src/HOL/Probability/Borel_Space.thy
 author hoelzl Tue Jan 18 21:37:23 2011 +0100 (2011-01-18) changeset 41654 32fe42892983 parent 41545 9c869baf1c66 child 41830 719b0a517c33 permissions -rw-r--r--
Gauge measure removed
```     1 (* Author: Armin Heller, Johannes Hoelzl, TU Muenchen *)
```
```     2
```
```     3 header {*Borel spaces*}
```
```     4
```
```     5 theory Borel_Space
```
```     6   imports Sigma_Algebra Positive_Extended_Real Multivariate_Analysis
```
```     7 begin
```
```     8
```
```     9 lemma LIMSEQ_max:
```
```    10   "u ----> (x::real) \<Longrightarrow> (\<lambda>i. max (u i) 0) ----> max x 0"
```
```    11   by (fastsimp intro!: LIMSEQ_I dest!: LIMSEQ_D)
```
```    12
```
```    13 section "Generic Borel spaces"
```
```    14
```
```    15 definition "borel = sigma \<lparr> space = UNIV::'a::topological_space set, sets = open\<rparr>"
```
```    16 abbreviation "borel_measurable M \<equiv> measurable M borel"
```
```    17
```
```    18 interpretation borel: sigma_algebra borel
```
```    19   by (auto simp: borel_def intro!: sigma_algebra_sigma)
```
```    20
```
```    21 lemma in_borel_measurable:
```
```    22    "f \<in> borel_measurable M \<longleftrightarrow>
```
```    23     (\<forall>S \<in> sets (sigma \<lparr> space = UNIV, sets = open\<rparr>).
```
```    24       f -` S \<inter> space M \<in> sets M)"
```
```    25   by (auto simp add: measurable_def borel_def)
```
```    26
```
```    27 lemma in_borel_measurable_borel:
```
```    28    "f \<in> borel_measurable M \<longleftrightarrow>
```
```    29     (\<forall>S \<in> sets borel.
```
```    30       f -` S \<inter> space M \<in> sets M)"
```
```    31   by (auto simp add: measurable_def borel_def)
```
```    32
```
```    33 lemma space_borel[simp]: "space borel = UNIV"
```
```    34   unfolding borel_def by auto
```
```    35
```
```    36 lemma borel_open[simp]:
```
```    37   assumes "open A" shows "A \<in> sets borel"
```
```    38 proof -
```
```    39   have "A \<in> open" unfolding mem_def using assms .
```
```    40   thus ?thesis unfolding borel_def sigma_def by (auto intro!: sigma_sets.Basic)
```
```    41 qed
```
```    42
```
```    43 lemma borel_closed[simp]:
```
```    44   assumes "closed A" shows "A \<in> sets borel"
```
```    45 proof -
```
```    46   have "space borel - (- A) \<in> sets borel"
```
```    47     using assms unfolding closed_def by (blast intro: borel_open)
```
```    48   thus ?thesis by simp
```
```    49 qed
```
```    50
```
```    51 lemma (in sigma_algebra) borel_measurable_vimage:
```
```    52   fixes f :: "'a \<Rightarrow> 'x::t2_space"
```
```    53   assumes borel: "f \<in> borel_measurable M"
```
```    54   shows "f -` {x} \<inter> space M \<in> sets M"
```
```    55 proof (cases "x \<in> f ` space M")
```
```    56   case True then obtain y where "x = f y" by auto
```
```    57   from closed_sing[of "f y"]
```
```    58   have "{f y} \<in> sets borel" by (rule borel_closed)
```
```    59   with assms show ?thesis
```
```    60     unfolding in_borel_measurable_borel `x = f y` by auto
```
```    61 next
```
```    62   case False hence "f -` {x} \<inter> space M = {}" by auto
```
```    63   thus ?thesis by auto
```
```    64 qed
```
```    65
```
```    66 lemma (in sigma_algebra) borel_measurableI:
```
```    67   fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space"
```
```    68   assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
```
```    69   shows "f \<in> borel_measurable M"
```
```    70   unfolding borel_def
```
```    71 proof (rule measurable_sigma, simp_all)
```
```    72   fix S :: "'x set" assume "S \<in> open" thus "f -` S \<inter> space M \<in> sets M"
```
```    73     using assms[of S] by (simp add: mem_def)
```
```    74 qed
```
```    75
```
```    76 lemma borel_singleton[simp, intro]:
```
```    77   fixes x :: "'a::t1_space"
```
```    78   shows "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets borel"
```
```    79   proof (rule borel.insert_in_sets)
```
```    80     show "{x} \<in> sets borel"
```
```    81       using closed_sing[of x] by (rule borel_closed)
```
```    82   qed simp
```
```    83
```
```    84 lemma (in sigma_algebra) borel_measurable_const[simp, intro]:
```
```    85   "(\<lambda>x. c) \<in> borel_measurable M"
```
```    86   by (auto intro!: measurable_const)
```
```    87
```
```    88 lemma (in sigma_algebra) borel_measurable_indicator[simp, intro!]:
```
```    89   assumes A: "A \<in> sets M"
```
```    90   shows "indicator A \<in> borel_measurable M"
```
```    91   unfolding indicator_def_raw using A
```
```    92   by (auto intro!: measurable_If_set borel_measurable_const)
```
```    93
```
```    94 lemma (in sigma_algebra) borel_measurable_indicator_iff:
```
```    95   "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
```
```    96     (is "?I \<in> borel_measurable M \<longleftrightarrow> _")
```
```    97 proof
```
```    98   assume "?I \<in> borel_measurable M"
```
```    99   then have "?I -` {1} \<inter> space M \<in> sets M"
```
```   100     unfolding measurable_def by auto
```
```   101   also have "?I -` {1} \<inter> space M = A \<inter> space M"
```
```   102     unfolding indicator_def_raw by auto
```
```   103   finally show "A \<inter> space M \<in> sets M" .
```
```   104 next
```
```   105   assume "A \<inter> space M \<in> sets M"
```
```   106   moreover have "?I \<in> borel_measurable M \<longleftrightarrow>
```
```   107     (indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"
```
```   108     by (intro measurable_cong) (auto simp: indicator_def)
```
```   109   ultimately show "?I \<in> borel_measurable M" by auto
```
```   110 qed
```
```   111
```
```   112 lemma borel_measurable_translate:
```
```   113   assumes "A \<in> sets borel" and trans: "\<And>B. open B \<Longrightarrow> f -` B \<in> sets borel"
```
```   114   shows "f -` A \<in> sets borel"
```
```   115 proof -
```
```   116   have "A \<in> sigma_sets UNIV open" using assms
```
```   117     by (simp add: borel_def sigma_def)
```
```   118   thus ?thesis
```
```   119   proof (induct rule: sigma_sets.induct)
```
```   120     case (Basic a) thus ?case using trans[of a] by (simp add: mem_def)
```
```   121   next
```
```   122     case (Compl a)
```
```   123     moreover have "UNIV \<in> sets borel"
```
```   124       using borel.top by simp
```
```   125     ultimately show ?case
```
```   126       by (auto simp: vimage_Diff borel.Diff)
```
```   127   qed (auto simp add: vimage_UN)
```
```   128 qed
```
```   129
```
```   130 lemma (in sigma_algebra) borel_measurable_restricted:
```
```   131   fixes f :: "'a \<Rightarrow> 'x\<Colon>{topological_space, semiring_1}" assumes "A \<in> sets M"
```
```   132   shows "f \<in> borel_measurable (restricted_space A) \<longleftrightarrow>
```
```   133     (\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
```
```   134     (is "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable M")
```
```   135 proof -
```
```   136   interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
```
```   137   have *: "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable ?R"
```
```   138     by (auto intro!: measurable_cong)
```
```   139   show ?thesis unfolding *
```
```   140     unfolding in_borel_measurable_borel
```
```   141   proof (simp, safe)
```
```   142     fix S :: "'x set" assume "S \<in> sets borel"
```
```   143       "\<forall>S\<in>sets borel. ?f -` S \<inter> A \<in> op \<inter> A ` sets M"
```
```   144     then have "?f -` S \<inter> A \<in> op \<inter> A ` sets M" by auto
```
```   145     then have f: "?f -` S \<inter> A \<in> sets M"
```
```   146       using `A \<in> sets M` sets_into_space by fastsimp
```
```   147     show "?f -` S \<inter> space M \<in> sets M"
```
```   148     proof cases
```
```   149       assume "0 \<in> S"
```
```   150       then have "?f -` S \<inter> space M = ?f -` S \<inter> A \<union> (space M - A)"
```
```   151         using `A \<in> sets M` sets_into_space by auto
```
```   152       then show ?thesis using f `A \<in> sets M` by (auto intro!: Un Diff)
```
```   153     next
```
```   154       assume "0 \<notin> S"
```
```   155       then have "?f -` S \<inter> space M = ?f -` S \<inter> A"
```
```   156         using `A \<in> sets M` sets_into_space
```
```   157         by (auto simp: indicator_def split: split_if_asm)
```
```   158       then show ?thesis using f by auto
```
```   159     qed
```
```   160   next
```
```   161     fix S :: "'x set" assume "S \<in> sets borel"
```
```   162       "\<forall>S\<in>sets borel. ?f -` S \<inter> space M \<in> sets M"
```
```   163     then have f: "?f -` S \<inter> space M \<in> sets M" by auto
```
```   164     then show "?f -` S \<inter> A \<in> op \<inter> A ` sets M"
```
```   165       using `A \<in> sets M` sets_into_space
```
```   166       apply (simp add: image_iff)
```
```   167       apply (rule bexI[OF _ f])
```
```   168       by auto
```
```   169   qed
```
```   170 qed
```
```   171
```
```   172 lemma (in sigma_algebra) borel_measurable_subalgebra:
```
```   173   assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
```
```   174   shows "f \<in> borel_measurable M"
```
```   175   using assms unfolding measurable_def by auto
```
```   176
```
```   177 section "Borel spaces on euclidean spaces"
```
```   178
```
```   179 lemma lessThan_borel[simp, intro]:
```
```   180   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   181   shows "{..< a} \<in> sets borel"
```
```   182   by (blast intro: borel_open)
```
```   183
```
```   184 lemma greaterThan_borel[simp, intro]:
```
```   185   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   186   shows "{a <..} \<in> sets borel"
```
```   187   by (blast intro: borel_open)
```
```   188
```
```   189 lemma greaterThanLessThan_borel[simp, intro]:
```
```   190   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   191   shows "{a<..<b} \<in> sets borel"
```
```   192   by (blast intro: borel_open)
```
```   193
```
```   194 lemma atMost_borel[simp, intro]:
```
```   195   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   196   shows "{..a} \<in> sets borel"
```
```   197   by (blast intro: borel_closed)
```
```   198
```
```   199 lemma atLeast_borel[simp, intro]:
```
```   200   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   201   shows "{a..} \<in> sets borel"
```
```   202   by (blast intro: borel_closed)
```
```   203
```
```   204 lemma atLeastAtMost_borel[simp, intro]:
