src/HOL/Probability/Borel_Space.thy
 author hoelzl Tue Mar 22 20:06:10 2011 +0100 (2011-03-22) changeset 42067 66c8281349ec parent 41981 cdf7693bbe08 child 42150 b0c0638c4aad permissions -rw-r--r--
```     1 (*  Title:      HOL/Probability/Lebesgue_Integration.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Armin Heller, TU München
```
```     4 *)
```
```     5
```
```     6 header {*Borel spaces*}
```
```     7
```
```     8 theory Borel_Space
```
```     9   imports Sigma_Algebra Multivariate_Analysis
```
```    10 begin
```
```    11
```
```    12 section "Generic Borel spaces"
```
```    13
```
```    14 definition "borel = sigma \<lparr> space = UNIV::'a::topological_space set, sets = open\<rparr>"
```
```    15 abbreviation "borel_measurable M \<equiv> measurable M borel"
```
```    16
```
```    17 interpretation borel: sigma_algebra borel
```
```    18   by (auto simp: borel_def intro!: sigma_algebra_sigma)
```
```    19
```
```    20 lemma in_borel_measurable:
```
```    21    "f \<in> borel_measurable M \<longleftrightarrow>
```
```    22     (\<forall>S \<in> sets (sigma \<lparr> space = UNIV, sets = open\<rparr>).
```
```    23       f -` S \<inter> space M \<in> sets M)"
```
```    24   by (auto simp add: measurable_def borel_def)
```
```    25
```
```    26 lemma in_borel_measurable_borel:
```
```    27    "f \<in> borel_measurable M \<longleftrightarrow>
```
```    28     (\<forall>S \<in> sets borel.
```
```    29       f -` S \<inter> space M \<in> sets M)"
```
```    30   by (auto simp add: measurable_def borel_def)
```
```    31
```
```    32 lemma space_borel[simp]: "space borel = UNIV"
```
```    33   unfolding borel_def by auto
```
```    34
```
```    35 lemma borel_open[simp]:
```
```    36   assumes "open A" shows "A \<in> sets borel"
```
```    37 proof -
```
```    38   have "A \<in> open" unfolding mem_def using assms .
```
```    39   thus ?thesis unfolding borel_def sigma_def by (auto intro!: sigma_sets.Basic)
```
```    40 qed
```
```    41
```
```    42 lemma borel_closed[simp]:
```
```    43   assumes "closed A" shows "A \<in> sets borel"
```
```    44 proof -
```
```    45   have "space borel - (- A) \<in> sets borel"
```
```    46     using assms unfolding closed_def by (blast intro: borel_open)
```
```    47   thus ?thesis by simp
```
```    48 qed
```
```    49
```
```    50 lemma borel_comp[intro,simp]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
```
```    51   unfolding Compl_eq_Diff_UNIV by (intro borel.Diff) auto
```
```    52
```
```    53 lemma (in sigma_algebra) borel_measurable_vimage:
```
```    54   fixes f :: "'a \<Rightarrow> 'x::t2_space"
```
```    55   assumes borel: "f \<in> borel_measurable M"
```
```    56   shows "f -` {x} \<inter> space M \<in> sets M"
```
```    57 proof (cases "x \<in> f ` space M")
```
```    58   case True then obtain y where "x = f y" by auto
```
```    59   from closed_singleton[of "f y"]
```
```    60   have "{f y} \<in> sets borel" by (rule borel_closed)
```
```    61   with assms show ?thesis
```
```    62     unfolding in_borel_measurable_borel `x = f y` by auto
```
```    63 next
```
```    64   case False hence "f -` {x} \<inter> space M = {}" by auto
```
```    65   thus ?thesis by auto
```
```    66 qed
```
```    67
```
```    68 lemma (in sigma_algebra) borel_measurableI:
```
```    69   fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space"
```
```    70   assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
```
```    71   shows "f \<in> borel_measurable M"
```
```    72   unfolding borel_def
```
```    73 proof (rule measurable_sigma, simp_all)
```
```    74   fix S :: "'x set" assume "S \<in> open" thus "f -` S \<inter> space M \<in> sets M"
```
```    75     using assms[of S] by (simp add: mem_def)
```
```    76 qed
```
```    77
```
```    78 lemma borel_singleton[simp, intro]:
```
```    79   fixes x :: "'a::t1_space"
```
```    80   shows "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets borel"
```
```    81   proof (rule borel.insert_in_sets)
```
```    82     show "{x} \<in> sets borel"
```
```    83       using closed_singleton[of x] by (rule borel_closed)
```
```    84   qed simp
```
```    85
```
```    86 lemma (in sigma_algebra) borel_measurable_const[simp, intro]:
```
```    87   "(\<lambda>x. c) \<in> borel_measurable M"
```
```    88   by (auto intro!: measurable_const)
```
```    89
```
```    90 lemma (in sigma_algebra) borel_measurable_indicator[simp, intro!]:
```
```    91   assumes A: "A \<in> sets M"
```
```    92   shows "indicator A \<in> borel_measurable M"
```
```    93   unfolding indicator_def_raw using A
```
```    94   by (auto intro!: measurable_If_set borel_measurable_const)
```
```    95
```
```    96 lemma (in sigma_algebra) borel_measurable_indicator_iff:
```
```    97   "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
```
```    98     (is "?I \<in> borel_measurable M \<longleftrightarrow> _")
```
```    99 proof
```
```   100   assume "?I \<in> borel_measurable M"
```
```   101   then have "?I -` {1} \<inter> space M \<in> sets M"
```
```   102     unfolding measurable_def by auto
```
```   103   also have "?I -` {1} \<inter> space M = A \<inter> space M"
```
```   104     unfolding indicator_def_raw by auto
```
```   105   finally show "A \<inter> space M \<in> sets M" .
```
```   106 next
```
```   107   assume "A \<inter> space M \<in> sets M"
```
```   108   moreover have "?I \<in> borel_measurable M \<longleftrightarrow>
```
```   109     (indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"
```
```   110     by (intro measurable_cong) (auto simp: indicator_def)
```
```   111   ultimately show "?I \<in> borel_measurable M" by auto
```
```   112 qed
```
```   113
```
```   114 lemma (in sigma_algebra) borel_measurable_restricted:
```
```   115   fixes f :: "'a \<Rightarrow> extreal" assumes "A \<in> sets M"
```
```   116   shows "f \<in> borel_measurable (restricted_space A) \<longleftrightarrow>
```
```   117     (\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
```
```   118     (is "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable M")
```
```   119 proof -
```
```   120   interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
```
```   121   have *: "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable ?R"
```
```   122     by (auto intro!: measurable_cong)
```
```   123   show ?thesis unfolding *
```
```   124     unfolding in_borel_measurable_borel
```
```   125   proof (simp, safe)
```
```   126     fix S :: "extreal set" assume "S \<in> sets borel"
```
```   127       "\<forall>S\<in>sets borel. ?f -` S \<inter> A \<in> op \<inter> A ` sets M"
```
```   128     then have "?f -` S \<inter> A \<in> op \<inter> A ` sets M" by auto
```
```   129     then have f: "?f -` S \<inter> A \<in> sets M"
```
```   130       using `A \<in> sets M` sets_into_space by fastsimp
```
```   131     show "?f -` S \<inter> space M \<in> sets M"
```
```   132     proof cases
```
```   133       assume "0 \<in> S"
```
```   134       then have "?f -` S \<inter> space M = ?f -` S \<inter> A \<union> (space M - A)"
```
```   135         using `A \<in> sets M` sets_into_space by auto
```
```   136       then show ?thesis using f `A \<in> sets M` by (auto intro!: Un Diff)
```
```   137     next
```
```   138       assume "0 \<notin> S"
```
```   139       then have "?f -` S \<inter> space M = ?f -` S \<inter> A"
```
```   140         using `A \<in> sets M` sets_into_space
```
```   141         by (auto simp: indicator_def split: split_if_asm)
```
```   142       then show ?thesis using f by auto
```
```   143     qed
```
```   144   next
```
```   145     fix S :: "extreal set" assume "S \<in> sets borel"
```
```   146       "\<forall>S\<in>sets borel. ?f -` S \<inter> space M \<in> sets M"
```
```   147     then have f: "?f -` S \<inter> space M \<in> sets M" by auto
```
```   148     then show "?f -` S \<inter> A \<in> op \<inter> A ` sets M"
```
```   149       using `A \<in> sets M` sets_into_space
```
```   150       apply (simp add: image_iff)
```
```   151       apply (rule bexI[OF _ f])
```
```   152       by auto
```
```   153   qed
```
```   154 qed
```
```   155
```
```   156 lemma (in sigma_algebra) borel_measurable_subalgebra:
```
```   157   assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
```
```   158   shows "f \<in> borel_measurable M"
```
```   159   using assms unfolding measurable_def by auto
```
```   160
```
```   161 section "Borel spaces on euclidean spaces"
```
```   162
```
```   163 lemma lessThan_borel[simp, intro]:
```
```   164   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   165   shows "{..< a} \<in> sets borel"
```
```   166   by (blast intro: borel_open)
```
```   167
```
```   168 lemma greaterThan_borel[simp, intro]:
```
```   169   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   170   shows "{a <..} \<in> sets borel"
```
```   171   by (blast intro: borel_open)
```
```   172
```
```   173 lemma greaterThanLessThan_borel[simp, intro]:
```
```   174   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   175   shows "{a<..<b} \<in> sets borel"
```
```   176   by (blast intro: borel_open)
```
```   177
```
```   178 lemma atMost_borel[simp, intro]:
```
```   179   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   180   shows "{..a} \<in> sets borel"
```
```   181   by (blast intro: borel_closed)
```
```   182
```
```   183 lemma atLeast_borel[simp, intro]:
```
```   184   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   185   shows "{a..} \<in> sets borel"
```
```   186   by (blast intro: borel_closed)
```
```   187
```
```   188 lemma atLeastAtMost_borel[simp, intro]:
```
```   189   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   190   shows "{a..b} \<in> sets borel"
```
```   191   by (blast intro: borel_closed)
```
```   192
```
```   193 lemma greaterThanAtMost_borel[simp, intro]:
```
```   194   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   195   shows "{a<..b} \<in> sets borel"
```
```   196   unfolding greaterThanAtMost_def by blast
```
```   197
```
```   198 lemma atLeastLessThan_borel[simp, intro]:
