (-\<infinity>)} \<in> sets borel"
1553     "{x \<in> space borel. H (ereal x) (\<infinity>)} \<in> sets borel"
1554   shows "{x \<in> space M. H (f x) (g x)} \<in> sets M"
1555 proof -
1556   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)))"
1557   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"
1558   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
1559   note * = this
1560   from assms show ?thesis
1561     by (subst *) (simp del: space_borel split del: if_split)
1562 qed
1564 lemma borel_measurable_ereal_iff:
1565   shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
1566 proof
1567   assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
1568   from borel_measurable_real_of_ereal[OF this]
1569   show "f \<in> borel_measurable M" by auto
1570 qed auto
1572 lemma borel_measurable_erealD[measurable_dest]:
1573   "(\<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"
1574   unfolding borel_measurable_ereal_iff by simp
1576 lemma borel_measurable_ereal_iff_real:
1577   fixes f :: "'a \<Rightarrow> ereal"
1578   shows "f \<in> borel_measurable M \<longleftrightarrow>
1579     ((\<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)"
1580 proof safe
1581   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"
1582   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
1583   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
1584   let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real_of_ereal (f x))"
1585   have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
1586   also have "?f = f" by (auto simp: fun_eq_iff ereal_real)
1587   finally show "f \<in> borel_measurable M" .
1588 qed simp_all
1590 lemma borel_measurable_ereal_iff_Iio:
1591   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
1592   by (auto simp: borel_Iio measurable_iff_measure_of)
1594 lemma borel_measurable_ereal_iff_Ioi:
1595   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
1596   by (auto simp: borel_Ioi measurable_iff_measure_of)
1598 lemma vimage_sets_compl_iff:
1599   "f -` A \<inter> space M \<in> sets M \<longleftrightarrow> f -` (- A) \<inter> space M \<in> sets M"
1600 proof -
1601   { fix A assume "f -` A \<inter> space M \<in> sets M"
1602     moreover have "f -` (- A) \<inter> space M = space M - f -` A \<inter> space M" by auto
1603     ultimately have "f -` (- A) \<inter> space M \<in> sets M" by auto }
1604   from this[of A] this[of "-A"] show ?thesis
1605     by (metis double_complement)
1606 qed
1608 lemma borel_measurable_iff_Iic_ereal:
1609   "(f::'a\<Rightarrow>ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
1610   unfolding borel_measurable_ereal_iff_Ioi vimage_sets_compl_iff[where A="{a <..}" for a] by simp
1612 lemma borel_measurable_iff_Ici_ereal:
1613   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
1614   unfolding borel_measurable_ereal_iff_Iio vimage_sets_compl_iff[where A="{..< a}" for a] by simp
1616 lemma borel_measurable_ereal2:
1617   fixes f g :: "'a \<Rightarrow> ereal"
1618   assumes f: "f \<in> borel_measurable M"
1619   assumes g: "g \<in> borel_measurable M"
1620   assumes H: "(\<lambda>x. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
1621     "(\<lambda>x. H (-\<infinity>) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
1622     "(\<lambda>x. H (\<infinity>) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
1623     "(\<lambda>x. H (ereal (real_of_ereal (f x))) (-\<infinity>)) \<in> borel_measurable M"
1624     "(\<lambda>x. H (ereal (real_of_ereal (f x))) (\<infinity>)) \<in> borel_measurable M"
1625   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
1626 proof -
1627   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)))"
1628   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"
1629   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
1630   note * = this
1631   from assms show ?thesis unfolding * by simp
1632 qed
1634 lemma [measurable(raw)]:
1635   fixes f :: "'a \<Rightarrow> ereal"
1636   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
1637   shows borel_measurable_ereal_add: "(\<lambda>x. f x + g x) \<in> borel_measurable M"
