src/HOL/Probability/Probability_Measure.thy
 author hoelzl Fri Nov 02 14:23:40 2012 +0100 (2012-11-02) changeset 50002 ce0d316b5b44 parent 50001 382bd3173584 child 50003 8c213922ed49 permissions -rw-r--r--
```     1 (*  Title:      HOL/Probability/Probability_Measure.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Armin Heller, TU München
```
```     4 *)
```
```     5
```
```     6 header {*Probability measure*}
```
```     7
```
```     8 theory Probability_Measure
```
```     9   imports Lebesgue_Measure Radon_Nikodym
```
```    10 begin
```
```    11
```
```    12 lemma funset_eq_UN_fun_upd_I:
```
```    13   assumes *: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A"
```
```    14   and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))"
```
```    15   and ***: "\<And>f x. \<lbrakk> f \<in> F A ; x \<in> G f \<rbrakk> \<Longrightarrow> f(a:=x) \<in> F (insert a A)"
```
```    16   shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` (G f))"
```
```    17 proof safe
```
```    18   fix f assume f: "f \<in> F (insert a A)"
```
```    19   show "f \<in> (\<Union>f\<in>F A. fun_upd f a ` G f)"
```
```    20   proof (rule UN_I[of "f(a := d)"])
```
```    21     show "f(a := d) \<in> F A" using *[OF f] .
```
```    22     show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))"
```
```    23     proof (rule image_eqI[of _ _ "f a"])
```
```    24       show "f a \<in> G (f(a := d))" using **[OF f] .
```
```    25     qed simp
```
```    26   qed
```
```    27 next
```
```    28   fix f x assume "f \<in> F A" "x \<in> G f"
```
```    29   from ***[OF this] show "f(a := x) \<in> F (insert a A)" .
```
```    30 qed
```
```    31
```
```    32 lemma extensional_funcset_insert_eq[simp]:
```
```    33   assumes "a \<notin> A"
```
```    34   shows "extensional (insert a A) \<inter> (insert a A \<rightarrow> B) = (\<Union>f \<in> extensional A \<inter> (A \<rightarrow> B). (\<lambda>b. f(a := b)) ` B)"
```
```    35   apply (rule funset_eq_UN_fun_upd_I)
```
```    36   using assms
```
```    37   by (auto intro!: inj_onI dest: inj_onD split: split_if_asm simp: extensional_def)
```
```    38
```
```    39 lemma finite_extensional_funcset[simp, intro]:
```
```    40   assumes "finite A" "finite B"
```
```    41   shows "finite (extensional A \<inter> (A \<rightarrow> B))"
```
```    42   using assms by induct auto
```
```    43
```
```    44 lemma finite_PiE[simp, intro]:
```
```    45   assumes fin: "finite A" "\<And>i. i \<in> A \<Longrightarrow> finite (B i)"
```
```    46   shows "finite (Pi\<^isub>E A B)"
```
```    47 proof -
```
```    48   have *: "(Pi\<^isub>E A B) \<subseteq> extensional A \<inter> (A \<rightarrow> (\<Union>i\<in>A. B i))" by auto
```
```    49   show ?thesis
```
```    50     using fin by (intro finite_subset[OF *] finite_extensional_funcset) auto
```
```    51 qed
```
```    52
```
```    53
```
```    54 lemma countably_additiveI[case_names countably]:
```
```    55   assumes "\<And>A. \<lbrakk> range A \<subseteq> M ; disjoint_family A ; (\<Union>i. A i) \<in> M\<rbrakk> \<Longrightarrow> (\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
```
```    56   shows "countably_additive M \<mu>"
```
```    57   using assms unfolding countably_additive_def by auto
```
```    58
```
```    59 lemma convex_le_Inf_differential:
```
```    60   fixes f :: "real \<Rightarrow> real"
```
```    61   assumes "convex_on I f"
```
```    62   assumes "x \<in> interior I" "y \<in> I"
```
```    63   shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
```
```    64     (is "_ \<ge> _ + Inf (?F x) * (y - x)")
```
```    65 proof -
```
```    66   show ?thesis
```
```    67   proof (cases rule: linorder_cases)
```
```    68     assume "x < y"
```
```    69     moreover
```
```    70     have "open (interior I)" by auto
```
```    71     from openE[OF this `x \<in> interior I`] guess e . note e = this
```
```    72     moreover def t \<equiv> "min (x + e / 2) ((x + y) / 2)"
```
```    73     ultimately have "x < t" "t < y" "t \<in> ball x e"
```
```    74       by (auto simp: dist_real_def field_simps split: split_min)
```
```    75     with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
```
```    76
```
```    77     have "open (interior I)" by auto
```
```    78     from openE[OF this `x \<in> interior I`] guess e .
```
```    79     moreover def K \<equiv> "x - e / 2"
```
```    80     with `0 < e` have "K \<in> ball x e" "K < x" by (auto simp: dist_real_def)
```
```    81     ultimately have "K \<in> I" "K < x" "x \<in> I"
```
```    82       using interior_subset[of I] `x \<in> interior I` by auto
```
```    83
```
```    84     have "Inf (?F x) \<le> (f x - f y) / (x - y)"
```
```    85     proof (rule Inf_lower2)
```
```    86       show "(f x - f t) / (x - t) \<in> ?F x"
```
```    87         using `t \<in> I` `x < t` by auto
```
```    88       show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
```
```    89         using `convex_on I f` `x \<in> I` `y \<in> I` `x < t` `t < y` by (rule convex_on_diff)
```
```    90     next
```
```    91       fix y assume "y \<in> ?F x"
```
```    92       with order_trans[OF convex_on_diff[OF `convex_on I f` `K \<in> I` _ `K < x` _]]
```
```    93       show "(f K - f x) / (K - x) \<le> y" by auto
```
```    94     qed
```
```    95     then show ?thesis
```
```    96       using `x < y` by (simp add: field_simps)
```
```    97   next
```
```    98     assume "y < x"
```
```    99     moreover
```
```   100     have "open (interior I)" by auto
```
```   101     from openE[OF this `x \<in> interior I`] guess e . note e = this
```
```   102     moreover def t \<equiv> "x + e / 2"
```
```   103     ultimately have "x < t" "t \<in> ball x e"
```
```   104       by (auto simp: dist_real_def field_simps)
```
```   105     with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
```
```   106
```
```   107     have "(f x - f y) / (x - y) \<le> Inf (?F x)"
```
```   108     proof (rule Inf_greatest)
```
```   109       have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
```
```   110         using `y < x` by (auto simp: field_simps)
```
```   111       also
```
```   112       fix z  assume "z \<in> ?F x"
```
```   113       with order_trans[OF convex_on_diff[OF `convex_on I f` `y \<in> I` _ `y < x`]]
```
```   114       have "(f y - f x) / (y - x) \<le> z" by auto
```
```   115       finally show "(f x - f y) / (x - y) \<le> z" .