```
```   199   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   200   shows "{a..<b} \<in> sets borel"
```
```   201   unfolding atLeastLessThan_def by blast
```
```   202
```
```   203 lemma hafspace_less_borel[simp, intro]:
```
```   204   fixes a :: real
```
```   205   shows "{x::'a::euclidean_space. a < x \$\$ i} \<in> sets borel"
```
```   206   by (auto intro!: borel_open open_halfspace_component_gt)
```
```   207
```
```   208 lemma hafspace_greater_borel[simp, intro]:
```
```   209   fixes a :: real
```
```   210   shows "{x::'a::euclidean_space. x \$\$ i < a} \<in> sets borel"
```
```   211   by (auto intro!: borel_open open_halfspace_component_lt)
```
```   212
```
```   213 lemma hafspace_less_eq_borel[simp, intro]:
```
```   214   fixes a :: real
```
```   215   shows "{x::'a::euclidean_space. a \<le> x \$\$ i} \<in> sets borel"
```
```   216   by (auto intro!: borel_closed closed_halfspace_component_ge)
```
```   217
```
```   218 lemma hafspace_greater_eq_borel[simp, intro]:
```
```   219   fixes a :: real
```
```   220   shows "{x::'a::euclidean_space. x \$\$ i \<le> a} \<in> sets borel"
```
```   221   by (auto intro!: borel_closed closed_halfspace_component_le)
```
```   222
```
```   223 lemma (in sigma_algebra) borel_measurable_less[simp, intro]:
```
```   224   fixes f :: "'a \<Rightarrow> real"
```
```   225   assumes f: "f \<in> borel_measurable M"
```
```   226   assumes g: "g \<in> borel_measurable M"
```
```   227   shows "{w \<in> space M. f w < g w} \<in> sets M"
```
```   228 proof -
```
```   229   have "{w \<in> space M. f w < g w} =
```
```   230         (\<Union>r. (f -` {..< of_rat r} \<inter> space M) \<inter> (g -` {of_rat r <..} \<inter> space M))"
```
```   231     using Rats_dense_in_real by (auto simp add: Rats_def)
```
```   232   then show ?thesis using f g
```
```   233     by simp (blast intro: measurable_sets)
```
```   234 qed
```
```   235
```
```   236 lemma (in sigma_algebra) borel_measurable_le[simp, intro]:
```
```   237   fixes f :: "'a \<Rightarrow> real"
```
```   238   assumes f: "f \<in> borel_measurable M"
```
```   239   assumes g: "g \<in> borel_measurable M"
```
```   240   shows "{w \<in> space M. f w \<le> g w} \<in> sets M"
```
```   241 proof -
```
```   242   have "{w \<in> space M. f w \<le> g w} = space M - {w \<in> space M. g w < f w}"
```
```   243     by auto
```
```   244   thus ?thesis using f g
```
```   245     by simp blast
```
```   246 qed
```
```   247
```
```   248 lemma (in sigma_algebra) borel_measurable_eq[simp, intro]:
```
```   249   fixes f :: "'a \<Rightarrow> real"
```
```   250   assumes f: "f \<in> borel_measurable M"
```
```   251   assumes g: "g \<in> borel_measurable M"
```
```   252   shows "{w \<in> space M. f w = g w} \<in> sets M"
```
```   253 proof -
```
```   254   have "{w \<in> space M. f w = g w} =
```
```   255         {w \<in> space M. f w \<le> g w} \<inter> {w \<in> space M. g w \<le> f w}"
```
```   256     by auto
```
```   257   thus ?thesis using f g by auto
```
```   258 qed
```
```   259
```
```   260 lemma (in sigma_algebra) borel_measurable_neq[simp, intro]:
```
```   261   fixes f :: "'a \<Rightarrow> real"
```
```   262   assumes f: "f \<in> borel_measurable M"
```
```   263   assumes g: "g \<in> borel_measurable M"
```
```   264   shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
```
```   265 proof -
```
```   266   have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}"
```
```   267     by auto
```
```   268   thus ?thesis using f g by auto
```
```   269 qed
```
```   270
```
```   271 subsection "Borel space equals sigma algebras over intervals"
```
```   272
```
```   273 lemma rational_boxes:
```
```   274   fixes x :: "'a\<Colon>ordered_euclidean_space"
```
```   275   assumes "0 < e"
```
```   276   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"
```
```   277 proof -
```
```   278   def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
```
```   279   then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos)
```
```   280   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \$\$ i \<and> x \$\$ i - y < e'" (is "\<forall>i. ?th i")
```
```   281   proof
```
```   282     fix i from Rats_dense_in_real[of "x \$\$ i - e'" "x \$\$ i"] e
```
```   283     show "?th i" by auto
```
```   284   qed
```
```   285   from choice[OF this] guess a .. note a = this
```
```   286   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \$\$ i < y \<and> y - x \$\$ i < e'" (is "\<forall>i. ?th i")
```
```   287   proof
```
```   288     fix i from Rats_dense_in_real[of "x \$\$ i" "x \$\$ i + e'"] e
```
```   289     show "?th i" by auto
```
```   290   qed
```
```   291   from choice[OF this] guess b .. note b = this
```
```   292   { fix y :: 'a assume *: "Chi a < y" "y < Chi b"
```
```   293     have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x \$\$ i) (y \$\$ i))\<twosuperior>)"
```
```   294       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
```
```   295     also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))"
```
```   296     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
```
```   297       fix i assume i: "i \<in> {..<DIM('a)}"
```
```   298       have "a i < y\$\$i \<and> y\$\$i < b i" using * i eucl_less[where 'a='a] by auto
```
```   299       moreover have "a i < x\$\$i" "x\$\$i - a i < e'" using a by auto
```
```   300       moreover have "x\$\$i < b i" "b i - x\$\$i < e'" using b by auto
```
```   301       ultimately have "\<bar>x\$\$i - y\$\$i\<bar> < 2 * e'" by auto
```
```   302       then have "dist (x \$\$ i) (y \$\$ i) < e/sqrt (real (DIM('a)))"
```
```   303         unfolding e'_def by (auto simp: dist_real_def)
```
```   304       then have "(dist (x \$\$ i) (y \$\$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
```
```   305         by (rule power_strict_mono) auto
```
```   306       then show "(dist (x \$\$ i) (y \$\$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
```
```   307         by (simp add: power_divide)
```
```   308     qed auto
```
```   309     also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive)
```
```   310     finally have "dist x y < e" . }
```
```   311   with a b show ?thesis
```
```   312     apply (rule_tac exI[of _ "Chi a"])
```
```   313     apply (rule_tac exI[of _ "Chi b"])
```
```   314     using eucl_less[where 'a='a] by auto
```
```   315 qed
```
```   316
```
```   317 lemma ex_rat_list:
```
```   318   fixes x :: "'a\<Colon>ordered_euclidean_space"
```
```   319   assumes "\<And> i. x \$\$ i \<in> \<rat>"
```
```   320   shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x \$\$ i)"
```
```   321 proof -
```
```   322   have "\<forall>i. \<exists>r. x \$\$ i = of_rat r" using assms unfolding Rats_def by blast
```
```   323   from choice[OF this] guess r ..
```
```   324   then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"])
```
```   325 qed
```
```   326
```
```   327 lemma open_UNION:
```
```   328   fixes M :: "'a\<Colon>ordered_euclidean_space set"
```
```   329   assumes "open M"
```
```   330   shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M}
```
```   331                    (\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})"
```
```   332     (is "M = UNION ?idx ?box")
```
```   333 proof safe
```
```   334   fix x assume "x \<in> M"
```
```   335   obtain e where e: "e > 0" "ball x e \<subseteq> M"
```
```   336     using openE[OF assms `x \<in> M`] by auto
```
```   337   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"
```
```   338     using rational_boxes[OF e(1)] by blast
```
```   339   then obtain p q where pq: "length p = DIM ('a)"
```
```   340                             "length q = DIM ('a)"
```
```   341                             "\<forall> i < DIM ('a). of_rat (p ! i) = a \$\$ i \<and> of_rat (q ! i) = b \$\$ i"
```
```   342     using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast
```
```   343   hence p: "Chi (of_rat \<circ> op ! p) = a"
```
```   344     using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a]
```
```   345     unfolding o_def by auto
```
```   346   from pq have q: "Chi (of_rat \<circ> op ! q) = b"
```
```   347     using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b]
```
```   348     unfolding o_def by auto
```
```   349   have "x \<in> ?box (p, q)"
```
```   350     using p q ab by auto
```
```   351   thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto
```
```   352 qed auto
```
```   353
```
```   354 lemma halfspace_span_open:
```
```   355   "sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a}))
```
```   356     \<subseteq> sets borel"
```
```   357   by (auto intro!: borel.sigma_sets_subset[simplified] borel_open
```
```   358                    open_halfspace_component_lt)
```
```   359
```
```   360 lemma halfspace_lt_in_halfspace:
```
```   361   "{x\<Colon>'a. x \$\$ i < a} \<in> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a})\<rparr>)"
```
```   362   by (auto intro!: sigma_sets.Basic simp: sets_sigma)
```
```   363
```
```   364 lemma halfspace_gt_in_halfspace:
```
```   365   "{x\<Colon>'a. a < x \$\$ i} \<in> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a})\<rparr>)"
```
```   366   (is "?set \<in> sets ?SIGMA")
```
```   367 proof -
```
```   368   interpret sigma_algebra "?SIGMA"
```
```   369     by (intro sigma_algebra_sigma_sets) (simp_all add: sets_sigma)
```
```   370   have *: "?set = (\<Union>n. space ?SIGMA - {x\<Colon>'a. x \$\$ i < a + 1 / real (Suc n)})"
```
```   371   proof (safe, simp_all add: not_less)
```
```   372     fix x assume "a < x \$\$ i"
```
```   373     with reals_Archimedean[of "x \$\$ i - a"]
```
```   374     obtain n where "a + 1 / real (Suc n) < x \$\$ i"
```
```   375       by (auto simp: inverse_eq_divide field_simps)
```
```   376     then show "\<exists>n. a + 1 / real (Suc n) \<le> x \$\$ i"
```
```   377       by (blast intro: less_imp_le)
```
```   378   next
```
```   379     fix x n
```
```   380     have "a < a + 1 / real (Suc n)" by auto
```
```   381     also assume "\<dots> \<le> x"
```
```   382     finally show "a < x" .
