src/HOL/Probability/Borel_Space.thy
 author hoelzl Wed Apr 25 19:26:00 2012 +0200 (2012-04-25) changeset 47761 dfe747e72fa8 parent 47694 05663f75964c child 49774 dfa8ddb874ce permissions -rw-r--r--
moved lemmas to appropriate places
```     1 (*  Title:      HOL/Probability/Borel_Space.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Armin Heller, TU München
```
```     4 *)
```
```     5
```
```     6 header {*Borel spaces*}
```
```     7
```
```     8 theory Borel_Space
```
```     9   imports Sigma_Algebra "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
```
```    10 begin
```
```    11
```
```    12 section "Generic Borel spaces"
```
```    13
```
```    14 definition borel :: "'a::topological_space measure" where
```
```    15   "borel = sigma UNIV {S. open S}"
```
```    16
```
```    17 abbreviation "borel_measurable M \<equiv> measurable M borel"
```
```    18
```
```    19 lemma in_borel_measurable:
```
```    20    "f \<in> borel_measurable M \<longleftrightarrow>
```
```    21     (\<forall>S \<in> sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)"
```
```    22   by (auto simp add: measurable_def borel_def)
```
```    23
```
```    24 lemma in_borel_measurable_borel:
```
```    25    "f \<in> borel_measurable M \<longleftrightarrow>
```
```    26     (\<forall>S \<in> sets borel.
```
```    27       f -` S \<inter> space M \<in> sets M)"
```
```    28   by (auto simp add: measurable_def borel_def)
```
```    29
```
```    30 lemma space_borel[simp]: "space borel = UNIV"
```
```    31   unfolding borel_def by auto
```
```    32
```
```    33 lemma borel_open[simp]:
```
```    34   assumes "open A" shows "A \<in> sets borel"
```
```    35 proof -
```
```    36   have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms .
```
```    37   thus ?thesis unfolding borel_def by auto
```
```    38 qed
```
```    39
```
```    40 lemma borel_closed[simp]:
```
```    41   assumes "closed A" shows "A \<in> sets borel"
```
```    42 proof -
```
```    43   have "space borel - (- A) \<in> sets borel"
```
```    44     using assms unfolding closed_def by (blast intro: borel_open)
```
```    45   thus ?thesis by simp
```
```    46 qed
```
```    47
```
```    48 lemma borel_comp[intro,simp]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
```
```    49   unfolding Compl_eq_Diff_UNIV by (intro Diff) auto
```
```    50
```
```    51 lemma 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_singleton[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 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_measure_of, simp_all)
```
```    72   fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"
```
```    73     using assms[of S] by simp
```
```    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 insert_in_sets)
```
```    80     show "{x} \<in> sets borel"
```
```    81       using closed_singleton[of x] by (rule borel_closed)
```
```    82   qed simp
```
```    83
```
```    84 lemma borel_measurable_const[simp, intro]:
```
```    85   "(\<lambda>x. c) \<in> borel_measurable M"
```
```    86   by auto
```
```    87
```
```    88 lemma borel_measurable_indicator[simp, intro!]:
```
```    89   assumes A: "A \<in> sets M"
```
```    90   shows "indicator A \<in> borel_measurable M"
```
```    91   unfolding indicator_def [abs_def] using A
```
```    92   by (auto intro!: measurable_If_set)
```
```    93
```
```    94 lemma 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 [abs_def] 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_subalgebra:
```
```   113   assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
```
```   114   shows "f \<in> borel_measurable M"
```
```   115   using assms unfolding measurable_def by auto
```
```   116
```
```   117 section "Borel spaces on euclidean spaces"
```
```   118
```
```   119 lemma lessThan_borel[simp, intro]:
```
```   120   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   121   shows "{..< a} \<in> sets borel"
```
```   122   by (blast intro: borel_open)
```
```   123
```
```   124 lemma greaterThan_borel[simp, intro]:
```
```   125   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   126   shows "{a <..} \<in> sets borel"
```
```   127   by (blast intro: borel_open)
```
```   128
```
```   129 lemma greaterThanLessThan_borel[simp, intro]:
```
```   130   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   131   shows "{a<..<b} \<in> sets borel"
```
```   132   by (blast intro: borel_open)
```
```   133
```
```   134 lemma atMost_borel[simp, intro]:
```
```   135   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   136   shows "{..a} \<in> sets borel"
```
```   137   by (blast intro: borel_closed)
```
```   138
```
```   139 lemma atLeast_borel[simp, intro]:
```
```   140   fixes a :: "'a\<Colon>ordered_euclidean_space"
```
```   141   shows "{a..} \<in> sets borel"
```
```   142   by (blast intro: borel_closed)
```
```   143
```
```   144 lemma atLeastAtMost_borel[simp, intro]:
```
```   145   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   146   shows "{a..b} \<in> sets borel"
```
```   147   by (blast intro: borel_closed)
```
```   148
```
```   149 lemma greaterThanAtMost_borel[simp, intro]:
```
```   150   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   151   shows "{a<..b} \<in> sets borel"
```
```   152   unfolding greaterThanAtMost_def by blast
```
```   153
```
```   154 lemma atLeastLessThan_borel[simp, intro]:
```
```   155   fixes a b :: "'a\<Colon>ordered_euclidean_space"
```
```   156   shows "{a..<b} \<in> sets borel"
```
```   157   unfolding atLeastLessThan_def by blast
```
```   158
```
```   159 lemma hafspace_less_borel[simp, intro]:
```
```   160   fixes a :: real
```
```   161   shows "{x::'a::euclidean_space. a < x \$\$ i} \<in> sets borel"
```
```   162   by (auto intro!: borel_open open_halfspace_component_gt)
```
```   163
```
```   164 lemma hafspace_greater_borel[simp, intro]:
```
```   165   fixes a :: real
```
```   166   shows "{x::'a::euclidean_space. x \$\$ i < a} \<in> sets borel"
```
```   167   by (auto intro!: borel_open open_halfspace_component_lt)
```
```   168
```
```   169 lemma hafspace_less_eq_borel[simp, intro]:
```
```   170   fixes a :: real
```
```   171   shows "{x::'a::euclidean_space. a \<le> x \$\$ i} \<in> sets borel"
```
```   172   by (auto intro!: borel_closed closed_halfspace_component_ge)
```
```   173
```
```   174 lemma hafspace_greater_eq_borel[simp, intro]:
```
```   175   fixes a :: real
```
```   176   shows "{x::'a::euclidean_space. x \$\$ i \<le> a} \<in> sets borel"
```
```   177   by (auto intro!: borel_closed closed_halfspace_component_le)
```
```   178
```
```   179 lemma borel_measurable_less[simp, intro]:
```
```   180   fixes f :: "'a \<Rightarrow> real"
```
```   181   assumes f: "f \<in> borel_measurable M"
```
```   182   assumes g: "g \<in> borel_measurable M"
```
```   183   shows "{w \<in> space M. f w < g w} \<in> sets M"
```
```   184 proof -
```
```   185   have "{w \<in> space M. f w < g w} =
```
```   186         (\<Union>r. (f -` {..< of_rat r} \<inter> space M) \<inter> (g -` {of_rat r <..} \<inter> space M))"
```
```   187     using Rats_dense_in_real by (auto simp add: Rats_def)
```
```   188   then show ?thesis using f g
```
```   189     by simp (blast intro: measurable_sets)
```
```   190 qed
```
```   191
```
```   192 lemma borel_measurable_le[simp, intro]:
```
```   193   fixes f :: "'a \<Rightarrow> real"
```
```   194   assumes f: "f \<in> borel_measurable M"
```
```   195   assumes g: "g \<in> borel_measurable M"
```
```   196   shows "{w \<in> space M. f w \<le> g w} \<in> sets M"
```
```   197 proof -
```
```   198   have "{w \<in> space M. f w \<le> g w} = space M - {w \<in> space M. g w < f w}"
```
```   199     by auto
```
```   200   thus ?thesis using f g
```
```   201     by simp blast
```
```   202 qed
```
```   203
```
```   204 lemma borel_measurable_eq[simp, intro]:
```
```   205   fixes f :: "'a \<Rightarrow> real"
```
```   206   assumes f: "f \<in> borel_measurable M"
```
```   207   assumes g: "g \<in> borel_measurable M"
```
```   208   shows "{w \<in> space M. f w = g w} \<in> sets M"
```
```   209 proof -
```
```   210   have "{w \<in> space M. f w = g w} =
```
```   211         {w \<in> space M. f w \<le> g w} \<inter> {w \<in> space M. g w \<le> f w}"
```
```   212     by auto
```
```   213   thus ?thesis using f g by auto
```
```   214 qed
```
```   215
```
```   216 lemma borel_measurable_neq[simp, intro]:
```
```   217   fixes f :: "'a \<Rightarrow> real"
```
```   218   assumes f: "f \<in> borel_measurable M"
```
```   219   assumes g: "g \<in> borel_measurable M"
```
```   220   shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
```
```   221 proof -
```
```   222   have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}"
```
```   223     by auto
```
```   224   thus ?thesis using f g by auto
```
```   225 qed
```
```   226
```
```   227 subsection "Borel space equals sigma algebras over intervals"
```
```   228
```
```   229 lemma rational_boxes:
```
```   230   fixes x :: "'a\<Colon>ordered_euclidean_space"
```
```   231   assumes "0 < e"
```
```   232   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"
