src/HOL/Probability/Borel_Space.thy
 author hoelzl Mon Mar 14 14:37:49 2011 +0100 (2011-03-14) changeset 41981 cdf7693bbe08 parent 41969 1cf3e4107a2a child 42067 66c8281349ec permissions -rw-r--r--
reworked Probability theory: measures are not type restricted to positive extended reals
```     1 (* Author: Armin Heller, Johannes Hoelzl, TU Muenchen *)
```
```     2
```
```     3 header {*Borel spaces*}
```
```     4
```
```     5 theory Borel_Space
```
```     6   imports Sigma_Algebra Multivariate_Analysis
```
```     7 begin
```
```     8
```
```     9 section "Generic Borel spaces"
```
```    10
```
```    11 definition "borel = sigma \<lparr> space = UNIV::'a::topological_space set, sets = open\<rparr>"
```
```    12 abbreviation "borel_measurable M \<equiv> measurable M borel"
```
```    13
```
```    14 interpretation borel: sigma_algebra borel
```
```    15   by (auto simp: borel_def intro!: sigma_algebra_sigma)
```
```    16
```
```    17 lemma in_borel_measurable:
```
```    18    "f \<in> borel_measurable M \<longleftrightarrow>
```
```    19     (\<forall>S \<in> sets (sigma \<lparr> space = UNIV, sets = open\<rparr>).
```
```    20       f -` S \<inter> space M \<in> sets M)"
```
```    21   by (auto simp add: measurable_def borel_def)
```
```    22
```
```    23 lemma in_borel_measurable_borel:
```
```    24    "f \<in> borel_measurable M \<longleftrightarrow>
```
```    25     (\<forall>S \<in> sets borel.
```
```    26       f -` S \<inter> space M \<in> sets M)"
```
```    27   by (auto simp add: measurable_def borel_def)
```
```    28
```
```    29 lemma space_borel[simp]: "space borel = UNIV"
```
```    30   unfolding borel_def by auto
```
```    31
```
```    32 lemma borel_open[simp]:
```
```    33   assumes "open A" shows "A \<in> sets borel"
```
```    34 proof -
```
```    35   have "A \<in> open" unfolding mem_def using assms .
```
```    36   thus ?thesis unfolding borel_def sigma_def by (auto intro!: sigma_sets.Basic)
```
```    37 qed
```
```    38
```
```    39 lemma borel_closed[simp]:
```
```    40   assumes "closed A" shows "A \<in> sets borel"
```
```    41 proof -
```
```    42   have "space borel - (- A) \<in> sets borel"
```
```    43     using assms unfolding closed_def by (blast intro: borel_open)
```
```    44   thus ?thesis by simp
```
```    45 qed
```
```    46
```
```    47 lemma borel_comp[intro,simp]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
```
```    48   unfolding Compl_eq_Diff_UNIV by (intro borel.Diff) auto
```
```    49
```
```    50 lemma (in sigma_algebra) borel_measurable_vimage:
```
```    51   fixes f :: "'a \<Rightarrow> 'x::t2_space"
```
```    52   assumes borel: "f \<in> borel_measurable M"
```
```    53   shows "f -` {x} \<inter> space M \<in> sets M"
```
```    54 proof (cases "x \<in> f ` space M")
```
```    55   case True then obtain y where "x = f y" by auto
```
```    56   from closed_singleton[of "f y"]
```
```    57   have "{f y} \<in> sets borel" by (rule borel_closed)
```
```    58   with assms show ?thesis
```
```    59     unfolding in_borel_measurable_borel `x = f y` by auto
```
```    60 next
```
```    61   case False hence "f -` {x} \<inter> space M = {}" by auto
```
```    62   thus ?thesis by auto
```
```    63 qed
```
```    64
```
```    65 lemma (in sigma_algebra) borel_measurableI:
```
```    66   fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space"
```
```    67   assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
```
```    68   shows "f \<in> borel_measurable M"
```
```    69   unfolding borel_def
```
```    70 proof (rule measurable_sigma, simp_all)
```
```    71   fix S :: "'x set" assume "S \<in> open" thus "f -` S \<inter> space M \<in> sets M"
```
```    72     using assms[of S] by (simp add: mem_def)
```
```    73 qed
```
```    74
```
```    75 lemma borel_singleton[simp, intro]:
```
```    76   fixes x :: "'a::t1_space"
```
```    77   shows "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets borel"
```
```    78   proof (rule borel.insert_in_sets)
```
```    79     show "{x} \<in> sets borel"
```
```    80       using closed_singleton[of x] by (rule borel_closed)
```
```    81   qed simp
```
```    82
```
```    83 lemma (in sigma_algebra) borel_measurable_const[simp, intro]:
```
```    84   "(\<lambda>x. c) \<in> borel_measurable M"
```
```    85   by (auto intro!: measurable_const)
```
```    86
```
```    87 lemma (in sigma_algebra) borel_measurable_indicator[simp, intro!]:
```
```    88   assumes A: "A \<in> sets M"
```
```    89   shows "indicator A \<in> borel_measurable M"
```
```    90   unfolding indicator_def_raw using A
```
```    91   by (auto intro!: measurable_If_set borel_measurable_const)
```
```    92
```
```    93 lemma (in sigma_algebra) borel_measurable_indicator_iff:
```
```    94   "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
```
```    95     (is "?I \<in> borel_measurable M \<longleftrightarrow> _")
```
```    96 proof
```
```    97   assume "?I \<in> borel_measurable M"
```
```    98   then have "?I -` {1} \<inter> space M \<in> sets M"
```
```    99     unfolding measurable_def by auto
```
```   100   also have "?I -` {1} \<inter> space M = A \<inter> space M"
```
```   101     unfolding indicator_def_raw by auto
```
```   102   finally show "A \<inter> space M \<in> sets M" .
```
```   103 next
```
```   104   assume "A \<inter> space M \<in> sets M"
```
```   105   moreover have "?I \<in> borel_measurable M \<longleftrightarrow>
```
```   106     (indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"
```
```   107     by (intro measurable_cong) (auto simp: indicator_def)
```
```   108   ultimately show "?I \<in> borel_measurable M" by auto
```
```   109 qed
```
```   110
```
```   111 lemma (in sigma_algebra) borel_measurable_restricted:
```
```   112   fixes f :: "'a \<Rightarrow> extreal" assumes "A \<in> sets M"
```
```   113   shows "f \<in> borel_measurable (restricted_space A) \<longleftrightarrow>
```
```   114     (\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
```
```   115     (is "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable M")
```
```   116 proof -
```
```   117   interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
```
```   118   have *: "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable ?R"
```
```   119     by (auto intro!: measurable_cong)
```
```   120   show ?thesis unfolding *
```
```   121     unfolding in_borel_measurable_borel
```
```   122   proof (simp, safe)
```
```   123     fix S :: "extreal set" assume "S \<in> sets borel"
```
```   124       "\<forall>S\<in>sets borel. ?f -` S \<inter> A \<in> op \<inter> A ` sets M"
```
```   125     then have "?f -` S \<inter> A \<in> op \<inter> A ` sets M" by auto
```
```   126     then have f: "?f -` S \<inter> A \<in> sets M"
```
```   127       using `A \<in> sets M` sets_into_space by fastsimp
```
```   128     show "?f -` S \<inter> space M \<in> sets M"
```
```   129     proof cases
```
```   130       assume "0 \<in> S"
```
```   131       then have "?f -` S \<inter> space M = ?f -` S \<inter> A \<union> (space M - A)"
```
```   132         using `A \<in> sets M` sets_into_space by auto
```
```   133       then show ?thesis using f `A \<in> sets M` by (auto intro!: Un Diff)
```
```   134     next
```
```   135       assume "0 \<notin> S"
```
```   136       then have "?f -` S \<inter> space M = ?f -` S \<inter> A"
```
```   137         using `A \<in> sets M` sets_into_space
```
```   138         by (auto simp: indicator_def split: split_if_asm)
```
```   139       then show ?thesis using f by auto
```
```   140     qed
```
```   141   next
```
```   142     fix S :: "extreal set" assume "S \<in> sets borel"
```
```   143       "\<forall>S\<in>sets borel. ?f -` S \<inter> space M \<in> sets M"
```
```   144     then have f: "?f -` S \<inter> space M \<in> sets M" by auto
```
```   145     then show "?f -` S \<inter> A \<in> op \<inter> A ` sets M"
```
```   146       using `A \<in> sets M` sets_into_space
```
```   147       apply (simp add: image_iff)
```
```   148       apply (rule bexI[OF _ f])
```
```   149       by auto
```
```   150   qed
```
```   151 qed
```
```   152
```
```   153 lemma (in sigma_algebra) borel_measurable_subalgebra:
```
```   154   assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
```
```   155   shows "f \<in> borel_measurable M"
```
```   156   using assms unfolding measurable_def by auto
```
```   157
```
```   158 section "Borel spaces on euclidean spaces"
```
```   159
```
```   160 lemma lessThan_borel[simp, intro]:
```
```   161   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   162   shows "{..< a} \<in> sets borel"
```
```   163   by (blast intro: borel_open)
```
```   164
```
```   165 lemma greaterThan_borel[simp, intro]:
```
```   166   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   167   shows "{a <..} \<in> sets borel"
```
```   168   by (blast intro: borel_open)
```
```   169
```
```   170 lemma greaterThanLessThan_borel[simp, intro]:
```
```   171   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   172   shows "{a<..<b} \<in> sets borel"
```
```   173   by (blast intro: borel_open)
```
```   174
```
```   175 lemma atMost_borel[simp, intro]:
```
```   176   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   177   shows "{..a} \<in> sets borel"
```
```   178   by (blast intro: borel_closed)
```
```   179
```
```   180 lemma atLeast_borel[simp, intro]:
```
```   181   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   182   shows "{a..} \<in> sets borel"
```
```   183   by (blast intro: borel_closed)
```
```   184
```
```   185 lemma atLeastAtMost_borel[simp, intro]:
```
```   186   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   187   shows "{a..b} \<in> sets borel"
```
```   188   by (blast intro: borel_closed)
```
```   189
```
```   190 lemma greaterThanAtMost_borel[simp, intro]:
```
```   191   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   192   shows "{a<..b} \<in> sets borel"
```
```   193   unfolding greaterThanAtMost_def by blast
```
```   194
```
