src/HOL/Probability/Borel_Space.thy
 author hoelzl Fri Nov 02 14:23:40 2012 +0100 (2012-11-02) changeset 50002 ce0d316b5b44 parent 50001 382bd3173584 child 50003 8c213922ed49 permissions -rw-r--r--
add measurability prover; add support for Borel sets
1 (*  Title:      HOL/Probability/Borel_Space.thy
2     Author:     Johannes Hölzl, TU München
3     Author:     Armin Heller, TU München
4 *)
6 header {*Borel spaces*}
8 theory Borel_Space
9   imports Sigma_Algebra "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
10 begin
12 section "Generic Borel spaces"
14 definition borel :: "'a::topological_space measure" where
15   "borel = sigma UNIV {S. open S}"
17 abbreviation "borel_measurable M \<equiv> measurable M borel"
19 lemma in_borel_measurable:
20    "f \<in> borel_measurable M \<longleftrightarrow>
21     (\<forall>S \<in> sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)"
22   by (auto simp add: measurable_def borel_def)
24 lemma in_borel_measurable_borel:
25    "f \<in> borel_measurable M \<longleftrightarrow>
26     (\<forall>S \<in> sets borel.
27       f -` S \<inter> space M \<in> sets M)"
28   by (auto simp add: measurable_def borel_def)
30 lemma space_borel[simp]: "space borel = UNIV"
31   unfolding borel_def by auto
33 lemma space_in_borel[measurable]: "UNIV \<in> sets borel"
34   unfolding borel_def by auto
36 lemma pred_Collect_borel[measurable (raw)]: "Sigma_Algebra.pred borel P \<Longrightarrow> {x. P x} \<in> sets borel"
37   unfolding borel_def pred_def by auto
39 lemma borel_open[simp, measurable (raw generic)]:
40   assumes "open A" shows "A \<in> sets borel"
41 proof -
42   have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms .
43   thus ?thesis unfolding borel_def by auto
44 qed
46 lemma borel_closed[simp, measurable (raw generic)]:
47   assumes "closed A" shows "A \<in> sets borel"
48 proof -
49   have "space borel - (- A) \<in> sets borel"
50     using assms unfolding closed_def by (blast intro: borel_open)
51   thus ?thesis by simp
52 qed
54 lemma borel_insert[measurable]:
55   "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets (borel :: 'a::t2_space measure)"
56   unfolding insert_def by (rule Un) auto
58 lemma borel_comp[intro, simp, measurable]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
59   unfolding Compl_eq_Diff_UNIV by simp
61 lemma borel_measurable_vimage:
62   fixes f :: "'a \<Rightarrow> 'x::t2_space"
63   assumes borel[measurable]: "f \<in> borel_measurable M"
64   shows "f -` {x} \<inter> space M \<in> sets M"
65   by simp
67 lemma borel_measurableI:
68   fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space"
69   assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
70   shows "f \<in> borel_measurable M"
71   unfolding borel_def
72 proof (rule measurable_measure_of, simp_all)
73   fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"
74     using assms[of S] by simp
75 qed
77 lemma borel_singleton[simp, intro]:
78   fixes x :: "'a::t1_space"
79   shows "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets borel"
80   proof (rule insert_in_sets)
81     show "{x} \<in> sets borel"
82       using closed_singleton[of x] by (rule borel_closed)
83   qed simp
85 lemma borel_measurable_const[simp, intro, measurable (raw)]:
86   "(\<lambda>x. c) \<in> borel_measurable M"
87   by auto
89 lemma borel_measurable_indicator[simp, intro!]:
90   assumes A: "A \<in> sets M"
91   shows "indicator A \<in> borel_measurable M"
92   unfolding indicator_def [abs_def] using A
93   by (auto intro!: measurable_If_set)
95 lemma borel_measurable_indicator'[measurable]:
96   "{x\<in>space M. x \<in> A} \<in> sets M \<Longrightarrow> indicator A \<in> borel_measurable M"
97   unfolding indicator_def[abs_def]
98   by (auto intro!: measurable_If)
100 lemma borel_measurable_indicator_iff:
101   "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
102     (is "?I \<in> borel_measurable M \<longleftrightarrow> _")
103 proof
104   assume "?I \<in> borel_measurable M"
105   then have "?I -` {1} \<inter> space M \<in> sets M"
106     unfolding measurable_def by auto
107   also have "?I -` {1} \<inter> space M = A \<inter> space M"
108     unfolding indicator_def [abs_def] by auto
109   finally show "A \<inter> space M \<in> sets M" .
110 next
111   assume "A \<inter> space M \<in> sets M"
112   moreover have "?I \<in> borel_measurable M \<longleftrightarrow>
113     (indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"
114     by (intro measurable_cong) (auto simp: indicator_def)
115   ultimately show "?I \<in> borel_measurable M" by auto
116 qed
118 lemma borel_measurable_subalgebra:
119   assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
120   shows "f \<in> borel_measurable M"
121   using assms unfolding measurable_def by auto
123 lemma borel_measurable_continuous_on1:
124   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
125   assumes "continuous_on UNIV f"
126   shows "f \<in> borel_measurable borel"
127   apply(rule borel_measurableI)
128   using continuous_open_preimage[OF assms] unfolding vimage_def by auto
130 section "Borel spaces on euclidean spaces"
132 lemma borel_measurable_euclidean_component'[measurable]:
133   "(\<lambda>x::'a::euclidean_space. x \$\$ i) \<in> borel_measurable borel"
134   by (intro continuous_on_euclidean_component continuous_on_id borel_measurable_continuous_on1)
136 lemma borel_measurable_euclidean_component:
137   "(f :: 'a \<Rightarrow> 'b::euclidean_space) \<in> borel_measurable M \<Longrightarrow>(\<lambda>x. f x \$\$ i) \<in> borel_measurable M"
138   by simp
140 lemma [simp, intro, measurable]:
141   fixes a b :: "'a\<Colon>ordered_euclidean_space"
142   shows lessThan_borel: "{..< a} \<in> sets borel"
143     and greaterThan_borel: "{a <..} \<in> sets borel"
144     and greaterThanLessThan_borel: "{a<..<b} \<in> sets borel"
145     and atMost_borel: "{..a} \<in> sets borel"
146     and atLeast_borel: "{a..} \<in> sets borel"
147     and atLeastAtMost_borel: "{a..b} \<in> sets borel"
148     and greaterThanAtMost_borel: "{a<..b} \<in> sets borel"
149     and atLeastLessThan_borel: "{a..<b} \<in> sets borel"
150   unfolding greaterThanAtMost_def atLeastLessThan_def
151   by (blast intro: borel_open borel_closed)+
153 lemma
154   shows hafspace_less_borel[simp, intro]: "{x::'a::euclidean_space. a < x \$\$ i} \<in> sets borel"
155     and hafspace_greater_borel[simp, intro]: "{x::'a::euclidean_space. x \$\$ i < a} \<in> sets borel"
156     and hafspace_less_eq_borel[simp, intro]: "{x::'a::euclidean_space. a \<le> x \$\$ i} \<in> sets borel"
157     and hafspace_greater_eq_borel[simp, intro]: "{x::'a::euclidean_space. x \$\$ i \<le> a} \<in> sets borel"
158   by simp_all
160 lemma borel_measurable_less[simp, intro, measurable]:
161   fixes f :: "'a \<Rightarrow> real"
162   assumes f: "f \<in> borel_measurable M"
163   assumes g: "g \<in> borel_measurable M"
164   shows "{w \<in> space M. f w < g w} \<in> sets M"
165 proof -
166   have "{w \<in> space M. f w < g w} = {x \<in> space M. \<exists>r. f x < of_rat r \<and> of_rat r < g x}"
167     using Rats_dense_in_real by (auto simp add: Rats_def)
168   with f g show ?thesis
169     by simp
170 qed
172 lemma [simp, intro]:
173   fixes f :: "'a \<Rightarrow> real"
174   assumes f[measurable]: "f \<in> borel_measurable M"
175   assumes g[measurable]: "g \<in> borel_measurable M"
176   shows borel_measurable_le[measurable]: "{w \<in> space M. f w \<le> g w} \<in> sets M"
177     and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M"
178     and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
179   unfolding eq_iff not_less[symmetric] by measurable+
181 subsection "Borel space equals sigma algebras over intervals"
183 lemma rational_boxes:
184   fixes x :: "'a\<Colon>ordered_euclidean_space"
185   assumes "0 < e"
