src/HOL/Probability/Probability_Measure.thy
author haftmann
Fri Jun 19 07:53:35 2015 +0200 (2015-06-19)
changeset 60517 f16e4fb20652
parent 59427 084330e2ec5e
child 61125 4c68426800de
permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
hoelzl@42148
     1
(*  Title:      HOL/Probability/Probability_Measure.thy
hoelzl@42067
     2
    Author:     Johannes Hölzl, TU München
hoelzl@42067
     3
    Author:     Armin Heller, TU München
hoelzl@42067
     4
*)
hoelzl@42067
     5
wenzelm@58876
     6
section {*Probability measure*}
hoelzl@42067
     7
hoelzl@42148
     8
theory Probability_Measure
hoelzl@47694
     9
  imports Lebesgue_Measure Radon_Nikodym
hoelzl@35582
    10
begin
hoelzl@35582
    11
hoelzl@45777
    12
locale prob_space = finite_measure +
hoelzl@47694
    13
  assumes emeasure_space_1: "emeasure M (space M) = 1"
hoelzl@38656
    14
hoelzl@45777
    15
lemma prob_spaceI[Pure.intro!]:
hoelzl@47694
    16
  assumes *: "emeasure M (space M) = 1"
hoelzl@45777
    17
  shows "prob_space M"
hoelzl@45777
    18
proof -
hoelzl@45777
    19
  interpret finite_measure M
hoelzl@45777
    20
  proof
hoelzl@47694
    21
    show "emeasure M (space M) \<noteq> \<infinity>" using * by simp 
hoelzl@45777
    22
  qed
hoelzl@45777
    23
  show "prob_space M" by default fact
hoelzl@38656
    24
qed
hoelzl@38656
    25
hoelzl@59425
    26
lemma prob_space_imp_sigma_finite: "prob_space M \<Longrightarrow> sigma_finite_measure M"
hoelzl@59425
    27
  unfolding prob_space_def finite_measure_def by simp
hoelzl@59425
    28
hoelzl@40859
    29
abbreviation (in prob_space) "events \<equiv> sets M"
hoelzl@47694
    30
abbreviation (in prob_space) "prob \<equiv> measure M"
hoelzl@47694
    31
abbreviation (in prob_space) "random_variable M' X \<equiv> X \<in> measurable M M'"
wenzelm@53015
    32
abbreviation (in prob_space) "expectation \<equiv> integral\<^sup>L M"
hoelzl@57235
    33
abbreviation (in prob_space) "variance X \<equiv> integral\<^sup>L M (\<lambda>x. (X x - expectation X)\<^sup>2)"
hoelzl@35582
    34
hoelzl@57447
    35
lemma (in prob_space) finite_measure [simp]: "finite_measure M"
hoelzl@57447
    36
  by unfold_locales
hoelzl@57447
    37
hoelzl@47694
    38
lemma (in prob_space) prob_space_distr:
hoelzl@47694
    39
  assumes f: "f \<in> measurable M M'" shows "prob_space (distr M M' f)"
hoelzl@47694
    40
proof (rule prob_spaceI)
hoelzl@47694
    41
  have "f -` space M' \<inter> space M = space M" using f by (auto dest: measurable_space)
hoelzl@47694
    42
  with f show "emeasure (distr M M' f) (space (distr M M' f)) = 1"
hoelzl@47694
    43
    by (auto simp: emeasure_distr emeasure_space_1)
hoelzl@43339
    44
qed
hoelzl@43339
    45
hoelzl@40859
    46
lemma (in prob_space) prob_space: "prob (space M) = 1"
hoelzl@47694
    47
  using emeasure_space_1 unfolding measure_def by (simp add: one_ereal_def)
hoelzl@41981
    48
hoelzl@41981
    49
lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1"
hoelzl@41981
    50
  using bounded_measure[of A] by (simp add: prob_space)
hoelzl@41981
    51
hoelzl@47694
    52
lemma (in prob_space) not_empty: "space M \<noteq> {}"
hoelzl@47694
    53
  using prob_space by auto
hoelzl@41981
    54
hoelzl@47694
    55
lemma (in prob_space) measure_le_1: "emeasure M X \<le> 1"
hoelzl@47694
    56
  using emeasure_space[of M X] by (simp add: emeasure_space_1)
hoelzl@42950
    57
hoelzl@43339
    58
lemma (in prob_space) AE_I_eq_1:
hoelzl@47694
    59
  assumes "emeasure M {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M"
hoelzl@47694
    60
  shows "AE x in M. P x"
hoelzl@43339
    61
proof (rule AE_I)
hoelzl@47694
    62
  show "emeasure M (space M - {x \<in> space M. P x}) = 0"
hoelzl@47694
    63
    using assms emeasure_space_1 by (simp add: emeasure_compl)
hoelzl@43339
    64
qed (insert assms, auto)
hoelzl@43339
    65
hoelzl@59000
    66
lemma prob_space_restrict_space:
hoelzl@59000
    67
  "S \<in> sets M \<Longrightarrow> emeasure M S = 1 \<Longrightarrow> prob_space (restrict_space M S)"
hoelzl@59000
    68
  by (intro prob_spaceI)
hoelzl@59000
    69
     (simp add: emeasure_restrict_space space_restrict_space)
hoelzl@59000
    70
hoelzl@40859
    71
lemma (in prob_space) prob_compl:
hoelzl@41981
    72
  assumes A: "A \<in> events"
hoelzl@38656
    73
  shows "prob (space M - A) = 1 - prob A"
hoelzl@41981
    74
  using finite_measure_compl[OF A] by (simp add: prob_space)
hoelzl@35582
    75
hoelzl@47694
    76
lemma (in prob_space) AE_in_set_eq_1:
hoelzl@47694
    77
  assumes "A \<in> events" shows "(AE x in M. x \<in> A) \<longleftrightarrow> prob A = 1"
hoelzl@47694
    78
proof
hoelzl@47694
    79
  assume ae: "AE x in M. x \<in> A"
hoelzl@47694
    80
  have "{x \<in> space M. x \<in> A} = A" "{x \<in> space M. x \<notin> A} = space M - A"
immler@50244
    81
    using `A \<in> events`[THEN sets.sets_into_space] by auto
hoelzl@47694
    82
  with AE_E2[OF ae] `A \<in> events` have "1 - emeasure M A = 0"
hoelzl@47694
    83
    by (simp add: emeasure_compl emeasure_space_1)
hoelzl@47694
    84
  then show "prob A = 1"
hoelzl@47694
    85
    using `A \<in> events` by (simp add: emeasure_eq_measure one_ereal_def)
hoelzl@47694
    86
next
hoelzl@47694
    87
  assume prob: "prob A = 1"
hoelzl@47694
    88
  show "AE x in M. x \<in> A"
hoelzl@47694
    89
  proof (rule AE_I)
hoelzl@47694
    90
    show "{x \<in> space M. x \<notin> A} \<subseteq> space M - A" by auto
hoelzl@47694
    91
    show "emeasure M (space M - A) = 0"
hoelzl@47694
    92
      using `A \<in> events` prob
hoelzl@47694
    93
      by (simp add: prob_compl emeasure_space_1 emeasure_eq_measure one_ereal_def)
hoelzl@47694
    94
    show "space M - A \<in> events"
hoelzl@47694
    95
      using `A \<in> events` by auto
hoelzl@47694
    96
  qed
hoelzl@47694
    97
qed
hoelzl@47694
    98
hoelzl@47694
    99
lemma (in prob_space) AE_False: "(AE x in M. False) \<longleftrightarrow> False"
hoelzl@47694
   100
proof
hoelzl@47694
   101
  assume "AE x in M. False"
hoelzl@47694
   102
  then have "AE x in M. x \<in> {}" by simp
hoelzl@47694
   103
  then show False
hoelzl@47694
   104
    by (subst (asm) AE_in_set_eq_1) auto
hoelzl@47694
   105
qed simp
hoelzl@47694
   106
hoelzl@47694
   107
lemma (in prob_space) AE_prob_1:
hoelzl@47694
   108
  assumes "prob A = 1" shows "AE x in M. x \<in> A"
hoelzl@47694
   109
proof -
hoelzl@47694
   110
  from `prob A = 1` have "A \<in> events"
hoelzl@47694
   111
    by (metis measure_notin_sets zero_neq_one)
hoelzl@47694
   112
  with AE_in_set_eq_1 assms show ?thesis by simp
hoelzl@47694
   113
qed
hoelzl@47694
   114
hoelzl@50098
   115
lemma (in prob_space) AE_const[simp]: "(AE x in M. P) \<longleftrightarrow> P"
hoelzl@50098
   116
  by (cases P) (auto simp: AE_False)
hoelzl@50098
   117
hoelzl@59000
   118
lemma (in prob_space) ae_filter_bot: "ae_filter M \<noteq> bot"
hoelzl@59000
   119
  by (simp add: trivial_limit_def)
hoelzl@59000
   120
hoelzl@50098
   121
lemma (in prob_space) AE_contr:
hoelzl@50098
   122
  assumes ae: "AE \<omega> in M. P \<omega>" "AE \<omega> in M. \<not> P \<omega>"
hoelzl@50098
   123
  shows False
hoelzl@50098
   124
proof -
hoelzl@50098
   125
  from ae have "AE \<omega> in M. False" by eventually_elim auto
hoelzl@50098
   126
  then show False by auto
hoelzl@50098
   127
qed
hoelzl@50098
   128
hoelzl@59000
   129
lemma (in prob_space) emeasure_eq_1_AE:
hoelzl@59000
   130
  "S \<in> sets M \<Longrightarrow> AE x in M. x \<in> S \<Longrightarrow> emeasure M S = 1"
hoelzl@59000
   131
  by (subst emeasure_eq_AE[where B="space M"]) (auto simp: emeasure_space_1)
hoelzl@59000
   132
hoelzl@57025
   133
lemma (in prob_space) integral_ge_const:
hoelzl@57025
   134
  fixes c :: real
hoelzl@57025
   135
  shows "integrable M f \<Longrightarrow> (AE x in M. c \<le> f x) \<Longrightarrow> c \<le> (\<integral>x. f x \<partial>M)"
hoelzl@57025
   136
  using integral_mono_AE[of M "\<lambda>x. c" f] prob_space by simp
hoelzl@57025
   137
hoelzl@57025
   138
lemma (in prob_space) integral_le_const:
hoelzl@57025
   139
  fixes c :: real
hoelzl@57025
   140
  shows "integrable M f \<Longrightarrow> (AE x in M. f x \<le> c) \<Longrightarrow> (\<integral>x. f x \<partial>M) \<le> c"
hoelzl@57025
   141
  using integral_mono_AE[of M f "\<lambda>x. c"] prob_space by simp
hoelzl@57025
   142
hoelzl@57025
   143
lemma (in prob_space) nn_integral_ge_const:
hoelzl@57025
   144
  "(AE x in M. c \<le> f x) \<Longrightarrow> c \<le> (\<integral>\<^sup>+x. f x \<partial>M)"
hoelzl@57025
   145
  using nn_integral_mono_AE[of "\<lambda>x. c" f M] emeasure_space_1
hoelzl@57025
   146
  by (simp add: nn_integral_const_If split: split_if_asm)
hoelzl@57025
   147
hoelzl@43339
   148
lemma (in prob_space) expectation_less:
hoelzl@56993
   149
  fixes X :: "_ \<Rightarrow> real"
hoelzl@43339
   150
  assumes [simp]: "integrable M X"
hoelzl@49786
   151