```   195 lemma atLeastLessThan_borel[simp, intro]:
```
```   196   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   197   shows "{a..<b} \<in> sets borel"
```
```   198   unfolding atLeastLessThan_def by blast
```
```   199
```
```   200 lemma hafspace_less_borel[simp, intro]:
```
```   201   fixes a :: real
```
```   202   shows "{x::'a::euclidean_space. a < x \$\$ i} \<in> sets borel"
```
```   203   by (auto intro!: borel_open open_halfspace_component_gt)
```
```   204
```
```   205 lemma hafspace_greater_borel[simp, intro]:
```
```   206   fixes a :: real
```
```   207   shows "{x::'a::euclidean_space. x \$\$ i < a} \<in> sets borel"
```
```   208   by (auto intro!: borel_open open_halfspace_component_lt)
```
```   209
```
```   210 lemma hafspace_less_eq_borel[simp, intro]:
```
```   211   fixes a :: real
```
```   212   shows "{x::'a::euclidean_space. a \<le> x \$\$ i} \<in> sets borel"
```
```   213   by (auto intro!: borel_closed closed_halfspace_component_ge)
```
```   214
```
```   215 lemma hafspace_greater_eq_borel[simp, intro]:
```
```   216   fixes a :: real
```
```   217   shows "{x::'a::euclidean_space. x \$\$ i \<le> a} \<in> sets borel"
```
```   218   by (auto intro!: borel_closed closed_halfspace_component_le)
```
```   219
```
```   220 lemma (in sigma_algebra) borel_measurable_less[simp, intro]:
```
```   221   fixes f :: "'a \<Rightarrow> real"
```
```   222   assumes f: "f \<in> borel_measurable M"
```
```   223   assumes g: "g \<in> borel_measurable M"
```
```   224   shows "{w \<in> space M. f w < g w} \<in> sets M"
```
```   225 proof -
```
```   226   have "{w \<in> space M. f w < g w} =
```
```   227         (\<Union>r. (f -` {..< of_rat r} \<inter> space M) \<inter> (g -` {of_rat r <..} \<inter> space M))"
```
```   228     using Rats_dense_in_real by (auto simp add: Rats_def)
```
```   229   then show ?thesis using f g
```
```   230     by simp (blast intro: measurable_sets)
```
```   231 qed
```
```   232
```
```   233 lemma (in sigma_algebra) borel_measurable_le[simp, intro]:
```
```   234   fixes f :: "'a \<Rightarrow> real"
```
```   235   assumes f: "f \<in> borel_measurable M"
```
```   236   assumes g: "g \<in> borel_measurable M"
```
```   237   shows "{w \<in> space M. f w \<le> g w} \<in> sets M"
```
```   238 proof -
```
```   239   have "{w \<in> space M. f w \<le> g w} = space M - {w \<in> space M. g w < f w}"
```
```   240     by auto
```
```   241   thus ?thesis using f g
```
```   242     by simp blast
```
```   243 qed
```
```   244
```
```   245 lemma (in sigma_algebra) borel_measurable_eq[simp, intro]:
```
```   246   fixes f :: "'a \<Rightarrow> real"
```
```   247   assumes f: "f \<in> borel_measurable M"
```
```   248   assumes g: "g \<in> borel_measurable M"
```
```   249   shows "{w \<in> space M. f w = g w} \<in> sets M"
```
```   250 proof -
```
```   251   have "{w \<in> space M. f w = g w} =
```
```   252         {w \<in> space M. f w \<le> g w} \<inter> {w \<in> space M. g w \<le> f w}"
```
```   253     by auto
```
```   254   thus ?thesis using f g by auto
```
```   255 qed
```
```   256
```
```   257 lemma (in sigma_algebra) borel_measurable_neq[simp, intro]:
```
```   258   fixes f :: "'a \<Rightarrow> real"
```
```   259   assumes f: "f \<in> borel_measurable M"
```
```   260   assumes g: "g \<in> borel_measurable M"
```
```   261   shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
```
```   262 proof -
```
```   263   have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}"
```
```   264     by auto
```
```   265   thus ?thesis using f g by auto
```
```   266 qed
```
```   267
```
```   268 subsection "Borel space equals sigma algebras over intervals"
```
```   269
```
```   270 lemma rational_boxes:
```
```   271   fixes x :: "'a\<Colon>ordered_euclidean_space"
```
```   272   assumes "0 < e"
```
```   273   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"
```
```   274 proof -
```
```   275   def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
```
```   276   then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos)
```
```   277   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \$\$ i \<and> x \$\$ i - y < e'" (is "\<forall>i. ?th i")
```
```   278   proof
```
```   279     fix i from Rats_dense_in_real[of "x \$\$ i - e'" "x \$\$ i"] e
```
```   280     show "?th i" by auto
```
```   281   qed
```
```   282   from choice[OF this] guess a .. note a = this
```
```   283   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \$\$ i < y \<and> y - x \$\$ i < e'" (is "\<forall>i. ?th i")
```
```   284   proof
```
```   285     fix i from Rats_dense_in_real[of "x \$\$ i" "x \$\$ i + e'"] e
```
```   286     show "?th i" by auto
```
```   287   qed
```
```   288   from choice[OF this] guess b .. note b = this
```
```   289   { fix y :: 'a assume *: "Chi a < y" "y < Chi b"
```
```   290     have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x \$\$ i) (y \$\$ i))\<twosuperior>)"
```
```   291       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
```
```   292     also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))"
```
```   293     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
```
```   294       fix i assume i: "i \<in> {..<DIM('a)}"
```
```   295       have "a i < y\$\$i \<and> y\$\$i < b i" using * i eucl_less[where 'a='a] by auto
```
```   296       moreover have "a i < x\$\$i" "x\$\$i - a i < e'" using a by auto
```
```   297       moreover have "x\$\$i < b i" "b i - x\$\$i < e'" using b by auto
```
```   298       ultimately have "\<bar>x\$\$i - y\$\$i\<bar> < 2 * e'" by auto
```
```   299       then have "dist (x \$\$ i) (y \$\$ i) < e/sqrt (real (DIM('a)))"
```
```   300         unfolding e'_def by (auto simp: dist_real_def)
```
```   301       then have "(dist (x \$\$ i) (y \$\$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
```
```   302         by (rule power_strict_mono) auto
```
```   303       then show "(dist (x \$\$ i) (y \$\$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
```
```   304         by (simp add: power_divide)
```
```   305     qed auto
```
```   306     also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive)
```
```   307     finally have "dist x y < e" . }
```
```   308   with a b show ?thesis
```
```   309     apply (rule_tac exI[of _ "Chi a"])
```
```   310     apply (rule_tac exI[of _ "Chi b"])
```
```   311     using eucl_less[where 'a='a] by auto
```
```   312 qed
```
```   313
```
```   314 lemma ex_rat_list:
```
```   315   fixes x :: "'a\<Colon>ordered_euclidean_space"
```
```   316   assumes "\<And> i. x \$\$ i \<in> \<rat>"
```
```   317   shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x \$\$ i)"
```
```   318 proof -
```
```   319   have "\<forall>i. \<exists>r. x \$\$ i = of_rat r" using assms unfolding Rats_def by blast
```
```   320   from choice[OF this] guess r ..
```
```   321   then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"])
```
```   322 qed
```
```   323
```
```   324 lemma open_UNION:
```
```   325   fixes M :: "'a\<Colon>ordered_euclidean_space set"
```
```   326   assumes "open M"
```
```   327   shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M}
```
```   328                    (\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})"
```
```   329     (is "M = UNION ?idx ?box")
```
```   330 proof safe
```
```   331   fix x assume "x \<in> M"
```
```   332   obtain e where e: "e > 0" "ball x e \<subseteq> M"
```
```   333     using openE[OF assms `x \<in> M`] by auto
```
```   334   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"
```
```   335     using rational_boxes[OF e(1)] by blast
```
```   336   then obtain p q where pq: "length p = DIM ('a)"
```
```   337                             "length q = DIM ('a)"
```
```   338                             "\<forall> i < DIM ('a). of_rat (p ! i) = a \$\$ i \<and> of_rat (q ! i) = b \$\$ i"
```
```   339     using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast
```
```   340   hence p: "Chi (of_rat \<circ> op ! p) = a"
```
```   341     using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a]
```
```   342     unfolding o_def by auto
```
```   343   from pq have q: "Chi (of_rat \<circ> op ! q) = b"
```
```   344     using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b]
```
```   345     unfolding o_def by auto
```
```   346   have "x \<in> ?box (p, q)"
```
```   347     using p q ab by auto
```
```   348   thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto
```
```   349 qed auto
```
```   350
```
```   351 lemma halfspace_span_open:
```
```   352   "sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a}))
```
```   353     \<subseteq> sets borel"
```
```   354   by (auto intro!: borel.sigma_sets_subset[simplified] borel_open
```
```   355                    open_halfspace_component_lt)
```
```   356
```
```   357 lemma halfspace_lt_in_halfspace:
```
```   358   "{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>)"
```
```   359   by (auto intro!: sigma_sets.Basic simp: sets_sigma)
```
```   360
```
```   361 lemma halfspace_gt_in_halfspace:
```
```   362   "{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>)"
```
```   363   (is "?set \<in> sets ?SIGMA")
```
```   364 proof -
```
```   365   interpret sigma_algebra "?SIGMA"
```
```   366     by (intro sigma_algebra_sigma_sets) (simp_all add: sets_sigma)
```
```   367   have *: "?set = (\<Union>n. space ?SIGMA - {x\<Colon>'a. x \$\$ i < a + 1 / real (Suc n)})"
```
```   368   proof (safe, simp_all add: not_less)
```
```   369     fix x assume "a < x \$\$ i"
```
```   370     with reals_Archimedean[of "x \$\$ i - a"]
```
```   371     obtain n where "a + 1 / real (Suc n) < x \$\$ i"
```
```   372       by (auto simp: inverse_eq_divide field_simps)
```
```   373     then show "\<exists>n. a + 1 / real (Suc n) \<le> x \$\$ i"
```
```   374       by (blast intro: less_imp_le)
```
```   375   next
```
```   376     fix x n
```
```   377     have "a < a + 1 / real (Suc n)" by auto
```
```   378     also assume "\<dots> \<le> x"
```
```   379     finally show "a < x" .