186   shows "\<exists>a b. (\<forall>i. a \$\$ i \<in> \<rat>) \<and> (\<forall>i. b \$\$ i \<in> \<rat>) \<and> x \<in> {a <..< b} \<and> {a <..< b} \<subseteq> ball x e"
187 proof -
188   def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
189   then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos)
190   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \$\$ i \<and> x \$\$ i - y < e'" (is "\<forall>i. ?th i")
191   proof
192     fix i from Rats_dense_in_real[of "x \$\$ i - e'" "x \$\$ i"] e
193     show "?th i" by auto
194   qed
195   from choice[OF this] guess a .. note a = this
196   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \$\$ i < y \<and> y - x \$\$ i < e'" (is "\<forall>i. ?th i")
197   proof
198     fix i from Rats_dense_in_real[of "x \$\$ i" "x \$\$ i + e'"] e
199     show "?th i" by auto
200   qed
201   from choice[OF this] guess b .. note b = this
202   { fix y :: 'a assume *: "Chi a < y" "y < Chi b"
203     have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x \$\$ i) (y \$\$ i))\<twosuperior>)"
204       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
205     also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))"
206     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
207       fix i assume i: "i \<in> {..<DIM('a)}"
208       have "a i < y\$\$i \<and> y\$\$i < b i" using * i eucl_less[where 'a='a] by auto
209       moreover have "a i < x\$\$i" "x\$\$i - a i < e'" using a by auto
210       moreover have "x\$\$i < b i" "b i - x\$\$i < e'" using b by auto
211       ultimately have "\<bar>x\$\$i - y\$\$i\<bar> < 2 * e'" by auto
212       then have "dist (x \$\$ i) (y \$\$ i) < e/sqrt (real (DIM('a)))"
213         unfolding e'_def by (auto simp: dist_real_def)
214       then have "(dist (x \$\$ i) (y \$\$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
215         by (rule power_strict_mono) auto
216       then show "(dist (x \$\$ i) (y \$\$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
217         by (simp add: power_divide)
218     qed auto
219     also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive)
220     finally have "dist x y < e" . }
221   with a b show ?thesis
222     apply (rule_tac exI[of _ "Chi a"])
223     apply (rule_tac exI[of _ "Chi b"])
224     using eucl_less[where 'a='a] by auto
225 qed
227 lemma ex_rat_list:
228   fixes x :: "'a\<Colon>ordered_euclidean_space"
229   assumes "\<And> i. x \$\$ i \<in> \<rat>"
230   shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x \$\$ i)"
231 proof -
232   have "\<forall>i. \<exists>r. x \$\$ i = of_rat r" using assms unfolding Rats_def by blast
233   from choice[OF this] guess r ..
234   then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"])
235 qed
237 lemma open_UNION:
238   fixes M :: "'a\<Colon>ordered_euclidean_space set"
239   assumes "open M"
240   shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M}
241                    (\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})"
242     (is "M = UNION ?idx ?box")
243 proof safe
244   fix x assume "x \<in> M"
245   obtain e where e: "e > 0" "ball x e \<subseteq> M"
246     using openE[OF assms `x \<in> M`] by auto
247   then obtain a b where ab: "x \<in> {a <..< b}" "\<And>i. a \$\$ i \<in> \<rat>" "\<And>i. b \$\$ i \<in> \<rat>" "{a <..< b} \<subseteq> ball x e"
248     using rational_boxes[OF e(1)] by blast
249   then obtain p q where pq: "length p = DIM ('a)"
250                             "length q = DIM ('a)"
251                             "\<forall> i < DIM ('a). of_rat (p ! i) = a \$\$ i \<and> of_rat (q ! i) = b \$\$ i"
252     using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast
253   hence p: "Chi (of_rat \<circ> op ! p) = a"
254     using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a]
255     unfolding o_def by auto
256   from pq have q: "Chi (of_rat \<circ> op ! q) = b"
257     using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b]
258     unfolding o_def by auto
259   have "x \<in> ?box (p, q)"
260     using p q ab by auto
261   thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto
262 qed auto
264 lemma borel_sigma_sets_subset:
265   "A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel"
266   using sigma_sets_subset[of A borel] by simp
268 lemma borel_eq_sigmaI1:
269   fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
270   assumes borel_eq: "borel = sigma UNIV X"
271   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (range F))"
272   assumes F: "\<And>i. F i \<in> sets borel"
273   shows "borel = sigma UNIV (range F)"
274   unfolding borel_def
275 proof (intro sigma_eqI antisym)
276   have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel"
277     unfolding borel_def by simp
278   also have "\<dots> = sigma_sets UNIV X"
279     unfolding borel_eq by simp
280   also have "\<dots> \<subseteq> sigma_sets UNIV (range F)"
281     using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto
282   finally show "sigma_sets UNIV {S. open S} \<subseteq> sigma_sets UNIV (range F)" .
283   show "sigma_sets UNIV (range F) \<subseteq> sigma_sets UNIV {S. open S}"
284     unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto
285 qed auto
287 lemma borel_eq_sigmaI2:
288   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set"
289     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
290   assumes borel_eq: "borel = sigma UNIV (range (\<lambda>(i, j). G i j))"
291   assumes X: "\<And>i j. G i j \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
292   assumes F: "\<And>i j. F i j \<in> sets borel"
293   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
294   using assms by (intro borel_eq_sigmaI1[where X="range (\<lambda>(i, j). G i j)" and F="(\<lambda>(i, j). F i j)"]) auto
296 lemma borel_eq_sigmaI3:
297   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
298   assumes borel_eq: "borel = sigma UNIV X"
299   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
300   assumes F: "\<And>i j. F i j \<in> sets borel"
301   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
302   using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto
304 lemma borel_eq_sigmaI4:
305   fixes F :: "'i \<Rightarrow> 'a::topological_space set"
306     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
307   assumes borel_eq: "borel = sigma UNIV (range (\<lambda>(i, j). G i j))"
308   assumes X: "\<And>i j. G i j \<in> sets (sigma UNIV (range F))"
309   assumes F: "\<And>i. F i \<in> sets borel"
310   shows "borel = sigma UNIV (range F)"
311   using assms by (intro borel_eq_sigmaI1[where X="range (\<lambda>(i, j). G i j)" and F=F]) auto
313 lemma borel_eq_sigmaI5:
314   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set"
315   assumes borel_eq: "borel = sigma UNIV (range G)"
316   assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
317   assumes F: "\<And>i j. F i j \<in> sets borel"
318   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
319   using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto
321 lemma halfspace_gt_in_halfspace:
322   "{x\<Colon>'a. a < x \$\$ i} \<in> sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a}))"
323   (is "?set \<in> ?SIGMA")
324 proof -
325   interpret sigma_algebra UNIV ?SIGMA
326     by (intro sigma_algebra_sigma_sets) simp_all
327   have *: "?set = (\<Union>n. UNIV - {x\<Colon>'a. x \$\$ i < a + 1 / real (Suc n)})"
328   proof (safe, simp_all add: not_less)
329     fix x :: 'a assume "a < x \$\$ i"
330     with reals_Archimedean[of "x \$\$ i - a"]
331     obtain n where "a + 1 / real (Suc n) < x \$\$ i"
332       by (auto simp: inverse_eq_divide field_simps)
333     then show "\<exists>n. a + 1 / real (Suc n) \<le> x \$\$ i"
334       by (blast intro: less_imp_le)
335   next
336     fix x n
337     have "a < a + 1 / real (Suc n)" by auto
338     also assume "\<dots> \<le> x"
339     finally show "a < x" .