```
```   116     next
```
```   117       have "open (interior I)" by auto
```
```   118       from openE[OF this `x \<in> interior I`] guess e . note e = this
```
```   119       then have "x + e / 2 \<in> ball x e" by (auto simp: dist_real_def)
```
```   120       with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I" by auto
```
```   121       then show "?F x \<noteq> {}" by blast
```
```   122     qed
```
```   123     then show ?thesis
```
```   124       using `y < x` by (simp add: field_simps)
```
```   125   qed simp
```
```   126 qed
```
```   127
```
```   128 lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N"
```
```   129   by (rule measure_eqI) (auto simp: emeasure_distr)
```
```   130
```
```   131 locale prob_space = finite_measure +
```
```   132   assumes emeasure_space_1: "emeasure M (space M) = 1"
```
```   133
```
```   134 lemma prob_spaceI[Pure.intro!]:
```
```   135   assumes *: "emeasure M (space M) = 1"
```
```   136   shows "prob_space M"
```
```   137 proof -
```
```   138   interpret finite_measure M
```
```   139   proof
```
```   140     show "emeasure M (space M) \<noteq> \<infinity>" using * by simp
```
```   141   qed
```
```   142   show "prob_space M" by default fact
```
```   143 qed
```
```   144
```
```   145 abbreviation (in prob_space) "events \<equiv> sets M"
```
```   146 abbreviation (in prob_space) "prob \<equiv> measure M"
```
```   147 abbreviation (in prob_space) "random_variable M' X \<equiv> X \<in> measurable M M'"
```
```   148 abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M"
```
```   149
```
```   150 lemma (in prob_space) prob_space_distr:
```
```   151   assumes f: "f \<in> measurable M M'" shows "prob_space (distr M M' f)"
```
```   152 proof (rule prob_spaceI)
```
```   153   have "f -` space M' \<inter> space M = space M" using f by (auto dest: measurable_space)
```
```   154   with f show "emeasure (distr M M' f) (space (distr M M' f)) = 1"
```
```   155     by (auto simp: emeasure_distr emeasure_space_1)
```
```   156 qed
```
```   157
```
```   158 lemma (in prob_space) prob_space: "prob (space M) = 1"
```
```   159   using emeasure_space_1 unfolding measure_def by (simp add: one_ereal_def)
```
```   160
```
```   161 lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1"
```
```   162   using bounded_measure[of A] by (simp add: prob_space)
```
```   163
```
```   164 lemma (in prob_space) not_empty: "space M \<noteq> {}"
```
```   165   using prob_space by auto
```
```   166
```
```   167 lemma (in prob_space) measure_le_1: "emeasure M X \<le> 1"
```
```   168   using emeasure_space[of M X] by (simp add: emeasure_space_1)
```
```   169
```
```   170 lemma (in prob_space) AE_I_eq_1:
```
```   171   assumes "emeasure M {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M"
```
```   172   shows "AE x in M. P x"
```
```   173 proof (rule AE_I)
```
```   174   show "emeasure M (space M - {x \<in> space M. P x}) = 0"
```
```   175     using assms emeasure_space_1 by (simp add: emeasure_compl)
```
```   176 qed (insert assms, auto)
```
```   177
```
```   178 lemma (in prob_space) prob_compl:
```
```   179   assumes A: "A \<in> events"
```
```   180   shows "prob (space M - A) = 1 - prob A"
```
```   181   using finite_measure_compl[OF A] by (simp add: prob_space)
```
```   182
```
```   183 lemma (in prob_space) AE_in_set_eq_1:
```
```   184   assumes "A \<in> events" shows "(AE x in M. x \<in> A) \<longleftrightarrow> prob A = 1"
```
```   185 proof
```
```   186   assume ae: "AE x in M. x \<in> A"
```
```   187   have "{x \<in> space M. x \<in> A} = A" "{x \<in> space M. x \<notin> A} = space M - A"
```
```   188     using `A \<in> events`[THEN sets_into_space] by auto
```
```   189   with AE_E2[OF ae] `A \<in> events` have "1 - emeasure M A = 0"
```
```   190     by (simp add: emeasure_compl emeasure_space_1)
```
```   191   then show "prob A = 1"
```
```   192     using `A \<in> events` by (simp add: emeasure_eq_measure one_ereal_def)
```
```   193 next
```
```   194   assume prob: "prob A = 1"
```
```   195   show "AE x in M. x \<in> A"
```
```   196   proof (rule AE_I)
```
```   197     show "{x \<in> space M. x \<notin> A} \<subseteq> space M - A" by auto
```
```   198     show "emeasure M (space M - A) = 0"
```
```   199       using `A \<in> events` prob
```
```   200       by (simp add: prob_compl emeasure_space_1 emeasure_eq_measure one_ereal_def)
```
```   201     show "space M - A \<in> events"
```
```   202       using `A \<in> events` by auto
```
```   203   qed
```
```   204 qed
```
```   205
```
```   206 lemma (in prob_space) AE_False: "(AE x in M. False) \<longleftrightarrow> False"
```
```   207 proof
```
```   208   assume "AE x in M. False"
```
```   209   then have "AE x in M. x \<in> {}" by simp
```
```   210   then show False
```
```   211     by (subst (asm) AE_in_set_eq_1) auto
```
```   212 qed simp
```
```   213
```
```   214 lemma (in prob_space) AE_prob_1:
```
```   215   assumes "prob A = 1" shows "AE x in M. x \<in> A"
```
```   216 proof -
```
```   217   from `prob A = 1` have "A \<in> events"
```
```   218     by (metis measure_notin_sets zero_neq_one)
```
```   219   with AE_in_set_eq_1 assms show ?thesis by simp
```
```   220 qed
```
```   221
```
```   222 lemma (in finite_measure) prob_space_increasing: "increasing M (measure M)"
```
```   223   by (auto intro!: finite_measure_mono simp: increasing_def)
```
```   224
```
```   225 lemma (in finite_measure) prob_zero_union:
```
```   226   assumes "s \<in> sets M" "t \<in> sets M" "measure M t = 0"
```
```   227   shows "measure M (s \<union> t) = measure M s"
```
```   228 using assms
```
```   229 proof -
```
```   230   have "measure M (s \<union> t) \<le> measure M s"
```
```   231     using finite_measure_subadditive[of s t] assms by auto
```
```   232   moreover have "measure M (s \<union> t) \<ge> measure M s"
```
```   233     using assms by (blast intro: finite_measure_mono)
```
```   234   ultimately show ?thesis by simp
```
```   235 qed
```
```   236
```
```   237 lemma (in finite_measure) prob_eq_compl:
```
```   238   assumes "s \<in> sets M" "t \<in> sets M"
```
```   239   assumes "measure M (space M - s) = measure M (space M - t)"
```
```   240   shows "measure M s = measure M t"
```
```   241   using assms finite_measure_compl by auto
```
```   242
```
```   243 lemma (in prob_space) prob_one_inter:
```
```   244   assumes events:"s \<in> events" "t \<in> events"
```
```   245   assumes "prob t = 1"
```
```   246   shows "prob (s \<inter> t) = prob s"
```
```   247 proof -
```
```   248   have "prob ((space M - s) \<union> (space M - t)) = prob (space M - s)"
```
```   249     using events assms  prob_compl[of "t"] by (auto intro!: prob_zero_union)
```
```   250   also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)"
```
```   251     by blast
```
```   252   finally show "prob (s \<inter> t) = prob s"
```
```   253     using events by (auto intro!: prob_eq_compl[of "s \<inter> t" s])
```
```   254 qed
```
```   255
```
```   256 lemma (in finite_measure) prob_eq_bigunion_image:
```
```   257   assumes "range f \<subseteq> sets M" "range g \<subseteq> sets M"
```