```
```   380   qed
```
```   381   show "?set \<in> sets ?SIGMA" unfolding *
```
```   382     by (safe intro!: countable_UN Diff halfspace_lt_in_halfspace)
```
```   383 qed
```
```   384
```
```   385 lemma open_span_halfspace:
```
```   386   "sets borel \<subseteq> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x \$\$ i < a})\<rparr>)"
```
```   387     (is "_ \<subseteq> sets ?SIGMA")
```
```   388 proof -
```
```   389   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) simp
```
```   390   then interpret sigma_algebra ?SIGMA .
```
```   391   { fix S :: "'a set" assume "S \<in> open" then have "open S" unfolding mem_def .
```
```   392     from open_UNION[OF this]
```
```   393     obtain I where *: "S =
```
```   394       (\<Union>(a, b)\<in>I.
```
```   395           (\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) \$\$ i < x \$\$ i}) \<inter>
```
```   396           (\<Inter> i<DIM('a). {x. x \$\$ i < (Chi (real_of_rat \<circ> op ! b)::'a) \$\$ i}))"
```
```   397       unfolding greaterThanLessThan_def
```
```   398       unfolding eucl_greaterThan_eq_halfspaces[where 'a='a]
```
```   399       unfolding eucl_lessThan_eq_halfspaces[where 'a='a]
```
```   400       by blast
```
```   401     have "S \<in> sets ?SIGMA"
```
```   402       unfolding *
```
```   403       by (auto intro!: countable_UN Int countable_INT halfspace_lt_in_halfspace halfspace_gt_in_halfspace) }
```
```   404   then show ?thesis unfolding borel_def
```
```   405     by (intro sets_sigma_subset) auto
```
```   406 qed
```
```   407
```
```   408 lemma halfspace_span_halfspace_le:
```
```   409   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a})\<rparr>) \<subseteq>
```
```   410    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. x \$\$ i \<le> a})\<rparr>)"
```
```   411   (is "_ \<subseteq> sets ?SIGMA")
```
```   412 proof -
```
```   413   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   414   then interpret sigma_algebra ?SIGMA .
```
```   415   { fix a i
```
```   416     have *: "{x::'a. x\$\$i < a} = (\<Union>n. {x. x\$\$i \<le> a - 1/real (Suc n)})"
```
```   417     proof (safe, simp_all)
```
```   418       fix x::'a assume *: "x\$\$i < a"
```
```   419       with reals_Archimedean[of "a - x\$\$i"]
```
```   420       obtain n where "x \$\$ i < a - 1 / (real (Suc n))"
```
```   421         by (auto simp: field_simps inverse_eq_divide)
```
```   422       then show "\<exists>n. x \$\$ i \<le> a - 1 / (real (Suc n))"
```
```   423         by (blast intro: less_imp_le)
```
```   424     next
```
```   425       fix x::'a and n
```
```   426       assume "x\$\$i \<le> a - 1 / real (Suc n)"
```
```   427       also have "\<dots> < a" by auto
```
```   428       finally show "x\$\$i < a" .
```
```   429     qed
```
```   430     have "{x. x\$\$i < a} \<in> sets ?SIGMA" unfolding *
```
```   431       by (safe intro!: countable_UN)
```
```   432          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   433   then show ?thesis by (intro sets_sigma_subset) auto
```
```   434 qed
```
```   435
```
```   436 lemma halfspace_span_halfspace_ge:
```
```   437   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a})\<rparr>) \<subseteq>
```
```   438    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a \<le> x \$\$ i})\<rparr>)"
```
```   439   (is "_ \<subseteq> sets ?SIGMA")
```
```   440 proof -
```
```   441   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   442   then interpret sigma_algebra ?SIGMA .
```
```   443   { fix a i have *: "{x::'a. x\$\$i < a} = space ?SIGMA - {x::'a. a \<le> x\$\$i}" by auto
```
```   444     have "{x. x\$\$i < a} \<in> sets ?SIGMA" unfolding *
```
```   445       by (safe intro!: Diff)
```
```   446          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   447   then show ?thesis by (intro sets_sigma_subset) auto
```
```   448 qed
```
```   449
```
```   450 lemma halfspace_le_span_halfspace_gt:
```
```   451   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
```
```   452    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a < x \$\$ i})\<rparr>)"
```
```   453   (is "_ \<subseteq> sets ?SIGMA")
```
```   454 proof -
```
```   455   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   456   then interpret sigma_algebra ?SIGMA .
```
```   457   { fix a i have *: "{x::'a. x\$\$i \<le> a} = space ?SIGMA - {x::'a. a < x\$\$i}" by auto
```
```   458     have "{x. x\$\$i \<le> a} \<in> sets ?SIGMA" unfolding *
```
```   459       by (safe intro!: Diff)
```
```   460          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   461   then show ?thesis by (intro sets_sigma_subset) auto
```
```   462 qed
```
```   463
```
```   464 lemma halfspace_le_span_atMost:
```
```   465   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
```
```   466    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>)"
```
```   467   (is "_ \<subseteq> sets ?SIGMA")
```
```   468 proof -
```
```   469   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   470   then interpret sigma_algebra ?SIGMA .
```
```   471   have "\<And>a i. {x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   472   proof cases
```
```   473     fix a i assume "i < DIM('a)"
```
```   474     then have *: "{x::'a. x\$\$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})"
```
```   475     proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm)
```
```   476       fix x
```
```   477       from real_arch_simple[of "Max ((\<lambda>i. x\$\$i)`{..<DIM('a)})"] guess k::nat ..
```
```   478       then have "\<And>i. i < DIM('a) \<Longrightarrow> x\$\$i \<le> real k"
```
```   479         by (subst (asm) Max_le_iff) auto
```
```   480       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x \$\$ ia \<le> real k"
```
```   481         by (auto intro!: exI[of _ k])
```
```   482     qed
```
```   483     show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA" unfolding *
```
```   484       by (safe intro!: countable_UN)
```
```   485          (auto simp: sets_sigma intro!: sigma_sets.Basic)
```
```   486   next
```
```   487     fix a i assume "\<not> i < DIM('a)"
```
```   488     then show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   489       using top by auto
```
```   490   qed
```
```   491   then show ?thesis by (intro sets_sigma_subset) auto
```
```   492 qed
```
```   493
```
```   494 lemma halfspace_le_span_greaterThan:
```
```   495   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
```
```   496    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {a<..})\<rparr>)"
```
```   497   (is "_ \<subseteq> sets ?SIGMA")
```
```   498 proof -
```
```   499   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   500   then interpret sigma_algebra ?SIGMA .
```
```   501   have "\<And>a i. {x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   502   proof cases
```
```   503     fix a i assume "i < DIM('a)"
```
```   504     have "{x::'a. x\$\$i \<le> a} = space ?SIGMA - {x::'a. a < x\$\$i}" by auto
```
```   505     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)`
```
```   506     proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
```
```   507       fix x
```
```   508       from real_arch_lt[of "Max ((\<lambda>i. -x\$\$i)`{..<DIM('a)})"]
```
```   509       guess k::nat .. note k = this
```
```   510       { fix i assume "i < DIM('a)"
```
```   511         then have "-x\$\$i < real k"
```
```   512           using k by (subst (asm) Max_less_iff) auto
```
```   513         then have "- real k < x\$\$i" by simp }
```
```   514       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x \$\$ ia"
```
```   515         by (auto intro!: exI[of _ k])
```
```   516     qed
```
```   517     finally show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   518       apply (simp only:)
```
```   519       apply (safe intro!: countable_UN Diff)
```
```   520       by (auto simp: sets_sigma intro!: sigma_sets.Basic)
```
```   521   next
```
```   522     fix a i assume "\<not> i < DIM('a)"
```
```   523     then show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   524       using top by auto
```
```   525   qed
```
```   526   then show ?thesis by (intro sets_sigma_subset) auto
```
```   527 qed
```
```   528
```
```   529 lemma halfspace_le_span_lessThan:
```
```   530   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x \$\$ i})\<rparr>) \<subseteq>
```
```   531    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..<a})\<rparr>)"
```
```   532   (is "_ \<subseteq> sets ?SIGMA")
```
```   533 proof -
```
```   534   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   535   then interpret sigma_algebra ?SIGMA .
```
```   536   have "\<And>a i. {x. a \<le> x\$\$i} \<in> sets ?SIGMA"
```
```   537   proof cases
```
```   538     fix a i assume "i < DIM('a)"
```
```   539     have "{x::'a. a \<le> x\$\$i} = space ?SIGMA - {x::'a. x\$\$i < a}" by auto
```
```   540     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)`
```
```   541     proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
```
```   542       fix x
```
```   543       from real_arch_lt[of "Max ((\<lambda>i. x\$\$i)`{..<DIM('a)})"]
```
```   544       guess k::nat .. note k = this
```
```   545       { fix i assume "i < DIM('a)"
```
```   546         then have "x\$\$i < real k"
```
```   547           using k by (subst (asm) Max_less_iff) auto
```
```   548         then have "x\$\$i < real k" by simp }
```
```   549       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x \$\$ ia < real k"
```
```   550         by (auto intro!: exI[of _ k])
```
```   551     qed
```
```   552     finally show "{x. a \<le> x\$\$i} \<in> sets ?SIGMA"
```
```   553       apply (simp only:)
```
```   554       apply (safe intro!: countable_UN Diff)
```
```   555       by (auto simp: sets_sigma intro!: sigma_sets.Basic)
```
```   556   next
```
```   557     fix a i assume "\<not> i < DIM('a)"
```
```   558     then show "{x. a \<le> x\$\$i} \<in> sets ?SIGMA"
```
```   559       using top by auto
```
```   560   qed
```
```   561   then show ?thesis by (intro sets_sigma_subset) auto
```
```   562 qed
```
```   563
```
```   564 lemma atMost_span_atLeastAtMost:
```
```   565   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>) \<subseteq>
```
```   566    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a,b). {a..b})\<rparr>)"
```
```   567   (is "_ \<subseteq> sets ?SIGMA")
```
```   568 proof -
```
```   569   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   570   then interpret sigma_algebra ?SIGMA .