```
```   205   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   206   shows "{a..b} \<in> sets borel"
```
```   207   by (blast intro: borel_closed)
```
```   208
```
```   209 lemma greaterThanAtMost_borel[simp, intro]:
```
```   210   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   211   shows "{a<..b} \<in> sets borel"
```
```   212   unfolding greaterThanAtMost_def by blast
```
```   213
```
```   214 lemma atLeastLessThan_borel[simp, intro]:
```
```   215   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   216   shows "{a..<b} \<in> sets borel"
```
```   217   unfolding atLeastLessThan_def by blast
```
```   218
```
```   219 lemma hafspace_less_borel[simp, intro]:
```
```   220   fixes a :: real
```
```   221   shows "{x::'a::euclidean_space. a < x \$\$ i} \<in> sets borel"
```
```   222   by (auto intro!: borel_open open_halfspace_component_gt)
```
```   223
```
```   224 lemma hafspace_greater_borel[simp, intro]:
```
```   225   fixes a :: real
```
```   226   shows "{x::'a::euclidean_space. x \$\$ i < a} \<in> sets borel"
```
```   227   by (auto intro!: borel_open open_halfspace_component_lt)
```
```   228
```
```   229 lemma hafspace_less_eq_borel[simp, intro]:
```
```   230   fixes a :: real
```
```   231   shows "{x::'a::euclidean_space. a \<le> x \$\$ i} \<in> sets borel"
```
```   232   by (auto intro!: borel_closed closed_halfspace_component_ge)
```
```   233
```
```   234 lemma hafspace_greater_eq_borel[simp, intro]:
```
```   235   fixes a :: real
```
```   236   shows "{x::'a::euclidean_space. x \$\$ i \<le> a} \<in> sets borel"
```
```   237   by (auto intro!: borel_closed closed_halfspace_component_le)
```
```   238
```
```   239 lemma (in sigma_algebra) borel_measurable_less[simp, intro]:
```
```   240   fixes f :: "'a \<Rightarrow> real"
```
```   241   assumes f: "f \<in> borel_measurable M"
```
```   242   assumes g: "g \<in> borel_measurable M"
```
```   243   shows "{w \<in> space M. f w < g w} \<in> sets M"
```
```   244 proof -
```
```   245   have "{w \<in> space M. f w < g w} =
```
```   246         (\<Union>r. (f -` {..< of_rat r} \<inter> space M) \<inter> (g -` {of_rat r <..} \<inter> space M))"
```
```   247     using Rats_dense_in_real by (auto simp add: Rats_def)
```
```   248   then show ?thesis using f g
```
```   249     by simp (blast intro: measurable_sets)
```
```   250 qed
```
```   251
```
```   252 lemma (in sigma_algebra) borel_measurable_le[simp, intro]:
```
```   253   fixes f :: "'a \<Rightarrow> real"
```
```   254   assumes f: "f \<in> borel_measurable M"
```
```   255   assumes g: "g \<in> borel_measurable M"
```
```   256   shows "{w \<in> space M. f w \<le> g w} \<in> sets M"
```
```   257 proof -
```
```   258   have "{w \<in> space M. f w \<le> g w} = space M - {w \<in> space M. g w < f w}"
```
```   259     by auto
```
```   260   thus ?thesis using f g
```
```   261     by simp blast
```
```   262 qed
```
```   263
```
```   264 lemma (in sigma_algebra) borel_measurable_eq[simp, intro]:
```
```   265   fixes f :: "'a \<Rightarrow> real"
```
```   266   assumes f: "f \<in> borel_measurable M"
```
```   267   assumes g: "g \<in> borel_measurable M"
```
```   268   shows "{w \<in> space M. f w = g w} \<in> sets M"
```
```   269 proof -
```
```   270   have "{w \<in> space M. f w = g w} =
```
```   271         {w \<in> space M. f w \<le> g w} \<inter> {w \<in> space M. g w \<le> f w}"
```
```   272     by auto
```
```   273   thus ?thesis using f g by auto
```
```   274 qed
```
```   275
```
```   276 lemma (in sigma_algebra) borel_measurable_neq[simp, intro]:
```
```   277   fixes f :: "'a \<Rightarrow> real"
```
```   278   assumes f: "f \<in> borel_measurable M"
```
```   279   assumes g: "g \<in> borel_measurable M"
```
```   280   shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
```
```   281 proof -
```
```   282   have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}"
```
```   283     by auto
```
```   284   thus ?thesis using f g by auto
```
```   285 qed
```
```   286
```
```   287 subsection "Borel space equals sigma algebras over intervals"
```
```   288
```
```   289 lemma rational_boxes:
```
```   290   fixes x :: "'a\<Colon>ordered_euclidean_space"
```
```   291   assumes "0 < e"
```
```   292   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"
```
```   293 proof -
```
```   294   def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
```
```   295   then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos)
```
```   296   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \$\$ i \<and> x \$\$ i - y < e'" (is "\<forall>i. ?th i")
```
```   297   proof
```
```   298     fix i from Rats_dense_in_real[of "x \$\$ i - e'" "x \$\$ i"] e
```
```   299     show "?th i" by auto
```
```   300   qed
```
```   301   from choice[OF this] guess a .. note a = this
```
```   302   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \$\$ i < y \<and> y - x \$\$ i < e'" (is "\<forall>i. ?th i")
```
```   303   proof
```
```   304     fix i from Rats_dense_in_real[of "x \$\$ i" "x \$\$ i + e'"] e
```
```   305     show "?th i" by auto
```
```   306   qed
```
```   307   from choice[OF this] guess b .. note b = this
```
```   308   { fix y :: 'a assume *: "Chi a < y" "y < Chi b"
```
```   309     have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x \$\$ i) (y \$\$ i))\<twosuperior>)"
```
```   310       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
```
```   311     also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))"
```
```   312     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
```
```   313       fix i assume i: "i \<in> {..<DIM('a)}"
```
```   314       have "a i < y\$\$i \<and> y\$\$i < b i" using * i eucl_less[where 'a='a] by auto
```
```   315       moreover have "a i < x\$\$i" "x\$\$i - a i < e'" using a by auto
```
```   316       moreover have "x\$\$i < b i" "b i - x\$\$i < e'" using b by auto
```
```   317       ultimately have "\<bar>x\$\$i - y\$\$i\<bar> < 2 * e'" by auto
```
```   318       then have "dist (x \$\$ i) (y \$\$ i) < e/sqrt (real (DIM('a)))"
```
```   319         unfolding e'_def by (auto simp: dist_real_def)
```
```   320       then have "(dist (x \$\$ i) (y \$\$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
```
```   321         by (rule power_strict_mono) auto
```
```   322       then show "(dist (x \$\$ i) (y \$\$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
```
```   323         by (simp add: power_divide)
```
```   324     qed auto
```
```   325     also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive)
```
```   326     finally have "dist x y < e" . }
```
```   327   with a b show ?thesis
```
```   328     apply (rule_tac exI[of _ "Chi a"])
```
```   329     apply (rule_tac exI[of _ "Chi b"])
```
```   330     using eucl_less[where 'a='a] by auto
```
```   331 qed
```
```   332
```
```   333 lemma ex_rat_list:
```
```   334   fixes x :: "'a\<Colon>ordered_euclidean_space"
```
```   335   assumes "\<And> i. x \$\$ i \<in> \<rat>"
```
```   336   shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x \$\$ i)"
```
```   337 proof -
```
```   338   have "\<forall>i. \<exists>r. x \$\$ i = of_rat r" using assms unfolding Rats_def by blast
```
```   339   from choice[OF this] guess r ..
```
```   340   then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"])
```
```   341 qed
```
```   342
```
```   343 lemma open_UNION:
```
```   344   fixes M :: "'a\<Colon>ordered_euclidean_space set"
```
```   345   assumes "open M"
```
```   346   shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M}
```
```   347                    (\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})"
```
```   348     (is "M = UNION ?idx ?box")
```
```   349 proof safe
```
```   350   fix x assume "x \<in> M"
```
```   351   obtain e where e: "e > 0" "ball x e \<subseteq> M"
```
```   352     using openE[OF assms `x \<in> M`] by auto
```
```   353   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"
```
```   354     using rational_boxes[OF e(1)] by blast
```
```   355   then obtain p q where pq: "length p = DIM ('a)"
```
```   356                             "length q = DIM ('a)"
```
```   357                             "\<forall> i < DIM ('a). of_rat (p ! i) = a \$\$ i \<and> of_rat (q ! i) = b \$\$ i"
```
```   358     using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast
```
```   359   hence p: "Chi (of_rat \<circ> op ! p) = a"
```
```   360     using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a]
```
```   361     unfolding o_def by auto
```
```   362   from pq have q: "Chi (of_rat \<circ> op ! q) = b"
```
```   363     using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b]
```
```   364     unfolding o_def by auto
```
```   365   have "x \<in> ?box (p, q)"
```
```   366     using p q ab by auto
```
```   367   thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto
```
```   368 qed auto
```
```   369
```
```   370 lemma halfspace_span_open:
```
```   371   "sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a}))
```
```   372     \<subseteq> sets borel"
```
```   373   by (auto intro!: borel.sigma_sets_subset[simplified] borel_open
```
```   374                    open_halfspace_component_lt)
```
```   375
```
```   376 lemma halfspace_lt_in_halfspace:
```
```   377   "{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>)"
```
```   378   by (auto intro!: sigma_sets.Basic simp: sets_sigma)
```
```   379
```
```   380 lemma halfspace_gt_in_halfspace:
```
```   381   "{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>)"
```
```   382   (is "?set \<in> sets ?SIGMA")
```
```   383 proof -
```
```   384   interpret sigma_algebra "?SIGMA"
```
```   385     by (intro sigma_algebra_sigma_sets) (simp_all add: sets_sigma)
```
```   386   have *: "?set = (\<Union>n. space ?SIGMA - {x\<Colon>'a. x \$\$ i < a + 1 / real (Suc n)})"
```
```   387   proof (safe, simp_all add: not_less)
```
```   388     fix x assume "a < x \$\$ i"
```
```   389     with reals_Archimedean[of "x \$\$ i - a"]
```
```   390     obtain n where "a + 1 / real (Suc n) < x \$\$ i"
```
```   391       by (auto simp: inverse_eq_divide field_simps)
```
```   392     then show "\<exists>n. a + 1 / real (Suc n) \<le> x \$\$ i"
```
```   393       by (blast intro: less_imp_le)
```
```   394   next
```
```   395     fix x n
```
```   396     have "a < a + 1 / real (Suc n)" by auto
```
```   397     also assume "\<dots> \<le> x"
```
```   398     finally show "a < x" .