```
```   383   qed
```
```   384   show "?set \<in> sets ?SIGMA" unfolding *
```
```   385     by (safe intro!: countable_UN Diff halfspace_lt_in_halfspace)
```
```   386 qed
```
```   387
```
```   388 lemma open_span_halfspace:
```
```   389   "sets borel \<subseteq> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x \$\$ i < a})\<rparr>)"
```
```   390     (is "_ \<subseteq> sets ?SIGMA")
```
```   391 proof -
```
```   392   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) simp
```
```   393   then interpret sigma_algebra ?SIGMA .
```
```   394   { fix S :: "'a set" assume "S \<in> open" then have "open S" unfolding mem_def .
```
```   395     from open_UNION[OF this]
```
```   396     obtain I where *: "S =
```
```   397       (\<Union>(a, b)\<in>I.
```
```   398           (\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) \$\$ i < x \$\$ i}) \<inter>
```
```   399           (\<Inter> i<DIM('a). {x. x \$\$ i < (Chi (real_of_rat \<circ> op ! b)::'a) \$\$ i}))"
```
```   400       unfolding greaterThanLessThan_def
```
```   401       unfolding eucl_greaterThan_eq_halfspaces[where 'a='a]
```
```   402       unfolding eucl_lessThan_eq_halfspaces[where 'a='a]
```
```   403       by blast
```
```   404     have "S \<in> sets ?SIGMA"
```
```   405       unfolding *
```
```   406       by (auto intro!: countable_UN Int countable_INT halfspace_lt_in_halfspace halfspace_gt_in_halfspace) }
```
```   407   then show ?thesis unfolding borel_def
```
```   408     by (intro sets_sigma_subset) auto
```
```   409 qed
```
```   410
```
```   411 lemma halfspace_span_halfspace_le:
```
```   412   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a})\<rparr>) \<subseteq>
```
```   413    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. x \$\$ i \<le> a})\<rparr>)"
```
```   414   (is "_ \<subseteq> sets ?SIGMA")
```
```   415 proof -
```
```   416   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   417   then interpret sigma_algebra ?SIGMA .
```
```   418   { fix a i
```
```   419     have *: "{x::'a. x\$\$i < a} = (\<Union>n. {x. x\$\$i \<le> a - 1/real (Suc n)})"
```
```   420     proof (safe, simp_all)
```
```   421       fix x::'a assume *: "x\$\$i < a"
```
```   422       with reals_Archimedean[of "a - x\$\$i"]
```
```   423       obtain n where "x \$\$ i < a - 1 / (real (Suc n))"
```
```   424         by (auto simp: field_simps inverse_eq_divide)
```
```   425       then show "\<exists>n. x \$\$ i \<le> a - 1 / (real (Suc n))"
```
```   426         by (blast intro: less_imp_le)
```
```   427     next
```
```   428       fix x::'a and n
```
```   429       assume "x\$\$i \<le> a - 1 / real (Suc n)"
```
```   430       also have "\<dots> < a" by auto
```
```   431       finally show "x\$\$i < a" .
```
```   432     qed
```
```   433     have "{x. x\$\$i < a} \<in> sets ?SIGMA" unfolding *
```
```   434       by (safe intro!: countable_UN)
```
```   435          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   436   then show ?thesis by (intro sets_sigma_subset) auto
```
```   437 qed
```
```   438
```
```   439 lemma halfspace_span_halfspace_ge:
```
```   440   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a})\<rparr>) \<subseteq>
```
```   441    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a \<le> x \$\$ i})\<rparr>)"
```
```   442   (is "_ \<subseteq> sets ?SIGMA")
```
```   443 proof -
```
```   444   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   445   then interpret sigma_algebra ?SIGMA .
```
```   446   { fix a i have *: "{x::'a. x\$\$i < a} = space ?SIGMA - {x::'a. a \<le> x\$\$i}" by auto
```
```   447     have "{x. x\$\$i < a} \<in> sets ?SIGMA" unfolding *
```
```   448       by (safe intro!: Diff)
```
```   449          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   450   then show ?thesis by (intro sets_sigma_subset) auto
```
```   451 qed
```
```   452
```
```   453 lemma halfspace_le_span_halfspace_gt:
```
```   454   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
```
```   455    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a < x \$\$ i})\<rparr>)"
```
```   456   (is "_ \<subseteq> sets ?SIGMA")
```
```   457 proof -
```
```   458   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   459   then interpret sigma_algebra ?SIGMA .
```
```   460   { fix a i have *: "{x::'a. x\$\$i \<le> a} = space ?SIGMA - {x::'a. a < x\$\$i}" by auto
```
```   461     have "{x. x\$\$i \<le> a} \<in> sets ?SIGMA" unfolding *
```
```   462       by (safe intro!: Diff)
```
```   463          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   464   then show ?thesis by (intro sets_sigma_subset) auto
```
```   465 qed
```
```   466
```
```   467 lemma halfspace_le_span_atMost:
```
```   468   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
```
```   469    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>)"
```
```   470   (is "_ \<subseteq> sets ?SIGMA")
```
```   471 proof -
```
```   472   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   473   then interpret sigma_algebra ?SIGMA .
```
```   474   have "\<And>a i. {x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   475   proof cases
```
```   476     fix a i assume "i < DIM('a)"
```
```   477     then have *: "{x::'a. x\$\$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})"
```
```   478     proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm)
```
```   479       fix x
```
```   480       from real_arch_simple[of "Max ((\<lambda>i. x\$\$i)`{..<DIM('a)})"] guess k::nat ..
```
```   481       then have "\<And>i. i < DIM('a) \<Longrightarrow> x\$\$i \<le> real k"
```
```   482         by (subst (asm) Max_le_iff) auto
```
```   483       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x \$\$ ia \<le> real k"
```
```   484         by (auto intro!: exI[of _ k])
```
```   485     qed
```
```   486     show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA" unfolding *
```
```   487       by (safe intro!: countable_UN)
```
```   488          (auto simp: sets_sigma intro!: sigma_sets.Basic)
```
```   489   next
```
```   490     fix a i assume "\<not> i < DIM('a)"
```
```   491     then show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   492       using top by auto
```
```   493   qed
```
```   494   then show ?thesis by (intro sets_sigma_subset) auto
```
```   495 qed
```
```   496
```
```   497 lemma halfspace_le_span_greaterThan:
```
```   498   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
```
```   499    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {a<..})\<rparr>)"
```
```   500   (is "_ \<subseteq> sets ?SIGMA")
```
```   501 proof -
```
```   502   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   503   then interpret sigma_algebra ?SIGMA .
```
```   504   have "\<And>a i. {x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   505   proof cases
```
```   506     fix a i assume "i < DIM('a)"
```
```   507     have "{x::'a. x\$\$i \<le> a} = space ?SIGMA - {x::'a. a < x\$\$i}" by auto
```
```   508     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)`
```
```   509     proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
```
```   510       fix x
```
```   511       from real_arch_lt[of "Max ((\<lambda>i. -x\$\$i)`{..<DIM('a)})"]
```
```   512       guess k::nat .. note k = this
```
```   513       { fix i assume "i < DIM('a)"
```
```   514         then have "-x\$\$i < real k"
```
```   515           using k by (subst (asm) Max_less_iff) auto
```
```   516         then have "- real k < x\$\$i" by simp }
```
```   517       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x \$\$ ia"
```
```   518         by (auto intro!: exI[of _ k])
```
```   519     qed
```
```   520     finally show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   521       apply (simp only:)
```
```   522       apply (safe intro!: countable_UN Diff)
```
```   523       by (auto simp: sets_sigma intro!: sigma_sets.Basic)
```
```   524   next
```
```   525     fix a i assume "\<not> i < DIM('a)"
```
```   526     then show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   527       using top by auto
```
```   528   qed
```
```   529   then show ?thesis by (intro sets_sigma_subset) auto
```
```   530 qed
```
```   531
```
```   532 lemma halfspace_le_span_lessThan:
```
```   533   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x \$\$ i})\<rparr>) \<subseteq>
```
```   534    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..<a})\<rparr>)"
```
```   535   (is "_ \<subseteq> sets ?SIGMA")
```
```   536 proof -
```
```   537   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   538   then interpret sigma_algebra ?SIGMA .
```
```   539   have "\<And>a i. {x. a \<le> x\$\$i} \<in> sets ?SIGMA"
```
```   540   proof cases
```
```   541     fix a i assume "i < DIM('a)"
```
```   542     have "{x::'a. a \<le> x\$\$i} = space ?SIGMA - {x::'a. x\$\$i < a}" by auto
```
```   543     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)`
```
```   544     proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
```
```   545       fix x
```
```   546       from real_arch_lt[of "Max ((\<lambda>i. x\$\$i)`{..<DIM('a)})"]
```
```   547       guess k::nat .. note k = this
```
```   548       { fix i assume "i < DIM('a)"
```
```   549         then have "x\$\$i < real k"
```
```   550           using k by (subst (asm) Max_less_iff) auto
```
```   551         then have "x\$\$i < real k" by simp }
```
```   552       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x \$\$ ia < real k"
```
```   553         by (auto intro!: exI[of _ k])
```
```   554     qed
```
```   555     finally show "{x. a \<le> x\$\$i} \<in> sets ?SIGMA"
```
```   556       apply (simp only:)
```
```   557       apply (safe intro!: countable_UN Diff)
```
```   558       by (auto simp: sets_sigma intro!: sigma_sets.Basic)
```
```   559   next
```
```   560     fix a i assume "\<not> i < DIM('a)"
```
```   561     then show "{x. a \<le> x\$\$i} \<in> sets ?SIGMA"
```
```   562       using top by auto
```
```   563   qed
```
```   564   then show ?thesis by (intro sets_sigma_subset) auto
```
```   565 qed
```
```   566
```
```   567 lemma atMost_span_atLeastAtMost:
```
```   568   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>) \<subseteq>
```
```   569    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a,b). {a..b})\<rparr>)"
```
```   570   (is "_ \<subseteq> sets ?SIGMA")
```
```   571 proof -
```
```   572   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   573   then interpret sigma_algebra ?SIGMA .