```
```   571   { fix a::'a
```
```   572     have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
```
```   573     proof (safe, simp_all add: eucl_le[where 'a='a])
```
```   574       fix x
```
```   575       from real_arch_simple[of "Max ((\<lambda>i. - x\$\$i)`{..<DIM('a)})"]
```
```   576       guess k::nat .. note k = this
```
```   577       { fix i assume "i < DIM('a)"
```
```   578         with k have "- x\$\$i \<le> real k"
```
```   579           by (subst (asm) Max_le_iff) (auto simp: field_simps)
```
```   580         then have "- real k \<le> x\$\$i" by simp }
```
```   581       then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x \$\$ i"
```
```   582         by (auto intro!: exI[of _ k])
```
```   583     qed
```
```   584     have "{..a} \<in> sets ?SIGMA" unfolding *
```
```   585       by (safe intro!: countable_UN)
```
```   586          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   587   then show ?thesis by (intro sets_sigma_subset) auto
```
```   588 qed
```
```   589
```
```   590 lemma borel_eq_atMost:
```
```   591   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> a. {.. a::'a\<Colon>ordered_euclidean_space})\<rparr>)"
```
```   592     (is "_ = ?SIGMA")
```
```   593 proof (intro algebra.equality antisym)
```
```   594   show "sets borel \<subseteq> sets ?SIGMA"
```
```   595     using halfspace_le_span_atMost halfspace_span_halfspace_le open_span_halfspace
```
```   596     by auto
```
```   597   show "sets ?SIGMA \<subseteq> sets borel"
```
```   598     by (rule borel.sets_sigma_subset) auto
```
```   599 qed auto
```
```   600
```
```   601 lemma borel_eq_atLeastAtMost:
```
```   602   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space, b). {a .. b})\<rparr>)"
```
```   603    (is "_ = ?SIGMA")
```
```   604 proof (intro algebra.equality antisym)
```
```   605   show "sets borel \<subseteq> sets ?SIGMA"
```
```   606     using atMost_span_atLeastAtMost halfspace_le_span_atMost
```
```   607       halfspace_span_halfspace_le open_span_halfspace
```
```   608     by auto
```
```   609   show "sets ?SIGMA \<subseteq> sets borel"
```
```   610     by (rule borel.sets_sigma_subset) auto
```
```   611 qed auto
```
```   612
```
```   613 lemma borel_eq_greaterThan:
```
```   614   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {a <..})\<rparr>)"
```
```   615    (is "_ = ?SIGMA")
```
```   616 proof (intro algebra.equality antisym)
```
```   617   show "sets borel \<subseteq> sets ?SIGMA"
```
```   618     using halfspace_le_span_greaterThan
```
```   619       halfspace_span_halfspace_le open_span_halfspace
```
```   620     by auto
```
```   621   show "sets ?SIGMA \<subseteq> sets borel"
```
```   622     by (rule borel.sets_sigma_subset) auto
```
```   623 qed auto
```
```   624
```
```   625 lemma borel_eq_lessThan:
```
```   626   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {..< a})\<rparr>)"
```
```   627    (is "_ = ?SIGMA")
```
```   628 proof (intro algebra.equality antisym)
```
```   629   show "sets borel \<subseteq> sets ?SIGMA"
```
```   630     using halfspace_le_span_lessThan
```
```   631       halfspace_span_halfspace_ge open_span_halfspace
```
```   632     by auto
```
```   633   show "sets ?SIGMA \<subseteq> sets borel"
```
```   634     by (rule borel.sets_sigma_subset) auto
```
```   635 qed auto
```
```   636
```
```   637 lemma borel_eq_greaterThanLessThan:
```
```   638   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, b). {a <..< (b :: 'a \<Colon> ordered_euclidean_space)})\<rparr>)"
```
```   639     (is "_ = ?SIGMA")
```
```   640 proof (intro algebra.equality antisym)
```
```   641   show "sets ?SIGMA \<subseteq> sets borel"
```
```   642     by (rule borel.sets_sigma_subset) auto
```
```   643   show "sets borel \<subseteq> sets ?SIGMA"
```
```   644   proof -
```
```   645     have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   646     then interpret sigma_algebra ?SIGMA .
```
```   647     { fix M :: "'a set" assume "M \<in> open"
```
```   648       then have "open M" by (simp add: mem_def)
```
```   649       have "M \<in> sets ?SIGMA"
```
```   650         apply (subst open_UNION[OF `open M`])
```
```   651         apply (safe intro!: countable_UN)
```
```   652         by (auto simp add: sigma_def intro!: sigma_sets.Basic) }
```
```   653     then show ?thesis
```
```   654       unfolding borel_def by (intro sets_sigma_subset) auto
```
```   655   qed
```
```   656 qed auto
```
```   657
```
```   658 lemma borel_eq_halfspace_le:
```
```   659   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x\$\$i \<le> a})\<rparr>)"
```
```   660    (is "_ = ?SIGMA")
```
```   661 proof (intro algebra.equality antisym)
```
```   662   show "sets borel \<subseteq> sets ?SIGMA"
```
```   663     using open_span_halfspace halfspace_span_halfspace_le by auto
```
```   664   show "sets ?SIGMA \<subseteq> sets borel"
```
```   665     by (rule borel.sets_sigma_subset) auto
```
```   666 qed auto
```
```   667
```
```   668 lemma borel_eq_halfspace_less:
```
```   669   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x\$\$i < a})\<rparr>)"
```
```   670    (is "_ = ?SIGMA")
```
```   671 proof (intro algebra.equality antisym)
```
```   672   show "sets borel \<subseteq> sets ?SIGMA"
```
```   673     using open_span_halfspace .
```
```   674   show "sets ?SIGMA \<subseteq> sets borel"
```
```   675     by (rule borel.sets_sigma_subset) auto
```
```   676 qed auto
```
```   677
```
```   678 lemma borel_eq_halfspace_gt:
```
```   679   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a < x\$\$i})\<rparr>)"
```
```   680    (is "_ = ?SIGMA")
```
```   681 proof (intro algebra.equality antisym)
```
```   682   show "sets borel \<subseteq> sets ?SIGMA"
```
```   683     using halfspace_le_span_halfspace_gt open_span_halfspace halfspace_span_halfspace_le by auto
```
```   684   show "sets ?SIGMA \<subseteq> sets borel"
```
```   685     by (rule borel.sets_sigma_subset) auto
```
```   686 qed auto
```
```   687
```
```   688 lemma borel_eq_halfspace_ge:
```
```   689   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a \<le> x\$\$i})\<rparr>)"
```
```   690    (is "_ = ?SIGMA")
```
```   691 proof (intro algebra.equality antisym)
```
```   692   show "sets borel \<subseteq> sets ?SIGMA"
```
```   693     using halfspace_span_halfspace_ge open_span_halfspace by auto
```
```   694   show "sets ?SIGMA \<subseteq> sets borel"
```
```   695     by (rule borel.sets_sigma_subset) auto
```
```   696 qed auto
```
```   697
```
```   698 lemma (in sigma_algebra) borel_measurable_halfspacesI:
```
```   699   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   700   assumes "borel = (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
```
```   701   and "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
```
```   702   and "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M"
```
```   703   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)"
```
```   704 proof safe
```
```   705   fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M"
```
```   706   then show "S a i \<in> sets M" unfolding assms
```
```   707     by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1) sigma_def)
```
```   708 next
```
```   709   assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M"
```
```   710   { fix a i have "S a i \<in> sets M"
```
```   711     proof cases
```
```   712       assume "i < DIM('c)"
```
```   713       with a show ?thesis unfolding assms(2) by simp
```
```   714     next
```
```   715       assume "\<not> i < DIM('c)"
```
```   716       from assms(3)[OF this] show ?thesis .
```
```   717     qed }
```
```   718   then have "f \<in> measurable M (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
```
```   719     by (auto intro!: measurable_sigma simp: assms(2))
```
```   720   then show "f \<in> borel_measurable M" unfolding measurable_def
```
```   721     unfolding assms(1) by simp
```
```   722 qed
```
```   723
```
```   724 lemma (in sigma_algebra) borel_measurable_iff_halfspace_le:
```
```   725   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   726   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w \$\$ i \<le> a} \<in> sets M)"
```
```   727   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
```
```   728
```
```   729 lemma (in sigma_algebra) borel_measurable_iff_halfspace_less:
```
```   730   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   731   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w \$\$ i < a} \<in> sets M)"
```
```   732   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
```
```   733
```
```   734 lemma (in sigma_algebra) borel_measurable_iff_halfspace_ge:
```
```   735   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   736   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w \$\$ i} \<in> sets M)"
```
```   737   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
```
```   738
```
```   739 lemma (in sigma_algebra) borel_measurable_iff_halfspace_greater:
```
```   740   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   741   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w \$\$ i} \<in> sets M)"
```
```   742   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_gt]) auto
```
```   743
```
```   744 lemma (in sigma_algebra) borel_measurable_iff_le:
```
```   745   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
```
```   746   using borel_measurable_iff_halfspace_le[where 'c=real] by simp
```
```   747
```
```   748 lemma (in sigma_algebra) borel_measurable_iff_less:
```
```   749   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
```
```   750   using borel_measurable_iff_halfspace_less[where 'c=real] by simp
```
```   751
```
```   752 lemma (in sigma_algebra) borel_measurable_iff_ge:
```
```   753   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
```
```   754   using borel_measurable_iff_halfspace_ge[where 'c=real] by simp
```
```   755
```
```   756 lemma (in sigma_algebra) borel_measurable_iff_greater:
```
```   757   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
```
```   758   using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
```
```   759
```
```   760 lemma borel_measurable_euclidean_component:
```
```   761   "(\<lambda>x::'a::euclidean_space. x \$\$ i) \<in> borel_measurable borel"
```
```   762   unfolding borel_def[where 'a=real]
```
```   763 proof (rule borel.measurable_sigma, simp_all)
```
```   764   fix S::"real set" assume "S \<in> open" then have "open S" unfolding mem_def .
