src/HOL/Probability/Probability_Measure.thy
 author hoelzl Fri Sep 16 13:56:51 2016 +0200 (2016-09-16) changeset 63886 685fb01256af parent 63627 6ddb43c6b711 child 64008 17a20ca86d62 permissions -rw-r--r--
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
```     1 (*  Title:      HOL/Probability/Probability_Measure.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Armin Heller, TU München
```
```     4 *)
```
```     5
```
```     6 section \<open>Probability measure\<close>
```
```     7
```
```     8 theory Probability_Measure
```
```     9   imports "~~/src/HOL/Analysis/Analysis"
```
```    10 begin
```
```    11
```
```    12 locale prob_space = finite_measure +
```
```    13   assumes emeasure_space_1: "emeasure M (space M) = 1"
```
```    14
```
```    15 lemma prob_spaceI[Pure.intro!]:
```
```    16   assumes *: "emeasure M (space M) = 1"
```
```    17   shows "prob_space M"
```
```    18 proof -
```
```    19   interpret finite_measure M
```
```    20   proof
```
```    21     show "emeasure M (space M) \<noteq> \<infinity>" using * by simp
```
```    22   qed
```
```    23   show "prob_space M" by standard fact
```
```    24 qed
```
```    25
```
```    26 lemma prob_space_imp_sigma_finite: "prob_space M \<Longrightarrow> sigma_finite_measure M"
```
```    27   unfolding prob_space_def finite_measure_def by simp
```
```    28
```
```    29 abbreviation (in prob_space) "events \<equiv> sets M"
```
```    30 abbreviation (in prob_space) "prob \<equiv> measure M"
```
```    31 abbreviation (in prob_space) "random_variable M' X \<equiv> X \<in> measurable M M'"
```
```    32 abbreviation (in prob_space) "expectation \<equiv> integral\<^sup>L M"
```
```    33 abbreviation (in prob_space) "variance X \<equiv> integral\<^sup>L M (\<lambda>x. (X x - expectation X)\<^sup>2)"
```
```    34
```
```    35 lemma (in prob_space) finite_measure [simp]: "finite_measure M"
```
```    36   by unfold_locales
```
```    37
```
```    38 lemma (in prob_space) prob_space_distr:
```
```    39   assumes f: "f \<in> measurable M M'" shows "prob_space (distr M M' f)"
```
```    40 proof (rule prob_spaceI)
```
```    41   have "f -` space M' \<inter> space M = space M" using f by (auto dest: measurable_space)
```
```    42   with f show "emeasure (distr M M' f) (space (distr M M' f)) = 1"
```
```    43     by (auto simp: emeasure_distr emeasure_space_1)
```
```    44 qed
```
```    45
```
```    46 lemma prob_space_distrD:
```
```    47   assumes f: "f \<in> measurable M N" and M: "prob_space (distr M N f)" shows "prob_space M"
```
```    48 proof
```
```    49   interpret M: prob_space "distr M N f" by fact
```
```    50   have "f -` space N \<inter> space M = space M"
```
```    51     using f[THEN measurable_space] by auto
```
```    52   then show "emeasure M (space M) = 1"
```
```    53     using M.emeasure_space_1 by (simp add: emeasure_distr[OF f])
```
```    54 qed
```
```    55
```
```    56 lemma (in prob_space) prob_space: "prob (space M) = 1"
```
```    57   using emeasure_space_1 unfolding measure_def by (simp add: one_ennreal.rep_eq)
```
```    58
```
```    59 lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1"
```
```    60   using bounded_measure[of A] by (simp add: prob_space)
```
```    61
```
```    62 lemma (in prob_space) not_empty: "space M \<noteq> {}"
```
```    63   using prob_space by auto
```
```    64
```
```    65 lemma (in prob_space) emeasure_eq_1_AE:
```
```    66   "S \<in> sets M \<Longrightarrow> AE x in M. x \<in> S \<Longrightarrow> emeasure M S = 1"
```
```    67   by (subst emeasure_eq_AE[where B="space M"]) (auto simp: emeasure_space_1)
```
```    68
```
```    69 lemma (in prob_space) emeasure_le_1: "emeasure M S \<le> 1"
```
```    70   unfolding ennreal_1[symmetric] emeasure_eq_measure by (subst ennreal_le_iff) auto
```
```    71
```
```    72 lemma (in prob_space) emeasure_ge_1_iff: "emeasure M A \<ge> 1 \<longleftrightarrow> emeasure M A = 1"
```
```    73   by (rule iffI, intro antisym emeasure_le_1) simp_all
```
```    74
```
```    75 lemma (in prob_space) AE_iff_emeasure_eq_1:
```
```    76   assumes [measurable]: "Measurable.pred M P"
```
```    77   shows "(AE x in M. P x) \<longleftrightarrow> emeasure M {x\<in>space M. P x} = 1"
```
```    78 proof -
```
```    79   have *: "{x \<in> space M. \<not> P x} = space M - {x\<in>space M. P x}"
```
```    80     by auto
```
```    81   show ?thesis
```
```    82     by (auto simp add: ennreal_minus_eq_0 * emeasure_compl emeasure_space_1 AE_iff_measurable[OF _ refl]
```
```    83              intro: antisym emeasure_le_1)
```
```    84 qed
```
```    85
```
```    86 lemma (in prob_space) measure_le_1: "emeasure M X \<le> 1"
```
```    87   using emeasure_space[of M X] by (simp add: emeasure_space_1)
```
```    88
```
```    89 lemma (in prob_space) measure_ge_1_iff: "measure M A \<ge> 1 \<longleftrightarrow> measure M A = 1"
```
```    90   by (auto intro!: antisym)
```
```    91
```
```    92 lemma (in prob_space) AE_I_eq_1:
```
```    93   assumes "emeasure M {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M"
```
```    94   shows "AE x in M. P x"
```
```    95 proof (rule AE_I)
```
```    96   show "emeasure M (space M - {x \<in> space M. P x}) = 0"
```
```    97     using assms emeasure_space_1 by (simp add: emeasure_compl)
```
```    98 qed (insert assms, auto)
```
```    99
```
```   100 lemma prob_space_restrict_space:
```
```   101   "S \<in> sets M \<Longrightarrow> emeasure M S = 1 \<Longrightarrow> prob_space (restrict_space M S)"
```
```   102   by (intro prob_spaceI)
```
```   103      (simp add: emeasure_restrict_space space_restrict_space)
```
```   104
```
```   105 lemma (in prob_space) prob_compl:
```
```   106   assumes A: "A \<in> events"
```
```   107   shows "prob (space M - A) = 1 - prob A"
```
```   108   using finite_measure_compl[OF A] by (simp add: prob_space)
```
```   109
```
```   110 lemma (in prob_space) AE_in_set_eq_1:
```
```   111   assumes A[measurable]: "A \<in> events" shows "(AE x in M. x \<in> A) \<longleftrightarrow> prob A = 1"
```
```   112 proof -
```
```   113   have *: "{x\<in>space M. x \<in> A} = A"
```
```   114     using A[THEN sets.sets_into_space] by auto
```
```   115   show ?thesis
```
```   116     by (subst AE_iff_emeasure_eq_1) (auto simp: emeasure_eq_measure *)
```
```   117 qed
```
```   118
```
```   119 lemma (in prob_space) AE_False: "(AE x in M. False) \<longleftrightarrow> False"
```
```   120 proof
```
```   121   assume "AE x in M. False"
```
```   122   then have "AE x in M. x \<in> {}" by simp
```
```   123   then show False
```
```   124     by (subst (asm) AE_in_set_eq_1) auto
```
```   125 qed simp
```
```   126
```
```   127 lemma (in prob_space) AE_prob_1:
```
```   128   assumes "prob A = 1" shows "AE x in M. x \<in> A"
```
```   129 proof -
```
```   130   from \<open>prob A = 1\<close> have "A \<in> events"
```
```   131     by (metis measure_notin_sets zero_neq_one)
```
```   132   with AE_in_set_eq_1 assms show ?thesis by simp
```
```   133 qed
```
```   134
```
```   135 lemma (in prob_space) AE_const[simp]: "(AE x in M. P) \<longleftrightarrow> P"
```
```   136   by (cases P) (auto simp: AE_False)
```
```   137
```
```   138 lemma (in prob_space) ae_filter_bot: "ae_filter M \<noteq> bot"
```
```   139   by (simp add: trivial_limit_def)
```
```   140
```
```   141 lemma (in prob_space) AE_contr:
```
```   142   assumes ae: "AE \<omega> in M. P \<omega>" "AE \<omega> in M. \<not> P \<omega>"
```
```   143   shows False
```
```   144 proof -
```
```   145   from ae have "AE \<omega> in M. False" by eventually_elim auto
```
```   146   then show False by auto
```
```   147 qed
```
```   148
```
```   149 lemma (in prob_space) integral_ge_const:
```
```   150   fixes c :: real
```
```   151   shows "integrable M f \<Longrightarrow> (AE x in M. c \<le> f x) \<Longrightarrow> c \<le> (\<integral>x. f x \<partial>M)"
```
```   152   using integral_mono_AE[of M "\<lambda>x. c" f] prob_space by simp
```
```   153
```
```   154 lemma (in prob_space) integral_le_const:
```
```   155   fixes c :: real
```
```   156   shows "integrable M f \<Longrightarrow> (AE x in M. f x \<le> c) \<Longrightarrow> (\<integral>x. f x \<partial>M) \<le> c"
```
```   157   using integral_mono_AE[of M f "\<lambda>x. c"] prob_space by simp
```
```   158
```
```   159 lemma (in prob_space) nn_integral_ge_const:
```