1638     and borel_measurable_ereal_times: "(\<lambda>x. f x * g x) \<in> borel_measurable M"
1639   by (simp_all add: borel_measurable_ereal2)
1641 lemma [measurable(raw)]:
1642   fixes f g :: "'a \<Rightarrow> ereal"
1643   assumes "f \<in> borel_measurable M"
1644   assumes "g \<in> borel_measurable M"
1645   shows borel_measurable_ereal_diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
1646     and borel_measurable_ereal_divide: "(\<lambda>x. f x / g x) \<in> borel_measurable M"
1647   using assms by (simp_all add: minus_ereal_def divide_ereal_def)
1649 lemma borel_measurable_ereal_sum[measurable (raw)]:
1650   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
1651   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
1652   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
1653   using assms by (induction S rule: infinite_finite_induct) auto
1655 lemma borel_measurable_ereal_prod[measurable (raw)]:
1656   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
1657   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
1658   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
1659   using assms by (induction S rule: infinite_finite_induct) auto
1661 lemma borel_measurable_extreal_suminf[measurable (raw)]:
1662   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
1663   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
1664   shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
1665   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
1667 subsection "Borel space on the extended non-negative reals"
1669 text \<open> @{type ennreal} is a topological monoid, so no rules for plus are required, also all order
1670   statements are usually done on type classes. \<close>
1672 lemma measurable_enn2ereal[measurable]: "enn2ereal \<in> borel \<rightarrow>\<^sub>M borel"
1673   by (intro borel_measurable_continuous_on1 continuous_on_enn2ereal)
1675 lemma measurable_e2ennreal[measurable]: "e2ennreal \<in> borel \<rightarrow>\<^sub>M borel"
1676   by (intro borel_measurable_continuous_on1 continuous_on_e2ennreal)
1678 lemma borel_measurable_enn2real[measurable (raw)]:
1679   "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. enn2real (f x)) \<in> M \<rightarrow>\<^sub>M borel"
1680   unfolding enn2real_def[abs_def] by measurable
1682 definition [simp]: "is_borel f M \<longleftrightarrow> f \<in> borel_measurable M"
1684 lemma is_borel_transfer[transfer_rule]: "rel_fun (rel_fun op = pcr_ennreal) op = is_borel is_borel"
1685   unfolding is_borel_def[abs_def]
1686 proof (safe intro!: rel_funI ext dest!: rel_fun_eq_pcr_ennreal[THEN iffD1])
1687   fix f and M :: "'a measure"
1688   show "f \<in> borel_measurable M" if f: "enn2ereal \<circ> f \<in> borel_measurable M"
1689     using measurable_compose[OF f measurable_e2ennreal] by simp
1690 qed simp
1692 context
1693   includes ennreal.lifting
1694 begin
1696 lemma measurable_ennreal[measurable]: "ennreal \<in> borel \<rightarrow>\<^sub>M borel"
1697   unfolding is_borel_def[symmetric]
1698   by transfer simp
1700 lemma borel_measurable_ennreal_iff[simp]:
1701   assumes [simp]: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
1702   shows "(\<lambda>x. ennreal (f x)) \<in> M \<rightarrow>\<^sub>M borel \<longleftrightarrow> f \<in> M \<rightarrow>\<^sub>M borel"
1703 proof safe
1704   assume "(\<lambda>x. ennreal (f x)) \<in> M \<rightarrow>\<^sub>M borel"
1705   then have "(\<lambda>x. enn2real (ennreal (f x))) \<in> M \<rightarrow>\<^sub>M borel"
1706     by measurable
1707   then show "f \<in> M \<rightarrow>\<^sub>M borel"
1708     by (rule measurable_cong[THEN iffD1, rotated]) auto
1709 qed measurable
1711 lemma borel_measurable_times_ennreal[measurable (raw)]:
1712   fixes f g :: "'a \<Rightarrow> ennreal"
1713   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"
1714   unfolding is_borel_def[symmetric] by transfer simp
1716 lemma borel_measurable_inverse_ennreal[measurable (raw)]:
1717   fixes f :: "'a \<Rightarrow> ennreal"
1718   shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. inverse (f x)) \<in> M \<rightarrow>\<^sub>M borel"