```
```   765   from open_vimage_euclidean_component[OF this]
```
```   766   show "(\<lambda>x. x \$\$ i) -` S \<in> sets borel"
```
```   767     by (auto intro: borel_open)
```
```   768 qed
```
```   769
```
```   770 lemma (in sigma_algebra) borel_measurable_euclidean_space:
```
```   771   fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
```
```   772   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x \$\$ i) \<in> borel_measurable M)"
```
```   773 proof safe
```
```   774   fix i assume "f \<in> borel_measurable M"
```
```   775   then show "(\<lambda>x. f x \$\$ i) \<in> borel_measurable M"
```
```   776     using measurable_comp[of f _ _ "\<lambda>x. x \$\$ i", unfolded comp_def]
```
```   777     by (auto intro: borel_measurable_euclidean_component)
```
```   778 next
```
```   779   assume f: "\<forall>i<DIM('c). (\<lambda>x. f x \$\$ i) \<in> borel_measurable M"
```
```   780   then show "f \<in> borel_measurable M"
```
```   781     unfolding borel_measurable_iff_halfspace_le by auto
```
```   782 qed
```
```   783
```
```   784 subsection "Borel measurable operators"
```
```   785
```
```   786 lemma (in sigma_algebra) affine_borel_measurable_vector:
```
```   787   fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
```
```   788   assumes "f \<in> borel_measurable M"
```
```   789   shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
```
```   790 proof (rule borel_measurableI)
```
```   791   fix S :: "'x set" assume "open S"
```
```   792   show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
```
```   793   proof cases
```
```   794     assume "b \<noteq> 0"
```
```   795     with `open S` have "((\<lambda>x. (- a + x) /\<^sub>R b) ` S) \<in> open" (is "?S \<in> open")
```
```   796       by (auto intro!: open_affinity simp: scaleR.add_right mem_def)
```
```   797     hence "?S \<in> sets borel"
```
```   798       unfolding borel_def by (auto simp: sigma_def intro!: sigma_sets.Basic)
```
```   799     moreover
```
```   800     from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
```
```   801       apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
```
```   802     ultimately show ?thesis using assms unfolding in_borel_measurable_borel
```
```   803       by auto
```
```   804   qed simp
```
```   805 qed
```
```   806
```
```   807 lemma (in sigma_algebra) affine_borel_measurable:
```
```   808   fixes g :: "'a \<Rightarrow> real"
```
```   809   assumes g: "g \<in> borel_measurable M"
```
```   810   shows "(\<lambda>x. a + (g x) * b) \<in> borel_measurable M"
```
```   811   using affine_borel_measurable_vector[OF assms] by (simp add: mult_commute)
```
```   812
```
```   813 lemma (in sigma_algebra) borel_measurable_add[simp, intro]:
```
```   814   fixes f :: "'a \<Rightarrow> real"
```
```   815   assumes f: "f \<in> borel_measurable M"
```
```   816   assumes g: "g \<in> borel_measurable M"
```
```   817   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```   818 proof -
```
```   819   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}"
```
```   820     by auto
```
```   821   have "\<And>a. (\<lambda>w. a + (g w) * -1) \<in> borel_measurable M"
```
```   822     by (rule affine_borel_measurable [OF g])
```
```   823   then have "\<And>a. {w \<in> space M. (\<lambda>w. a + (g w) * -1)(w) \<le> f w} \<in> sets M" using f
```
```   824     by auto
```
```   825   then have "\<And>a. {w \<in> space M. a \<le> f w + g w} \<in> sets M"
```
```   826     by (simp add: 1)
```
```   827   then show ?thesis
```
```   828     by (simp add: borel_measurable_iff_ge)
```
```   829 qed
```
```   830
```
```   831 lemma (in sigma_algebra) borel_measurable_setsum[simp, intro]:
```
```   832   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
```
```   833   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   834   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```   835 proof cases
```
```   836   assume "finite S"
```
```   837   thus ?thesis using assms by induct auto
```
```   838 qed simp
```
```   839
```
```   840 lemma (in sigma_algebra) borel_measurable_square:
```
```   841   fixes f :: "'a \<Rightarrow> real"
```
```   842   assumes f: "f \<in> borel_measurable M"
```
```   843   shows "(\<lambda>x. (f x)^2) \<in> borel_measurable M"
```
```   844 proof -
```
```   845   {
```
```   846     fix a
```
```   847     have "{w \<in> space M. (f w)\<twosuperior> \<le> a} \<in> sets M"
```
```   848     proof (cases rule: linorder_cases [of a 0])
```
```   849       case less
```
```   850       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = {}"
```
```   851         by auto (metis less order_le_less_trans power2_less_0)
```
```   852       also have "... \<in> sets M"
```
```   853         by (rule empty_sets)
```
```   854       finally show ?thesis .
```
```   855     next
```
```   856       case equal
```
```   857       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
```
```   858              {w \<in> space M. f w \<le> 0} \<inter> {w \<in> space M. 0 \<le> f w}"
```
```   859         by auto
```
```   860       also have "... \<in> sets M"
```
```   861         apply (insert f)
```
```   862         apply (rule Int)
```
```   863         apply (simp add: borel_measurable_iff_le)
```
```   864         apply (simp add: borel_measurable_iff_ge)
```
```   865         done
```
```   866       finally show ?thesis .
```
```   867     next
```
```   868       case greater
```
```   869       have "\<forall>x. (f x ^ 2 \<le> sqrt a ^ 2) = (- sqrt a  \<le> f x & f x \<le> sqrt a)"
```
```   870         by (metis abs_le_interval_iff abs_of_pos greater real_sqrt_abs
```
```   871                   real_sqrt_le_iff real_sqrt_power)
```
```   872       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
```
```   873              {w \<in> space M. -(sqrt a) \<le> f w} \<inter> {w \<in> space M. f w \<le> sqrt a}"
```
```   874         using greater by auto
```
```   875       also have "... \<in> sets M"
```
```   876         apply (insert f)
```
```   877         apply (rule Int)
```
```   878         apply (simp add: borel_measurable_iff_ge)
```
```   879         apply (simp add: borel_measurable_iff_le)
```
```   880         done
```
```   881       finally show ?thesis .
```
```   882     qed
```
```   883   }
```
```   884   thus ?thesis by (auto simp add: borel_measurable_iff_le)
```
```   885 qed
```
```   886
```
```   887 lemma times_eq_sum_squares:
```
```   888    fixes x::real
```
```   889    shows"x*y = ((x+y)^2)/4 - ((x-y)^ 2)/4"
```
```   890 by (simp add: power2_eq_square ring_distribs diff_divide_distrib [symmetric])
```
```   891
```
```   892 lemma (in sigma_algebra) borel_measurable_uminus[simp, intro]:
```
```   893   fixes g :: "'a \<Rightarrow> real"
```
```   894   assumes g: "g \<in> borel_measurable M"
```
```   895   shows "(\<lambda>x. - g x) \<in> borel_measurable M"
```
```   896 proof -
```
```   897   have "(\<lambda>x. - g x) = (\<lambda>x. 0 + (g x) * -1)"
```
```   898     by simp
```
```   899   also have "... \<in> borel_measurable M"
```
```   900     by (fast intro: affine_borel_measurable g)
```
```   901   finally show ?thesis .
```
```   902 qed
```
```   903
```
```   904 lemma (in sigma_algebra) borel_measurable_times[simp, intro]:
```
```   905   fixes f :: "'a \<Rightarrow> real"
```
```   906   assumes f: "f \<in> borel_measurable M"
```
```   907   assumes g: "g \<in> borel_measurable M"
```
```   908   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```   909 proof -
```
```   910   have 1: "(\<lambda>x. 0 + (f x + g x)\<twosuperior> * inverse 4) \<in> borel_measurable M"
```
```   911     using assms by (fast intro: affine_borel_measurable borel_measurable_square)
```
```   912   have "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) =
```
```   913         (\<lambda>x. 0 + ((f x + -g x) ^ 2 * inverse -4))"
```
```   914     by (simp add: minus_divide_right)
```
```   915   also have "... \<in> borel_measurable M"
```
```   916     using f g by (fast intro: affine_borel_measurable borel_measurable_square f g)
```
```   917   finally have 2: "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) \<in> borel_measurable M" .