```
```   574   { fix a::'a
```
```   575     have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
```
```   576     proof (safe, simp_all add: eucl_le[where 'a='a])
```
```   577       fix x
```
```   578       from real_arch_simple[of "Max ((\<lambda>i. - x\$\$i)`{..<DIM('a)})"]
```
```   579       guess k::nat .. note k = this
```
```   580       { fix i assume "i < DIM('a)"
```
```   581         with k have "- x\$\$i \<le> real k"
```
```   582           by (subst (asm) Max_le_iff) (auto simp: field_simps)
```
```   583         then have "- real k \<le> x\$\$i" by simp }
```
```   584       then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x \$\$ i"
```
```   585         by (auto intro!: exI[of _ k])
```
```   586     qed
```
```   587     have "{..a} \<in> sets ?SIGMA" unfolding *
```
```   588       by (safe intro!: countable_UN)
```
```   589          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   590   then show ?thesis by (intro sets_sigma_subset) auto
```
```   591 qed
```
```   592
```
```   593 lemma borel_eq_atMost:
```
```   594   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> a. {.. a::'a\<Colon>ordered_euclidean_space})\<rparr>)"
```
```   595     (is "_ = ?SIGMA")
```
```   596 proof (intro algebra.equality antisym)
```
```   597   show "sets borel \<subseteq> sets ?SIGMA"
```
```   598     using halfspace_le_span_atMost halfspace_span_halfspace_le open_span_halfspace
```
```   599     by auto
```
```   600   show "sets ?SIGMA \<subseteq> sets borel"
```
```   601     by (rule borel.sets_sigma_subset) auto
```
```   602 qed auto
```
```   603
```
```   604 lemma borel_eq_atLeastAtMost:
```
```   605   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space, b). {a .. b})\<rparr>)"
```
```   606    (is "_ = ?SIGMA")
```
```   607 proof (intro algebra.equality antisym)
```
```   608   show "sets borel \<subseteq> sets ?SIGMA"
```
```   609     using atMost_span_atLeastAtMost halfspace_le_span_atMost
```
```   610       halfspace_span_halfspace_le open_span_halfspace
```
```   611     by auto
```
```   612   show "sets ?SIGMA \<subseteq> sets borel"
```
```   613     by (rule borel.sets_sigma_subset) auto
```
```   614 qed auto
```
```   615
```
```   616 lemma borel_eq_greaterThan:
```
```   617   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {a <..})\<rparr>)"
```
```   618    (is "_ = ?SIGMA")
```
```   619 proof (intro algebra.equality antisym)
```
```   620   show "sets borel \<subseteq> sets ?SIGMA"
```
```   621     using halfspace_le_span_greaterThan
```
```   622       halfspace_span_halfspace_le open_span_halfspace
```
```   623     by auto
```
```   624   show "sets ?SIGMA \<subseteq> sets borel"
```
```   625     by (rule borel.sets_sigma_subset) auto
```
```   626 qed auto
```
```   627
```
```   628 lemma borel_eq_lessThan:
```
```   629   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {..< a})\<rparr>)"
```
```   630    (is "_ = ?SIGMA")
```
```   631 proof (intro algebra.equality antisym)
```
```   632   show "sets borel \<subseteq> sets ?SIGMA"
```
```   633     using halfspace_le_span_lessThan
```
```   634       halfspace_span_halfspace_ge open_span_halfspace
```
```   635     by auto
```
```   636   show "sets ?SIGMA \<subseteq> sets borel"
```
```   637     by (rule borel.sets_sigma_subset) auto
```
```   638 qed auto
```
```   639
```
```   640 lemma borel_eq_greaterThanLessThan:
```
```   641   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, b). {a <..< (b :: 'a \<Colon> ordered_euclidean_space)})\<rparr>)"
```
```   642     (is "_ = ?SIGMA")
```
```   643 proof (intro algebra.equality antisym)
```
```   644   show "sets ?SIGMA \<subseteq> sets borel"
```
```   645     by (rule borel.sets_sigma_subset) auto
```
```   646   show "sets borel \<subseteq> sets ?SIGMA"
```
```   647   proof -
```
```   648     have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   649     then interpret sigma_algebra ?SIGMA .
```
```   650     { fix M :: "'a set" assume "M \<in> open"
```
```   651       then have "open M" by (simp add: mem_def)
```
```   652       have "M \<in> sets ?SIGMA"
```
```   653         apply (subst open_UNION[OF `open M`])
```
```   654         apply (safe intro!: countable_UN)
```
```   655         by (auto simp add: sigma_def intro!: sigma_sets.Basic) }
```
```   656     then show ?thesis
```
```   657       unfolding borel_def by (intro sets_sigma_subset) auto
```
```   658   qed
```
```   659 qed auto
```
```   660
```
```   661 lemma borel_eq_halfspace_le:
```
```   662   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x\$\$i \<le> a})\<rparr>)"
```
```   663    (is "_ = ?SIGMA")
```
```   664 proof (intro algebra.equality antisym)
```
```   665   show "sets borel \<subseteq> sets ?SIGMA"
```
```   666     using open_span_halfspace halfspace_span_halfspace_le by auto
```
```   667   show "sets ?SIGMA \<subseteq> sets borel"
```
```   668     by (rule borel.sets_sigma_subset) auto
```
```   669 qed auto
```
```   670
```
```   671 lemma borel_eq_halfspace_less:
```
```   672   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x\$\$i < a})\<rparr>)"
```
```   673    (is "_ = ?SIGMA")
```
```   674 proof (intro algebra.equality antisym)
```
```   675   show "sets borel \<subseteq> sets ?SIGMA"
```
```   676     using open_span_halfspace .
```
```   677   show "sets ?SIGMA \<subseteq> sets borel"
```
```   678     by (rule borel.sets_sigma_subset) auto
```
```   679 qed auto
```
```   680
```
```   681 lemma borel_eq_halfspace_gt:
```
```   682   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a < x\$\$i})\<rparr>)"
```
```   683    (is "_ = ?SIGMA")
```
```   684 proof (intro algebra.equality antisym)
```
```   685   show "sets borel \<subseteq> sets ?SIGMA"
```
```   686     using halfspace_le_span_halfspace_gt open_span_halfspace halfspace_span_halfspace_le by auto
```
```   687   show "sets ?SIGMA \<subseteq> sets borel"
```
```   688     by (rule borel.sets_sigma_subset) auto
```
```   689 qed auto
```
```   690
```
```   691 lemma borel_eq_halfspace_ge:
```
```   692   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a \<le> x\$\$i})\<rparr>)"
```
```   693    (is "_ = ?SIGMA")
```
```   694 proof (intro algebra.equality antisym)
```
```   695   show "sets borel \<subseteq> sets ?SIGMA"
```
```   696     using halfspace_span_halfspace_ge open_span_halfspace by auto
```
```   697   show "sets ?SIGMA \<subseteq> sets borel"
```
```   698     by (rule borel.sets_sigma_subset) auto
```
```   699 qed auto
```
```   700
```
```   701 lemma (in sigma_algebra) borel_measurable_halfspacesI:
```
```   702   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   703   assumes "borel = (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
```
```   704   and "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
```
```   705   and "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M"
```
```   706   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)"
```
```   707 proof safe
```
```   708   fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M"
```
```   709   then show "S a i \<in> sets M" unfolding assms
```
```   710     by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1) sigma_def)
```
```   711 next
```
```   712   assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M"
```
```   713   { fix a i have "S a i \<in> sets M"
```
```   714     proof cases
```
```   715       assume "i < DIM('c)"
```
```   716       with a show ?thesis unfolding assms(2) by simp
```
```   717     next
```
```   718       assume "\<not> i < DIM('c)"
```
```   719       from assms(3)[OF this] show ?thesis .
```
```   720     qed }
```
```   721   then have "f \<in> measurable M (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
```
```   722     by (auto intro!: measurable_sigma simp: assms(2))
```
```   723   then show "f \<in> borel_measurable M" unfolding measurable_def
```
```   724     unfolding assms(1) by simp
```
```   725 qed
```
```   726
```
```   727 lemma (in sigma_algebra) borel_measurable_iff_halfspace_le:
```
```   728   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   729   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w \$\$ i \<le> a} \<in> sets M)"
```
```   730   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
```
```   731
```
```   732 lemma (in sigma_algebra) borel_measurable_iff_halfspace_less:
```
```   733   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   734   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w \$\$ i < a} \<in> sets M)"
```
```   735   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
```
```   736
```
```   737 lemma (in sigma_algebra) borel_measurable_iff_halfspace_ge:
```
```   738   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   739   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w \$\$ i} \<in> sets M)"
```
```   740   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
```
```   741
```
```   742 lemma (in sigma_algebra) borel_measurable_iff_halfspace_greater:
```
```   743   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   744   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w \$\$ i} \<in> sets M)"
```
```   745   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_gt]) auto
```
```   746
```
```   747 lemma (in sigma_algebra) borel_measurable_iff_le:
```
```   748   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
```
```   749   using borel_measurable_iff_halfspace_le[where 'c=real] by simp
```
```   750
```
```   751 lemma (in sigma_algebra) borel_measurable_iff_less:
```
```   752   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
```
```   753   using borel_measurable_iff_halfspace_less[where 'c=real] by simp
```
```   754
```
```   755 lemma (in sigma_algebra) borel_measurable_iff_ge:
```
```   756   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
```
```   757   using borel_measurable_iff_halfspace_ge[where 'c=real] by simp
```
```   758
```
```   759 lemma (in sigma_algebra) borel_measurable_iff_greater:
```
```   760   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
```
```   761   using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
```
```   762
```
```   763 lemma borel_measurable_euclidean_component:
```
```   764   "(\<lambda>x::'a::euclidean_space. x \$\$ i) \<in> borel_measurable borel"
```
```   765   unfolding borel_def[where 'a=real]
```
```   766 proof (rule borel.measurable_sigma, simp_all)
```
```   767   fix S::"real set" assume "S \<in> open" then have "open S" unfolding mem_def .
