src/HOL/Probability/Borel_Space.thy
 author hoelzl Wed Feb 23 11:33:45 2011 +0100 (2011-02-23) changeset 41830 719b0a517c33 parent 41545 9c869baf1c66 child 41969 1cf3e4107a2a permissions -rw-r--r--
log is borel measurable
```     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 borel_comp[intro,simp]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
```
```    52   unfolding Compl_eq_Diff_UNIV by (intro borel.Diff) auto
```
```    53
```
```    54 lemma (in sigma_algebra) borel_measurable_vimage:
```
```    55   fixes f :: "'a \<Rightarrow> 'x::t2_space"
```
```    56   assumes borel: "f \<in> borel_measurable M"
```
```    57   shows "f -` {x} \<inter> space M \<in> sets M"
```
```    58 proof (cases "x \<in> f ` space M")
```
```    59   case True then obtain y where "x = f y" by auto
```
```    60   from closed_sing[of "f y"]
```
```    61   have "{f y} \<in> sets borel" by (rule borel_closed)
```
```    62   with assms show ?thesis
```
```    63     unfolding in_borel_measurable_borel `x = f y` by auto
```
```    64 next
```
```    65   case False hence "f -` {x} \<inter> space M = {}" by auto
```
```    66   thus ?thesis by auto
```
```    67 qed
```
```    68
```
```    69 lemma (in sigma_algebra) borel_measurableI:
```
```    70   fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space"
```
```    71   assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
```
```    72   shows "f \<in> borel_measurable M"
```
```    73   unfolding borel_def
```
```    74 proof (rule measurable_sigma, simp_all)
```
```    75   fix S :: "'x set" assume "S \<in> open" thus "f -` S \<inter> space M \<in> sets M"
```
```    76     using assms[of S] by (simp add: mem_def)
```
```    77 qed
```
```    78
```
```    79 lemma borel_singleton[simp, intro]:
```
```    80   fixes x :: "'a::t1_space"
```
```    81   shows "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets borel"
```
```    82   proof (rule borel.insert_in_sets)
```
```    83     show "{x} \<in> sets borel"
```
```    84       using closed_sing[of x] by (rule borel_closed)
```
```    85   qed simp
```
```    86
```
```    87 lemma (in sigma_algebra) borel_measurable_const[simp, intro]:
```
```    88   "(\<lambda>x. c) \<in> borel_measurable M"
```
```    89   by (auto intro!: measurable_const)
```
```    90
```
```    91 lemma (in sigma_algebra) borel_measurable_indicator[simp, intro!]:
```
```    92   assumes A: "A \<in> sets M"
```
```    93   shows "indicator A \<in> borel_measurable M"
```
```    94   unfolding indicator_def_raw using A
```
```    95   by (auto intro!: measurable_If_set borel_measurable_const)
```
```    96
```
```    97 lemma (in sigma_algebra) borel_measurable_indicator_iff:
```
```    98   "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
```
```    99     (is "?I \<in> borel_measurable M \<longleftrightarrow> _")
```
```   100 proof
```
```   101   assume "?I \<in> borel_measurable M"
```
```   102   then have "?I -` {1} \<inter> space M \<in> sets M"
```
```   103     unfolding measurable_def by auto
```
```   104   also have "?I -` {1} \<inter> space M = A \<inter> space M"
```
```   105     unfolding indicator_def_raw by auto
```
```   106   finally show "A \<inter> space M \<in> sets M" .
```
```   107 next
```
```   108   assume "A \<inter> space M \<in> sets M"
```
```   109   moreover have "?I \<in> borel_measurable M \<longleftrightarrow>
```
```   110     (indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"
```
```   111     by (intro measurable_cong) (auto simp: indicator_def)
```
```   112   ultimately show "?I \<in> borel_measurable M" by auto
```
```   113 qed
```
```   114
```
```   115 lemma borel_measurable_translate:
```
```   116   assumes "A \<in> sets borel" and trans: "\<And>B. open B \<Longrightarrow> f -` B \<in> sets borel"
```
```   117   shows "f -` A \<in> sets borel"
```
```   118 proof -
```
```   119   have "A \<in> sigma_sets UNIV open" using assms
```
```   120     by (simp add: borel_def sigma_def)
```
```   121   thus ?thesis
```
```   122   proof (induct rule: sigma_sets.induct)
```
```   123     case (Basic a) thus ?case using trans[of a] by (simp add: mem_def)
```
```   124   next
```
```   125     case (Compl a)
```
```   126     moreover have "UNIV \<in> sets borel"
```
```   127       using borel.top by simp
```
```   128     ultimately show ?case
```
```   129       by (auto simp: vimage_Diff borel.Diff)
```
```   130   qed (auto simp add: vimage_UN)
```
```   131 qed
```
```   132
```
```   133 lemma (in sigma_algebra) borel_measurable_restricted:
```
```   134   fixes f :: "'a \<Rightarrow> 'x\<Colon>{topological_space, semiring_1}" assumes "A \<in> sets M"
```
```   135   shows "f \<in> borel_measurable (restricted_space A) \<longleftrightarrow>
```
```   136     (\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
```
```   137     (is "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable M")
```
```   138 proof -
```
```   139   interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
```
```   140   have *: "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable ?R"
```
```   141     by (auto intro!: measurable_cong)
```
```   142   show ?thesis unfolding *
```
```   143     unfolding in_borel_measurable_borel
```
```   144   proof (simp, safe)
```
```   145     fix S :: "'x set" assume "S \<in> sets borel"
```
```   146       "\<forall>S\<in>sets borel. ?f -` S \<inter> A \<in> op \<inter> A ` sets M"
```
```   147     then have "?f -` S \<inter> A \<in> op \<inter> A ` sets M" by auto
```
```   148     then have f: "?f -` S \<inter> A \<in> sets M"
```
```   149       using `A \<in> sets M` sets_into_space by fastsimp
```
```   150     show "?f -` S \<inter> space M \<in> sets M"
```
```   151     proof cases
```
```   152       assume "0 \<in> S"
```
```   153       then have "?f -` S \<inter> space M = ?f -` S \<inter> A \<union> (space M - A)"
```
```   154         using `A \<in> sets M` sets_into_space by auto
```
```   155       then show ?thesis using f `A \<in> sets M` by (auto intro!: Un Diff)
```
```   156     next
```
```   157       assume "0 \<notin> S"
```
```   158       then have "?f -` S \<inter> space M = ?f -` S \<inter> A"
```
```   159         using `A \<in> sets M` sets_into_space
```
```   160         by (auto simp: indicator_def split: split_if_asm)
```
```   161       then show ?thesis using f by auto
```
```   162     qed
```
```   163   next
```
```   164     fix S :: "'x set" assume "S \<in> sets borel"
```
```   165       "\<forall>S\<in>sets borel. ?f -` S \<inter> space M \<in> sets M"
```
```   166     then have f: "?f -` S \<inter> space M \<in> sets M" by auto
```
```   167     then show "?f -` S \<inter> A \<in> op \<inter> A ` sets M"
```
```   168       using `A \<in> sets M` sets_into_space
```
```   169       apply (simp add: image_iff)
```
```   170       apply (rule bexI[OF _ f])
```
```   171       by auto
```
```   172   qed
```
```   173 qed
```
```   174
```
```   175 lemma (in sigma_algebra) borel_measurable_subalgebra:
```
```   176   assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
```
```   177   shows "f \<in> borel_measurable M"
```
```   178   using assms unfolding measurable_def by auto
```
```   179
```
```   180 section "Borel spaces on euclidean spaces"
```
```   181
```
```   182 lemma lessThan_borel[simp, intro]:
```
```   183   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   184   shows "{..< a} \<in> sets borel"
```
```   185   by (blast intro: borel_open)
```
```   186
```
```   187 lemma greaterThan_borel[simp, intro]:
```
```   188   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   189   shows "{a <..} \<in> sets borel"
```
```   190   by (blast intro: borel_open)
```
```   191
```
```   192 lemma greaterThanLessThan_borel[simp, intro]:
```
```   193   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   194   shows "{a<..<b} \<in> sets borel"
```
```   195   by (blast intro: borel_open)
```
```   196
```
```   197 lemma atMost_borel[simp, intro]:
```
```   198   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   199   shows "{..a} \<in> sets borel"
```
```   200   by (blast intro: borel_closed)
```
```   201
```
```   202 lemma atLeast_borel[simp, intro]:
```
```   203   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   204   shows "{a..} \<in> sets borel"
```
```   205   by (blast intro: borel_closed)
```
```   206
```
```   207 lemma atLeastAtMost_borel[simp, intro]:
```
```   208   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   209   shows "{a..b} \<in> sets borel"
```
```   210   by (blast intro: borel_closed)
```
```   211
```
```   212 lemma greaterThanAtMost_borel[simp, intro]:
```
```   213   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   214   shows "{a<..b} \<in> sets borel"
```
```   215   unfolding greaterThanAtMost_def by blast
```
```   216
```
```   217 lemma atLeastLessThan_borel[simp, intro]:
```
```   218   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   219   shows "{a..<b} \<in> sets borel"
```
```   220   unfolding atLeastLessThan_def by blast
```
```   221
```
```   222 lemma hafspace_less_borel[simp, intro]:
```
```   223   fixes a :: real
```
```   224   shows "{x::'a::euclidean_space. a < x \$\$ i} \<in> sets borel"
```
```   225   by (auto intro!: borel_open open_halfspace_component_gt)
```
```   226
```
```   227 lemma hafspace_greater_borel[simp, intro]:
```
```   228   fixes a :: real
```
```   229   shows "{x::'a::euclidean_space. x \$\$ i < a} \<in> sets borel"
```
```   230   by (auto intro!: borel_open open_halfspace_component_lt)
```
```   231
```
```   232 lemma hafspace_less_eq_borel[simp, intro]:
```
```   233   fixes a :: real
```
```   234   shows "{x::'a::euclidean_space. a \<le> x \$\$ i} \<in> sets borel"
```
```   235   by (auto intro!: borel_closed closed_halfspace_component_ge)
```
```   236
```
```   237 lemma hafspace_greater_eq_borel[simp, intro]:
```
```   238   fixes a :: real
```
```   239   shows "{x::'a::euclidean_space. x \$\$ i \<le> a} \<in> sets borel"
```
```   240   by (auto intro!: borel_closed closed_halfspace_component_le)
```
```   241
```
```   242 lemma (in sigma_algebra) borel_measurable_less[simp, intro]:
```
```   243   fixes f :: "'a \<Rightarrow> real"
```
```   244   assumes f: "f \<in> borel_measurable M"
```
```   245   assumes g: "g \<in> borel_measurable M"
```
```   246   shows "{w \<in> space M. f w < g w} \<in> sets M"
```
```   247 proof -
```