```   258   assumes "disjoint_family f" "disjoint_family g"
```
```   259   assumes "\<And> n :: nat. measure M (f n) = measure M (g n)"
```
```   260   shows "measure M (\<Union> i. f i) = measure M (\<Union> i. g i)"
```
```   261 using assms
```
```   262 proof -
```
```   263   have a: "(\<lambda> i. measure M (f i)) sums (measure M (\<Union> i. f i))"
```
```   264     by (rule finite_measure_UNION[OF assms(1,3)])
```
```   265   have b: "(\<lambda> i. measure M (g i)) sums (measure M (\<Union> i. g i))"
```
```   266     by (rule finite_measure_UNION[OF assms(2,4)])
```
```   267   show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp
```
```   268 qed
```
```   269
```
```   270 lemma (in finite_measure) prob_countably_zero:
```
```   271   assumes "range c \<subseteq> sets M"
```
```   272   assumes "\<And> i. measure M (c i) = 0"
```
```   273   shows "measure M (\<Union> i :: nat. c i) = 0"
```
```   274 proof (rule antisym)
```
```   275   show "measure M (\<Union> i :: nat. c i) \<le> 0"
```
```   276     using finite_measure_subadditive_countably[OF assms(1)] by (simp add: assms(2))
```
```   277 qed (simp add: measure_nonneg)
```
```   278
```
```   279 lemma (in prob_space) prob_equiprobable_finite_unions:
```
```   280   assumes "s \<in> events"
```
```   281   assumes s_finite: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> events"
```
```   282   assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> (prob {x} = prob {y})"
```
```   283   shows "prob s = real (card s) * prob {SOME x. x \<in> s}"
```
```   284 proof (cases "s = {}")
```
```   285   case False hence "\<exists> x. x \<in> s" by blast
```
```   286   from someI_ex[OF this] assms
```
```   287   have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast
```
```   288   have "prob s = (\<Sum> x \<in> s. prob {x})"
```
```   289     using finite_measure_eq_setsum_singleton[OF s_finite] by simp
```
```   290   also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto
```
```   291   also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}"
```
```   292     using setsum_constant assms by (simp add: real_eq_of_nat)
```
```   293   finally show ?thesis by simp
```
```   294 qed simp
```
```   295
```
```   296 lemma (in prob_space) prob_real_sum_image_fn:
```
```   297   assumes "e \<in> events"
```
```   298   assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> events"
```
```   299   assumes "finite s"
```
```   300   assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"
```
```   301   assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)"
```
```   302   shows "prob e = (\<Sum> x \<in> s. prob (e \<inter> f x))"
```
```   303 proof -
```
```   304   have e: "e = (\<Union> i \<in> s. e \<inter> f i)"
```
```   305     using `e \<in> events` sets_into_space upper by blast
```
```   306   hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp
```
```   307   also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))"
```
```   308   proof (rule finite_measure_finite_Union)
```
```   309     show "finite s" by fact
```
```   310     show "(\<lambda>i. e \<inter> f i)`s \<subseteq> events" using assms(2) by auto
```
```   311     show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"
```
```   312       using disjoint by (auto simp: disjoint_family_on_def)
```
```   313   qed
```
```   314   finally show ?thesis .
```
```   315 qed
```
```   316
```
```   317 lemma (in prob_space) expectation_less:
```
```   318   assumes [simp]: "integrable M X"
```
```   319   assumes gt: "AE x in M. X x < b"
```
```   320   shows "expectation X < b"
```
```   321 proof -
```
```   322   have "expectation X < expectation (\<lambda>x. b)"
```
```   323     using gt emeasure_space_1
```
```   324     by (intro integral_less_AE_space) auto
```
```   325   then show ?thesis using prob_space by simp
```
```   326 qed
```
```   327
```
```   328 lemma (in prob_space) expectation_greater:
```
```   329   assumes [simp]: "integrable M X"
```
```   330   assumes gt: "AE x in M. a < X x"
```
```   331   shows "a < expectation X"
```
```   332 proof -
```
```   333   have "expectation (\<lambda>x. a) < expectation X"
```
```   334     using gt emeasure_space_1
```
```   335     by (intro integral_less_AE_space) auto
```
```   336   then show ?thesis using prob_space by simp
```
```   337 qed
```
```   338
```
```   339 lemma (in prob_space) jensens_inequality:
```
```   340   fixes a b :: real
```
```   341   assumes X: "integrable M X" "AE x in M. X x \<in> I"
```
```   342   assumes I: "I = {a <..< b} \<or> I = {a <..} \<or> I = {..< b} \<or> I = UNIV"
```
```   343   assumes q: "integrable M (\<lambda>x. q (X x))" "convex_on I q"
```
```   344   shows "q (expectation X) \<le> expectation (\<lambda>x. q (X x))"
```
```   345 proof -
```
```   346   let ?F = "\<lambda>x. Inf ((\<lambda>t. (q x - q t) / (x - t)) ` ({x<..} \<inter> I))"
```
```   347   from X(2) AE_False have "I \<noteq> {}" by auto
```
```   348
```
```   349   from I have "open I" by auto
```
```   350
```
```   351   note I
```
```   352   moreover
```
```   353   { assume "I \<subseteq> {a <..}"
```
```   354     with X have "a < expectation X"
```
```   355       by (intro expectation_greater) auto }
```
```   356   moreover
```
```   357   { assume "I \<subseteq> {..< b}"
```
```   358     with X have "expectation X < b"
```
```   359       by (intro expectation_less) auto }
```
```   360   ultimately have "expectation X \<in> I"
```
```   361     by (elim disjE)  (auto simp: subset_eq)
```
```   362   moreover
```
```   363   { fix y assume y: "y \<in> I"
```
```   364     with q(2) `open I` have "Sup ((\<lambda>x. q x + ?F x * (y - x)) ` I) = q y"
```
```   365       by (auto intro!: Sup_eq_maximum convex_le_Inf_differential image_eqI[OF _ y] simp: interior_open) }
```
```   366   ultimately have "q (expectation X) = Sup ((\<lambda>x. q x + ?F x * (expectation X - x)) ` I)"
```
```   367     by simp
```
```   368   also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
```
```   369   proof (rule Sup_least)
```
```   370     show "(\<lambda>x. q x + ?F x * (expectation X - x)) ` I \<noteq> {}"
```
```   371       using `I \<noteq> {}` by auto
```
```   372   next
```
```   373     fix k assume "k \<in> (\<lambda>x. q x + ?F x * (expectation X - x)) ` I"
```
```   374     then guess x .. note x = this
```
```   375     have "q x + ?F x * (expectation X  - x) = expectation (\<lambda>w. q x + ?F x * (X w - x))"
```
```   376       using prob_space by (simp add: X)
```
```   377     also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
```
```   378       using `x \<in> I` `open I` X(2)
```
```   379       apply (intro integral_mono_AE integral_add integral_cmult integral_diff
```
```   380                 lebesgue_integral_const X q)
```
```   381       apply (elim eventually_elim1)
```
```   382       apply (intro convex_le_Inf_differential)
```
```   383       apply (auto simp: interior_open q)
```
```   384       done
```
```   385     finally show "k \<le> expectation (\<lambda>w. q (X w))" using x by auto
```
```   386   qed
```
```   387   finally show "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" .