```
```   233 proof -
```
```   234   def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
```
```   235   then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos)
```
```   236   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \$\$ i \<and> x \$\$ i - y < e'" (is "\<forall>i. ?th i")
```
```   237   proof
```
```   238     fix i from Rats_dense_in_real[of "x \$\$ i - e'" "x \$\$ i"] e
```
```   239     show "?th i" by auto
```
```   240   qed
```
```   241   from choice[OF this] guess a .. note a = this
```
```   242   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \$\$ i < y \<and> y - x \$\$ i < e'" (is "\<forall>i. ?th i")
```
```   243   proof
```
```   244     fix i from Rats_dense_in_real[of "x \$\$ i" "x \$\$ i + e'"] e
```
```   245     show "?th i" by auto
```
```   246   qed
```
```   247   from choice[OF this] guess b .. note b = this
```
```   248   { fix y :: 'a assume *: "Chi a < y" "y < Chi b"
```
```   249     have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x \$\$ i) (y \$\$ i))\<twosuperior>)"
```
```   250       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
```
```   251     also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))"
```
```   252     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
```
```   253       fix i assume i: "i \<in> {..<DIM('a)}"
```
```   254       have "a i < y\$\$i \<and> y\$\$i < b i" using * i eucl_less[where 'a='a] by auto
```
```   255       moreover have "a i < x\$\$i" "x\$\$i - a i < e'" using a by auto
```
```   256       moreover have "x\$\$i < b i" "b i - x\$\$i < e'" using b by auto
```
```   257       ultimately have "\<bar>x\$\$i - y\$\$i\<bar> < 2 * e'" by auto
```
```   258       then have "dist (x \$\$ i) (y \$\$ i) < e/sqrt (real (DIM('a)))"
```
```   259         unfolding e'_def by (auto simp: dist_real_def)
```
```   260       then have "(dist (x \$\$ i) (y \$\$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
```
```   261         by (rule power_strict_mono) auto
```
```   262       then show "(dist (x \$\$ i) (y \$\$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
```
```   263         by (simp add: power_divide)
```
```   264     qed auto
```
```   265     also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive)
```
```   266     finally have "dist x y < e" . }
```
```   267   with a b show ?thesis
```
```   268     apply (rule_tac exI[of _ "Chi a"])
```
```   269     apply (rule_tac exI[of _ "Chi b"])
```
```   270     using eucl_less[where 'a='a] by auto
```
```   271 qed
```
```   272
```
```   273 lemma ex_rat_list:
```
```   274   fixes x :: "'a\<Colon>ordered_euclidean_space"
```
```   275   assumes "\<And> i. x \$\$ i \<in> \<rat>"
```
```   276   shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x \$\$ i)"
```
```   277 proof -
```
```   278   have "\<forall>i. \<exists>r. x \$\$ i = of_rat r" using assms unfolding Rats_def by blast
```
```   279   from choice[OF this] guess r ..
```
```   280   then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"])
```
```   281 qed
```
```   282
```
```   283 lemma open_UNION:
```
```   284   fixes M :: "'a\<Colon>ordered_euclidean_space set"
```
```   285   assumes "open M"
```
```   286   shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M}
```
```   287                    (\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})"
```
```   288     (is "M = UNION ?idx ?box")
```
```   289 proof safe
```
```   290   fix x assume "x \<in> M"
```
```   291   obtain e where e: "e > 0" "ball x e \<subseteq> M"
```
```   292     using openE[OF assms `x \<in> M`] by auto
```
```   293   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"
```
```   294     using rational_boxes[OF e(1)] by blast
```
```   295   then obtain p q where pq: "length p = DIM ('a)"
```
```   296                             "length q = DIM ('a)"
```
```   297                             "\<forall> i < DIM ('a). of_rat (p ! i) = a \$\$ i \<and> of_rat (q ! i) = b \$\$ i"
```
```   298     using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast
```
```   299   hence p: "Chi (of_rat \<circ> op ! p) = a"
```
```   300     using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a]
```
```   301     unfolding o_def by auto
```
```   302   from pq have q: "Chi (of_rat \<circ> op ! q) = b"
```
```   303     using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b]
```
```   304     unfolding o_def by auto
```
```   305   have "x \<in> ?box (p, q)"
```
```   306     using p q ab by auto
```
```   307   thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto
```
```   308 qed auto
```
```   309
```
```   310 lemma borel_sigma_sets_subset:
```
```   311   "A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel"
```
```   312   using sigma_sets_subset[of A borel] by simp
```
```   313
```
```   314 lemma borel_eq_sigmaI1:
```
```   315   fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
```
```   316   assumes borel_eq: "borel = sigma UNIV X"
```
```   317   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (range F))"
```
```   318   assumes F: "\<And>i. F i \<in> sets borel"
```
```   319   shows "borel = sigma UNIV (range F)"
```
```   320   unfolding borel_def
```
```   321 proof (intro sigma_eqI antisym)
```
```   322   have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel"
```
```   323     unfolding borel_def by simp
```
```   324   also have "\<dots> = sigma_sets UNIV X"
```
```   325     unfolding borel_eq by simp
```
```   326   also have "\<dots> \<subseteq> sigma_sets UNIV (range F)"
```
```   327     using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto
```
```   328   finally show "sigma_sets UNIV {S. open S} \<subseteq> sigma_sets UNIV (range F)" .
```
```   329   show "sigma_sets UNIV (range F) \<subseteq> sigma_sets UNIV {S. open S}"
```
```   330     unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto
```
```   331 qed auto
```
```   332
```
```   333 lemma borel_eq_sigmaI2:
```
```   334   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set"
```
```   335     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
```
```   336   assumes borel_eq: "borel = sigma UNIV (range (\<lambda>(i, j). G i j))"
```
```   337   assumes X: "\<And>i j. G i j \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
```
```   338   assumes F: "\<And>i j. F i j \<in> sets borel"
```
```   339   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
```
```   340   using assms by (intro borel_eq_sigmaI1[where X="range (\<lambda>(i, j). G i j)" and F="(\<lambda>(i, j). F i j)"]) auto
```
```   341
```
```   342 lemma borel_eq_sigmaI3:
```
```   343   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
```
```   344   assumes borel_eq: "borel = sigma UNIV X"
```
```   345   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
```
```   346   assumes F: "\<And>i j. F i j \<in> sets borel"
```
```   347   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
```
```   348   using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto
```
```   349
```
```   350 lemma borel_eq_sigmaI4:
```
```   351   fixes F :: "'i \<Rightarrow> 'a::topological_space set"
```
```   352     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
```
```   353   assumes borel_eq: "borel = sigma UNIV (range (\<lambda>(i, j). G i j))"
```
```   354   assumes X: "\<And>i j. G i j \<in> sets (sigma UNIV (range F))"
```
```   355   assumes F: "\<And>i. F i \<in> sets borel"
```
```   356   shows "borel = sigma UNIV (range F)"
```
```   357   using assms by (intro borel_eq_sigmaI1[where X="range (\<lambda>(i, j). G i j)" and F=F]) auto
```
```   358
```
```   359 lemma borel_eq_sigmaI5:
```
```   360   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set"
```
```   361   assumes borel_eq: "borel = sigma UNIV (range G)"
```
```   362   assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
```
```   363   assumes F: "\<And>i j. F i j \<in> sets borel"
```
```   364   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
```
```   365   using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto
```
```   366
```
```   367 lemma halfspace_gt_in_halfspace:
```
```   368   "{x\<Colon>'a. a < x \$\$ i} \<in> sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a}))"
```
```   369   (is "?set \<in> ?SIGMA")
```
```   370 proof -
```
```   371   interpret sigma_algebra UNIV ?SIGMA
```
```   372     by (intro sigma_algebra_sigma_sets) simp_all
```
```   373   have *: "?set = (\<Union>n. UNIV - {x\<Colon>'a. x \$\$ i < a + 1 / real (Suc n)})"
```
```   374   proof (safe, simp_all add: not_less)
```
```   375     fix x assume "a < x \$\$ i"
```
```   376     with reals_Archimedean[of "x \$\$ i - a"]
```
```   377     obtain n where "a + 1 / real (Suc n) < x \$\$ i"
```
```   378       by (auto simp: inverse_eq_divide field_simps)
```
```   379     then show "\<exists>n. a + 1 / real (Suc n) \<le> x \$\$ i"
```
```   380       by (blast intro: less_imp_le)
```
```   381   next
```
```   382     fix x n
```
```   383     have "a < a + 1 / real (Suc n)" by auto
```
```   384     also assume "\<dots> \<le> x"
```
```   385     finally show "a < x" .