```   248   have "{w \<in> space M. f w < g w} =
```
```   249         (\<Union>r. (f -` {..< of_rat r} \<inter> space M) \<inter> (g -` {of_rat r <..} \<inter> space M))"
```
```   250     using Rats_dense_in_real by (auto simp add: Rats_def)
```
```   251   then show ?thesis using f g
```
```   252     by simp (blast intro: measurable_sets)
```
```   253 qed
```
```   254
```
```   255 lemma (in sigma_algebra) borel_measurable_le[simp, intro]:
```
```   256   fixes f :: "'a \<Rightarrow> real"
```
```   257   assumes f: "f \<in> borel_measurable M"
```
```   258   assumes g: "g \<in> borel_measurable M"
```
```   259   shows "{w \<in> space M. f w \<le> g w} \<in> sets M"
```
```   260 proof -
```
```   261   have "{w \<in> space M. f w \<le> g w} = space M - {w \<in> space M. g w < f w}"
```
```   262     by auto
```
```   263   thus ?thesis using f g
```
```   264     by simp blast
```
```   265 qed
```
```   266
```
```   267 lemma (in sigma_algebra) borel_measurable_eq[simp, intro]:
```
```   268   fixes f :: "'a \<Rightarrow> real"
```
```   269   assumes f: "f \<in> borel_measurable M"
```
```   270   assumes g: "g \<in> borel_measurable M"
```
```   271   shows "{w \<in> space M. f w = g w} \<in> sets M"
```
```   272 proof -
```
```   273   have "{w \<in> space M. f w = g w} =
```
```   274         {w \<in> space M. f w \<le> g w} \<inter> {w \<in> space M. g w \<le> f w}"
```
```   275     by auto
```
```   276   thus ?thesis using f g by auto
```
```   277 qed
```
```   278
```
```   279 lemma (in sigma_algebra) borel_measurable_neq[simp, intro]:
```
```   280   fixes f :: "'a \<Rightarrow> real"
```
```   281   assumes f: "f \<in> borel_measurable M"
```
```   282   assumes g: "g \<in> borel_measurable M"
```
```   283   shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
```
```   284 proof -
```
```   285   have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}"
```
```   286     by auto
```
```   287   thus ?thesis using f g by auto
```
```   288 qed
```
```   289
```
```   290 subsection "Borel space equals sigma algebras over intervals"
```
```   291
```
```   292 lemma rational_boxes:
```
```   293   fixes x :: "'a\<Colon>ordered_euclidean_space"
```
```   294   assumes "0 < e"
```
```   295   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"
```
```   296 proof -
```
```   297   def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
```
```   298   then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos)
```
```   299   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \$\$ i \<and> x \$\$ i - y < e'" (is "\<forall>i. ?th i")
```
```   300   proof
```
```   301     fix i from Rats_dense_in_real[of "x \$\$ i - e'" "x \$\$ i"] e
```
```   302     show "?th i" by auto
```
```   303   qed
```
```   304   from choice[OF this] guess a .. note a = this
```
```   305   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \$\$ i < y \<and> y - x \$\$ i < e'" (is "\<forall>i. ?th i")
```
```   306   proof
```
```   307     fix i from Rats_dense_in_real[of "x \$\$ i" "x \$\$ i + e'"] e
```
```   308     show "?th i" by auto
```
```   309   qed
```
```   310   from choice[OF this] guess b .. note b = this
```
```   311   { fix y :: 'a assume *: "Chi a < y" "y < Chi b"
```
```   312     have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x \$\$ i) (y \$\$ i))\<twosuperior>)"
```
```   313       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
```
```   314     also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))"
```
```   315     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
```
```   316       fix i assume i: "i \<in> {..<DIM('a)}"
```
```   317       have "a i < y\$\$i \<and> y\$\$i < b i" using * i eucl_less[where 'a='a] by auto
```
```   318       moreover have "a i < x\$\$i" "x\$\$i - a i < e'" using a by auto
```
```   319       moreover have "x\$\$i < b i" "b i - x\$\$i < e'" using b by auto
```
```   320       ultimately have "\<bar>x\$\$i - y\$\$i\<bar> < 2 * e'" by auto
```
```   321       then have "dist (x \$\$ i) (y \$\$ i) < e/sqrt (real (DIM('a)))"
```
```   322         unfolding e'_def by (auto simp: dist_real_def)
```
```   323       then have "(dist (x \$\$ i) (y \$\$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
```
```   324         by (rule power_strict_mono) auto
```
```   325       then show "(dist (x \$\$ i) (y \$\$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
```
```   326         by (simp add: power_divide)
```
```   327     qed auto
```
```   328     also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive)
```
```   329     finally have "dist x y < e" . }
```
```   330   with a b show ?thesis
```
```   331     apply (rule_tac exI[of _ "Chi a"])
```
```   332     apply (rule_tac exI[of _ "Chi b"])
```
```   333     using eucl_less[where 'a='a] by auto
```
```   334 qed
```
```   335
```
```   336 lemma ex_rat_list:
```
```   337   fixes x :: "'a\<Colon>ordered_euclidean_space"
```
```   338   assumes "\<And> i. x \$\$ i \<in> \<rat>"
```
```   339   shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x \$\$ i)"
```
```   340 proof -
```
```   341   have "\<forall>i. \<exists>r. x \$\$ i = of_rat r" using assms unfolding Rats_def by blast
```
```   342   from choice[OF this] guess r ..
```
```   343   then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"])
```
```   344 qed
```
```   345
```
```   346 lemma open_UNION:
```
```   347   fixes M :: "'a\<Colon>ordered_euclidean_space set"
```
```   348   assumes "open M"
```
```   349   shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M}
```
```   350                    (\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})"
```
```   351     (is "M = UNION ?idx ?box")
```
```   352 proof safe
```
```   353   fix x assume "x \<in> M"
```
```   354   obtain e where e: "e > 0" "ball x e \<subseteq> M"
```
```   355     using openE[OF assms `x \<in> M`] by auto
```
```   356   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"
```
```   357     using rational_boxes[OF e(1)] by blast
```
```   358   then obtain p q where pq: "length p = DIM ('a)"
```
```   359                             "length q = DIM ('a)"
```
```   360                             "\<forall> i < DIM ('a). of_rat (p ! i) = a \$\$ i \<and> of_rat (q ! i) = b \$\$ i"
```
```   361     using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast
```
```   362   hence p: "Chi (of_rat \<circ> op ! p) = a"
```
```   363     using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a]
```
```   364     unfolding o_def by auto
```
```   365   from pq have q: "Chi (of_rat \<circ> op ! q) = b"
```
```   366     using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b]
```
```   367     unfolding o_def by auto
```
```   368   have "x \<in> ?box (p, q)"
```
```   369     using p q ab by auto
```
```   370   thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto
```
```   371 qed auto
```
```   372
```
```   373 lemma halfspace_span_open:
```
```   374   "sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a}))
```
```   375     \<subseteq> sets borel"
```
```   376   by (auto intro!: borel.sigma_sets_subset[simplified] borel_open
```
```   377                    open_halfspace_component_lt)
```
```   378
```
```   379 lemma halfspace_lt_in_halfspace:
```
```   380   "{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>)"
```
```   381   by (auto intro!: sigma_sets.Basic simp: sets_sigma)
```
```   382
```
```   383 lemma halfspace_gt_in_halfspace:
```
```   384   "{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>)"
```
```   385   (is "?set \<in> sets ?SIGMA")
```
```   386 proof -
```
```   387   interpret sigma_algebra "?SIGMA"
```
```   388     by (intro sigma_algebra_sigma_sets) (simp_all add: sets_sigma)
```
```   389   have *: "?set = (\<Union>n. space ?SIGMA - {x\<Colon>'a. x \$\$ i < a + 1 / real (Suc n)})"
```
```   390   proof (safe, simp_all add: not_less)
```
```   391     fix x assume "a < x \$\$ i"
```
```   392     with reals_Archimedean[of "x \$\$ i - a"]
```
```   393     obtain n where "a + 1 / real (Suc n) < x \$\$ i"
```
```   394       by (auto simp: inverse_eq_divide field_simps)
```
```   395     then show "\<exists>n. a + 1 / real (Suc n) \<le> x \$\$ i"
```
```   396       by (blast intro: less_imp_le)
```
```   397   next
```
```   398     fix x n
```
```   399     have "a < a + 1 / real (Suc n)" by auto
```
```   400     also assume "\<dots> \<le> x"
```
```   401     finally show "a < x" .
```
```   402   qed
```
```   403   show "?set \<in> sets ?SIGMA" unfolding *
```
```   404     by (safe intro!: countable_UN Diff halfspace_lt_in_halfspace)
```
```   405 qed
```
```   406
```
```   407 lemma open_span_halfspace:
```
```   408   "sets borel \<subseteq> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x \$\$ i < a})\<rparr>)"
```
```   409     (is "_ \<subseteq> sets ?SIGMA")
```
```   410 proof -
```
```   411   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) simp
```
```   412   then interpret sigma_algebra ?SIGMA .
```
```   413   { fix S :: "'a set" assume "S \<in> open" then have "open S" unfolding mem_def .
```
```   414     from open_UNION[OF this]
```
```   415     obtain I where *: "S =
```
```   416       (\<Union>(a, b)\<in>I.
```
```   417           (\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) \$\$ i < x \$\$ i}) \<inter>
```
```   418           (\<Inter> i<DIM('a). {x. x \$\$ i < (Chi (real_of_rat \<circ> op ! b)::'a) \$\$ i}))"
```
```   419       unfolding greaterThanLessThan_def
```
```   420       unfolding eucl_greaterThan_eq_halfspaces[where 'a='a]
```
```   421       unfolding eucl_lessThan_eq_halfspaces[where 'a='a]
```
```   422       by blast
```
```   423     have "S \<in> sets ?SIGMA"
```
```   424       unfolding *
```
```   425       by (auto intro!: countable_UN Int countable_INT halfspace_lt_in_halfspace halfspace_gt_in_halfspace) }
```
```   426   then show ?thesis unfolding borel_def
```
```   427     by (intro sets_sigma_subset) auto
```
```   428 qed
```
```   429
```
```   430 lemma halfspace_span_halfspace_le:
```
```   431   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a})\<rparr>) \<subseteq>
```
```   432    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. x \$\$ i \<le> a})\<rparr>)"
```
```   433   (is "_ \<subseteq> sets ?SIGMA")
```
```   434 proof -
```
```   435   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   436   then interpret sigma_algebra ?SIGMA .