```   160   "(AE x in M. c \<le> f x) \<Longrightarrow> c \<le> (\<integral>\<^sup>+x. f x \<partial>M)"
```
```   161   using nn_integral_mono_AE[of "\<lambda>x. c" f M] emeasure_space_1
```
```   162   by (simp split: if_split_asm)
```
```   163
```
```   164 lemma (in prob_space) expectation_less:
```
```   165   fixes X :: "_ \<Rightarrow> real"
```
```   166   assumes [simp]: "integrable M X"
```
```   167   assumes gt: "AE x in M. X x < b"
```
```   168   shows "expectation X < b"
```
```   169 proof -
```
```   170   have "expectation X < expectation (\<lambda>x. b)"
```
```   171     using gt emeasure_space_1
```
```   172     by (intro integral_less_AE_space) auto
```
```   173   then show ?thesis using prob_space by simp
```
```   174 qed
```
```   175
```
```   176 lemma (in prob_space) expectation_greater:
```
```   177   fixes X :: "_ \<Rightarrow> real"
```
```   178   assumes [simp]: "integrable M X"
```
```   179   assumes gt: "AE x in M. a < X x"
```
```   180   shows "a < expectation X"
```
```   181 proof -
```
```   182   have "expectation (\<lambda>x. a) < expectation X"
```
```   183     using gt emeasure_space_1
```
```   184     by (intro integral_less_AE_space) auto
```
```   185   then show ?thesis using prob_space by simp
```
```   186 qed
```
```   187
```
```   188 lemma (in prob_space) jensens_inequality:
```
```   189   fixes q :: "real \<Rightarrow> real"
```
```   190   assumes X: "integrable M X" "AE x in M. X x \<in> I"
```
```   191   assumes I: "I = {a <..< b} \<or> I = {a <..} \<or> I = {..< b} \<or> I = UNIV"
```
```   192   assumes q: "integrable M (\<lambda>x. q (X x))" "convex_on I q"
```
```   193   shows "q (expectation X) \<le> expectation (\<lambda>x. q (X x))"
```
```   194 proof -
```
```   195   let ?F = "\<lambda>x. Inf ((\<lambda>t. (q x - q t) / (x - t)) ` ({x<..} \<inter> I))"
```
```   196   from X(2) AE_False have "I \<noteq> {}" by auto
```
```   197
```
```   198   from I have "open I" by auto
```
```   199
```
```   200   note I
```
```   201   moreover
```
```   202   { assume "I \<subseteq> {a <..}"
```
```   203     with X have "a < expectation X"
```
```   204       by (intro expectation_greater) auto }
```
```   205   moreover
```
```   206   { assume "I \<subseteq> {..< b}"
```
```   207     with X have "expectation X < b"
```
```   208       by (intro expectation_less) auto }
```
```   209   ultimately have "expectation X \<in> I"
```
```   210     by (elim disjE)  (auto simp: subset_eq)
```
```   211   moreover
```
```   212   { fix y assume y: "y \<in> I"
```
```   213     with q(2) \<open>open I\<close> have "Sup ((\<lambda>x. q x + ?F x * (y - x)) ` I) = q y"
```
```   214       by (auto intro!: cSup_eq_maximum convex_le_Inf_differential image_eqI [OF _ y] simp: interior_open) }
```
```   215   ultimately have "q (expectation X) = Sup ((\<lambda>x. q x + ?F x * (expectation X - x)) ` I)"
```
```   216     by simp
```
```   217   also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
```
```   218   proof (rule cSup_least)
```
```   219     show "(\<lambda>x. q x + ?F x * (expectation X - x)) ` I \<noteq> {}"
```
```   220       using \<open>I \<noteq> {}\<close> by auto
```
```   221   next
```
```   222     fix k assume "k \<in> (\<lambda>x. q x + ?F x * (expectation X - x)) ` I"
```
```   223     then guess x .. note x = this
```
```   224     have "q x + ?F x * (expectation X  - x) = expectation (\<lambda>w. q x + ?F x * (X w - x))"
```
```   225       using prob_space by (simp add: X)
```
```   226     also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
```
```   227       using \<open>x \<in> I\<close> \<open>open I\<close> X(2)
```
```   228       apply (intro integral_mono_AE Bochner_Integration.integrable_add Bochner_Integration.integrable_mult_right Bochner_Integration.integrable_diff
```
```   229                 integrable_const X q)
```
```   230       apply (elim eventually_mono)
```
```   231       apply (intro convex_le_Inf_differential)
```
```   232       apply (auto simp: interior_open q)
```
```   233       done
```
```   234     finally show "k \<le> expectation (\<lambda>w. q (X w))" using x by auto
```
```   235   qed
```
```   236   finally show "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" .
```
```   237 qed
```
```   238
```
```   239 subsection  \<open>Introduce binder for probability\<close>
```
```   240
```
```   241 syntax
```
```   242   "_prob" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("('\<P>'((/_ in _./ _)'))")
```
```   243
```
```   244 translations
```
```   245   "\<P>(x in M. P)" => "CONST measure M {x \<in> CONST space M. P}"
```
```   246
```
```   247 print_translation \<open>
```
```   248   let
```
```   249     fun to_pattern (Const (@{const_syntax Pair}, _) \$ l \$ r) =
```
```   250       Syntax.const @{const_syntax Pair} :: to_pattern l @ to_pattern r
```
```   251     | to_pattern (t as (Const (@{syntax_const "_bound"}, _)) \$ _) = [t]
```
```   252
```
```   253     fun mk_pattern ((t, n) :: xs) = mk_patterns n xs |>> curry list_comb t
```
```   254     and mk_patterns 0 xs = ([], xs)
```
```   255     | mk_patterns n xs =
```
```   256       let
```
```   257         val (t, xs') = mk_pattern xs
```
```   258         val (ts, xs'') = mk_patterns (n - 1) xs'
```
```   259       in
```
```   260         (t :: ts, xs'')
```
```   261       end
```
```   262
```
```   263     fun unnest_tuples
```
```   264       (Const (@{syntax_const "_pattern"}, _) \$
```
```   265         t1 \$
```
```   266         (t as (Const (@{syntax_const "_pattern"}, _) \$ _ \$ _)))
```
```   267       = let
```
```   268         val (_ \$ t2 \$ t3) = unnest_tuples t
```
```   269       in
```
```   270         Syntax.const @{syntax_const "_pattern"} \$
```
```   271           unnest_tuples t1 \$
```
```   272           (Syntax.const @{syntax_const "_patterns"} \$ t2 \$ t3)
```
```   273       end
```
```   274     | unnest_tuples pat = pat
```
```   275
```
```   276     fun tr' [sig_alg, Const (@{const_syntax Collect}, _) \$ t] =
```
```   277       let
```
```   278         val bound_dummyT = Const (@{syntax_const "_bound"}, dummyT)
```
```   279
```
```   280         fun go pattern elem
```
```   281           (Const (@{const_syntax "conj"}, _) \$
```
```   282             (Const (@{const_syntax Set.member}, _) \$ elem' \$ (Const (@{const_syntax space}, _) \$ sig_alg')) \$
```
```   283             u)
```
```   284           = let
```
```   285               val _ = if sig_alg aconv sig_alg' andalso to_pattern elem' = rev elem then () else raise Match;
```
```   286               val (pat, rest) = mk_pattern (rev pattern);
```
```   287               val _ = case rest of [] => () | _ => raise Match
```
```   288             in
```
```   289               Syntax.const @{syntax_const "_prob"} \$ unnest_tuples pat \$ sig_alg \$ u
```
```   290             end
```
```   291         | go pattern elem (Abs abs) =
```
```   292             let
```
```   293               val (x as (_ \$ tx), t) = Syntax_Trans.atomic_abs_tr' abs
```
```   294             in
```
```   295               go ((x, 0) :: pattern) (bound_dummyT \$ tx :: elem) t
```
```   296             end
```
```   297         | go pattern elem (Const (@{const_syntax case_prod}, _) \$ t) =
```
```   298             go
```
```   299               ((Syntax.const @{syntax_const "_pattern"}, 2) :: pattern)
```
```   300               (Syntax.const @{const_syntax Pair} :: elem)
```
```   301               t
```
```   302       in
```
```   303         go [] [] t
```
```   304       end
```
```   305   in
```
```   306     [(@{const_syntax Sigma_Algebra.measure}, K tr')]
```
```   307   end
```
```   308 \<close>
```
```   309
```
```   310 definition
```
```   311   "cond_prob M P Q = \<P>(\<omega> in M. P \<omega> \<and> Q \<omega>) / \<P>(\<omega> in M. Q \<omega>)"
```
```   312
```
```   313 syntax
```
```   314   "_conditional_prob" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("('\<P>'(_ in _. _ \<bar>/ _'))")
```
```   315
```
```   316 translations
```
```   317   "\<P>(x in M. P \<bar> Q)" => "CONST cond_prob M (\<lambda>x. P) (\<lambda>x. Q)"
```
```   318
```
```   319 lemma (in prob_space) AE_E_prob:
```
```   320   assumes ae: "AE x in M. P x"
```
```   321   obtains S where "S \<subseteq> {x \<in> space M. P x}" "S \<in> events" "prob S = 1"
```
```   322 proof -
```
```   323   from ae[THEN AE_E] guess N .