```
```   388 qed
```
```   389
```
```   390 lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0:
```
```   391   assumes "{x} \<in> events"
```
```   392   assumes "prob {x} = 1"
```
```   393   assumes "{y} \<in> events"
```
```   394   assumes "y \<noteq> x"
```
```   395   shows "prob {y} = 0"
```
```   396   using prob_one_inter[of "{y}" "{x}"] assms by auto
```
```   397
```
```   398 subsection  {* Introduce binder for probability *}
```
```   399
```
```   400 syntax
```
```   401   "_prob" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("('\<P>'(_ in _. _'))")
```
```   402
```
```   403 translations
```
```   404   "\<P>(x in M. P)" => "CONST measure M {x \<in> CONST space M. P}"
```
```   405
```
```   406 definition
```
```   407   "cond_prob M P Q = \<P>(\<omega> in M. P \<omega> \<and> Q \<omega>) / \<P>(\<omega> in M. Q \<omega>)"
```
```   408
```
```   409 syntax
```
```   410   "_conditional_prob" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("('\<P>'(_ in _. _ \<bar>/ _'))")
```
```   411
```
```   412 translations
```
```   413   "\<P>(x in M. P \<bar> Q)" => "CONST cond_prob M (\<lambda>x. P) (\<lambda>x. Q)"
```
```   414
```
```   415 lemma (in prob_space) AE_E_prob:
```
```   416   assumes ae: "AE x in M. P x"
```
```   417   obtains S where "S \<subseteq> {x \<in> space M. P x}" "S \<in> events" "prob S = 1"
```
```   418 proof -
```
```   419   from ae[THEN AE_E] guess N .
```
```   420   then show thesis
```
```   421     by (intro that[of "space M - N"])
```
```   422        (auto simp: prob_compl prob_space emeasure_eq_measure)
```
```   423 qed
```
```   424
```
```   425 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)"
```
```   426   by (auto intro!: arg_cong[where f=prob] simp add: prob_compl[symmetric])
```
```   427
```
```   428 lemma (in prob_space) prob_eq_AE:
```
```   429   "(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)"
```
```   430   by (rule finite_measure_eq_AE) auto
```
```   431
```
```   432 lemma (in prob_space) prob_eq_0_AE:
```
```   433   assumes not: "AE x in M. \<not> P x" shows "\<P>(x in M. P x) = 0"
```
```   434 proof cases
```
```   435   assume "{x\<in>space M. P x} \<in> events"
```
```   436   with not have "\<P>(x in M. P x) = \<P>(x in M. False)"
```
```   437     by (intro prob_eq_AE) auto
```
```   438   then show ?thesis by simp
```
```   439 qed (simp add: measure_notin_sets)
```
```   440
```
```   441 lemma (in prob_space) prob_sums:
```
```   442   assumes P: "\<And>n. {x\<in>space M. P n x} \<in> events"
```
```   443   assumes Q: "{x\<in>space M. Q x} \<in> events"
```
```   444   assumes ae: "AE x in M. (\<forall>n. P n x \<longrightarrow> Q x) \<and> (Q x \<longrightarrow> (\<exists>!n. P n x))"
```
```   445   shows "(\<lambda>n. \<P>(x in M. P n x)) sums \<P>(x in M. Q x)"
```
```   446 proof -
```
```   447   from ae[THEN AE_E_prob] guess S . note S = this
```
```   448   then have disj: "disjoint_family (\<lambda>n. {x\<in>space M. P n x} \<inter> S)"
```
```   449     by (auto simp: disjoint_family_on_def)
```
```   450   from S have ae_S:
```
```   451     "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)"
```
```   452     "\<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"
```
```   453     using ae by (auto dest!: AE_prob_1)
```
```   454   from ae_S have *:
```
```   455     "\<P>(x in M. Q x) = prob (\<Union>n. {x\<in>space M. P n x} \<inter> S)"
```
```   456     using P Q S by (intro finite_measure_eq_AE) auto
```
```   457   from ae_S have **:
```
```   458     "\<And>n. \<P>(x in M. P n x) = prob ({x\<in>space M. P n x} \<inter> S)"
```
```   459     using P Q S by (intro finite_measure_eq_AE) auto
```
```   460   show ?thesis
```
```   461     unfolding * ** using S P disj
```
```   462     by (intro finite_measure_UNION) auto
```
```   463 qed
```
```   464
```
```   465 lemma (in prob_space) cond_prob_eq_AE:
```
```   466   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"
```
```   467   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"
```
```   468   shows "cond_prob M P Q = cond_prob M P' Q'"
```
```   469   using P Q
```
```   470   by (auto simp: cond_prob_def intro!: arg_cong2[where f="op /"] prob_eq_AE sets_Collect_conj)
```
```   471
```
```   472
```
```   473 lemma (in prob_space) joint_distribution_Times_le_fst:
```
```   474   "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
```
```   475     \<Longrightarrow> emeasure (distr M (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))) (A \<times> B) \<le> emeasure (distr M MX X) A"
```
```   476   by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
```
```   477
```
```   478 lemma (in prob_space) joint_distribution_Times_le_snd:
```
```   479   "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
```
```   480     \<Longrightarrow> emeasure (distr M (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))) (A \<times> B) \<le> emeasure (distr M MY Y) B"
```
```   481   by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
```
```   482
```
```   483 locale pair_prob_space = pair_sigma_finite M1 M2 + M1: prob_space M1 + M2: prob_space M2 for M1 M2
```
```   484
```
```   485 sublocale pair_prob_space \<subseteq> P: prob_space "M1 \<Otimes>\<^isub>M M2"
```
```   486 proof
```
```   487   show "emeasure (M1 \<Otimes>\<^isub>M M2) (space (M1 \<Otimes>\<^isub>M M2)) = 1"
```
```   488     by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_space_1 M2.emeasure_space_1 space_pair_measure)
```
```   489 qed
```
```   490
```
```   491 locale product_prob_space = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
```
```   492   fixes I :: "'i set"
```
```   493   assumes prob_space: "\<And>i. prob_space (M i)"
```
```   494
```
```   495 sublocale product_prob_space \<subseteq> M: prob_space "M i" for i
```
```   496   by (rule prob_space)
```
```   497
```
```   498 locale finite_product_prob_space = finite_product_sigma_finite M I + product_prob_space M I for M I
```
```   499
```
```   500 sublocale finite_product_prob_space \<subseteq> prob_space "\<Pi>\<^isub>M i\<in>I. M i"
```
```   501 proof
```
```   502   show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (space (\<Pi>\<^isub>M i\<in>I. M i)) = 1"
```
```   503     by (simp add: measure_times M.emeasure_space_1 setprod_1 space_PiM)
```
```   504 qed
```
```   505
```
```   506 lemma (in finite_product_prob_space) prob_times:
```
```   507   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets (M i)"
```
```   508   shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
```
```   509 proof -
```
```   510   have "ereal (measure (\<Pi>\<^isub>M i\<in>I. M i) (\<Pi>\<^isub>E i\<in>I. X i)) = emeasure (\<Pi>\<^isub>M i\<in>I. M i) (\<Pi>\<^isub>E i\<in>I. X i)"
```