```
```   386   qed
```
```   387   show "?set \<in> ?SIGMA" unfolding *
```
```   388     by (auto intro!: Diff)
```
```   389 qed
```
```   390
```
```   391 lemma borel_eq_halfspace_less:
```
```   392   "borel = sigma UNIV (range (\<lambda>(a, i). {x::'a::ordered_euclidean_space. x \$\$ i < a}))"
```
```   393   (is "_ = ?SIGMA")
```
```   394 proof (rule borel_eq_sigmaI3[OF borel_def])
```
```   395   fix S :: "'a set" assume "S \<in> {S. open S}"
```
```   396   then have "open S" by simp
```
```   397   from open_UNION[OF this]
```
```   398   obtain I where *: "S =
```
```   399     (\<Union>(a, b)\<in>I.
```
```   400         (\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) \$\$ i < x \$\$ i}) \<inter>
```
```   401         (\<Inter> i<DIM('a). {x. x \$\$ i < (Chi (real_of_rat \<circ> op ! b)::'a) \$\$ i}))"
```
```   402     unfolding greaterThanLessThan_def
```
```   403     unfolding eucl_greaterThan_eq_halfspaces[where 'a='a]
```
```   404     unfolding eucl_lessThan_eq_halfspaces[where 'a='a]
```
```   405     by blast
```
```   406   show "S \<in> ?SIGMA"
```
```   407     unfolding *
```
```   408     by (safe intro!: countable_UN Int countable_INT) (auto intro!: halfspace_gt_in_halfspace)
```
```   409 qed auto
```
```   410
```
```   411 lemma borel_eq_halfspace_le:
```
```   412   "borel = sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x \$\$ i \<le> a}))"
```
```   413   (is "_ = ?SIGMA")
```
```   414 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
```
```   415   fix a i
```
```   416   have *: "{x::'a. x\$\$i < a} = (\<Union>n. {x. x\$\$i \<le> a - 1/real (Suc n)})"
```
```   417   proof (safe, simp_all)
```
```   418     fix x::'a assume *: "x\$\$i < a"
```
```   419     with reals_Archimedean[of "a - x\$\$i"]
```
```   420     obtain n where "x \$\$ i < a - 1 / (real (Suc n))"
```
```   421       by (auto simp: field_simps inverse_eq_divide)
```
```   422     then show "\<exists>n. x \$\$ i \<le> a - 1 / (real (Suc n))"
```
```   423       by (blast intro: less_imp_le)
```
```   424   next
```
```   425     fix x::'a and n
```
```   426     assume "x\$\$i \<le> a - 1 / real (Suc n)"
```
```   427     also have "\<dots> < a" by auto
```
```   428     finally show "x\$\$i < a" .
```
```   429   qed
```
```   430   show "{x. x\$\$i < a} \<in> ?SIGMA" unfolding *
```
```   431     by (safe intro!: countable_UN) auto
```
```   432 qed auto
```
```   433
```
```   434 lemma borel_eq_halfspace_ge:
```
```   435   "borel = sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x \$\$ i}))"
```
```   436   (is "_ = ?SIGMA")
```
```   437 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
```
```   438   fix a i have *: "{x::'a. x\$\$i < a} = space ?SIGMA - {x::'a. a \<le> x\$\$i}" by auto
```
```   439   show "{x. x\$\$i < a} \<in> ?SIGMA" unfolding *
```
```   440       by (safe intro!: compl_sets) auto
```
```   441 qed auto
```
```   442
```
```   443 lemma borel_eq_halfspace_greater:
```
```   444   "borel = sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a < x \$\$ i}))"
```
```   445   (is "_ = ?SIGMA")
```
```   446 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])
```
```   447   fix a i have *: "{x::'a. x\$\$i \<le> a} = space ?SIGMA - {x::'a. a < x\$\$i}" by auto
```
```   448   show "{x. x\$\$i \<le> a} \<in> ?SIGMA" unfolding *
```
```   449     by (safe intro!: compl_sets) auto
```
```   450 qed auto
```
```   451
```
```   452 lemma borel_eq_atMost:
```
```   453   "borel = sigma UNIV (range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space}))"
```
```   454   (is "_ = ?SIGMA")
```
```   455 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
```
```   456   fix a i show "{x. x\$\$i \<le> a} \<in> ?SIGMA"
```
```   457   proof cases
```
```   458     assume "i < DIM('a)"
```
```   459     then have *: "{x::'a. x\$\$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})"
```
```   460     proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm)
```
```   461       fix x
```
```   462       from real_arch_simple[of "Max ((\<lambda>i. x\$\$i)`{..<DIM('a)})"] guess k::nat ..
```
```   463       then have "\<And>i. i < DIM('a) \<Longrightarrow> x\$\$i \<le> real k"
```
```   464         by (subst (asm) Max_le_iff) auto
```
```   465       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x \$\$ ia \<le> real k"
```
```   466         by (auto intro!: exI[of _ k])
```
```   467     qed
```
```   468     show "{x. x\$\$i \<le> a} \<in> ?SIGMA" unfolding *
```
```   469       by (safe intro!: countable_UN) auto
```
```   470   qed (auto intro: sigma_sets_top sigma_sets.Empty)
```
```   471 qed auto
```
```   472
```
```   473 lemma borel_eq_greaterThan:
```
```   474   "borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {a<..}))"
```
```   475   (is "_ = ?SIGMA")
```
```   476 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
```
```   477   fix a i show "{x. x\$\$i \<le> a} \<in> ?SIGMA"
```
```   478   proof cases
```
```   479     assume "i < DIM('a)"
```
```   480     have "{x::'a. x\$\$i \<le> a} = UNIV - {x::'a. a < x\$\$i}" by auto
```
```   481     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)`
```
```   482     proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
```
```   483       fix x
```
```   484       from reals_Archimedean2[of "Max ((\<lambda>i. -x\$\$i)`{..<DIM('a)})"]
```
```   485       guess k::nat .. note k = this
```
```   486       { fix i assume "i < DIM('a)"
```
```   487         then have "-x\$\$i < real k"
```
```   488           using k by (subst (asm) Max_less_iff) auto
```
```   489         then have "- real k < x\$\$i" by simp }
```
```   490       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x \$\$ ia"
```
```   491         by (auto intro!: exI[of _ k])
```
```   492     qed
```
```   493     finally show "{x. x\$\$i \<le> a} \<in> ?SIGMA"
```
```   494       apply (simp only:)
```
```   495       apply (safe intro!: countable_UN Diff)
```
```   496       apply (auto intro: sigma_sets_top)
```
```   497       done
```
```   498   qed (auto intro: sigma_sets_top sigma_sets.Empty)
```
```   499 qed auto
```
```   500
```
```   501 lemma borel_eq_lessThan:
```
```   502   "borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {..<a}))"
```
```   503   (is "_ = ?SIGMA")
```
```   504 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge])
```
```   505   fix a i show "{x. a \<le> x\$\$i} \<in> ?SIGMA"
```
```   506   proof cases
```
```   507     fix a i assume "i < DIM('a)"
```
```   508     have "{x::'a. a \<le> x\$\$i} = UNIV - {x::'a. x\$\$i < a}" by auto
```
```   509     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)`
```
```   510     proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
```
```   511       fix x
```
```   512       from reals_Archimedean2[of "Max ((\<lambda>i. x\$\$i)`{..<DIM('a)})"]
```
```   513       guess k::nat .. note k = this
```
```   514       { fix i assume "i < DIM('a)"
```
```   515         then have "x\$\$i < real k"
```
```   516           using k by (subst (asm) Max_less_iff) auto
```
```   517         then have "x\$\$i < real k" by simp }
```
```   518       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x \$\$ ia < real k"
```
```   519         by (auto intro!: exI[of _ k])
```
```   520     qed
```
```   521     finally show "{x. a \<le> x\$\$i} \<in> ?SIGMA"
```
```   522       apply (simp only:)
```
```   523       apply (safe intro!: countable_UN Diff)
```
```   524       apply (auto intro: sigma_sets_top)
```
```   525       done
```
```   526   qed (auto intro: sigma_sets_top sigma_sets.Empty)
```
```   527 qed auto
```
```   528
```
```   529 lemma borel_eq_atLeastAtMost:
```
```   530   "borel = sigma UNIV (range (\<lambda>(a,b). {a..b} \<Colon>'a\<Colon>ordered_euclidean_space set))"
```
```   531   (is "_ = ?SIGMA")
```
```   532 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
```
```   533   fix a::'a
```
```   534   have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
```
```   535   proof (safe, simp_all add: eucl_le[where 'a='a])
```
```   536     fix x
```
```   537     from real_arch_simple[of "Max ((\<lambda>i. - x\$\$i)`{..<DIM('a)})"]
```
```   538     guess k::nat .. note k = this
```
```   539     { fix i assume "i < DIM('a)"
```
```   540       with k have "- x\$\$i \<le> real k"
```
```   541         by (subst (asm) Max_le_iff) (auto simp: field_simps)
```
```   542       then have "- real k \<le> x\$\$i" by simp }
```
```   543     then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x \$\$ i"
```
```   544       by (auto intro!: exI[of _ k])
```
```   545   qed
```
```   546   show "{..a} \<in> ?SIGMA" unfolding *
```
```   547     by (safe intro!: countable_UN)
```
```   548        (auto intro!: sigma_sets_top)
```
```   549 qed auto
```
```   550
```
```   551 lemma borel_eq_greaterThanLessThan:
```
```   552   "borel = sigma UNIV (range (\<lambda> (a, b). {a <..< b} :: 'a \<Colon> ordered_euclidean_space set))"
```
```   553     (is "_ = ?SIGMA")
```
```   554 proof (rule borel_eq_sigmaI1[OF borel_def])
```
```   555   fix M :: "'a set" assume "M \<in> {S. open S}"
```
```   556   then have "open M" by simp
```
```   557   show "M \<in> ?SIGMA"
```
```   558     apply (subst open_UNION[OF `open M`])
```
```   559     apply (safe intro!: countable_UN)
```
```   560     apply auto
```
```   561     done
```
```   562 qed auto
```
```   563
```
```   564 lemma borel_eq_atLeastLessThan:
```
```   565   "borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA")
```
```   566 proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan])
```
```   567   have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto
```
```   568   fix x :: real
```
```   569   have "{..<x} = (\<Union>i::nat. {-real i ..< x})"
```
```   570     by (auto simp: move_uminus real_arch_simple)
```
```   571   then show "{..< x} \<in> ?SIGMA"
```
```   572     by (auto intro: sigma_sets.intros)
```
```   573 qed auto
```
```   574
```
```   575 lemma borel_measurable_halfspacesI:
```
```   576   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   577   assumes F: "borel = sigma UNIV (range F)"
```
```   578   and S_eq: "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
```
```   579   and S: "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M"
```
```   580   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)"
```
```   581 proof safe
```
```   582   fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M"
```
```   583   then show "S a i \<in> sets M" unfolding assms
```
```   584     by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1))
```
```   585 next
```
```   586   assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M"
```
```   587   { fix a i have "S a i \<in> sets M"
```
```   588     proof cases
```
```   589       assume "i < DIM('c)"
```
```   590       with a show ?thesis unfolding assms(2) by simp
```
```   591     next
```
```   592       assume "\<not> i < DIM('c)"
```
```   593       from S[OF this] show ?thesis .