```
```   324   then show thesis
```
```   325     by (intro that[of "space M - N"])
```
```   326        (auto simp: prob_compl prob_space emeasure_eq_measure measure_nonneg)
```
```   327 qed
```
```   328
```
```   329 lemma (in prob_space) prob_neg: "{x\<in>space M. P x} \<in> events \<Longrightarrow> \<P>(x in M. \<not> P x) = 1 - \<P>(x in M. P x)"
```
```   330   by (auto intro!: arg_cong[where f=prob] simp add: prob_compl[symmetric])
```
```   331
```
```   332 lemma (in prob_space) prob_eq_AE:
```
```   333   "(AE x in M. P x \<longleftrightarrow> Q x) \<Longrightarrow> {x\<in>space M. P x} \<in> events \<Longrightarrow> {x\<in>space M. Q x} \<in> events \<Longrightarrow> \<P>(x in M. P x) = \<P>(x in M. Q x)"
```
```   334   by (rule finite_measure_eq_AE) auto
```
```   335
```
```   336 lemma (in prob_space) prob_eq_0_AE:
```
```   337   assumes not: "AE x in M. \<not> P x" shows "\<P>(x in M. P x) = 0"
```
```   338 proof cases
```
```   339   assume "{x\<in>space M. P x} \<in> events"
```
```   340   with not have "\<P>(x in M. P x) = \<P>(x in M. False)"
```
```   341     by (intro prob_eq_AE) auto
```
```   342   then show ?thesis by simp
```
```   343 qed (simp add: measure_notin_sets)
```
```   344
```
```   345 lemma (in prob_space) prob_Collect_eq_0:
```
```   346   "{x \<in> space M. P x} \<in> sets M \<Longrightarrow> \<P>(x in M. P x) = 0 \<longleftrightarrow> (AE x in M. \<not> P x)"
```
```   347   using AE_iff_measurable[OF _ refl, of M "\<lambda>x. \<not> P x"] by (simp add: emeasure_eq_measure measure_nonneg)
```
```   348
```
```   349 lemma (in prob_space) prob_Collect_eq_1:
```
```   350   "{x \<in> space M. P x} \<in> sets M \<Longrightarrow> \<P>(x in M. P x) = 1 \<longleftrightarrow> (AE x in M. P x)"
```
```   351   using AE_in_set_eq_1[of "{x\<in>space M. P x}"] by simp
```
```   352
```
```   353 lemma (in prob_space) prob_eq_0:
```
```   354   "A \<in> sets M \<Longrightarrow> prob A = 0 \<longleftrightarrow> (AE x in M. x \<notin> A)"
```
```   355   using AE_iff_measurable[OF _ refl, of M "\<lambda>x. x \<notin> A"]
```
```   356   by (auto simp add: emeasure_eq_measure Int_def[symmetric] measure_nonneg)
```
```   357
```
```   358 lemma (in prob_space) prob_eq_1:
```
```   359   "A \<in> sets M \<Longrightarrow> prob A = 1 \<longleftrightarrow> (AE x in M. x \<in> A)"
```
```   360   using AE_in_set_eq_1[of A] by simp
```
```   361
```
```   362 lemma (in prob_space) prob_sums:
```
```   363   assumes P: "\<And>n. {x\<in>space M. P n x} \<in> events"
```
```   364   assumes Q: "{x\<in>space M. Q x} \<in> events"
```
```   365   assumes ae: "AE x in M. (\<forall>n. P n x \<longrightarrow> Q x) \<and> (Q x \<longrightarrow> (\<exists>!n. P n x))"
```
```   366   shows "(\<lambda>n. \<P>(x in M. P n x)) sums \<P>(x in M. Q x)"
```
```   367 proof -
```
```   368   from ae[THEN AE_E_prob] guess S . note S = this
```
```   369   then have disj: "disjoint_family (\<lambda>n. {x\<in>space M. P n x} \<inter> S)"
```
```   370     by (auto simp: disjoint_family_on_def)
```
```   371   from S have ae_S:
```
```   372     "AE x in M. x \<in> {x\<in>space M. Q x} \<longleftrightarrow> x \<in> (\<Union>n. {x\<in>space M. P n x} \<inter> S)"
```
```   373     "\<And>n. AE x in M. x \<in> {x\<in>space M. P n x} \<longleftrightarrow> x \<in> {x\<in>space M. P n x} \<inter> S"
```
```   374     using ae by (auto dest!: AE_prob_1)
```
```   375   from ae_S have *:
```
```   376     "\<P>(x in M. Q x) = prob (\<Union>n. {x\<in>space M. P n x} \<inter> S)"
```
```   377     using P Q S by (intro finite_measure_eq_AE) auto
```
```   378   from ae_S have **:
```
```   379     "\<And>n. \<P>(x in M. P n x) = prob ({x\<in>space M. P n x} \<inter> S)"
```
```   380     using P Q S by (intro finite_measure_eq_AE) auto
```
```   381   show ?thesis
```
```   382     unfolding * ** using S P disj
```
```   383     by (intro finite_measure_UNION) auto
```
```   384 qed
```
```   385
```
```   386 lemma (in prob_space) prob_setsum:
```
```   387   assumes [simp, intro]: "finite I"
```
```   388   assumes P: "\<And>n. n \<in> I \<Longrightarrow> {x\<in>space M. P n x} \<in> events"
```
```   389   assumes Q: "{x\<in>space M. Q x} \<in> events"
```
```   390   assumes ae: "AE x in M. (\<forall>n\<in>I. P n x \<longrightarrow> Q x) \<and> (Q x \<longrightarrow> (\<exists>!n\<in>I. P n x))"
```
```   391   shows "\<P>(x in M. Q x) = (\<Sum>n\<in>I. \<P>(x in M. P n x))"
```
```   392 proof -
```
```   393   from ae[THEN AE_E_prob] guess S . note S = this
```
```   394   then have disj: "disjoint_family_on (\<lambda>n. {x\<in>space M. P n x} \<inter> S) I"
```
```   395     by (auto simp: disjoint_family_on_def)
```
```   396   from S have ae_S:
```
```   397     "AE x in M. x \<in> {x\<in>space M. Q x} \<longleftrightarrow> x \<in> (\<Union>n\<in>I. {x\<in>space M. P n x} \<inter> S)"
```
```   398     "\<And>n. n \<in> I \<Longrightarrow> AE x in M. x \<in> {x\<in>space M. P n x} \<longleftrightarrow> x \<in> {x\<in>space M. P n x} \<inter> S"
```
```   399     using ae by (auto dest!: AE_prob_1)
```
```   400   from ae_S have *:
```
```   401     "\<P>(x in M. Q x) = prob (\<Union>n\<in>I. {x\<in>space M. P n x} \<inter> S)"
```
```   402     using P Q S by (intro finite_measure_eq_AE) (auto intro!: sets.Int)
```
```   403   from ae_S have **:
```
```   404     "\<And>n. n \<in> I \<Longrightarrow> \<P>(x in M. P n x) = prob ({x\<in>space M. P n x} \<inter> S)"
```
```   405     using P Q S by (intro finite_measure_eq_AE) auto
```
```   406   show ?thesis
```
```   407     using S P disj
```
```   408     by (auto simp add: * ** simp del: UN_simps intro!: finite_measure_finite_Union)
```
```   409 qed
```
```   410
```
```   411 lemma (in prob_space) prob_EX_countable:
```
```   412   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> {x\<in>space M. P i x} \<in> sets M" and I: "countable I"
```
```   413   assumes disj: "AE x in M. \<forall>i\<in>I. \<forall>j\<in>I. P i x \<longrightarrow> P j x \<longrightarrow> i = j"
```
```   414   shows "\<P>(x in M. \<exists>i\<in>I. P i x) = (\<integral>\<^sup>+i. \<P>(x in M. P i x) \<partial>count_space I)"
```
```   415 proof -
```
```   416   let ?N= "\<lambda>x. \<exists>!i\<in>I. P i x"
```
```   417   have "ennreal (\<P>(x in M. \<exists>i\<in>I. P i x)) = \<P>(x in M. (\<exists>i\<in>I. P i x \<and> ?N x))"
```
```   418     unfolding ennreal_inj[OF measure_nonneg measure_nonneg]
```
```   419   proof (rule prob_eq_AE)
```
```   420     show "AE x in M. (\<exists>i\<in>I. P i x) = (\<exists>i\<in>I. P i x \<and> ?N x)"
```
```   421       using disj by eventually_elim blast
```
```   422   qed (auto intro!: sets.sets_Collect_countable_Ex' sets.sets_Collect_conj sets.sets_Collect_countable_Ex1' I sets)+
```
```   423   also have "\<P>(x in M. (\<exists>i\<in>I. P i x \<and> ?N x)) = emeasure M (\<Union>i\<in>I. {x\<in>space M. P i x \<and> ?N x})"
```
```   424     unfolding emeasure_eq_measure by (auto intro!: arg_cong[where f=prob] simp: measure_nonneg)
```
```   425   also have "\<dots> = (\<integral>\<^sup>+i. emeasure M {x\<in>space M. P i x \<and> ?N x} \<partial>count_space I)"
```
```   426     by (rule emeasure_UN_countable)
```
```   427        (auto intro!: sets.sets_Collect_countable_Ex' sets.sets_Collect_conj sets.sets_Collect_countable_Ex1' I sets
```
```   428              simp: disjoint_family_on_def)
```
```   429   also have "\<dots> = (\<integral>\<^sup>+i. \<P>(x in M. P i x) \<partial>count_space I)"
```
```   430     unfolding emeasure_eq_measure using disj
```
```   431     by (intro nn_integral_cong ennreal_inj[THEN iffD2] prob_eq_AE)
```
```   432        (auto intro!: sets.sets_Collect_countable_Ex' sets.sets_Collect_conj sets.sets_Collect_countable_Ex1' I sets measure_nonneg)+
```
```   433   finally show ?thesis .
```
```   434 qed
```
```   435
```
```   436 lemma (in prob_space) cond_prob_eq_AE:
```
```   437   assumes P: "AE x in M. Q x \<longrightarrow> P x \<longleftrightarrow> P' x" "{x\<in>space M. P x} \<in> events" "{x\<in>space M. P' x} \<in> events"
```
```   438   assumes Q: "AE x in M. Q x \<longleftrightarrow> Q' x" "{x\<in>space M. Q x} \<in> events" "{x\<in>space M. Q' x} \<in> events"
```
```   439   shows "cond_prob M P Q = cond_prob M P' Q'"
```
```   440   using P Q
```
```   441   by (auto simp: cond_prob_def intro!: arg_cong2[where f="op /"] prob_eq_AE sets.sets_Collect_conj)
```
```   442
```
```   443
```
```   444 lemma (in prob_space) joint_distribution_Times_le_fst:
```
```   445   "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
```
```   446     \<Longrightarrow> emeasure (distr M (MX \<Otimes>\<^sub>M MY) (\<lambda>x. (X x, Y x))) (A \<times> B) \<le> emeasure (distr M MX X) A"
```
```   447   by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
```
```   448
```
```   449 lemma (in prob_space) joint_distribution_Times_le_snd:
```
```   450   "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
```
```   451     \<Longrightarrow> emeasure (distr M (MX \<Otimes>\<^sub>M MY) (\<lambda>x. (X x, Y x))) (A \<times> B) \<le> emeasure (distr M MY Y) B"
```
```   452   by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
```
```   453
```
```   454 lemma (in prob_space) variance_eq:
```
```   455   fixes X :: "'a \<Rightarrow> real"
```
```   456   assumes [simp]: "integrable M X"
```
```   457   assumes [simp]: "integrable M (\<lambda>x. (X x)\<^sup>2)"
```
```   458   shows "variance X = expectation (\<lambda>x. (X x)\<^sup>2) - (expectation X)\<^sup>2"
```
```   459   by (simp add: field_simps prob_space power2_diff power2_eq_square[symmetric])
```
```   460
```
```   461 lemma (in prob_space) variance_positive: "0 \<le> variance (X::'a \<Rightarrow> real)"
```
```   462   by (intro integral_nonneg_AE) (auto intro!: integral_nonneg_AE)
```
```   463
```
```   464 lemma (in prob_space) variance_mean_zero:
```
```   465   "expectation X = 0 \<Longrightarrow> variance X = expectation (\<lambda>x. (X x)^2)"
```
```   466   by simp
```
```   467
```
```   468 locale pair_prob_space = pair_sigma_finite M1 M2 + M1: prob_space M1 + M2: prob_space M2 for M1 M2
```
```   469
```
```   470 sublocale pair_prob_space \<subseteq> P?: prob_space "M1 \<Otimes>\<^sub>M M2"
```
```   471 proof
```
```   472   show "emeasure (M1 \<Otimes>\<^sub>M M2) (space (M1 \<Otimes>\<^sub>M M2)) = 1"
```
```   473     by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_space_1 M2.emeasure_space_1 space_pair_measure)
```
```   474 qed
```
```   475
```
```   476 locale product_prob_space = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
```
```   477   fixes I :: "'i set"