```   511     using X by (simp add: emeasure_eq_measure)
```
```   512   also have "\<dots> = (\<Prod>i\<in>I. emeasure (M i) (X i))"
```
```   513     using measure_times X by simp
```
```   514   also have "\<dots> = ereal (\<Prod>i\<in>I. measure (M i) (X i))"
```
```   515     using X by (simp add: M.emeasure_eq_measure setprod_ereal)
```
```   516   finally show ?thesis by simp
```
```   517 qed
```
```   518
```
```   519 section {* Distributions *}
```
```   520
```
```   521 definition "distributed M N X f \<longleftrightarrow> distr M N X = density N f \<and>
```
```   522   f \<in> borel_measurable N \<and> (AE x in N. 0 \<le> f x) \<and> X \<in> measurable M N"
```
```   523
```
```   524 lemma
```
```   525   shows distributed_distr_eq_density: "distributed M N X f \<Longrightarrow> distr M N X = density N f"
```
```   526     and distributed_measurable: "distributed M N X f \<Longrightarrow> X \<in> measurable M N"
```
```   527     and distributed_borel_measurable: "distributed M N X f \<Longrightarrow> f \<in> borel_measurable N"
```
```   528     and distributed_AE: "distributed M N X f \<Longrightarrow> (AE x in N. 0 \<le> f x)"
```
```   529   by (simp_all add: distributed_def)
```
```   530
```
```   531 lemma
```
```   532   shows distributed_real_measurable: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> f \<in> borel_measurable N"
```
```   533     and distributed_real_AE: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> (AE x in N. 0 \<le> f x)"
```
```   534   by (simp_all add: distributed_def borel_measurable_ereal_iff)
```
```   535
```
```   536 lemma distributed_count_space:
```
```   537   assumes X: "distributed M (count_space A) X P" and a: "a \<in> A" and A: "finite A"
```
```   538   shows "P a = emeasure M (X -` {a} \<inter> space M)"
```
```   539 proof -
```
```   540   have "emeasure M (X -` {a} \<inter> space M) = emeasure (distr M (count_space A) X) {a}"
```
```   541     using X a A by (simp add: distributed_measurable emeasure_distr)
```
```   542   also have "\<dots> = emeasure (density (count_space A) P) {a}"
```
```   543     using X by (simp add: distributed_distr_eq_density)
```
```   544   also have "\<dots> = (\<integral>\<^isup>+x. P a * indicator {a} x \<partial>count_space A)"
```
```   545     using X a by (auto simp add: emeasure_density distributed_def indicator_def intro!: positive_integral_cong)
```
```   546   also have "\<dots> = P a"
```
```   547     using X a by (subst positive_integral_cmult_indicator) (auto simp: distributed_def one_ereal_def[symmetric] AE_count_space)
```
```   548   finally show ?thesis ..
```
```   549 qed
```
```   550
```
```   551 lemma distributed_cong_density:
```
```   552   "(AE x in N. f x = g x) \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable N \<Longrightarrow>
```
```   553     distributed M N X f \<longleftrightarrow> distributed M N X g"
```
```   554   by (auto simp: distributed_def intro!: density_cong)
```
```   555
```
```   556 lemma subdensity:
```
```   557   assumes T: "T \<in> measurable P Q"
```
```   558   assumes f: "distributed M P X f"
```
```   559   assumes g: "distributed M Q Y g"
```
```   560   assumes Y: "Y = T \<circ> X"
```
```   561   shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
```
```   562 proof -
```
```   563   have "{x\<in>space Q. g x = 0} \<in> null_sets (distr M Q (T \<circ> X))"
```
```   564     using g Y by (auto simp: null_sets_density_iff distributed_def)
```
```   565   also have "distr M Q (T \<circ> X) = distr (distr M P X) Q T"
```
```   566     using T f[THEN distributed_measurable] by (rule distr_distr[symmetric])
```
```   567   finally have "T -` {x\<in>space Q. g x = 0} \<inter> space P \<in> null_sets (distr M P X)"
```
```   568     using T by (subst (asm) null_sets_distr_iff) auto
```
```   569   also have "T -` {x\<in>space Q. g x = 0} \<inter> space P = {x\<in>space P. g (T x) = 0}"
```
```   570     using T by (auto dest: measurable_space)
```
```   571   finally show ?thesis
```
```   572     using f g by (auto simp add: null_sets_density_iff distributed_def)
```
```   573 qed
```
```   574
```
```   575 lemma subdensity_real:
```
```   576   fixes g :: "'a \<Rightarrow> real" and f :: "'b \<Rightarrow> real"
```
```   577   assumes T: "T \<in> measurable P Q"
```
```   578   assumes f: "distributed M P X f"
```
```   579   assumes g: "distributed M Q Y g"
```
```   580   assumes Y: "Y = T \<circ> X"
```
```   581   shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
```
```   582   using subdensity[OF T, of M X "\<lambda>x. ereal (f x)" Y "\<lambda>x. ereal (g x)"] assms by auto
```
```   583
```
```   584 lemma distributed_emeasure:
```
```   585   "distributed M N X f \<Longrightarrow> A \<in> sets N \<Longrightarrow> emeasure M (X -` A \<inter> space M) = (\<integral>\<^isup>+x. f x * indicator A x \<partial>N)"
```
```   586   by (auto simp: distributed_measurable distributed_AE distributed_borel_measurable
```
```   587                  distributed_distr_eq_density[symmetric] emeasure_density[symmetric] emeasure_distr)
```
```   588
```
```   589 lemma distributed_positive_integral:
```
```   590   "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> (\<integral>\<^isup>+x. f x * g x \<partial>N) = (\<integral>\<^isup>+x. g (X x) \<partial>M)"
```
```   591   by (auto simp: distributed_measurable distributed_AE distributed_borel_measurable
```
```   592                  distributed_distr_eq_density[symmetric] positive_integral_density[symmetric] positive_integral_distr)
```
```   593
```
```   594 lemma distributed_integral:
```
```   595   "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)"
```
```   596   by (auto simp: distributed_real_measurable distributed_real_AE distributed_measurable
```
```   597                  distributed_distr_eq_density[symmetric] integral_density[symmetric] integral_distr)
```
```   598
```
```   599 lemma distributed_transform_integral:
```
```   600   assumes Px: "distributed M N X Px"
```
```   601   assumes "distributed M P Y Py"
```
```   602   assumes Y: "Y = T \<circ> X" and T: "T \<in> measurable N P" and f: "f \<in> borel_measurable P"
```
```   603   shows "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. Px x * f (T x) \<partial>N)"
```
```   604 proof -
```
```   605   have "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. f (Y x) \<partial>M)"
```
```   606     by (rule distributed_integral) fact+
```
```   607   also have "\<dots> = (\<integral>x. f (T (X x)) \<partial>M)"
```
```   608     using Y by simp
```
```   609   also have "\<dots> = (\<integral>x. Px x * f (T x) \<partial>N)"
```
```   610     using measurable_comp[OF T f] Px by (intro distributed_integral[symmetric]) (auto simp: comp_def)
```
```   611   finally show ?thesis .