```
```   594     qed }
```
```   595   then show "f \<in> borel_measurable M"
```
```   596     by (auto intro!: measurable_measure_of simp: S_eq F)
```
```   597 qed
```
```   598
```
```   599 lemma borel_measurable_iff_halfspace_le:
```
```   600   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   601   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w \$\$ i \<le> a} \<in> sets M)"
```
```   602   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
```
```   603
```
```   604 lemma borel_measurable_iff_halfspace_less:
```
```   605   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   606   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w \$\$ i < a} \<in> sets M)"
```
```   607   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
```
```   608
```
```   609 lemma borel_measurable_iff_halfspace_ge:
```
```   610   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   611   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w \$\$ i} \<in> sets M)"
```
```   612   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
```
```   613
```
```   614 lemma borel_measurable_iff_halfspace_greater:
```
```   615   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
```
```   616   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w \$\$ i} \<in> sets M)"
```
```   617   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto
```
```   618
```
```   619 lemma borel_measurable_iff_le:
```
```   620   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
```
```   621   using borel_measurable_iff_halfspace_le[where 'c=real] by simp
```
```   622
```
```   623 lemma borel_measurable_iff_less:
```
```   624   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
```
```   625   using borel_measurable_iff_halfspace_less[where 'c=real] by simp
```
```   626
```
```   627 lemma borel_measurable_iff_ge:
```
```   628   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
```
```   629   using borel_measurable_iff_halfspace_ge[where 'c=real] by simp
```
```   630
```
```   631 lemma borel_measurable_iff_greater:
```
```   632   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
```
```   633   using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
```
```   634
```
```   635 lemma borel_measurable_euclidean_component:
```
```   636   "(\<lambda>x::'a::euclidean_space. x \$\$ i) \<in> borel_measurable borel"
```
```   637 proof (rule borel_measurableI)
```
```   638   fix S::"real set" assume "open S"
```
```   639   from open_vimage_euclidean_component[OF this]
```
```   640   show "(\<lambda>x. x \$\$ i) -` S \<inter> space borel \<in> sets borel"
```
```   641     by (auto intro: borel_open)
```
```   642 qed
```
```   643
```
```   644 lemma borel_measurable_euclidean_space:
```
```   645   fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
```
```   646   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x \$\$ i) \<in> borel_measurable M)"
```
```   647 proof safe
```
```   648   fix i assume "f \<in> borel_measurable M"
```
```   649   then show "(\<lambda>x. f x \$\$ i) \<in> borel_measurable M"
```
```   650     using measurable_comp[of f _ _ "\<lambda>x. x \$\$ i", unfolded comp_def]
```
```   651     by (auto intro: borel_measurable_euclidean_component)
```
```   652 next
```
```   653   assume f: "\<forall>i<DIM('c). (\<lambda>x. f x \$\$ i) \<in> borel_measurable M"
```
```   654   then show "f \<in> borel_measurable M"
```
```   655     unfolding borel_measurable_iff_halfspace_le by auto
```
```   656 qed
```
```   657
```
```   658 subsection "Borel measurable operators"
```
```   659
```
```   660 lemma affine_borel_measurable_vector:
```
```   661   fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
```
```   662   assumes "f \<in> borel_measurable M"
```
```   663   shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
```
```   664 proof (rule borel_measurableI)
```
```   665   fix S :: "'x set" assume "open S"
```
```   666   show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
```
```   667   proof cases
```
```   668     assume "b \<noteq> 0"
```
```   669     with `open S` have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")
```
```   670       by (auto intro!: open_affinity simp: scaleR_add_right)
```
```   671     hence "?S \<in> sets borel" by auto
```
```   672     moreover
```
```   673     from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
```
```   674       apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
```
```   675     ultimately show ?thesis using assms unfolding in_borel_measurable_borel
```
```   676       by auto
```
```   677   qed simp
```
```   678 qed
```
```   679
```
```   680 lemma affine_borel_measurable:
```
```   681   fixes g :: "'a \<Rightarrow> real"
```
```   682   assumes g: "g \<in> borel_measurable M"
```
```   683   shows "(\<lambda>x. a + (g x) * b) \<in> borel_measurable M"
```
```   684   using affine_borel_measurable_vector[OF assms] by (simp add: mult_commute)
```
```   685
```
```   686 lemma borel_measurable_add[simp, intro]:
```
```   687   fixes f :: "'a \<Rightarrow> real"
```
```   688   assumes f: "f \<in> borel_measurable M"
```
```   689   assumes g: "g \<in> borel_measurable M"
```
```   690   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```   691 proof -
```
```   692   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}"
```
```   693     by auto
```
```   694   have "\<And>a. (\<lambda>w. a + (g w) * -1) \<in> borel_measurable M"
```
```   695     by (rule affine_borel_measurable [OF g])
```
```   696   then have "\<And>a. {w \<in> space M. (\<lambda>w. a + (g w) * -1)(w) \<le> f w} \<in> sets M" using f
```
```   697     by auto
```
```   698   then have "\<And>a. {w \<in> space M. a \<le> f w + g w} \<in> sets M"
```
```   699     by (simp add: 1)
```
```   700   then show ?thesis
```
```   701     by (simp add: borel_measurable_iff_ge)
```
```   702 qed
```
```   703
```
```   704 lemma borel_measurable_setsum[simp, intro]:
```
```   705   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
```
```   706   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   707   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```   708 proof cases
```
```   709   assume "finite S"
```
```   710   thus ?thesis using assms by induct auto
```
```   711 qed simp
```
```   712
```
```   713 lemma borel_measurable_square:
```
```   714   fixes f :: "'a \<Rightarrow> real"
```
```   715   assumes f: "f \<in> borel_measurable M"
```
```   716   shows "(\<lambda>x. (f x)^2) \<in> borel_measurable M"
```
```   717 proof -
```
```   718   {
```
```   719     fix a
```
```   720     have "{w \<in> space M. (f w)\<twosuperior> \<le> a} \<in> sets M"
```
```   721     proof (cases rule: linorder_cases [of a 0])
```
```   722       case less
```
```   723       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = {}"
```
```   724         by auto (metis less order_le_less_trans power2_less_0)
```
```   725       also have "... \<in> sets M"
```
```   726         by (rule empty_sets)
```
```   727       finally show ?thesis .
```
```   728     next
```
```   729       case equal
```
```   730       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
```
```   731              {w \<in> space M. f w \<le> 0} \<inter> {w \<in> space M. 0 \<le> f w}"
```
```   732         by auto
```
```   733       also have "... \<in> sets M"
```
```   734         apply (insert f)
```
```   735         apply (rule Int)
```
```   736         apply (simp add: borel_measurable_iff_le)
```
```   737         apply (simp add: borel_measurable_iff_ge)
```
```   738         done
```
```   739       finally show ?thesis .