  assumes gt: "AE x in M. X x < b"
hoelzl@43339
   152
  shows "expectation X < b"
hoelzl@43339
   153
proof -
hoelzl@43339
   154
  have "expectation X < expectation (\<lambda>x. b)"
hoelzl@47694
   155
    using gt emeasure_space_1
hoelzl@43340
   156
    by (intro integral_less_AE_space) auto
hoelzl@43339
   157
  then show ?thesis using prob_space by simp
hoelzl@43339
   158
qed
hoelzl@43339
   159
hoelzl@43339
   160
lemma (in prob_space) expectation_greater:
hoelzl@56993
   161
  fixes X :: "_ \<Rightarrow> real"
hoelzl@43339
   162
  assumes [simp]: "integrable M X"
hoelzl@49786
   163
  assumes gt: "AE x in M. a < X x"
hoelzl@43339
   164
  shows "a < expectation X"
hoelzl@43339
   165
proof -
hoelzl@43339
   166
  have "expectation (\<lambda>x. a) < expectation X"
hoelzl@47694
   167
    using gt emeasure_space_1
hoelzl@43340
   168
    by (intro integral_less_AE_space) auto
hoelzl@43339
   169
  then show ?thesis using prob_space by simp
hoelzl@43339
   170
qed
hoelzl@43339
   171
hoelzl@43339
   172
lemma (in prob_space) jensens_inequality:
hoelzl@56993
   173
  fixes q :: "real \<Rightarrow> real"
hoelzl@49786
   174
  assumes X: "integrable M X" "AE x in M. X x \<in> I"
hoelzl@43339
   175
  assumes I: "I = {a <..< b} \<or> I = {a <..} \<or> I = {..< b} \<or> I = UNIV"
hoelzl@43339
   176
  assumes q: "integrable M (\<lambda>x. q (X x))" "convex_on I q"
hoelzl@43339
   177
  shows "q (expectation X) \<le> expectation (\<lambda>x. q (X x))"
hoelzl@43339
   178
proof -
wenzelm@46731
   179
  let ?F = "\<lambda>x. Inf ((\<lambda>t. (q x - q t) / (x - t)) ` ({x<..} \<inter> I))"
hoelzl@49786
   180
  from X(2) AE_False have "I \<noteq> {}" by auto
hoelzl@43339
   181
hoelzl@43339
   182
  from I have "open I" by auto
hoelzl@43339
   183
hoelzl@43339
   184
  note I
hoelzl@43339
   185
  moreover
hoelzl@43339
   186
  { assume "I \<subseteq> {a <..}"
hoelzl@43339
   187
    with X have "a < expectation X"
hoelzl@43339
   188
      by (intro expectation_greater) auto }
hoelzl@43339
   189
  moreover
hoelzl@43339
   190
  { assume "I \<subseteq> {..< b}"
hoelzl@43339
   191
    with X have "expectation X < b"
hoelzl@43339
   192
      by (intro expectation_less) auto }
hoelzl@43339
   193
  ultimately have "expectation X \<in> I"
hoelzl@43339
   194
    by (elim disjE)  (auto simp: subset_eq)
hoelzl@43339
   195
  moreover
hoelzl@43339
   196
  { fix y assume y: "y \<in> I"
hoelzl@43339
   197
    with q(2) `open I` have "Sup ((\<lambda>x. q x + ?F x * (y - x)) ` I) = q y"
haftmann@56166
   198
      by (auto intro!: cSup_eq_maximum convex_le_Inf_differential image_eqI [OF _ y] simp: interior_open simp del: Sup_image_eq Inf_image_eq) }
hoelzl@43339
   199
  ultimately have "q (expectation X) = Sup ((\<lambda>x. q x + ?F x * (expectation X - x)) ` I)"
hoelzl@43339
   200
    by simp
hoelzl@43339
   201
  also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
hoelzl@51475
   202
  proof (rule cSup_least)
hoelzl@43339
   203
    show "(\<lambda>x. q x + ?F x * (expectation X - x)) ` I \<noteq> {}"
hoelzl@43339
   204
      using `I \<noteq> {}` by auto
hoelzl@43339
   205
  next
hoelzl@43339
   206
    fix k assume "k \<in> (\<lambda>x. q x + ?F x * (expectation X - x)) ` I"
hoelzl@43339
   207
    then guess x .. note x = this
hoelzl@43339
   208
    have "q x + ?F x * (expectation X  - x) = expectation (\<lambda>w. q x + ?F x * (X w - x))"
hoelzl@47694
   209
      using prob_space by (simp add: X)
hoelzl@43339
   210
    also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
hoelzl@43339
   211
      using `x \<in> I` `open I` X(2)
hoelzl@56993
   212
      apply (intro integral_mono_AE integrable_add integrable_mult_right integrable_diff
hoelzl@56993
   213
                integrable_const X q)
hoelzl@49786
   214
      apply (elim eventually_elim1)
hoelzl@49786
   215
      apply (intro convex_le_Inf_differential)
hoelzl@49786
   216
      apply (auto simp: interior_open q)
hoelzl@49786
   217
      done
hoelzl@43339
   218
    finally show "k \<le> expectation (\<lambda>w. q (X w))" using x by auto
hoelzl@43339
   219
  qed
hoelzl@43339
   220
  finally show "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" .
hoelzl@43339
   221
qed
hoelzl@43339
   222
hoelzl@50001
   223
subsection  {* Introduce binder for probability *}
hoelzl@50001
   224
hoelzl@50001
   225
syntax
Andreas@59356
   226
  "_prob" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("('\<P>'((/_ in _./ _)'))")
hoelzl@50001
   227
hoelzl@50001
   228
translations
hoelzl@50001
   229
  "\<P>(x in M. P)" => "CONST measure M {x \<in> CONST space M. P}"
hoelzl@50001
   230
Andreas@58764
   231
print_translation {*
Andreas@58764
   232
  let
Andreas@58764
   233
    fun to_pattern (Const (@{const_syntax Pair}, _) $ l $ r) =
Andreas@58764
   234
      Syntax.const @{const_syntax Pair} :: to_pattern l @ to_pattern r
Andreas@58764
   235
    | to_pattern (t as (Const (@{syntax_const "_bound"}, _)) $ _) = [t]
Andreas@58764
   236
Andreas@58764
   237
    fun mk_pattern ((t, n) :: xs) = mk_patterns n xs |>> curry list_comb t
Andreas@58764
   238
    and mk_patterns 0 xs = ([], xs)
Andreas@58764
   239
    | mk_patterns n xs =
Andreas@58764
   240
      let
Andreas@58764
   241
        val (t, xs') = mk_pattern xs
Andreas@58764
   242
        val (ts, xs'') = mk_patterns (n - 1) xs'
Andreas@58764
   243
      in
Andreas@58764
   244
        (t :: ts, xs'')
Andreas@58764
   245
      end
Andreas@58764
   246
Andreas@58764
   247
    fun unnest_tuples
Andreas@58764
   248
      (Const (@{syntax_const "_pattern"}, _) $ 
Andreas@58764
   249
        t1 $
Andreas@58764
   250
        (t as (Const (@{syntax_const "_pattern"}, _) $ _ $ _)))
Andreas@58764
   251
      = let
Andreas@58764
   252
        val (_ $ t2 $ t3) = unnest_tuples t
Andreas@58764
   253
      in
Andreas@58764
   254
        Syntax.const @{syntax_const "_pattern"} $ 
Andreas@58764
   255
          unnest_tuples t1 $
Andreas@58764
   256
          (Syntax.const @{syntax_const "_patterns"} $ t2 $ t3)
Andreas@58764
   257
      end
Andreas@58764
   258
    | unnest_tuples pat = pat
Andreas@58764
   259
Andreas@58764
   260
    fun tr' [sig_alg, Const (@{const_syntax Collect}, _) $ t] = 
Andreas@58764
   261
      let
Andreas@58764
   262
        val bound_dummyT = Const (@{syntax_const "_bound"}, dummyT)
Andreas@58764
   263
Andreas@58764
   264
        fun go pattern elem
Andreas@58764
   265
          (Const (@{const_syntax "conj"}, _) $ 
Andreas@58764
   266
            (Const (@{const_syntax Set.member}, _) $ elem' $ (Const (@{const_syntax space}, _) $ sig_alg')) $
Andreas@58764
   267
            u)
Andreas@58764
   268
          = let
Andreas@58764
   269
              val _ = if sig_alg aconv sig_alg' andalso to_pattern elem' = rev elem then () else raise Match;
Andreas@58764
   270
              val (pat, rest) = mk_pattern (rev pattern);
Andreas@58764
   271
              val _ = case rest of [] => () | _ => raise Match
Andreas@58764
   272
            in
Andreas@58764
   273
              Syntax.const @{syntax_const "_prob"} $ unnest_tuples pat $ sig_alg $ u
Andreas@58764
   274
            end
Andreas@58764
   275
        | go pattern elem (Abs abs) =
Andreas@58764
   276
            let
Andreas@58764
   277
              val (x as (_ $ tx), t) = Syntax_Trans.atomic_abs_tr' abs
Andreas@58764
   278
            in
Andreas@58764
   279
              go ((x, 0) :: pattern) (bound_dummyT $ tx :: elem) t
Andreas@58764
   280
            end
Andreas@58764
   281
        | go pattern elem (Const (@{const_syntax case_prod}, _) $ t) =
Andreas@58764
   282
            go 
Andreas@58764
   283
              ((Syntax.const @{syntax_const "_pattern"}, 2) :: pattern)
Andreas@58764
   284
              (Syntax.const @{const_syntax Pair} :: elem)
Andreas@58764
   285
              t
Andreas@58764
   286
      in
Andreas@58764
   287
        go [] [] t
Andreas@58764
   288
      end
Andreas@58764
   289
  in
Andreas@58764
   290
    [(@{const_syntax Sigma_Algebra.measure}, K tr')]
Andreas@58764
   291
  end
Andreas@58764
   292
*}
Andreas@58764
   293
hoelzl@50001
   294
definition
hoelzl@50001
   295
  "cond_prob M P Q = \<P>(\<omega> in M. P \<omega> \<and> Q \<omega>) / \<P>(\<omega> in M. Q \<omega>)"
hoelzl@50001
   296
hoelzl@50001
   297
syntax
hoelzl@50001
   298
  "_conditional_prob" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("('\<P>'(_ in _. _ \<bar>/ _'))")
hoelzl@50001
   299
hoelzl@50001
   300
translations
hoelzl@50001
   301
  "\<P>(x in M. P \<bar> Q)" => "CONST cond_prob M (\<lambda>x. P) (\<lambda>x. Q)"
hoelzl@50001
   302
hoelzl@50001
   303
lemma (in prob_space) AE_E_prob:
hoelzl@50001
   304
  assumes ae: "AE x in M. P x"
hoelzl@50001
   305
  obtains S where "S \<subseteq> {x \<in> space M. P x}" "S \<in> events" "prob S = 1"
hoelzl@50001
   306
proof -
hoelzl@50001
   307
  from ae[THEN AE_E] guess N .