```
```   612 qed
```
```   613
```
```   614 lemma (in prob_space) distributed_unique:
```
```   615   assumes Px: "distributed M S X Px"
```
```   616   assumes Py: "distributed M S X Py"
```
```   617   shows "AE x in S. Px x = Py x"
```
```   618 proof -
```
```   619   interpret X: prob_space "distr M S X"
```
```   620     using distributed_measurable[OF Px] by (rule prob_space_distr)
```
```   621   have "sigma_finite_measure (distr M S X)" ..
```
```   622   with sigma_finite_density_unique[of Px S Py ] Px Py
```
```   623   show ?thesis
```
```   624     by (auto simp: distributed_def)
```
```   625 qed
```
```   626
```
```   627 lemma (in prob_space) distributed_jointI:
```
```   628   assumes "sigma_finite_measure S" "sigma_finite_measure T"
```
```   629   assumes X[simp]: "X \<in> measurable M S" and Y[simp]: "Y \<in> measurable M T"
```
```   630   assumes f[simp]: "f \<in> borel_measurable (S \<Otimes>\<^isub>M T)" "AE x in S \<Otimes>\<^isub>M T. 0 \<le> f x"
```
```   631   assumes eq: "\<And>A B. A \<in> sets S \<Longrightarrow> B \<in> sets T \<Longrightarrow>
```
```   632     emeasure M {x \<in> space M. X x \<in> A \<and> Y x \<in> B} = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. f (x, y) * indicator B y \<partial>T) * indicator A x \<partial>S)"
```
```   633   shows "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) f"
```
```   634   unfolding distributed_def
```
```   635 proof safe
```
```   636   interpret S: sigma_finite_measure S by fact
```
```   637   interpret T: sigma_finite_measure T by fact
```
```   638   interpret ST: pair_sigma_finite S T by default
```
```   639
```
```   640   from ST.sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('b \<times> 'c) set" .. note F = this
```
```   641   let ?E = "{a \<times> b |a b. a \<in> sets S \<and> b \<in> sets T}"
```
```   642   let ?P = "S \<Otimes>\<^isub>M T"
```
```   643   show "distr M ?P (\<lambda>x. (X x, Y x)) = density ?P f" (is "?L = ?R")
```
```   644   proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of S T]])
```
```   645     show "?E \<subseteq> Pow (space ?P)"
```
```   646       using space_closed[of S] space_closed[of T] by (auto simp: space_pair_measure)
```
```   647     show "sets ?L = sigma_sets (space ?P) ?E"
```
```   648       by (simp add: sets_pair_measure space_pair_measure)
```
```   649     then show "sets ?R = sigma_sets (space ?P) ?E"
```
```   650       by simp
```
```   651   next
```
```   652     interpret L: prob_space ?L
```
```   653       by (rule prob_space_distr) (auto intro!: measurable_Pair)
```
```   654     show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?L (F i) \<noteq> \<infinity>"
```
```   655       using F by (auto simp: space_pair_measure)
```
```   656   next
```
```   657     fix E assume "E \<in> ?E"
```
```   658     then obtain A B where E[simp]: "E = A \<times> B" and A[simp]: "A \<in> sets S" and B[simp]: "B \<in> sets T" by auto
```
```   659     have "emeasure ?L E = emeasure M {x \<in> space M. X x \<in> A \<and> Y x \<in> B}"
```
```   660       by (auto intro!: arg_cong[where f="emeasure M"] simp add: emeasure_distr measurable_Pair)
```
```   661     also have "\<dots> = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. (f (x, y) * indicator B y) * indicator A x \<partial>T) \<partial>S)"
```
```   662       by (auto simp add: eq measurable_Pair measurable_compose[OF _ f(1)] positive_integral_multc
```
```   663                intro!: positive_integral_cong)
```
```   664     also have "\<dots> = emeasure ?R E"
```
```   665       by (auto simp add: emeasure_density T.positive_integral_fst_measurable(2)[symmetric]
```
```   666                intro!: positive_integral_cong split: split_indicator)
```
```   667     finally show "emeasure ?L E = emeasure ?R E" .