```
```   740     next
```
```   741       case greater
```
```   742       have "\<forall>x. (f x ^ 2 \<le> sqrt a ^ 2) = (- sqrt a  \<le> f x & f x \<le> sqrt a)"
```
```   743         by (metis abs_le_interval_iff abs_of_pos greater real_sqrt_abs
```
```   744                   real_sqrt_le_iff real_sqrt_power)
```
```   745       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
```
```   746              {w \<in> space M. -(sqrt a) \<le> f w} \<inter> {w \<in> space M. f w \<le> sqrt a}"
```
```   747         using greater by auto
```
```   748       also have "... \<in> sets M"
```
```   749         apply (insert f)
```
```   750         apply (rule Int)
```
```   751         apply (simp add: borel_measurable_iff_ge)
```
```   752         apply (simp add: borel_measurable_iff_le)
```
```   753         done
```
```   754       finally show ?thesis .
```
```   755     qed
```
```   756   }
```
```   757   thus ?thesis by (auto simp add: borel_measurable_iff_le)
```
```   758 qed
```
```   759
```
```   760 lemma times_eq_sum_squares:
```
```   761    fixes x::real
```
```   762    shows"x*y = ((x+y)^2)/4 - ((x-y)^ 2)/4"
```
```   763 by (simp add: power2_eq_square ring_distribs diff_divide_distrib [symmetric])
```
```   764
```
```   765 lemma borel_measurable_uminus[simp, intro]:
```
```   766   fixes g :: "'a \<Rightarrow> real"
```
```   767   assumes g: "g \<in> borel_measurable M"
```
```   768   shows "(\<lambda>x. - g x) \<in> borel_measurable M"
```
```   769 proof -
```
```   770   have "(\<lambda>x. - g x) = (\<lambda>x. 0 + (g x) * -1)"
```
```   771     by simp
```
```   772   also have "... \<in> borel_measurable M"
```
```   773     by (fast intro: affine_borel_measurable g)
```
```   774   finally show ?thesis .
```
```   775 qed
```
```   776
```
```   777 lemma borel_measurable_times[simp, intro]:
```
```   778   fixes f :: "'a \<Rightarrow> real"
```
```   779   assumes f: "f \<in> borel_measurable M"
```
```   780   assumes g: "g \<in> borel_measurable M"
```
```   781   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```   782 proof -
```
```   783   have 1: "(\<lambda>x. 0 + (f x + g x)\<twosuperior> * inverse 4) \<in> borel_measurable M"
```
```   784     using assms by (fast intro: affine_borel_measurable borel_measurable_square)
```
```   785   have "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) =
```
```   786         (\<lambda>x. 0 + ((f x + -g x) ^ 2 * inverse -4))"
```
```   787     by (simp add: minus_divide_right)
```
```   788   also have "... \<in> borel_measurable M"
```
```   789     using f g by (fast intro: affine_borel_measurable borel_measurable_square f g)
```
```   790   finally have 2: "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) \<in> borel_measurable M" .
```
```   791   show ?thesis
```
```   792     apply (simp add: times_eq_sum_squares diff_minus)
```
```   793     using 1 2 by simp
```
```   794 qed
```
```   795
```
```   796 lemma borel_measurable_setprod[simp, intro]:
```
```   797   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
```
```   798   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```   799   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```   800 proof cases
```
```   801   assume "finite S"
```
```   802   thus ?thesis using assms by induct auto
```
```   803 qed simp
```
```   804
```
```   805 lemma borel_measurable_diff[simp, intro]:
```
```   806   fixes f :: "'a \<Rightarrow> real"
```
```   807   assumes f: "f \<in> borel_measurable M"
```
```   808   assumes g: "g \<in> borel_measurable M"
```
```   809   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```   810   unfolding diff_minus using assms by fast
```
```   811
```
```   812 lemma borel_measurable_inverse[simp, intro]:
```
```   813   fixes f :: "'a \<Rightarrow> real"
```
```   814   assumes "f \<in> borel_measurable M"
```
```   815   shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
```
```   816   unfolding borel_measurable_iff_ge unfolding inverse_eq_divide
```
```   817 proof safe
```
```   818   fix a :: real
```
```   819   have *: "{w \<in> space M. a \<le> 1 / f w} =
```
```   820       ({w \<in> space M. 0 < f w} \<inter> {w \<in> space M. a * f w \<le> 1}) \<union>
```
```   821       ({w \<in> space M. f w < 0} \<inter> {w \<in> space M. 1 \<le> a * f w}) \<union>
```
```   822       ({w \<in> space M. f w = 0} \<inter> {w \<in> space M. a \<le> 0})" by (auto simp: le_divide_eq)
```
```   823   show "{w \<in> space M. a \<le> 1 / f w} \<in> sets M" using assms unfolding *
```
```   824     by (auto intro!: Int Un)
```
```   825 qed
```
```   826
```
```   827 lemma borel_measurable_divide[simp, intro]:
```
```   828   fixes f :: "'a \<Rightarrow> real"
```
```   829   assumes "f \<in> borel_measurable M"
```
```   830   and "g \<in> borel_measurable M"
```
```   831   shows "(\<lambda>x. f x / g x) \<in> borel_measurable M"
```
```   832   unfolding field_divide_inverse
```
```   833   by (rule borel_measurable_inverse borel_measurable_times assms)+
```
```   834
```
```   835 lemma borel_measurable_max[intro, simp]:
```
```   836   fixes f g :: "'a \<Rightarrow> real"
```
```   837   assumes "f \<in> borel_measurable M"
```
```   838   assumes "g \<in> borel_measurable M"
```
```   839   shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
```
```   840   unfolding borel_measurable_iff_le
```
```   841 proof safe
```
```   842   fix a
```
```   843   have "{x \<in> space M. max (g x) (f x) \<le> a} =
```
```   844     {x \<in> space M. g x \<le> a} \<inter> {x \<in> space M. f x \<le> a}" by auto
```
```   845   thus "{x \<in> space M. max (g x) (f x) \<le> a} \<in> sets M"
```
```   846     using assms unfolding borel_measurable_iff_le
```
```   847     by (auto intro!: Int)
```
```   848 qed
```
```   849
```
```   850 lemma borel_measurable_min[intro, simp]:
```
```   851   fixes f g :: "'a \<Rightarrow> real"
```
```   852   assumes "f \<in> borel_measurable M"
```
```   853   assumes "g \<in> borel_measurable M"
```
```   854   shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
```
```   855   unfolding borel_measurable_iff_ge
```
```   856 proof safe
```
```   857   fix a
```
```   858   have "{x \<in> space M. a \<le> min (g x) (f x)} =
```
```   859     {x \<in> space M. a \<le> g x} \<inter> {x \<in> space M. a \<le> f x}" by auto
```
```   860   thus "{x \<in> space M. a \<le> min (g x) (f x)} \<in> sets M"
```
```   861     using assms unfolding borel_measurable_iff_ge
```
```   862     by (auto intro!: Int)
```
```   863 qed
```
```   864
```
```   865 lemma borel_measurable_abs[simp, intro]:
```
```   866   assumes "f \<in> borel_measurable M"
```
```   867   shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
```
```   868 proof -
```
```   869   have *: "\<And>x. \<bar>f x\<bar> = max 0 (f x) + max 0 (- f x)" by (simp add: max_def)
```
```   870   show ?thesis unfolding * using assms by auto
```
```   871 qed
```
```   872
```
```   873 lemma borel_measurable_nth[simp, intro]:
```
```   874   "(\<lambda>x::real^'n. x \$ i) \<in> borel_measurable borel"
```
```   875   using borel_measurable_euclidean_component
```
```   876   unfolding nth_conv_component by auto
```
```   877
```
```   878 lemma borel_measurable_continuous_on1:
```
```   879   fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
```
```   880   assumes "continuous_on UNIV f"
```
```   881   shows "f \<in> borel_measurable borel"
```
```   882   apply(rule borel_measurableI)
```
```   883   using continuous_open_preimage[OF assms] unfolding vimage_def by auto
```
```   884
```
```   885 lemma borel_measurable_continuous_on:
```
```   886   fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
```
```   887   assumes cont: "continuous_on A f" "open A"
```
```   888   shows "(\<lambda>x. if x \<in> A then f x else c) \<in> borel_measurable borel" (is "?f \<in> _")
```
```   889 proof (rule borel_measurableI)
```
```   890   fix S :: "'b set" assume "open S"
```
```   891   then have "open {x\<in>A. f x \<in> S}"
```
```   892     by (intro continuous_open_preimage[OF cont]) auto
```
```   893   then have *: "{x\<in>A. f x \<in> S} \<in> sets borel" by auto
```
```   894   have "?f -` S \<inter> space borel =
```
```   895     {x\<in>A. f x \<in> S} \<union> (if c \<in> S then space borel - A else {})"
```
```   896     by (auto split: split_if_asm)
```
```   897   also have "\<dots> \<in> sets borel"
```
```   898     using * `open A` by (auto simp del: space_borel intro!: Un)
```
```   899   finally show "?f -` S \<inter> space borel \<in> sets borel" .