```
```   399   qed
```
```   400   show "?set \<in> sets ?SIGMA" unfolding *
```
```   401     by (safe intro!: countable_UN Diff halfspace_lt_in_halfspace)
```
```   402 qed
```
```   403
```
```   404 lemma open_span_halfspace:
```
```   405   "sets borel \<subseteq> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x \$\$ i < a})\<rparr>)"
```
```   406     (is "_ \<subseteq> sets ?SIGMA")
```
```   407 proof -
```
```   408   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) simp
```
```   409   then interpret sigma_algebra ?SIGMA .
```
```   410   { fix S :: "'a set" assume "S \<in> open" then have "open S" unfolding mem_def .
```
```   411     from open_UNION[OF this]
```
```   412     obtain I where *: "S =
```
```   413       (\<Union>(a, b)\<in>I.
```
```   414           (\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) \$\$ i < x \$\$ i}) \<inter>
```
```   415           (\<Inter> i<DIM('a). {x. x \$\$ i < (Chi (real_of_rat \<circ> op ! b)::'a) \$\$ i}))"
```
```   416       unfolding greaterThanLessThan_def
```
```   417       unfolding eucl_greaterThan_eq_halfspaces[where 'a='a]
```
```   418       unfolding eucl_lessThan_eq_halfspaces[where 'a='a]
```
```   419       by blast
```
```   420     have "S \<in> sets ?SIGMA"
```
```   421       unfolding *
```
```   422       by (auto intro!: countable_UN Int countable_INT halfspace_lt_in_halfspace halfspace_gt_in_halfspace) }
```
```   423   then show ?thesis unfolding borel_def
```
```   424     by (intro sets_sigma_subset) auto
```
```   425 qed
```
```   426
```
```   427 lemma halfspace_span_halfspace_le:
```
```   428   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a})\<rparr>) \<subseteq>
```
```   429    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. x \$\$ i \<le> a})\<rparr>)"
```
```   430   (is "_ \<subseteq> sets ?SIGMA")
```
```   431 proof -
```
```   432   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   433   then interpret sigma_algebra ?SIGMA .
```
```   434   { fix a i
```
```   435     have *: "{x::'a. x\$\$i < a} = (\<Union>n. {x. x\$\$i \<le> a - 1/real (Suc n)})"
```
```   436     proof (safe, simp_all)
```
```   437       fix x::'a assume *: "x\$\$i < a"
```
```   438       with reals_Archimedean[of "a - x\$\$i"]
```
```   439       obtain n where "x \$\$ i < a - 1 / (real (Suc n))"
```
```   440         by (auto simp: field_simps inverse_eq_divide)
```
```   441       then show "\<exists>n. x \$\$ i \<le> a - 1 / (real (Suc n))"
```
```   442         by (blast intro: less_imp_le)
```
```   443     next
```
```   444       fix x::'a and n
```
```   445       assume "x\$\$i \<le> a - 1 / real (Suc n)"
```
```   446       also have "\<dots> < a" by auto
```
```   447       finally show "x\$\$i < a" .
```
```   448     qed
```
```   449     have "{x. x\$\$i < a} \<in> sets ?SIGMA" unfolding *
```
```   450       by (safe intro!: countable_UN)
```
```   451          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   452   then show ?thesis by (intro sets_sigma_subset) auto
```
```   453 qed
```
```   454
```
```   455 lemma halfspace_span_halfspace_ge:
```
```   456   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a})\<rparr>) \<subseteq>
```
```   457    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a \<le> x \$\$ i})\<rparr>)"
```
```   458   (is "_ \<subseteq> sets ?SIGMA")
```
```   459 proof -
```
```   460   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   461   then interpret sigma_algebra ?SIGMA .
```
```   462   { fix a i have *: "{x::'a. x\$\$i < a} = space ?SIGMA - {x::'a. a \<le> x\$\$i}" by auto
```
```   463     have "{x. x\$\$i < a} \<in> sets ?SIGMA" unfolding *
```
```   464       by (safe intro!: Diff)
```
```   465          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   466   then show ?thesis by (intro sets_sigma_subset) auto
```
```   467 qed
```
```   468
```
```   469 lemma halfspace_le_span_halfspace_gt:
```
```   470   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
```
```   471    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a < x \$\$ i})\<rparr>)"
```
```   472   (is "_ \<subseteq> sets ?SIGMA")
```
```   473 proof -
```
```   474   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   475   then interpret sigma_algebra ?SIGMA .
```
```   476   { fix a i have *: "{x::'a. x\$\$i \<le> a} = space ?SIGMA - {x::'a. a < x\$\$i}" by auto
```
```   477     have "{x. x\$\$i \<le> a} \<in> sets ?SIGMA" unfolding *
```
```   478       by (safe intro!: Diff)
```
```   479          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   480   then show ?thesis by (intro sets_sigma_subset) auto
```
```   481 qed
```
```   482
```
```   483 lemma halfspace_le_span_atMost:
```
```   484   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
```
```   485    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>)"
```
```   486   (is "_ \<subseteq> sets ?SIGMA")
```
```   487 proof -
```
```   488   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   489   then interpret sigma_algebra ?SIGMA .
```
```   490   have "\<And>a i. {x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   491   proof cases
```
```   492     fix a i assume "i < DIM('a)"
```
```   493     then have *: "{x::'a. x\$\$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})"
```
```   494     proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm)
```
```   495       fix x
```
```   496       from real_arch_simple[of "Max ((\<lambda>i. x\$\$i)`{..<DIM('a)})"] guess k::nat ..
```
```   497       then have "\<And>i. i < DIM('a) \<Longrightarrow> x\$\$i \<le> real k"
```
```   498         by (subst (asm) Max_le_iff) auto
```
```   499       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x \$\$ ia \<le> real k"
```
```   500         by (auto intro!: exI[of _ k])
```
```   501     qed
```
```   502     show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA" unfolding *
```
```   503       by (safe intro!: countable_UN)
```
```   504          (auto simp: sets_sigma intro!: sigma_sets.Basic)
```
```   505   next
```
```   506     fix a i assume "\<not> i < DIM('a)"
```
```   507     then show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   508       using top by auto
```
```   509   qed
```
```   510   then show ?thesis by (intro sets_sigma_subset) auto
```
```   511 qed
```
```   512
```
```   513 lemma halfspace_le_span_greaterThan:
```
```   514   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
```
```   515    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {a<..})\<rparr>)"
```
```   516   (is "_ \<subseteq> sets ?SIGMA")
```
```   517 proof -
```
```   518   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   519   then interpret sigma_algebra ?SIGMA .
```
```   520   have "\<And>a i. {x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   521   proof cases
```
```   522     fix a i assume "i < DIM('a)"
```
```   523     have "{x::'a. x\$\$i \<le> a} = space ?SIGMA - {x::'a. a < x\$\$i}" by auto
```
```   524     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)`
```
```   525     proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
```
```   526       fix x
```
```   527       from real_arch_lt[of "Max ((\<lambda>i. -x\$\$i)`{..<DIM('a)})"]
```
```   528       guess k::nat .. note k = this
```
```   529       { fix i assume "i < DIM('a)"
```
```   530         then have "-x\$\$i < real k"
```
```   531           using k by (subst (asm) Max_less_iff) auto
```
```   532         then have "- real k < x\$\$i" by simp }
```
```   533       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x \$\$ ia"
```
```   534         by (auto intro!: exI[of _ k])
```
```   535     qed
```
```   536     finally show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   537       apply (simp only:)
```
```   538       apply (safe intro!: countable_UN Diff)
```
```   539       by (auto simp: sets_sigma intro!: sigma_sets.Basic)
```
```   540   next
```
```   541     fix a i assume "\<not> i < DIM('a)"
```
```   542     then show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   543       using top by auto
```
```   544   qed
```
```   545   then show ?thesis by (intro sets_sigma_subset) auto
```
```   546 qed
```
```   547
```
```   548 lemma halfspace_le_span_lessThan:
```
```   549   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x \$\$ i})\<rparr>) \<subseteq>
```
```   550    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..<a})\<rparr>)"
```
```   551   (is "_ \<subseteq> sets ?SIGMA")
```
```   552 proof -
```
```   553   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   554   then interpret sigma_algebra ?SIGMA .
```
```   555   have "\<And>a i. {x. a \<le> x\$\$i} \<in> sets ?SIGMA"
```
```   556   proof cases
```
```   557     fix a i assume "i < DIM('a)"
```
```   558     have "{x::'a. a \<le> x\$\$i} = space ?SIGMA - {x::'a. x\$\$i < a}" by auto
```
```   559     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)`
```
```   560     proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
```
```   561       fix x
```
```   562       from real_arch_lt[of "Max ((\<lambda>i. x\$\$i)`{..<DIM('a)})"]
```
```   563       guess k::nat .. note k = this
```
```   564       { fix i assume "i < DIM('a)"
```
```   565         then have "x\$\$i < real k"
```
```   566           using k by (subst (asm) Max_less_iff) auto
```
```   567         then have "x\$\$i < real k" by simp }
```
```   568       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x \$\$ ia < real k"
```
```   569         by (auto intro!: exI[of _ k])
```
```   570     qed
```
```   571     finally show "{x. a \<le> x\$\$i} \<in> sets ?SIGMA"
```
```   572       apply (simp only:)
```
```   573       apply (safe intro!: countable_UN Diff)
```
```   574       by (auto simp: sets_sigma intro!: sigma_sets.Basic)
```
```   575   next
```
```   576     fix a i assume "\<not> i < DIM('a)"
```
```   577     then show "{x. a \<le> x\$\$i} \<in> sets ?SIGMA"
```
```   578       using top by auto
```
```   579   qed
```
```   580   then show ?thesis by (intro sets_sigma_subset) auto
```
```   581 qed
```
```   582
```
```   583 lemma atMost_span_atLeastAtMost:
```
```   584   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>) \<subseteq>
```
```   585    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a,b). {a..b})\<rparr>)"
```
```   586   (is "_ \<subseteq> sets ?SIGMA")
```
```   587 proof -
```
```   588   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   589   then interpret sigma_algebra ?SIGMA .