```
```   768   from open_vimage_euclidean_component[OF this]
```
```   769   show "(\<lambda>x. x \$\$ i) -` S \<in> sets borel"
```
```   770     by (auto intro: borel_open)
```
```   771 qed
```
```   772
```
```   773 lemma (in sigma_algebra) borel_measurable_euclidean_space:
```
```   774   fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
```
```   775   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x \$\$ i) \<in> borel_measurable M)"
```
```   776 proof safe
```
```   777   fix i assume "f \<in> borel_measurable M"
```
```   778   then show "(\<lambda>x. f x \$\$ i) \<in> borel_measurable M"
```
```   779     using measurable_comp[of f _ _ "\<lambda>x. x \$\$ i", unfolded comp_def]
```
```   780     by (auto intro: borel_measurable_euclidean_component)
```
```   781 next
```
```   782   assume f: "\<forall>i<DIM('c). (\<lambda>x. f x \$\$ i) \<in> borel_measurable M"
```
```   783   then show "f \<in> borel_measurable M"
```
```   784     unfolding borel_measurable_iff_halfspace_le by auto
```
```   785 qed
```
```   786
```
```   787 subsection "Borel measurable operators"
```
```   788
```
```   789 lemma (in sigma_algebra) affine_borel_measurable_vector:
```
```   790   fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
```
```   791   assumes "f \<in> borel_measurable M"
```
```   792   shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
```
```   793 proof (rule borel_measurableI)
```
```   794   fix S :: "'x set" assume "open S"
```
```   795   show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
```
```   796   proof cases
```
```   797     assume "b \<noteq> 0"
```
```   798     with `open S` have "((\<lambda>x. (- a + x) /\<^sub>R b) ` S) \<in> open" (is "?S \<in> open")
```
```   799       by (auto intro!: open_affinity simp: scaleR.add_right mem_def)
```
```   800     hence "?S \<in> sets borel"
```
```   801       unfolding borel_def by (auto simp: sigma_def intro!: sigma_sets.Basic)
```
```   802     moreover
```
```   803     from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
```
```   804       apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
```
```   805     ultimately show ?thesis using assms unfolding in_borel_measurable_borel
```
```   806       by auto
```
```   807   qed simp
```
```   808 qed
```
```   809
```
```   810 lemma (in sigma_algebra) affine_borel_measurable:
```
```   811   fixes g :: "'a \<Rightarrow> real"
```
```   812   assumes g: "g \<in> borel_measurable M"
```
```   813   shows "(\<lambda>x. a + (g x) * b) \<in> borel_measurable M"
```
```   814   using affine_borel_measurable_vector[OF assms] by (simp add: mult_commute)
```
```   815
```
```   816 lemma (in sigma_algebra) borel_measurable_add[simp, intro]:
```
```   817   fixes f :: "'a \<Rightarrow> real"
```
```   818   assumes f: "f \<in> borel_measurable M"
```
```   819   assumes g: "g \<in> borel_measurable M"
```
```   820   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```   821 proof -
```
```   822   have 1: "\<And>a. {w\<in>space M. a \<le> f w + g w} = {w \<in> space M. a + g w * -1 \<le> f w}"
```
```   823     by auto
```
```   824   have "\<And>a. (\<lambda>w. a + (g w) * -1) \<in> borel_measurable M"
```
```   825     by (rule affine_borel_measurable [OF g])
```
```   826   then have "\<And>a. {w \<in> space M. (\<lambda>w. a + (g w) * -1)(w) \<le> f w} \<in> sets M" using f
```
```   827     by auto
```
```   828   then have "\<And>a. {w \<in> space M. a \<le> f w + g w} \<in> sets M"
```
```   829     by (simp add: 1)
```
```   830   then show ?thesis
```
```   831     by (simp add: borel_measurable_iff_ge)
```
```   832 qed
```
```   833
```
```   834 lemma (in sigma_algebra) borel_measurable_setsum[simp, intro]:
```
```   835   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
```
```   836   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   837   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```   838 proof cases
```
```   839   assume "finite S"
```
```   840   thus ?thesis using assms by induct auto
```
```   841 qed simp
```
```   842
```
```   843 lemma (in sigma_algebra) borel_measurable_square:
```
```   844   fixes f :: "'a \<Rightarrow> real"
```
```   845   assumes f: "f \<in> borel_measurable M"
```
```   846   shows "(\<lambda>x. (f x)^2) \<in> borel_measurable M"
```
```   847 proof -
```
```   848   {
```
```   849     fix a
```
```   850     have "{w \<in> space M. (f w)\<twosuperior> \<le> a} \<in> sets M"
```
```   851     proof (cases rule: linorder_cases [of a 0])
```
```   852       case less
```
```   853       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = {}"
```
```   854         by auto (metis less order_le_less_trans power2_less_0)
```
```   855       also have "... \<in> sets M"
```
```   856         by (rule empty_sets)
```
```   857       finally show ?thesis .
```
```   858     next
```
```   859       case equal
```
```   860       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
```
```   861              {w \<in> space M. f w \<le> 0} \<inter> {w \<in> space M. 0 \<le> f w}"
```
```   862         by auto
```
```   863       also have "... \<in> sets M"
```
```   864         apply (insert f)
```
```   865         apply (rule Int)
```
```   866         apply (simp add: borel_measurable_iff_le)
```
```   867         apply (simp add: borel_measurable_iff_ge)
```
```   868         done
```
```   869       finally show ?thesis .
```
```   870     next
```
```   871       case greater
```
```   872       have "\<forall>x. (f x ^ 2 \<le> sqrt a ^ 2) = (- sqrt a  \<le> f x & f x \<le> sqrt a)"
```
```   873         by (metis abs_le_interval_iff abs_of_pos greater real_sqrt_abs
```
```   874                   real_sqrt_le_iff real_sqrt_power)
```
```   875       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
```
```   876              {w \<in> space M. -(sqrt a) \<le> f w} \<inter> {w \<in> space M. f w \<le> sqrt a}"
```
```   877         using greater by auto
```
```   878       also have "... \<in> sets M"
```
```   879         apply (insert f)
```
```   880         apply (rule Int)
```
```   881         apply (simp add: borel_measurable_iff_ge)
```
```   882         apply (simp add: borel_measurable_iff_le)
```
```   883         done
```
```   884       finally show ?thesis .
```
```   885     qed
```
```   886   }
```
```   887   thus ?thesis by (auto simp add: borel_measurable_iff_le)
```
```   888 qed
```
```   889
```
```   890 lemma times_eq_sum_squares:
```
```   891    fixes x::real
```
```   892    shows"x*y = ((x+y)^2)/4 - ((x-y)^ 2)/4"
```
```   893 by (simp add: power2_eq_square ring_distribs diff_divide_distrib [symmetric])
```
```   894
```
```   895 lemma (in sigma_algebra) borel_measurable_uminus[simp, intro]:
```
```   896   fixes g :: "'a \<Rightarrow> real"
```
```   897   assumes g: "g \<in> borel_measurable M"
```
```   898   shows "(\<lambda>x. - g x) \<in> borel_measurable M"
```
```   899 proof -
```
```   900   have "(\<lambda>x. - g x) = (\<lambda>x. 0 + (g x) * -1)"
```
```   901     by simp
```
```   902   also have "... \<in> borel_measurable M"
```
```   903     by (fast intro: affine_borel_measurable g)
```
```   904   finally show ?thesis .
```
```   905 qed
```
```   906
```
```   907 lemma (in sigma_algebra) borel_measurable_times[simp, intro]:
```
```   908   fixes f :: "'a \<Rightarrow> real"
```
```   909   assumes f: "f \<in> borel_measurable M"
```
```   910   assumes g: "g \<in> borel_measurable M"
```
```   911   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```   912 proof -
```
```   913   have 1: "(\<lambda>x. 0 + (f x + g x)\<twosuperior> * inverse 4) \<in> borel_measurable M"
```
```   914     using assms by (fast intro: affine_borel_measurable borel_measurable_square)
```
```   915   have "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) =
```
```   916         (\<lambda>x. 0 + ((f x + -g x) ^ 2 * inverse -4))"
```
```   917     by (simp add: minus_divide_right)
```
```   918   also have "... \<in> borel_measurable M"
```
```   919     using f g by (fast intro: affine_borel_measurable borel_measurable_square f g)
```
```   920   finally have 2: "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) \<in> borel_measurable M" .