```
```   437   { fix a i
```
```   438     have *: "{x::'a. x\$\$i < a} = (\<Union>n. {x. x\$\$i \<le> a - 1/real (Suc n)})"
```
```   439     proof (safe, simp_all)
```
```   440       fix x::'a assume *: "x\$\$i < a"
```
```   441       with reals_Archimedean[of "a - x\$\$i"]
```
```   442       obtain n where "x \$\$ i < a - 1 / (real (Suc n))"
```
```   443         by (auto simp: field_simps inverse_eq_divide)
```
```   444       then show "\<exists>n. x \$\$ i \<le> a - 1 / (real (Suc n))"
```
```   445         by (blast intro: less_imp_le)
```
```   446     next
```
```   447       fix x::'a and n
```
```   448       assume "x\$\$i \<le> a - 1 / real (Suc n)"
```
```   449       also have "\<dots> < a" by auto
```
```   450       finally show "x\$\$i < a" .
```
```   451     qed
```
```   452     have "{x. x\$\$i < a} \<in> sets ?SIGMA" unfolding *
```
```   453       by (safe intro!: countable_UN)
```
```   454          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   455   then show ?thesis by (intro sets_sigma_subset) auto
```
```   456 qed
```
```   457
```
```   458 lemma halfspace_span_halfspace_ge:
```
```   459   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a})\<rparr>) \<subseteq>
```
```   460    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a \<le> x \$\$ i})\<rparr>)"
```
```   461   (is "_ \<subseteq> sets ?SIGMA")
```
```   462 proof -
```
```   463   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   464   then interpret sigma_algebra ?SIGMA .
```
```   465   { fix a i have *: "{x::'a. x\$\$i < a} = space ?SIGMA - {x::'a. a \<le> x\$\$i}" by auto
```
```   466     have "{x. x\$\$i < a} \<in> sets ?SIGMA" unfolding *
```
```   467       by (safe intro!: Diff)
```
```   468          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   469   then show ?thesis by (intro sets_sigma_subset) auto
```
```   470 qed
```
```   471
```
```   472 lemma halfspace_le_span_halfspace_gt:
```
```   473   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
```
```   474    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a < x \$\$ i})\<rparr>)"
```
```   475   (is "_ \<subseteq> sets ?SIGMA")
```
```   476 proof -
```
```   477   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   478   then interpret sigma_algebra ?SIGMA .
```
```   479   { fix a i have *: "{x::'a. x\$\$i \<le> a} = space ?SIGMA - {x::'a. a < x\$\$i}" by auto
```
```   480     have "{x. x\$\$i \<le> a} \<in> sets ?SIGMA" unfolding *
```
```   481       by (safe intro!: Diff)
```
```   482          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   483   then show ?thesis by (intro sets_sigma_subset) auto
```
```   484 qed
```
```   485
```
```   486 lemma halfspace_le_span_atMost:
```
```   487   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
```
```   488    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>)"
```
```   489   (is "_ \<subseteq> sets ?SIGMA")
```
```   490 proof -
```
```   491   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   492   then interpret sigma_algebra ?SIGMA .
```
```   493   have "\<And>a i. {x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   494   proof cases
```
```   495     fix a i assume "i < DIM('a)"
```
```   496     then have *: "{x::'a. x\$\$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})"
```
```   497     proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm)
```
```   498       fix x
```
```   499       from real_arch_simple[of "Max ((\<lambda>i. x\$\$i)`{..<DIM('a)})"] guess k::nat ..
```
```   500       then have "\<And>i. i < DIM('a) \<Longrightarrow> x\$\$i \<le> real k"
```
```   501         by (subst (asm) Max_le_iff) auto
```
```   502       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x \$\$ ia \<le> real k"
```
```   503         by (auto intro!: exI[of _ k])
```
```   504     qed
```
```   505     show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA" unfolding *
```
```   506       by (safe intro!: countable_UN)
```
```   507          (auto simp: sets_sigma intro!: sigma_sets.Basic)
```
```   508   next
```
```   509     fix a i assume "\<not> i < DIM('a)"
```
```   510     then show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   511       using top by auto
```
```   512   qed
```
```   513   then show ?thesis by (intro sets_sigma_subset) auto
```
```   514 qed
```
```   515
```
```   516 lemma halfspace_le_span_greaterThan:
```
```   517   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
```
```   518    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {a<..})\<rparr>)"
```
```   519   (is "_ \<subseteq> sets ?SIGMA")
```
```   520 proof -
```
```   521   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   522   then interpret sigma_algebra ?SIGMA .
```
```   523   have "\<And>a i. {x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   524   proof cases
```
```   525     fix a i assume "i < DIM('a)"
```
```   526     have "{x::'a. x\$\$i \<le> a} = space ?SIGMA - {x::'a. a < x\$\$i}" by auto
```
```   527     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)`
```
```   528     proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
```
```   529       fix x
```
```   530       from real_arch_lt[of "Max ((\<lambda>i. -x\$\$i)`{..<DIM('a)})"]
```
```   531       guess k::nat .. note k = this
```
```   532       { fix i assume "i < DIM('a)"
```
```   533         then have "-x\$\$i < real k"
```
```   534           using k by (subst (asm) Max_less_iff) auto
```
```   535         then have "- real k < x\$\$i" by simp }
```
```   536       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x \$\$ ia"
```
```   537         by (auto intro!: exI[of _ k])
```
```   538     qed
```
```   539     finally show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   540       apply (simp only:)
```
```   541       apply (safe intro!: countable_UN Diff)
```
```   542       by (auto simp: sets_sigma intro!: sigma_sets.Basic)
```
```   543   next
```
```   544     fix a i assume "\<not> i < DIM('a)"
```
```   545     then show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA"
```
```   546       using top by auto
```
```   547   qed
```
```   548   then show ?thesis by (intro sets_sigma_subset) auto
```
```   549 qed
```
```   550
```
```   551 lemma halfspace_le_span_lessThan:
```
```   552   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x \$\$ i})\<rparr>) \<subseteq>
```
```   553    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..<a})\<rparr>)"
```
```   554   (is "_ \<subseteq> sets ?SIGMA")
```
```   555 proof -
```
```   556   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   557   then interpret sigma_algebra ?SIGMA .
```
```   558   have "\<And>a i. {x. a \<le> x\$\$i} \<in> sets ?SIGMA"
```
```   559   proof cases
```
```   560     fix a i assume "i < DIM('a)"
```
```   561     have "{x::'a. a \<le> x\$\$i} = space ?SIGMA - {x::'a. x\$\$i < a}" by auto
```
```   562     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)`
```
```   563     proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
```
```   564       fix x
```
```   565       from real_arch_lt[of "Max ((\<lambda>i. x\$\$i)`{..<DIM('a)})"]
```
```   566       guess k::nat .. note k = this
```
```   567       { fix i assume "i < DIM('a)"
```
```   568         then have "x\$\$i < real k"
```
```   569           using k by (subst (asm) Max_less_iff) auto
```
```   570         then have "x\$\$i < real k" by simp }
```
```   571       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x \$\$ ia < real k"
```
```   572         by (auto intro!: exI[of _ k])
```
```   573     qed
```
```   574     finally show "{x. a \<le> x\$\$i} \<in> sets ?SIGMA"
```
```   575       apply (simp only:)
```
```   576       apply (safe intro!: countable_UN Diff)
```
```   577       by (auto simp: sets_sigma intro!: sigma_sets.Basic)
```
```   578   next
```
```   579     fix a i assume "\<not> i < DIM('a)"
```
```   580     then show "{x. a \<le> x\$\$i} \<in> sets ?SIGMA"
```
```   581       using top by auto
```
```   582   qed
```
```   583   then show ?thesis by (intro sets_sigma_subset) auto
```
```   584 qed
```
```   585
```
```   586 lemma atMost_span_atLeastAtMost:
```
```   587   "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>) \<subseteq>
```
```   588    sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a,b). {a..b})\<rparr>)"
```
```   589   (is "_ \<subseteq> sets ?SIGMA")
```
```   590 proof -
```
```   591   have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   592   then interpret sigma_algebra ?SIGMA .
```
```   593   { fix a::'a
```
```   594     have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
```
```   595     proof (safe, simp_all add: eucl_le[where 'a='a])
```
```   596       fix x
```
```   597       from real_arch_simple[of "Max ((\<lambda>i. - x\$\$i)`{..<DIM('a)})"]
```
```   598       guess k::nat .. note k = this
```
```   599       { fix i assume "i < DIM('a)"
```
```   600         with k have "- x\$\$i \<le> real k"
```
```   601           by (subst (asm) Max_le_iff) (auto simp: field_simps)
```
```   602         then have "- real k \<le> x\$\$i" by simp }
```
```   603       then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x \$\$ i"
```
```   604         by (auto intro!: exI[of _ k])
```
```   605     qed
```
```   606     have "{..a} \<in> sets ?SIGMA" unfolding *
```
```   607       by (safe intro!: countable_UN)
```
```   608          (auto simp: sets_sigma intro!: sigma_sets.Basic) }
```
```   609   then show ?thesis by (intro sets_sigma_subset) auto
```
```   610 qed
```
```   611
```
```   612 lemma borel_eq_atMost:
```
```   613   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> a. {.. a::'a\<Colon>ordered_euclidean_space})\<rparr>)"
```
```   614     (is "_ = ?SIGMA")
```
```   615 proof (intro algebra.equality antisym)
```
```   616   show "sets borel \<subseteq> sets ?SIGMA"
```
```   617     using halfspace_le_span_atMost halfspace_span_halfspace_le open_span_halfspace
```
```   618     by auto
```
```   619   show "sets ?SIGMA \<subseteq> sets borel"
```
```   620     by (rule borel.sets_sigma_subset) auto
```
```   621 qed auto
```
```   622
```
```   623 lemma borel_eq_atLeastAtMost:
```
```   624   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space, b). {a .. b})\<rparr>)"
```
```   625    (is "_ = ?SIGMA")
```
```   626 proof (intro algebra.equality antisym)
```
```   627   show "sets borel \<subseteq> sets ?SIGMA"
```
```   628     using atMost_span_atLeastAtMost halfspace_le_span_atMost
```
```   629       halfspace_span_halfspace_le open_span_halfspace
```
```   630     by auto
```
```   631   show "sets ?SIGMA \<subseteq> sets borel"
```
```   632     by (rule borel.sets_sigma_subset) auto
```
```   633 qed auto
```
```   634
```
```   635 lemma borel_eq_greaterThan:
```
```   636   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {a <..})\<rparr>)"
```
```   637    (is "_ = ?SIGMA")
```
```   638 proof (intro algebra.equality antisym)
```
```   639   show "sets borel \<subseteq> sets ?SIGMA"
```
```   640     using halfspace_le_span_greaterThan
```
```   641       halfspace_span_halfspace_le open_span_halfspace
```
```   642     by auto
```
```   643   show "sets ?SIGMA \<subseteq> sets borel"
```
```   644     by (rule borel.sets_sigma_subset) auto
```
```   645 qed auto
```
```   646
```
```   647 lemma borel_eq_lessThan:
```
```   648   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {..< a})\<rparr>)"
```
```   649    (is "_ = ?SIGMA")
```
```   650 proof (intro algebra.equality antisym)
```
```   651   show "sets borel \<subseteq> sets ?SIGMA"
```
```   652     using halfspace_le_span_lessThan
```
```   653       halfspace_span_halfspace_ge open_span_halfspace
```
```   654     by auto
```
```   655   show "sets ?SIGMA \<subseteq> sets borel"
```
```   656     by (rule borel.sets_sigma_subset) auto
```
```   657 qed auto
```
```   658
```
```   659 lemma borel_eq_greaterThanLessThan:
```
```   660   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, b). {a <..< (b :: 'a \<Colon> ordered_euclidean_space)})\<rparr>)"
```
```   661     (is "_ = ?SIGMA")
```
```   662 proof (intro algebra.equality antisym)
```
```   663   show "sets ?SIGMA \<subseteq> sets borel"
```
```   664     by (rule borel.sets_sigma_subset) auto
```
```   665   show "sets borel \<subseteq> sets ?SIGMA"
```
```   666   proof -
```
```   667     have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
```
```   668     then interpret sigma_algebra ?SIGMA .