```
```   590   { fix a::'a
```
```   591     have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
```
```   592     proof (safe, simp_all add: eucl_le[where 'a='a])
```
```   593       fix x
```
```   594       from real_arch_simple[of "Max ((\<lambda>i. - x\$\$i)`{..<DIM('a)})"]
```
```   595       guess k::nat .. note k = this
```
```   596       { fix i assume "i < DIM('a)"
```
```   597         with k have "- x\$\$i \<le> real k"
```
```   598           by (subst (asm) Max_le_iff) (auto simp: field_simps)
```
```   599         then have "- real k \<le> x\$\$i" by simp }
```
```   600       then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x \$\$ i"
```
```   601         by (auto intro!: exI[of _ k])
```
```   602     qed
```
```   603     have "{..a} \<in> sets ?SIGMA" unfolding *
```
```   604       by (safe intro!: countable_UN)
```
```   605          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   606   then show ?thesis by (intro sets_sigma_subset) auto
```
```   607 qed
```
```   608
```
```   609 lemma borel_eq_atMost:
```
```   610   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> a. {.. a::'a\<Colon>ordered_euclidean_space})\<rparr>)"
```
```   611     (is "_ = ?SIGMA")
```
```   612 proof (intro algebra.equality antisym)
```
```   613   show "sets borel \<subseteq> sets ?SIGMA"
```
```   614     using halfspace_le_span_atMost halfspace_span_halfspace_le open_span_halfspace
```
```   615     by auto
```
```   616   show "sets ?SIGMA \<subseteq> sets borel"
```
```   617     by (rule borel.sets_sigma_subset) auto
```
```   618 qed auto
```
```   619
```
```   620 lemma borel_eq_atLeastAtMost:
```
```   621   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space, b). {a .. b})\<rparr>)"
```
```   622    (is "_ = ?SIGMA")
```
```   623 proof (intro algebra.equality antisym)
```
```   624   show "sets borel \<subseteq> sets ?SIGMA"
```
```   625     using atMost_span_atLeastAtMost halfspace_le_span_atMost
```
```   626       halfspace_span_halfspace_le open_span_halfspace
```
```   627     by auto
```
```   628   show "sets ?SIGMA \<subseteq> sets borel"
```
```   629     by (rule borel.sets_sigma_subset) auto
```
```   630 qed auto
```
```   631
```
```   632 lemma borel_eq_greaterThan:
```
```   633   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {a <..})\<rparr>)"
```
```   634    (is "_ = ?SIGMA")
```
```   635 proof (intro algebra.equality antisym)
```
```   636   show "sets borel \<subseteq> sets ?SIGMA"
```
```   637     using halfspace_le_span_greaterThan
```
```   638       halfspace_span_halfspace_le open_span_halfspace
```
```   639     by auto
```
```   640   show "sets ?SIGMA \<subseteq> sets borel"
```
```   641     by (rule borel.sets_sigma_subset) auto
```
```   642 qed auto
```
```   643
```
```   644 lemma borel_eq_lessThan:
```
```   645   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {..< a})\<rparr>)"
```
```   646    (is "_ = ?SIGMA")
```
```   647 proof (intro algebra.equality antisym)
```
```   648   show "sets borel \<subseteq> sets ?SIGMA"
```
```   649     using halfspace_le_span_lessThan
```
```   650       halfspace_span_halfspace_ge open_span_halfspace
```
```   651     by auto
```
```   652   show "sets ?SIGMA \<subseteq> sets borel"
```
```   653     by (rule borel.sets_sigma_subset) auto
```
```   654 qed auto
```
```   655
```
```   656 lemma borel_eq_greaterThanLessThan:
```
```   657   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, b). {a <..< (b :: 'a \<Colon> ordered_euclidean_space)})\<rparr>)"
```
```   658     (is "_ = ?SIGMA")
```
```   659 proof (intro algebra.equality antisym)
```
```   660   show "sets ?SIGMA \<subseteq> sets borel"
```
```   661     by (rule borel.sets_sigma_subset) auto
```
```   662   show "sets borel \<subseteq> sets ?SIGMA"
```
```   663   proof -
```
```   664     have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   665     then interpret sigma_algebra ?SIGMA .
```
```   666     { fix M :: "'a set" assume "M \<in> open"
```
```   667       then have "open M" by (simp add: mem_def)
```
```   668       have "M \<in> sets ?SIGMA"
```
```   669         apply (subst open_UNION[OF `open M`])
```
```   670         apply (safe intro!: countable_UN)
```
```   671         by (auto simp add: sigma_def intro!: sigma_sets.Basic) }
```
```   672     then show ?thesis
```
```   673       unfolding borel_def by (intro sets_sigma_subset) auto
```
```   674   qed
```
```   675 qed auto
```
```   676
```
```   677 lemma borel_eq_halfspace_le:
```
```   678   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x\$\$i \<le> a})\<rparr>)"
```
```   679    (is "_ = ?SIGMA")
```
```   680 proof (intro algebra.equality antisym)
```
```   681   show "sets borel \<subseteq> sets ?SIGMA"
```
```   682     using open_span_halfspace halfspace_span_halfspace_le by auto
```
```   683   show "sets ?SIGMA \<subseteq> sets borel"
```
```   684     by (rule borel.sets_sigma_subset) auto
```
```   685 qed auto
```
```   686
```
```   687 lemma borel_eq_halfspace_less:
```
```   688   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x\$\$i < a})\<rparr>)"
```
```   689    (is "_ = ?SIGMA")
```
```   690 proof (intro algebra.equality antisym)
```
```   691   show "sets borel \<subseteq> sets ?SIGMA"
```
```   692     using open_span_halfspace .
```
```   693   show "sets ?SIGMA \<subseteq> sets borel"
```
```   694     by (rule borel.sets_sigma_subset) auto
```
```   695 qed auto
```
```   696
```
```   697 lemma borel_eq_halfspace_gt:
```
```   698   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a < x\$\$i})\<rparr>)"
```
```   699    (is "_ = ?SIGMA")
```
```   700 proof (intro algebra.equality antisym)
```
```   701   show "sets borel \<subseteq> sets ?SIGMA"
```
```   702     using halfspace_le_span_halfspace_gt open_span_halfspace halfspace_span_halfspace_le by auto
```
```   703   show "sets ?SIGMA \<subseteq> sets borel"
```
```   704     by (rule borel.sets_sigma_subset) auto
```
```   705 qed auto
```
```   706
```
```   707 lemma borel_eq_halfspace_ge:
```
```   708   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a \<le> x\$\$i})\<rparr>)"
```
```   709    (is "_ = ?SIGMA")
```
```   710 proof (intro algebra.equality antisym)
```
```   711   show "sets borel \<subseteq> sets ?SIGMA"
```
```   712     using halfspace_span_halfspace_ge open_span_halfspace by auto
```
```   713   show "sets ?SIGMA \<subseteq> sets borel"
```
```   714     by (rule borel.sets_sigma_subset) auto
```
```   715 qed auto
```
```   716
```
```   717 lemma (in sigma_algebra) borel_measurable_halfspacesI:
```
```   718   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   719   assumes "borel = (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
```
```   720   and "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
```
```   721   and "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M"
```
```   722   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)"
```
```   723 proof safe
```
```   724   fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M"
```
```   725   then show "S a i \<in> sets M" unfolding assms
```
```   726     by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1) sigma_def)
```
```   727 next
```
```   728   assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M"
```
```   729   { fix a i have "S a i \<in> sets M"
```
```   730     proof cases
```
```   731       assume "i < DIM('c)"
```
```   732       with a show ?thesis unfolding assms(2) by simp
```
```   733     next
```
```   734       assume "\<not> i < DIM('c)"
```
```   735       from assms(3)[OF this] show ?thesis .
```
```   736     qed }
```
```   737   then have "f \<in> measurable M (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
```
```   738     by (auto intro!: measurable_sigma simp: assms(2))
```
```   739   then show "f \<in> borel_measurable M" unfolding measurable_def
```
```   740     unfolding assms(1) by simp
```
```   741 qed
```
```   742
```
```   743 lemma (in sigma_algebra) borel_measurable_iff_halfspace_le:
```
```   744   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   745   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w \$\$ i \<le> a} \<in> sets M)"
```
```   746   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
```
```   747
```
```   748 lemma (in sigma_algebra) borel_measurable_iff_halfspace_less:
```
```   749   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   750   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w \$\$ i < a} \<in> sets M)"
```
```   751   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
```
```   752
```
```   753 lemma (in sigma_algebra) borel_measurable_iff_halfspace_ge:
```
```   754   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   755   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w \$\$ i} \<in> sets M)"
```
```   756   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
```
```   757
```
```   758 lemma (in sigma_algebra) borel_measurable_iff_halfspace_greater:
```
```   759   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   760   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w \$\$ i} \<in> sets M)"
```
```   761   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_gt]) auto
```
```   762
```
```   763 lemma (in sigma_algebra) borel_measurable_iff_le:
```
```   764   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
```
```   765   using borel_measurable_iff_halfspace_le[where 'c=real] by simp
```
```   766
```
```   767 lemma (in sigma_algebra) borel_measurable_iff_less:
```
```   768   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
```
```   769   using borel_measurable_iff_halfspace_less[where 'c=real] by simp
```
```   770
```
```   771 lemma (in sigma_algebra) borel_measurable_iff_ge:
```
```   772   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
```
```   773   using borel_measurable_iff_halfspace_ge[where 'c=real] by simp
```
```   774
```
```   775 lemma (in sigma_algebra) borel_measurable_iff_greater:
```
```   776   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
```
```   777   using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
```
```   778
```
```   779 lemma borel_measurable_euclidean_component:
```
```   780   "(\<lambda>x::'a::euclidean_space. x \$\$ i) \<in> borel_measurable borel"
```
```   781   unfolding borel_def[where 'a=real]
```
```   782 proof (rule borel.measurable_sigma, simp_all)
```
```   783   fix S::"real set" assume "S \<in> open" then have "open S" unfolding mem_def .