hoelzl@50001
   308
  then show thesis
hoelzl@50001
   309
    by (intro that[of "space M - N"])
hoelzl@50001
   310
       (auto simp: prob_compl prob_space emeasure_eq_measure)
hoelzl@50001
   311
qed
hoelzl@50001
   312
hoelzl@50001
   313
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)"
hoelzl@50001
   314
  by (auto intro!: arg_cong[where f=prob] simp add: prob_compl[symmetric])
hoelzl@50001
   315
hoelzl@50001
   316
lemma (in prob_space) prob_eq_AE:
hoelzl@50001
   317
  "(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)"
hoelzl@50001
   318
  by (rule finite_measure_eq_AE) auto
hoelzl@50001
   319
hoelzl@50001
   320
lemma (in prob_space) prob_eq_0_AE:
hoelzl@50001
   321
  assumes not: "AE x in M. \<not> P x" shows "\<P>(x in M. P x) = 0"
hoelzl@50001
   322
proof cases
hoelzl@50001
   323
  assume "{x\<in>space M. P x} \<in> events"
hoelzl@50001
   324
  with not have "\<P>(x in M. P x) = \<P>(x in M. False)"
hoelzl@50001
   325
    by (intro prob_eq_AE) auto
hoelzl@50001
   326
  then show ?thesis by simp
hoelzl@50001
   327
qed (simp add: measure_notin_sets)
hoelzl@50001
   328
hoelzl@50098
   329
lemma (in prob_space) prob_Collect_eq_0:
hoelzl@50098
   330
  "{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)"
hoelzl@50098
   331
  using AE_iff_measurable[OF _ refl, of M "\<lambda>x. \<not> P x"] by (simp add: emeasure_eq_measure)
hoelzl@50098
   332
hoelzl@50098
   333
lemma (in prob_space) prob_Collect_eq_1:
hoelzl@50098
   334
  "{x \<in> space M. P x} \<in> sets M \<Longrightarrow> \<P>(x in M. P x) = 1 \<longleftrightarrow> (AE x in M. P x)"
hoelzl@50098
   335
  using AE_in_set_eq_1[of "{x\<in>space M. P x}"] by simp
hoelzl@50098
   336
hoelzl@50098
   337
lemma (in prob_space) prob_eq_0:
hoelzl@50098
   338
  "A \<in> sets M \<Longrightarrow> prob A = 0 \<longleftrightarrow> (AE x in M. x \<notin> A)"
hoelzl@50098
   339
  using AE_iff_measurable[OF _ refl, of M "\<lambda>x. x \<notin> A"]
hoelzl@50098
   340
  by (auto simp add: emeasure_eq_measure Int_def[symmetric])
hoelzl@50098
   341
hoelzl@50098
   342
lemma (in prob_space) prob_eq_1:
hoelzl@50098
   343
  "A \<in> sets M \<Longrightarrow> prob A = 1 \<longleftrightarrow> (AE x in M. x \<in> A)"
hoelzl@50098
   344
  using AE_in_set_eq_1[of A] by simp
hoelzl@50098
   345
hoelzl@50001
   346
lemma (in prob_space) prob_sums:
hoelzl@50001
   347
  assumes P: "\<And>n. {x\<in>space M. P n x} \<in> events"
hoelzl@50001
   348
  assumes Q: "{x\<in>space M. Q x} \<in> events"
hoelzl@50001
   349
  assumes ae: "AE x in M. (\<forall>n. P n x \<longrightarrow> Q x) \<and> (Q x \<longrightarrow> (\<exists>!n. P n x))"
hoelzl@50001
   350
  shows "(\<lambda>n. \<P>(x in M. P n x)) sums \<P>(x in M. Q x)"
hoelzl@50001
   351
proof -
hoelzl@50001
   352
  from ae[THEN AE_E_prob] guess S . note S = this
hoelzl@50001
   353
  then have disj: "disjoint_family (\<lambda>n. {x\<in>space M. P n x} \<inter> S)"
hoelzl@50001
   354
    by (auto simp: disjoint_family_on_def)
hoelzl@50001
   355
  from S have ae_S:
hoelzl@50001
   356
    "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)"
hoelzl@50001
   357
    "\<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"
hoelzl@50001
   358
    using ae by (auto dest!: AE_prob_1)
hoelzl@50001
   359
  from ae_S have *:
hoelzl@50001
   360
    "\<P>(x in M. Q x) = prob (\<Union>n. {x\<in>space M. P n x} \<inter> S)"
hoelzl@50001
   361
    using P Q S by (intro finite_measure_eq_AE) auto
hoelzl@50001
   362
  from ae_S have **:
hoelzl@50001
   363
    "\<And>n. \<P>(x in M. P n x) = prob ({x\<in>space M. P n x} \<inter> S)"
hoelzl@50001
   364
    using P Q S by (intro finite_measure_eq_AE) auto
hoelzl@50001
   365
  show ?thesis
hoelzl@50001
   366
    unfolding * ** using S P disj
hoelzl@50001
   367
    by (intro finite_measure_UNION) auto
hoelzl@50001
   368
qed
hoelzl@50001
   369
hoelzl@59000
   370
lemma (in prob_space) prob_setsum:
hoelzl@59000
   371
  assumes [simp, intro]: "finite I"
hoelzl@59000
   372
  assumes P: "\<And>n. n \<in> I \<Longrightarrow> {x\<in>space M. P n x} \<in> events"
hoelzl@59000
   373
  assumes Q: "{x\<in>space M. Q x} \<in> events"
hoelzl@59000
   374
  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))"
hoelzl@59000
   375
  shows "\<P>(x in M. Q x) = (\<Sum>n\<in>I. \<P>(x in M. P n x))"
hoelzl@59000
   376
proof -
hoelzl@59000
   377
  from ae[THEN AE_E_prob] guess S . note S = this
hoelzl@59000
   378
  then have disj: "disjoint_family_on (\<lambda>n. {x\<in>space M. P n x} \<inter> S) I"
hoelzl@59000
   379
    by (auto simp: disjoint_family_on_def)
hoelzl@59000
   380
  from S have ae_S:
hoelzl@59000
   381
    "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)"
hoelzl@59000
   382
    "\<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"
hoelzl@59000
   383
    using ae by (auto dest!: AE_prob_1)
hoelzl@59000
   384
  from ae_S have *:
hoelzl@59000
   385
    "\<P>(x in M. Q x) = prob (\<Union>n\<in>I. {x\<in>space M. P n x} \<inter> S)"
hoelzl@59000
   386
    using P Q S by (intro finite_measure_eq_AE) (auto intro!: sets.Int)
hoelzl@59000
   387
  from ae_S have **:
hoelzl@59000
   388
    "\<And>n. n \<in> I \<Longrightarrow> \<P>(x in M. P n x) = prob ({x\<in>space M. P n x} \<inter> S)"
hoelzl@59000
   389
    using P Q S by (intro finite_measure_eq_AE) auto
hoelzl@59000
   390
  show ?thesis
hoelzl@59000
   391
    using S P disj
hoelzl@59000
   392
    by (auto simp add: * ** simp del: UN_simps intro!: finite_measure_finite_Union)
hoelzl@59000
   393
qed
hoelzl@59000
   394
hoelzl@54418
   395
lemma (in prob_space) prob_EX_countable:
hoelzl@54418
   396
  assumes sets: "\<And>i. i \<in> I \<Longrightarrow> {x\<in>space M. P i x} \<in> sets M" and I: "countable I" 
hoelzl@54418
   397
  assumes disj: "AE x in M. \<forall>i\<in>I. \<forall>j\<in>I. P i x \<longrightarrow> P j x \<longrightarrow> i = j"
hoelzl@54418
   398
  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)"
hoelzl@54418
   399
proof -
hoelzl@54418
   400
  let ?N= "\<lambda>x. \<exists>!i\<in>I. P i x"
hoelzl@54418
   401
  have "ereal (\<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))"
hoelzl@54418
   402
    unfolding ereal.inject
hoelzl@54418
   403
  proof (rule prob_eq_AE)
hoelzl@54418
   404
    show "AE x in M. (\<exists>i\<in>I. P i x) = (\<exists>i\<in>I. P i x \<and> ?N x)"
hoelzl@54418
   405
      using disj by eventually_elim blast
hoelzl@54418
   406
  qed (auto intro!: sets.sets_Collect_countable_Ex' sets.sets_Collect_conj sets.sets_Collect_countable_Ex1' I sets)+
hoelzl@54418
   407
  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})"
hoelzl@54418
   408
    unfolding emeasure_eq_measure by (auto intro!: arg_cong[where f=prob])
hoelzl@54418
   409
  also have "\<dots> = (\<integral>\<^sup>+i. emeasure M {x\<in>space M. P i x \<and> ?N x} \<partial>count_space I)"
hoelzl@54418
   410
    by (rule emeasure_UN_countable)
hoelzl@54418
   411
       (auto intro!: sets.sets_Collect_countable_Ex' sets.sets_Collect_conj sets.sets_Collect_countable_Ex1' I sets
hoelzl@54418
   412
             simp: disjoint_family_on_def)
hoelzl@54418
   413
  also have "\<dots> = (\<integral>\<^sup>+i. \<P>(x in M. P i x) \<partial>count_space I)"
hoelzl@54418
   414
    unfolding emeasure_eq_measure using disj
hoelzl@56996
   415
    by (intro nn_integral_cong ereal.inject[THEN iffD2] prob_eq_AE)
hoelzl@54418
   416
       (auto intro!: sets.sets_Collect_countable_Ex' sets.sets_Collect_conj sets.sets_Collect_countable_Ex1' I sets)+
hoelzl@54418
   417
  finally show ?thesis .