```
```   669     { fix M :: "'a set" assume "M \<in> open"
```
```   670       then have "open M" by (simp add: mem_def)
```
```   671       have "M \<in> sets ?SIGMA"
```
```   672         apply (subst open_UNION[OF `open M`])
```
```   673         apply (safe intro!: countable_UN)
```
```   674         by (auto simp add: sigma_def intro!: sigma_sets.Basic) }
```
```   675     then show ?thesis
```
```   676       unfolding borel_def by (intro sets_sigma_subset) auto
```
```   677   qed
```
```   678 qed auto
```
```   679
```
```   680 lemma borel_eq_halfspace_le:
```
```   681   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x\$\$i \<le> a})\<rparr>)"
```
```   682    (is "_ = ?SIGMA")
```
```   683 proof (intro algebra.equality antisym)
```
```   684   show "sets borel \<subseteq> sets ?SIGMA"
```
```   685     using open_span_halfspace halfspace_span_halfspace_le by auto
```
```   686   show "sets ?SIGMA \<subseteq> sets borel"
```
```   687     by (rule borel.sets_sigma_subset) auto
```
```   688 qed auto
```
```   689
```
```   690 lemma borel_eq_halfspace_less:
```
```   691   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x\$\$i < a})\<rparr>)"
```
```   692    (is "_ = ?SIGMA")
```
```   693 proof (intro algebra.equality antisym)
```
```   694   show "sets borel \<subseteq> sets ?SIGMA"
```
```   695     using open_span_halfspace .
```
```   696   show "sets ?SIGMA \<subseteq> sets borel"
```
```   697     by (rule borel.sets_sigma_subset) auto
```
```   698 qed auto
```
```   699
```
```   700 lemma borel_eq_halfspace_gt:
```
```   701   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a < x\$\$i})\<rparr>)"
```
```   702    (is "_ = ?SIGMA")
```
```   703 proof (intro algebra.equality antisym)
```
```   704   show "sets borel \<subseteq> sets ?SIGMA"
```
```   705     using halfspace_le_span_halfspace_gt open_span_halfspace halfspace_span_halfspace_le by auto
```
```   706   show "sets ?SIGMA \<subseteq> sets borel"
```
```   707     by (rule borel.sets_sigma_subset) auto
```
```   708 qed auto
```
```   709
```
```   710 lemma borel_eq_halfspace_ge:
```
```   711   "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a \<le> x\$\$i})\<rparr>)"
```
```   712    (is "_ = ?SIGMA")
```
```   713 proof (intro algebra.equality antisym)
```
```   714   show "sets borel \<subseteq> sets ?SIGMA"
```
```   715     using halfspace_span_halfspace_ge open_span_halfspace by auto
```
```   716   show "sets ?SIGMA \<subseteq> sets borel"
```
```   717     by (rule borel.sets_sigma_subset) auto
```
```   718 qed auto
```
```   719
```
```   720 lemma (in sigma_algebra) borel_measurable_halfspacesI:
```
```   721   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   722   assumes "borel = (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
```
```   723   and "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
```
```   724   and "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M"
```
```   725   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)"
```
```   726 proof safe
```
```   727   fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M"
```
```   728   then show "S a i \<in> sets M" unfolding assms
```
```   729     by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1) sigma_def)
```
```   730 next
```
```   731   assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M"
```
```   732   { fix a i have "S a i \<in> sets M"
```
```   733     proof cases
```
```   734       assume "i < DIM('c)"
```
```   735       with a show ?thesis unfolding assms(2) by simp
```
```   736     next
```
```   737       assume "\<not> i < DIM('c)"
```
```   738       from assms(3)[OF this] show ?thesis .
```
```   739     qed }
```
```   740   then have "f \<in> measurable M (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
```
```   741     by (auto intro!: measurable_sigma simp: assms(2))
```
```   742   then show "f \<in> borel_measurable M" unfolding measurable_def
```
```   743     unfolding assms(1) by simp
```
```   744 qed
```
```   745
```
```   746 lemma (in sigma_algebra) borel_measurable_iff_halfspace_le:
```
```   747   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   748   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w \$\$ i \<le> a} \<in> sets M)"
```
```   749   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
```
```   750
```
```   751 lemma (in sigma_algebra) borel_measurable_iff_halfspace_less:
```
```   752   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   753   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w \$\$ i < a} \<in> sets M)"
```
```   754   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
```
```   755
```
```   756 lemma (in sigma_algebra) borel_measurable_iff_halfspace_ge:
```
```   757   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   758   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w \$\$ i} \<in> sets M)"
```
```   759   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
```
```   760
```
```   761 lemma (in sigma_algebra) borel_measurable_iff_halfspace_greater:
```
```   762   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   763   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w \$\$ i} \<in> sets M)"
```
```   764   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_gt]) auto
```
```   765
```
```   766 lemma (in sigma_algebra) borel_measurable_iff_le:
```
```   767   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
```
```   768   using borel_measurable_iff_halfspace_le[where 'c=real] by simp
```
```   769
```
```   770 lemma (in sigma_algebra) borel_measurable_iff_less:
```
```   771   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
```
```   772   using borel_measurable_iff_halfspace_less[where 'c=real] by simp
```
```   773
```
```   774 lemma (in sigma_algebra) borel_measurable_iff_ge:
```
```   775   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
```
```   776   using borel_measurable_iff_halfspace_ge[where 'c=real] by simp
```
```   777
```
```   778 lemma (in sigma_algebra) borel_measurable_iff_greater:
```
```   779   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
```
```   780   using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
```
```   781
```
```   782 lemma borel_measurable_euclidean_component:
```
```   783   "(\<lambda>x::'a::euclidean_space. x \$\$ i) \<in> borel_measurable borel"
```
```   784   unfolding borel_def[where 'a=real]
```
```   785 proof (rule borel.measurable_sigma, simp_all)
```
```   786   fix S::"real set" assume "S \<in> open" then have "open S" unfolding mem_def .
```
```   787   from open_vimage_euclidean_component[OF this]
```
```   788   show "(\<lambda>x. x \$\$ i) -` S \<in> sets borel"
```
```   789     by (auto intro: borel_open)
```
```   790 qed
```
```   791
```
```   792 lemma (in sigma_algebra) borel_measurable_euclidean_space:
```
```   793   fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
```
```   794   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x \$\$ i) \<in> borel_measurable M)"
```
```   795 proof safe
```
```   796   fix i assume "f \<in> borel_measurable M"
```
```   797   then show "(\<lambda>x. f x \$\$ i) \<in> borel_measurable M"
```
```   798     using measurable_comp[of f _ _ "\<lambda>x. x \$\$ i", unfolded comp_def]
```
```   799     by (auto intro: borel_measurable_euclidean_component)
```
```   800 next
```
```   801   assume f: "\<forall>i<DIM('c). (\<lambda>x. f x \$\$ i) \<in> borel_measurable M"
```
```   802   then show "f \<in> borel_measurable M"
```
```   803     unfolding borel_measurable_iff_halfspace_le by auto
```
```   804 qed
```
```   805
```
```   806 subsection "Borel measurable operators"
```
```   807
```
```   808 lemma (in sigma_algebra) affine_borel_measurable_vector:
```
```   809   fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
```
```   810   assumes "f \<in> borel_measurable M"
```
```   811   shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
```
```   812 proof (rule borel_measurableI)
```
```   813   fix S :: "'x set" assume "open S"
```
```   814   show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
```
```   815   proof cases
```
```   816     assume "b \<noteq> 0"
```
```   817     with `open S` have "((\<lambda>x. (- a + x) /\<^sub>R b) ` S) \<in> open" (is "?S \<in> open")
```
```   818       by (auto intro!: open_affinity simp: scaleR.add_right mem_def)
```
```   819     hence "?S \<in> sets borel"
```
```   820       unfolding borel_def by (auto simp: sigma_def intro!: sigma_sets.Basic)
```
```   821     moreover
```
```   822     from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
```
```   823       apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
```
```   824     ultimately show ?thesis using assms unfolding in_borel_measurable_borel
```
```   825       by auto
```
```   826   qed simp
```
```   827 qed
```
```   828
```
```   829 lemma (in sigma_algebra) affine_borel_measurable:
```
```   830   fixes g :: "'a \<Rightarrow> real"
```
```   831   assumes g: "g \<in> borel_measurable M"
```
```   832   shows "(\<lambda>x. a + (g x) * b) \<in> borel_measurable M"
```
```   833   using affine_borel_measurable_vector[OF assms] by (simp add: mult_commute)
```
```   834
```
```   835 lemma (in sigma_algebra) borel_measurable_add[simp, intro]:
```
```   836   fixes f :: "'a \<Rightarrow> real"
```
```   837   assumes f: "f \<in> borel_measurable M"
```
```   838   assumes g: "g \<in> borel_measurable M"
```
```   839   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```   840 proof -
```
```   841   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}"
```
```   842     by auto
```
```   843   have "\<And>a. (\<lambda>w. a + (g w) * -1) \<in> borel_measurable M"
```
```   844     by (rule affine_borel_measurable [OF g])
```
```   845   then have "\<And>a. {w \<in> space M. (\<lambda>w. a + (g w) * -1)(w) \<le> f w} \<in> sets M" using f
```
```   846     by auto
```
```   847   then have "\<And>a. {w \<in> space M. a \<le> f w + g w} \<in> sets M"
```
```   848     by (simp add: 1)
```
```   849   then show ?thesis
```
```   850     by (simp add: borel_measurable_iff_ge)
```
```   851 qed
```
```   852
```
```   853 lemma (in sigma_algebra) borel_measurable_setsum[simp, intro]:
```
```   854   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
```
```   855   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   856   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```   857 proof cases
```
```   858   assume "finite S"
```
```   859   thus ?thesis using assms by induct auto
```
```   860 qed simp
```
```   861
```
```   862 lemma (in sigma_algebra) borel_measurable_square:
```
```   863   fixes f :: "'a \<Rightarrow> real"
```
```   864   assumes f: "f \<in> borel_measurable M"
```
```   865   shows "(\<lambda>x. (f x)^2) \<in> borel_measurable M"
```
```   866 proof -
```
```   867   {
```
```   868     fix a
```
```   869     have "{w \<in> space M. (f w)\<twosuperior> \<le> a} \<in> sets M"
```
```   870     proof (cases rule: linorder_cases [of a 0])
```
```   871       case less
```
```   872       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = {}"
```
```   873         by auto (metis less order_le_less_trans power2_less_0)
```
```   874       also have "... \<in> sets M"
```
```   875         by (rule empty_sets)
```
```   876       finally show ?thesis .