```
```   784   from open_vimage_euclidean_component[OF this]
```
```   785   show "(\<lambda>x. x \$\$ i) -` S \<in> sets borel"
```
```   786     by (auto intro: borel_open)
```
```   787 qed
```
```   788
```
```   789 lemma (in sigma_algebra) borel_measurable_euclidean_space:
```
```   790   fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
```
```   791   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x \$\$ i) \<in> borel_measurable M)"
```
```   792 proof safe
```
```   793   fix i assume "f \<in> borel_measurable M"
```
```   794   then show "(\<lambda>x. f x \$\$ i) \<in> borel_measurable M"
```
```   795     using measurable_comp[of f _ _ "\<lambda>x. x \$\$ i", unfolded comp_def]
```
```   796     by (auto intro: borel_measurable_euclidean_component)
```
```   797 next
```
```   798   assume f: "\<forall>i<DIM('c). (\<lambda>x. f x \$\$ i) \<in> borel_measurable M"
```
```   799   then show "f \<in> borel_measurable M"
```
```   800     unfolding borel_measurable_iff_halfspace_le by auto
```
```   801 qed
```
```   802
```
```   803 subsection "Borel measurable operators"
```
```   804
```
```   805 lemma (in sigma_algebra) affine_borel_measurable_vector:
```
```   806   fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
```
```   807   assumes "f \<in> borel_measurable M"
```
```   808   shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
```
```   809 proof (rule borel_measurableI)
```
```   810   fix S :: "'x set" assume "open S"
```
```   811   show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
```
```   812   proof cases
```
```   813     assume "b \<noteq> 0"
```
```   814     with `open S` have "((\<lambda>x. (- a + x) /\<^sub>R b) ` S) \<in> open" (is "?S \<in> open")
```
```   815       by (auto intro!: open_affinity simp: scaleR.add_right mem_def)
```
```   816     hence "?S \<in> sets borel"
```
```   817       unfolding borel_def by (auto simp: sigma_def intro!: sigma_sets.Basic)
```
```   818     moreover
```
```   819     from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
```
```   820       apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
```
```   821     ultimately show ?thesis using assms unfolding in_borel_measurable_borel
```
```   822       by auto
```
```   823   qed simp
```
```   824 qed
```
```   825
```
```   826 lemma (in sigma_algebra) affine_borel_measurable:
```
```   827   fixes g :: "'a \<Rightarrow> real"
```
```   828   assumes g: "g \<in> borel_measurable M"
```
```   829   shows "(\<lambda>x. a + (g x) * b) \<in> borel_measurable M"
```
```   830   using affine_borel_measurable_vector[OF assms] by (simp add: mult_commute)
```
```   831
```
```   832 lemma (in sigma_algebra) borel_measurable_add[simp, intro]:
```
```   833   fixes f :: "'a \<Rightarrow> real"
```
```   834   assumes f: "f \<in> borel_measurable M"
```
```   835   assumes g: "g \<in> borel_measurable M"
```
```   836   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```   837 proof -
```
```   838   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}"
```
```   839     by auto
```
```   840   have "\<And>a. (\<lambda>w. a + (g w) * -1) \<in> borel_measurable M"
```
```   841     by (rule affine_borel_measurable [OF g])
```
```   842   then have "\<And>a. {w \<in> space M. (\<lambda>w. a + (g w) * -1)(w) \<le> f w} \<in> sets M" using f
```
```   843     by auto
```
```   844   then have "\<And>a. {w \<in> space M. a \<le> f w + g w} \<in> sets M"
```
```   845     by (simp add: 1)
```
```   846   then show ?thesis
```
```   847     by (simp add: borel_measurable_iff_ge)
```
```   848 qed
```
```   849
```
```   850 lemma (in sigma_algebra) borel_measurable_setsum[simp, intro]:
```
```   851   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
```
```   852   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   853   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```   854 proof cases
```
```   855   assume "finite S"
```
```   856   thus ?thesis using assms by induct auto
```
```   857 qed simp
```
```   858
```
```   859 lemma (in sigma_algebra) borel_measurable_square:
```
```   860   fixes f :: "'a \<Rightarrow> real"
```
```   861   assumes f: "f \<in> borel_measurable M"
```
```   862   shows "(\<lambda>x. (f x)^2) \<in> borel_measurable M"
```
```   863 proof -
```
```   864   {
```
```   865     fix a
```
```   866     have "{w \<in> space M. (f w)\<twosuperior> \<le> a} \<in> sets M"
```
```   867     proof (cases rule: linorder_cases [of a 0])
```
```   868       case less
```
```   869       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = {}"
```
```   870         by auto (metis less order_le_less_trans power2_less_0)
```
```   871       also have "... \<in> sets M"
```
```   872         by (rule empty_sets)
```
```   873       finally show ?thesis .
```
```   874     next
```
```   875       case equal
```
```   876       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
```
```   877              {w \<in> space M. f w \<le> 0} \<inter> {w \<in> space M. 0 \<le> f w}"
```
```   878         by auto
```
```   879       also have "... \<in> sets M"
```
```   880         apply (insert f)
```
```   881         apply (rule Int)
```
```   882         apply (simp add: borel_measurable_iff_le)
```
```   883         apply (simp add: borel_measurable_iff_ge)
```
```   884         done
```
```   885       finally show ?thesis .
```
```   886     next
```
```   887       case greater
```
```   888       have "\<forall>x. (f x ^ 2 \<le> sqrt a ^ 2) = (- sqrt a  \<le> f x & f x \<le> sqrt a)"
```
```   889         by (metis abs_le_interval_iff abs_of_pos greater real_sqrt_abs
```
```   890                   real_sqrt_le_iff real_sqrt_power)
```
```   891       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
```
```   892              {w \<in> space M. -(sqrt a) \<le> f w} \<inter> {w \<in> space M. f w \<le> sqrt a}"
```
```   893         using greater by auto
```
```   894       also have "... \<in> sets M"
```
```   895         apply (insert f)
```
```   896         apply (rule Int)
```
```   897         apply (simp add: borel_measurable_iff_ge)
```
```   898         apply (simp add: borel_measurable_iff_le)
```
```   899         done
```
```   900       finally show ?thesis .
```
```   901     qed
```
```   902   }
```
```   903   thus ?thesis by (auto simp add: borel_measurable_iff_le)
```
```   904 qed
```
```   905
```
```   906 lemma times_eq_sum_squares:
```
```   907    fixes x::real
```
```   908    shows"x*y = ((x+y)^2)/4 - ((x-y)^ 2)/4"
```
```   909 by (simp add: power2_eq_square ring_distribs diff_divide_distrib [symmetric])
```
```   910
```
```   911 lemma (in sigma_algebra) borel_measurable_uminus[simp, intro]:
```
```   912   fixes g :: "'a \<Rightarrow> real"
```
```   913   assumes g: "g \<in> borel_measurable M"
```
```   914   shows "(\<lambda>x. - g x) \<in> borel_measurable M"
```
```   915 proof -
```
```   916   have "(\<lambda>x. - g x) = (\<lambda>x. 0 + (g x) * -1)"
```
```   917     by simp
```
```   918   also have "... \<in> borel_measurable M"
```
```   919     by (fast intro: affine_borel_measurable g)
```
```   920   finally show ?thesis .
```
```   921 qed
```
```   922
```
```   923 lemma (in sigma_algebra) borel_measurable_times[simp, intro]:
```
```   924   fixes f :: "'a \<Rightarrow> real"
```
```   925   assumes f: "f \<in> borel_measurable M"
```
```   926   assumes g: "g \<in> borel_measurable M"
```
```   927   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```   928 proof -
```
```   929   have 1: "(\<lambda>x. 0 + (f x + g x)\<twosuperior> * inverse 4) \<in> borel_measurable M"
```
```   930     using assms by (fast intro: affine_borel_measurable borel_measurable_square)
```
```   931   have "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) =
```
```   932         (\<lambda>x. 0 + ((f x + -g x) ^ 2 * inverse -4))"
```
```   933     by (simp add: minus_divide_right)
```
```   934   also have "... \<in> borel_measurable M"
```
```   935     using f g by (fast intro: affine_borel_measurable borel_measurable_square f g)
```
```   936   finally have 2: "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) \<in> borel_measurable M" .