```
```   900 qed
```
```   901
```
```   902 lemma convex_measurable:
```
```   903   fixes a b :: real
```
```   904   assumes X: "X \<in> borel_measurable M" "X ` space M \<subseteq> { a <..< b}"
```
```   905   assumes q: "convex_on { a <..< b} q"
```
```   906   shows "q \<circ> X \<in> borel_measurable M"
```
```   907 proof -
```
```   908   have "(\<lambda>x. if x \<in> {a <..< b} then q x else 0) \<in> borel_measurable borel"
```
```   909   proof (rule borel_measurable_continuous_on)
```
```   910     show "open {a<..<b}" by auto
```
```   911     from this q show "continuous_on {a<..<b} q"
```
```   912       by (rule convex_on_continuous)
```
```   913   qed
```
```   914   then have "(\<lambda>x. if x \<in> {a <..< b} then q x else 0) \<circ> X \<in> borel_measurable M" (is ?qX)
```
```   915     using X by (intro measurable_comp) auto
```
```   916   moreover have "?qX \<longleftrightarrow> q \<circ> X \<in> borel_measurable M"
```
```   917     using X by (intro measurable_cong) auto
```
```   918   ultimately show ?thesis by simp
```
```   919 qed
```
```   920
```
```   921 lemma borel_measurable_borel_log: assumes "1 < b" shows "log b \<in> borel_measurable borel"
```
```   922 proof -
```
```   923   { fix x :: real assume x: "x \<le> 0"
```
```   924     { fix x::real assume "x \<le> 0" then have "\<And>u. exp u = x \<longleftrightarrow> False" by auto }
```
```   925     from this[of x] x this[of 0] have "log b 0 = log b x"
```
```   926       by (auto simp: ln_def log_def) }
```
```   927   note log_imp = this
```
```   928   have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) \<in> borel_measurable borel"
```
```   929   proof (rule borel_measurable_continuous_on)
```
```   930     show "continuous_on {0<..} (log b)"
```
```   931       by (auto intro!: continuous_at_imp_continuous_on DERIV_log DERIV_isCont
```
```   932                simp: continuous_isCont[symmetric])
```
```   933     show "open ({0<..}::real set)" by auto
```
```   934   qed
```
```   935   also have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) = log b"
```
```   936     by (simp add: fun_eq_iff not_less log_imp)
```
```   937   finally show ?thesis .
```
```   938 qed
```
```   939
```
```   940 lemma borel_measurable_log[simp,intro]:
```
```   941   assumes f: "f \<in> borel_measurable M" and "1 < b"
```
```   942   shows "(\<lambda>x. log b (f x)) \<in> borel_measurable M"
```
```   943   using measurable_comp[OF f borel_measurable_borel_log[OF `1 < b`]]
```
```   944   by (simp add: comp_def)
```
```   945
```
```   946 lemma borel_measurable_real_floor:
```
```   947   "(\<lambda>x::real. real \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
```
```   948   unfolding borel_measurable_iff_ge
```
```   949 proof (intro allI)
```
```   950   fix a :: real
```
```   951   { fix x have "a \<le> real \<lfloor>x\<rfloor> \<longleftrightarrow> real \<lceil>a\<rceil> \<le> x"
```
```   952       using le_floor_eq[of "\<lceil>a\<rceil>" x] ceiling_le_iff[of a "\<lfloor>x\<rfloor>"]
```
```   953       unfolding real_eq_of_int by simp }
```
```   954   then have "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} = {real \<lceil>a\<rceil>..}" by auto
```
```   955   then show "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} \<in> sets borel" by auto
```
```   956 qed
```
```   957
```
```   958 lemma borel_measurable_real_natfloor[intro, simp]:
```
```   959   assumes "f \<in> borel_measurable M"
```
```   960   shows "(\<lambda>x. real (natfloor (f x))) \<in> borel_measurable M"
```
```   961 proof -
```
```   962   have "\<And>x. real (natfloor (f x)) = max 0 (real \<lfloor>f x\<rfloor>)"
```
```   963     by (auto simp: max_def natfloor_def)
```
```   964   with borel_measurable_max[OF measurable_comp[OF assms borel_measurable_real_floor] borel_measurable_const]
```
```   965   show ?thesis by (simp add: comp_def)
```
```   966 qed
```
```   967
```
```   968 subsection "Borel space on the extended reals"
```
```   969
```
```   970 lemma borel_measurable_ereal_borel:
```
```   971   "ereal \<in> borel_measurable borel"
```
```   972 proof (rule borel_measurableI)
```
```   973   fix X :: "ereal set" assume "open X"
```
```   974   then have "open (ereal -` X \<inter> space borel)"
```
```   975     by (simp add: open_ereal_vimage)
```
```   976   then show "ereal -` X \<inter> space borel \<in> sets borel" by auto
```
```   977 qed
```
```   978
```
```   979 lemma borel_measurable_ereal[simp, intro]:
```
```   980   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
```
```   981   using measurable_comp[OF f borel_measurable_ereal_borel] unfolding comp_def .
```
```   982
```
```   983 lemma borel_measurable_real_of_ereal_borel:
```
```   984   "(real :: ereal \<Rightarrow> real) \<in> borel_measurable borel"
```
```   985 proof (rule borel_measurableI)
```
```   986   fix B :: "real set" assume "open B"
```
```   987   have *: "ereal -` real -` (B - {0}) = B - {0}" by auto
```
```   988   have open_real: "open (real -` (B - {0}) :: ereal set)"
```
```   989     unfolding open_ereal_def * using `open B` by auto
```
```   990   show "(real -` B \<inter> space borel :: ereal set) \<in> sets borel"
```
```   991   proof cases
```
```   992     assume "0 \<in> B"
```
```   993     then have *: "real -` B = real -` (B - {0}) \<union> {-\<infinity>, \<infinity>, 0::ereal}"
```
```   994       by (auto simp add: real_of_ereal_eq_0)
```
```   995     then show "(real -` B :: ereal set) \<inter> space borel \<in> sets borel"
```
```   996       using open_real by auto
```
```   997   next
```
```   998     assume "0 \<notin> B"
```
```   999     then have *: "(real -` B :: ereal set) = real -` (B - {0})"
```
```  1000       by (auto simp add: real_of_ereal_eq_0)
```
```  1001     then show "(real -` B :: ereal set) \<inter> space borel \<in> sets borel"
```
```  1002       using open_real by auto
```
```  1003   qed
```
```  1004 qed
```
```  1005
```
```  1006 lemma borel_measurable_real_of_ereal[simp, intro]:
```
```  1007   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. real (f x :: ereal)) \<in> borel_measurable M"
```
```  1008   using measurable_comp[OF f borel_measurable_real_of_ereal_borel] unfolding comp_def .
```
```  1009
```
```  1010 lemma borel_measurable_ereal_iff:
```
```  1011   shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
```
```  1012 proof
```
```  1013   assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
```
```  1014   from borel_measurable_real_of_ereal[OF this]
```
```  1015   show "f \<in> borel_measurable M" by auto
```
```  1016 qed auto
```
```  1017
```
```  1018 lemma borel_measurable_ereal_iff_real:
```
```  1019   fixes f :: "'a \<Rightarrow> ereal"
```
```  1020   shows "f \<in> borel_measurable M \<longleftrightarrow>
```
```  1021     ((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)"
```
```  1022 proof safe
```
```  1023   assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M"
```
```  1024   have "f -` {\<infinity>} \<inter> space M = {x\<in>space M. f x = \<infinity>}" "f -` {-\<infinity>} \<inter> space M = {x\<in>space M. f x = -\<infinity>}" by auto
```
```  1025   with * have **: "{x\<in>space M. f x = \<infinity>} \<in> sets M" "{x\<in>space M. f x = -\<infinity>} \<in> sets M" by simp_all
```
```  1026   let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real (f x))"
```
```  1027   have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
```
```  1028   also have "?f = f" by (auto simp: fun_eq_iff ereal_real)
```
```  1029   finally show "f \<in> borel_measurable M" .