1719   unfolding is_borel_def[symmetric] by transfer simp
1721 lemma borel_measurable_divide_ennreal[measurable (raw)]:
1722   fixes f :: "'a \<Rightarrow> ennreal"
1723   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"
1724   unfolding divide_ennreal_def by simp
1726 lemma borel_measurable_minus_ennreal[measurable (raw)]:
1727   fixes f :: "'a \<Rightarrow> ennreal"
1728   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"
1729   unfolding is_borel_def[symmetric] by transfer simp
1731 lemma borel_measurable_prod_ennreal[measurable (raw)]:
1732   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ennreal"
1733   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
1734   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
1735   using assms by (induction S rule: infinite_finite_induct) auto
1737 end
1739 hide_const (open) is_borel
1741 subsection \<open>LIMSEQ is borel measurable\<close>
1743 lemma borel_measurable_LIMSEQ_real:
1744   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
1745   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
1746   and u: "\<And>i. u i \<in> borel_measurable M"
1747   shows "u' \<in> borel_measurable M"
1748 proof -
1749   have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)"
1750     using u' by (simp add: lim_imp_Liminf)
1751   moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M"
1752     by auto
1753   ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff)
1754 qed
1756 lemma borel_measurable_LIMSEQ_metric:
1757   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b :: metric_space"
1758   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
1759   assumes lim: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. f i x) \<longlonglongrightarrow> g x"
1760   shows "g \<in> borel_measurable M"
1761   unfolding borel_eq_closed
1762 proof (safe intro!: measurable_measure_of)
1763   fix A :: "'b set" assume "closed A"
1765   have [measurable]: "(\<lambda>x. infdist (g x) A) \<in> borel_measurable M"
1766   proof (rule borel_measurable_LIMSEQ_real)
1767     show "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. infdist (f i x) A) \<longlonglongrightarrow> infdist (g x) A"
1768       by (intro tendsto_infdist lim)
1769     show "\<And>i. (\<lambda>x. infdist (f i x) A) \<in> borel_measurable M"
1770       by (intro borel_measurable_continuous_on[where f="\<lambda>x. infdist x A"]
1771         continuous_at_imp_continuous_on ballI continuous_infdist continuous_ident) auto
1772   qed
1774   show "g -` A \<inter> space M \<in> sets M"
1775   proof cases
1776     assume "A \<noteq> {}"
1777     then have "\<And>x. infdist x A = 0 \<longleftrightarrow> x \<in> A"
1778       using \<open>closed A\<close> by (simp add: in_closed_iff_infdist_zero)
1779     then have "g -` A \<inter> space M = {x\<in>space M. infdist (g x) A = 0}"
1780       by auto
1781     also have "\<dots> \<in> sets M"
1782       by measurable
1783     finally show ?thesis .
1784   qed simp
1785 qed auto
1787 lemma sets_Collect_Cauchy[measurable]:
1788   fixes f :: "nat \<Rightarrow> 'a => 'b::{metric_space, second_countable_topology}"
1789   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
1790   shows "{x\<in>space M. Cauchy (\<lambda>i. f i x)} \<in> sets M"
1791   unfolding metric_Cauchy_iff2 using f by auto
1793 lemma borel_measurable_lim_metric[measurable (raw)]:
1794   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"
1795   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
1796   shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
1797 proof -
1798   define u' where "u' x = lim (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)" for x
1799   then have *: "\<And>x. lim (\<lambda>i. f i x) = (if Cauchy (\<lambda>i. f i x) then u' x else (THE x. False))"
1800     by (auto simp: lim_def convergent_eq_cauchy[symmetric])
1801   have "u' \<in> borel_measurable M"
1802   proof (rule borel_measurable_LIMSEQ_metric)
1803     fix x
1804     have "convergent (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)"