```
```   937   show ?thesis
```
```   938     apply (simp add: times_eq_sum_squares diff_minus)
```
```   939     using 1 2 by simp
```
```   940 qed
```
```   941
```
```   942 lemma (in sigma_algebra) borel_measurable_setprod[simp, intro]:
```
```   943   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
```
```   944   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   945   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```   946 proof cases
```
```   947   assume "finite S"
```
```   948   thus ?thesis using assms by induct auto
```
```   949 qed simp
```
```   950
```
```   951 lemma (in sigma_algebra) borel_measurable_diff[simp, intro]:
```
```   952   fixes f :: "'a \<Rightarrow> real"
```
```   953   assumes f: "f \<in> borel_measurable M"
```
```   954   assumes g: "g \<in> borel_measurable M"
```
```   955   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```   956   unfolding diff_minus using assms by fast
```
```   957
```
```   958 lemma (in sigma_algebra) borel_measurable_inverse[simp, intro]:
```
```   959   fixes f :: "'a \<Rightarrow> real"
```
```   960   assumes "f \<in> borel_measurable M"
```
```   961   shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
```
```   962   unfolding borel_measurable_iff_ge unfolding inverse_eq_divide
```
```   963 proof safe
```
```   964   fix a :: real
```
```   965   have *: "{w \<in> space M. a \<le> 1 / f w} =
```
```   966       ({w \<in> space M. 0 < f w} \<inter> {w \<in> space M. a * f w \<le> 1}) \<union>
```
```   967       ({w \<in> space M. f w < 0} \<inter> {w \<in> space M. 1 \<le> a * f w}) \<union>
```
```   968       ({w \<in> space M. f w = 0} \<inter> {w \<in> space M. a \<le> 0})" by (auto simp: le_divide_eq)
```
```   969   show "{w \<in> space M. a \<le> 1 / f w} \<in> sets M" using assms unfolding *
```
```   970     by (auto intro!: Int Un)
```
```   971 qed
```
```   972
```
```   973 lemma (in sigma_algebra) borel_measurable_divide[simp, intro]:
```
```   974   fixes f :: "'a \<Rightarrow> real"
```
```   975   assumes "f \<in> borel_measurable M"
```
```   976   and "g \<in> borel_measurable M"
```
```   977   shows "(\<lambda>x. f x / g x) \<in> borel_measurable M"
```
```   978   unfolding field_divide_inverse
```
```   979   by (rule borel_measurable_inverse borel_measurable_times assms)+
```
```   980
```
```   981 lemma (in sigma_algebra) borel_measurable_max[intro, simp]:
```
```   982   fixes f g :: "'a \<Rightarrow> real"
```
```   983   assumes "f \<in> borel_measurable M"
```
```   984   assumes "g \<in> borel_measurable M"
```
```   985   shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
```
```   986   unfolding borel_measurable_iff_le
```
```   987 proof safe
```
```   988   fix a
```
```   989   have "{x \<in> space M. max (g x) (f x) \<le> a} =
```
```   990     {x \<in> space M. g x \<le> a} \<inter> {x \<in> space M. f x \<le> a}" by auto
```
```   991   thus "{x \<in> space M. max (g x) (f x) \<le> a} \<in> sets M"
```
```   992     using assms unfolding borel_measurable_iff_le
```
```   993     by (auto intro!: Int)
```
```   994 qed
```
```   995
```
```   996 lemma (in sigma_algebra) borel_measurable_min[intro, simp]:
```
```   997   fixes f g :: "'a \<Rightarrow> real"
```
```   998   assumes "f \<in> borel_measurable M"
```
```   999   assumes "g \<in> borel_measurable M"
```
```  1000   shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
```
```  1001   unfolding borel_measurable_iff_ge
```
```  1002 proof safe
```
```  1003   fix a
```
```  1004   have "{x \<in> space M. a \<le> min (g x) (f x)} =
```
```  1005     {x \<in> space M. a \<le> g x} \<inter> {x \<in> space M. a \<le> f x}" by auto
```
```  1006   thus "{x \<in> space M. a \<le> min (g x) (f x)} \<in> sets M"
```
```  1007     using assms unfolding borel_measurable_iff_ge
```
```  1008     by (auto intro!: Int)
```
```  1009 qed
```
```  1010
```
```  1011 lemma (in sigma_algebra) borel_measurable_abs[simp, intro]:
```
```  1012   assumes "f \<in> borel_measurable M"
```
```  1013   shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
```
```  1014 proof -
```
```  1015   have *: "\<And>x. \<bar>f x\<bar> = max 0 (f x) + max 0 (- f x)" by (simp add: max_def)
```
```  1016   show ?thesis unfolding * using assms by auto
```
```  1017 qed
```
```  1018
```
```  1019 lemma borel_measurable_nth[simp, intro]:
```
```  1020   "(\<lambda>x::real^'n. x \$ i) \<in> borel_measurable borel"
```
```  1021   using borel_measurable_euclidean_component
```
```  1022   unfolding nth_conv_component by auto
```
```  1023
```
```  1024 section "Borel space over the real line with infinity"
```
```  1025
```
```  1026 lemma borel_Real_measurable:
```
```  1027   "A \<in> sets borel \<Longrightarrow> Real -` A \<in> sets borel"
```
```  1028 proof (rule borel_measurable_translate)
```
```  1029   fix B :: "pextreal set" assume "open B"
```
```  1030   then obtain T x where T: "open T" "Real ` (T \<inter> {0..}) = B - {\<omega>}" and
```
```  1031     x: "\<omega> \<in> B \<Longrightarrow> 0 \<le> x" "\<omega> \<in> B \<Longrightarrow> {Real x <..} \<subseteq> B"
```
```  1032     unfolding open_pextreal_def by blast
```
```  1033   have "Real -` B = Real -` (B - {\<omega>})" by auto
```
```  1034   also have "\<dots> = Real -` (Real ` (T \<inter> {0..}))" using T by simp
```
```  1035   also have "\<dots> = (if 0 \<in> T then T \<union> {.. 0} else T \<inter> {0..})"
```
```  1036     apply (auto simp add: Real_eq_Real image_iff)
```
```  1037     apply (rule_tac x="max 0 x" in bexI)
```
```  1038     by (auto simp: max_def)
```
```  1039   finally show "Real -` B \<in> sets borel"
```
```  1040     using `open T` by auto
```
```  1041 qed simp
```
```  1042
```
```  1043 lemma borel_real_measurable:
```
```  1044   "A \<in> sets borel \<Longrightarrow> (real -` A :: pextreal set) \<in> sets borel"
```
```  1045 proof (rule borel_measurable_translate)
```
```  1046   fix B :: "real set" assume "open B"
```
```  1047   { fix x have "0 < real x \<longleftrightarrow> (\<exists>r>0. x = Real r)" by (cases x) auto }
```
```  1048   note Ex_less_real = this
```
```  1049   have *: "real -` B = (if 0 \<in> B then real -` (B \<inter> {0 <..}) \<union> {0, \<omega>} else real -` (B \<inter> {0 <..}))"
```
```  1050     by (force simp: Ex_less_real)
```
```  1051
```
```  1052   have "open (real -` (B \<inter> {0 <..}) :: pextreal set)"
```
```  1053     unfolding open_pextreal_def using `open B`
```
```  1054     by (auto intro!: open_Int exI[of _ "B \<inter> {0 <..}"] simp: image_iff Ex_less_real)
```
```  1055   then show "(real -` B :: pextreal set) \<in> sets borel" unfolding * by auto
```
```  1056 qed simp
```
```  1057
```
```  1058 lemma (in sigma_algebra) borel_measurable_Real[intro, simp]:
```
```  1059   assumes "f \<in> borel_measurable M"
```
```  1060   shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
```
```  1061   unfolding in_borel_measurable_borel
```
```  1062 proof safe
```
```  1063   fix S :: "pextreal set" assume "S \<in> sets borel"
```
```  1064   from borel_Real_measurable[OF this]
```
```  1065   have "(Real \<circ> f) -` S \<inter> space M \<in> sets M"
```
```  1066     using assms
```
```  1067     unfolding vimage_compose in_borel_measurable_borel
```
```  1068     by auto
```
```  1069   thus "(\<lambda>x. Real (f x)) -` S \<inter> space M \<in> sets M" by (simp add: comp_def)
```
```  1070 qed
```
```  1071
```
```  1072 lemma (in sigma_algebra) borel_measurable_real[intro, simp]:
```
```  1073   fixes f :: "'a \<Rightarrow> pextreal"
```
```  1074   assumes "f \<in> borel_measurable M"
```
```  1075   shows "(\<lambda>x. real (f x)) \<in> borel_measurable M"
```
```  1076   unfolding in_borel_measurable_borel
```
```  1077 proof safe
```
```  1078   fix S :: "real set" assume "S \<in> sets borel"
```
```  1079   from borel_real_measurable[OF this]
```
```  1080   have "(real \<circ> f) -` S \<inter> space M \<in> sets M"
```
```  1081     using assms
```
```  1082     unfolding vimage_compose in_borel_measurable_borel
```
```  1083     by auto
```
```  1084   thus "(\<lambda>x. real (f x)) -` S \<inter> space M \<in> sets M" by (simp add: comp_def)
```
```  1085 qed
```
```  1086
```
```  1087 lemma (in sigma_algebra) borel_measurable_Real_eq:
```
```  1088   assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
```
```  1089   shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
```
```  1090 proof
```
```  1091   have [simp]: "(\<lambda>x. Real (f x)) -` {\<omega>} \<inter> space M = {}"
```
```  1092     by auto
```
```  1093   assume "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
```
```  1094   hence "(\<lambda>x. real (Real (f x))) \<in> borel_measurable M"
```
```  1095     by (rule borel_measurable_real)
```
```  1096   moreover have "\<And>x. x \<in> space M \<Longrightarrow> real (Real (f x)) = f x"
```
```  1097     using assms by auto
```
```  1098   ultimately show "f \<in> borel_measurable M"
```
```  1099     by (simp cong: measurable_cong)
```
```  1100 qed auto
```
```  1101
```
```  1102 lemma (in sigma_algebra) borel_measurable_pextreal_eq_real:
```
```  1103   "f \<in> borel_measurable M \<longleftrightarrow>
```
```  1104     ((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<omega>} \<inter> space M \<in> sets M)"
```
```  1105 proof safe
```
```  1106   assume "f \<in> borel_measurable M"
```
```  1107   then show "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<omega>} \<inter> space M \<in> sets M"
```
```  1108     by (auto intro: borel_measurable_vimage borel_measurable_real)
```
```  1109 next
```
```  1110   assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<omega>} \<inter> space M \<in> sets M"
```
```  1111   have "f -` {\<omega>} \<inter> space M = {x\<in>space M. f x = \<omega>}" by auto
```
```  1112   with * have **: "{x\<in>space M. f x = \<omega>} \<in> sets M" by simp
```
```  1113   have f: "f = (\<lambda>x. if f x = \<omega> then \<omega> else Real (real (f x)))"
```
```  1114     by (simp add: fun_eq_iff Real_real)
```
```  1115   show "f \<in> borel_measurable M"
```
```  1116     apply (subst f)
```
```  1117     apply (rule measurable_If)
```
```  1118     using * ** by auto
```
```  1119 qed
```
```  1120
```
```  1121 lemma (in sigma_algebra) less_eq_ge_measurable:
```
```  1122   fixes f :: "'a \<Rightarrow> 'c::linorder"
```
```  1123   shows "{x\<in>space M. a < f x} \<in> sets M \<longleftrightarrow> {x\<in>space M. f x \<le> a} \<in> sets M"
```
```  1124 proof
```
```  1125   assume "{x\<in>space M. f x \<le> a} \<in> sets M"
```
```  1126   moreover have "{x\<in>space M. a < f x} = space M - {x\<in>space M. f x \<le> a}" by auto
```
```  1127   ultimately show "{x\<in>space M. a < f x} \<in> sets M" by auto
```
```  1128 next
```
```  1129   assume "{x\<in>space M. a < f x} \<in> sets M"
```
```  1130   moreover have "{x\<in>space M. f x \<le> a} = space M - {x\<in>space M. a < f x}" by auto
```
```  1131   ultimately show "{x\<in>space M. f x \<le> a} \<in> sets M" by auto
```
```  1132 qed
```
```  1133
```
```  1134 lemma (in sigma_algebra) greater_eq_le_measurable:
```
```  1135   fixes f :: "'a \<Rightarrow> 'c::linorder"
```
```  1136   shows "{x\<in>space M. f x < a} \<in> sets M \<longleftrightarrow> {x\<in>space M. a \<le> f x} \<in> sets M"
```
```  1137 proof
```
```  1138   assume "{x\<in>space M. a \<le> f x} \<in> sets M"
```
```  1139   moreover have "{x\<in>space M. f x < a} = space M - {x\<in>space M. a \<le> f x}" by auto
```
```  1140   ultimately show "{x\<in>space M. f x < a} \<in> sets M" by auto
```
```  1141 next
```
```  1142   assume "{x\<in>space M. f x < a} \<in> sets M"
```
```  1143   moreover have "{x\<in>space M. a \<le> f x} = space M - {x\<in>space M. f x < a}" by auto
```
```  1144   ultimately show "{x\<in>space M. a \<le> f x} \<in> sets M" by auto
```
```  1145 qed
```
```  1146
```
```  1147 lemma (in sigma_algebra) less_eq_le_pextreal_measurable:
```
```  1148   fixes f :: "'a \<Rightarrow> pextreal"
```
```  1149   shows "(\<forall>a. {x\<in>space M. a < f x} \<in> sets M) \<longleftrightarrow> (\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M)"
```
```  1150 proof
```
```  1151   assume a: "\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M"
```
```  1152   show "\<forall>a. {x \<in> space M. a < f x} \<in> sets M"
```
```  1153   proof
```
```  1154     fix a show "{x \<in> space M. a < f x} \<in> sets M"
```
```  1155     proof (cases a)
```
```  1156       case (preal r)
```
```  1157       have "{x\<in>space M. a < f x} = (\<Union>i. {x\<in>space M. a + inverse (of_nat (Suc i)) \<le> f x})"
```
```  1158       proof safe
```
```  1159         fix x assume "a < f x" and [simp]: "x \<in> space M"
```
```  1160         with ex_pextreal_inverse_of_nat_Suc_less[of "f x - a"]
```
```  1161         obtain n where "a + inverse (of_nat (Suc n)) < f x"
```
```  1162           by (cases "f x", auto simp: pextreal_minus_order)
```
```  1163         then have "a + inverse (of_nat (Suc n)) \<le> f x" by simp
```
```  1164         then show "x \<in> (\<Union>i. {x \<in> space M. a + inverse (of_nat (Suc i)) \<le> f x})"
```
```  1165           by auto
```
```  1166       next
```
```  1167         fix i x assume [simp]: "x \<in> space M"
```
```  1168         have "a < a + inverse (of_nat (Suc i))" using preal by auto
```
```  1169         also assume "a + inverse (of_nat (Suc i)) \<le> f x"
```
```  1170         finally show "a < f x" .