340   qed
341   show "?set \<in> ?SIGMA" unfolding *
342     by (auto del: Diff intro!: Diff)
343 qed
345 lemma borel_eq_halfspace_less:
346   "borel = sigma UNIV (range (\<lambda>(a, i). {x::'a::ordered_euclidean_space. x \$\$ i < a}))"
347   (is "_ = ?SIGMA")
348 proof (rule borel_eq_sigmaI3[OF borel_def])
349   fix S :: "'a set" assume "S \<in> {S. open S}"
350   then have "open S" by simp
351   from open_UNION[OF this]
352   obtain I where *: "S =
353     (\<Union>(a, b)\<in>I.
354         (\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) \$\$ i < x \$\$ i}) \<inter>
355         (\<Inter> i<DIM('a). {x. x \$\$ i < (Chi (real_of_rat \<circ> op ! b)::'a) \$\$ i}))"
356     unfolding greaterThanLessThan_def
357     unfolding eucl_greaterThan_eq_halfspaces[where 'a='a]
358     unfolding eucl_lessThan_eq_halfspaces[where 'a='a]
359     by blast
360   show "S \<in> ?SIGMA"
361     unfolding *
362     by (safe intro!: countable_UN Int countable_INT) (auto intro!: halfspace_gt_in_halfspace)
363 qed auto
365 lemma borel_eq_halfspace_le:
366   "borel = sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x \$\$ i \<le> a}))"
367   (is "_ = ?SIGMA")
368 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
369   fix a i
370   have *: "{x::'a. x\$\$i < a} = (\<Union>n. {x. x\$\$i \<le> a - 1/real (Suc n)})"
371   proof (safe, simp_all)
372     fix x::'a assume *: "x\$\$i < a"
373     with reals_Archimedean[of "a - x\$\$i"]
374     obtain n where "x \$\$ i < a - 1 / (real (Suc n))"
375       by (auto simp: field_simps inverse_eq_divide)
376     then show "\<exists>n. x \$\$ i \<le> a - 1 / (real (Suc n))"
377       by (blast intro: less_imp_le)
378   next
379     fix x::'a and n
380     assume "x\$\$i \<le> a - 1 / real (Suc n)"
381     also have "\<dots> < a" by auto
382     finally show "x\$\$i < a" .
383   qed
384   show "{x. x\$\$i < a} \<in> ?SIGMA" unfolding *
385     by (safe intro!: countable_UN) auto
386 qed auto
388 lemma borel_eq_halfspace_ge:
389   "borel = sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x \$\$ i}))"
390   (is "_ = ?SIGMA")
391 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
392   fix a i have *: "{x::'a. x\$\$i < a} = space ?SIGMA - {x::'a. a \<le> x\$\$i}" by auto
393   show "{x. x\$\$i < a} \<in> ?SIGMA" unfolding *
394       by (safe intro!: compl_sets) auto
395 qed auto
397 lemma borel_eq_halfspace_greater:
398   "borel = sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a < x \$\$ i}))"
399   (is "_ = ?SIGMA")
400 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])
401   fix a i have *: "{x::'a. x\$\$i \<le> a} = space ?SIGMA - {x::'a. a < x\$\$i}" by auto
402   show "{x. x\$\$i \<le> a} \<in> ?SIGMA" unfolding *
403     by (safe intro!: compl_sets) auto
404 qed auto
406 lemma borel_eq_atMost:
407   "borel = sigma UNIV (range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space}))"
408   (is "_ = ?SIGMA")
409 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
410   fix a i show "{x. x\$\$i \<le> a} \<in> ?SIGMA"
411   proof cases
412     assume "i < DIM('a)"
413     then have *: "{x::'a. x\$\$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})"
414     proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm)
415       fix x
416       from real_arch_simple[of "Max ((\<lambda>i. x\$\$i)`{..<DIM('a)})"] guess k::nat ..
417       then have "\<And>i. i < DIM('a) \<Longrightarrow> x\$\$i \<le> real k"
418         by (subst (asm) Max_le_iff) auto
419       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x \$\$ ia \<le> real k"
420         by (auto intro!: exI[of _ k])
421     qed
422     show "{x. x\$\$i \<le> a} \<in> ?SIGMA" unfolding *
423       by (safe intro!: countable_UN) auto
424   qed (auto intro: sigma_sets_top sigma_sets.Empty)
425 qed auto
427 lemma borel_eq_greaterThan:
428   "borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {a<..}))"
429   (is "_ = ?SIGMA")
430 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
431   fix a i show "{x. x\$\$i \<le> a} \<in> ?SIGMA"
432   proof cases
433     assume "i < DIM('a)"
434     have "{x::'a. x\$\$i \<le> a} = UNIV - {x::'a. a < x\$\$i}" by auto
435     also have *: "{x::'a. a < x\$\$i} = (\<Union>k::nat. {(\<chi>\<chi> n. if n = i then a else -real k) <..})" using `i <DIM('a)`
436     proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
437       fix x
438       from reals_Archimedean2[of "Max ((\<lambda>i. -x\$\$i)`{..<DIM('a)})"]
439       guess k::nat .. note k = this
440       { fix i assume "i < DIM('a)"
441         then have "-x\$\$i < real k"
442           using k by (subst (asm) Max_less_iff) auto
443         then have "- real k < x\$\$i" by simp }
444       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x \$\$ ia"
445         by (auto intro!: exI[of _ k])
446     qed
447     finally show "{x. x\$\$i \<le> a} \<in> ?SIGMA"
448       apply (simp only:)
449       apply (safe intro!: countable_UN Diff)
450       apply (auto intro: sigma_sets_top)
451       done
452   qed (auto intro: sigma_sets_top sigma_sets.Empty)
453 qed auto
455 lemma borel_eq_lessThan:
456   "borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {..<a}))"
457   (is "_ = ?SIGMA")
458 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge])
459   fix a i show "{x. a \<le> x\$\$i} \<in> ?SIGMA"
460   proof cases
461     fix a i assume "i < DIM('a)"
462     have "{x::'a. a \<le> x\$\$i} = UNIV - {x::'a. x\$\$i < a}" by auto
463     also have *: "{x::'a. x\$\$i < a} = (\<Union>k::nat. {..< (\<chi>\<chi> n. if n = i then a else real k)})" using `i <DIM('a)`
464     proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
465       fix x
466       from reals_Archimedean2[of "Max ((\<lambda>i. x\$\$i)`{..<DIM('a)})"]
467       guess k::nat .. note k = this
468       { fix i assume "i < DIM('a)"
469         then have "x\$\$i < real k"
470           using k by (subst (asm) Max_less_iff) auto
471         then have "x\$\$i < real k" by simp }
472       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x \$\$ ia < real k"
473         by (auto intro!: exI[of _ k])
474     qed
475     finally show "{x. a \<le> x\$\$i} \<in> ?SIGMA"
476       apply (simp only:)
477       apply (safe intro!: countable_UN Diff)
478       apply (auto intro: sigma_sets_top)
479       done
480   qed (auto intro: sigma_sets_top sigma_sets.Empty)
481 qed auto