1805       by (cases "Cauchy (\<lambda>i. f i x)")
1806          (auto simp add: convergent_eq_cauchy[symmetric] convergent_def)
1807     then show "(\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0) \<longlonglongrightarrow> u' x"
1808       unfolding u'_def
1809       by (rule convergent_LIMSEQ_iff[THEN iffD1])
1810   qed measurable
1811   then show ?thesis
1812     unfolding * by measurable
1813 qed
1815 lemma borel_measurable_suminf[measurable (raw)]:
1816   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"
1817   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
1818   shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M"
1819   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
1821 lemma Collect_closed_imp_pred_borel: "closed {x. P x} \<Longrightarrow> Measurable.pred borel P"
1822   by (simp add: pred_def)
1824 (* Proof by Jeremy Avigad and Luke Serafin *)
1825 lemma isCont_borel_pred[measurable]:
1826   fixes f :: "'b::metric_space \<Rightarrow> 'a::metric_space"
1827   shows "Measurable.pred borel (isCont f)"
1828 proof (subst measurable_cong)
1829   let ?I = "\<lambda>j. inverse(real (Suc j))"
1830   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
1831     unfolding continuous_at_eps_delta
1832   proof safe
1833     fix i assume "\<forall>e>0. \<exists>d>0. \<forall>y. dist y x < d \<longrightarrow> dist (f y) (f x) < e"
1834     moreover have "0 < ?I i / 2"
1835       by simp
1836     ultimately obtain d where d: "0 < d" "\<And>y. dist x y < d \<Longrightarrow> dist (f y) (f x) < ?I i / 2"
1837       by (metis dist_commute)
1838     then obtain j where j: "?I j < d"
1839       by (metis reals_Archimedean)
1841     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"
1842     proof (safe intro!: exI[where x=j])
1843       fix y z assume *: "dist x y < ?I j" "dist x z < ?I j"
1844       have "dist (f y) (f z) \<le> dist (f y) (f x) + dist (f z) (f x)"
1845         by (rule dist_triangle2)
1846       also have "\<dots> < ?I i / 2 + ?I i / 2"
1847         by (intro add_strict_mono d less_trans[OF _ j] *)
1848       also have "\<dots> \<le> ?I i"
1849         by (simp add: field_simps of_nat_Suc)
1850       finally show "dist (f y) (f z) \<le> ?I i"
1851         by simp
1852     qed
1853   next
1854     fix e::real assume "0 < e"
1855     then obtain n where n: "?I n < e"
1856       by (metis reals_Archimedean)
1857     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"
1858     from this[THEN spec, of "Suc n"]
1859     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)"
1860       by auto
1862     show "\<exists>d>0. \<forall>y. dist y x < d \<longrightarrow> dist (f y) (f x) < e"
1863     proof (safe intro!: exI[of _ "?I j"])
1864       fix y assume "dist y x < ?I j"
1865       then have "dist (f y) (f x) \<le> ?I (Suc n)"
1866         by (intro j) (auto simp: dist_commute)
1867       also have "?I (Suc n) < ?I n"
1868         by simp
1869       also note n
1870       finally show "dist (f y) (f x) < e" .
1871     qed simp
1872   qed
1873 qed (intro pred_intros_countable closed_Collect_all closed_Collect_le open_Collect_less
1874            Collect_closed_imp_pred_borel closed_Collect_imp open_Collect_conj continuous_intros)
1876 lemma isCont_borel:
1877   fixes f :: "'b::metric_space \<Rightarrow> 'a::metric_space"
1878   shows "{x. isCont f x} \<in> sets borel"
1879   by simp
1881 lemma is_real_interval:
1882   assumes S: "is_interval S"
1883   shows "\<exists>a b::real. S = {} \<or> S = UNIV \<or> S = {..<b} \<or> S = {..b} \<or> S = {a<..} \<or> S = {a..} \<or>
1884     S = {a<..<b} \<or> S = {a<..b} \<or> S = {a..<b} \<or> S = {a..b}"
1885   using S unfolding is_interval_1 by (blast intro: interval_cases)
1887 lemma real_interval_borel_measurable:
1888   assumes "is_interval (S::real set)"
1889   shows "S \<in> sets borel"
1890 proof -
1891   from assms is_real_interval have "\<exists>a b::real. S = {} \<or> S = UNIV \<or> S = {..<b} \<or> S = {..b} \<or>
1892     S = {a<..} \<or> S = {a..} \<or> S = {a<..<b} \<or> S = {a<..b} \<or> S = {a..<b} \<or> S = {a..b}" by auto
1893   then guess a ..