```
```   668   qed
```
```   669 qed (auto intro!: measurable_Pair)
```
```   670
```
```   671 lemma (in prob_space) distributed_swap:
```
```   672   assumes "sigma_finite_measure S" "sigma_finite_measure T"
```
```   673   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
```
```   674   shows "distributed M (T \<Otimes>\<^isub>M S) (\<lambda>x. (Y x, X x)) (\<lambda>(x, y). Pxy (y, x))"
```
```   675 proof -
```
```   676   interpret S: sigma_finite_measure S by fact
```
```   677   interpret T: sigma_finite_measure T by fact
```
```   678   interpret ST: pair_sigma_finite S T by default
```
```   679   interpret TS: pair_sigma_finite T S by default
```
```   680
```
```   681   note measurable_Pxy = measurable_compose[OF _ distributed_borel_measurable[OF Pxy]]
```
```   682   show ?thesis
```
```   683     apply (subst TS.distr_pair_swap)
```
```   684     unfolding distributed_def
```
```   685   proof safe
```
```   686     let ?D = "distr (S \<Otimes>\<^isub>M T) (T \<Otimes>\<^isub>M S) (\<lambda>(x, y). (y, x))"
```
```   687     show 1: "(\<lambda>(x, y). Pxy (y, x)) \<in> borel_measurable ?D"
```
```   688       by (auto simp: measurable_split_conv intro!: measurable_Pair measurable_Pxy)
```
```   689     with Pxy
```
```   690     show "AE x in distr (S \<Otimes>\<^isub>M T) (T \<Otimes>\<^isub>M S) (\<lambda>(x, y). (y, x)). 0 \<le> (case x of (x, y) \<Rightarrow> Pxy (y, x))"
```
```   691       by (subst AE_distr_iff)
```
```   692          (auto dest!: distributed_AE
```
```   693                simp: measurable_split_conv split_beta
```
```   694                intro!: measurable_Pair borel_measurable_ereal_le)
```
```   695     show 2: "random_variable (distr (S \<Otimes>\<^isub>M T) (T \<Otimes>\<^isub>M S) (\<lambda>(x, y). (y, x))) (\<lambda>x. (Y x, X x))"
```
```   696       using measurable_compose[OF distributed_measurable[OF Pxy] measurable_fst]
```
```   697       using measurable_compose[OF distributed_measurable[OF Pxy] measurable_snd]
```
```   698       by (auto intro!: measurable_Pair)
```
```   699     { fix A assume A: "A \<in> sets (T \<Otimes>\<^isub>M S)"
```
```   700       let ?B = "(\<lambda>(x, y). (y, x)) -` A \<inter> space (S \<Otimes>\<^isub>M T)"
```
```   701       from sets_into_space[OF A]
```
```   702       have "emeasure M ((\<lambda>x. (Y x, X x)) -` A \<inter> space M) =
```
```   703         emeasure M ((\<lambda>x. (X x, Y x)) -` ?B \<inter> space M)"
```
```   704         by (auto intro!: arg_cong2[where f=emeasure] simp: space_pair_measure)
```
```   705       also have "\<dots> = (\<integral>\<^isup>+ x. Pxy x * indicator ?B x \<partial>(S \<Otimes>\<^isub>M T))"
```
```   706         using Pxy A by (intro distributed_emeasure measurable_sets) (auto simp: measurable_split_conv measurable_Pair)
```
```   707       finally have "emeasure M ((\<lambda>x. (Y x, X x)) -` A \<inter> space M) =
```
```   708         (\<integral>\<^isup>+ x. Pxy x * indicator A (snd x, fst x) \<partial>(S \<Otimes>\<^isub>M T))"
```
```   709         by (auto intro!: positive_integral_cong split: split_indicator) }
```
```   710     note * = this
```
```   711     show "distr M ?D (\<lambda>x. (Y x, X x)) = density ?D (\<lambda>(x, y). Pxy (y, x))"
```
```   712       apply (intro measure_eqI)
```
```   713       apply (simp_all add: emeasure_distr[OF 2] emeasure_density[OF 1])
```
```   714       apply (subst positive_integral_distr)
```
```   715       apply (auto intro!: measurable_pair measurable_Pxy * simp: comp_def split_beta)
```
```   716       done
```
```   717   qed
```
```   718 qed
```
```   719
```
```   720 lemma (in prob_space) distr_marginal1:
```
```   721   assumes "sigma_finite_measure S" "sigma_finite_measure T"
```
```   722   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
```
```   723   defines "Px \<equiv> \<lambda>x. (\<integral>\<^isup>+z. Pxy (x, z) \<partial>T)"
```
```   724   shows "distributed M S X Px"
```
```   725   unfolding distributed_def
```
```   726 proof safe
```
```   727   interpret S: sigma_finite_measure S by fact
```
```   728   interpret T: sigma_finite_measure T by fact
```
```   729   interpret ST: pair_sigma_finite S T by default
```
```   730
```
```   731   have XY: "(\<lambda>x. (X x, Y x)) \<in> measurable M (S \<Otimes>\<^isub>M T)"
```
```   732     using Pxy by (rule distributed_measurable)
```
```   733   then show X: "X \<in> measurable M S"
```
```   734     unfolding measurable_pair_iff by (simp add: comp_def)
```
```   735   from XY have Y: "Y \<in> measurable M T"
```
```   736     unfolding measurable_pair_iff by (simp add: comp_def)
```
```   737
```
```   738   from Pxy show borel: "Px \<in> borel_measurable S"
```
```   739     by (auto intro!: T.positive_integral_fst_measurable dest!: distributed_borel_measurable simp: Px_def)
```
```   740
```
```   741   interpret Pxy: prob_space "distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
```
```   742     using XY by (rule prob_space_distr)
```
```   743   have "(\<integral>\<^isup>+ x. max 0 (- Pxy x) \<partial>(S \<Otimes>\<^isub>M T)) = (\<integral>\<^isup>+ x. 0 \<partial>(S \<Otimes>\<^isub>M T))"
```
```   744     using Pxy
```
```   745     by (intro positive_integral_cong_AE) (auto simp: max_def dest: distributed_borel_measurable distributed_AE)
```
```   746
```
```   747   show "distr M S X = density S Px"
```
```   748   proof (rule measure_eqI)
```
```   749     fix A assume A: "A \<in> sets (distr M S X)"
```
```   750     with X Y XY have "emeasure (distr M S X) A = emeasure (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) (A \<times> space T)"
```
```   751       by (auto simp add: emeasure_distr
```
```   752                intro!: arg_cong[where f="emeasure M"] dest: measurable_space)
```
```   753     also have "\<dots> = emeasure (density (S \<Otimes>\<^isub>M T) Pxy) (A \<times> space T)"
```
```   754       using Pxy by (simp add: distributed_def)
```
```   755     also have "\<dots> = \<integral>\<^isup>+ x. \<integral>\<^isup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T \<partial>S"
```
```   756       using A borel Pxy
```
```   757       by (simp add: emeasure_density T.positive_integral_fst_measurable(2)[symmetric] distributed_def)
```
```   758     also have "\<dots> = \<integral>\<^isup>+ x. Px x * indicator A x \<partial>S"
```
```   759       apply (rule positive_integral_cong_AE)
```
```   760       using Pxy[THEN distributed_AE, THEN ST.AE_pair] AE_space
```
```   761     proof eventually_elim
```
```   762       fix x assume "x \<in> space S" "AE y in T. 0 \<le> Pxy (x, y)"
```
```   763       moreover have eq: "\<And>y. y \<in> space T \<Longrightarrow> indicator (A \<times> space T) (x, y) = indicator A x"
```
```   764         by (auto simp: indicator_def)
```
```   765       ultimately have "(\<integral>\<^isup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T) = (\<integral>\<^isup>+ y. Pxy (x, y) \<partial>T) * indicator A x"
```
```   766         using Pxy[THEN distributed_borel_measurable] by (simp add: eq positive_integral_multc measurable_Pair2 cong: positive_integral_cong)
```
```   767       also have "(\<integral>\<^isup>+ y. Pxy (x, y) \<partial>T) = Px x"
```
```   768         by (simp add: Px_def ereal_real positive_integral_positive)
```
```   769       finally show "(\<integral>\<^isup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T) = Px x * indicator A x" .