483 lemma borel_eq_atLeastAtMost:
484   "borel = sigma UNIV (range (\<lambda>(a,b). {a..b} \<Colon>'a\<Colon>ordered_euclidean_space set))"
485   (is "_ = ?SIGMA")
486 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
487   fix a::'a
488   have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
489   proof (safe, simp_all add: eucl_le[where 'a='a])
490     fix x
491     from real_arch_simple[of "Max ((\<lambda>i. - x\$\$i)`{..<DIM('a)})"]
492     guess k::nat .. note k = this
493     { fix i assume "i < DIM('a)"
494       with k have "- x\$\$i \<le> real k"
495         by (subst (asm) Max_le_iff) (auto simp: field_simps)
496       then have "- real k \<le> x\$\$i" by simp }
497     then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x \$\$ i"
498       by (auto intro!: exI[of _ k])
499   qed
500   show "{..a} \<in> ?SIGMA" unfolding *
501     by (safe intro!: countable_UN)
502        (auto intro!: sigma_sets_top)
503 qed auto
505 lemma borel_eq_greaterThanLessThan:
506   "borel = sigma UNIV (range (\<lambda> (a, b). {a <..< b} :: 'a \<Colon> ordered_euclidean_space set))"
507     (is "_ = ?SIGMA")
508 proof (rule borel_eq_sigmaI1[OF borel_def])
509   fix M :: "'a set" assume "M \<in> {S. open S}"
510   then have "open M" by simp
511   show "M \<in> ?SIGMA"
512     apply (subst open_UNION[OF `open M`])
513     apply (safe intro!: countable_UN)
514     apply auto
515     done
516 qed auto
518 lemma borel_eq_atLeastLessThan:
519   "borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA")
520 proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan])
521   have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto
522   fix x :: real
523   have "{..<x} = (\<Union>i::nat. {-real i ..< x})"
524     by (auto simp: move_uminus real_arch_simple)
525   then show "{..< x} \<in> ?SIGMA"
526     by (auto intro: sigma_sets.intros)
527 qed auto
529 lemma borel_measurable_halfspacesI:
530   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
531   assumes F: "borel = sigma UNIV (range F)"
532   and S_eq: "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
533   and S: "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M"
534   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)"
535 proof safe
536   fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M"
537   then show "S a i \<in> sets M" unfolding assms
538     by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1))
539 next
540   assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M"
541   { fix a i have "S a i \<in> sets M"
542     proof cases
543       assume "i < DIM('c)"
544       with a show ?thesis unfolding assms(2) by simp
545     next
546       assume "\<not> i < DIM('c)"
547       from S[OF this] show ?thesis .
548     qed }
549   then show "f \<in> borel_measurable M"
550     by (auto intro!: measurable_measure_of simp: S_eq F)
551 qed
553 lemma borel_measurable_iff_halfspace_le:
554   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
555   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w \$\$ i \<le> a} \<in> sets M)"
556   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
558 lemma borel_measurable_iff_halfspace_less:
559   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
560   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w \$\$ i < a} \<in> sets M)"
561   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
563 lemma borel_measurable_iff_halfspace_ge:
564   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
565   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w \$\$ i} \<in> sets M)"
566   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
568 lemma borel_measurable_iff_halfspace_greater:
569   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
570   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w \$\$ i} \<in> sets M)"
571   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto
573 lemma borel_measurable_iff_le:
574   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
575   using borel_measurable_iff_halfspace_le[where 'c=real] by simp
577 lemma borel_measurable_iff_less:
578   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
579   using borel_measurable_iff_halfspace_less[where 'c=real] by simp
581 lemma borel_measurable_iff_ge:
582   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
583   using borel_measurable_iff_halfspace_ge[where 'c=real]
584   by simp
586 lemma borel_measurable_iff_greater:
587   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
588   using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
590 lemma borel_measurable_euclidean_space:
591   fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
592   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x \$\$ i) \<in> borel_measurable M)"
593 proof safe
594   fix i assume "f \<in> borel_measurable M"
595   then show "(\<lambda>x. f x \$\$ i) \<in> borel_measurable M"
596     by (auto intro: borel_measurable_euclidean_component)
597 next
598   assume f: "\<forall>i<DIM('c). (\<lambda>x. f x \$\$ i) \<in> borel_measurable M"
599   then show "f \<in> borel_measurable M"
600     unfolding borel_measurable_iff_halfspace_le by auto
601 qed
603 subsection "Borel measurable operators"
605 lemma borel_measurable_continuous_on:
606   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
607   assumes f: "continuous_on UNIV f" and g: "g \<in> borel_measurable M"
608   shows "(\<lambda>x. f (g x)) \<in> borel_measurable M"
609   using measurable_comp[OF g borel_measurable_continuous_on1[OF f]] by (simp add: comp_def)
611 lemma borel_measurable_continuous_on_open':
612   fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
613   assumes cont: "continuous_on A f" "open A"
614   shows "(\<lambda>x. if x \<in> A then f x else c) \<in> borel_measurable borel" (is "?f \<in> _")
615 proof (rule borel_measurableI)
616   fix S :: "'b set" assume "open S"
617   then have "open {x\<in>A. f x \<in> S}"
618     by (intro continuous_open_preimage[OF cont]) auto
619   then have *: "{x\<in>A. f x \<in> S} \<in> sets borel" by auto
620   have "?f -` S \<inter> space borel =
621     {x\<in>A. f x \<in> S} \<union> (if c \<in> S then space borel - A else {})"
622     by (auto split: split_if_asm)
623   also have "\<dots> \<in> sets borel"
624     using * `open A` by auto
625   finally show "?f -` S \<inter> space borel \<in> sets borel" .