```
```   918   show ?thesis
```
```   919     apply (simp add: times_eq_sum_squares diff_minus)
```
```   920     using 1 2 by simp
```
```   921 qed
```
```   922
```
```   923 lemma (in sigma_algebra) borel_measurable_setprod[simp, intro]:
```
```   924   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
```
```   925   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   926   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```   927 proof cases
```
```   928   assume "finite S"
```
```   929   thus ?thesis using assms by induct auto
```
```   930 qed simp
```
```   931
```
```   932 lemma (in sigma_algebra) borel_measurable_diff[simp, intro]:
```
```   933   fixes f :: "'a \<Rightarrow> real"
```
```   934   assumes f: "f \<in> borel_measurable M"
```
```   935   assumes g: "g \<in> borel_measurable M"
```
```   936   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```   937   unfolding diff_minus using assms by fast
```
```   938
```
```   939 lemma (in sigma_algebra) borel_measurable_inverse[simp, intro]:
```
```   940   fixes f :: "'a \<Rightarrow> real"
```
```   941   assumes "f \<in> borel_measurable M"
```
```   942   shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
```
```   943   unfolding borel_measurable_iff_ge unfolding inverse_eq_divide
```
```   944 proof safe
```
```   945   fix a :: real
```
```   946   have *: "{w \<in> space M. a \<le> 1 / f w} =
```
```   947       ({w \<in> space M. 0 < f w} \<inter> {w \<in> space M. a * f w \<le> 1}) \<union>
```
```   948       ({w \<in> space M. f w < 0} \<inter> {w \<in> space M. 1 \<le> a * f w}) \<union>
```
```   949       ({w \<in> space M. f w = 0} \<inter> {w \<in> space M. a \<le> 0})" by (auto simp: le_divide_eq)
```
```   950   show "{w \<in> space M. a \<le> 1 / f w} \<in> sets M" using assms unfolding *
```
```   951     by (auto intro!: Int Un)
```
```   952 qed
```
```   953
```
```   954 lemma (in sigma_algebra) borel_measurable_divide[simp, intro]:
```
```   955   fixes f :: "'a \<Rightarrow> real"
```
```   956   assumes "f \<in> borel_measurable M"
```
```   957   and "g \<in> borel_measurable M"
```
```   958   shows "(\<lambda>x. f x / g x) \<in> borel_measurable M"
```
```   959   unfolding field_divide_inverse
```
```   960   by (rule borel_measurable_inverse borel_measurable_times assms)+
```
```   961
```
```   962 lemma (in sigma_algebra) borel_measurable_max[intro, simp]:
```
```   963   fixes f g :: "'a \<Rightarrow> real"
```
```   964   assumes "f \<in> borel_measurable M"
```
```   965   assumes "g \<in> borel_measurable M"
```
```   966   shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
```
```   967   unfolding borel_measurable_iff_le
```
```   968 proof safe
```
```   969   fix a
```
```   970   have "{x \<in> space M. max (g x) (f x) \<le> a} =
```
```   971     {x \<in> space M. g x \<le> a} \<inter> {x \<in> space M. f x \<le> a}" by auto
```
```   972   thus "{x \<in> space M. max (g x) (f x) \<le> a} \<in> sets M"
```
```   973     using assms unfolding borel_measurable_iff_le
```
```   974     by (auto intro!: Int)
```
```   975 qed
```
```   976
```
```   977 lemma (in sigma_algebra) borel_measurable_min[intro, simp]:
```
```   978   fixes f g :: "'a \<Rightarrow> real"
```
```   979   assumes "f \<in> borel_measurable M"
```
```   980   assumes "g \<in> borel_measurable M"
```
```   981   shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
```
```   982   unfolding borel_measurable_iff_ge
```
```   983 proof safe
```
```   984   fix a
```
```   985   have "{x \<in> space M. a \<le> min (g x) (f x)} =
```
```   986     {x \<in> space M. a \<le> g x} \<inter> {x \<in> space M. a \<le> f x}" by auto
```
```   987   thus "{x \<in> space M. a \<le> min (g x) (f x)} \<in> sets M"
```
```   988     using assms unfolding borel_measurable_iff_ge
```
```   989     by (auto intro!: Int)
```
```   990 qed
```
```   991
```
```   992 lemma (in sigma_algebra) borel_measurable_abs[simp, intro]:
```
```   993   assumes "f \<in> borel_measurable M"
```
```   994   shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
```
```   995 proof -
```
```   996   have *: "\<And>x. \<bar>f x\<bar> = max 0 (f x) + max 0 (- f x)" by (simp add: max_def)
```
```   997   show ?thesis unfolding * using assms by auto
```
```   998 qed
```
```   999
```
```  1000 lemma borel_measurable_nth[simp, intro]:
```
```  1001   "(\<lambda>x::real^'n. x \$ i) \<in> borel_measurable borel"
```
```  1002   using borel_measurable_euclidean_component
```
```  1003   unfolding nth_conv_component by auto
```
```  1004
```
```  1005 lemma borel_measurable_continuous_on1:
```
```  1006   fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
```
```  1007   assumes "continuous_on UNIV f"
```
```  1008   shows "f \<in> borel_measurable borel"
```
```  1009   apply(rule borel.borel_measurableI)
```
```  1010   using continuous_open_preimage[OF assms] unfolding vimage_def by auto
```
```  1011
```
```  1012 lemma borel_measurable_continuous_on:
```
```  1013   fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
```
```  1014   assumes cont: "continuous_on A f" "open A" and f: "f -` {c} \<inter> A \<in> sets borel"
```
```  1015   shows "(\<lambda>x. if x \<in> A then f x else c) \<in> borel_measurable borel" (is "?f \<in> _")
```
```  1016 proof (rule borel.borel_measurableI)
```
```  1017   fix S :: "'b set" assume "open S"
```
```  1018   then have "open {x\<in>A. f x \<in> S - {c}}"
```
```  1019     by (intro continuous_open_preimage[OF cont]) auto
```
```  1020   then have *: "{x\<in>A. f x \<in> S - {c}} \<in> sets borel" by auto
```
```  1021   show "?f -` S \<inter> space borel \<in> sets borel"
```
```  1022   proof cases
```
```  1023     assume "c \<in> S"
```
```  1024     then have "?f -` S = {x\<in>A. f x \<in> S - {c}} \<union> (f -` {c} \<inter> A) \<union> -A"
```
```  1025       by auto
```
```  1026     with * show "?f -` S \<inter> space borel \<in> sets borel"
```
```  1027       using `open A` f by (auto intro!: borel.Un)
```
```  1028   next
```
```  1029     assume "c \<notin> S"
```
```  1030     then have "?f -` S = {x\<in>A. f x \<in> S - {c}}" by (auto split: split_if_asm)
```
```  1031     with * show "?f -` S \<inter> space borel \<in> sets borel" by auto
```
```  1032   qed
```
```  1033 qed
```
```  1034
```
```  1035 lemma borel_measurable_borel_log: assumes "1 < b" shows "log b \<in> borel_measurable borel"
```
```  1036 proof -
```
```  1037   { fix x :: real assume x: "x \<le> 0"
```
```  1038     { fix x::real assume "x \<le> 0" then have "\<And>u. exp u = x \<longleftrightarrow> False" by auto }
```
```  1039     from this[of x] x this[of 0] have "log b 0 = log b x"
```
```  1040       by (auto simp: ln_def log_def) }
```
```  1041   note log_imp = this
```
```  1042   have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) \<in> borel_measurable borel"
```
```  1043   proof (rule borel_measurable_continuous_on)
```
```  1044     show "continuous_on {0<..} (log b)"
```
```  1045       by (auto intro!: continuous_at_imp_continuous_on DERIV_log DERIV_isCont
```
```  1046                simp: continuous_isCont[symmetric])
```
```  1047     show "open ({0<..}::real set)" by auto
```
```  1048     show "log b -` {log b 0} \<inter> {0<..} \<in> sets borel"
```
```  1049     proof cases
```
```  1050       assume "log b -` {log b 0} \<inter> {0<..} = {}"
```
```  1051       then show ?thesis by simp
```
```  1052     next
```
```  1053       assume "log b -` {log b 0} \<inter> {0<..} \<noteq> {}"
```
```  1054       then obtain x where "0 < x" "log b x = log b 0" by auto
```
```  1055       with log_inj[OF `1 < b`] have "log b -` {log b 0} \<inter> {0<..} = {x}"
```
```  1056         by (auto simp: inj_on_def)
```
```  1057       then show ?thesis by simp
```
```  1058     qed
```
```  1059   qed
```
```  1060   also have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) = log b"
```
```  1061     by (simp add: fun_eq_iff not_less log_imp)
```
```  1062   finally show ?thesis .
```
```  1063 qed
```
```  1064
```
```  1065 lemma (in sigma_algebra) borel_measurable_log[simp,intro]:
```
```  1066   assumes f: "f \<in> borel_measurable M" and "1 < b"
```
```  1067   shows "(\<lambda>x. log b (f x)) \<in> borel_measurable M"
```
```  1068   using measurable_comp[OF f borel_measurable_borel_log[OF `1 < b`]]
```
```  1069   by (simp add: comp_def)
```
```  1070
```
```  1071 subsection "Borel space on the extended reals"
```
```  1072
```
```  1073 lemma borel_measurable_extreal_borel:
```
```  1074   "extreal \<in> borel_measurable borel"
```
```  1075   unfolding borel_def[where 'a=extreal]
```
```  1076 proof (rule borel.measurable_sigma)
```
```  1077   fix X :: "extreal set" assume "X \<in> sets \<lparr>space = UNIV, sets = open \<rparr>"
```
```  1078   then have "open X" by (auto simp: mem_def)
```
```  1079   then have "open (extreal -` X \<inter> space borel)"
```
```  1080     by (simp add: open_extreal_vimage)
```
```  1081   then show "extreal -` X \<inter> space borel \<in> sets borel" by auto
```
```  1082 qed auto
```
```  1083
```
```  1084 lemma (in sigma_algebra) borel_measurable_extreal[simp, intro]:
```
```  1085   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. extreal (f x)) \<in> borel_measurable M"
```
```  1086   using measurable_comp[OF f borel_measurable_extreal_borel] unfolding comp_def .
```
```  1087
```
```  1088 lemma borel_measurable_real_of_extreal_borel:
```
```  1089   "(real :: extreal \<Rightarrow> real) \<in> borel_measurable borel"
```
```  1090   unfolding borel_def[where 'a=real]
```
```  1091 proof (rule borel.measurable_sigma)
```
```  1092   fix B :: "real set" assume "B \<in> sets \<lparr>space = UNIV, sets = open \<rparr>"
```
```  1093   then have "open B" by (auto simp: mem_def)
```
```  1094   have *: "extreal -` real -` (B - {0}) = B - {0}" by auto
```
```  1095   have open_real: "open (real -` (B - {0}) :: extreal set)"
```
```  1096     unfolding open_extreal_def * using `open B` by auto
```
```  1097   show "(real -` B \<inter> space borel :: extreal set) \<in> sets borel"
```
```  1098   proof cases
```
```  1099     assume "0 \<in> B"
```
```  1100     then have *: "real -` B = real -` (B - {0}) \<union> {-\<infinity>, \<infinity>, 0}"
```
```  1101       by (auto simp add: real_of_extreal_eq_0)
```
```  1102     then show "(real -` B :: extreal set) \<inter> space borel \<in> sets borel"
```
```  1103       using open_real by auto
```
```  1104   next
```
```  1105     assume "0 \<notin> B"
```
```  1106     then have *: "(real -` B :: extreal set) = real -` (B - {0})"
```
```  1107       by (auto simp add: real_of_extreal_eq_0)
```
```  1108     then show "(real -` B :: extreal set) \<inter> space borel \<in> sets borel"
```
```  1109       using open_real by auto
```
```  1110   qed
```
```  1111 qed auto
```
```  1112
```
```  1113 lemma (in sigma_algebra) borel_measurable_real_of_extreal[simp, intro]:
```
```  1114   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. real (f x :: extreal)) \<in> borel_measurable M"
```
```  1115   using measurable_comp[OF f borel_measurable_real_of_extreal_borel] unfolding comp_def .