hoelzl@54418
   418
qed
hoelzl@54418
   419
hoelzl@50001
   420
lemma (in prob_space) cond_prob_eq_AE:
hoelzl@50001
   421
  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"
hoelzl@50001
   422
  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"
hoelzl@50001
   423
  shows "cond_prob M P Q = cond_prob M P' Q'"
hoelzl@50001
   424
  using P Q
immler@50244
   425
  by (auto simp: cond_prob_def intro!: arg_cong2[where f="op /"] prob_eq_AE sets.sets_Collect_conj)
hoelzl@50001
   426
hoelzl@50001
   427
hoelzl@40859
   428
lemma (in prob_space) joint_distribution_Times_le_fst:
hoelzl@47694
   429
  "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
wenzelm@53015
   430
    \<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"
hoelzl@47694
   431
  by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
hoelzl@40859
   432
hoelzl@40859
   433
lemma (in prob_space) joint_distribution_Times_le_snd:
hoelzl@47694
   434
  "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
wenzelm@53015
   435
    \<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"
hoelzl@47694
   436
  by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
hoelzl@40859
   437
hoelzl@57235
   438
lemma (in prob_space) variance_eq:
hoelzl@57235
   439
  fixes X :: "'a \<Rightarrow> real"
hoelzl@57235
   440
  assumes [simp]: "integrable M X"
hoelzl@57235
   441
  assumes [simp]: "integrable M (\<lambda>x. (X x)\<^sup>2)"
hoelzl@57235
   442
  shows "variance X = expectation (\<lambda>x. (X x)\<^sup>2) - (expectation X)\<^sup>2"
hoelzl@57235
   443
  by (simp add: field_simps prob_space power2_diff power2_eq_square[symmetric])
hoelzl@57235
   444
hoelzl@57235
   445
lemma (in prob_space) variance_positive: "0 \<le> variance (X::'a \<Rightarrow> real)"
hoelzl@57235
   446
  by (intro integral_nonneg_AE) (auto intro!: integral_nonneg_AE)
hoelzl@57235
   447
hoelzl@57447
   448
lemma (in prob_space) variance_mean_zero:
hoelzl@57447
   449
  "expectation X = 0 \<Longrightarrow> variance X = expectation (\<lambda>x. (X x)^2)"
hoelzl@57447
   450
  by simp
hoelzl@57447
   451
hoelzl@45777
   452
locale pair_prob_space = pair_sigma_finite M1 M2 + M1: prob_space M1 + M2: prob_space M2 for M1 M2
hoelzl@41689
   453
wenzelm@53015
   454
sublocale pair_prob_space \<subseteq> P: prob_space "M1 \<Otimes>\<^sub>M M2"
hoelzl@45777
   455
proof
wenzelm@53015
   456
  show "emeasure (M1 \<Otimes>\<^sub>M M2) (space (M1 \<Otimes>\<^sub>M M2)) = 1"
hoelzl@49776
   457
    by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_space_1 M2.emeasure_space_1 space_pair_measure)
hoelzl@45777
   458
qed
hoelzl@40859
   459
hoelzl@47694
   460
locale product_prob_space = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
hoelzl@45777
   461
  fixes I :: "'i set"
hoelzl@45777
   462
  assumes prob_space: "\<And>i. prob_space (M i)"
hoelzl@42988
   463
hoelzl@45777
   464
sublocale product_prob_space \<subseteq> M: prob_space "M i" for i
hoelzl@42988
   465
  by (rule prob_space)
hoelzl@42988
   466
hoelzl@45777
   467
locale finite_product_prob_space = finite_product_sigma_finite M I + product_prob_space M I for M I
hoelzl@42988
   468
wenzelm@53015
   469
sublocale finite_product_prob_space \<subseteq> prob_space "\<Pi>\<^sub>M i\<in>I. M i"
hoelzl@45777
   470
proof
wenzelm@53015
   471
  show "emeasure (\<Pi>\<^sub>M i\<in>I. M i) (space (\<Pi>\<^sub>M i\<in>I. M i)) = 1"
haftmann@57418
   472
    by (simp add: measure_times M.emeasure_space_1 setprod.neutral_const space_PiM)
hoelzl@45777
   473
qed
hoelzl@42988
   474
hoelzl@42988
   475
lemma (in finite_product_prob_space) prob_times:
hoelzl@42988
   476
  assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets (M i)"
wenzelm@53015
   477
  shows "prob (\<Pi>\<^sub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
hoelzl@42988
   478
proof -
wenzelm@53015
   479
  have "ereal (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)"
hoelzl@47694
   480
    using X by (simp add: emeasure_eq_measure)
hoelzl@47694
   481
  also have "\<dots> = (\<Prod>i\<in>I. emeasure (M i) (X i))"
hoelzl@42988
   482
    using measure_times X by simp
hoelzl@47694
   483
  also have "\<dots> = ereal (\<Prod>i\<in>I. measure (M i) (X i))"
hoelzl@47694
   484
    using X by (simp add: M.emeasure_eq_measure setprod_ereal)
hoelzl@42859
   485
  finally show ?thesis by simp
hoelzl@42859
   486
qed
hoelzl@42859
   487
hoelzl@56994
   488
subsection {* Distributions *}
hoelzl@42892
   489
hoelzl@47694
   490
definition "distributed M N X f \<longleftrightarrow> distr M N X = density N f \<and> 
hoelzl@47694
   491
  f \<in> borel_measurable N \<and> (AE x in N. 0 \<le> f x) \<and> X \<in> measurable M N"
hoelzl@36624
   492
hoelzl@47694
   493
lemma
hoelzl@50003
   494
  assumes "distributed M N X f"
hoelzl@50003
   495
  shows distributed_distr_eq_density: "distr M N X = density N f"
hoelzl@50003
   496
    and distributed_measurable: "X \<in> measurable M N"
hoelzl@50003
   497
    and distributed_borel_measurable: "f \<in> borel_measurable N"
hoelzl@50003
   498
    and distributed_AE: "(AE x in N. 0 \<le> f x)"
hoelzl@50003
   499
  using assms by (simp_all add: distributed_def)
hoelzl@50003
   500
hoelzl@50003
   501
lemma
hoelzl@50003
   502
  assumes D: "distributed M N X f"
hoelzl@50003
   503
  shows distributed_measurable'[measurable_dest]:
hoelzl@50003
   504
      "g \<in> measurable L M \<Longrightarrow> (\<lambda>x. X (g x)) \<in> measurable L N"
hoelzl@50003
   505
    and distributed_borel_measurable'[measurable_dest]:
hoelzl@50003
   506
      "h \<in> measurable L N \<Longrightarrow> (\<lambda>x. f (h x)) \<in> borel_measurable L"
hoelzl@50003
   507
  using distributed_measurable[OF D] distributed_borel_measurable[OF D]
hoelzl@50003
   508
  by simp_all
hoelzl@39097
   509
hoelzl@47694
   510
lemma
hoelzl@47694
   511
  shows distributed_real_measurable: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> f \<in> borel_measurable N"
hoelzl@47694
   512
    and distributed_real_AE: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> (AE x in N. 0 \<le> f x)"
hoelzl@47694
   513
  by (simp_all add: distributed_def borel_measurable_ereal_iff)
hoelzl@35977
   514
hoelzl@59353
   515
lemma distributed_real_measurable':
hoelzl@59353
   516
  "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> h \<in> measurable L N \<Longrightarrow> (\<lambda>x. f (h x)) \<in> borel_measurable L"
hoelzl@59353
   517
  by simp
hoelzl@50003
   518
hoelzl@59353
   519
lemma joint_distributed_measurable1:
hoelzl@59353
   520
  "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"
hoelzl@59353
   521
  by simp
hoelzl@59353
   522
hoelzl@59353
   523
lemma joint_distributed_measurable2:
hoelzl@59353
   524
  "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"
hoelzl@59353
   525
  by simp
hoelzl@50003
   526
hoelzl@47694
   527
lemma distributed_count_space:
hoelzl@47694
   528
  assumes X: "distributed M (count_space A) X P" and a: "a \<in> A" and A: "finite A"
hoelzl@47694
   529
  shows "P a = emeasure M (X -` {a} \<inter> space M)"
hoelzl@39097
   530
proof -
hoelzl@47694
   531
  have "emeasure M (X -` {a} \<inter> space M) = emeasure (distr M (count_space A) X) {a}"
hoelzl@50003
   532
    using X a A by (simp add: emeasure_distr)
hoelzl@47694
   533
  also have "\<dots> = emeasure (density (count_space A) P) {a}"
hoelzl@47694
   534
    using X by (simp add: distributed_distr_eq_density)
wenzelm@53015
   535
  also have "\<dots> = (\<integral>\<^sup>+x. P a * indicator {a} x \<partial>count_space A)"
hoelzl@56996
   536
    using X a by (auto simp add: emeasure_density distributed_def indicator_def intro!: nn_integral_cong)
hoelzl@47694
   537
  also have "\<dots> = P a"
hoelzl@56996
   538
    using X a by (subst nn_integral_cmult_indicator) (auto simp: distributed_def one_ereal_def[symmetric] AE_count_space)
hoelzl@47694
   539
  finally show ?thesis ..
hoelzl@39092
   540
qed
hoelzl@35977
   541
hoelzl@47694
   542
lemma distributed_cong_density:
hoelzl@47694
   543
  "(AE x in N. f x = g x) \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable N \<Longrightarrow>
hoelzl@47694
   544
    distributed M N X f \<longleftrightarrow> distributed M N X g"
hoelzl@47694
   545
  by (auto simp: distributed_def intro!: density_cong)
hoelzl@47694
   546
hoelzl@47694
   547
lemma subdensity:
hoelzl@47694
   548
  assumes T: "T \<in> measurable P Q"
hoelzl@47694
   549
  assumes f: "distributed M P X f"
hoelzl@47694
   550
  assumes g: "distributed M Q Y g"
hoelzl@47694
   551
  assumes Y: "Y = T \<circ> X"
hoelzl@47694
   552
  shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
hoelzl@47694
   553
proof -
hoelzl@47694
   554
  have "{x\<in>space Q. g x = 0} \<in> null_sets (distr M Q (T \<circ> X))"
hoelzl@47694
   555
    using g Y by (auto simp: null_sets_density_iff distributed_def)
hoelzl@47694
   556
  also have "distr M Q (T \<circ> X) = distr (distr M P X) Q T"
hoelzl@47694
   557
    using T f[THEN distributed_measurable] by (rule distr_distr[symmetric])
hoelzl@47694
   558
  finally have "T -` {x\<in>space Q. g x = 0} \<inter> space P \<in> null_sets (distr M P X)"
hoelzl@47694
   559
    using T by (subst (asm) null_sets_distr_iff) auto
hoelzl@47694
   560
  also have "T -` {x\<in>space Q. g x = 0} \<inter> space P = {x\<in>space P. g (T x) = 0}"
hoelzl@47694
   561
    using T by (auto dest: measurable_space)
hoelzl@47694
   562
  finally show ?thesis
hoelzl@47694
   563
    using f g by (auto simp add: null_sets_density_iff distributed_def)
hoelzl@35977
   564
qed
hoelzl@35977
   565
hoelzl@47694
   566
lemma subdensity_real:
hoelzl@47694
   567
  fixes g :: "'a \<Rightarrow> real" and f :: "'b \<Rightarrow> real"
hoelzl@47694
   568
  assumes T: "T \<in> measurable P Q"
hoelzl@47694
   569
  assumes f: "distributed M P X f"
hoelzl@47694
   570
  assumes g: "distributed M Q Y g"
hoelzl@47694
   571
  assumes Y: "Y = T \<circ> X"
hoelzl@47694
   572
  shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
hoelzl@47694
   573
  using subdensity[OF T, of M X "\<lambda>x. ereal (f x)" Y "\<lambda>x. ereal (g x)"] assms by auto
hoelzl@47694
   574
hoelzl@49788
   575
lemma distributed_emeasure:
wenzelm@53015
   576
  "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)"
hoelzl@50003
   577
  by (auto simp: distributed_AE
hoelzl@49788
   578
                 distributed_distr_eq_density[symmetric] emeasure_density[symmetric] emeasure_distr)
hoelzl@49788
   579
hoelzl@56996
   580
lemma distributed_nn_integral:
wenzelm@53015
   581
  "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)"
hoelzl@50003
   582
  by (auto simp: distributed_AE
hoelzl@56996
   583
                 distributed_distr_eq_density[symmetric] nn_integral_density[symmetric] nn_integral_distr)
hoelzl@49788
   584
hoelzl@47694
   585
lemma distributed_integral:
hoelzl@47694
   586
  "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> (\<integral>x. f x * g x \<partial>N) = (\<integral>x. g (X x) \<partial>M)"
hoelzl@50003
   587
  by (auto simp: distributed_real_AE
hoelzl@56993
   588
                 distributed_distr_eq_density[symmetric] integral_real_density[symmetric] integral_distr)
hoelzl@47694
   589
  
hoelzl@47694
   590
lemma distributed_transform_integral:
hoelzl@47694
   591
  assumes Px: "distributed M N X Px"
hoelzl@47694
   592
  assumes "distributed M P Y Py"
hoelzl@47694
   593
  assumes Y: "Y = T \<circ> X" and T: "T \<in> measurable N P" and f: "f \<in> borel_measurable P"
hoelzl@47694
   594
  shows "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. Px x * f (T x) \<partial>N)"
hoelzl@41689
   595
proof -
hoelzl@47694
   596
  have "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. f (Y x) \<partial>M)"
hoelzl@47694
   597
    by (rule distributed_integral) fact+
hoelzl@47694
   598
  also have "\<dots> = (\<integral>x. f (T (X x)) \<partial>M)"
hoelzl@47694
   599
    using Y by simp
hoelzl@47694
   600
  also have "\<dots> = (\<integral>x. Px x * f (T x) \<partial>N)"
hoelzl@47694
   601
    using measurable_comp[OF T f] Px by (intro distributed_integral[symmetric]) (auto simp: comp_def)
hoelzl@45777
   602
  finally show ?thesis .