1894   then guess b ..
1895   thus ?thesis
1896     by auto
1897 qed
1899 text \<open>The next lemmas hold in any second countable linorder (including ennreal or ereal for instance),
1900 but in the current state they are restricted to reals.\<close>
1902 lemma borel_measurable_mono_on_fnc:
1903   fixes f :: "real \<Rightarrow> real" and A :: "real set"
1904   assumes "mono_on f A"
1905   shows "f \<in> borel_measurable (restrict_space borel A)"
1906   apply (rule measurable_restrict_countable[OF mono_on_ctble_discont[OF assms]])
1907   apply (auto intro!: image_eqI[where x="{x}" for x] simp: sets_restrict_space)
1908   apply (auto simp add: sets_restrict_restrict_space continuous_on_eq_continuous_within
1909               cong: measurable_cong_sets
1910               intro!: borel_measurable_continuous_on_restrict intro: continuous_within_subset)
1911   done
1913 lemma borel_measurable_piecewise_mono:
1914   fixes f::"real \<Rightarrow> real" and C::"real set set"
1915   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"
1916   shows "f \<in> borel_measurable borel"
1917 by (rule measurable_piecewise_restrict[of C], auto intro: borel_measurable_mono_on_fnc simp: assms)
1919 lemma borel_measurable_mono:
1920   fixes f :: "real \<Rightarrow> real"
1921   shows "mono f \<Longrightarrow> f \<in> borel_measurable borel"
1922   using borel_measurable_mono_on_fnc[of f UNIV] by (simp add: mono_def mono_on_def)
1924 lemma measurable_bdd_below_real[measurable (raw)]:
1925   fixes F :: "'a \<Rightarrow> 'i \<Rightarrow> real"
1926   assumes [simp]: "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> M \<rightarrow>\<^sub>M borel"
1927   shows "Measurable.pred M (\<lambda>x. bdd_below ((\<lambda>i. F i x)`I))"
1928 proof (subst measurable_cong)
1929   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
1930     by (auto simp: bdd_below_def intro!: bexI[of _ "of_int (floor _)"] intro: order_trans of_int_floor_le)
1931   show "Measurable.pred M (\<lambda>w. \<exists>q\<in>\<int>. \<forall>i\<in>I. q \<le> F i w)"
1932     using countable_int by measurable
1933 qed
1935 lemma borel_measurable_cINF_real[measurable (raw)]:
1936   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> real"
1937   assumes [simp]: "countable I"
1938   assumes F[measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
1939   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
1940 proof (rule measurable_piecewise_restrict)
1941   let ?\<Omega> = "{x\<in>space M. bdd_below ((\<lambda>i. F i x)`I)}"
1942   show "countable {?\<Omega>, - ?\<Omega>}" "space M \<subseteq> \<Union>{?\<Omega>, - ?\<Omega>}" "\<And>X. X \<in> {?\<Omega>, - ?\<Omega>} \<Longrightarrow> X \<inter> space M \<in> sets M"
1943     by auto
1944   fix X assume "X \<in> {?\<Omega>, - ?\<Omega>}" then show "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable (restrict_space M X)"
1945   proof safe
1946     show "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable (restrict_space M ?\<Omega>)"
1947       by (intro borel_measurable_cINF measurable_restrict_space1 F)
1948          (auto simp: space_restrict_space)
1949     show "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable (restrict_space M (-?\<Omega>))"
1950     proof (subst measurable_cong)
1951       fix x assume "x \<in> space (restrict_space M (-?\<Omega>))"
1952       then have "\<not> (\<forall>i\<in>I. - F i x \<le> y)" for y
1953         by (auto simp: space_restrict_space bdd_above_def bdd_above_uminus[symmetric])
1954       then show "(INF i:I. F i x) = - (THE x. False)"
1955         by (auto simp: space_restrict_space Inf_real_def Sup_real_def Least_def simp del: Set.ball_simps(10))
1956     qed simp
1957   qed
1958 qed
1960 lemma borel_Ici: "borel = sigma UNIV (range (\<lambda>x::real. {x ..}))"
1961 proof (safe intro!: borel_eq_sigmaI1[OF borel_Iio])