```
```   921   show ?thesis
```
```   922     apply (simp add: times_eq_sum_squares diff_minus)
```
```   923     using 1 2 by simp
```
```   924 qed
```
```   925
```
```   926 lemma (in sigma_algebra) borel_measurable_setprod[simp, intro]:
```
```   927   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
```
```   928   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   929   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```   930 proof cases
```
```   931   assume "finite S"
```
```   932   thus ?thesis using assms by induct auto
```
```   933 qed simp
```
```   934
```
```   935 lemma (in sigma_algebra) borel_measurable_diff[simp, intro]:
```
```   936   fixes f :: "'a \<Rightarrow> real"
```
```   937   assumes f: "f \<in> borel_measurable M"
```
```   938   assumes g: "g \<in> borel_measurable M"
```
```   939   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```   940   unfolding diff_minus using assms by fast
```
```   941
```
```   942 lemma (in sigma_algebra) borel_measurable_inverse[simp, intro]:
```
```   943   fixes f :: "'a \<Rightarrow> real"
```
```   944   assumes "f \<in> borel_measurable M"
```
```   945   shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
```
```   946   unfolding borel_measurable_iff_ge unfolding inverse_eq_divide
```
```   947 proof safe
```
```   948   fix a :: real
```
```   949   have *: "{w \<in> space M. a \<le> 1 / f w} =
```
```   950       ({w \<in> space M. 0 < f w} \<inter> {w \<in> space M. a * f w \<le> 1}) \<union>
```
```   951       ({w \<in> space M. f w < 0} \<inter> {w \<in> space M. 1 \<le> a * f w}) \<union>
```
```   952       ({w \<in> space M. f w = 0} \<inter> {w \<in> space M. a \<le> 0})" by (auto simp: le_divide_eq)
```
```   953   show "{w \<in> space M. a \<le> 1 / f w} \<in> sets M" using assms unfolding *
```
```   954     by (auto intro!: Int Un)
```
```   955 qed
```
```   956
```
```   957 lemma (in sigma_algebra) borel_measurable_divide[simp, intro]:
```
```   958   fixes f :: "'a \<Rightarrow> real"
```
```   959   assumes "f \<in> borel_measurable M"
```
```   960   and "g \<in> borel_measurable M"
```
```   961   shows "(\<lambda>x. f x / g x) \<in> borel_measurable M"
```
```   962   unfolding field_divide_inverse
```
```   963   by (rule borel_measurable_inverse borel_measurable_times assms)+
```
```   964
```
```   965 lemma (in sigma_algebra) borel_measurable_max[intro, simp]:
```
```   966   fixes f g :: "'a \<Rightarrow> real"
```
```   967   assumes "f \<in> borel_measurable M"
```
```   968   assumes "g \<in> borel_measurable M"
```
```   969   shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
```
```   970   unfolding borel_measurable_iff_le
```
```   971 proof safe
```
```   972   fix a
```
```   973   have "{x \<in> space M. max (g x) (f x) \<le> a} =
```
```   974     {x \<in> space M. g x \<le> a} \<inter> {x \<in> space M. f x \<le> a}" by auto
```
```   975   thus "{x \<in> space M. max (g x) (f x) \<le> a} \<in> sets M"
```
```   976     using assms unfolding borel_measurable_iff_le
```
```   977     by (auto intro!: Int)
```
```   978 qed
```
```   979
```
```   980 lemma (in sigma_algebra) borel_measurable_min[intro, simp]:
```
```   981   fixes f g :: "'a \<Rightarrow> real"
```
```   982   assumes "f \<in> borel_measurable M"
```
```   983   assumes "g \<in> borel_measurable M"
```
```   984   shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
```
```   985   unfolding borel_measurable_iff_ge
```
```   986 proof safe
```
```   987   fix a
```
```   988   have "{x \<in> space M. a \<le> min (g x) (f x)} =
```
```   989     {x \<in> space M. a \<le> g x} \<inter> {x \<in> space M. a \<le> f x}" by auto
```
```   990   thus "{x \<in> space M. a \<le> min (g x) (f x)} \<in> sets M"
```
```   991     using assms unfolding borel_measurable_iff_ge
```
```   992     by (auto intro!: Int)
```
```   993 qed
```
```   994
```
```   995 lemma (in sigma_algebra) borel_measurable_abs[simp, intro]:
```
```   996   assumes "f \<in> borel_measurable M"
```
```   997   shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
```
```   998 proof -
```
```   999   have *: "\<And>x. \<bar>f x\<bar> = max 0 (f x) + max 0 (- f x)" by (simp add: max_def)
```
```  1000   show ?thesis unfolding * using assms by auto
```
```  1001 qed
```
```  1002
```
```  1003 lemma borel_measurable_nth[simp, intro]:
```
```  1004   "(\<lambda>x::real^'n. x \$ i) \<in> borel_measurable borel"
```
```  1005   using borel_measurable_euclidean_component
```
```  1006   unfolding nth_conv_component by auto
```
```  1007
```
```  1008 lemma borel_measurable_continuous_on1:
```
```  1009   fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
```
```  1010   assumes "continuous_on UNIV f"
```
```  1011   shows "f \<in> borel_measurable borel"
```
```  1012   apply(rule borel.borel_measurableI)
```
```  1013   using continuous_open_preimage[OF assms] unfolding vimage_def by auto
```
```  1014
```
```  1015 lemma borel_measurable_continuous_on:
```
```  1016   fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
```
```  1017   assumes cont: "continuous_on A f" "open A" and f: "f -` {c} \<inter> A \<in> sets borel"
```
```  1018   shows "(\<lambda>x. if x \<in> A then f x else c) \<in> borel_measurable borel" (is "?f \<in> _")
```
```  1019 proof (rule borel.borel_measurableI)
```
```  1020   fix S :: "'b set" assume "open S"
```
```  1021   then have "open {x\<in>A. f x \<in> S - {c}}"
```
```  1022     by (intro continuous_open_preimage[OF cont]) auto
```
```  1023   then have *: "{x\<in>A. f x \<in> S - {c}} \<in> sets borel" by auto
```
```  1024   show "?f -` S \<inter> space borel \<in> sets borel"
```
```  1025   proof cases
```
```  1026     assume "c \<in> S"
```
```  1027     then have "?f -` S = {x\<in>A. f x \<in> S - {c}} \<union> (f -` {c} \<inter> A) \<union> -A"
```
```  1028       by auto
```
```  1029     with * show "?f -` S \<inter> space borel \<in> sets borel"
```
```  1030       using `open A` f by (auto intro!: borel.Un)
```
```  1031   next
```
```  1032     assume "c \<notin> S"
```
```  1033     then have "?f -` S = {x\<in>A. f x \<in> S - {c}}" by (auto split: split_if_asm)
```
```  1034     with * show "?f -` S \<inter> space borel \<in> sets borel" by auto
```
```  1035   qed
```
```  1036 qed
```
```  1037
```
```  1038 lemma borel_measurable_borel_log: assumes "1 < b" shows "log b \<in> borel_measurable borel"
```
```  1039 proof -
```
```  1040   { fix x :: real assume x: "x \<le> 0"
```
```  1041     { fix x::real assume "x \<le> 0" then have "\<And>u. exp u = x \<longleftrightarrow> False" by auto }
```
```  1042     from this[of x] x this[of 0] have "log b 0 = log b x"
```
```  1043       by (auto simp: ln_def log_def) }
```
```  1044   note log_imp = this
```
```  1045   have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) \<in> borel_measurable borel"
```
```  1046   proof (rule borel_measurable_continuous_on)
```
```  1047     show "continuous_on {0<..} (log b)"
```
```  1048       by (auto intro!: continuous_at_imp_continuous_on DERIV_log DERIV_isCont
```
```  1049                simp: continuous_isCont[symmetric])
```
```  1050     show "open ({0<..}::real set)" by auto
```
```  1051     show "log b -` {log b 0} \<inter> {0<..} \<in> sets borel"
```
```  1052     proof cases
```
```  1053       assume "log b -` {log b 0} \<inter> {0<..} = {}"
```
```  1054       then show ?thesis by simp
```
```  1055     next
```
```  1056       assume "log b -` {log b 0} \<inter> {0<..} \<noteq> {}"
```
```  1057       then obtain x where "0 < x" "log b x = log b 0" by auto
```
```  1058       with log_inj[OF `1 < b`] have "log b -` {log b 0} \<inter> {0<..} = {x}"
```
```  1059         by (auto simp: inj_on_def)
```
```  1060       then show ?thesis by simp
```
```  1061     qed
```
```  1062   qed
```
```  1063   also have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) = log b"
```
```  1064     by (simp add: fun_eq_iff not_less log_imp)
```
```  1065   finally show ?thesis .
```
```  1066 qed
```
```  1067
```
```  1068 lemma (in sigma_algebra) borel_measurable_log[simp,intro]:
```
```  1069   assumes f: "f \<in> borel_measurable M" and "1 < b"
```
```  1070   shows "(\<lambda>x. log b (f x)) \<in> borel_measurable M"
```
```  1071   using measurable_comp[OF f borel_measurable_borel_log[OF `1 < b`]]
```
```  1072   by (simp add: comp_def)
```
```  1073
```
```  1074 subsection "Borel space on the extended reals"
```
```  1075
```
```  1076 lemma borel_measurable_extreal_borel:
```
```  1077   "extreal \<in> borel_measurable borel"
```
```  1078   unfolding borel_def[where 'a=extreal]
```
```  1079 proof (rule borel.measurable_sigma)
```
```  1080   fix X :: "extreal set" assume "X \<in> sets \<lparr>space = UNIV, sets = open \<rparr>"
```
```  1081   then have "open X" by (auto simp: mem_def)
```
```  1082   then have "open (extreal -` X \<inter> space borel)"
```
```  1083     by (simp add: open_extreal_vimage)
```
```  1084   then show "extreal -` X \<inter> space borel \<in> sets borel" by auto
```
```  1085 qed auto
```
```  1086
```
```  1087 lemma (in sigma_algebra) borel_measurable_extreal[simp, intro]:
```
```  1088   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. extreal (f x)) \<in> borel_measurable M"
```
```  1089   using measurable_comp[OF f borel_measurable_extreal_borel] unfolding comp_def .
```
```  1090
```
```  1091 lemma borel_measurable_real_of_extreal_borel:
```
```  1092   "(real :: extreal \<Rightarrow> real) \<in> borel_measurable borel"
```
```  1093   unfolding borel_def[where 'a=real]
```
```  1094 proof (rule borel.measurable_sigma)
```
```  1095   fix B :: "real set" assume "B \<in> sets \<lparr>space = UNIV, sets = open \<rparr>"
```
```  1096   then have "open B" by (auto simp: mem_def)
```
```  1097   have *: "extreal -` real -` (B - {0}) = B - {0}" by auto
```
```  1098   have open_real: "open (real -` (B - {0}) :: extreal set)"
```
```  1099     unfolding open_extreal_def * using `open B` by auto
```
```  1100   show "(real -` B \<inter> space borel :: extreal set) \<in> sets borel"
```
```  1101   proof cases
```
```  1102     assume "0 \<in> B"
```
```  1103     then have *: "real -` B = real -` (B - {0}) \<union> {-\<infinity>, \<infinity>, 0}"
```
```  1104       by (auto simp add: real_of_extreal_eq_0)
```
```  1105     then show "(real -` B :: extreal set) \<inter> space borel \<in> sets borel"
```
```  1106       using open_real by auto
```
```  1107   next
```
```  1108     assume "0 \<notin> B"
```
```  1109     then have *: "(real -` B :: extreal set) = real -` (B - {0})"
```
```  1110       by (auto simp add: real_of_extreal_eq_0)
```
```  1111     then show "(real -` B :: extreal set) \<inter> space borel \<in> sets borel"
```
```  1112       using open_real by auto
```
```  1113   qed
```
```  1114 qed auto
```
```  1115
```
```  1116 lemma (in sigma_algebra) borel_measurable_real_of_extreal[simp, intro]:
```
```  1117   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. real (f x :: extreal)) \<in> borel_measurable M"
```
```  1118   using measurable_comp[OF f borel_measurable_real_of_extreal_borel] unfolding comp_def .
```
```  1119
```
```  1120 lemma (in sigma_algebra) borel_measurable_extreal_iff:
```
```  1121   shows "(\<lambda>x. extreal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
```
```  1122 proof
```
```  1123   assume "(\<lambda>x. extreal (f x)) \<in> borel_measurable M"
```
```  1124   from borel_measurable_real_of_extreal[OF this]
```
```  1125   show "f \<in> borel_measurable M" by auto
```
```  1126 qed auto
```
```  1127
```
```  1128 lemma (in sigma_algebra) borel_measurable_extreal_iff_real:
```
```  1129   "f \<in> borel_measurable M \<longleftrightarrow>
```
```  1130     ((\<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)"
```
```  1131 proof safe
```
```  1132   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"
```
```  1133   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
```
```  1134   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
```
```  1135   let "?f x" = "if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else extreal (real (f x))"
```
```  1136   have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
```
```  1137   also have "?f = f" by (auto simp: fun_eq_iff extreal_real)
```
```  1138   finally show "f \<in> borel_measurable M" .