hoelzl@39092
   603
qed
hoelzl@36624
   604
hoelzl@49788
   605
lemma (in prob_space) distributed_unique:
hoelzl@47694
   606
  assumes Px: "distributed M S X Px"
hoelzl@49788
   607
  assumes Py: "distributed M S X Py"
hoelzl@49788
   608
  shows "AE x in S. Px x = Py x"
hoelzl@49788
   609
proof -
hoelzl@49788
   610
  interpret X: prob_space "distr M S X"
hoelzl@50003
   611
    using Px by (intro prob_space_distr) simp
hoelzl@49788
   612
  have "sigma_finite_measure (distr M S X)" ..
hoelzl@49788
   613
  with sigma_finite_density_unique[of Px S Py ] Px Py
hoelzl@49788
   614
  show ?thesis
hoelzl@49788
   615
    by (auto simp: distributed_def)
hoelzl@49788
   616
qed
hoelzl@49788
   617
hoelzl@49788
   618
lemma (in prob_space) distributed_jointI:
hoelzl@49788
   619
  assumes "sigma_finite_measure S" "sigma_finite_measure T"
hoelzl@50003
   620
  assumes X[measurable]: "X \<in> measurable M S" and Y[measurable]: "Y \<in> measurable M T"
wenzelm@53015
   621
  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"
hoelzl@49788
   622
  assumes eq: "\<And>A B. A \<in> sets S \<Longrightarrow> B \<in> sets T \<Longrightarrow> 
wenzelm@53015
   623
    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)"
wenzelm@53015
   624
  shows "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) f"
hoelzl@49788
   625
  unfolding distributed_def
hoelzl@49788
   626
proof safe
hoelzl@49788
   627
  interpret S: sigma_finite_measure S by fact
hoelzl@49788
   628
  interpret T: sigma_finite_measure T by fact
hoelzl@49788
   629
  interpret ST: pair_sigma_finite S T by default
hoelzl@47694
   630
hoelzl@49788
   631
  from ST.sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('b \<times> 'c) set" .. note F = this
hoelzl@49788
   632
  let ?E = "{a \<times> b |a b. a \<in> sets S \<and> b \<in> sets T}"
wenzelm@53015
   633
  let ?P = "S \<Otimes>\<^sub>M T"
hoelzl@49788
   634
  show "distr M ?P (\<lambda>x. (X x, Y x)) = density ?P f" (is "?L = ?R")
hoelzl@49788
   635
  proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of S T]])
hoelzl@49788
   636
    show "?E \<subseteq> Pow (space ?P)"
immler@50244
   637
      using sets.space_closed[of S] sets.space_closed[of T] by (auto simp: space_pair_measure)
hoelzl@49788
   638
    show "sets ?L = sigma_sets (space ?P) ?E"
hoelzl@49788
   639
      by (simp add: sets_pair_measure space_pair_measure)
hoelzl@49788
   640
    then show "sets ?R = sigma_sets (space ?P) ?E"
hoelzl@49788
   641
      by simp
hoelzl@49788
   642
  next
hoelzl@49788
   643
    interpret L: prob_space ?L
hoelzl@49788
   644
      by (rule prob_space_distr) (auto intro!: measurable_Pair)
hoelzl@49788
   645
    show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?L (F i) \<noteq> \<infinity>"
hoelzl@49788
   646
      using F by (auto simp: space_pair_measure)
hoelzl@49788
   647
  next
hoelzl@49788
   648
    fix E assume "E \<in> ?E"
hoelzl@50003
   649
    then obtain A B where E[simp]: "E = A \<times> B"
hoelzl@50003
   650
      and A[measurable]: "A \<in> sets S" and B[measurable]: "B \<in> sets T" by auto
hoelzl@49788
   651
    have "emeasure ?L E = emeasure M {x \<in> space M. X x \<in> A \<and> Y x \<in> B}"
hoelzl@49788
   652
      by (auto intro!: arg_cong[where f="emeasure M"] simp add: emeasure_distr measurable_Pair)
wenzelm@53015
   653
    also have "\<dots> = (\<integral>\<^sup>+x. (\<integral>\<^sup>+y. (f (x, y) * indicator B y) * indicator A x \<partial>T) \<partial>S)"
hoelzl@56996
   654
      using f by (auto simp add: eq nn_integral_multc intro!: nn_integral_cong)
hoelzl@49788
   655
    also have "\<dots> = emeasure ?R E"
hoelzl@56996
   656
      by (auto simp add: emeasure_density T.nn_integral_fst[symmetric]
hoelzl@56996
   657
               intro!: nn_integral_cong split: split_indicator)
hoelzl@49788
   658
    finally show "emeasure ?L E = emeasure ?R E" .
hoelzl@49788
   659
  qed
hoelzl@50003
   660
qed (auto simp: f)
hoelzl@49788
   661
hoelzl@49788
   662
lemma (in prob_space) distributed_swap:
hoelzl@49788
   663
  assumes "sigma_finite_measure S" "sigma_finite_measure T"
wenzelm@53015
   664
  assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
wenzelm@53015
   665
  shows "distributed M (T \<Otimes>\<^sub>M S) (\<lambda>x. (Y x, X x)) (\<lambda>(x, y). Pxy (y, x))"
hoelzl@49788
   666
proof -
hoelzl@49788
   667
  interpret S: sigma_finite_measure S by fact
hoelzl@49788
   668
  interpret T: sigma_finite_measure T by fact
hoelzl@49788
   669
  interpret ST: pair_sigma_finite S T by default
hoelzl@49788
   670
  interpret TS: pair_sigma_finite T S by default
hoelzl@49788
   671
hoelzl@50003
   672
  note Pxy[measurable]
hoelzl@49788
   673
  show ?thesis 
hoelzl@49788
   674
    apply (subst TS.distr_pair_swap)
hoelzl@49788
   675
    unfolding distributed_def
hoelzl@49788
   676
  proof safe
wenzelm@53015
   677
    let ?D = "distr (S \<Otimes>\<^sub>M T) (T \<Otimes>\<^sub>M S) (\<lambda>(x, y). (y, x))"
hoelzl@49788
   678
    show 1: "(\<lambda>(x, y). Pxy (y, x)) \<in> borel_measurable ?D"
hoelzl@50003
   679
      by auto
hoelzl@49788
   680
    with Pxy
wenzelm@53015
   681
    show "AE x in distr (S \<Otimes>\<^sub>M T) (T \<Otimes>\<^sub>M S) (\<lambda>(x, y). (y, x)). 0 \<le> (case x of (x, y) \<Rightarrow> Pxy (y, x))"
hoelzl@49788
   682
      by (subst AE_distr_iff)
hoelzl@49788
   683
         (auto dest!: distributed_AE
hoelzl@49788
   684
               simp: measurable_split_conv split_beta
hoelzl@51683
   685
               intro!: measurable_Pair)
wenzelm@53015
   686
    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))"
hoelzl@50003
   687
      using Pxy by auto
wenzelm@53015
   688
    { fix A assume A: "A \<in> sets (T \<Otimes>\<^sub>M S)"
wenzelm@53015
   689
      let ?B = "(\<lambda>(x, y). (y, x)) -` A \<inter> space (S \<Otimes>\<^sub>M T)"
immler@50244
   690
      from sets.sets_into_space[OF A]
hoelzl@49788
   691
      have "emeasure M ((\<lambda>x. (Y x, X x)) -` A \<inter> space M) =
hoelzl@49788
   692
        emeasure M ((\<lambda>x. (X x, Y x)) -` ?B \<inter> space M)"
hoelzl@49788
   693
        by (auto intro!: arg_cong2[where f=emeasure] simp: space_pair_measure)
wenzelm@53015
   694
      also have "\<dots> = (\<integral>\<^sup>+ x. Pxy x * indicator ?B x \<partial>(S \<Otimes>\<^sub>M T))"
hoelzl@50003
   695
        using Pxy A by (intro distributed_emeasure) auto
hoelzl@49788
   696
      finally have "emeasure M ((\<lambda>x. (Y x, X x)) -` A \<inter> space M) =
wenzelm@53015
   697
        (\<integral>\<^sup>+ x. Pxy x * indicator A (snd x, fst x) \<partial>(S \<Otimes>\<^sub>M T))"
hoelzl@56996
   698
        by (auto intro!: nn_integral_cong split: split_indicator) }
hoelzl@49788
   699
    note * = this
hoelzl@49788
   700
    show "distr M ?D (\<lambda>x. (Y x, X x)) = density ?D (\<lambda>(x, y). Pxy (y, x))"
hoelzl@49788
   701
      apply (intro measure_eqI)
hoelzl@49788
   702
      apply (simp_all add: emeasure_distr[OF 2] emeasure_density[OF 1])
hoelzl@56996
   703
      apply (subst nn_integral_distr)
hoelzl@50003
   704
      apply (auto intro!: * simp: comp_def split_beta)
hoelzl@49788
   705
      done
hoelzl@49788
   706
  qed
hoelzl@36624
   707
qed
hoelzl@36624
   708
hoelzl@47694
   709
lemma (in prob_space) distr_marginal1:
hoelzl@47694
   710
  assumes "sigma_finite_measure S" "sigma_finite_measure T"
wenzelm@53015
   711
  assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
wenzelm@53015
   712
  defines "Px \<equiv> \<lambda>x. (\<integral>\<^sup>+z. Pxy (x, z) \<partial>T)"
hoelzl@47694
   713
  shows "distributed M S X Px"
hoelzl@47694
   714
  unfolding distributed_def
hoelzl@47694
   715
proof safe
hoelzl@47694
   716
  interpret S: sigma_finite_measure S by fact
hoelzl@47694
   717
  interpret T: sigma_finite_measure T by fact
hoelzl@47694
   718
  interpret ST: pair_sigma_finite S T by default
hoelzl@47694
   719
hoelzl@50003
   720
  note Pxy[measurable]
hoelzl@50003
   721
  show X: "X \<in> measurable M S" by simp
hoelzl@47694
   722
hoelzl@50003
   723
  show borel: "Px \<in> borel_measurable S"
hoelzl@56996
   724
    by (auto intro!: T.nn_integral_fst simp: Px_def)
hoelzl@39097
   725
wenzelm@53015
   726
  interpret Pxy: prob_space "distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
hoelzl@50003
   727
    by (intro prob_space_distr) simp
wenzelm@53015
   728
  have "(\<integral>\<^sup>+ x. max 0 (- Pxy x) \<partial>(S \<Otimes>\<^sub>M T)) = (\<integral>\<^sup>+ x. 0 \<partial>(S \<Otimes>\<^sub>M T))"
hoelzl@47694
   729
    using Pxy
hoelzl@56996
   730
    by (intro nn_integral_cong_AE) (auto simp: max_def dest: distributed_AE)
hoelzl@49788
   731
hoelzl@47694
   732
  show "distr M S X = density S Px"
hoelzl@47694
   733
  proof (rule measure_eqI)
hoelzl@47694
   734
    fix A assume A: "A \<in> sets (distr M S X)"
hoelzl@50003
   735
    with X measurable_space[of Y M T]
wenzelm@53015
   736
    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)"
hoelzl@50003
   737
      by (auto simp add: emeasure_distr intro!: arg_cong[where f="emeasure M"])
wenzelm@53015
   738
    also have "\<dots> = emeasure (density (S \<Otimes>\<^sub>M T) Pxy) (A \<times> space T)"
hoelzl@47694
   739
      using Pxy by (simp add: distributed_def)
wenzelm@53015
   740
    also have "\<dots> = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T \<partial>S"
hoelzl@47694
   741
      using A borel Pxy
hoelzl@56996
   742
      by (simp add: emeasure_density T.nn_integral_fst[symmetric])
wenzelm@53015
   743
    also have "\<dots> = \<integral>\<^sup>+ x. Px x * indicator A x \<partial>S"
hoelzl@56996
   744
      apply (rule nn_integral_cong_AE)
hoelzl@49788
   745
      using Pxy[THEN distributed_AE, THEN ST.AE_pair] AE_space
hoelzl@47694
   746
    proof eventually_elim
hoelzl@49788
   747
      fix x assume "x \<in> space S" "AE y in T. 0 \<le> Pxy (x, y)"
hoelzl@47694
   748
      moreover have eq: "\<And>y. y \<in> space T \<Longrightarrow> indicator (A \<times> space T) (x, y) = indicator A x"
hoelzl@47694
   749
        by (auto simp: indicator_def)
wenzelm@53015
   750
      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"
hoelzl@56996
   751
        by (simp add: eq nn_integral_multc cong: nn_integral_cong)
wenzelm@53015
   752
      also have "(\<integral>\<^sup>+ y. Pxy (x, y) \<partial>T) = Px x"
hoelzl@56996
   753
        by (simp add: Px_def ereal_real nn_integral_nonneg)
wenzelm@53015
   754
      finally show "(\<integral>\<^sup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T) = Px x * indicator A x" .