```
```   478   assumes prob_space: "\<And>i. prob_space (M i)"
```
```   479
```
```   480 sublocale product_prob_space \<subseteq> M?: prob_space "M i" for i
```
```   481   by (rule prob_space)
```
```   482
```
```   483 locale finite_product_prob_space = finite_product_sigma_finite M I + product_prob_space M I for M I
```
```   484
```
```   485 sublocale finite_product_prob_space \<subseteq> prob_space "\<Pi>\<^sub>M i\<in>I. M i"
```
```   486 proof
```
```   487   show "emeasure (\<Pi>\<^sub>M i\<in>I. M i) (space (\<Pi>\<^sub>M i\<in>I. M i)) = 1"
```
```   488     by (simp add: measure_times M.emeasure_space_1 setprod.neutral_const space_PiM)
```
```   489 qed
```
```   490
```
```   491 lemma (in finite_product_prob_space) prob_times:
```
```   492   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets (M i)"
```
```   493   shows "prob (\<Pi>\<^sub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
```
```   494 proof -
```
```   495   have "ennreal (measure (\<Pi>\<^sub>M i\<in>I. M i) (\<Pi>\<^sub>E i\<in>I. X i)) = emeasure (\<Pi>\<^sub>M i\<in>I. M i) (\<Pi>\<^sub>E i\<in>I. X i)"
```
```   496     using X by (simp add: emeasure_eq_measure)
```
```   497   also have "\<dots> = (\<Prod>i\<in>I. emeasure (M i) (X i))"
```
```   498     using measure_times X by simp
```
```   499   also have "\<dots> = ennreal (\<Prod>i\<in>I. measure (M i) (X i))"
```
```   500     using X by (simp add: M.emeasure_eq_measure setprod_ennreal measure_nonneg)
```
```   501   finally show ?thesis by (simp add: measure_nonneg setprod_nonneg)
```
```   502 qed
```
```   503
```
```   504 subsection \<open>Distributions\<close>
```
```   505
```
```   506 definition distributed :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> ennreal) \<Rightarrow> bool"
```
```   507 where
```
```   508   "distributed M N X f \<longleftrightarrow>
```
```   509   distr M N X = density N f \<and> f \<in> borel_measurable N \<and> X \<in> measurable M N"
```
```   510
```
```   511 term distributed
```
```   512
```
```   513 lemma
```
```   514   assumes "distributed M N X f"
```
```   515   shows distributed_distr_eq_density: "distr M N X = density N f"
```
```   516     and distributed_measurable: "X \<in> measurable M N"
```
```   517     and distributed_borel_measurable: "f \<in> borel_measurable N"
```
```   518   using assms by (simp_all add: distributed_def)
```
```   519
```
```   520 lemma
```
```   521   assumes D: "distributed M N X f"
```
```   522   shows distributed_measurable'[measurable_dest]:
```
```   523       "g \<in> measurable L M \<Longrightarrow> (\<lambda>x. X (g x)) \<in> measurable L N"
```
```   524     and distributed_borel_measurable'[measurable_dest]:
```
```   525       "h \<in> measurable L N \<Longrightarrow> (\<lambda>x. f (h x)) \<in> borel_measurable L"
```
```   526   using distributed_measurable[OF D] distributed_borel_measurable[OF D]
```
```   527   by simp_all
```
```   528
```
```   529 lemma distributed_real_measurable:
```
```   530   "(\<And>x. x \<in> space N \<Longrightarrow> 0 \<le> f x) \<Longrightarrow> distributed M N X (\<lambda>x. ennreal (f x)) \<Longrightarrow> f \<in> borel_measurable N"
```
```   531   by (simp_all add: distributed_def)
```
```   532
```
```   533 lemma distributed_real_measurable':
```
```   534   "(\<And>x. x \<in> space N \<Longrightarrow> 0 \<le> f x) \<Longrightarrow> distributed M N X (\<lambda>x. ennreal (f x)) \<Longrightarrow>
```
```   535     h \<in> measurable L N \<Longrightarrow> (\<lambda>x. f (h x)) \<in> borel_measurable L"
```
```   536   using distributed_real_measurable[measurable] by simp
```
```   537
```
```   538 lemma joint_distributed_measurable1:
```
```   539   "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) f \<Longrightarrow> h1 \<in> measurable N M \<Longrightarrow> (\<lambda>x. X (h1 x)) \<in> measurable N S"
```
```   540   by simp
```
```   541
```
```   542 lemma joint_distributed_measurable2:
```
```   543   "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) f \<Longrightarrow> h2 \<in> measurable N M \<Longrightarrow> (\<lambda>x. Y (h2 x)) \<in> measurable N T"
```
```   544   by simp
```
```   545
```
```   546 lemma distributed_count_space:
```
```   547   assumes X: "distributed M (count_space A) X P" and a: "a \<in> A" and A: "finite A"
```
```   548   shows "P a = emeasure M (X -` {a} \<inter> space M)"
```
```   549 proof -
```
```   550   have "emeasure M (X -` {a} \<inter> space M) = emeasure (distr M (count_space A) X) {a}"
```
```   551     using X a A by (simp add: emeasure_distr)
```
```   552   also have "\<dots> = emeasure (density (count_space A) P) {a}"
```
```   553     using X by (simp add: distributed_distr_eq_density)
```
```   554   also have "\<dots> = (\<integral>\<^sup>+x. P a * indicator {a} x \<partial>count_space A)"
```
```   555     using X a by (auto simp add: emeasure_density distributed_def indicator_def intro!: nn_integral_cong)
```
```   556   also have "\<dots> = P a"
```
```   557     using X a by (subst nn_integral_cmult_indicator) (auto simp: distributed_def one_ennreal_def[symmetric] AE_count_space)
```
```   558   finally show ?thesis ..
```
```   559 qed
```
```   560
```
```   561 lemma distributed_cong_density:
```
```   562   "(AE x in N. f x = g x) \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable N \<Longrightarrow>
```
```   563     distributed M N X f \<longleftrightarrow> distributed M N X g"
```
```   564   by (auto simp: distributed_def intro!: density_cong)
```
```   565
```
```   566 lemma (in prob_space) distributed_imp_emeasure_nonzero:
```
```   567   assumes X: "distributed M MX X Px"
```
```   568   shows "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> 0"
```
```   569 proof
```
```   570   note Px = distributed_borel_measurable[OF X]
```
```   571   interpret X: prob_space "distr M MX X"
```
```   572     using distributed_measurable[OF X] by (rule prob_space_distr)
```
```   573
```
```   574   assume "emeasure MX {x \<in> space MX. Px x \<noteq> 0} = 0"
```
```   575   with Px have "AE x in MX. Px x = 0"
```
```   576     by (intro AE_I[OF subset_refl]) (auto simp: borel_measurable_ennreal_iff)
```
```   577   moreover
```
```   578   from X.emeasure_space_1 have "(\<integral>\<^sup>+x. Px x \<partial>MX) = 1"
```
```   579     unfolding distributed_distr_eq_density[OF X] using Px
```
```   580     by (subst (asm) emeasure_density)
```
```   581        (auto simp: borel_measurable_ennreal_iff intro!: integral_cong cong: nn_integral_cong)
```
```   582   ultimately show False
```
```   583     by (simp add: nn_integral_cong_AE)
```
```   584 qed
```
```   585
```
```   586 lemma subdensity:
```
```   587   assumes T: "T \<in> measurable P Q"
```
```   588   assumes f: "distributed M P X f"
```
```   589   assumes g: "distributed M Q Y g"
```
```   590   assumes Y: "Y = T \<circ> X"
```
```   591   shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
```
```   592 proof -
```
```   593   have "{x\<in>space Q. g x = 0} \<in> null_sets (distr M Q (T \<circ> X))"
```
```   594     using g Y by (auto simp: null_sets_density_iff distributed_def)
```
```   595   also have "distr M Q (T \<circ> X) = distr (distr M P X) Q T"
```
```   596     using T f[THEN distributed_measurable] by (rule distr_distr[symmetric])
```
```   597   finally have "T -` {x\<in>space Q. g x = 0} \<inter> space P \<in> null_sets (distr M P X)"
```
```   598     using T by (subst (asm) null_sets_distr_iff) auto
```
```   599   also have "T -` {x\<in>space Q. g x = 0} \<inter> space P = {x\<in>space P. g (T x) = 0}"
```
```   600     using T by (auto dest: measurable_space)
```
```   601   finally show ?thesis
```
```   602     using f g by (auto simp add: null_sets_density_iff distributed_def)
```
```   603 qed
```
```   604
```
```   605 lemma subdensity_real:
```
```   606   fixes g :: "'a \<Rightarrow> real" and f :: "'b \<Rightarrow> real"
```
```   607   assumes T: "T \<in> measurable P Q"
```
```   608   assumes f: "distributed M P X f"
```
```   609   assumes g: "distributed M Q Y g"
```
```   610   assumes Y: "Y = T \<circ> X"
```
```   611   shows "(AE x in P. 0 \<le> g (T x)) \<Longrightarrow> (AE x in P. 0 \<le> f x) \<Longrightarrow> AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
```
```   612   using subdensity[OF T, of M X "\<lambda>x. ennreal (f x)" Y "\<lambda>x. ennreal (g x)"] assms
```
```   613   by auto
```
```   614
```
```   615 lemma distributed_emeasure:
```
```   616   "distributed M N X f \<Longrightarrow> A \<in> sets N \<Longrightarrow> emeasure M (X -` A \<inter> space M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>N)"
```
```   617   by (auto simp: distributed_distr_eq_density[symmetric] emeasure_density[symmetric] emeasure_distr)
```
```   618
```
```   619 lemma distributed_nn_integral:
```
```   620   "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> (\<integral>\<^sup>+x. f x * g x \<partial>N) = (\<integral>\<^sup>+x. g (X x) \<partial>M)"
```
```   621   by (auto simp: distributed_distr_eq_density[symmetric] nn_integral_density[symmetric] nn_integral_distr)
```
```   622
```
```   623 lemma distributed_integral:
```
```   624   "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> (\<And>x. x \<in> space N \<Longrightarrow> 0 \<le> f x) \<Longrightarrow>
```
```   625     (\<integral>x. f x * g x \<partial>N) = (\<integral>x. g (X x) \<partial>M)"
```
```   626   supply distributed_real_measurable[measurable]
```
```   627   by (auto simp: distributed_distr_eq_density[symmetric] integral_real_density[symmetric] integral_distr)
```
```   628
```
```   629 lemma distributed_transform_integral:
```
```   630   assumes Px: "distributed M N X Px" "\<And>x. x \<in> space N \<Longrightarrow> 0 \<le> Px x"
```
```   631   assumes "distributed M P Y Py" "\<And>x. x \<in> space P \<Longrightarrow> 0 \<le> Py x"
```
```   632   assumes Y: "Y = T \<circ> X" and T: "T \<in> measurable N P" and f: "f \<in> borel_measurable P"
```
```   633   shows "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. Px x * f (T x) \<partial>N)"
```
```   634 proof -
```
```   635   have "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. f (Y x) \<partial>M)"
```
```   636     by (rule distributed_integral) fact+
```
```   637   also have "\<dots> = (\<integral>x. f (T (X x)) \<partial>M)"
```
```   638     using Y by simp
```
```   639   also have "\<dots> = (\<integral>x. Px x * f (T x) \<partial>N)"
```
```   640     using measurable_comp[OF T f] Px by (intro distributed_integral[symmetric]) (auto simp: comp_def)
```
```   641   finally show ?thesis .