626 qed
628 lemma borel_measurable_continuous_on_open:
629   fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
630   assumes cont: "continuous_on A f" "open A"
631   assumes g: "g \<in> borel_measurable M"
632   shows "(\<lambda>x. if g x \<in> A then f (g x) else c) \<in> borel_measurable M"
633   using measurable_comp[OF g borel_measurable_continuous_on_open'[OF cont], of c]
634   by (simp add: comp_def)
636 lemma borel_measurable_uminus[simp, intro, measurable (raw)]:
637   fixes g :: "'a \<Rightarrow> real"
638   assumes g: "g \<in> borel_measurable M"
639   shows "(\<lambda>x. - g x) \<in> borel_measurable M"
640   by (rule borel_measurable_continuous_on[OF _ g]) (auto intro: continuous_on_minus continuous_on_id)
642 lemma euclidean_component_prod:
643   fixes x :: "'a :: euclidean_space \<times> 'b :: euclidean_space"
644   shows "x \$\$ i = (if i < DIM('a) then fst x \$\$ i else snd x \$\$ (i - DIM('a)))"
645   unfolding euclidean_component_def basis_prod_def inner_prod_def by auto
647 lemma borel_measurable_Pair[simp, intro, measurable (raw)]:
648   fixes f :: "'a \<Rightarrow> 'b::ordered_euclidean_space" and g :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
649   assumes f: "f \<in> borel_measurable M"
650   assumes g: "g \<in> borel_measurable M"
651   shows "(\<lambda>x. (f x, g x)) \<in> borel_measurable M"
652 proof (intro borel_measurable_iff_halfspace_le[THEN iffD2] allI impI)
653   fix i and a :: real assume i: "i < DIM('b \<times> 'c)"
654   have [simp]: "\<And>P A B C. {w. (P \<longrightarrow> A w \<and> B w) \<and> (\<not> P \<longrightarrow> A w \<and> C w)} =
655     {w. A w \<and> (P \<longrightarrow> B w) \<and> (\<not> P \<longrightarrow> C w)}" by auto
656   from i f g show "{w \<in> space M. (f w, g w) \$\$ i \<le> a} \<in> sets M"
657     by (auto simp: euclidean_component_prod)
658 qed
660 lemma continuous_on_fst: "continuous_on UNIV fst"
661 proof -
662   have [simp]: "range fst = UNIV" by (auto simp: image_iff)
663   show ?thesis
664     using closed_vimage_fst
665     by (auto simp: continuous_on_closed closed_closedin vimage_def)
666 qed
668 lemma continuous_on_snd: "continuous_on UNIV snd"
669 proof -
670   have [simp]: "range snd = UNIV" by (auto simp: image_iff)
671   show ?thesis
672     using closed_vimage_snd
673     by (auto simp: continuous_on_closed closed_closedin vimage_def)
674 qed
676 lemma borel_measurable_continuous_Pair:
677   fixes f :: "'a \<Rightarrow> 'b::ordered_euclidean_space" and g :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
678   assumes [simp]: "f \<in> borel_measurable M"
679   assumes [simp]: "g \<in> borel_measurable M"
680   assumes H: "continuous_on UNIV (\<lambda>x. H (fst x) (snd x))"
681   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
682 proof -
683   have eq: "(\<lambda>x. H (f x) (g x)) = (\<lambda>x. (\<lambda>x. H (fst x) (snd x)) (f x, g x))" by auto
684   show ?thesis
685     unfolding eq by (rule borel_measurable_continuous_on[OF H]) auto
686 qed
688 lemma borel_measurable_add[simp, intro, measurable (raw)]:
689   fixes f g :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
690   assumes f: "f \<in> borel_measurable M"
691   assumes g: "g \<in> borel_measurable M"
692   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
693   using f g
694   by (rule borel_measurable_continuous_Pair)
695      (auto intro: continuous_on_fst continuous_on_snd continuous_on_add)
697 lemma borel_measurable_setsum[simp, intro, measurable (raw)]:
698   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
699   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
700   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
701 proof cases
702   assume "finite S"
703   thus ?thesis using assms by induct auto
704 qed simp
706 lemma borel_measurable_diff[simp, intro, measurable (raw)]:
707   fixes f :: "'a \<Rightarrow> real"
708   assumes f: "f \<in> borel_measurable M"
709   assumes g: "g \<in> borel_measurable M"
710   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
711   unfolding diff_minus using assms by fast
713 lemma borel_measurable_times[simp, intro, measurable (raw)]:
714   fixes f :: "'a \<Rightarrow> real"
715   assumes f: "f \<in> borel_measurable M"
716   assumes g: "g \<in> borel_measurable M"
717   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
718   using f g
719   by (rule borel_measurable_continuous_Pair)
720      (auto intro: continuous_on_fst continuous_on_snd continuous_on_mult)
722 lemma continuous_on_dist:
723   fixes f :: "'a :: t2_space \<Rightarrow> 'b :: metric_space"
724   shows "continuous_on A f \<Longrightarrow> continuous_on A g \<Longrightarrow> continuous_on A (\<lambda>x. dist (f x) (g x))"
725   unfolding continuous_on_eq_continuous_within by (auto simp: continuous_dist)
727 lemma borel_measurable_dist[simp, intro, measurable (raw)]:
728   fixes g f :: "'a \<Rightarrow> 'b::ordered_euclidean_space"
729   assumes f: "f \<in> borel_measurable M"
730   assumes g: "g \<in> borel_measurable M"
731   shows "(\<lambda>x. dist (f x) (g x)) \<in> borel_measurable M"
732   using f g
733   by (rule borel_measurable_continuous_Pair)
734      (intro continuous_on_dist continuous_on_fst continuous_on_snd)
736 lemma borel_measurable_scaleR[measurable (raw)]:
737   fixes g :: "'a \<Rightarrow> 'b::ordered_euclidean_space"
738   assumes f: "f \<in> borel_measurable M"
739   assumes g: "g \<in> borel_measurable M"
740   shows "(\<lambda>x. f x *\<^sub>R g x) \<in> borel_measurable M"
741   by (rule borel_measurable_continuous_Pair[OF f g])
742      (auto intro!: continuous_on_scaleR continuous_on_fst continuous_on_snd)
744 lemma affine_borel_measurable_vector:
745   fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
746   assumes "f \<in> borel_measurable M"
747   shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
748 proof (rule borel_measurableI)
749   fix S :: "'x set" assume "open S"
750   show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
751   proof cases
752     assume "b \<noteq> 0"
753     with `open S` have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")
754       by (auto intro!: open_affinity simp: scaleR_add_right)
755     hence "?S \<in> sets borel" by auto
756     moreover
757     from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
758       apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
759     ultimately show ?thesis using assms unfolding in_borel_measurable_borel
760       by auto
761   qed simp
762 qed
764 lemma borel_measurable_const_scaleR[measurable (raw)]:
765   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. b *\<^sub>R f x ::'a::real_normed_vector) \<in> borel_measurable M"
766   using affine_borel_measurable_vector[of f M 0 b] by simp
768 lemma borel_measurable_const_add[measurable (raw)]:
769   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. a + f x ::'a::real_normed_vector) \<in> borel_measurable M"
770   using affine_borel_measurable_vector[of f M a 1] by simp
772 lemma borel_measurable_setprod[simp, intro, measurable (raw)]:
773   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
774   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
775   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
776 proof cases
777   assume "finite S"
778   thus ?thesis using assms by induct auto
779 qed simp
781 lemma borel_measurable_inverse[simp, intro, measurable (raw)]:
782   fixes f :: "'a \<Rightarrow> real"
783   assumes f: "f \<in> borel_measurable M"
784   shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
785 proof -
786   have *: "\<And>x::real. inverse x = (if x \<in> UNIV - {0} then inverse x else 0)" by auto
787   show ?thesis
788     apply (subst *)
789     apply (rule borel_measurable_continuous_on_open)
790     apply (auto intro!: f continuous_on_inverse continuous_on_id)
791     done
792 qed
794 lemma borel_measurable_divide[simp, intro, measurable (raw)]:
795   fixes f :: "'a \<Rightarrow> real"
796   assumes "f \<in> borel_measurable M"
797   and "g \<in> borel_measurable M"
798   shows "(\<lambda>x. f x / g x) \<in> borel_measurable M"
799   unfolding field_divide_inverse
800   by (rule borel_measurable_inverse borel_measurable_times assms)+
802 lemma borel_measurable_max[intro, simp, measurable (raw)]:
803   fixes f g :: "'a \<Rightarrow> real"
804   assumes "f \<in> borel_measurable M"
805   assumes "g \<in> borel_measurable M"
806   shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
807   unfolding max_def by (auto intro!: assms measurable_If)
809 lemma borel_measurable_min[intro, simp, measurable (raw)]:
810   fixes f g :: "'a \<Rightarrow> real"
811   assumes "f \<in> borel_measurable M"
812   assumes "g \<in> borel_measurable M"
813   shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
814   unfolding min_def by (auto intro!: assms measurable_If)
816 lemma borel_measurable_abs[simp, intro, measurable (raw)]:
817   assumes "f \<in> borel_measurable M"
818   shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
819 proof -
820   have *: "\<And>x. \<bar>f x\<bar> = max 0 (f x) + max 0 (- f x)" by (simp add: max_def)
821   show ?thesis unfolding * using assms by auto
822 qed
824 lemma borel_measurable_nth[simp, intro, measurable (raw)]:
825   "(\<lambda>x::real^'n. x \$ i) \<in> borel_measurable borel"
826   using borel_measurable_euclidean_component'
827   unfolding nth_conv_component by auto
829 lemma convex_measurable:
830   fixes a b :: real
831   assumes X: "X \<in> borel_measurable M" "X ` space M \<subseteq> { a <..< b}"
832   assumes q: "convex_on { a <..< b} q"
833   shows "(\<lambda>x. q (X x)) \<in> borel_measurable M"
834 proof -
835   have "(\<lambda>x. if X x \<in> {a <..< b} then q (X x) else 0) \<in> borel_measurable M" (is "?qX")
836   proof (rule borel_measurable_continuous_on_open[OF _ _ X(1)])
837     show "open {a<..<b}" by auto
838     from this q show "continuous_on {a<..<b} q"
839       by (rule convex_on_continuous)
840   qed
841   also have "?qX \<longleftrightarrow> (\<lambda>x. q (X x)) \<in> borel_measurable M"
842     using X by (intro measurable_cong) auto
843   finally show ?thesis .