1962   fix x :: real
1963   have eq: "{..<x} = space (sigma UNIV (range atLeast)) - {x ..}"
1964     by auto
1965   show "{..<x} \<in> sets (sigma UNIV (range atLeast))"
1966     unfolding eq by (intro sets.compl_sets) auto
1967 qed auto
1969 lemma borel_measurable_pred_less[measurable (raw)]:
1970   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
1971   shows "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> Measurable.pred M (\<lambda>w. f w < g w)"
1972   unfolding Measurable.pred_def by (rule borel_measurable_less)
1974 no_notation
1975   eucl_less (infix "<e" 50)
1977 lemma borel_measurable_Max2[measurable (raw)]:
1978   fixes f::"_ \<Rightarrow> _ \<Rightarrow> 'a::{second_countable_topology, dense_linorder, linorder_topology}"
1979   assumes "finite I"
1980     and [measurable]: "\<And>i. f i \<in> borel_measurable M"
1981   shows "(\<lambda>x. Max{f i x |i. i \<in> I}) \<in> borel_measurable M"
1982 by (simp add: borel_measurable_Max[OF assms(1), where ?f=f and ?M=M] Setcompr_eq_image)
1984 lemma measurable_compose_n [measurable (raw)]:
1985   assumes "T \<in> measurable M M"
1986   shows "(T^^n) \<in> measurable M M"
1987 by (induction n, auto simp add: measurable_compose[OF _ assms])
1989 lemma measurable_real_imp_nat:
1990   fixes f::"'a \<Rightarrow> nat"
1991   assumes [measurable]: "(\<lambda>x. real(f x)) \<in> borel_measurable M"
1992   shows "f \<in> measurable M (count_space UNIV)"
1993 proof -
1994   let ?g = "(\<lambda>x. real(f x))"
1995   have "\<And>(n::nat). ?g-`({real n}) \<inter> space M = f-`{n} \<inter> space M" by auto
1996   moreover have "\<And>(n::nat). ?g-`({real n}) \<inter> space M \<in> sets M" using assms by measurable
1997   ultimately have "\<And>(n::nat). f-`{n} \<inter> space M \<in> sets M" by simp
1998   then show ?thesis using measurable_count_space_eq2_countable by blast
1999 qed
2001 lemma measurable_equality_set [measurable]:
2002   fixes f g::"_\<Rightarrow> 'a::{second_countable_topology, t2_space}"
2003   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
2004   shows "{x \<in> space M. f x = g x} \<in> sets M"
2006 proof -
2007   define A where "A = {x \<in> space M. f x = g x}"
2008   define B where "B = {y. \<exists>x::'a. y = (x,x)}"
2009   have "A = (\<lambda>x. (f x, g x))-`B \<inter> space M" unfolding A_def B_def by auto
2010   moreover have "(\<lambda>x. (f x, g x)) \<in> borel_measurable M" by simp
2011   moreover have "B \<in> sets borel" unfolding B_def by (simp add: closed_diagonal)
2012   ultimately have "A \<in> sets M" by simp
2013   then show ?thesis unfolding A_def by simp
2014 qed
2016 lemma measurable_inequality_set [measurable]:
2017   fixes f g::"_ \<Rightarrow> 'a::{second_countable_topology, linorder_topology}"
2018   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
2019   shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
2020         "{x \<in> space M. f x < g x} \<in> sets M"
2021         "{x \<in> space M. f x \<ge> g x} \<in> sets M"
2022         "{x \<in> space M. f x > g x} \<in> sets M"
2023 proof -
2024   define F where "F = (\<lambda>x. (f x, g x))"
2025   have * [measurable]: "F \<in> borel_measurable M" unfolding F_def by simp
2027   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
2028   moreover have "{(x, y) | x y. x \<le> (y::'a)} \<in> sets borel" using closed_subdiagonal borel_closed by blast
2029   ultimately show "{x \<in> space M. f x \<le> g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
2031   have "{x \<in> space M. f x < g x} = F-`{(x, y) | x y. x < y} \<inter> space M" unfolding F_def by auto
2032   moreover have "{(x, y) | x y. x < (y::'a)} \<in> sets borel" using open_subdiagonal borel_open by blast
2033   ultimately show "{x \<in> space M. f x < g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
2035   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