```
```   877     next
```
```   878       case equal
```
```   879       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
```
```   880              {w \<in> space M. f w \<le> 0} \<inter> {w \<in> space M. 0 \<le> f w}"
```
```   881         by auto
```
```   882       also have "... \<in> sets M"
```
```   883         apply (insert f)
```
```   884         apply (rule Int)
```
```   885         apply (simp add: borel_measurable_iff_le)
```
```   886         apply (simp add: borel_measurable_iff_ge)
```
```   887         done
```
```   888       finally show ?thesis .
```
```   889     next
```
```   890       case greater
```
```   891       have "\<forall>x. (f x ^ 2 \<le> sqrt a ^ 2) = (- sqrt a  \<le> f x & f x \<le> sqrt a)"
```
```   892         by (metis abs_le_interval_iff abs_of_pos greater real_sqrt_abs
```
```   893                   real_sqrt_le_iff real_sqrt_power)
```
```   894       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
```
```   895              {w \<in> space M. -(sqrt a) \<le> f w} \<inter> {w \<in> space M. f w \<le> sqrt a}"
```
```   896         using greater by auto
```
```   897       also have "... \<in> sets M"
```
```   898         apply (insert f)
```
```   899         apply (rule Int)
```
```   900         apply (simp add: borel_measurable_iff_ge)
```
```   901         apply (simp add: borel_measurable_iff_le)
```
```   902         done
```
```   903       finally show ?thesis .
```
```   904     qed
```
```   905   }
```
```   906   thus ?thesis by (auto simp add: borel_measurable_iff_le)
```
```   907 qed
```
```   908
```
```   909 lemma times_eq_sum_squares:
```
```   910    fixes x::real
```
```   911    shows"x*y = ((x+y)^2)/4 - ((x-y)^ 2)/4"
```
```   912 by (simp add: power2_eq_square ring_distribs diff_divide_distrib [symmetric])
```
```   913
```
```   914 lemma (in sigma_algebra) borel_measurable_uminus[simp, intro]:
```
```   915   fixes g :: "'a \<Rightarrow> real"
```
```   916   assumes g: "g \<in> borel_measurable M"
```
```   917   shows "(\<lambda>x. - g x) \<in> borel_measurable M"
```
```   918 proof -
```
```   919   have "(\<lambda>x. - g x) = (\<lambda>x. 0 + (g x) * -1)"
```
```   920     by simp
```
```   921   also have "... \<in> borel_measurable M"
```
```   922     by (fast intro: affine_borel_measurable g)
```
```   923   finally show ?thesis .
```
```   924 qed
```
```   925
```
```   926 lemma (in sigma_algebra) borel_measurable_times[simp, intro]:
```
```   927   fixes f :: "'a \<Rightarrow> real"
```
```   928   assumes f: "f \<in> borel_measurable M"
```
```   929   assumes g: "g \<in> borel_measurable M"
```
```   930   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```   931 proof -
```
```   932   have 1: "(\<lambda>x. 0 + (f x + g x)\<twosuperior> * inverse 4) \<in> borel_measurable M"
```
```   933     using assms by (fast intro: affine_borel_measurable borel_measurable_square)
```
```   934   have "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) =
```
```   935         (\<lambda>x. 0 + ((f x + -g x) ^ 2 * inverse -4))"
```
```   936     by (simp add: minus_divide_right)
```
```   937   also have "... \<in> borel_measurable M"
```
```   938     using f g by (fast intro: affine_borel_measurable borel_measurable_square f g)
```
```   939   finally have 2: "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) \<in> borel_measurable M" .
```
```   940   show ?thesis
```
```   941     apply (simp add: times_eq_sum_squares diff_minus)
```
```   942     using 1 2 by simp
```
```   943 qed
```
```   944
```
```   945 lemma (in sigma_algebra) borel_measurable_setprod[simp, intro]:
```
```   946   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
```
```   947   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   948   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```   949 proof cases
```
```   950   assume "finite S"
```
```   951   thus ?thesis using assms by induct auto
```
```   952 qed simp
```
```   953
```
```   954 lemma (in sigma_algebra) borel_measurable_diff[simp, intro]:
```
```   955   fixes f :: "'a \<Rightarrow> real"
```
```   956   assumes f: "f \<in> borel_measurable M"
```
```   957   assumes g: "g \<in> borel_measurable M"
```
```   958   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```   959   unfolding diff_minus using assms by fast
```
```   960
```
```   961 lemma (in sigma_algebra) borel_measurable_inverse[simp, intro]:
```
```   962   fixes f :: "'a \<Rightarrow> real"
```
```   963   assumes "f \<in> borel_measurable M"
```
```   964   shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
```
```   965   unfolding borel_measurable_iff_ge unfolding inverse_eq_divide
```
```   966 proof safe
```
```   967   fix a :: real
```
```   968   have *: "{w \<in> space M. a \<le> 1 / f w} =
```
```   969       ({w \<in> space M. 0 < f w} \<inter> {w \<in> space M. a * f w \<le> 1}) \<union>
```
```   970       ({w \<in> space M. f w < 0} \<inter> {w \<in> space M. 1 \<le> a * f w}) \<union>
```
```   971       ({w \<in> space M. f w = 0} \<inter> {w \<in> space M. a \<le> 0})" by (auto simp: le_divide_eq)
```
```   972   show "{w \<in> space M. a \<le> 1 / f w} \<in> sets M" using assms unfolding *
```
```   973     by (auto intro!: Int Un)
```
```   974 qed
```
```   975
```
```   976 lemma (in sigma_algebra) borel_measurable_divide[simp, intro]:
```
```   977   fixes f :: "'a \<Rightarrow> real"
```
```   978   assumes "f \<in> borel_measurable M"
```
```   979   and "g \<in> borel_measurable M"
```
```   980   shows "(\<lambda>x. f x / g x) \<in> borel_measurable M"
```
```   981   unfolding field_divide_inverse
```
```   982   by (rule borel_measurable_inverse borel_measurable_times assms)+
```
```   983
```
```   984 lemma (in sigma_algebra) borel_measurable_max[intro, simp]:
```
```   985   fixes f g :: "'a \<Rightarrow> real"
```
```   986   assumes "f \<in> borel_measurable M"
```
```   987   assumes "g \<in> borel_measurable M"
```
```   988   shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
```
```   989   unfolding borel_measurable_iff_le
```
```   990 proof safe
```
```   991   fix a
```
```   992   have "{x \<in> space M. max (g x) (f x) \<le> a} =
```
```   993     {x \<in> space M. g x \<le> a} \<inter> {x \<in> space M. f x \<le> a}" by auto
```
```   994   thus "{x \<in> space M. max (g x) (f x) \<le> a} \<in> sets M"
```
```   995     using assms unfolding borel_measurable_iff_le
```
```   996     by (auto intro!: Int)
```
```   997 qed
```
```   998
```
```   999 lemma (in sigma_algebra) borel_measurable_min[intro, simp]:
```
```  1000   fixes f g :: "'a \<Rightarrow> real"
```
```  1001   assumes "f \<in> borel_measurable M"
```
```  1002   assumes "g \<in> borel_measurable M"
```
```  1003   shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
```
```  1004   unfolding borel_measurable_iff_ge
```
```  1005 proof safe
```
```  1006   fix a
```
```  1007   have "{x \<in> space M. a \<le> min (g x) (f x)} =
```
```  1008     {x \<in> space M. a \<le> g x} \<inter> {x \<in> space M. a \<le> f x}" by auto
```
```  1009   thus "{x \<in> space M. a \<le> min (g x) (f x)} \<in> sets M"
```
```  1010     using assms unfolding borel_measurable_iff_ge
```
```  1011     by (auto intro!: Int)
```
```  1012 qed
```
```  1013
```
```  1014 lemma (in sigma_algebra) borel_measurable_abs[simp, intro]:
```
```  1015   assumes "f \<in> borel_measurable M"
```
```  1016   shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
```
```  1017 proof -
```
```  1018   have *: "\<And>x. \<bar>f x\<bar> = max 0 (f x) + max 0 (- f x)" by (simp add: max_def)
```
```  1019   show ?thesis unfolding * using assms by auto
```
```  1020 qed
```
```  1021
```
```  1022 lemma borel_measurable_nth[simp, intro]:
```
```  1023   "(\<lambda>x::real^'n. x \$ i) \<in> borel_measurable borel"
```
```  1024   using borel_measurable_euclidean_component
```
```  1025   unfolding nth_conv_component by auto
```
```  1026
```
```  1027 section "Borel space over the real line with infinity"
```
```  1028
```
```  1029 lemma borel_Real_measurable:
```
```  1030   "A \<in> sets borel \<Longrightarrow> Real -` A \<in> sets borel"
```
```  1031 proof (rule borel_measurable_translate)
```
```  1032   fix B :: "pextreal set" assume "open B"
```
```  1033   then obtain T x where T: "open T" "Real ` (T \<inter> {0..}) = B - {\<omega>}" and
```
```  1034     x: "\<omega> \<in> B \<Longrightarrow> 0 \<le> x" "\<omega> \<in> B \<Longrightarrow> {Real x <..} \<subseteq> B"
```
```  1035     unfolding open_pextreal_def by blast