```
```  1116
```
```  1117 lemma (in sigma_algebra) borel_measurable_extreal_iff:
```
```  1118   shows "(\<lambda>x. extreal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
```
```  1119 proof
```
```  1120   assume "(\<lambda>x. extreal (f x)) \<in> borel_measurable M"
```
```  1121   from borel_measurable_real_of_extreal[OF this]
```
```  1122   show "f \<in> borel_measurable M" by auto
```
```  1123 qed auto
```
```  1124
```
```  1125 lemma (in sigma_algebra) borel_measurable_extreal_iff_real:
```
```  1126   "f \<in> borel_measurable M \<longleftrightarrow>
```
```  1127     ((\<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)"
```
```  1128 proof safe
```
```  1129   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"
```
```  1130   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
```
```  1131   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
```
```  1132   let "?f x" = "if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else extreal (real (f x))"
```
```  1133   have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
```
```  1134   also have "?f = f" by (auto simp: fun_eq_iff extreal_real)
```
```  1135   finally show "f \<in> borel_measurable M" .
```
```  1136 qed (auto intro: measurable_sets borel_measurable_real_of_extreal)
```
```  1137
```
```  1138 lemma (in sigma_algebra) less_eq_ge_measurable:
```
```  1139   fixes f :: "'a \<Rightarrow> 'c::linorder"
```
```  1140   shows "f -` {a <..} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {..a} \<inter> space M \<in> sets M"
```
```  1141 proof
```
```  1142   assume "f -` {a <..} \<inter> space M \<in> sets M"
```
```  1143   moreover have "f -` {..a} \<inter> space M = space M - f -` {a <..} \<inter> space M" by auto
```
```  1144   ultimately show "f -` {..a} \<inter> space M \<in> sets M" by auto
```
```  1145 next
```
```  1146   assume "f -` {..a} \<inter> space M \<in> sets M"
```
```  1147   moreover have "f -` {a <..} \<inter> space M = space M - f -` {..a} \<inter> space M" by auto
```
```  1148   ultimately show "f -` {a <..} \<inter> space M \<in> sets M" by auto
```
```  1149 qed
```
```  1150
```
```  1151 lemma (in sigma_algebra) greater_eq_le_measurable:
```
```  1152   fixes f :: "'a \<Rightarrow> 'c::linorder"
```
```  1153   shows "f -` {..< a} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {a ..} \<inter> space M \<in> sets M"
```
```  1154 proof
```
```  1155   assume "f -` {a ..} \<inter> space M \<in> sets M"
```
```  1156   moreover have "f -` {..< a} \<inter> space M = space M - f -` {a ..} \<inter> space M" by auto
```
```  1157   ultimately show "f -` {..< a} \<inter> space M \<in> sets M" by auto
```
```  1158 next
```
```  1159   assume "f -` {..< a} \<inter> space M \<in> sets M"
```
```  1160   moreover have "f -` {a ..} \<inter> space M = space M - f -` {..< a} \<inter> space M" by auto
```
```  1161   ultimately show "f -` {a ..} \<inter> space M \<in> sets M" by auto
```
```  1162 qed
```
```  1163
```
```  1164 lemma (in sigma_algebra) borel_measurable_uminus_borel_extreal:
```
```  1165   "(uminus :: extreal \<Rightarrow> extreal) \<in> borel_measurable borel"
```
```  1166 proof (subst borel_def, rule borel.measurable_sigma)
```
```  1167   fix X :: "extreal set" assume "X \<in> sets \<lparr>space = UNIV, sets = open\<rparr>"
```
```  1168   then have "open X" by (simp add: mem_def)
```
```  1169   have "uminus -` X = uminus ` X" by (force simp: image_iff)
```
```  1170   then have "open (uminus -` X)" using `open X` extreal_open_uminus by auto
```
```  1171   then show "uminus -` X \<inter> space borel \<in> sets borel" by auto
```
```  1172 qed auto
```
```  1173
```
```  1174 lemma (in sigma_algebra) borel_measurable_uminus_extreal[intro]:
```
```  1175   assumes "f \<in> borel_measurable M"
```
```  1176   shows "(\<lambda>x. - f x :: extreal) \<in> borel_measurable M"
```
```  1177   using measurable_comp[OF assms borel_measurable_uminus_borel_extreal] by (simp add: comp_def)
```
```  1178
```
```  1179 lemma (in sigma_algebra) borel_measurable_uminus_eq_extreal[simp]:
```
```  1180   "(\<lambda>x. - f x :: extreal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
```
```  1181 proof
```
```  1182   assume ?l from borel_measurable_uminus_extreal[OF this] show ?r by simp
```
```  1183 qed auto
```
```  1184
```
```  1185 lemma (in sigma_algebra) borel_measurable_eq_atMost_extreal:
```
```  1186   "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
```
```  1187 proof (intro iffI allI)
```
```  1188   assume pos[rule_format]: "\<forall>a. f -` {..a} \<inter> space M \<in> sets M"
```
```  1189   show "f \<in> borel_measurable M"
```
```  1190     unfolding borel_measurable_extreal_iff_real borel_measurable_iff_le
```
```  1191   proof (intro conjI allI)
```
```  1192     fix a :: real
```
```  1193     { fix x :: extreal assume *: "\<forall>i::nat. real i < x"
```
```  1194       have "x = \<infinity>"
```
```  1195       proof (rule extreal_top)
```
```  1196         fix B from real_arch_lt[of B] guess n ..
```
```  1197         then have "extreal B < real n" by auto
```
```  1198         with * show "B \<le> x" by (metis less_trans less_imp_le)
```
```  1199       qed }
```
```  1200     then have "f -` {\<infinity>} \<inter> space M = space M - (\<Union>i::nat. f -` {.. real i} \<inter> space M)"
```
```  1201       by (auto simp: not_le)
```
```  1202     then show "f -` {\<infinity>} \<inter> space M \<in> sets M" using pos by (auto simp del: UN_simps intro!: Diff)
```
```  1203     moreover
```
```  1204     have "{-\<infinity>} = {..-\<infinity>}" by auto
```
```  1205     then show "f -` {-\<infinity>} \<inter> space M \<in> sets M" using pos by auto
```
```  1206     moreover have "{x\<in>space M. f x \<le> extreal a} \<in> sets M"
```
```  1207       using pos[of "extreal a"] by (simp add: vimage_def Int_def conj_commute)
```
```  1208     moreover have "{w \<in> space M. real (f w) \<le> a} =
```
```  1209       (if a < 0 then {w \<in> space M. f w \<le> extreal a} - f -` {-\<infinity>} \<inter> space M
```
```  1210       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")
```
```  1211       proof (intro set_eqI) fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases "f x") auto qed
```
```  1212     ultimately show "{w \<in> space M. real (f w) \<le> a} \<in> sets M" by auto
```
```  1213   qed
```
```  1214 qed (simp add: measurable_sets)
```
```  1215
```
```  1216 lemma (in sigma_algebra) borel_measurable_eq_atLeast_extreal:
```
```  1217   "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
```
```  1218 proof
```
```  1219   assume pos: "\<forall>a. f -` {a..} \<inter> space M \<in> sets M"
```
```  1220   moreover have "\<And>a. (\<lambda>x. - f x) -` {..a} = f -` {-a ..}"
```
```  1221     by (auto simp: extreal_uminus_le_reorder)
```
```  1222   ultimately have "(\<lambda>x. - f x) \<in> borel_measurable M"
```
```  1223     unfolding borel_measurable_eq_atMost_extreal by auto
```
```  1224   then show "f \<in> borel_measurable M" by simp
```
```  1225 qed (simp add: measurable_sets)
```
```  1226
```
```  1227 lemma (in sigma_algebra) borel_measurable_extreal_iff_less:
```
```  1228   "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
```
```  1229   unfolding borel_measurable_eq_atLeast_extreal greater_eq_le_measurable ..
```
```  1230
```
```  1231 lemma (in sigma_algebra) borel_measurable_extreal_iff_ge:
```
```  1232   "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
```
```  1233   unfolding borel_measurable_eq_atMost_extreal less_eq_ge_measurable ..