hoelzl@47694
   755
    qed
hoelzl@47694
   756
    finally show "emeasure (distr M S X) A = emeasure (density S Px) A"
hoelzl@47694
   757
      using A borel Pxy by (simp add: emeasure_density)
hoelzl@47694
   758
  qed simp
hoelzl@47694
   759
  
hoelzl@49788
   760
  show "AE x in S. 0 \<le> Px x"
hoelzl@56996
   761
    by (simp add: Px_def nn_integral_nonneg real_of_ereal_pos)
hoelzl@40859
   762
qed
hoelzl@40859
   763
hoelzl@49788
   764
lemma (in prob_space) distr_marginal2:
hoelzl@49788
   765
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
wenzelm@53015
   766
  assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
wenzelm@53015
   767
  shows "distributed M T Y (\<lambda>y. (\<integral>\<^sup>+x. Pxy (x, y) \<partial>S))"
hoelzl@49788
   768
  using distr_marginal1[OF T S distributed_swap[OF S T]] Pxy by simp
hoelzl@49788
   769
hoelzl@49788
   770
lemma (in prob_space) distributed_marginal_eq_joint1:
hoelzl@49788
   771
  assumes T: "sigma_finite_measure T"
hoelzl@49788
   772
  assumes S: "sigma_finite_measure S"
hoelzl@49788
   773
  assumes Px: "distributed M S X Px"
wenzelm@53015
   774
  assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
wenzelm@53015
   775
  shows "AE x in S. Px x = (\<integral>\<^sup>+y. Pxy (x, y) \<partial>T)"
hoelzl@49788
   776
  using Px distr_marginal1[OF S T Pxy] by (rule distributed_unique)
hoelzl@49788
   777
hoelzl@49788
   778
lemma (in prob_space) distributed_marginal_eq_joint2:
hoelzl@49788
   779
  assumes T: "sigma_finite_measure T"
hoelzl@49788
   780
  assumes S: "sigma_finite_measure S"
hoelzl@49788
   781
  assumes Py: "distributed M T Y Py"
wenzelm@53015
   782
  assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
wenzelm@53015
   783
  shows "AE y in T. Py y = (\<integral>\<^sup>+x. Pxy (x, y) \<partial>S)"
hoelzl@49788
   784
  using Py distr_marginal2[OF S T Pxy] by (rule distributed_unique)
hoelzl@49788
   785
hoelzl@49795
   786
lemma (in prob_space) distributed_joint_indep':
hoelzl@49795
   787
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@50003
   788
  assumes X[measurable]: "distributed M S X Px" and Y[measurable]: "distributed M T Y Py"
wenzelm@53015
   789
  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))"
wenzelm@53015
   790
  shows "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>(x, y). Px x * Py y)"
hoelzl@49795
   791
  unfolding distributed_def
hoelzl@49795
   792
proof safe
hoelzl@49795
   793
  interpret S: sigma_finite_measure S by fact
hoelzl@49795
   794
  interpret T: sigma_finite_measure T by fact
hoelzl@49795
   795
  interpret ST: pair_sigma_finite S T by default
hoelzl@49795
   796
hoelzl@49795
   797
  interpret X: prob_space "density S Px"
hoelzl@49795
   798
    unfolding distributed_distr_eq_density[OF X, symmetric]
hoelzl@50003
   799
    by (rule prob_space_distr) simp
hoelzl@49795
   800
  have sf_X: "sigma_finite_measure (density S Px)" ..
hoelzl@49795
   801
hoelzl@49795
   802
  interpret Y: prob_space "density T Py"
hoelzl@49795
   803
    unfolding distributed_distr_eq_density[OF Y, symmetric]
hoelzl@50003
   804
    by (rule prob_space_distr) simp
hoelzl@49795
   805
  have sf_Y: "sigma_finite_measure (density T Py)" ..
hoelzl@49795
   806
wenzelm@53015
   807
  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)"
hoelzl@49795
   808
    unfolding indep[symmetric] distributed_distr_eq_density[OF X] distributed_distr_eq_density[OF Y]
hoelzl@49795
   809
    using distributed_borel_measurable[OF X] distributed_AE[OF X]
hoelzl@49795
   810
    using distributed_borel_measurable[OF Y] distributed_AE[OF Y]
hoelzl@50003
   811
    by (rule pair_measure_density[OF _ _ _ _ T sf_Y])
hoelzl@49795
   812
wenzelm@53015
   813
  show "random_variable (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))" by auto
hoelzl@49795
   814
wenzelm@53015
   815
  show Pxy: "(\<lambda>(x, y). Px x * Py y) \<in> borel_measurable (S \<Otimes>\<^sub>M T)" by auto
hoelzl@49795
   816
wenzelm@53015
   817
  show "AE x in S \<Otimes>\<^sub>M T. 0 \<le> (case x of (x, y) \<Rightarrow> Px x * Py y)"
hoelzl@51683
   818
    apply (intro ST.AE_pair_measure borel_measurable_le Pxy borel_measurable_const)
hoelzl@49795
   819
    using distributed_AE[OF X]
hoelzl@49795
   820
    apply eventually_elim
hoelzl@49795
   821
    using distributed_AE[OF Y]
hoelzl@49795
   822
    apply eventually_elim
hoelzl@49795
   823
    apply auto
hoelzl@49795
   824
    done
hoelzl@49795
   825
qed
hoelzl@49795
   826
hoelzl@57235
   827
lemma distributed_integrable:
hoelzl@57235
   828
  "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow>
hoelzl@57235
   829
    integrable N (\<lambda>x. f x * g x) \<longleftrightarrow> integrable M (\<lambda>x. g (X x))"
hoelzl@57235
   830
  by (auto simp: distributed_real_AE
hoelzl@57235
   831
                    distributed_distr_eq_density[symmetric] integrable_real_density[symmetric] integrable_distr_eq)
hoelzl@57235
   832
  
hoelzl@57235
   833
lemma distributed_transform_integrable:
hoelzl@57235
   834
  assumes Px: "distributed M N X Px"
hoelzl@57235
   835
  assumes "distributed M P Y Py"
hoelzl@57235
   836
  assumes Y: "Y = (\<lambda>x. T (X x))" and T: "T \<in> measurable N P" and f: "f \<in> borel_measurable P"
hoelzl@57235
   837
  shows "integrable P (\<lambda>x. Py x * f x) \<longleftrightarrow> integrable N (\<lambda>x. Px x * f (T x))"
hoelzl@57235
   838
proof -
hoelzl@57235
   839
  have "integrable P (\<lambda>x. Py x * f x) \<longleftrightarrow> integrable M (\<lambda>x. f (Y x))"
hoelzl@57235
   840
    by (rule distributed_integrable) fact+
hoelzl@57235
   841
  also have "\<dots> \<longleftrightarrow> integrable M (\<lambda>x. f (T (X x)))"
hoelzl@57235
   842
    using Y by simp
hoelzl@57235
   843
  also have "\<dots> \<longleftrightarrow> integrable N (\<lambda>x. Px x * f (T x))"
hoelzl@57235
   844
    using measurable_comp[OF T f] Px by (intro distributed_integrable[symmetric]) (auto simp: comp_def)
hoelzl@57235
   845
  finally show ?thesis .