```
```  1036   have "Real -` B = Real -` (B - {\<omega>})" by auto
```
```  1037   also have "\<dots> = Real -` (Real ` (T \<inter> {0..}))" using T by simp
```
```  1038   also have "\<dots> = (if 0 \<in> T then T \<union> {.. 0} else T \<inter> {0..})"
```
```  1039     apply (auto simp add: Real_eq_Real image_iff)
```
```  1040     apply (rule_tac x="max 0 x" in bexI)
```
```  1041     by (auto simp: max_def)
```
```  1042   finally show "Real -` B \<in> sets borel"
```
```  1043     using `open T` by auto
```
```  1044 qed simp
```
```  1045
```
```  1046 lemma borel_real_measurable:
```
```  1047   "A \<in> sets borel \<Longrightarrow> (real -` A :: pextreal set) \<in> sets borel"
```
```  1048 proof (rule borel_measurable_translate)
```
```  1049   fix B :: "real set" assume "open B"
```
```  1050   { fix x have "0 < real x \<longleftrightarrow> (\<exists>r>0. x = Real r)" by (cases x) auto }
```
```  1051   note Ex_less_real = this
```
```  1052   have *: "real -` B = (if 0 \<in> B then real -` (B \<inter> {0 <..}) \<union> {0, \<omega>} else real -` (B \<inter> {0 <..}))"
```
```  1053     by (force simp: Ex_less_real)
```
```  1054
```
```  1055   have "open (real -` (B \<inter> {0 <..}) :: pextreal set)"
```
```  1056     unfolding open_pextreal_def using `open B`
```
```  1057     by (auto intro!: open_Int exI[of _ "B \<inter> {0 <..}"] simp: image_iff Ex_less_real)
```
```  1058   then show "(real -` B :: pextreal set) \<in> sets borel" unfolding * by auto
```
```  1059 qed simp
```
```  1060
```
```  1061 lemma (in sigma_algebra) borel_measurable_Real[intro, simp]:
```
```  1062   assumes "f \<in> borel_measurable M"
```
```  1063   shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
```
```  1064   unfolding in_borel_measurable_borel
```
```  1065 proof safe
```
```  1066   fix S :: "pextreal set" assume "S \<in> sets borel"
```
```  1067   from borel_Real_measurable[OF this]
```
```  1068   have "(Real \<circ> f) -` S \<inter> space M \<in> sets M"
```
```  1069     using assms
```
```  1070     unfolding vimage_compose in_borel_measurable_borel
```
```  1071     by auto
```
```  1072   thus "(\<lambda>x. Real (f x)) -` S \<inter> space M \<in> sets M" by (simp add: comp_def)
```
```  1073 qed
```
```  1074
```
```  1075 lemma (in sigma_algebra) borel_measurable_real[intro, simp]:
```
```  1076   fixes f :: "'a \<Rightarrow> pextreal"
```
```  1077   assumes "f \<in> borel_measurable M"
```
```  1078   shows "(\<lambda>x. real (f x)) \<in> borel_measurable M"
```
```  1079   unfolding in_borel_measurable_borel
```
```  1080 proof safe
```
```  1081   fix S :: "real set" assume "S \<in> sets borel"
```
```  1082   from borel_real_measurable[OF this]
```
```  1083   have "(real \<circ> f) -` S \<inter> space M \<in> sets M"
```
```  1084     using assms
```
```  1085     unfolding vimage_compose in_borel_measurable_borel
```
```  1086     by auto
```
```  1087   thus "(\<lambda>x. real (f x)) -` S \<inter> space M \<in> sets M" by (simp add: comp_def)
```
```  1088 qed
```
```  1089
```
```  1090 lemma (in sigma_algebra) borel_measurable_Real_eq:
```
```  1091   assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
```
```  1092   shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
```
```  1093 proof
```
```  1094   have [simp]: "(\<lambda>x. Real (f x)) -` {\<omega>} \<inter> space M = {}"
```
```  1095     by auto
```
```  1096   assume "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
```
```  1097   hence "(\<lambda>x. real (Real (f x))) \<in> borel_measurable M"
```
```  1098     by (rule borel_measurable_real)
```
```  1099   moreover have "\<And>x. x \<in> space M \<Longrightarrow> real (Real (f x)) = f x"
```
```  1100     using assms by auto
```
```  1101   ultimately show "f \<in> borel_measurable M"
```
```  1102     by (simp cong: measurable_cong)
```
```  1103 qed auto
```
```  1104
```
```  1105 lemma (in sigma_algebra) borel_measurable_pextreal_eq_real:
```
```  1106   "f \<in> borel_measurable M \<longleftrightarrow>
```
```  1107     ((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<omega>} \<inter> space M \<in> sets M)"
```
```  1108 proof safe
```
```  1109   assume "f \<in> borel_measurable M"
```
```  1110   then show "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<omega>} \<inter> space M \<in> sets M"
```
```  1111     by (auto intro: borel_measurable_vimage borel_measurable_real)
```
```  1112 next
```
```  1113   assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<omega>} \<inter> space M \<in> sets M"
```
```  1114   have "f -` {\<omega>} \<inter> space M = {x\<in>space M. f x = \<omega>}" by auto
```
```  1115   with * have **: "{x\<in>space M. f x = \<omega>} \<in> sets M" by simp
```
```  1116   have f: "f = (\<lambda>x. if f x = \<omega> then \<omega> else Real (real (f x)))"
```
```  1117     by (simp add: fun_eq_iff Real_real)
```
```  1118   show "f \<in> borel_measurable M"
```
```  1119     apply (subst f)
```
```  1120     apply (rule measurable_If)
```
```  1121     using * ** by auto
```
```  1122 qed
```
```  1123
```
```  1124 lemma borel_measurable_continuous_on1:
```
```  1125   fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
```
```  1126   assumes "continuous_on UNIV f"
```
```  1127   shows "f \<in> borel_measurable borel"
```
```  1128   apply(rule borel.borel_measurableI)
```
```  1129   using continuous_open_preimage[OF assms] unfolding vimage_def by auto
```
```  1130
```
```  1131 lemma borel_measurable_continuous_on:
```
```  1132   fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
```
```  1133   assumes cont: "continuous_on A f" "open A" and f: "f -` {c} \<inter> A \<in> sets borel"
```
```  1134   shows "(\<lambda>x. if x \<in> A then f x else c) \<in> borel_measurable borel" (is "?f \<in> _")
```
```  1135 proof (rule borel.borel_measurableI)
```
```  1136   fix S :: "'b set" assume "open S"
```
```  1137   then have "open {x\<in>A. f x \<in> S - {c}}"
```
```  1138     by (intro continuous_open_preimage[OF cont]) auto
```
```  1139   then have *: "{x\<in>A. f x \<in> S - {c}} \<in> sets borel" by auto
```
```  1140   show "?f -` S \<inter> space borel \<in> sets borel"
```
```  1141   proof cases
```
```  1142     assume "c \<in> S"
```
```  1143     then have "?f -` S = {x\<in>A. f x \<in> S - {c}} \<union> (f -` {c} \<inter> A) \<union> -A"
```
```  1144       by auto
```
```  1145     with * show "?f -` S \<inter> space borel \<in> sets borel"
```
```  1146       using `open A` f by (auto intro!: borel.Un)
```
```  1147   next
```
```  1148     assume "c \<notin> S"
```
```  1149     then have "?f -` S = {x\<in>A. f x \<in> S - {c}}" by (auto split: split_if_asm)
```
```  1150     with * show "?f -` S \<inter> space borel \<in> sets borel" by auto
```
```  1151   qed
```
```  1152 qed
```
```  1153
```
```  1154 lemma borel_measurable_borel_log: assumes "1 < b" shows "log b \<in> borel_measurable borel"
```
```  1155 proof -
```
```  1156   { fix x :: real assume x: "x \<le> 0"
```
```  1157     { fix x::real assume "x \<le> 0" then have "\<And>u. exp u = x \<longleftrightarrow> False" by auto }
```
```  1158     from this[of x] x this[of 0] have "log b 0 = log b x"
```
```  1159       by (auto simp: ln_def log_def) }
```
```  1160   note log_imp = this
```
```  1161   have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) \<in> borel_measurable borel"
```
```  1162   proof (rule borel_measurable_continuous_on)
```
```  1163     show "continuous_on {0<..} (log b)"
```
```  1164       by (auto intro!: continuous_at_imp_continuous_on DERIV_log DERIV_isCont
```
```  1165                simp: continuous_isCont[symmetric])
```
```  1166     show "open ({0<..}::real set)" by auto
```
```  1167     show "log b -` {log b 0} \<inter> {0<..} \<in> sets borel"
```
```  1168     proof cases
```
```  1169       assume "log b -` {log b 0} \<inter> {0<..} = {}"
```
```  1170       then show ?thesis by simp
```
```  1171     next
```
```  1172       assume "log b -` {log b 0} \<inter> {0<..} \<noteq> {}"
```
```  1173       then obtain x where "0 < x" "log b x = log b 0" by auto
```
```  1174       with log_inj[OF `1 < b`] have "log b -` {log b 0} \<inter> {0<..} = {x}"
```
```  1175         by (auto simp: inj_on_def)
```
```  1176       then show ?thesis by simp
```
```  1177     qed
```
```  1178   qed
```
```  1179   also have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) = log b"
```
```  1180     by (simp add: fun_eq_iff not_less log_imp)
```
```  1181   finally show ?thesis .