```
```   642 qed
```
```   643
```
```   644 lemma (in prob_space) distributed_unique:
```
```   645   assumes Px: "distributed M S X Px"
```
```   646   assumes Py: "distributed M S X Py"
```
```   647   shows "AE x in S. Px x = Py x"
```
```   648 proof -
```
```   649   interpret X: prob_space "distr M S X"
```
```   650     using Px by (intro prob_space_distr) simp
```
```   651   have "sigma_finite_measure (distr M S X)" ..
```
```   652   with sigma_finite_density_unique[of Px S Py ] Px Py
```
```   653   show ?thesis
```
```   654     by (auto simp: distributed_def)
```
```   655 qed
```
```   656
```
```   657 lemma (in prob_space) distributed_jointI:
```
```   658   assumes "sigma_finite_measure S" "sigma_finite_measure T"
```
```   659   assumes X[measurable]: "X \<in> measurable M S" and Y[measurable]: "Y \<in> measurable M T"
```
```   660   assumes [measurable]: "f \<in> borel_measurable (S \<Otimes>\<^sub>M T)" and f: "AE x in S \<Otimes>\<^sub>M T. 0 \<le> f x"
```
```   661   assumes eq: "\<And>A B. A \<in> sets S \<Longrightarrow> B \<in> sets T \<Longrightarrow>
```
```   662     emeasure M {x \<in> space M. X x \<in> A \<and> Y x \<in> B} = (\<integral>\<^sup>+x. (\<integral>\<^sup>+y. f (x, y) * indicator B y \<partial>T) * indicator A x \<partial>S)"
```
```   663   shows "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) f"
```
```   664   unfolding distributed_def
```
```   665 proof safe
```
```   666   interpret S: sigma_finite_measure S by fact
```
```   667   interpret T: sigma_finite_measure T by fact
```
```   668   interpret ST: pair_sigma_finite S T ..
```
```   669
```
```   670   from ST.sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('b \<times> 'c) set" .. note F = this
```
```   671   let ?E = "{a \<times> b |a b. a \<in> sets S \<and> b \<in> sets T}"
```
```   672   let ?P = "S \<Otimes>\<^sub>M T"
```
```   673   show "distr M ?P (\<lambda>x. (X x, Y x)) = density ?P f" (is "?L = ?R")
```
```   674   proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of S T]])
```
```   675     show "?E \<subseteq> Pow (space ?P)"
```
```   676       using sets.space_closed[of S] sets.space_closed[of T] by (auto simp: space_pair_measure)
```
```   677     show "sets ?L = sigma_sets (space ?P) ?E"
```
```   678       by (simp add: sets_pair_measure space_pair_measure)
```
```   679     then show "sets ?R = sigma_sets (space ?P) ?E"
```
```   680       by simp
```
```   681   next
```
```   682     interpret L: prob_space ?L
```
```   683       by (rule prob_space_distr) (auto intro!: measurable_Pair)
```
```   684     show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?L (F i) \<noteq> \<infinity>"
```
```   685       using F by (auto simp: space_pair_measure)
```
```   686   next
```
```   687     fix E assume "E \<in> ?E"
```
```   688     then obtain A B where E[simp]: "E = A \<times> B"
```
```   689       and A[measurable]: "A \<in> sets S" and B[measurable]: "B \<in> sets T" by auto
```
```   690     have "emeasure ?L E = emeasure M {x \<in> space M. X x \<in> A \<and> Y x \<in> B}"
```
```   691       by (auto intro!: arg_cong[where f="emeasure M"] simp add: emeasure_distr measurable_Pair)
```
```   692     also have "\<dots> = (\<integral>\<^sup>+x. (\<integral>\<^sup>+y. (f (x, y) * indicator B y) * indicator A x \<partial>T) \<partial>S)"
```
```   693       using f by (auto simp add: eq nn_integral_multc intro!: nn_integral_cong)
```
```   694     also have "\<dots> = emeasure ?R E"
```
```   695       by (auto simp add: emeasure_density T.nn_integral_fst[symmetric]
```
```   696                intro!: nn_integral_cong split: split_indicator)
```
```   697     finally show "emeasure ?L E = emeasure ?R E" .
```
```   698   qed
```
```   699 qed (auto simp: f)
```
```   700
```
```   701 lemma (in prob_space) distributed_swap:
```
```   702   assumes "sigma_finite_measure S" "sigma_finite_measure T"
```
```   703   assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
```
```   704   shows "distributed M (T \<Otimes>\<^sub>M S) (\<lambda>x. (Y x, X x)) (\<lambda>(x, y). Pxy (y, x))"
```
```   705 proof -
```
```   706   interpret S: sigma_finite_measure S by fact
```
```   707   interpret T: sigma_finite_measure T by fact
```
```   708   interpret ST: pair_sigma_finite S T ..
```
```   709   interpret TS: pair_sigma_finite T S ..
```
```   710
```
```   711   note Pxy[measurable]
```
```   712   show ?thesis
```
```   713     apply (subst TS.distr_pair_swap)
```
```   714     unfolding distributed_def
```
```   715   proof safe
```
```   716     let ?D = "distr (S \<Otimes>\<^sub>M T) (T \<Otimes>\<^sub>M S) (\<lambda>(x, y). (y, x))"
```
```   717     show 1: "(\<lambda>(x, y). Pxy (y, x)) \<in> borel_measurable ?D"
```
```   718       by auto
```
```   719     show 2: "random_variable (distr (S \<Otimes>\<^sub>M T) (T \<Otimes>\<^sub>M S) (\<lambda>(x, y). (y, x))) (\<lambda>x. (Y x, X x))"
```
```   720       using Pxy by auto
```
```   721     { fix A assume A: "A \<in> sets (T \<Otimes>\<^sub>M S)"
```
```   722       let ?B = "(\<lambda>(x, y). (y, x)) -` A \<inter> space (S \<Otimes>\<^sub>M T)"
```
```   723       from sets.sets_into_space[OF A]
```
```   724       have "emeasure M ((\<lambda>x. (Y x, X x)) -` A \<inter> space M) =
```
```   725         emeasure M ((\<lambda>x. (X x, Y x)) -` ?B \<inter> space M)"
```
```   726         by (auto intro!: arg_cong2[where f=emeasure] simp: space_pair_measure)
```
```   727       also have "\<dots> = (\<integral>\<^sup>+ x. Pxy x * indicator ?B x \<partial>(S \<Otimes>\<^sub>M T))"
```
```   728         using Pxy A by (intro distributed_emeasure) auto
```
```   729       finally have "emeasure M ((\<lambda>x. (Y x, X x)) -` A \<inter> space M) =
```
```   730         (\<integral>\<^sup>+ x. Pxy x * indicator A (snd x, fst x) \<partial>(S \<Otimes>\<^sub>M T))"
```
```   731         by (auto intro!: nn_integral_cong split: split_indicator) }
```
```   732     note * = this
```
```   733     show "distr M ?D (\<lambda>x. (Y x, X x)) = density ?D (\<lambda>(x, y). Pxy (y, x))"
```
```   734       apply (intro measure_eqI)
```
```   735       apply (simp_all add: emeasure_distr[OF 2] emeasure_density[OF 1])
```
```   736       apply (subst nn_integral_distr)
```
```   737       apply (auto intro!: * simp: comp_def split_beta)
```
```   738       done
```
```   739   qed
```
```   740 qed
```
```   741
```
```   742 lemma (in prob_space) distr_marginal1:
```
```   743   assumes "sigma_finite_measure S" "sigma_finite_measure T"
```
```   744   assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
```
```   745   defines "Px \<equiv> \<lambda>x. (\<integral>\<^sup>+z. Pxy (x, z) \<partial>T)"
```
```   746   shows "distributed M S X Px"
```
```   747   unfolding distributed_def
```
```   748 proof safe
```
```   749   interpret S: sigma_finite_measure S by fact
```
```   750   interpret T: sigma_finite_measure T by fact
```
```   751   interpret ST: pair_sigma_finite S T ..
```
```   752
```
```   753   note Pxy[measurable]
```
```   754   show X: "X \<in> measurable M S" by simp
```
```   755
```
```   756   show borel: "Px \<in> borel_measurable S"
```
```   757     by (auto intro!: T.nn_integral_fst simp: Px_def)
```
```   758
```
```   759   interpret Pxy: prob_space "distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
```
```   760     by (intro prob_space_distr) simp
```
```   761
```
```   762   show "distr M S X = density S Px"
```
```   763   proof (rule measure_eqI)
```
```   764     fix A assume A: "A \<in> sets (distr M S X)"
```
```   765     with X measurable_space[of Y M T]
```
```   766     have "emeasure (distr M S X) A = emeasure (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))) (A \<times> space T)"
```
```   767       by (auto simp add: emeasure_distr intro!: arg_cong[where f="emeasure M"])
```
```   768     also have "\<dots> = emeasure (density (S \<Otimes>\<^sub>M T) Pxy) (A \<times> space T)"
```
```   769       using Pxy by (simp add: distributed_def)
```
```   770     also have "\<dots> = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T \<partial>S"
```
```   771       using A borel Pxy
```
```   772       by (simp add: emeasure_density T.nn_integral_fst[symmetric])
```
```   773     also have "\<dots> = \<integral>\<^sup>+ x. Px x * indicator A x \<partial>S"
```
```   774     proof (rule nn_integral_cong)
```
```   775       fix x assume "x \<in> space S"
```
```   776       moreover have eq: "\<And>y. y \<in> space T \<Longrightarrow> indicator (A \<times> space T) (x, y) = indicator A x"
```
```   777         by (auto simp: indicator_def)
```
```   778       ultimately have "(\<integral>\<^sup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T) = (\<integral>\<^sup>+ y. Pxy (x, y) \<partial>T) * indicator A x"
```
```   779         by (simp add: eq nn_integral_multc cong: nn_integral_cong)
```
```   780       also have "(\<integral>\<^sup>+ y. Pxy (x, y) \<partial>T) = Px x"
```
```   781         by (simp add: Px_def)
```
```   782       finally show "(\<integral>\<^sup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T) = Px x * indicator A x" .