2036   moreover have "{(x, y) | x y. x \<ge> (y::'a)} \<in> sets borel" using closed_superdiagonal borel_closed by blast
2037   ultimately show "{x \<in> space M. f x \<ge> g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
2039   have "{x \<in> space M. f x > g x} = F-`{(x, y) | x y. x > y} \<inter> space M" unfolding F_def by auto
2040   moreover have "{(x, y) | x y. x > (y::'a)} \<in> sets borel" using open_superdiagonal borel_open by blast
2041   ultimately show "{x \<in> space M. f x > g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
2042 qed
2044 lemma measurable_limit [measurable]:
2045   fixes f::"nat \<Rightarrow> 'a \<Rightarrow> 'b::first_countable_topology"
2046   assumes [measurable]: "\<And>n::nat. f n \<in> borel_measurable M"
2047   shows "Measurable.pred M (\<lambda>x. (\<lambda>n. f n x) \<longlonglongrightarrow> c)"
2048 proof -
2049   obtain A :: "nat \<Rightarrow> 'b set" where A:
2050     "\<And>i. open (A i)"
2051     "\<And>i. c \<in> A i"
2052     "\<And>S. open S \<Longrightarrow> c \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
2053   by (rule countable_basis_at_decseq) blast
2055   have [measurable]: "\<And>N i. (f N)-`(A i) \<inter> space M \<in> sets M" using A(1) by auto
2056   then have mes: "(\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. (f N)-`(A i) \<inter> space M) \<in> sets M" by blast
2058   have "(u \<longlonglongrightarrow> c) \<longleftrightarrow> (\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)" for u::"nat \<Rightarrow> 'b"
2059   proof
2060     assume "u \<longlonglongrightarrow> c"
2061     then have "eventually (\<lambda>n. u n \<in> A i) sequentially" for i using A(1)[of i] A(2)[of i]
2062       by (simp add: topological_tendstoD)
2063     then show "(\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)" by auto
2064   next
2065     assume H: "(\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)"
2066     show "(u \<longlonglongrightarrow> c)"
2067     proof (rule topological_tendstoI)
2068       fix S assume "open S" "c \<in> S"
2069       with A(3)[OF this] obtain i where "A i \<subseteq> S"
2070         using eventually_False_sequentially eventually_mono by blast
2071       moreover have "eventually (\<lambda>n. u n \<in> A i) sequentially" using H by simp
2072       ultimately show "\<forall>\<^sub>F n in sequentially. u n \<in> S"
2073         by (simp add: eventually_mono subset_eq)
2074     qed
2075   qed
2076   then have "{x. (\<lambda>n. f n x) \<longlonglongrightarrow> c} = (\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. (f N)-`(A i))"
2077     by (auto simp add: atLeast_def eventually_at_top_linorder)
2078   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)"
2079     by auto
2080   then have "{x \<in> space M. (\<lambda>n. f n x) \<longlonglongrightarrow> c} \<in> sets M" using mes by simp
2081   then show ?thesis by auto
2082 qed
2084 lemma measurable_limit2 [measurable]:
2085   fixes u::"nat \<Rightarrow> 'a \<Rightarrow> real"
2086   assumes [measurable]: "\<And>n. u n \<in> borel_measurable M" "v \<in> borel_measurable M"
2087   shows "Measurable.pred M (\<lambda>x. (\<lambda>n. u n x) \<longlonglongrightarrow> v x)"
2088 proof -
2089   define w where "w = (\<lambda>n x. u n x - v x)"
2090   have [measurable]: "w n \<in> borel_measurable M" for n unfolding w_def by auto
2091   have "((\<lambda>n. u n x) \<longlonglongrightarrow> v x) \<longleftrightarrow> ((\<lambda>n. w n x) \<longlonglongrightarrow> 0)" for x
2092     unfolding w_def using Lim_null by auto
2093   then show ?thesis using measurable_limit by auto
2094 qed
2096 lemma measurable_P_restriction [measurable (raw)]:
2097   assumes [measurable]: "Measurable.pred M P" "A \<in> sets M"
2098   shows "{x \<in> A. P x} \<in> sets M"
2099 proof -
2100   have "A \<subseteq> space M" using sets.sets_into_space[OF assms(2)].