```
```  1182 qed
```
```  1183
```
```  1184 lemma (in sigma_algebra) borel_measurable_log[simp,intro]:
```
```  1185   assumes f: "f \<in> borel_measurable M" and "1 < b"
```
```  1186   shows "(\<lambda>x. log b (f x)) \<in> borel_measurable M"
```
```  1187   using measurable_comp[OF f borel_measurable_borel_log[OF `1 < b`]]
```
```  1188   by (simp add: comp_def)
```
```  1189
```
```  1190
```
```  1191 lemma (in sigma_algebra) less_eq_ge_measurable:
```
```  1192   fixes f :: "'a \<Rightarrow> 'c::linorder"
```
```  1193   shows "{x\<in>space M. a < f x} \<in> sets M \<longleftrightarrow> {x\<in>space M. f x \<le> a} \<in> sets M"
```
```  1194 proof
```
```  1195   assume "{x\<in>space M. f x \<le> a} \<in> sets M"
```
```  1196   moreover have "{x\<in>space M. a < f x} = space M - {x\<in>space M. f x \<le> a}" by auto
```
```  1197   ultimately show "{x\<in>space M. a < f x} \<in> sets M" by auto
```
```  1198 next
```
```  1199   assume "{x\<in>space M. a < f x} \<in> sets M"
```
```  1200   moreover have "{x\<in>space M. f x \<le> a} = space M - {x\<in>space M. a < f x}" by auto
```
```  1201   ultimately show "{x\<in>space M. f x \<le> a} \<in> sets M" by auto
```
```  1202 qed
```
```  1203
```
```  1204 lemma (in sigma_algebra) greater_eq_le_measurable:
```
```  1205   fixes f :: "'a \<Rightarrow> 'c::linorder"
```
```  1206   shows "{x\<in>space M. f x < a} \<in> sets M \<longleftrightarrow> {x\<in>space M. a \<le> f x} \<in> sets M"
```
```  1207 proof
```
```  1208   assume "{x\<in>space M. a \<le> f x} \<in> sets M"
```
```  1209   moreover have "{x\<in>space M. f x < a} = space M - {x\<in>space M. a \<le> f x}" by auto
```
```  1210   ultimately show "{x\<in>space M. f x < a} \<in> sets M" by auto
```
```  1211 next
```
```  1212   assume "{x\<in>space M. f x < a} \<in> sets M"
```
```  1213   moreover have "{x\<in>space M. a \<le> f x} = space M - {x\<in>space M. f x < a}" by auto
```
```  1214   ultimately show "{x\<in>space M. a \<le> f x} \<in> sets M" by auto
```
```  1215 qed
```
```  1216
```
```  1217 lemma (in sigma_algebra) less_eq_le_pextreal_measurable:
```
```  1218   fixes f :: "'a \<Rightarrow> pextreal"
```
```  1219   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)"
```
```  1220 proof
```
```  1221   assume a: "\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M"
```
```  1222   show "\<forall>a. {x \<in> space M. a < f x} \<in> sets M"
```
```  1223   proof
```
```  1224     fix a show "{x \<in> space M. a < f x} \<in> sets M"
```
```  1225     proof (cases a)
```
```  1226       case (preal r)
```
```  1227       have "{x\<in>space M. a < f x} = (\<Union>i. {x\<in>space M. a + inverse (of_nat (Suc i)) \<le> f x})"
```
```  1228       proof safe
```
```  1229         fix x assume "a < f x" and [simp]: "x \<in> space M"
```
```  1230         with ex_pextreal_inverse_of_nat_Suc_less[of "f x - a"]
```
```  1231         obtain n where "a + inverse (of_nat (Suc n)) < f x"
```
```  1232           by (cases "f x", auto simp: pextreal_minus_order)
```
```  1233         then have "a + inverse (of_nat (Suc n)) \<le> f x" by simp
```
```  1234         then show "x \<in> (\<Union>i. {x \<in> space M. a + inverse (of_nat (Suc i)) \<le> f x})"
```
```  1235           by auto
```
```  1236       next
```
```  1237         fix i x assume [simp]: "x \<in> space M"
```
```  1238         have "a < a + inverse (of_nat (Suc i))" using preal by auto
```
```  1239         also assume "a + inverse (of_nat (Suc i)) \<le> f x"
```
```  1240         finally show "a < f x" .
```
```  1241       qed
```
```  1242       with a show ?thesis by auto
```
```  1243     qed simp
```
```  1244   qed
```
```  1245 next
```
```  1246   assume a': "\<forall>a. {x \<in> space M. a < f x} \<in> sets M"
```
```  1247   then have a: "\<forall>a. {x \<in> space M. f x \<le> a} \<in> sets M" unfolding less_eq_ge_measurable .
```
```  1248   show "\<forall>a. {x \<in> space M. a \<le> f x} \<in> sets M" unfolding greater_eq_le_measurable[symmetric]
```
```  1249   proof
```
```  1250     fix a show "{x \<in> space M. f x < a} \<in> sets M"
```
```  1251     proof (cases a)
```
```  1252       case (preal r)
```
```  1253       show ?thesis
```
```  1254       proof cases
```
```  1255         assume "a = 0" then show ?thesis by simp
```
```  1256       next
```
```  1257         assume "a \<noteq> 0"
```
```  1258         have "{x\<in>space M. f x < a} = (\<Union>i. {x\<in>space M. f x \<le> a - inverse (of_nat (Suc i))})"
```
```  1259         proof safe
```
```  1260           fix x assume "f x < a" and [simp]: "x \<in> space M"
```
```  1261           with ex_pextreal_inverse_of_nat_Suc_less[of "a - f x"]
```
```  1262           obtain n where "inverse (of_nat (Suc n)) < a - f x"
```
```  1263             using preal by (cases "f x") auto
```
```  1264           then have "f x \<le> a - inverse (of_nat (Suc n)) "
```
```  1265             using preal by (cases "f x") (auto split: split_if_asm)
```
```  1266           then show "x \<in> (\<Union>i. {x \<in> space M. f x \<le> a - inverse (of_nat (Suc i))})"
```
```  1267             by auto
```
```  1268         next
```
```  1269           fix i x assume [simp]: "x \<in> space M"
```
```  1270           assume "f x \<le> a - inverse (of_nat (Suc i))"
```
```  1271           also have "\<dots> < a" using `a \<noteq> 0` preal by auto
```
```  1272           finally show "f x < a" .
```
```  1273         qed
```
```  1274         with a show ?thesis by auto
```
```  1275       qed
```
```  1276     next
```
```  1277       case infinite
```
```  1278       have "f -` {\<omega>} \<inter> space M = (\<Inter>n. {x\<in>space M. of_nat n < f x})"
```
```  1279       proof (safe, simp_all, safe)
```
```  1280         fix x assume *: "\<forall>n::nat. Real (real n) < f x"
```
```  1281         show "f x = \<omega>"    proof (rule ccontr)
```
```  1282           assume "f x \<noteq> \<omega>"
```
```  1283           with real_arch_lt[of "real (f x)"] obtain n where "f x < of_nat n"
```
```  1284             by (auto simp: pextreal_noteq_omega_Ex)
```
```  1285           with *[THEN spec, of n] show False by auto
```
```  1286         qed
```
```  1287       qed
```
```  1288       with a' have \<omega>: "f -` {\<omega>} \<inter> space M \<in> sets M" by auto
```
```  1289       moreover have "{x \<in> space M. f x < a} = space M - f -` {\<omega>} \<inter> space M"
```
```  1290         using infinite by auto
```
```  1291       ultimately show ?thesis by auto
```
```  1292     qed
```
```  1293   qed
```
```  1294 qed
```
```  1295
```
```  1296 lemma (in sigma_algebra) borel_measurable_pextreal_iff_greater:
```
```  1297   "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. a < f x} \<in> sets M)"
```
```  1298 proof safe
```
```  1299   fix a assume f: "f \<in> borel_measurable M"
```
```  1300   have "{x\<in>space M. a < f x} = f -` {a <..} \<inter> space M" by auto
```
```  1301   with f show "{x\<in>space M. a < f x} \<in> sets M"
```
```  1302     by (auto intro!: measurable_sets)
```
```  1303 next
```
```  1304   assume *: "\<forall>a. {x\<in>space M. a < f x} \<in> sets M"
```
```  1305   hence **: "\<forall>a. {x\<in>space M. f x < a} \<in> sets M"
```
```  1306     unfolding less_eq_le_pextreal_measurable
```
```  1307     unfolding greater_eq_le_measurable .
```
```  1308   show "f \<in> borel_measurable M" unfolding borel_measurable_pextreal_eq_real borel_measurable_iff_greater
```
```  1309   proof safe
```
```  1310     have "f -` {\<omega>} \<inter> space M = space M - {x\<in>space M. f x < \<omega>}" by auto
```
```  1311     then show \<omega>: "f -` {\<omega>} \<inter> space M \<in> sets M" using ** by auto
```
```  1312     fix a
```
```  1313     have "{w \<in> space M. a < real (f w)} =
```
```  1314       (if 0 \<le> a then {w\<in>space M. Real a < f w} - (f -` {\<omega>} \<inter> space M) else space M)"
```
```  1315     proof (split split_if, safe del: notI)
```
```  1316       fix x assume "0 \<le> a"
```
```  1317       { assume "a < real (f x)" then show "Real a < f x" "x \<notin> f -` {\<omega>} \<inter> space M"
```
```  1318           using `0 \<le> a` by (cases "f x", auto) }
```
```  1319       { assume "Real a < f x" "x \<notin> f -` {\<omega>}" then show "a < real (f x)"
```
```  1320           using `0 \<le> a` by (cases "f x", auto) }
```
```  1321     next
```
```  1322       fix x assume "\<not> 0 \<le> a" then show "a < real (f x)" by (cases "f x") auto
```
```  1323     qed
```
```  1324     then show "{w \<in> space M. a < real (f w)} \<in> sets M"
```
```  1325       using \<omega> * by (auto intro!: Diff)
```
```  1326   qed
```
```  1327 qed
```
```  1328
```
```  1329 lemma (in sigma_algebra) borel_measurable_pextreal_iff_less:
```
```  1330   "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. f x < a} \<in> sets M)"
```
```  1331   using borel_measurable_pextreal_iff_greater unfolding less_eq_le_pextreal_measurable greater_eq_le_measurable .
```
```  1332
```
```  1333 lemma (in sigma_algebra) borel_measurable_pextreal_iff_le:
```
```  1334   "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. f x \<le> a} \<in> sets M)"
```
```  1335   using borel_measurable_pextreal_iff_greater unfolding less_eq_ge_measurable .
```
```  1336
```
```  1337 lemma (in sigma_algebra) borel_measurable_pextreal_iff_ge:
```
```  1338   "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M)"
```
```  1339   using borel_measurable_pextreal_iff_greater unfolding less_eq_le_pextreal_measurable .
```
```  1340
```
```  1341 lemma (in sigma_algebra) borel_measurable_pextreal_eq_const:
```
```  1342   fixes f :: "'a \<Rightarrow> pextreal" assumes "f \<in> borel_measurable M"
```
```  1343   shows "{x\<in>space M. f x = c} \<in> sets M"
```
```  1344 proof -
```
```  1345   have "{x\<in>space M. f x = c} = (f -` {c} \<inter> space M)" by auto
```
```  1346   then show ?thesis using assms by (auto intro!: measurable_sets)
```
```  1347 qed
```
```  1348
```
```  1349 lemma (in sigma_algebra) borel_measurable_pextreal_neq_const:
```
```  1350   fixes f :: "'a \<Rightarrow> pextreal"
```
```  1351   assumes "f \<in> borel_measurable M"
```
```  1352   shows "{x\<in>space M. f x \<noteq> c} \<in> sets M"
```
```  1353 proof -
```
```  1354   have "{x\<in>space M. f x \<noteq> c} = space M - (f -` {c} \<inter> space M)" by auto
```
```  1355   then show ?thesis using assms by (auto intro!: measurable_sets)
```
```  1356 qed
```
```  1357
```
```  1358 lemma (in sigma_algebra) borel_measurable_pextreal_less[intro,simp]:
```
```  1359   fixes f g :: "'a \<Rightarrow> pextreal"
```
```  1360   assumes f: "f \<in> borel_measurable M"
```
```  1361   assumes g: "g \<in> borel_measurable M"
```
```  1362   shows "{x \<in> space M. f x < g x} \<in> sets M"
```
```  1363 proof -
```
```  1364   have "(\<lambda>x. real (f x)) \<in> borel_measurable M"
```
```  1365     "(\<lambda>x. real (g x)) \<in> borel_measurable M"
```
```  1366     using assms by (auto intro!: borel_measurable_real)
```
```  1367   from borel_measurable_less[OF this]
```
```  1368   have "{x \<in> space M. real (f x) < real (g x)} \<in> sets M" .