```
```  1234
```
```  1235 lemma (in sigma_algebra) borel_measurable_extreal_eq_const:
```
```  1236   fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M"
```
```  1237   shows "{x\<in>space M. f x = c} \<in> sets M"
```
```  1238 proof -
```
```  1239   have "{x\<in>space M. f x = c} = (f -` {c} \<inter> space M)" by auto
```
```  1240   then show ?thesis using assms by (auto intro!: measurable_sets)
```
```  1241 qed
```
```  1242
```
```  1243 lemma (in sigma_algebra) borel_measurable_extreal_neq_const:
```
```  1244   fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M"
```
```  1245   shows "{x\<in>space M. f x \<noteq> c} \<in> sets M"
```
```  1246 proof -
```
```  1247   have "{x\<in>space M. f x \<noteq> c} = space M - (f -` {c} \<inter> space M)" by auto
```
```  1248   then show ?thesis using assms by (auto intro!: measurable_sets)
```
```  1249 qed
```
```  1250
```
```  1251 lemma (in sigma_algebra) borel_measurable_extreal_le[intro,simp]:
```
```  1252   fixes f g :: "'a \<Rightarrow> extreal"
```
```  1253   assumes f: "f \<in> borel_measurable M"
```
```  1254   assumes g: "g \<in> borel_measurable M"
```
```  1255   shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
```
```  1256 proof -
```
```  1257   have "{x \<in> space M. f x \<le> g x} =
```
```  1258     {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>
```
```  1259     f -` {-\<infinity>} \<inter> space M \<union> g -` {\<infinity>} \<inter> space M" (is "?l = ?r")
```
```  1260   proof (intro set_eqI)
```
```  1261     fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases rule: extreal2_cases[of "f x" "g x"]) auto
```
```  1262   qed
```
```  1263   with f g show ?thesis by (auto intro!: Un simp: measurable_sets)
```
```  1264 qed
```
```  1265
```
```  1266 lemma (in sigma_algebra) borel_measurable_extreal_less[intro,simp]:
```
```  1267   fixes f :: "'a \<Rightarrow> extreal"
```
```  1268   assumes f: "f \<in> borel_measurable M"
```
```  1269   assumes g: "g \<in> borel_measurable M"
```
```  1270   shows "{x \<in> space M. f x < g x} \<in> sets M"
```
```  1271 proof -
```
```  1272   have "{x \<in> space M. f x < g x} = space M - {x \<in> space M. g x \<le> f x}" by auto
```
```  1273   then show ?thesis using g f by auto
```
```  1274 qed
```
```  1275
```
```  1276 lemma (in sigma_algebra) borel_measurable_extreal_eq[intro,simp]:
```
```  1277   fixes f :: "'a \<Rightarrow> extreal"
```
```  1278   assumes f: "f \<in> borel_measurable M"
```
```  1279   assumes g: "g \<in> borel_measurable M"
```
```  1280   shows "{w \<in> space M. f w = g w} \<in> sets M"
```
```  1281 proof -
```
```  1282   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
```
```  1283   then show ?thesis using g f by auto
```
```  1284 qed
```
```  1285
```
```  1286 lemma (in sigma_algebra) borel_measurable_extreal_neq[intro,simp]:
```
```  1287   fixes f :: "'a \<Rightarrow> extreal"
```
```  1288   assumes f: "f \<in> borel_measurable M"
```
```  1289   assumes g: "g \<in> borel_measurable M"
```
```  1290   shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
```
```  1291 proof -
```
```  1292   have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}" by auto
```
```  1293   thus ?thesis using f g by auto
```
```  1294 qed
```
```  1295
```
```  1296 lemma (in sigma_algebra) split_sets:
```
```  1297   "{x\<in>space M. P x \<or> Q x} = {x\<in>space M. P x} \<union> {x\<in>space M. Q x}"
```
```  1298   "{x\<in>space M. P x \<and> Q x} = {x\<in>space M. P x} \<inter> {x\<in>space M. Q x}"
```
```  1299   by auto
```
```  1300
```
```  1301 lemma (in sigma_algebra) borel_measurable_extreal_add[intro, simp]:
```
```  1302   fixes f :: "'a \<Rightarrow> extreal"
```
```  1303   assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1304   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```  1305 proof -
```
```  1306   { fix x assume "x \<in> space M" then have "f x + g x =
```
```  1307       (if f x = \<infinity> \<or> g x = \<infinity> then \<infinity>
```
```  1308         else if f x = -\<infinity> \<or> g x = -\<infinity> then -\<infinity>
```
```  1309         else extreal (real (f x) + real (g x)))"
```
```  1310       by (cases rule: extreal2_cases[of "f x" "g x"]) auto }
```
```  1311   with assms show ?thesis
```
```  1312     by (auto cong: measurable_cong simp: split_sets
```
```  1313              intro!: Un measurable_If measurable_sets)
```
```  1314 qed
```
```  1315
```
```  1316 lemma (in sigma_algebra) borel_measurable_extreal_setsum[simp, intro]:
```
```  1317   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> extreal"
```
```  1318   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1319   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1320 proof cases
```
```  1321   assume "finite S"
```
```  1322   thus ?thesis using assms
```
```  1323     by induct auto
```
```  1324 qed (simp add: borel_measurable_const)
```
```  1325
```
```  1326 lemma abs_extreal_ge0[simp]: "0 \<le> x \<Longrightarrow> \<bar>x :: extreal\<bar> = x"
```
```  1327   by (cases x) auto
```
```  1328
```
```  1329 lemma abs_extreal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: extreal\<bar> = -x"
```
```  1330   by (cases x) auto
```
```  1331
```
```  1332 lemma abs_extreal_pos[simp]: "0 \<le> \<bar>x :: extreal\<bar>"
```
```  1333   by (cases x) auto
```
```  1334
```
```  1335 lemma (in sigma_algebra) borel_measurable_extreal_abs[intro, simp]:
```
```  1336   fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M"
```
```  1337   shows "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
```
```  1338 proof -
```
```  1339   { fix x have "\<bar>f x\<bar> = (if 0 \<le> f x then f x else - f x)" by auto }
```
```  1340   then show ?thesis using assms by (auto intro!: measurable_If)
```
```  1341 qed
```
```  1342
```
```  1343 lemma (in sigma_algebra) borel_measurable_extreal_times[intro, simp]:
```
```  1344   fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1345   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1346 proof -
```
```  1347   { fix f g :: "'a \<Rightarrow> extreal"
```
```  1348     assume b: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1349       and pos: "\<And>x. 0 \<le> f x" "\<And>x. 0 \<le> g x"
```
```  1350     { fix x have *: "f x * g x = (if f x = 0 \<or> g x = 0 then 0
```
```  1351         else if f x = \<infinity> \<or> g x = \<infinity> then \<infinity>
```
```  1352         else extreal (real (f x) * real (g x)))"
```
```  1353       apply (cases rule: extreal2_cases[of "f x" "g x"])
```
```  1354       using pos[of x] by auto }
```
```  1355     with b have "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1356       by (auto cong: measurable_cong simp: split_sets
```
```  1357                intro!: Un measurable_If measurable_sets) }
```
```  1358   note pos_times = this
```
```  1359   have *: "(\<lambda>x. f x * g x) =
```
```  1360     (\<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>))"
```
```  1361     by (auto simp: fun_eq_iff)
```
```  1362   show ?thesis using assms unfolding *
```
```  1363     by (intro measurable_If pos_times borel_measurable_uminus_extreal)
```
```  1364        (auto simp: split_sets intro!: Int)
```
```  1365 qed
```
```  1366
```
```  1367 lemma (in sigma_algebra) borel_measurable_extreal_setprod[simp, intro]:
```
```  1368   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> extreal"
```
```  1369   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1370   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1371 proof cases
```
```  1372   assume "finite S"
```
```  1373   thus ?thesis using assms by induct auto
```
```  1374 qed simp
```
```  1375
```
```  1376 lemma (in sigma_algebra) borel_measurable_extreal_min[simp, intro]:
```
```  1377   fixes f g :: "'a \<Rightarrow> extreal"
```
```  1378   assumes "f \<in> borel_measurable M"
```
```  1379   assumes "g \<in> borel_measurable M"
```
```  1380   shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
```
```  1381   using assms unfolding min_def by (auto intro!: measurable_If)
```
```  1382
```
```  1383 lemma (in sigma_algebra) borel_measurable_extreal_max[simp, intro]:
```
```  1384   fixes f g :: "'a \<Rightarrow> extreal"
```
```  1385   assumes "f \<in> borel_measurable M"
```
```  1386   and "g \<in> borel_measurable M"
```
```  1387   shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
```
```  1388   using assms unfolding max_def by (auto intro!: measurable_If)
```
```  1389
```
```  1390 lemma (in sigma_algebra) borel_measurable_SUP[simp, intro]:
```
```  1391   fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> extreal"
```
```  1392   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1393   shows "(\<lambda>x. SUP i : A. f i x) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M")
```
```  1394   unfolding borel_measurable_extreal_iff_ge
```
```  1395 proof
```
```  1396   fix a
```
```  1397   have "?sup -` {a<..} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"
```
```  1398     by (auto simp: less_SUP_iff SUPR_apply)
```
```  1399   then show "?sup -` {a<..} \<inter> space M \<in> sets M"
```
```  1400     using assms by auto
```
```  1401 qed
```
```  1402
```
```  1403 lemma (in sigma_algebra) borel_measurable_INF[simp, intro]:
```
```  1404   fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> extreal"
```
```  1405   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1406   shows "(\<lambda>x. INF i : A. f i x) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M")
```
```  1407   unfolding borel_measurable_extreal_iff_less
```
```  1408 proof
```
```  1409   fix a
```
```  1410   have "?inf -` {..<a} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"
```
```  1411     by (auto simp: INF_less_iff INFI_apply)
```
```  1412   then show "?inf -` {..<a} \<inter> space M \<in> sets M"
```
```  1413     using assms by auto
```
```  1414 qed
```
```  1415
```
```  1416 lemma (in sigma_algebra) borel_measurable_liminf[simp, intro]:
```
```  1417   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
```
```  1418   assumes "\<And>i. f i \<in> borel_measurable M"
```
```  1419   shows "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1420   unfolding liminf_SUPR_INFI using assms by auto
```
```  1421
```
```  1422 lemma (in sigma_algebra) borel_measurable_limsup[simp, intro]:
```
```  1423   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
```
```  1424   assumes "\<And>i. f i \<in> borel_measurable M"
```
```  1425   shows "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1426   unfolding limsup_INFI_SUPR using assms by auto
```
```  1427
```
```  1428 lemma (in sigma_algebra) borel_measurable_extreal_diff[simp, intro]:
```
```  1429   fixes f g :: "'a \<Rightarrow> extreal"
```
```  1430   assumes "f \<in> borel_measurable M"
```
```  1431   assumes "g \<in> borel_measurable M"
```
```  1432   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```  1433   unfolding minus_extreal_def using assms by auto
```
```  1434
```
```  1435 lemma (in sigma_algebra) borel_measurable_psuminf[simp, intro]:
```
```  1436   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
```
```  1437   assumes "\<And>i. f i \<in> borel_measurable M" and pos: "\<And>i x. x \<in> space M \<Longrightarrow> 0 \<le> f i x"
```
```  1438   shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
```
```  1439   apply (subst measurable_cong)
```
```  1440   apply (subst suminf_extreal_eq_SUPR)
```
```  1441   apply (rule pos)
```
```  1442   using assms by auto
```
```  1443
```
```  1444 section "LIMSEQ is borel measurable"
```
```  1445
```
```  1446 lemma (in sigma_algebra) borel_measurable_LIMSEQ:
```
```  1447   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
```
```  1448   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
```
```  1449   and u: "\<And>i. u i \<in> borel_measurable M"
```
```  1450   shows "u' \<in> borel_measurable M"
```
```  1451 proof -
```
```  1452   have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. extreal (u n x)) = extreal (u' x)"
```
```  1453     using u' by (simp add: lim_imp_Liminf trivial_limit_sequentially lim_extreal)
```
```  1454   moreover from u have "(\<lambda>x. liminf (\<lambda>n. extreal (u n x))) \<in> borel_measurable M"
```
```  1455     by auto
```
```  1456   ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_extreal_iff)
```
```  1457 qed
```
```  1458
```
```  1459 end
```