```
```  1139 qed (auto intro: measurable_sets borel_measurable_real_of_extreal)
```
```  1140
```
```  1141 lemma (in sigma_algebra) less_eq_ge_measurable:
```
```  1142   fixes f :: "'a \<Rightarrow> 'c::linorder"
```
```  1143   shows "f -` {a <..} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {..a} \<inter> space M \<in> sets M"
```
```  1144 proof
```
```  1145   assume "f -` {a <..} \<inter> space M \<in> sets M"
```
```  1146   moreover have "f -` {..a} \<inter> space M = space M - f -` {a <..} \<inter> space M" by auto
```
```  1147   ultimately show "f -` {..a} \<inter> space M \<in> sets M" by auto
```
```  1148 next
```
```  1149   assume "f -` {..a} \<inter> space M \<in> sets M"
```
```  1150   moreover have "f -` {a <..} \<inter> space M = space M - f -` {..a} \<inter> space M" by auto
```
```  1151   ultimately show "f -` {a <..} \<inter> space M \<in> sets M" by auto
```
```  1152 qed
```
```  1153
```
```  1154 lemma (in sigma_algebra) greater_eq_le_measurable:
```
```  1155   fixes f :: "'a \<Rightarrow> 'c::linorder"
```
```  1156   shows "f -` {..< a} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {a ..} \<inter> space M \<in> sets M"
```
```  1157 proof
```
```  1158   assume "f -` {a ..} \<inter> space M \<in> sets M"
```
```  1159   moreover have "f -` {..< a} \<inter> space M = space M - f -` {a ..} \<inter> space M" by auto
```
```  1160   ultimately show "f -` {..< a} \<inter> space M \<in> sets M" by auto
```
```  1161 next
```
```  1162   assume "f -` {..< a} \<inter> space M \<in> sets M"
```
```  1163   moreover have "f -` {a ..} \<inter> space M = space M - f -` {..< a} \<inter> space M" by auto
```
```  1164   ultimately show "f -` {a ..} \<inter> space M \<in> sets M" by auto
```
```  1165 qed
```
```  1166
```
```  1167 lemma (in sigma_algebra) borel_measurable_uminus_borel_extreal:
```
```  1168   "(uminus :: extreal \<Rightarrow> extreal) \<in> borel_measurable borel"
```
```  1169 proof (subst borel_def, rule borel.measurable_sigma)
```
```  1170   fix X :: "extreal set" assume "X \<in> sets \<lparr>space = UNIV, sets = open\<rparr>"
```
```  1171   then have "open X" by (simp add: mem_def)
```
```  1172   have "uminus -` X = uminus ` X" by (force simp: image_iff)
```
```  1173   then have "open (uminus -` X)" using `open X` extreal_open_uminus by auto
```
```  1174   then show "uminus -` X \<inter> space borel \<in> sets borel" by auto
```
```  1175 qed auto
```
```  1176
```
```  1177 lemma (in sigma_algebra) borel_measurable_uminus_extreal[intro]:
```
```  1178   assumes "f \<in> borel_measurable M"
```
```  1179   shows "(\<lambda>x. - f x :: extreal) \<in> borel_measurable M"
```
```  1180   using measurable_comp[OF assms borel_measurable_uminus_borel_extreal] by (simp add: comp_def)
```
```  1181
```
```  1182 lemma (in sigma_algebra) borel_measurable_uminus_eq_extreal[simp]:
```
```  1183   "(\<lambda>x. - f x :: extreal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
```
```  1184 proof
```
```  1185   assume ?l from borel_measurable_uminus_extreal[OF this] show ?r by simp
```
```  1186 qed auto
```
```  1187
```
```  1188 lemma (in sigma_algebra) borel_measurable_eq_atMost_extreal:
```
```  1189   "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
```
```  1190 proof (intro iffI allI)
```
```  1191   assume pos[rule_format]: "\<forall>a. f -` {..a} \<inter> space M \<in> sets M"
```
```  1192   show "f \<in> borel_measurable M"
```
```  1193     unfolding borel_measurable_extreal_iff_real borel_measurable_iff_le
```
```  1194   proof (intro conjI allI)
```
```  1195     fix a :: real
```
```  1196     { fix x :: extreal assume *: "\<forall>i::nat. real i < x"
```
```  1197       have "x = \<infinity>"
```
```  1198       proof (rule extreal_top)
```
```  1199         fix B from real_arch_lt[of B] guess n ..
```
```  1200         then have "extreal B < real n" by auto
```
```  1201         with * show "B \<le> x" by (metis less_trans less_imp_le)
```
```  1202       qed }
```
```  1203     then have "f -` {\<infinity>} \<inter> space M = space M - (\<Union>i::nat. f -` {.. real i} \<inter> space M)"
```
```  1204       by (auto simp: not_le)
```
```  1205     then show "f -` {\<infinity>} \<inter> space M \<in> sets M" using pos by (auto simp del: UN_simps intro!: Diff)
```
```  1206     moreover
```
```  1207     have "{-\<infinity>} = {..-\<infinity>}" by auto
```
```  1208     then show "f -` {-\<infinity>} \<inter> space M \<in> sets M" using pos by auto
```
```  1209     moreover have "{x\<in>space M. f x \<le> extreal a} \<in> sets M"
```
```  1210       using pos[of "extreal a"] by (simp add: vimage_def Int_def conj_commute)
```
```  1211     moreover have "{w \<in> space M. real (f w) \<le> a} =
```
```  1212       (if a < 0 then {w \<in> space M. f w \<le> extreal a} - f -` {-\<infinity>} \<inter> space M
```
```  1213       else {w \<in> space M. f w \<le> extreal a} \<union> (f -` {\<infinity>} \<inter> space M) \<union> (f -` {-\<infinity>} \<inter> space M))" (is "?l = ?r")
```
```  1214       proof (intro set_eqI) fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases "f x") auto qed
```
```  1215     ultimately show "{w \<in> space M. real (f w) \<le> a} \<in> sets M" by auto
```
```  1216   qed
```
```  1217 qed (simp add: measurable_sets)
```
```  1218
```
```  1219 lemma (in sigma_algebra) borel_measurable_eq_atLeast_extreal:
```
```  1220   "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
```
```  1221 proof
```
```  1222   assume pos: "\<forall>a. f -` {a..} \<inter> space M \<in> sets M"
```
```  1223   moreover have "\<And>a. (\<lambda>x. - f x) -` {..a} = f -` {-a ..}"
```
```  1224     by (auto simp: extreal_uminus_le_reorder)
```
```  1225   ultimately have "(\<lambda>x. - f x) \<in> borel_measurable M"
```
```  1226     unfolding borel_measurable_eq_atMost_extreal by auto
```
```  1227   then show "f \<in> borel_measurable M" by simp
```
```  1228 qed (simp add: measurable_sets)
```
```  1229
```
```  1230 lemma (in sigma_algebra) borel_measurable_extreal_iff_less:
```
```  1231   "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
```
```  1232   unfolding borel_measurable_eq_atLeast_extreal greater_eq_le_measurable ..
```
```  1233
```
```  1234 lemma (in sigma_algebra) borel_measurable_extreal_iff_ge:
```
```  1235   "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
```
```  1236   unfolding borel_measurable_eq_atMost_extreal less_eq_ge_measurable ..