```
```  1171       qed
```
```  1172       with a show ?thesis by auto
```
```  1173     qed simp
```
```  1174   qed
```
```  1175 next
```
```  1176   assume a': "\<forall>a. {x \<in> space M. a < f x} \<in> sets M"
```
```  1177   then have a: "\<forall>a. {x \<in> space M. f x \<le> a} \<in> sets M" unfolding less_eq_ge_measurable .
```
```  1178   show "\<forall>a. {x \<in> space M. a \<le> f x} \<in> sets M" unfolding greater_eq_le_measurable[symmetric]
```
```  1179   proof
```
```  1180     fix a show "{x \<in> space M. f x < a} \<in> sets M"
```
```  1181     proof (cases a)
```
```  1182       case (preal r)
```
```  1183       show ?thesis
```
```  1184       proof cases
```
```  1185         assume "a = 0" then show ?thesis by simp
```
```  1186       next
```
```  1187         assume "a \<noteq> 0"
```
```  1188         have "{x\<in>space M. f x < a} = (\<Union>i. {x\<in>space M. f x \<le> a - inverse (of_nat (Suc i))})"
```
```  1189         proof safe
```
```  1190           fix x assume "f x < a" and [simp]: "x \<in> space M"
```
```  1191           with ex_pextreal_inverse_of_nat_Suc_less[of "a - f x"]
```
```  1192           obtain n where "inverse (of_nat (Suc n)) < a - f x"
```
```  1193             using preal by (cases "f x") auto
```
```  1194           then have "f x \<le> a - inverse (of_nat (Suc n)) "
```
```  1195             using preal by (cases "f x") (auto split: split_if_asm)
```
```  1196           then show "x \<in> (\<Union>i. {x \<in> space M. f x \<le> a - inverse (of_nat (Suc i))})"
```
```  1197             by auto
```
```  1198         next
```
```  1199           fix i x assume [simp]: "x \<in> space M"
```
```  1200           assume "f x \<le> a - inverse (of_nat (Suc i))"
```
```  1201           also have "\<dots> < a" using `a \<noteq> 0` preal by auto
```
```  1202           finally show "f x < a" .
```
```  1203         qed
```
```  1204         with a show ?thesis by auto
```
```  1205       qed
```
```  1206     next
```
```  1207       case infinite
```
```  1208       have "f -` {\<omega>} \<inter> space M = (\<Inter>n. {x\<in>space M. of_nat n < f x})"
```
```  1209       proof (safe, simp_all, safe)
```
```  1210         fix x assume *: "\<forall>n::nat. Real (real n) < f x"
```
```  1211         show "f x = \<omega>"    proof (rule ccontr)
```
```  1212           assume "f x \<noteq> \<omega>"
```
```  1213           with real_arch_lt[of "real (f x)"] obtain n where "f x < of_nat n"
```
```  1214             by (auto simp: pextreal_noteq_omega_Ex)
```
```  1215           with *[THEN spec, of n] show False by auto
```
```  1216         qed
```
```  1217       qed
```
```  1218       with a' have \<omega>: "f -` {\<omega>} \<inter> space M \<in> sets M" by auto
```
```  1219       moreover have "{x \<in> space M. f x < a} = space M - f -` {\<omega>} \<inter> space M"
```
```  1220         using infinite by auto
```
```  1221       ultimately show ?thesis by auto
```
```  1222     qed
```
```  1223   qed
```
```  1224 qed
```
```  1225
```
```  1226 lemma (in sigma_algebra) borel_measurable_pextreal_iff_greater:
```
```  1227   "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. a < f x} \<in> sets M)"
```
```  1228 proof safe
```
```  1229   fix a assume f: "f \<in> borel_measurable M"
```
```  1230   have "{x\<in>space M. a < f x} = f -` {a <..} \<inter> space M" by auto
```
```  1231   with f show "{x\<in>space M. a < f x} \<in> sets M"
```
```  1232     by (auto intro!: measurable_sets)
```
```  1233 next
```
```  1234   assume *: "\<forall>a. {x\<in>space M. a < f x} \<in> sets M"
```
```  1235   hence **: "\<forall>a. {x\<in>space M. f x < a} \<in> sets M"
```
```  1236     unfolding less_eq_le_pextreal_measurable
```
```  1237     unfolding greater_eq_le_measurable .
```
```  1238   show "f \<in> borel_measurable M" unfolding borel_measurable_pextreal_eq_real borel_measurable_iff_greater
```
```  1239   proof safe
```
```  1240     have "f -` {\<omega>} \<inter> space M = space M - {x\<in>space M. f x < \<omega>}" by auto
```
```  1241     then show \<omega>: "f -` {\<omega>} \<inter> space M \<in> sets M" using ** by auto
```
```  1242     fix a
```
```  1243     have "{w \<in> space M. a < real (f w)} =
```
```  1244       (if 0 \<le> a then {w\<in>space M. Real a < f w} - (f -` {\<omega>} \<inter> space M) else space M)"
```
```  1245     proof (split split_if, safe del: notI)
```
```  1246       fix x assume "0 \<le> a"
```
```  1247       { assume "a < real (f x)" then show "Real a < f x" "x \<notin> f -` {\<omega>} \<inter> space M"
```
```  1248           using `0 \<le> a` by (cases "f x", auto) }
```
```  1249       { assume "Real a < f x" "x \<notin> f -` {\<omega>}" then show "a < real (f x)"
```
```  1250           using `0 \<le> a` by (cases "f x", auto) }
```
```  1251     next
```
```  1252       fix x assume "\<not> 0 \<le> a" then show "a < real (f x)" by (cases "f x") auto
```
```  1253     qed
```
```  1254     then show "{w \<in> space M. a < real (f w)} \<in> sets M"
```
```  1255       using \<omega> * by (auto intro!: Diff)
```
```  1256   qed
```
```  1257 qed
```
```  1258
```
```  1259 lemma (in sigma_algebra) borel_measurable_pextreal_iff_less:
```
```  1260   "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. f x < a} \<in> sets M)"
```
```  1261   using borel_measurable_pextreal_iff_greater unfolding less_eq_le_pextreal_measurable greater_eq_le_measurable .
```
```  1262
```
```  1263 lemma (in sigma_algebra) borel_measurable_pextreal_iff_le:
```
```  1264   "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. f x \<le> a} \<in> sets M)"
```
```  1265   using borel_measurable_pextreal_iff_greater unfolding less_eq_ge_measurable .
```
```  1266
```
```  1267 lemma (in sigma_algebra) borel_measurable_pextreal_iff_ge:
```
```  1268   "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M)"
```
```  1269   using borel_measurable_pextreal_iff_greater unfolding less_eq_le_pextreal_measurable .
```
```  1270
```
```  1271 lemma (in sigma_algebra) borel_measurable_pextreal_eq_const:
```
```  1272   fixes f :: "'a \<Rightarrow> pextreal" assumes "f \<in> borel_measurable M"
```
```  1273   shows "{x\<in>space M. f x = c} \<in> sets M"
```
```  1274 proof -
```
```  1275   have "{x\<in>space M. f x = c} = (f -` {c} \<inter> space M)" by auto
```
```  1276   then show ?thesis using assms by (auto intro!: measurable_sets)
```
```  1277 qed
```
```  1278
```
```  1279 lemma (in sigma_algebra) borel_measurable_pextreal_neq_const:
```
```  1280   fixes f :: "'a \<Rightarrow> pextreal"
```
```  1281   assumes "f \<in> borel_measurable M"
```
```  1282   shows "{x\<in>space M. f x \<noteq> c} \<in> sets M"
```
```  1283 proof -
```
```  1284   have "{x\<in>space M. f x \<noteq> c} = space M - (f -` {c} \<inter> space M)" by auto
```
```  1285   then show ?thesis using assms by (auto intro!: measurable_sets)
```
```  1286 qed
```
```  1287
```
```  1288 lemma (in sigma_algebra) borel_measurable_pextreal_less[intro,simp]:
```
```  1289   fixes f g :: "'a \<Rightarrow> pextreal"
```
```  1290   assumes f: "f \<in> borel_measurable M"
```
```  1291   assumes g: "g \<in> borel_measurable M"
```
```  1292   shows "{x \<in> space M. f x < g x} \<in> sets M"
```
```  1293 proof -
```
```  1294   have "(\<lambda>x. real (f x)) \<in> borel_measurable M"
```
```  1295     "(\<lambda>x. real (g x)) \<in> borel_measurable M"
```
```  1296     using assms by (auto intro!: borel_measurable_real)
```
```  1297   from borel_measurable_less[OF this]
```
```  1298   have "{x \<in> space M. real (f x) < real (g x)} \<in> sets M" .