```
```  1369   moreover have "{x \<in> space M. f x \<noteq> \<omega>} \<in> sets M" using f by (rule borel_measurable_pextreal_neq_const)
```
```  1370   moreover have "{x \<in> space M. g x = \<omega>} \<in> sets M" using g by (rule borel_measurable_pextreal_eq_const)
```
```  1371   moreover have "{x \<in> space M. g x \<noteq> \<omega>} \<in> sets M" using g by (rule borel_measurable_pextreal_neq_const)
```
```  1372   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>
```
```  1373     ({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)})"
```
```  1374     by (auto simp: real_of_pextreal_strict_mono_iff)
```
```  1375   ultimately show ?thesis by auto
```
```  1376 qed
```
```  1377
```
```  1378 lemma (in sigma_algebra) borel_measurable_pextreal_le[intro,simp]:
```
```  1379   fixes f :: "'a \<Rightarrow> pextreal"
```
```  1380   assumes f: "f \<in> borel_measurable M"
```
```  1381   assumes g: "g \<in> borel_measurable M"
```
```  1382   shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
```
```  1383 proof -
```
```  1384   have "{x \<in> space M. f x \<le> g x} = space M - {x \<in> space M. g x < f x}" by auto
```
```  1385   then show ?thesis using g f by auto
```
```  1386 qed
```
```  1387
```
```  1388 lemma (in sigma_algebra) borel_measurable_pextreal_eq[intro,simp]:
```
```  1389   fixes f :: "'a \<Rightarrow> pextreal"
```
```  1390   assumes f: "f \<in> borel_measurable M"
```
```  1391   assumes g: "g \<in> borel_measurable M"
```
```  1392   shows "{w \<in> space M. f w = g w} \<in> sets M"
```
```  1393 proof -
```
```  1394   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
```
```  1395   then show ?thesis using g f by auto
```
```  1396 qed
```
```  1397
```
```  1398 lemma (in sigma_algebra) borel_measurable_pextreal_neq[intro,simp]:
```
```  1399   fixes f :: "'a \<Rightarrow> pextreal"
```
```  1400   assumes f: "f \<in> borel_measurable M"
```
```  1401   assumes g: "g \<in> borel_measurable M"
```
```  1402   shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
```
```  1403 proof -
```
```  1404   have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}" by auto
```
```  1405   thus ?thesis using f g by auto
```
```  1406 qed
```
```  1407
```
```  1408 lemma (in sigma_algebra) borel_measurable_pextreal_add[intro, simp]:
```
```  1409   fixes f :: "'a \<Rightarrow> pextreal"
```
```  1410   assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1411   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```  1412 proof -
```
```  1413   have *: "(\<lambda>x. f x + g x) =
```
```  1414      (\<lambda>x. if f x = \<omega> then \<omega> else if g x = \<omega> then \<omega> else Real (real (f x) + real (g x)))"
```
```  1415      by (auto simp: fun_eq_iff pextreal_noteq_omega_Ex)
```
```  1416   show ?thesis using assms unfolding *
```
```  1417     by (auto intro!: measurable_If)
```
```  1418 qed
```
```  1419
```
```  1420 lemma (in sigma_algebra) borel_measurable_pextreal_setsum[simp, intro]:
```
```  1421   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> pextreal"
```
```  1422   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1423   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1424 proof cases
```
```  1425   assume "finite S"
```
```  1426   thus ?thesis using assms
```
```  1427     by induct auto
```
```  1428 qed (simp add: borel_measurable_const)
```
```  1429
```
```  1430 lemma (in sigma_algebra) borel_measurable_pextreal_times[intro, simp]:
```
```  1431   fixes f :: "'a \<Rightarrow> pextreal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1432   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1433 proof -
```
```  1434   have *: "(\<lambda>x. f x * g x) =
```
```  1435      (\<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
```
```  1436       Real (real (f x) * real (g x)))"
```
```  1437      by (auto simp: fun_eq_iff pextreal_noteq_omega_Ex)
```
```  1438   show ?thesis using assms unfolding *
```
```  1439     by (auto intro!: measurable_If)
```
```  1440 qed
```
```  1441
```
```  1442 lemma (in sigma_algebra) borel_measurable_pextreal_setprod[simp, intro]:
```
```  1443   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> pextreal"
```
```  1444   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1445   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1446 proof cases
```
```  1447   assume "finite S"
```
```  1448   thus ?thesis using assms by induct auto
```
```  1449 qed simp
```
```  1450
```
```  1451 lemma (in sigma_algebra) borel_measurable_pextreal_min[simp, intro]:
```
```  1452   fixes f g :: "'a \<Rightarrow> pextreal"
```
```  1453   assumes "f \<in> borel_measurable M"
```
```  1454   assumes "g \<in> borel_measurable M"
```
```  1455   shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
```
```  1456   using assms unfolding min_def by (auto intro!: measurable_If)
```
```  1457
```
```  1458 lemma (in sigma_algebra) borel_measurable_pextreal_max[simp, intro]:
```
```  1459   fixes f g :: "'a \<Rightarrow> pextreal"
```
```  1460   assumes "f \<in> borel_measurable M"
```
```  1461   and "g \<in> borel_measurable M"
```
```  1462   shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
```
```  1463   using assms unfolding max_def by (auto intro!: measurable_If)
```
```  1464
```
```  1465 lemma (in sigma_algebra) borel_measurable_SUP[simp, intro]:
```
```  1466   fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> pextreal"
```
```  1467   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1468   shows "(\<lambda>x. SUP i : A. f i x) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M")
```
```  1469   unfolding borel_measurable_pextreal_iff_greater
```
```  1470 proof safe
```
```  1471   fix a
```
```  1472   have "{x\<in>space M. a < ?sup x} = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"
```
```  1473     by (auto simp: less_SUP_iff SUPR_apply)
```
```  1474   then show "{x\<in>space M. a < ?sup x} \<in> sets M"
```
```  1475     using assms by auto
```
```  1476 qed
```
```  1477
```
```  1478 lemma (in sigma_algebra) borel_measurable_INF[simp, intro]:
```
```  1479   fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> pextreal"
```
```  1480   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1481   shows "(\<lambda>x. INF i : A. f i x) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M")
```
```  1482   unfolding borel_measurable_pextreal_iff_less
```
```  1483 proof safe
```
```  1484   fix a
```
```  1485   have "{x\<in>space M. ?inf x < a} = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"
```
```  1486     by (auto simp: INF_less_iff INFI_apply)
```
```  1487   then show "{x\<in>space M. ?inf x < a} \<in> sets M"
```
```  1488     using assms by auto
```
```  1489 qed
```
```  1490
```
```  1491 lemma (in sigma_algebra) borel_measurable_pextreal_diff[simp, intro]:
```
```  1492   fixes f g :: "'a \<Rightarrow> pextreal"
```
```  1493   assumes "f \<in> borel_measurable M"
```
```  1494   assumes "g \<in> borel_measurable M"
```
```  1495   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```  1496   unfolding borel_measurable_pextreal_iff_greater
```
```  1497 proof safe
```
```  1498   fix a
```
```  1499   have "{x \<in> space M. a < f x - g x} = {x \<in> space M. g x + a < f x}"
```
```  1500     by (simp add: pextreal_less_minus_iff)
```
```  1501   then show "{x \<in> space M. a < f x - g x} \<in> sets M"
```
```  1502     using assms by auto
```
```  1503 qed
```
```  1504
```
```  1505 lemma (in sigma_algebra) borel_measurable_psuminf[simp, intro]:
```
```  1506   assumes "\<And>i. f i \<in> borel_measurable M"
```
```  1507   shows "(\<lambda>x. (\<Sum>\<^isub>\<infinity> i. f i x)) \<in> borel_measurable M"
```
```  1508   using assms unfolding psuminf_def by auto
```
```  1509
```
```  1510 section "LIMSEQ is borel measurable"
```
```  1511
```
```  1512 lemma (in sigma_algebra) borel_measurable_LIMSEQ:
```
```  1513   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
```
```  1514   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
```
```  1515   and u: "\<And>i. u i \<in> borel_measurable M"
```
```  1516   shows "u' \<in> borel_measurable M"
```
```  1517 proof -
```
```  1518   let "?pu x i" = "max (u i x) 0"
```
```  1519   let "?nu x i" = "max (- u i x) 0"
```
```  1520   { fix x assume x: "x \<in> space M"
```
```  1521     have "(?pu x) ----> max (u' x) 0"
```
```  1522       "(?nu x) ----> max (- u' x) 0"
```
```  1523       using u'[OF x] by (auto intro!: LIMSEQ_max LIMSEQ_minus)
```
```  1524     from LIMSEQ_imp_lim_INF[OF _ this(1)] LIMSEQ_imp_lim_INF[OF _ this(2)]
```
```  1525     have "(SUP n. INF m. Real (u (n + m) x)) = Real (u' x)"
```
```  1526       "(SUP n. INF m. Real (- u (n + m) x)) = Real (- u' x)"
```
```  1527       by (simp_all add: Real_max'[symmetric]) }
```
```  1528   note eq = this
```
```  1529   have *: "\<And>x. real (Real (u' x)) - real (Real (- u' x)) = u' x"
```
```  1530     by auto
```
```  1531   have "(\<lambda>x. SUP n. INF m. Real (u (n + m) x)) \<in> borel_measurable M"
```
```  1532        "(\<lambda>x. SUP n. INF m. Real (- u (n + m) x)) \<in> borel_measurable M"
```
```  1533     using u by auto
```
```  1534   with eq[THEN measurable_cong, of M "\<lambda>x. x" borel]
```
```  1535   have "(\<lambda>x. Real (u' x)) \<in> borel_measurable M"
```
```  1536        "(\<lambda>x. Real (- u' x)) \<in> borel_measurable M" by auto
```
```  1537   note this[THEN borel_measurable_real]
```
```  1538   from borel_measurable_diff[OF this]
```
```  1539   show ?thesis unfolding * .
```
```  1540 qed
```
```  1541
```
```  1542 end
```