844 qed
846 lemma borel_measurable_ln[simp, intro, measurable (raw)]:
847   assumes f: "f \<in> borel_measurable M"
848   shows "(\<lambda>x. ln (f x)) \<in> borel_measurable M"
849 proof -
850   { fix x :: real assume x: "x \<le> 0"
851     { fix x::real assume "x \<le> 0" then have "\<And>u. exp u = x \<longleftrightarrow> False" by auto }
852     from this[of x] x this[of 0] have "ln 0 = ln x"
853       by (auto simp: ln_def) }
854   note ln_imp = this
855   have "(\<lambda>x. if f x \<in> {0<..} then ln (f x) else ln 0) \<in> borel_measurable M"
856   proof (rule borel_measurable_continuous_on_open[OF _ _ f])
857     show "continuous_on {0<..} ln"
858       by (auto intro!: continuous_at_imp_continuous_on DERIV_ln DERIV_isCont
859                simp: continuous_isCont[symmetric])
860     show "open ({0<..}::real set)" by auto
861   qed
862   also have "(\<lambda>x. if x \<in> {0<..} then ln x else ln 0) = ln"
863     by (simp add: fun_eq_iff not_less ln_imp)
864   finally show ?thesis .
865 qed
867 lemma borel_measurable_log[simp, intro, measurable (raw)]:
868   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. log (g x) (f x)) \<in> borel_measurable M"
869   unfolding log_def by auto
871 lemma measurable_count_space_eq2_countable:
872   fixes f :: "'a => 'c::countable"
873   shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
874 proof -
875   { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
876     then have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)"
877       by auto
878     moreover assume "\<And>a. a\<in>A \<Longrightarrow> f -` {a} \<inter> space M \<in> sets M"
879     ultimately have "f -` X \<inter> space M \<in> sets M"
880       using `X \<subseteq> A` by (simp add: subset_eq del: UN_simps) }
881   then show ?thesis
882     unfolding measurable_def by auto
883 qed
885 lemma measurable_real_floor[measurable]:
886   "(floor :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
887 proof -
888   have "\<And>a x. \<lfloor>x\<rfloor> = a \<longleftrightarrow> (real a \<le> x \<and> x < real (a + 1))"
889     by (auto intro: floor_eq2)
890   then show ?thesis
891     by (auto simp: vimage_def measurable_count_space_eq2_countable)
892 qed
894 lemma measurable_real_natfloor[measurable]:
895   "(natfloor :: real \<Rightarrow> nat) \<in> measurable borel (count_space UNIV)"
896   by (simp add: natfloor_def[abs_def])
898 lemma measurable_real_ceiling[measurable]:
899   "(ceiling :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
900   unfolding ceiling_def[abs_def] by simp
902 lemma borel_measurable_real_floor: "(\<lambda>x::real. real \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
903   by simp
905 lemma borel_measurable_real_natfloor[intro, simp]:
906   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. real (natfloor (f x))) \<in> borel_measurable M"
907   by simp
909 subsection "Borel space on the extended reals"
911 lemma borel_measurable_ereal[simp, intro, measurable (raw)]:
912   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
913   using continuous_on_ereal f by (rule borel_measurable_continuous_on)
915 lemma borel_measurable_real_of_ereal[simp, intro, measurable (raw)]:
916   fixes f :: "'a \<Rightarrow> ereal"
917   assumes f: "f \<in> borel_measurable M"
918   shows "(\<lambda>x. real (f x)) \<in> borel_measurable M"
919 proof -
920   have "(\<lambda>x. if f x \<in> UNIV - { \<infinity>, - \<infinity> } then real (f x) else 0) \<in> borel_measurable M"
921     using continuous_on_real
922     by (rule borel_measurable_continuous_on_open[OF _ _ f]) auto
923   also have "(\<lambda>x. if f x \<in> UNIV - { \<infinity>, - \<infinity> } then real (f x) else 0) = (\<lambda>x. real (f x))"
924     by auto
925   finally show ?thesis .