```
```  1299   moreover have "{x \<in> space M. f x \<noteq> \<omega>} \<in> sets M" using f by (rule borel_measurable_pextreal_neq_const)
```
```  1300   moreover have "{x \<in> space M. g x = \<omega>} \<in> sets M" using g by (rule borel_measurable_pextreal_eq_const)
```
```  1301   moreover have "{x \<in> space M. g x \<noteq> \<omega>} \<in> sets M" using g by (rule borel_measurable_pextreal_neq_const)
```
```  1302   moreover have "{x \<in> space M. f x < g x} = ({x \<in> space M. g x = \<omega>} \<inter> {x \<in> space M. f x \<noteq> \<omega>}) \<union>
```
```  1303     ({x \<in> space M. g x \<noteq> \<omega>} \<inter> {x \<in> space M. f x \<noteq> \<omega>} \<inter> {x \<in> space M. real (f x) < real (g x)})"
```
```  1304     by (auto simp: real_of_pextreal_strict_mono_iff)
```
```  1305   ultimately show ?thesis by auto
```
```  1306 qed
```
```  1307
```
```  1308 lemma (in sigma_algebra) borel_measurable_pextreal_le[intro,simp]:
```
```  1309   fixes f :: "'a \<Rightarrow> pextreal"
```
```  1310   assumes f: "f \<in> borel_measurable M"
```
```  1311   assumes g: "g \<in> borel_measurable M"
```
```  1312   shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
```
```  1313 proof -
```
```  1314   have "{x \<in> space M. f x \<le> g x} = space M - {x \<in> space M. g x < f x}" by auto
```
```  1315   then show ?thesis using g f by auto
```
```  1316 qed
```
```  1317
```
```  1318 lemma (in sigma_algebra) borel_measurable_pextreal_eq[intro,simp]:
```
```  1319   fixes f :: "'a \<Rightarrow> pextreal"
```
```  1320   assumes f: "f \<in> borel_measurable M"
```
```  1321   assumes g: "g \<in> borel_measurable M"
```
```  1322   shows "{w \<in> space M. f w = g w} \<in> sets M"
```
```  1323 proof -
```
```  1324   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
```
```  1325   then show ?thesis using g f by auto
```
```  1326 qed
```
```  1327
```
```  1328 lemma (in sigma_algebra) borel_measurable_pextreal_neq[intro,simp]:
```
```  1329   fixes f :: "'a \<Rightarrow> pextreal"
```
```  1330   assumes f: "f \<in> borel_measurable M"
```
```  1331   assumes g: "g \<in> borel_measurable M"
```
```  1332   shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
```
```  1333 proof -
```
```  1334   have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}" by auto
```
```  1335   thus ?thesis using f g by auto
```
```  1336 qed
```
```  1337
```
```  1338 lemma (in sigma_algebra) borel_measurable_pextreal_add[intro, simp]:
```
```  1339   fixes f :: "'a \<Rightarrow> pextreal"
```
```  1340   assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1341   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```  1342 proof -
```
```  1343   have *: "(\<lambda>x. f x + g x) =
```
```  1344      (\<lambda>x. if f x = \<omega> then \<omega> else if g x = \<omega> then \<omega> else Real (real (f x) + real (g x)))"
```
```  1345      by (auto simp: fun_eq_iff pextreal_noteq_omega_Ex)
```
```  1346   show ?thesis using assms unfolding *
```
```  1347     by (auto intro!: measurable_If)
```
```  1348 qed
```
```  1349
```
```  1350 lemma (in sigma_algebra) borel_measurable_pextreal_setsum[simp, intro]:
```
```  1351   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> pextreal"
```
```  1352   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1353   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1354 proof cases
```
```  1355   assume "finite S"
```
```  1356   thus ?thesis using assms
```
```  1357     by induct auto
```
```  1358 qed (simp add: borel_measurable_const)
```
```  1359
```
```  1360 lemma (in sigma_algebra) borel_measurable_pextreal_times[intro, simp]:
```
```  1361   fixes f :: "'a \<Rightarrow> pextreal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1362   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1363 proof -
```
```  1364   have *: "(\<lambda>x. f x * g x) =
```
```  1365      (\<lambda>x. if f x = 0 then 0 else if g x = 0 then 0 else if f x = \<omega> then \<omega> else if g x = \<omega> then \<omega> else
```
```  1366       Real (real (f x) * real (g x)))"
```
```  1367      by (auto simp: fun_eq_iff pextreal_noteq_omega_Ex)
```
```  1368   show ?thesis using assms unfolding *
```
```  1369     by (auto intro!: measurable_If)
```
```  1370 qed
```
```  1371
```
```  1372 lemma (in sigma_algebra) borel_measurable_pextreal_setprod[simp, intro]:
```
```  1373   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> pextreal"
```
```  1374   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1375   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1376 proof cases
```
```  1377   assume "finite S"
```
```  1378   thus ?thesis using assms by induct auto
```
```  1379 qed simp
```
```  1380
```
```  1381 lemma (in sigma_algebra) borel_measurable_pextreal_min[simp, intro]:
```
```  1382   fixes f g :: "'a \<Rightarrow> pextreal"
```
```  1383   assumes "f \<in> borel_measurable M"
```
```  1384   assumes "g \<in> borel_measurable M"
```
```  1385   shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
```
```  1386   using assms unfolding min_def by (auto intro!: measurable_If)
```
```  1387
```
```  1388 lemma (in sigma_algebra) borel_measurable_pextreal_max[simp, intro]:
```
```  1389   fixes f g :: "'a \<Rightarrow> pextreal"
```
```  1390   assumes "f \<in> borel_measurable M"
```
```  1391   and "g \<in> borel_measurable M"
```
```  1392   shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
```
```  1393   using assms unfolding max_def by (auto intro!: measurable_If)
```
```  1394
```
```  1395 lemma (in sigma_algebra) borel_measurable_SUP[simp, intro]:
```
```  1396   fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> pextreal"
```
```  1397   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1398   shows "(\<lambda>x. SUP i : A. f i x) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M")
```
```  1399   unfolding borel_measurable_pextreal_iff_greater
```
```  1400 proof safe
```
```  1401   fix a
```
```  1402   have "{x\<in>space M. a < ?sup x} = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"
```
```  1403     by (auto simp: less_SUP_iff SUPR_apply)
```
```  1404   then show "{x\<in>space M. a < ?sup x} \<in> sets M"
```
```  1405     using assms by auto
```
```  1406 qed
```
```  1407
```
```  1408 lemma (in sigma_algebra) borel_measurable_INF[simp, intro]:
```
```  1409   fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> pextreal"
```
```  1410   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1411   shows "(\<lambda>x. INF i : A. f i x) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M")
```
```  1412   unfolding borel_measurable_pextreal_iff_less
```
```  1413 proof safe
```
```  1414   fix a
```
```  1415   have "{x\<in>space M. ?inf x < a} = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"
```
```  1416     by (auto simp: INF_less_iff INFI_apply)
```
```  1417   then show "{x\<in>space M. ?inf x < a} \<in> sets M"
```
```  1418     using assms by auto
```
```  1419 qed
```
```  1420
```
```  1421 lemma (in sigma_algebra) borel_measurable_pextreal_diff[simp, intro]:
```
```  1422   fixes f g :: "'a \<Rightarrow> pextreal"
```
```  1423   assumes "f \<in> borel_measurable M"
```
```  1424   assumes "g \<in> borel_measurable M"
```
```  1425   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```  1426   unfolding borel_measurable_pextreal_iff_greater
```
```  1427 proof safe
```
```  1428   fix a
```
```  1429   have "{x \<in> space M. a < f x - g x} = {x \<in> space M. g x + a < f x}"
```
```  1430     by (simp add: pextreal_less_minus_iff)
```
```  1431   then show "{x \<in> space M. a < f x - g x} \<in> sets M"
```
```  1432     using assms by auto
```
```  1433 qed
```
```  1434
```
```  1435 lemma (in sigma_algebra) borel_measurable_psuminf[simp, intro]:
```
```  1436   assumes "\<And>i. f i \<in> borel_measurable M"
```
```  1437   shows "(\<lambda>x. (\<Sum>\<^isub>\<infinity> i. f i x)) \<in> borel_measurable M"
```
```  1438   using assms unfolding psuminf_def by auto
```
```  1439
```
```  1440 section "LIMSEQ is borel measurable"
```
```  1441
```
```  1442 lemma (in sigma_algebra) borel_measurable_LIMSEQ:
```
```  1443   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
```
```  1444   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
```
```  1445   and u: "\<And>i. u i \<in> borel_measurable M"
```
```  1446   shows "u' \<in> borel_measurable M"
```
```  1447 proof -
```
```  1448   let "?pu x i" = "max (u i x) 0"
```
```  1449   let "?nu x i" = "max (- u i x) 0"
```
```  1450   { fix x assume x: "x \<in> space M"
```
```  1451     have "(?pu x) ----> max (u' x) 0"
```
```  1452       "(?nu x) ----> max (- u' x) 0"
```
```  1453       using u'[OF x] by (auto intro!: LIMSEQ_max LIMSEQ_minus)
```
```  1454     from LIMSEQ_imp_lim_INF[OF _ this(1)] LIMSEQ_imp_lim_INF[OF _ this(2)]
```
```  1455     have "(SUP n. INF m. Real (u (n + m) x)) = Real (u' x)"
```
```  1456       "(SUP n. INF m. Real (- u (n + m) x)) = Real (- u' x)"
```
```  1457       by (simp_all add: Real_max'[symmetric]) }
```
```  1458   note eq = this
```
```  1459   have *: "\<And>x. real (Real (u' x)) - real (Real (- u' x)) = u' x"
```
```  1460     by auto
```
```  1461   have "(\<lambda>x. SUP n. INF m. Real (u (n + m) x)) \<in> borel_measurable M"
```
```  1462        "(\<lambda>x. SUP n. INF m. Real (- u (n + m) x)) \<in> borel_measurable M"
```
```  1463     using u by auto
```
```  1464   with eq[THEN measurable_cong, of M "\<lambda>x. x" borel]
```
```  1465   have "(\<lambda>x. Real (u' x)) \<in> borel_measurable M"
```
```  1466        "(\<lambda>x. Real (- u' x)) \<in> borel_measurable M" by auto
```
```  1467   note this[THEN borel_measurable_real]
```
```  1468   from borel_measurable_diff[OF this]
```
```  1469   show ?thesis unfolding * .
```
```  1470 qed
```
```  1471
```
```  1472 end
```