```
```  1030 qed (auto intro: measurable_sets borel_measurable_real_of_ereal)
```
```  1031
```
```  1032 lemma less_eq_ge_measurable:
```
```  1033   fixes f :: "'a \<Rightarrow> 'c::linorder"
```
```  1034   shows "f -` {a <..} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {..a} \<inter> space M \<in> sets M"
```
```  1035 proof
```
```  1036   assume "f -` {a <..} \<inter> space M \<in> sets M"
```
```  1037   moreover have "f -` {..a} \<inter> space M = space M - f -` {a <..} \<inter> space M" by auto
```
```  1038   ultimately show "f -` {..a} \<inter> space M \<in> sets M" by auto
```
```  1039 next
```
```  1040   assume "f -` {..a} \<inter> space M \<in> sets M"
```
```  1041   moreover have "f -` {a <..} \<inter> space M = space M - f -` {..a} \<inter> space M" by auto
```
```  1042   ultimately show "f -` {a <..} \<inter> space M \<in> sets M" by auto
```
```  1043 qed
```
```  1044
```
```  1045 lemma greater_eq_le_measurable:
```
```  1046   fixes f :: "'a \<Rightarrow> 'c::linorder"
```
```  1047   shows "f -` {..< a} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {a ..} \<inter> space M \<in> sets M"
```
```  1048 proof
```
```  1049   assume "f -` {a ..} \<inter> space M \<in> sets M"
```
```  1050   moreover have "f -` {..< a} \<inter> space M = space M - f -` {a ..} \<inter> space M" by auto
```
```  1051   ultimately show "f -` {..< a} \<inter> space M \<in> sets M" by auto
```
```  1052 next
```
```  1053   assume "f -` {..< a} \<inter> space M \<in> sets M"
```
```  1054   moreover have "f -` {a ..} \<inter> space M = space M - f -` {..< a} \<inter> space M" by auto
```
```  1055   ultimately show "f -` {a ..} \<inter> space M \<in> sets M" by auto
```
```  1056 qed
```
```  1057
```
```  1058 lemma borel_measurable_uminus_borel_ereal:
```
```  1059   "(uminus :: ereal \<Rightarrow> ereal) \<in> borel_measurable borel"
```
```  1060 proof (rule borel_measurableI)
```
```  1061   fix X :: "ereal set" assume "open X"
```
```  1062   have "uminus -` X = uminus ` X" by (force simp: image_iff)
```
```  1063   then have "open (uminus -` X)" using `open X` ereal_open_uminus by auto
```
```  1064   then show "uminus -` X \<inter> space borel \<in> sets borel" by auto
```
```  1065 qed
```
```  1066
```
```  1067 lemma borel_measurable_uminus_ereal[intro]:
```
```  1068   assumes "f \<in> borel_measurable M"
```
```  1069   shows "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M"
```
```  1070   using measurable_comp[OF assms borel_measurable_uminus_borel_ereal] by (simp add: comp_def)
```
```  1071
```
```  1072 lemma borel_measurable_uminus_eq_ereal[simp]:
```
```  1073   "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
```
```  1074 proof
```
```  1075   assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp
```
```  1076 qed auto
```
```  1077
```
```  1078 lemma borel_measurable_eq_atMost_ereal:
```
```  1079   fixes f :: "'a \<Rightarrow> ereal"
```
```  1080   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
```
```  1081 proof (intro iffI allI)
```
```  1082   assume pos[rule_format]: "\<forall>a. f -` {..a} \<inter> space M \<in> sets M"
```
```  1083   show "f \<in> borel_measurable M"
```
```  1084     unfolding borel_measurable_ereal_iff_real borel_measurable_iff_le
```
```  1085   proof (intro conjI allI)
```
```  1086     fix a :: real
```
```  1087     { fix x :: ereal assume *: "\<forall>i::nat. real i < x"
```
```  1088       have "x = \<infinity>"
```
```  1089       proof (rule ereal_top)
```
```  1090         fix B from reals_Archimedean2[of B] guess n ..
```
```  1091         then have "ereal B < real n" by auto
```
```  1092         with * show "B \<le> x" by (metis less_trans less_imp_le)
```
```  1093       qed }
```
```  1094     then have "f -` {\<infinity>} \<inter> space M = space M - (\<Union>i::nat. f -` {.. real i} \<inter> space M)"
```
```  1095       by (auto simp: not_le)
```
```  1096     then show "f -` {\<infinity>} \<inter> space M \<in> sets M" using pos by (auto simp del: UN_simps intro!: Diff)
```
```  1097     moreover
```
```  1098     have "{-\<infinity>::ereal} = {..-\<infinity>}" by auto
```
```  1099     then show "f -` {-\<infinity>} \<inter> space M \<in> sets M" using pos by auto
```
```  1100     moreover have "{x\<in>space M. f x \<le> ereal a} \<in> sets M"
```
```  1101       using pos[of "ereal a"] by (simp add: vimage_def Int_def conj_commute)
```
```  1102     moreover have "{w \<in> space M. real (f w) \<le> a} =
```
```  1103       (if a < 0 then {w \<in> space M. f w \<le> ereal a} - f -` {-\<infinity>} \<inter> space M
```
```  1104       else {w \<in> space M. f w \<le> ereal a} \<union> (f -` {\<infinity>} \<inter> space M) \<union> (f -` {-\<infinity>} \<inter> space M))" (is "?l = ?r")
```
```  1105       proof (intro set_eqI) fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases "f x") auto qed
```
```  1106     ultimately show "{w \<in> space M. real (f w) \<le> a} \<in> sets M" by auto
```
```  1107   qed
```
```  1108 qed (simp add: measurable_sets)
```
```  1109
```
```  1110 lemma borel_measurable_eq_atLeast_ereal:
```
```  1111   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
```
```  1112 proof
```
```  1113   assume pos: "\<forall>a. f -` {a..} \<inter> space M \<in> sets M"
```
```  1114   moreover have "\<And>a. (\<lambda>x. - f x) -` {..a} = f -` {-a ..}"
```
```  1115     by (auto simp: ereal_uminus_le_reorder)
```
```  1116   ultimately have "(\<lambda>x. - f x) \<in> borel_measurable M"
```
```  1117     unfolding borel_measurable_eq_atMost_ereal by auto
```
```  1118   then show "f \<in> borel_measurable M" by simp
```
```  1119 qed (simp add: measurable_sets)
```
```  1120
```
```  1121 lemma borel_measurable_ereal_iff_less:
```
```  1122   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
```
```  1123   unfolding borel_measurable_eq_atLeast_ereal greater_eq_le_measurable ..
```
```  1124
```
```  1125 lemma borel_measurable_ereal_iff_ge:
```
```  1126   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
```
```  1127   unfolding borel_measurable_eq_atMost_ereal less_eq_ge_measurable ..
```
```  1128
```
```  1129 lemma borel_measurable_ereal_eq_const:
```
```  1130   fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M"
```
```  1131   shows "{x\<in>space M. f x = c} \<in> sets M"
```
```  1132 proof -
```
```  1133   have "{x\<in>space M. f x = c} = (f -` {c} \<inter> space M)" by auto
```
```  1134   then show ?thesis using assms by (auto intro!: measurable_sets)
```
```  1135 qed
```
```  1136
```
```  1137 lemma borel_measurable_ereal_neq_const:
```
```  1138   fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M"
```
```  1139   shows "{x\<in>space M. f x \<noteq> c} \<in> sets M"
```
```  1140 proof -
```
```  1141   have "{x\<in>space M. f x \<noteq> c} = space M - (f -` {c} \<inter> space M)" by auto
```
```  1142   then show ?thesis using assms by (auto intro!: measurable_sets)
```
```  1143 qed
```
```  1144
```
```  1145 lemma borel_measurable_ereal_le[intro,simp]:
```
```  1146   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1147   assumes f: "f \<in> borel_measurable M"
```
```  1148   assumes g: "g \<in> borel_measurable M"
```
```  1149   shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
```
```  1150 proof -
```
```  1151   have "{x \<in> space M. f x \<le> g x} =
```
```  1152     {x \<in> space M. real (f x) \<le> real (g x)} - (f -` {\<infinity>, -\<infinity>} \<inter> space M \<union> g -` {\<infinity>, -\<infinity>} \<inter> space M) \<union>
```
```  1153     f -` {-\<infinity>} \<inter> space M \<union> g -` {\<infinity>} \<inter> space M" (is "?l = ?r")
```
```  1154   proof (intro set_eqI)
```
```  1155     fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases rule: ereal2_cases[of "f x" "g x"]) auto
```
```  1156   qed
```
```  1157   with f g show ?thesis by (auto intro!: Un simp: measurable_sets)
```
```  1158 qed
```
```  1159
```
```  1160 lemma borel_measurable_ereal_less[intro,simp]:
```
```  1161   fixes f :: "'a \<Rightarrow> ereal"
```
```  1162   assumes f: "f \<in> borel_measurable M"
```
```  1163   assumes g: "g \<in> borel_measurable M"
```
```  1164   shows "{x \<in> space M. f x < g x} \<in> sets M"
```
```  1165 proof -
```
```  1166   have "{x \<in> space M. f x < g x} = space M - {x \<in> space M. g x \<le> f x}" by auto
```
```  1167   then show ?thesis using g f by auto
```
```  1168 qed
```
```  1169
```
```  1170 lemma borel_measurable_ereal_eq[intro,simp]:
```
```  1171   fixes f :: "'a \<Rightarrow> ereal"
```
```  1172   assumes f: "f \<in> borel_measurable M"
```
```  1173   assumes g: "g \<in> borel_measurable M"
```
```  1174   shows "{w \<in> space M. f w = g w} \<in> sets M"
```
```  1175 proof -
```
```  1176   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
```
```  1177   then show ?thesis using g f by auto
```
```  1178 qed
```
```  1179
```
```  1180 lemma borel_measurable_ereal_neq[intro,simp]:
```
```  1181   fixes f :: "'a \<Rightarrow> ereal"
```
```  1182   assumes f: "f \<in> borel_measurable M"
```
```  1183   assumes g: "g \<in> borel_measurable M"
```
```  1184   shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