```
```   783     qed
```
```   784     finally show "emeasure (distr M S X) A = emeasure (density S Px) A"
```
```   785       using A borel Pxy by (simp add: emeasure_density)
```
```   786   qed simp
```
```   787 qed
```
```   788
```
```   789 lemma (in prob_space) distr_marginal2:
```
```   790   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
```
```   791   assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
```
```   792   shows "distributed M T Y (\<lambda>y. (\<integral>\<^sup>+x. Pxy (x, y) \<partial>S))"
```
```   793   using distr_marginal1[OF T S distributed_swap[OF S T]] Pxy by simp
```
```   794
```
```   795 lemma (in prob_space) distributed_marginal_eq_joint1:
```
```   796   assumes T: "sigma_finite_measure T"
```
```   797   assumes S: "sigma_finite_measure S"
```
```   798   assumes Px: "distributed M S X Px"
```
```   799   assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
```
```   800   shows "AE x in S. Px x = (\<integral>\<^sup>+y. Pxy (x, y) \<partial>T)"
```
```   801   using Px distr_marginal1[OF S T Pxy] by (rule distributed_unique)
```
```   802
```
```   803 lemma (in prob_space) distributed_marginal_eq_joint2:
```
```   804   assumes T: "sigma_finite_measure T"
```
```   805   assumes S: "sigma_finite_measure S"
```
```   806   assumes Py: "distributed M T Y Py"
```
```   807   assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
```
```   808   shows "AE y in T. Py y = (\<integral>\<^sup>+x. Pxy (x, y) \<partial>S)"
```
```   809   using Py distr_marginal2[OF S T Pxy] by (rule distributed_unique)
```
```   810
```
```   811 lemma (in prob_space) distributed_joint_indep':
```
```   812   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
```
```   813   assumes X[measurable]: "distributed M S X Px" and Y[measurable]: "distributed M T Y Py"
```
```   814   assumes indep: "distr M S X \<Otimes>\<^sub>M distr M T Y = distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
```
```   815   shows "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>(x, y). Px x * Py y)"
```
```   816   unfolding distributed_def
```
```   817 proof safe
```
```   818   interpret S: sigma_finite_measure S by fact
```
```   819   interpret T: sigma_finite_measure T by fact
```
```   820   interpret ST: pair_sigma_finite S T ..
```
```   821
```
```   822   interpret X: prob_space "density S Px"
```
```   823     unfolding distributed_distr_eq_density[OF X, symmetric]
```
```   824     by (rule prob_space_distr) simp
```
```   825   have sf_X: "sigma_finite_measure (density S Px)" ..
```
```   826
```
```   827   interpret Y: prob_space "density T Py"
```
```   828     unfolding distributed_distr_eq_density[OF Y, symmetric]
```
```   829     by (rule prob_space_distr) simp
```
```   830   have sf_Y: "sigma_finite_measure (density T Py)" ..
```
```   831
```
```   832   show "distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) = density (S \<Otimes>\<^sub>M T) (\<lambda>(x, y). Px x * Py y)"
```
```   833     unfolding indep[symmetric] distributed_distr_eq_density[OF X] distributed_distr_eq_density[OF Y]
```
```   834     using distributed_borel_measurable[OF X]
```
```   835     using distributed_borel_measurable[OF Y]
```
```   836     by (rule pair_measure_density[OF _ _ T sf_Y])
```
```   837
```
```   838   show "random_variable (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))" by auto
```
```   839
```
```   840   show Pxy: "(\<lambda>(x, y). Px x * Py y) \<in> borel_measurable (S \<Otimes>\<^sub>M T)" by auto
```
```   841 qed
```
```   842
```
```   843 lemma distributed_integrable:
```
```   844   "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> (\<And>x. x \<in> space N \<Longrightarrow> 0 \<le> f x) \<Longrightarrow>
```
```   845     integrable N (\<lambda>x. f x * g x) \<longleftrightarrow> integrable M (\<lambda>x. g (X x))"
```
```   846   supply distributed_real_measurable[measurable]
```
```   847   by (auto simp: distributed_distr_eq_density[symmetric] integrable_real_density[symmetric] integrable_distr_eq)
```
```   848
```
```   849 lemma distributed_transform_integrable:
```
```   850   assumes Px: "distributed M N X Px" "\<And>x. x \<in> space N \<Longrightarrow> 0 \<le> Px x"
```
```   851   assumes "distributed M P Y Py" "\<And>x. x \<in> space P \<Longrightarrow> 0 \<le> Py x"
```
```   852   assumes Y: "Y = (\<lambda>x. T (X x))" and T: "T \<in> measurable N P" and f: "f \<in> borel_measurable P"
```
```   853   shows "integrable P (\<lambda>x. Py x * f x) \<longleftrightarrow> integrable N (\<lambda>x. Px x * f (T x))"
```
```   854 proof -
```
```   855   have "integrable P (\<lambda>x. Py x * f x) \<longleftrightarrow> integrable M (\<lambda>x. f (Y x))"
```
```   856     by (rule distributed_integrable) fact+
```
```   857   also have "\<dots> \<longleftrightarrow> integrable M (\<lambda>x. f (T (X x)))"
```
```   858     using Y by simp
```
```   859   also have "\<dots> \<longleftrightarrow> integrable N (\<lambda>x. Px x * f (T x))"
```
```   860     using measurable_comp[OF T f] Px by (intro distributed_integrable[symmetric]) (auto simp: comp_def)
```
```   861   finally show ?thesis .
```
```   862 qed
```
```   863
```
```   864 lemma distributed_integrable_var:
```
```   865   fixes X :: "'a \<Rightarrow> real"
```
```   866   shows "distributed M lborel X (\<lambda>x. ennreal (f x)) \<Longrightarrow> (\<And>x. 0 \<le> f x) \<Longrightarrow>
```
```   867     integrable lborel (\<lambda>x. f x * x) \<Longrightarrow> integrable M X"
```
```   868   using distributed_integrable[of M lborel X f "\<lambda>x. x"] by simp
```
```   869
```
```   870 lemma (in prob_space) distributed_variance:
```
```   871   fixes f::"real \<Rightarrow> real"
```
```   872   assumes D: "distributed M lborel X f" and [simp]: "\<And>x. 0 \<le> f x"
```
```   873   shows "variance X = (\<integral>x. x\<^sup>2 * f (x + expectation X) \<partial>lborel)"
```
```   874 proof (subst distributed_integral[OF D, symmetric])
```
```   875   show "(\<integral> x. f x * (x - expectation X)\<^sup>2 \<partial>lborel) = (\<integral> x. x\<^sup>2 * f (x + expectation X) \<partial>lborel)"
```
```   876     by (subst lborel_integral_real_affine[where c=1 and t="expectation X"])  (auto simp: ac_simps)
```
```   877 qed simp_all
```
```   878
```
```   879 lemma (in prob_space) variance_affine:
```
```   880   fixes f::"real \<Rightarrow> real"
```
```   881   assumes [arith]: "b \<noteq> 0"
```
```   882   assumes D[intro]: "distributed M lborel X f"
```
```   883   assumes [simp]: "prob_space (density lborel f)"
```
```   884   assumes I[simp]: "integrable M X"
```
```   885   assumes I2[simp]: "integrable M (\<lambda>x. (X x)\<^sup>2)"
```
```   886   shows "variance (\<lambda>x. a + b * X x) = b\<^sup>2 * variance X"
```
```   887   by (subst variance_eq)
```
```   888      (auto simp: power2_sum power_mult_distrib prob_space variance_eq right_diff_distrib)
```
```   889
```
```   890 definition
```
```   891   "simple_distributed M X f \<longleftrightarrow>
```
```   892     (\<forall>x. 0 \<le> f x) \<and>
```
```   893     distributed M (count_space (X`space M)) X (\<lambda>x. ennreal (f x)) \<and>
```
```   894     finite (X`space M)"
```
```   895
```
```   896 lemma simple_distributed_nonneg[dest]: "simple_distributed M X f \<Longrightarrow> 0 \<le> f x"
```
```   897   by (auto simp: simple_distributed_def)
```
```   898
```
```   899 lemma simple_distributed:
```
```   900   "simple_distributed M X Px \<Longrightarrow> distributed M (count_space (X`space M)) X Px"
```
```   901   unfolding simple_distributed_def by auto
```
```   902
```
```   903 lemma simple_distributed_finite[dest]: "simple_distributed M X P \<Longrightarrow> finite (X`space M)"
```
```   904   by (simp add: simple_distributed_def)
```
```   905
```
```   906 lemma (in prob_space) distributed_simple_function_superset:
```
```   907   assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
```
```   908   assumes A: "X`space M \<subseteq> A" "finite A"
```
```   909   defines "S \<equiv> count_space A" and "P' \<equiv> (\<lambda>x. if x \<in> X`space M then P x else 0)"
```
```   910   shows "distributed M S X P'"
```
```   911   unfolding distributed_def
```
```   912 proof safe
```
```   913   show "(\<lambda>x. ennreal (P' x)) \<in> borel_measurable S" unfolding S_def by simp
```
```   914   show "distr M S X = density S P'"
```
```   915   proof (rule measure_eqI_finite)
```
```   916     show "sets (distr M S X) = Pow A" "sets (density S P') = Pow A"
```
```   917       using A unfolding S_def by auto
```
```   918     show "finite A" by fact
```
```   919     fix a assume a: "a \<in> A"
```
```   920     then have "a \<notin> X`space M \<Longrightarrow> X -` {a} \<inter> space M = {}" by auto
```
```   921     with A a X have "emeasure (distr M S X) {a} = P' a"
```
```   922       by (subst emeasure_distr)
```
```   923          (auto simp add: S_def P'_def simple_functionD emeasure_eq_measure measurable_count_space_eq2
```
```   924                intro!: arg_cong[where f=prob])
```
```   925     also have "\<dots> = (\<integral>\<^sup>+x. ennreal (P' a) * indicator {a} x \<partial>S)"
```
```   926       using A X a
```
```   927       by (subst nn_integral_cmult_indicator)
```
```   928          (auto simp: S_def P'_def simple_distributed_def simple_functionD measure_nonneg)
```
```   929     also have "\<dots> = (\<integral>\<^sup>+x. ennreal (P' x) * indicator {a} x \<partial>S)"
```
```   930       by (auto simp: indicator_def intro!: nn_integral_cong)
```
```   931     also have "\<dots> = emeasure (density S P') {a}"
```
```   932       using a A by (intro emeasure_density[symmetric]) (auto simp: S_def)
```
```   933     finally show "emeasure (distr M S X) {a} = emeasure (density S P') {a}" .