926 qed
928 lemma borel_measurable_ereal_cases:
929   fixes f :: "'a \<Rightarrow> ereal"
930   assumes f: "f \<in> borel_measurable M"
931   assumes H: "(\<lambda>x. H (ereal (real (f x)))) \<in> borel_measurable M"
932   shows "(\<lambda>x. H (f x)) \<in> borel_measurable M"
933 proof -
934   let ?F = "\<lambda>x. if f x = \<infinity> then H \<infinity> else if f x = - \<infinity> then H (-\<infinity>) else H (ereal (real (f x)))"
935   { fix x have "H (f x) = ?F x" by (cases "f x") auto }
936   with f H show ?thesis by simp
937 qed
939 lemma
940   fixes f :: "'a \<Rightarrow> ereal" assumes f[simp]: "f \<in> borel_measurable M"
941   shows borel_measurable_ereal_abs[intro, simp, measurable(raw)]: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
942     and borel_measurable_ereal_inverse[simp, intro, measurable(raw)]: "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M"
943     and borel_measurable_uminus_ereal[intro, measurable(raw)]: "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M"
944   by (auto simp del: abs_real_of_ereal simp: borel_measurable_ereal_cases[OF f] measurable_If)
946 lemma borel_measurable_uminus_eq_ereal[simp]:
947   "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
948 proof
949   assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp
950 qed auto
952 lemma set_Collect_ereal2:
953   fixes f g :: "'a \<Rightarrow> ereal"
954   assumes f: "f \<in> borel_measurable M"
955   assumes g: "g \<in> borel_measurable M"
956   assumes H: "{x \<in> space M. H (ereal (real (f x))) (ereal (real (g x)))} \<in> sets M"
957     "{x \<in> space borel. H (-\<infinity>) (ereal x)} \<in> sets borel"
958     "{x \<in> space borel. H (\<infinity>) (ereal x)} \<in> sets borel"
959     "{x \<in> space borel. H (ereal x) (-\<infinity>)} \<in> sets borel"
960     "{x \<in> space borel. H (ereal x) (\<infinity>)} \<in> sets borel"
961   shows "{x \<in> space M. H (f x) (g x)} \<in> sets M"
962 proof -
963   let ?G = "\<lambda>y x. if g x = \<infinity> then H y \<infinity> else if g x = -\<infinity> then H y (-\<infinity>) else H y (ereal (real (g x)))"
964   let ?F = "\<lambda>x. if f x = \<infinity> then ?G \<infinity> x else if f x = -\<infinity> then ?G (-\<infinity>) x else ?G (ereal (real (f x))) x"
965   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
966   note * = this
967   from assms show ?thesis
968     by (subst *) (simp del: space_borel split del: split_if)
969 qed
971 lemma
972   fixes f g :: "'a \<Rightarrow> ereal"
973   assumes f: "f \<in> borel_measurable M"
974   assumes g: "g \<in> borel_measurable M"
975   shows borel_measurable_ereal_le[intro,simp,measurable(raw)]: "{x \<in> space M. f x \<le> g x} \<in> sets M"
976     and borel_measurable_ereal_less[intro,simp,measurable(raw)]: "{x \<in> space M. f x < g x} \<in> sets M"
977     and borel_measurable_ereal_eq[intro,simp,measurable(raw)]: "{w \<in> space M. f w = g w} \<in> sets M"
978     and borel_measurable_ereal_neq[intro,simp]: "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
979   using f g by (auto simp: f g set_Collect_ereal2[OF f g] intro!: sets_Collect_neg)
981 lemma borel_measurable_ereal_iff:
982   shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
983 proof
984   assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
985   from borel_measurable_real_of_ereal[OF this]
986   show "f \<in> borel_measurable M" by auto
987 qed auto
989 lemma borel_measurable_ereal_iff_real:
990   fixes f :: "'a \<Rightarrow> ereal"
991   shows "f \<in> borel_measurable M \<longleftrightarrow>
992     ((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)"
993 proof safe
994   assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M"
995   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
996   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
997   let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real (f x))"
998   have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
999   also have "?f = f" by (auto simp: fun_eq_iff ereal_real)
1000   finally show "f \<in> borel_measurable M" .
1001 qed simp_all
1003 lemma borel_measurable_eq_atMost_ereal:
1004   fixes f :: "'a \<Rightarrow> ereal"
1005   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
1006 proof (intro iffI allI)
1007   assume pos[rule_format]: "\<forall>a. f -` {..a} \<inter> space M \<in> sets M"
1008   show "f \<in> borel_measurable M"
1009     unfolding borel_measurable_ereal_iff_real borel_measurable_iff_le
1010   proof (intro conjI allI)
1011     fix a :: real
1012     { fix x :: ereal assume *: "\<forall>i::nat. real i < x"
1013       have "x = \<infinity>"
1014       proof (rule ereal_top)
1015         fix B from reals_Archimedean2[of B] guess n ..
1016         then have "ereal B < real n" by auto
1017         with * show "B \<le> x" by (metis less_trans less_imp_le)
1018       qed }
1019     then have "f -` {\<infinity>} \<inter> space M = space M - (\<Union>i::nat. f -` {.. real i} \<inter> space M)"
1020       by (auto simp: not_le)
1021     then show "f -` {\<infinity>} \<inter> space M \<in> sets M" using pos
1022       by (auto simp del: UN_simps)
1023     moreover
1024     have "{-\<infinity>::ereal} = {..-\<infinity>}" by auto
1025     then show "f -` {-\<infinity>} \<inter> space M \<in> sets M" using pos by auto
1026     moreover have "{x\<in>space M. f x \<le> ereal a} \<in> sets M"
1027       using pos[of "ereal a"] by (simp add: vimage_def Int_def conj_commute)
1028     moreover have "{w \<in> space M. real (f w) \<le> a} =
1029       (if a < 0 then {w \<in> space M. f w \<le> ereal a} - f -` {-\<infinity>} \<inter> space M
1030       else {w \<in> space M. f w \<le> ereal a} \<union> (f -` {\<infinity>} \<inter> space M) \<union> (f -` {-\<infinity>} \<inter> space M))" (is "?l = ?r")
1031       proof (intro set_eqI) fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases "f x") auto qed
1032     ultimately show "{w \<in> space M. real (f w) \<le> a} \<in> sets M" by auto
1033   qed
1034 qed (simp add: measurable_sets)
1036 lemma borel_measurable_eq_atLeast_ereal:
1037   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
1038 proof
1039   assume pos: "\<forall>a. f -` {a..} \<inter> space M \<in> sets M"
1040   moreover have "\<And>a. (\<lambda>x. - f x) -` {..a} = f -` {-a ..}"
1041     by (auto simp: ereal_uminus_le_reorder)
1042   ultimately have "(\<lambda>x. - f x) \<in> borel_measurable M"
1043     unfolding borel_measurable_eq_atMost_ereal by auto
1044   then show "f \<in> borel_measurable M" by simp
1045 qed (simp add: measurable_sets)
1047 lemma greater_eq_le_measurable:
1048   fixes f :: "'a \<Rightarrow> 'c::linorder"
1049   shows "f -` {..< a} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {a ..} \<inter> space M \<in> sets M"
1050 proof
1051   assume "f -` {a ..} \<inter> space M \<in> sets M"
1052   moreover have "f -` {..< a} \<inter> space M = space M - f -` {a ..} \<inter> space M" by auto
1053   ultimately show "f -` {..< a} \<inter> space M \<in> sets M" by auto
1054 next
1055   assume "f -` {..< a} \<inter> space M \<in> sets M"
1056   moreover have "f -` {a ..} \<inter> space M = space M - f -` {..< a} \<inter> space M" by auto
1057   ultimately show "f -` {a ..} \<inter> space M \<in> sets M" by auto
1058 qed
1060 lemma borel_measurable_ereal_iff_less:
1061   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
1062   unfolding borel_measurable_eq_atLeast_ereal greater_eq_le_measurable ..
1064 lemma less_eq_ge_measurable:
1065   fixes f :: "'a \<Rightarrow> 'c::linorder"
1066   shows "f -` {a <..} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {..a} \<inter> space M \<in> sets M"
1067 proof
1068   assume "f -` {a <..} \<inter> space M \<in> sets M"
1069   moreover have "f -` {..a} \<inter> space M = space M - f -` {a <..} \<inter> space M" by auto
1070   ultimately show "f -` {..a} \<inter> space M \<in> sets M" by auto
1071 next
1072   assume "f -` {..a} \<inter> space M \<in> sets M"
1073   moreover have "f -` {a <..} \<inter> space M = space M - f -` {..a} \<inter> space M" by auto
1074   ultimately show "f -` {a <..} \<inter> space M \<in> sets M" by auto
1075 qed
1077 lemma borel_measurable_ereal_iff_ge:
1078   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
1079   unfolding borel_measurable_eq_atMost_ereal less_eq_ge_measurable ..