hoelzl@57235
   846
qed
hoelzl@57235
   847
hoelzl@57275
   848
lemma distributed_integrable_var:
hoelzl@57275
   849
  fixes X :: "'a \<Rightarrow> real"
hoelzl@57275
   850
  shows "distributed M lborel X (\<lambda>x. ereal (f x)) \<Longrightarrow> integrable lborel (\<lambda>x. f x * x) \<Longrightarrow> integrable M X"
hoelzl@57275
   851
  using distributed_integrable[of M lborel X f "\<lambda>x. x"] by simp
hoelzl@57275
   852
hoelzl@57235
   853
lemma (in prob_space) distributed_variance:
hoelzl@57235
   854
  fixes f::"real \<Rightarrow> real"
hoelzl@57235
   855
  assumes D: "distributed M lborel X f"
hoelzl@57235
   856
  shows "variance X = (\<integral>x. x\<^sup>2 * f (x + expectation X) \<partial>lborel)"
hoelzl@57235
   857
proof (subst distributed_integral[OF D, symmetric])
hoelzl@57235
   858
  show "(\<integral> x. f x * (x - expectation X)\<^sup>2 \<partial>lborel) = (\<integral> x. x\<^sup>2 * f (x + expectation X) \<partial>lborel)"
hoelzl@57235
   859
    by (subst lborel_integral_real_affine[where c=1 and t="expectation X"])  (auto simp: ac_simps)
hoelzl@57235
   860
qed simp
hoelzl@57235
   861
hoelzl@57235
   862
lemma (in prob_space) variance_affine:
hoelzl@57235
   863
  fixes f::"real \<Rightarrow> real"
hoelzl@57235
   864
  assumes [arith]: "b \<noteq> 0"
hoelzl@57235
   865
  assumes D[intro]: "distributed M lborel X f"
hoelzl@57235
   866
  assumes [simp]: "prob_space (density lborel f)"
hoelzl@57235
   867
  assumes I[simp]: "integrable M X"
hoelzl@57235
   868
  assumes I2[simp]: "integrable M (\<lambda>x. (X x)\<^sup>2)" 
hoelzl@57235
   869
  shows "variance (\<lambda>x. a + b * X x) = b\<^sup>2 * variance X"
hoelzl@57235
   870
  by (subst variance_eq)
hoelzl@57235
   871
     (auto simp: power2_sum power_mult_distrib prob_space variance_eq right_diff_distrib)
hoelzl@57235
   872
hoelzl@47694
   873
definition
hoelzl@47694
   874
  "simple_distributed M X f \<longleftrightarrow> distributed M (count_space (X`space M)) X (\<lambda>x. ereal (f x)) \<and>
hoelzl@47694
   875
    finite (X`space M)"
hoelzl@42902
   876
hoelzl@47694
   877
lemma simple_distributed:
hoelzl@47694
   878
  "simple_distributed M X Px \<Longrightarrow> distributed M (count_space (X`space M)) X Px"
hoelzl@47694
   879
  unfolding simple_distributed_def by auto
hoelzl@42902
   880
hoelzl@47694
   881
lemma simple_distributed_finite[dest]: "simple_distributed M X P \<Longrightarrow> finite (X`space M)"
hoelzl@47694
   882
  by (simp add: simple_distributed_def)
hoelzl@42902
   883
hoelzl@47694
   884
lemma (in prob_space) distributed_simple_function_superset:
hoelzl@47694
   885
  assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
hoelzl@47694
   886
  assumes A: "X`space M \<subseteq> A" "finite A"
hoelzl@47694
   887
  defines "S \<equiv> count_space A" and "P' \<equiv> (\<lambda>x. if x \<in> X`space M then P x else 0)"
hoelzl@47694
   888
  shows "distributed M S X P'"
hoelzl@47694
   889
  unfolding distributed_def
hoelzl@47694
   890
proof safe
hoelzl@47694
   891
  show "(\<lambda>x. ereal (P' x)) \<in> borel_measurable S" unfolding S_def by simp
hoelzl@47694
   892
  show "AE x in S. 0 \<le> ereal (P' x)"
hoelzl@47694
   893
    using X by (auto simp: S_def P'_def simple_distributed_def intro!: measure_nonneg)
hoelzl@47694
   894
  show "distr M S X = density S P'"
hoelzl@47694
   895
  proof (rule measure_eqI_finite)
hoelzl@47694
   896
    show "sets (distr M S X) = Pow A" "sets (density S P') = Pow A"
hoelzl@47694
   897
      using A unfolding S_def by auto
hoelzl@47694
   898
    show "finite A" by fact
hoelzl@47694
   899
    fix a assume a: "a \<in> A"
hoelzl@47694
   900
    then have "a \<notin> X`space M \<Longrightarrow> X -` {a} \<inter> space M = {}" by auto
hoelzl@47694
   901
    with A a X have "emeasure (distr M S X) {a} = P' a"
hoelzl@47694
   902
      by (subst emeasure_distr)
hoelzl@50002
   903
         (auto simp add: S_def P'_def simple_functionD emeasure_eq_measure measurable_count_space_eq2
hoelzl@47694
   904
               intro!: arg_cong[where f=prob])
wenzelm@53015
   905
    also have "\<dots> = (\<integral>\<^sup>+x. ereal (P' a) * indicator {a} x \<partial>S)"
hoelzl@47694
   906
      using A X a
hoelzl@56996
   907
      by (subst nn_integral_cmult_indicator)
hoelzl@47694
   908
         (auto simp: S_def P'_def simple_distributed_def simple_functionD measure_nonneg)
wenzelm@53015
   909
    also have "\<dots> = (\<integral>\<^sup>+x. ereal (P' x) * indicator {a} x \<partial>S)"
hoelzl@56996
   910
      by (auto simp: indicator_def intro!: nn_integral_cong)
hoelzl@47694
   911
    also have "\<dots> = emeasure (density S P') {a}"
hoelzl@47694
   912
      using a A by (intro emeasure_density[symmetric]) (auto simp: S_def)
hoelzl@47694
   913
    finally show "emeasure (distr M S X) {a} = emeasure (density S P') {a}" .
hoelzl@47694
   914
  qed
hoelzl@47694
   915
  show "random_variable S X"
hoelzl@47694
   916
    using X(1) A by (auto simp: measurable_def simple_functionD S_def)
hoelzl@47694
   917
qed
hoelzl@42902
   918
hoelzl@47694
   919
lemma (in prob_space) simple_distributedI:
hoelzl@47694
   920
  assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
hoelzl@47694
   921
  shows "simple_distributed M X P"
hoelzl@47694
   922
  unfolding simple_distributed_def
hoelzl@47694
   923
proof
hoelzl@47694
   924
  have "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (if x \<in> X`space M then P x else 0))"
hoelzl@47694
   925
    (is "?A")
hoelzl@47694
   926
    using simple_functionD[OF X(1)] by (intro distributed_simple_function_superset[OF X]) auto
hoelzl@47694
   927
  also have "?A \<longleftrightarrow> distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (P x))"
hoelzl@47694
   928
    by (rule distributed_cong_density) auto
hoelzl@47694
   929
  finally show "\<dots>" .
hoelzl@47694
   930
qed (rule simple_functionD[OF X(1)])
hoelzl@47694
   931
hoelzl@47694
   932
lemma simple_distributed_joint_finite:
hoelzl@47694
   933
  assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
hoelzl@47694
   934
  shows "finite (X ` space M)" "finite (Y ` space M)"
hoelzl@42902
   935
proof -
hoelzl@47694
   936
  have "finite ((\<lambda>x. (X x, Y x)) ` space M)"
hoelzl@47694
   937
    using X by (auto simp: simple_distributed_def simple_functionD)
hoelzl@47694
   938
  then have "finite (fst ` (\<lambda>x. (X x, Y x)) ` space M)" "finite (snd ` (\<lambda>x. (X x, Y x)) ` space M)"
hoelzl@47694
   939
    by auto
hoelzl@47694
   940
  then show fin: "finite (X ` space M)" "finite (Y ` space M)"
hoelzl@47694
   941
    by (auto simp: image_image)
hoelzl@47694
   942
qed
hoelzl@47694
   943
hoelzl@47694
   944
lemma simple_distributed_joint2_finite:
hoelzl@47694
   945
  assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
hoelzl@47694
   946
  shows "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
hoelzl@47694
   947
proof -
hoelzl@47694
   948
  have "finite ((\<lambda>x. (X x, Y x, Z x)) ` space M)"
hoelzl@47694
   949
    using X by (auto simp: simple_distributed_def simple_functionD)
hoelzl@47694
   950
  then have "finite (fst ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
hoelzl@47694
   951
    "finite ((fst \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
hoelzl@47694
   952
    "finite ((snd \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
hoelzl@47694
   953
    by auto
hoelzl@47694
   954
  then show fin: "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
hoelzl@47694
   955
    by (auto simp: image_image)
hoelzl@42902
   956
qed
hoelzl@42902
   957
hoelzl@47694
   958
lemma simple_distributed_simple_function:
hoelzl@47694
   959
  "simple_distributed M X Px \<Longrightarrow> simple_function M X"
hoelzl@47694
   960
  unfolding simple_distributed_def distributed_def
hoelzl@50002
   961
  by (auto simp: simple_function_def measurable_count_space_eq2)
hoelzl@47694
   962
hoelzl@47694
   963
lemma simple_distributed_measure:
hoelzl@47694
   964
  "simple_distributed M X P \<Longrightarrow> a \<in> X`space M \<Longrightarrow> P a = measure M (X -` {a} \<inter> space M)"
hoelzl@47694
   965
  using distributed_count_space[of M "X`space M" X P a, symmetric]
hoelzl@47694
   966
  by (auto simp: simple_distributed_def measure_def)
hoelzl@47694
   967
hoelzl@47694
   968
lemma simple_distributed_nonneg: "simple_distributed M X f \<Longrightarrow> x \<in> space M \<Longrightarrow> 0 \<le> f (X x)"
hoelzl@47694
   969
  by (auto simp: simple_distributed_measure measure_nonneg)
hoelzl@42860
   970
hoelzl@47694
   971
lemma (in prob_space) simple_distributed_joint:
hoelzl@47694
   972
  assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
wenzelm@53015
   973
  defines "S \<equiv> count_space (X`space M) \<Otimes>\<^sub>M count_space (Y`space M)"
hoelzl@47694
   974
  defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x))`space M then Px x else 0)"
hoelzl@47694
   975
  shows "distributed M S (\<lambda>x. (X x, Y x)) P"
hoelzl@47694
   976
proof -
hoelzl@47694
   977
  from simple_distributed_joint_finite[OF X, simp]
hoelzl@47694
   978
  have S_eq: "S = count_space (X`space M \<times> Y`space M)"
hoelzl@47694
   979
    by (simp add: S_def pair_measure_count_space)
hoelzl@47694
   980
  show ?thesis
hoelzl@47694
   981
    unfolding S_eq P_def
hoelzl@47694
   982
  proof (rule distributed_simple_function_superset)
hoelzl@47694
   983
    show "simple_function M (\<lambda>x. (X x, Y x))"
hoelzl@47694
   984
      using X by (rule simple_distributed_simple_function)
hoelzl@47694
   985
    fix x assume "x \<in> (\<lambda>x. (X x, Y x)) ` space M"
hoelzl@47694
   986
    from simple_distributed_measure[OF X this]
hoelzl@47694
   987
    show "Px x = prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)" .
hoelzl@47694
   988
  qed auto
hoelzl@47694
   989
qed
hoelzl@42860
   990
hoelzl@47694
   991
lemma (in prob_space) simple_distributed_joint2:
hoelzl@47694
   992
  assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
wenzelm@53015
   993
  defines "S \<equiv> count_space (X`space M) \<Otimes>\<^sub>M count_space (Y`space M) \<Otimes>\<^sub>M count_space (Z`space M)"
hoelzl@47694
   994
  defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x))`space M then Px x else 0)"
hoelzl@47694
   995
  shows "distributed M S (\<lambda>x. (X x, Y x, Z x)) P"
hoelzl@47694
   996
proof -
hoelzl@47694
   997
  from simple_distributed_joint2_finite[OF X, simp]
hoelzl@47694
   998
  have S_eq: "S = count_space (X`space M \<times> Y`space M \<times> Z`space M)"
hoelzl@47694
   999
    by (simp add: S_def pair_measure_count_space)
hoelzl@47694
  1000
  show ?thesis
hoelzl@47694
  1001
    unfolding S_eq P_def
hoelzl@47694
  1002
  proof (rule distributed_simple_function_superset)
hoelzl@47694
  1003
    show "simple_function M (\<lambda>x. (X x, Y x, Z x))"
hoelzl@47694
  1004
      using X by (rule simple_distributed_simple_function)
hoelzl@47694
  1005
    fix x assume "x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M"
hoelzl@47694
  1006
    from simple_distributed_measure[OF X this]
hoelzl@47694
  1007
    show "Px x = prob ((\<lambda>x. (X x, Y x, Z x)) -` {x} \<inter> space M)" .