```
```   934   qed
```
```   935   show "random_variable S X"
```
```   936     using X(1) A by (auto simp: measurable_def simple_functionD S_def)
```
```   937 qed
```
```   938
```
```   939 lemma (in prob_space) simple_distributedI:
```
```   940   assumes X: "simple_function M X"
```
```   941     "\<And>x. 0 \<le> P x"
```
```   942     "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
```
```   943   shows "simple_distributed M X P"
```
```   944   unfolding simple_distributed_def
```
```   945 proof (safe intro!: X)
```
```   946   have "distributed M (count_space (X ` space M)) X (\<lambda>x. ennreal (if x \<in> X`space M then P x else 0))"
```
```   947     (is "?A")
```
```   948     using simple_functionD[OF X(1)] by (intro distributed_simple_function_superset[OF X(1,3)]) auto
```
```   949   also have "?A \<longleftrightarrow> distributed M (count_space (X ` space M)) X (\<lambda>x. ennreal (P x))"
```
```   950     by (rule distributed_cong_density) auto
```
```   951   finally show "\<dots>" .
```
```   952 qed (rule simple_functionD[OF X(1)])
```
```   953
```
```   954 lemma simple_distributed_joint_finite:
```
```   955   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
```
```   956   shows "finite (X ` space M)" "finite (Y ` space M)"
```
```   957 proof -
```
```   958   have "finite ((\<lambda>x. (X x, Y x)) ` space M)"
```
```   959     using X by (auto simp: simple_distributed_def simple_functionD)
```
```   960   then have "finite (fst ` (\<lambda>x. (X x, Y x)) ` space M)" "finite (snd ` (\<lambda>x. (X x, Y x)) ` space M)"
```
```   961     by auto
```
```   962   then show fin: "finite (X ` space M)" "finite (Y ` space M)"
```
```   963     by (auto simp: image_image)
```
```   964 qed
```
```   965
```
```   966 lemma simple_distributed_joint2_finite:
```
```   967   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
```
```   968   shows "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
```
```   969 proof -
```
```   970   have "finite ((\<lambda>x. (X x, Y x, Z x)) ` space M)"
```
```   971     using X by (auto simp: simple_distributed_def simple_functionD)
```
```   972   then have "finite (fst ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
```
```   973     "finite ((fst \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
```
```   974     "finite ((snd \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
```
```   975     by auto
```
```   976   then show fin: "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
```
```   977     by (auto simp: image_image)
```
```   978 qed
```
```   979
```
```   980 lemma simple_distributed_simple_function:
```
```   981   "simple_distributed M X Px \<Longrightarrow> simple_function M X"
```
```   982   unfolding simple_distributed_def distributed_def
```
```   983   by (auto simp: simple_function_def measurable_count_space_eq2)
```
```   984
```
```   985 lemma simple_distributed_measure:
```
```   986   "simple_distributed M X P \<Longrightarrow> a \<in> X`space M \<Longrightarrow> P a = measure M (X -` {a} \<inter> space M)"
```
```   987   using distributed_count_space[of M "X`space M" X P a, symmetric]
```
```   988   by (auto simp: simple_distributed_def measure_def)
```
```   989
```
```   990 lemma (in prob_space) simple_distributed_joint:
```
```   991   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
```
```   992   defines "S \<equiv> count_space (X`space M) \<Otimes>\<^sub>M count_space (Y`space M)"
```
```   993   defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x))`space M then Px x else 0)"
```
```   994   shows "distributed M S (\<lambda>x. (X x, Y x)) P"
```
```   995 proof -
```
```   996   from simple_distributed_joint_finite[OF X, simp]
```
```   997   have S_eq: "S = count_space (X`space M \<times> Y`space M)"
```
```   998     by (simp add: S_def pair_measure_count_space)
```
```   999   show ?thesis
```
```  1000     unfolding S_eq P_def
```
```  1001   proof (rule distributed_simple_function_superset)
```
```  1002     show "simple_function M (\<lambda>x. (X x, Y x))"
```
```  1003       using X by (rule simple_distributed_simple_function)
```
```  1004     fix x assume "x \<in> (\<lambda>x. (X x, Y x)) ` space M"
```
```  1005     from simple_distributed_measure[OF X this]
```
```  1006     show "Px x = prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)" .
```
```  1007   qed auto
```
```  1008 qed
```
```  1009
```
```  1010 lemma (in prob_space) simple_distributed_joint2:
```
```  1011   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
```
```  1012   defines "S \<equiv> count_space (X`space M) \<Otimes>\<^sub>M count_space (Y`space M) \<Otimes>\<^sub>M count_space (Z`space M)"
```
```  1013   defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x))`space M then Px x else 0)"
```
```  1014   shows "distributed M S (\<lambda>x. (X x, Y x, Z x)) P"
```
```  1015 proof -
```
```  1016   from simple_distributed_joint2_finite[OF X, simp]
```
```  1017   have S_eq: "S = count_space (X`space M \<times> Y`space M \<times> Z`space M)"
```
```  1018     by (simp add: S_def pair_measure_count_space)
```
```  1019   show ?thesis
```
```  1020     unfolding S_eq P_def
```
```  1021   proof (rule distributed_simple_function_superset)
```
```  1022     show "simple_function M (\<lambda>x. (X x, Y x, Z x))"
```
```  1023       using X by (rule simple_distributed_simple_function)
```
```  1024     fix x assume "x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M"
```
```  1025     from simple_distributed_measure[OF X this]
```
```  1026     show "Px x = prob ((\<lambda>x. (X x, Y x, Z x)) -` {x} \<inter> space M)" .
```
```  1027   qed auto
```
```  1028 qed
```
```  1029
```
```  1030 lemma (in prob_space) simple_distributed_setsum_space:
```
```  1031   assumes X: "simple_distributed M X f"
```
```  1032   shows "setsum f (X`space M) = 1"
```
```  1033 proof -
```
```  1034   from X have "setsum f (X`space M) = prob (\<Union>i\<in>X`space M. X -` {i} \<inter> space M)"
```
```  1035     by (subst finite_measure_finite_Union)
```
```  1036        (auto simp add: disjoint_family_on_def simple_distributed_measure simple_distributed_simple_function simple_functionD
```
```  1037              intro!: setsum.cong arg_cong[where f="prob"])
```
```  1038   also have "\<dots> = prob (space M)"
```
```  1039     by (auto intro!: arg_cong[where f=prob])
```
```  1040   finally show ?thesis
```
```  1041     using emeasure_space_1 by (simp add: emeasure_eq_measure)
```
```  1042 qed
```
```  1043
```
```  1044 lemma (in prob_space) distributed_marginal_eq_joint_simple:
```
```  1045   assumes Px: "simple_function M X"
```
```  1046   assumes Py: "simple_distributed M Y Py"
```
```  1047   assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
```
```  1048   assumes y: "y \<in> Y`space M"
```
```  1049   shows "Py y = (\<Sum>x\<in>X`space M. if (x, y) \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy (x, y) else 0)"
```
```  1050 proof -
```
```  1051   note Px = simple_distributedI[OF Px measure_nonneg refl]
```
```  1052   have "AE y in count_space (Y ` space M). ennreal (Py y) =
```
```  1053        \<integral>\<^sup>+ x. ennreal (if (x, y) \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy (x, y) else 0) \<partial>count_space (X ` space M)"
```
```  1054     using sigma_finite_measure_count_space_finite sigma_finite_measure_count_space_finite
```
```  1055       simple_distributed[OF Py] simple_distributed_joint[OF Pxy]
```
```  1056     by (rule distributed_marginal_eq_joint2)
```
```  1057        (auto intro: Py Px simple_distributed_finite)
```
```  1058   then have "ennreal (Py y) =
```
```  1059     (\<Sum>x\<in>X`space M. ennreal (if (x, y) \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy (x, y) else 0))"
```
```  1060     using y Px[THEN simple_distributed_finite]
```
```  1061     by (auto simp: AE_count_space nn_integral_count_space_finite)
```
```  1062   also have "\<dots> = (\<Sum>x\<in>X`space M. if (x, y) \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy (x, y) else 0)"
```
```  1063     using Pxy by (intro setsum_ennreal) auto
```
```  1064   finally show ?thesis
```
```  1065     using simple_distributed_nonneg[OF Py] simple_distributed_nonneg[OF Pxy]
```
```  1066     by (subst (asm) ennreal_inj) (auto intro!: setsum_nonneg)
```
```  1067 qed
```
```  1068
```
```  1069 lemma distributedI_real:
```
```  1070   fixes f :: "'a \<Rightarrow> real"
```
```  1071   assumes gen: "sets M1 = sigma_sets (space M1) E" and "Int_stable E"
```
```  1072     and A: "range A \<subseteq> E" "(\<Union>i::nat. A i) = space M1" "\<And>i. emeasure (distr M M1 X) (A i) \<noteq> \<infinity>"
```
```  1073     and X: "X \<in> measurable M M1"
```
```  1074     and f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x"
```
```  1075     and eq: "\<And>A. A \<in> E \<Longrightarrow> emeasure M (X -` A \<inter> space M) = (\<integral>\<^sup>+ x. f x * indicator A x \<partial>M1)"
```
```  1076   shows "distributed M M1 X f"
```
```  1077   unfolding distributed_def
```
```  1078 proof (intro conjI)
```
```  1079   show "distr M M1 X = density M1 f"
```
```  1080   proof (rule measure_eqI_generator_eq[where A=A])
```
```  1081     { fix A assume A: "A \<in> E"
```
```  1082       then have "A \<in> sigma_sets (space M1) E" by auto
```
```  1083       then have "A \<in> sets M1"
```
```  1084         using gen by simp
```
```  1085       with f A eq[of A] X show "emeasure (distr M M1 X) A = emeasure (density M1 f) A"
```
```  1086         by (auto simp add: emeasure_distr emeasure_density ennreal_indicator
```
```  1087                  intro!: nn_integral_cong split: split_indicator) }
```
```  1088     note eq_E = this
```
```  1089     show "Int_stable E" by fact
```
```  1090     { fix e assume "e \<in> E"
```
```  1091       then have "e \<in> sigma_sets (space M1) E" by auto
```
```  1092       then have "e \<in> sets M1" unfolding gen .