2101   then have "{x \<in> A. P x} = A \<inter> {x \<in> space M. P x}" by blast
2102   then show ?thesis by auto
2103 qed
2105 lemma measurable_sum_nat [measurable (raw)]:
2106   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> nat"
2107   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> measurable M (count_space UNIV)"
2108   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> measurable M (count_space UNIV)"
2109 proof cases
2110   assume "finite S"
2111   then show ?thesis using assms by induct auto
2112 qed simp
2115 lemma measurable_abs_powr [measurable]:
2116   fixes p::real
2117   assumes [measurable]: "f \<in> borel_measurable M"
2118   shows "(\<lambda>x. \<bar>f x\<bar> powr p) \<in> borel_measurable M"
2119 unfolding powr_def by auto
2121 text {* The next one is a variation around \verb+measurable_restrict_space+.*}
2123 lemma measurable_restrict_space3:
2124   assumes "f \<in> measurable M N" and
2125           "f \<in> A \<rightarrow> B"
2126   shows "f \<in> measurable (restrict_space M A) (restrict_space N B)"
2127 proof -
2128   have "f \<in> measurable (restrict_space M A) N" using assms(1) measurable_restrict_space1 by auto
2129   then show ?thesis by (metis Int_iff funcsetI funcset_mem
2130       measurable_restrict_space2[of f, of "restrict_space M A", of B, of N] assms(2) space_restrict_space)
2131 qed
2133 text {* The next one is a variation around \verb+measurable_piecewise_restrict+.*}
2135 lemma measurable_piecewise_restrict2:
2136   assumes [measurable]: "\<And>n. A n \<in> sets M"
2137       and "space M = (\<Union>(n::nat). A n)"
2138           "\<And>n. \<exists>h \<in> measurable M N. (\<forall>x \<in> A n. f x = h x)"
2139   shows "f \<in> measurable M N"
2140 proof (rule measurableI)
2141   fix B assume [measurable]: "B \<in> sets N"
2142   {
2143     fix n::nat
2144     obtain h where [measurable]: "h \<in> measurable M N" and "\<forall>x \<in> A n. f x = h x" using assms(3) by blast
2145     then have *: "f-`B \<inter> A n = h-`B \<inter> A n" by auto
2146     have "h-`B \<inter> A n = h-`B \<inter> space M \<inter> A n" using assms(2) sets.sets_into_space by auto
2147     then have "h-`B \<inter> A n \<in> sets M" by simp
2148     then have "f-`B \<inter> A n \<in> sets M" using * by simp
2149   }
2150   then have "(\<Union>n. f-`B \<inter> A n) \<in> sets M" by measurable
2151   moreover have "f-`B \<inter> space M = (\<Union>n. f-`B \<inter> A n)" using assms(2) by blast
2152   ultimately show "f-`B \<inter> space M \<in> sets M" by simp
2153 next
2154   fix x assume "x \<in> space M"
2155   then obtain n where "x \<in> A n" using assms(2) by blast
2156   obtain h where [measurable]: "h \<in> measurable M N" and "\<forall>x \<in> A n. f x = h x" using assms(3) by blast
2157   then have "f x = h x" using `x \<in> A n` by blast
2158   moreover have "h x \<in> space N" by (metis measurable_space `x \<in> space M` `h \<in> measurable M N`)
2159   ultimately show "f x \<in> space N" by simp
2160 qed
2162 end