```
```  1185 proof -
```
```  1186   have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}" by auto
```
```  1187   thus ?thesis using f g by auto
```
```  1188 qed
```
```  1189
```
```  1190 lemma split_sets:
```
```  1191   "{x\<in>space M. P x \<or> Q x} = {x\<in>space M. P x} \<union> {x\<in>space M. Q x}"
```
```  1192   "{x\<in>space M. P x \<and> Q x} = {x\<in>space M. P x} \<inter> {x\<in>space M. Q x}"
```
```  1193   by auto
```
```  1194
```
```  1195 lemma borel_measurable_ereal_add[intro, simp]:
```
```  1196   fixes f :: "'a \<Rightarrow> ereal"
```
```  1197   assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1198   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
```
```  1199 proof -
```
```  1200   { fix x assume "x \<in> space M" then have "f x + g x =
```
```  1201       (if f x = \<infinity> \<or> g x = \<infinity> then \<infinity>
```
```  1202         else if f x = -\<infinity> \<or> g x = -\<infinity> then -\<infinity>
```
```  1203         else ereal (real (f x) + real (g x)))"
```
```  1204       by (cases rule: ereal2_cases[of "f x" "g x"]) auto }
```
```  1205   with assms show ?thesis
```
```  1206     by (auto cong: measurable_cong simp: split_sets
```
```  1207              intro!: Un measurable_If measurable_sets)
```
```  1208 qed
```
```  1209
```
```  1210 lemma borel_measurable_ereal_setsum[simp, intro]:
```
```  1211   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1212   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1213   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1214 proof cases
```
```  1215   assume "finite S"
```
```  1216   thus ?thesis using assms
```
```  1217     by induct auto
```
```  1218 qed (simp add: borel_measurable_const)
```
```  1219
```
```  1220 lemma borel_measurable_ereal_abs[intro, simp]:
```
```  1221   fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M"
```
```  1222   shows "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
```
```  1223 proof -
```
```  1224   { fix x have "\<bar>f x\<bar> = (if 0 \<le> f x then f x else - f x)" by auto }
```
```  1225   then show ?thesis using assms by (auto intro!: measurable_If)
```
```  1226 qed
```
```  1227
```
```  1228 lemma borel_measurable_ereal_times[intro, simp]:
```
```  1229   fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1230   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1231 proof -
```
```  1232   { fix f g :: "'a \<Rightarrow> ereal"
```
```  1233     assume b: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1234       and pos: "\<And>x. 0 \<le> f x" "\<And>x. 0 \<le> g x"
```
```  1235     { fix x have *: "f x * g x = (if f x = 0 \<or> g x = 0 then 0
```
```  1236         else if f x = \<infinity> \<or> g x = \<infinity> then \<infinity>
```
```  1237         else ereal (real (f x) * real (g x)))"
```
```  1238       apply (cases rule: ereal2_cases[of "f x" "g x"])
```
```  1239       using pos[of x] by auto }
```
```  1240     with b have "(\<lambda>x. f x * g x) \<in> borel_measurable M"
```
```  1241       by (auto cong: measurable_cong simp: split_sets
```
```  1242                intro!: Un measurable_If measurable_sets) }
```
```  1243   note pos_times = this
```
```  1244   have *: "(\<lambda>x. f x * g x) =
```
```  1245     (\<lambda>x. if 0 \<le> f x \<and> 0 \<le> g x \<or> f x < 0 \<and> g x < 0 then \<bar>f x\<bar> * \<bar>g x\<bar> else - (\<bar>f x\<bar> * \<bar>g x\<bar>))"
```
```  1246     by (auto simp: fun_eq_iff)
```
```  1247   show ?thesis using assms unfolding *
```
```  1248     by (intro measurable_If pos_times borel_measurable_uminus_ereal)
```
```  1249        (auto simp: split_sets intro!: Int)
```
```  1250 qed
```
```  1251
```
```  1252 lemma borel_measurable_ereal_setprod[simp, intro]:
```
```  1253   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1254   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1255   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
```
```  1256 proof cases
```
```  1257   assume "finite S"
```
```  1258   thus ?thesis using assms by induct auto
```
```  1259 qed simp
```
```  1260
```
```  1261 lemma borel_measurable_ereal_min[simp, intro]:
```
```  1262   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1263   assumes "f \<in> borel_measurable M"
```
```  1264   assumes "g \<in> borel_measurable M"
```
```  1265   shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
```
```  1266   using assms unfolding min_def by (auto intro!: measurable_If)
```
```  1267
```
```  1268 lemma borel_measurable_ereal_max[simp, intro]:
```
```  1269   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1270   assumes "f \<in> borel_measurable M"
```
```  1271   and "g \<in> borel_measurable M"
```
```  1272   shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
```
```  1273   using assms unfolding max_def by (auto intro!: measurable_If)
```
```  1274
```
```  1275 lemma borel_measurable_SUP[simp, intro]:
```
```  1276   fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1277   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1278   shows "(\<lambda>x. SUP i : A. f i x) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M")
```
```  1279   unfolding borel_measurable_ereal_iff_ge
```
```  1280 proof
```
```  1281   fix a
```
```  1282   have "?sup -` {a<..} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"
```
```  1283     by (auto simp: less_SUP_iff)
```
```  1284   then show "?sup -` {a<..} \<inter> space M \<in> sets M"
```
```  1285     using assms by auto
```
```  1286 qed
```
```  1287
```
```  1288 lemma borel_measurable_INF[simp, intro]:
```
```  1289   fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1290   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1291   shows "(\<lambda>x. INF i : A. f i x) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M")
```
```  1292   unfolding borel_measurable_ereal_iff_less
```
```  1293 proof
```
```  1294   fix a
```
```  1295   have "?inf -` {..<a} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"
```
```  1296     by (auto simp: INF_less_iff)
```
```  1297   then show "?inf -` {..<a} \<inter> space M \<in> sets M"
```
```  1298     using assms by auto
```
```  1299 qed
```
```  1300
```
```  1301 lemma borel_measurable_liminf[simp, intro]:
```
```  1302   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1303   assumes "\<And>i. f i \<in> borel_measurable M"
```
```  1304   shows "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1305   unfolding liminf_SUPR_INFI using assms by auto
```
```  1306
```
```  1307 lemma borel_measurable_limsup[simp, intro]:
```
```  1308   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1309   assumes "\<And>i. f i \<in> borel_measurable M"
```
```  1310   shows "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
```
```  1311   unfolding limsup_INFI_SUPR using assms by auto
```
```  1312
```
```  1313 lemma borel_measurable_ereal_diff[simp, intro]:
```
```  1314   fixes f g :: "'a \<Rightarrow> ereal"
```
```  1315   assumes "f \<in> borel_measurable M"
```
```  1316   assumes "g \<in> borel_measurable M"
```
```  1317   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```  1318   unfolding minus_ereal_def using assms by auto
```
```  1319
```
```  1320 lemma borel_measurable_ereal_inverse[simp, intro]:
```
```  1321   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M"
```
```  1322 proof -
```
```  1323   { fix x have "inverse (f x) = (if f x = 0 then \<infinity> else ereal (inverse (real (f x))))"
```
```  1324       by (cases "f x") auto }
```
```  1325   with f show ?thesis
```
```  1326     by (auto intro!: measurable_If)
```
```  1327 qed
```
```  1328
```
```  1329 lemma borel_measurable_ereal_divide[simp, intro]:
```
```  1330   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. f x / g x :: ereal) \<in> borel_measurable M"
```
```  1331   unfolding divide_ereal_def by auto
```
```  1332
```
```  1333 lemma borel_measurable_psuminf[simp, intro]:
```
```  1334   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1335   assumes "\<And>i. f i \<in> borel_measurable M" and pos: "\<And>i x. x \<in> space M \<Longrightarrow> 0 \<le> f i x"
```
```  1336   shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
```
```  1337   apply (subst measurable_cong)
```
```  1338   apply (subst suminf_ereal_eq_SUPR)
```
```  1339   apply (rule pos)
```
```  1340   using assms by auto
```
```  1341
```
```  1342 section "LIMSEQ is borel measurable"
```
```  1343
```
```  1344 lemma borel_measurable_LIMSEQ:
```
```  1345   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
```
```  1346   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
```
```  1347   and u: "\<And>i. u i \<in> borel_measurable M"
```
```  1348   shows "u' \<in> borel_measurable M"
```
```  1349 proof -
```
```  1350   have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)"
```
```  1351     using u' by (simp add: lim_imp_Liminf)
```
```  1352   moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M"
```
```  1353     by auto
```
```  1354   ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff)
```
```  1355 qed
```
```  1356
```
```  1357 end
```