hoelzl@47694
  1008
  qed auto
hoelzl@47694
  1009
qed
hoelzl@47694
  1010
hoelzl@47694
  1011
lemma (in prob_space) simple_distributed_setsum_space:
hoelzl@47694
  1012
  assumes X: "simple_distributed M X f"
hoelzl@47694
  1013
  shows "setsum f (X`space M) = 1"
hoelzl@47694
  1014
proof -
hoelzl@47694
  1015
  from X have "setsum f (X`space M) = prob (\<Union>i\<in>X`space M. X -` {i} \<inter> space M)"
hoelzl@47694
  1016
    by (subst finite_measure_finite_Union)
hoelzl@47694
  1017
       (auto simp add: disjoint_family_on_def simple_distributed_measure simple_distributed_simple_function simple_functionD
haftmann@57418
  1018
             intro!: setsum.cong arg_cong[where f="prob"])
hoelzl@47694
  1019
  also have "\<dots> = prob (space M)"
hoelzl@47694
  1020
    by (auto intro!: arg_cong[where f=prob])
hoelzl@47694
  1021
  finally show ?thesis
hoelzl@47694
  1022
    using emeasure_space_1 by (simp add: emeasure_eq_measure one_ereal_def)
hoelzl@47694
  1023
qed
hoelzl@42860
  1024
hoelzl@47694
  1025
lemma (in prob_space) distributed_marginal_eq_joint_simple:
hoelzl@47694
  1026
  assumes Px: "simple_function M X"
hoelzl@47694
  1027
  assumes Py: "simple_distributed M Y Py"
hoelzl@47694
  1028
  assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694
  1029
  assumes y: "y \<in> Y`space M"
hoelzl@47694
  1030
  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)"
hoelzl@47694
  1031
proof -
hoelzl@47694
  1032
  note Px = simple_distributedI[OF Px refl]
hoelzl@47694
  1033
  have *: "\<And>f A. setsum (\<lambda>x. max 0 (ereal (f x))) A = ereal (setsum (\<lambda>x. max 0 (f x)) A)"
hoelzl@47694
  1034
    by (simp add: setsum_ereal[symmetric] zero_ereal_def)
hoelzl@49788
  1035
  from distributed_marginal_eq_joint2[OF
hoelzl@49788
  1036
    sigma_finite_measure_count_space_finite
hoelzl@49788
  1037
    sigma_finite_measure_count_space_finite
hoelzl@49788
  1038
    simple_distributed[OF Py] simple_distributed_joint[OF Pxy],
hoelzl@47694
  1039
    OF Py[THEN simple_distributed_finite] Px[THEN simple_distributed_finite]]
hoelzl@49788
  1040
    y
hoelzl@49788
  1041
    Px[THEN simple_distributed_finite]
hoelzl@49788
  1042
    Py[THEN simple_distributed_finite]
hoelzl@47694
  1043
    Pxy[THEN simple_distributed, THEN distributed_real_AE]
hoelzl@47694
  1044
  show ?thesis
hoelzl@47694
  1045
    unfolding AE_count_space
haftmann@57418
  1046
    apply (auto simp add: nn_integral_count_space_finite * intro!: setsum.cong split: split_max)
hoelzl@47694
  1047
    done
hoelzl@47694
  1048
qed
hoelzl@42860
  1049
hoelzl@50419
  1050
lemma distributedI_real:
hoelzl@50419
  1051
  fixes f :: "'a \<Rightarrow> real"
hoelzl@50419
  1052
  assumes gen: "sets M1 = sigma_sets (space M1) E" and "Int_stable E"
hoelzl@50419
  1053
    and A: "range A \<subseteq> E" "(\<Union>i::nat. A i) = space M1" "\<And>i. emeasure (distr M M1 X) (A i) \<noteq> \<infinity>"
hoelzl@50419
  1054
    and X: "X \<in> measurable M M1"
hoelzl@50419
  1055
    and f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x"
wenzelm@53015
  1056
    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)"
hoelzl@50419
  1057
  shows "distributed M M1 X f"
hoelzl@50419
  1058
  unfolding distributed_def
hoelzl@50419
  1059
proof (intro conjI)
hoelzl@50419
  1060
  show "distr M M1 X = density M1 f"
hoelzl@50419
  1061
  proof (rule measure_eqI_generator_eq[where A=A])
hoelzl@50419
  1062
    { fix A assume A: "A \<in> E"
hoelzl@50419
  1063
      then have "A \<in> sigma_sets (space M1) E" by auto
hoelzl@50419
  1064
      then have "A \<in> sets M1"
hoelzl@50419
  1065
        using gen by simp
hoelzl@50419
  1066
      with f A eq[of A] X show "emeasure (distr M M1 X) A = emeasure (density M1 f) A"
hoelzl@50419
  1067
        by (simp add: emeasure_distr emeasure_density borel_measurable_ereal
hoelzl@50419
  1068
                      times_ereal.simps[symmetric] ereal_indicator
hoelzl@50419
  1069
                 del: times_ereal.simps) }
hoelzl@50419
  1070
    note eq_E = this
hoelzl@50419
  1071
    show "Int_stable E" by fact
hoelzl@50419
  1072
    { fix e assume "e \<in> E"
hoelzl@50419
  1073
      then have "e \<in> sigma_sets (space M1) E" by auto
hoelzl@50419
  1074
      then have "e \<in> sets M1" unfolding gen .
hoelzl@50419
  1075
      then have "e \<subseteq> space M1" by (rule sets.sets_into_space) }
hoelzl@50419
  1076
    then show "E \<subseteq> Pow (space M1)" by auto
hoelzl@50419
  1077
    show "sets (distr M M1 X) = sigma_sets (space M1) E"
hoelzl@50419
  1078
      "sets (density M1 (\<lambda>x. ereal (f x))) = sigma_sets (space M1) E"
hoelzl@50419
  1079
      unfolding gen[symmetric] by auto
hoelzl@50419
  1080
  qed fact+
hoelzl@50419
  1081
qed (insert X f, auto)
hoelzl@50419
  1082
hoelzl@50419
  1083
lemma distributedI_borel_atMost:
hoelzl@50419
  1084
  fixes f :: "real \<Rightarrow> real"
hoelzl@50419
  1085
  assumes [measurable]: "X \<in> borel_measurable M"
hoelzl@50419
  1086
    and [measurable]: "f \<in> borel_measurable borel" and f[simp]: "AE x in lborel. 0 \<le> f x"
wenzelm@53015
  1087
    and g_eq: "\<And>a. (\<integral>\<^sup>+x. f x * indicator {..a} x \<partial>lborel)  = ereal (g a)"
hoelzl@50419
  1088
    and M_eq: "\<And>a. emeasure M {x\<in>space M. X x \<le> a} = ereal (g a)"
hoelzl@50419
  1089
  shows "distributed M lborel X f"
hoelzl@50419
  1090
proof (rule distributedI_real)
hoelzl@57447
  1091
  show "sets (lborel::real measure) = sigma_sets (space lborel) (range atMost)"
hoelzl@50419
  1092
    by (simp add: borel_eq_atMost)
hoelzl@50419
  1093
  show "Int_stable (range atMost :: real set set)"
hoelzl@50419
  1094
    by (auto simp: Int_stable_def)
hoelzl@50419
  1095
  have vimage_eq: "\<And>a. (X -` {..a} \<inter> space M) = {x\<in>space M. X x \<le> a}" by auto
hoelzl@50419
  1096
  def A \<equiv> "\<lambda>i::nat. {.. real i}"
hoelzl@50419
  1097
  then show "range A \<subseteq> range atMost" "(\<Union>i. A i) = space lborel"
hoelzl@50419
  1098
    "\<And>i. emeasure (distr M lborel X) (A i) \<noteq> \<infinity>"
hoelzl@50419
  1099
    by (auto simp: real_arch_simple emeasure_distr vimage_eq M_eq)
hoelzl@50419
  1100
hoelzl@50419
  1101
  fix A :: "real set" assume "A \<in> range atMost"
hoelzl@50419
  1102
  then obtain a where A: "A = {..a}" by auto
wenzelm@53015
  1103
  show "emeasure M (X -` A \<inter> space M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>lborel)"
hoelzl@50419
  1104
    unfolding vimage_eq A M_eq g_eq ..
hoelzl@50419
  1105
qed auto
hoelzl@50419
  1106
hoelzl@50419
  1107
lemma (in prob_space) uniform_distributed_params:
hoelzl@50419
  1108
  assumes X: "distributed M MX X (\<lambda>x. indicator A x / measure MX A)"
hoelzl@50419
  1109
  shows "A \<in> sets MX" "measure MX A \<noteq> 0"
hoelzl@50419
  1110
proof -
hoelzl@50419
  1111
  interpret X: prob_space "distr M MX X"
hoelzl@50419
  1112
    using distributed_measurable[OF X] by (rule prob_space_distr)
hoelzl@50419
  1113
hoelzl@50419
  1114
  show "measure MX A \<noteq> 0"
hoelzl@50419
  1115
  proof
hoelzl@50419
  1116
    assume "measure MX A = 0"
hoelzl@50419
  1117
    with X.emeasure_space_1 X.prob_space distributed_distr_eq_density[OF X]
hoelzl@50419
  1118
    show False
hoelzl@50419
  1119
      by (simp add: emeasure_density zero_ereal_def[symmetric])
hoelzl@50419
  1120
  qed
hoelzl@50419
  1121
  with measure_notin_sets[of A MX] show "A \<in> sets MX"
hoelzl@50419
  1122
    by blast
hoelzl@50419
  1123
qed
hoelzl@50419
  1124
hoelzl@47694
  1125
lemma prob_space_uniform_measure:
hoelzl@47694
  1126
  assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>"
hoelzl@47694
  1127
  shows "prob_space (uniform_measure M A)"
hoelzl@47694
  1128
proof
hoelzl@47694
  1129
  show "emeasure (uniform_measure M A) (space (uniform_measure M A)) = 1"
hoelzl@47694
  1130
    using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)], of "space M"]
immler@50244
  1131
    using sets.sets_into_space[OF emeasure_neq_0_sets[OF A(1)]] A
hoelzl@47694
  1132
    by (simp add: Int_absorb2 emeasure_nonneg)
hoelzl@47694
  1133
qed
hoelzl@47694
  1134
hoelzl@47694
  1135
lemma prob_space_uniform_count_measure: "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> prob_space (uniform_count_measure A)"
hoelzl@47694
  1136
  by default (auto simp: emeasure_uniform_count_measure space_uniform_count_measure one_ereal_def)
hoelzl@42860
  1137
hoelzl@59000
  1138
lemma (in prob_space) measure_uniform_measure_eq_cond_prob:
hoelzl@59000
  1139
  assumes [measurable]: "Measurable.pred M P" "Measurable.pred M Q"
hoelzl@59000
  1140
  shows "\<P>(x in uniform_measure M {x\<in>space M. Q x}. P x) = \<P>(x in M. P x \<bar> Q x)"
hoelzl@59000
  1141
proof cases
hoelzl@59000
  1142
  assume Q: "measure M {x\<in>space M. Q x} = 0"
hoelzl@59000
  1143
  then have "AE x in M. \<not> Q x"
hoelzl@59000
  1144
    by (simp add: prob_eq_0)
hoelzl@59000
  1145
  then have "AE x in M. indicator {x\<in>space M. Q x} x / ereal 0 = 0"
hoelzl@59000
  1146
    by (auto split: split_indicator)
hoelzl@59000
  1147
  from density_cong[OF _ _ this] show ?thesis
hoelzl@59000
  1148
    by (simp add: uniform_measure_def emeasure_eq_measure cond_prob_def Q measure_density_const)
hoelzl@59000
  1149
qed (auto simp add: emeasure_eq_measure cond_prob_def intro!: arg_cong[where f=prob])
hoelzl@59000
  1150
hoelzl@59000
  1151
lemma prob_space_point_measure:
hoelzl@59000
  1152
  "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)"
hoelzl@59000
  1153
  by (rule prob_spaceI) (simp add: space_point_measure emeasure_point_measure_finite)
hoelzl@59000
  1154
hoelzl@35582
  1155
end