```
```  1093       then have "e \<subseteq> space M1" by (rule sets.sets_into_space) }
```
```  1094     then show "E \<subseteq> Pow (space M1)" by auto
```
```  1095     show "sets (distr M M1 X) = sigma_sets (space M1) E"
```
```  1096       "sets (density M1 (\<lambda>x. ennreal (f x))) = sigma_sets (space M1) E"
```
```  1097       unfolding gen[symmetric] by auto
```
```  1098   qed fact+
```
```  1099 qed (insert X f, auto)
```
```  1100
```
```  1101 lemma distributedI_borel_atMost:
```
```  1102   fixes f :: "real \<Rightarrow> real"
```
```  1103   assumes [measurable]: "X \<in> borel_measurable M"
```
```  1104     and [measurable]: "f \<in> borel_measurable borel" and f[simp]: "AE x in lborel. 0 \<le> f x"
```
```  1105     and g_eq: "\<And>a. (\<integral>\<^sup>+x. f x * indicator {..a} x \<partial>lborel)  = ennreal (g a)"
```
```  1106     and M_eq: "\<And>a. emeasure M {x\<in>space M. X x \<le> a} = ennreal (g a)"
```
```  1107   shows "distributed M lborel X f"
```
```  1108 proof (rule distributedI_real)
```
```  1109   show "sets (lborel::real measure) = sigma_sets (space lborel) (range atMost)"
```
```  1110     by (simp add: borel_eq_atMost)
```
```  1111   show "Int_stable (range atMost :: real set set)"
```
```  1112     by (auto simp: Int_stable_def)
```
```  1113   have vimage_eq: "\<And>a. (X -` {..a} \<inter> space M) = {x\<in>space M. X x \<le> a}" by auto
```
```  1114   define A where "A i = {.. real i}" for i :: nat
```
```  1115   then show "range A \<subseteq> range atMost" "(\<Union>i. A i) = space lborel"
```
```  1116     "\<And>i. emeasure (distr M lborel X) (A i) \<noteq> \<infinity>"
```
```  1117     by (auto simp: real_arch_simple emeasure_distr vimage_eq M_eq)
```
```  1118
```
```  1119   fix A :: "real set" assume "A \<in> range atMost"
```
```  1120   then obtain a where A: "A = {..a}" by auto
```
```  1121   show "emeasure M (X -` A \<inter> space M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>lborel)"
```
```  1122     unfolding vimage_eq A M_eq g_eq ..
```
```  1123 qed auto
```
```  1124
```
```  1125 lemma (in prob_space) uniform_distributed_params:
```
```  1126   assumes X: "distributed M MX X (\<lambda>x. indicator A x / measure MX A)"
```
```  1127   shows "A \<in> sets MX" "measure MX A \<noteq> 0"
```
```  1128 proof -
```
```  1129   interpret X: prob_space "distr M MX X"
```
```  1130     using distributed_measurable[OF X] by (rule prob_space_distr)
```
```  1131
```
```  1132   show "measure MX A \<noteq> 0"
```
```  1133   proof
```
```  1134     assume "measure MX A = 0"
```
```  1135     with X.emeasure_space_1 X.prob_space distributed_distr_eq_density[OF X]
```
```  1136     show False
```
```  1137       by (simp add: emeasure_density zero_ennreal_def[symmetric])
```
```  1138   qed
```
```  1139   with measure_notin_sets[of A MX] show "A \<in> sets MX"
```
```  1140     by blast
```
```  1141 qed
```
```  1142
```
```  1143 lemma prob_space_uniform_measure:
```
```  1144   assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>"
```
```  1145   shows "prob_space (uniform_measure M A)"
```
```  1146 proof
```
```  1147   show "emeasure (uniform_measure M A) (space (uniform_measure M A)) = 1"
```
```  1148     using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)], of "space M"]
```
```  1149     using sets.sets_into_space[OF emeasure_neq_0_sets[OF A(1)]] A
```
```  1150     by (simp add: Int_absorb2 less_top)
```
```  1151 qed
```
```  1152
```
```  1153 lemma prob_space_uniform_count_measure: "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> prob_space (uniform_count_measure A)"
```
```  1154   by standard (auto simp: emeasure_uniform_count_measure space_uniform_count_measure one_ennreal_def)
```
```  1155
```
```  1156 lemma (in prob_space) measure_uniform_measure_eq_cond_prob:
```
```  1157   assumes [measurable]: "Measurable.pred M P" "Measurable.pred M Q"
```
```  1158   shows "\<P>(x in uniform_measure M {x\<in>space M. Q x}. P x) = \<P>(x in M. P x \<bar> Q x)"
```
```  1159 proof cases
```
```  1160   assume Q: "measure M {x\<in>space M. Q x} = 0"
```
```  1161   then have *: "AE x in M. \<not> Q x"
```
```  1162     by (simp add: prob_eq_0)
```
```  1163   then have "density M (\<lambda>x. indicator {x \<in> space M. Q x} x / emeasure M {x \<in> space M. Q x}) = density M (\<lambda>x. 0)"
```
```  1164     by (intro density_cong) auto
```
```  1165   with * show ?thesis
```
```  1166     unfolding uniform_measure_def
```
```  1167     by (simp add: emeasure_density measure_def cond_prob_def emeasure_eq_0_AE)
```
```  1168 next
```
```  1169   assume Q: "measure M {x\<in>space M. Q x} \<noteq> 0"
```
```  1170   then show "\<P>(x in uniform_measure M {x \<in> space M. Q x}. P x) = cond_prob M P Q"
```
```  1171     by (subst measure_uniform_measure)
```
```  1172        (auto simp: emeasure_eq_measure cond_prob_def measure_nonneg intro!: arg_cong[where f=prob])
```
```  1173 qed
```
```  1174
```
```  1175 lemma prob_space_point_measure:
```
```  1176   "finite S \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> 0 \<le> p s) \<Longrightarrow> (\<Sum>s\<in>S. p s) = 1 \<Longrightarrow> prob_space (point_measure S p)"
```
```  1177   by (rule prob_spaceI) (simp add: space_point_measure emeasure_point_measure_finite)
```
```  1178
```
```  1179 lemma (in prob_space) distr_pair_fst: "distr (N \<Otimes>\<^sub>M M) N fst = N"
```
```  1180 proof (intro measure_eqI)
```
```  1181   fix A assume A: "A \<in> sets (distr (N \<Otimes>\<^sub>M M) N fst)"
```
```  1182   from A have "emeasure (distr (N \<Otimes>\<^sub>M M) N fst) A = emeasure (N \<Otimes>\<^sub>M M) (A \<times> space M)"
```
```  1183     by (auto simp add: emeasure_distr space_pair_measure dest: sets.sets_into_space intro!: arg_cong2[where f=emeasure])
```
```  1184   with A show "emeasure (distr (N \<Otimes>\<^sub>M M) N fst) A = emeasure N A"
```
```  1185     by (simp add: emeasure_pair_measure_Times emeasure_space_1)
```
```  1186 qed simp
```
```  1187
```
```  1188 lemma (in product_prob_space) distr_reorder:
```
```  1189   assumes "inj_on t J" "t \<in> J \<rightarrow> K" "finite K"
```
```  1190   shows "distr (PiM K M) (Pi\<^sub>M J (\<lambda>x. M (t x))) (\<lambda>\<omega>. \<lambda>n\<in>J. \<omega> (t n)) = PiM J (\<lambda>x. M (t x))"
```
```  1191 proof (rule product_sigma_finite.PiM_eqI)
```
```  1192   show "product_sigma_finite (\<lambda>x. M (t x))" ..
```
```  1193   have "t`J \<subseteq> K" using assms by auto
```
```  1194   then show [simp]: "finite J"
```
```  1195     by (rule finite_imageD[OF finite_subset]) fact+
```
```  1196   fix A assume A: "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M (t i))"
```
```  1197   moreover have "((\<lambda>\<omega>. \<lambda>n\<in>J. \<omega> (t n)) -` Pi\<^sub>E J A \<inter> space (Pi\<^sub>M K M)) =
```
```  1198     (\<Pi>\<^sub>E i\<in>K. if i \<in> t`J then A (the_inv_into J t i) else space (M i))"
```
```  1199     using A A[THEN sets.sets_into_space] \<open>t \<in> J \<rightarrow> K\<close> \<open>inj_on t J\<close>
```
```  1200     by (subst prod_emb_Pi[symmetric]) (auto simp: space_PiM PiE_iff the_inv_into_f_f prod_emb_def)
```
```  1201   ultimately show "distr (Pi\<^sub>M K M) (Pi\<^sub>M J (\<lambda>x. M (t x))) (\<lambda>\<omega>. \<lambda>n\<in>J. \<omega> (t n)) (Pi\<^sub>E J A) = (\<Prod>i\<in>J. M (t i) (A i))"
```
```  1202     using assms
```
```  1203     apply (subst emeasure_distr)
```
```  1204     apply (auto intro!: sets_PiM_I_finite simp: Pi_iff)
```
```  1205     apply (subst emeasure_PiM)
```
```  1206     apply (auto simp: the_inv_into_f_f \<open>inj_on t J\<close> setprod.reindex[OF \<open>inj_on t J\<close>]
```
```  1207       if_distrib[where f="emeasure (M _)"] setprod.If_cases emeasure_space_1 Int_absorb1 \<open>t`J \<subseteq> K\<close>)
```
```  1208     done
```
```  1209 qed simp
```
```  1210
```
```  1211 lemma (in product_prob_space) distr_restrict:
```
```  1212   "J \<subseteq> K \<Longrightarrow> finite K \<Longrightarrow> (\<Pi>\<^sub>M i\<in>J. M i) = distr (\<Pi>\<^sub>M i\<in>K. M i) (\<Pi>\<^sub>M i\<in>J. M i) (\<lambda>f. restrict f J)"
```
```  1213   using distr_reorder[of "\<lambda>x. x" J K] by (simp add: Pi_iff subset_eq)
```
```  1214
```
```  1215 lemma (in product_prob_space) emeasure_prod_emb[simp]:
```
```  1216   assumes L: "J \<subseteq> L" "finite L" and X: "X \<in> sets (Pi\<^sub>M J M)"
```
```  1217   shows "emeasure (Pi\<^sub>M L M) (prod_emb L M J X) = emeasure (Pi\<^sub>M J M) X"
```
```  1218   by (subst distr_restrict[OF L])
```
```  1219      (simp add: prod_emb_def space_PiM emeasure_distr measurable_restrict_subset L X)
```
```  1220
```
```  1221 lemma emeasure_distr_restrict:
```
```  1222   assumes "I \<subseteq> K" and Q[measurable_cong]: "sets Q = sets (PiM K M)" and A[measurable]: "A \<in> sets (PiM I M)"
```
```  1223   shows "emeasure (distr Q (PiM I M) (\<lambda>\<omega>. restrict \<omega> I)) A = emeasure Q (prod_emb K M I A)"
```
```  1224   using \<open>I\<subseteq>K\<close> sets_eq_imp_space_eq[OF Q]
```
```  1225   by (subst emeasure_distr)
```
```  1226      (auto simp: measurable_cong_sets[OF Q] prod_emb_def space_PiM[symmetric] intro!: measurable_restrict)
```
```  1227
```
```  1228 lemma (in prob_space) prob_space_completion: "prob_space (completion M)"
```
```  1229   by (rule prob_spaceI) (simp add: emeasure_space_1)
```
```  1230
```
```  1231 end
```