```
```  1237
```
```  1238 lemma (in sigma_algebra) borel_measurable_extreal_eq_const:
```
```  1239   fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M"
```
```  1240   shows "{x\<in>space M. f x = c} \<in> sets M"
```
```  1241 proof -
```
```  1242   have "{x\<in>space M. f x = c} = (f -` {c} \<inter> space M)" by auto
```
```  1243   then show ?thesis using assms by (auto intro!: measurable_sets)
```
```  1244 qed
```
```  1245
```
```  1246 lemma (in sigma_algebra) borel_measurable_extreal_neq_const:
```
```  1247   fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M"
```
```  1248   shows "{x\<in>space M. f x \<noteq> c} \<in> sets M"
```
```  1249 proof -
```
```  1250   have "{x\<in>space M. f x \<noteq> c} = space M - (f -` {c} \<inter> space M)" by auto
```
```  1251   then show ?thesis using assms by (auto intro!: measurable_sets)
```
```  1252 qed
```
```  1253
```
```  1254 lemma (in sigma_algebra) borel_measurable_extreal_le[intro,simp]:
```
```  1255   fixes f g :: "'a \<Rightarrow> extreal"
```
```  1256   assumes f: "f \<in> borel_measurable M"
```
```  1257   assumes g: "g \<in> borel_measurable M"
```
```  1258   shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
```
```  1259 proof -
```
```  1260   have "{x \<in> space M. f x \<le> g x} =
```
```  1261     {x \<in> space M. real (f x) \<le> real (g x)} - (f -` {\<infinity>, -\<infinity>} \<inter> space M \<union> g -` {\<infinity>, -\<infinity>} \<inter> space M) \<union>
```
```  1262     f -` {-\<infinity>} \<inter> space M \<union> g -` {\<infinity>} \<inter> space M" (is "?l = ?r")
```
```  1263   proof (intro set_eqI)
```
```  1264     fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases rule: extreal2_cases[of "f x" "g x"]) auto
```
```  1265   qed
```
```  1266   with f g show ?thesis by (auto intro!: Un simp: measurable_sets)
```
```  1267 qed
```
```  1268
```
```  1269 lemma (in sigma_algebra) borel_measurable_extreal_less[intro,simp]:
```
```  1270   fixes f :: "'a \<Rightarrow> extreal"
```
```  1271   assumes f: "f \<in> borel_measurable M"
```
```  1272   assumes g: "g \<in> borel_measurable M"
```
```  1273   shows "{x \<in> space M. f x < g x} \<in> sets M"
```
```  1274 proof -
```
```  1275   have "{x \<in> space M. f x < g x} = space M - {x \<in> space M. g x \<le> f x}" by auto
```
```  1276   then show ?thesis using g f by auto
```
```  1277 qed
```
```  1278
```
```  1279 lemma (in sigma_algebra) borel_measurable_extreal_eq[intro,simp]:
```
```  1280   fixes f :: "'a \<Rightarrow> extreal"
```
```  1281   assumes f: "f \<in> borel_measurable M"
```
```  1282   assumes g: "g \<in> borel_measurable M"
```
```  1283   shows "{w \<in> space M. f w = g w} \<in> sets M"
```
```  1284 proof -
```
```  1285   have "{x \<in> space M. f x = g x} = {x \<in> space M. g x \<le> f x} \<inter> {x \<in> space M. f x \<le> g x}" by auto
```
```  1286   then show ?thesis using g f by auto
```
```  1287 qed
```
```  1288
```
```  1289 lemma (in sigma_algebra) borel_measurable_extreal_neq[intro,simp]:
```
```  1290   fixes f :: "'a \<Rightarrow> extreal"
```
```  1291   assumes f: "f \<in> borel_measurable M"
```
```  1292   assumes g: "g \<in> borel_measurable M"
```
```  1293   shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
```
```  1294 proof -
```
```  1295   have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}" by auto
```
```  1296   thus ?thesis using f g by auto
```
```  1297 qed
```
```  1298
```
```  1299 lemma (in sigma_algebra) split_sets:
```
```  1300   "{x\<in>space M. P x \<or> Q x} = {x\<in>space M. P x} \<union> {x\<in>space M. Q x}"
```
```  1301   "{x\<in>space M. P x \<and> Q x} = {x\<in>space M. P x} \<inter> {x\<in>space M. Q x}"
```
```  1302   by auto
```
```  1303
```
```  1304 lemma (in sigma_algebra) borel_measurable_extreal_add[intro, simp]:
```
```  1305   fixes f :: "'a \<Rightarrow> extreal"
```
```  1306   assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1307   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```  1308 proof -
```
```  1309   { fix x assume "x \<in> space M" then have "f x + g x =
```
```  1310       (if f x = \<infinity> \<or> g x = \<infinity> then \<infinity>
```
```  1311         else if f x = -\<infinity> \<or> g x = -\<infinity> then -\<infinity>
```
```  1312         else extreal (real (f x) + real (g x)))"
```
```  1313       by (cases rule: extreal2_cases[of "f x" "g x"]) auto }
```
```  1314   with assms show ?thesis
```
```  1315     by (auto cong: measurable_cong simp: split_sets
```
```  1316              intro!: Un measurable_If measurable_sets)
```
```  1317 qed
```
```  1318
```
```  1319 lemma (in sigma_algebra) borel_measurable_extreal_setsum[simp, intro]:
```
```  1320   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> extreal"
```
```  1321   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1322   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1323 proof cases
```
```  1324   assume "finite S"
```
```  1325   thus ?thesis using assms
```
```  1326     by induct auto
```
```  1327 qed (simp add: borel_measurable_const)
```
```  1328
```
```  1329 lemma abs_extreal_ge0[simp]: "0 \<le> x \<Longrightarrow> \<bar>x :: extreal\<bar> = x"
```
```  1330   by (cases x) auto
```
```  1331
```
```  1332 lemma abs_extreal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: extreal\<bar> = -x"
```
```  1333   by (cases x) auto
```
```  1334
```
```  1335 lemma abs_extreal_pos[simp]: "0 \<le> \<bar>x :: extreal\<bar>"
```
```  1336   by (cases x) auto
```
```  1337
```
```  1338 lemma (in sigma_algebra) borel_measurable_extreal_abs[intro, simp]:
```
```  1339   fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M"
```
```  1340   shows "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
```
```  1341 proof -
```
```  1342   { fix x have "\<bar>f x\<bar> = (if 0 \<le> f x then f x else - f x)" by auto }
```
```  1343   then show ?thesis using assms by (auto intro!: measurable_If)
```
```  1344 qed
```
```  1345
```
```  1346 lemma (in sigma_algebra) borel_measurable_extreal_times[intro, simp]:
```
```  1347   fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1348   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1349 proof -
```
```  1350   { fix f g :: "'a \<Rightarrow> extreal"
```
```  1351     assume b: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1352       and pos: "\<And>x. 0 \<le> f x" "\<And>x. 0 \<le> g x"
```
```  1353     { fix x have *: "f x * g x = (if f x = 0 \<or> g x = 0 then 0
```
```  1354         else if f x = \<infinity> \<or> g x = \<infinity> then \<infinity>
```
```  1355         else extreal (real (f x) * real (g x)))"
```
```  1356       apply (cases rule: extreal2_cases[of "f x" "g x"])
```
```  1357       using pos[of x] by auto }
```
```  1358     with b have "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1359       by (auto cong: measurable_cong simp: split_sets
```
```  1360                intro!: Un measurable_If measurable_sets) }
```
```  1361   note pos_times = this
```
```  1362   have *: "(\<lambda>x. f x * g x) =
```
```  1363     (\<lambda>x. if 0 \<le> f x \<and> 0 \<le> g x \<or> f x < 0 \<and> g x < 0 then \<bar>f x\<bar> * \<bar>g x\<bar> else - (\<bar>f x\<bar> * \<bar>g x\<bar>))"
```
```  1364     by (auto simp: fun_eq_iff)
```
```  1365   show ?thesis using assms unfolding *
```
```  1366     by (intro measurable_If pos_times borel_measurable_uminus_extreal)
```
```  1367        (auto simp: split_sets intro!: Int)
```
```  1368 qed
```
```  1369
```
```  1370 lemma (in sigma_algebra) borel_measurable_extreal_setprod[simp, intro]:
```
```  1371   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> extreal"
```
```  1372   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1373   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1374 proof cases
```
```  1375   assume "finite S"
```
```  1376   thus ?thesis using assms by induct auto
```
```  1377 qed simp
```
```  1378
```
```  1379 lemma (in sigma_algebra) borel_measurable_extreal_min[simp, intro]:
```
```  1380   fixes f g :: "'a \<Rightarrow> extreal"
```
```  1381   assumes "f \<in> borel_measurable M"
```
```  1382   assumes "g \<in> borel_measurable M"
```
```  1383   shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
```
```  1384   using assms unfolding min_def by (auto intro!: measurable_If)
```
```  1385
```
```  1386 lemma (in sigma_algebra) borel_measurable_extreal_max[simp, intro]:
```
```  1387   fixes f g :: "'a \<Rightarrow> extreal"
```
```  1388   assumes "f \<in> borel_measurable M"
```
```  1389   and "g \<in> borel_measurable M"
```
```  1390   shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
```
```  1391   using assms unfolding max_def by (auto intro!: measurable_If)
```
```  1392
```
```  1393 lemma (in sigma_algebra) borel_measurable_SUP[simp, intro]:
```
```  1394   fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> extreal"
```
```  1395   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1396   shows "(\<lambda>x. SUP i : A. f i x) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M")
```
```  1397   unfolding borel_measurable_extreal_iff_ge
```
```  1398 proof
```
```  1399   fix a
```
```  1400   have "?sup -` {a<..} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"
```
```  1401     by (auto simp: less_SUP_iff SUPR_apply)
```
```  1402   then show "?sup -` {a<..} \<inter> space M \<in> sets M"
```
```  1403     using assms by auto
```
```  1404 qed
```
```  1405
```
```  1406 lemma (in sigma_algebra) borel_measurable_INF[simp, intro]:
```
```  1407   fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> extreal"
```
```  1408   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1409   shows "(\<lambda>x. INF i : A. f i x) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M")
```
```  1410   unfolding borel_measurable_extreal_iff_less
```
```  1411 proof
```
```  1412   fix a
```
```  1413   have "?inf -` {..<a} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"
```
```  1414     by (auto simp: INF_less_iff INFI_apply)
```
```  1415   then show "?inf -` {..<a} \<inter> space M \<in> sets M"
```
```  1416     using assms by auto
```
```  1417 qed
```
```  1418
```
```  1419 lemma (in sigma_algebra) borel_measurable_liminf[simp, intro]:
```
```  1420   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
```
```  1421   assumes "\<And>i. f i \<in> borel_measurable M"
```
```  1422   shows "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1423   unfolding liminf_SUPR_INFI using assms by auto
```
```  1424
```
```  1425 lemma (in sigma_algebra) borel_measurable_limsup[simp, intro]:
```
```  1426   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
```
```  1427   assumes "\<And>i. f i \<in> borel_measurable M"
```
```  1428   shows "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1429   unfolding limsup_INFI_SUPR using assms by auto
```
```  1430
```
```  1431 lemma (in sigma_algebra) borel_measurable_extreal_diff[simp, intro]:
```
```  1432   fixes f g :: "'a \<Rightarrow> extreal"
```
```  1433   assumes "f \<in> borel_measurable M"
```
```  1434   assumes "g \<in> borel_measurable M"
```
```  1435   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```  1436   unfolding minus_extreal_def using assms by auto
```
```  1437
```
```  1438 lemma (in sigma_algebra) borel_measurable_psuminf[simp, intro]:
```
```  1439   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
```
```  1440   assumes "\<And>i. f i \<in> borel_measurable M" and pos: "\<And>i x. x \<in> space M \<Longrightarrow> 0 \<le> f i x"
```
```  1441   shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
```
```  1442   apply (subst measurable_cong)
```
```  1443   apply (subst suminf_extreal_eq_SUPR)
```
```  1444   apply (rule pos)
```
```  1445   using assms by auto
```
```  1446
```
```  1447 section "LIMSEQ is borel measurable"
```
```  1448
```
```  1449 lemma (in sigma_algebra) borel_measurable_LIMSEQ:
```
```  1450   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
```
```  1451   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
```
```  1452   and u: "\<And>i. u i \<in> borel_measurable M"
```
```  1453   shows "u' \<in> borel_measurable M"
```
```  1454 proof -
```
```  1455   have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. extreal (u n x)) = extreal (u' x)"
```
```  1456     using u' by (simp add: lim_imp_Liminf trivial_limit_sequentially lim_extreal)
```
```  1457   moreover from u have "(\<lambda>x. liminf (\<lambda>n. extreal (u n x))) \<in> borel_measurable M"
```
```  1458     by auto
```
```  1459   ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_extreal_iff)
```
```  1460 qed
```
```  1461
```
```  1462 end
```