1081 lemma borel_measurable_ereal2:
1082   fixes f g :: "'a \<Rightarrow> ereal"
1083   assumes f: "f \<in> borel_measurable M"
1084   assumes g: "g \<in> borel_measurable M"
1085   assumes H: "(\<lambda>x. H (ereal (real (f x))) (ereal (real (g x)))) \<in> borel_measurable M"
1086     "(\<lambda>x. H (-\<infinity>) (ereal (real (g x)))) \<in> borel_measurable M"
1087     "(\<lambda>x. H (\<infinity>) (ereal (real (g x)))) \<in> borel_measurable M"
1088     "(\<lambda>x. H (ereal (real (f x))) (-\<infinity>)) \<in> borel_measurable M"
1089     "(\<lambda>x. H (ereal (real (f x))) (\<infinity>)) \<in> borel_measurable M"
1090   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
1091 proof -
1092   let ?G = "\<lambda>y x. if g x = \<infinity> then H y \<infinity> else if g x = - \<infinity> then H y (-\<infinity>) else H y (ereal (real (g x)))"
1093   let ?F = "\<lambda>x. if f x = \<infinity> then ?G \<infinity> x else if f x = - \<infinity> then ?G (-\<infinity>) x else ?G (ereal (real (f x))) x"
1094   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
1095   note * = this
1096   from assms show ?thesis unfolding * by simp
1097 qed
1099 lemma
1100   fixes f :: "'a \<Rightarrow> ereal" assumes f: "f \<in> borel_measurable M"
1101   shows borel_measurable_ereal_eq_const: "{x\<in>space M. f x = c} \<in> sets M"
1102     and borel_measurable_ereal_neq_const: "{x\<in>space M. f x \<noteq> c} \<in> sets M"
1103   using f by auto
1105 lemma [intro, simp, measurable(raw)]:
1106   fixes f :: "'a \<Rightarrow> ereal"
1107   assumes [simp]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
1108   shows borel_measurable_ereal_add: "(\<lambda>x. f x + g x) \<in> borel_measurable M"
1109     and borel_measurable_ereal_times: "(\<lambda>x. f x * g x) \<in> borel_measurable M"
1110     and borel_measurable_ereal_min: "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
1111     and borel_measurable_ereal_max: "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
1112   by (auto simp add: borel_measurable_ereal2 measurable_If min_def max_def)
1114 lemma [simp, intro, measurable(raw)]:
1115   fixes f g :: "'a \<Rightarrow> ereal"
1116   assumes "f \<in> borel_measurable M"
1117   assumes "g \<in> borel_measurable M"
1118   shows borel_measurable_ereal_diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
1119     and borel_measurable_ereal_divide: "(\<lambda>x. f x / g x) \<in> borel_measurable M"
1120   unfolding minus_ereal_def divide_ereal_def using assms by auto
1122 lemma borel_measurable_ereal_setsum[simp, intro,measurable (raw)]:
1123   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
1124   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
1125   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
1126 proof cases
1127   assume "finite S"
1128   thus ?thesis using assms
1129     by induct auto
1130 qed simp
1132 lemma borel_measurable_ereal_setprod[simp, intro,measurable (raw)]:
1133   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
1134   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
1135   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
1136 proof cases
1137   assume "finite S"
1138   thus ?thesis using assms by induct auto
1139 qed simp
1141 lemma borel_measurable_SUP[simp, intro,measurable (raw)]:
1142   fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> ereal"
1143   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
1144   shows "(\<lambda>x. SUP i : A. f i x) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M")
1145   unfolding borel_measurable_ereal_iff_ge
1146 proof
1147   fix a
1148   have "?sup -` {a<..} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"
1149     by (auto simp: less_SUP_iff)
1150   then show "?sup -` {a<..} \<inter> space M \<in> sets M"
1151     using assms by auto
1152 qed
1154 lemma borel_measurable_INF[simp, intro,measurable (raw)]:
1155   fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> ereal"
1156   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
1157   shows "(\<lambda>x. INF i : A. f i x) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M")
1158   unfolding borel_measurable_ereal_iff_less
1159 proof
1160   fix a
1161   have "?inf -` {..<a} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"
1162     by (auto simp: INF_less_iff)
1163   then show "?inf -` {..<a} \<inter> space M \<in> sets M"
1164     using assms by auto
1165 qed
1167 lemma [simp, intro, measurable (raw)]:
1168   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
1169   assumes "\<And>i. f i \<in> borel_measurable M"
1170   shows borel_measurable_liminf: "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
1171     and borel_measurable_limsup: "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
1172   unfolding liminf_SUPR_INFI limsup_INFI_SUPR using assms by auto
1174 lemma borel_measurable_ereal_LIMSEQ:
1175   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
1176   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
1177   and u: "\<And>i. u i \<in> borel_measurable M"
1178   shows "u' \<in> borel_measurable M"
1179 proof -
1180   have "\<And>x. x \<in> space M \<Longrightarrow> u' x = liminf (\<lambda>n. u n x)"
1181     using u' by (simp add: lim_imp_Liminf[symmetric])
1182   then show ?thesis by (simp add: u cong: measurable_cong)
1183 qed
1185 lemma borel_measurable_psuminf[simp, intro, measurable (raw)]:
1186   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
1187   assumes "\<And>i. f i \<in> borel_measurable M" and pos: "\<And>i x. x \<in> space M \<Longrightarrow> 0 \<le> f i x"
1188   shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
1189   apply (subst measurable_cong)
1190   apply (subst suminf_ereal_eq_SUPR)
1191   apply (rule pos)
1192   using assms by auto
1194 section "LIMSEQ is borel measurable"
1196 lemma borel_measurable_LIMSEQ:
1197   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
1198   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
1199   and u: "\<And>i. u i \<in> borel_measurable M"
1200   shows "u' \<in> borel_measurable M"
1201 proof -
1202   have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)"
1203     using u' by (simp add: lim_imp_Liminf)
1204   moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M"
1205     by auto
1206   ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff)
1207 qed
1209 lemma sets_Collect_Cauchy[measurable]:
1210   fixes f :: "nat \<Rightarrow> 'a => real"
1211   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
1212   shows "{x\<in>space M. Cauchy (\<lambda>i. f i x)} \<in> sets M"
1213   unfolding Cauchy_iff2 using f by auto
1215 lemma borel_measurable_lim[measurable (raw)]:
1216   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real"
1217   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
1218   shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
1219 proof -
1220   def u' \<equiv> "\<lambda>x. lim (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)"
1221   then have *: "\<And>x. lim (\<lambda>i. f i x) = (if Cauchy (\<lambda>i. f i x) then u' x else (THE x. False))"
1222     by (auto simp: lim_def convergent_eq_cauchy[symmetric])
1223   have "u' \<in> borel_measurable M"
1224   proof (rule borel_measurable_LIMSEQ)
1225     fix x
1226     have "convergent (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)"
1227       by (cases "Cauchy (\<lambda>i. f i x)")
1228          (auto simp add: convergent_eq_cauchy[symmetric] convergent_def)
1229     then show "(\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0) ----> u' x"
1230       unfolding u'_def
1231       by (rule convergent_LIMSEQ_iff[THEN iffD1])
1232   qed measurable
1233   then show ?thesis
1234     unfolding * by measurable
1235 qed
1237 lemma borel_measurable_suminf[measurable (raw)]:
1238   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real"
1239   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
1240   shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M"
1241   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
1243 end