```
```   770     qed
```
```   771     finally show "emeasure (distr M S X) A = emeasure (density S Px) A"
```
```   772       using A borel Pxy by (simp add: emeasure_density)
```
```   773   qed simp
```
```   774
```
```   775   show "AE x in S. 0 \<le> Px x"
```
```   776     by (simp add: Px_def positive_integral_positive real_of_ereal_pos)
```
```   777 qed
```
```   778
```
```   779 lemma (in prob_space) distr_marginal2:
```
```   780   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
```
```   781   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
```
```   782   shows "distributed M T Y (\<lambda>y. (\<integral>\<^isup>+x. Pxy (x, y) \<partial>S))"
```
```   783   using distr_marginal1[OF T S distributed_swap[OF S T]] Pxy by simp
```
```   784
```
```   785 lemma (in prob_space) distributed_marginal_eq_joint1:
```
```   786   assumes T: "sigma_finite_measure T"
```
```   787   assumes S: "sigma_finite_measure S"
```
```   788   assumes Px: "distributed M S X Px"
```
```   789   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
```
```   790   shows "AE x in S. Px x = (\<integral>\<^isup>+y. Pxy (x, y) \<partial>T)"
```
```   791   using Px distr_marginal1[OF S T Pxy] by (rule distributed_unique)
```
```   792
```
```   793 lemma (in prob_space) distributed_marginal_eq_joint2:
```
```   794   assumes T: "sigma_finite_measure T"
```
```   795   assumes S: "sigma_finite_measure S"
```
```   796   assumes Py: "distributed M T Y Py"
```
```   797   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
```
```   798   shows "AE y in T. Py y = (\<integral>\<^isup>+x. Pxy (x, y) \<partial>S)"
```
```   799   using Py distr_marginal2[OF S T Pxy] by (rule distributed_unique)
```
```   800
```
```   801 lemma (in prob_space) distributed_joint_indep':
```
```   802   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
```
```   803   assumes X: "distributed M S X Px" and Y: "distributed M T Y Py"
```
```   804   assumes indep: "distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
```
```   805   shows "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>(x, y). Px x * Py y)"
```
```   806   unfolding distributed_def
```
```   807 proof safe
```
```   808   interpret S: sigma_finite_measure S by fact
```
```   809   interpret T: sigma_finite_measure T by fact
```
```   810   interpret ST: pair_sigma_finite S T by default
```
```   811
```
```   812   interpret X: prob_space "density S Px"
```
```   813     unfolding distributed_distr_eq_density[OF X, symmetric]
```
```   814     using distributed_measurable[OF X]
```
```   815     by (rule prob_space_distr)
```
```   816   have sf_X: "sigma_finite_measure (density S Px)" ..
```
```   817
```
```   818   interpret Y: prob_space "density T Py"
```
```   819     unfolding distributed_distr_eq_density[OF Y, symmetric]
```
```   820     using distributed_measurable[OF Y]
```
```   821     by (rule prob_space_distr)
```
```   822   have sf_Y: "sigma_finite_measure (density T Py)" ..
```
```   823
```
```   824   show "distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) = density (S \<Otimes>\<^isub>M T) (\<lambda>(x, y). Px x * Py y)"
```
```   825     unfolding indep[symmetric] distributed_distr_eq_density[OF X] distributed_distr_eq_density[OF Y]
```
```   826     using distributed_borel_measurable[OF X] distributed_AE[OF X]
```
```   827     using distributed_borel_measurable[OF Y] distributed_AE[OF Y]
```
```   828     by (rule pair_measure_density[OF _ _ _ _ S T sf_X sf_Y])
```
```   829
```
```   830   show "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
```
```   831     using distributed_measurable[OF X] distributed_measurable[OF Y]
```
```   832     by (auto intro: measurable_Pair)
```
```   833
```
```   834   show Pxy: "(\<lambda>(x, y). Px x * Py y) \<in> borel_measurable (S \<Otimes>\<^isub>M T)"
```
```   835     by (auto simp: split_beta'
```
```   836              intro!: measurable_compose[OF _ distributed_borel_measurable[OF X]]
```
```   837                      measurable_compose[OF _ distributed_borel_measurable[OF Y]])
```
```   838
```
```   839   show "AE x in S \<Otimes>\<^isub>M T. 0 \<le> (case x of (x, y) \<Rightarrow> Px x * Py y)"
```
```   840     apply (intro ST.AE_pair_measure borel_measurable_ereal_le Pxy borel_measurable_const)
```
```   841     using distributed_AE[OF X]
```
```   842     apply eventually_elim
```
```   843     using distributed_AE[OF Y]
```
```   844     apply eventually_elim
```
```   845     apply auto
```
```   846     done
```
```   847 qed
```
```   848
```
```   849 definition
```
```   850   "simple_distributed M X f \<longleftrightarrow> distributed M (count_space (X`space M)) X (\<lambda>x. ereal (f x)) \<and>
```
```   851     finite (X`space M)"
```
```   852
```
```   853 lemma simple_distributed:
```
```   854   "simple_distributed M X Px \<Longrightarrow> distributed M (count_space (X`space M)) X Px"
```
```   855   unfolding simple_distributed_def by auto
```
```   856
```
```   857 lemma simple_distributed_finite[dest]: "simple_distributed M X P \<Longrightarrow> finite (X`space M)"
```
```   858   by (simp add: simple_distributed_def)
```
```   859
```
```   860 lemma (in prob_space) distributed_simple_function_superset:
```
```   861   assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
```
```   862   assumes A: "X`space M \<subseteq> A" "finite A"
```
```   863   defines "S \<equiv> count_space A" and "P' \<equiv> (\<lambda>x. if x \<in> X`space M then P x else 0)"
```
```   864   shows "distributed M S X P'"
```
```   865   unfolding distributed_def
```
```   866 proof safe
```
```   867   show "(\<lambda>x. ereal (P' x)) \<in> borel_measurable S" unfolding S_def by simp
```
```   868   show "AE x in S. 0 \<le> ereal (P' x)"
```
```   869     using X by (auto simp: S_def P'_def simple_distributed_def intro!: measure_nonneg)
```
```   870   show "distr M S X = density S P'"
```
```   871   proof (rule measure_eqI_finite)
```
```   872     show "sets (distr M S X) = Pow A" "sets (density S P') = Pow A"
```
```   873       using A unfolding S_def by auto
```
```   874     show "finite A" by fact
```
```   875     fix a assume a: "a \<in> A"
```
```   876     then have "a \<notin> X`space M \<Longrightarrow> X -` {a} \<inter> space M = {}" by auto
```
```   877     with A a X have "emeasure (distr M S X) {a} = P' a"
```
```   878       by (subst emeasure_distr)
```
```   879          (auto simp add: S_def P'_def simple_functionD emeasure_eq_measure measurable_count_space_eq2
```
```   880                intro!: arg_cong[where f=prob])
```
```   881     also have "\<dots> = (\<integral>\<^isup>+x. ereal (P' a) * indicator {a} x \<partial>S)"
```
```   882       using A X a
```
```   883       by (subst positive_integral_cmult_indicator)
```
```   884          (auto simp: S_def P'_def simple_distributed_def simple_functionD measure_nonneg)
```
```   885     also have "\<dots> = (\<integral>\<^isup>+x. ereal (P' x) * indicator {a} x \<partial>S)"
```
```   886       by (auto simp: indicator_def intro!: positive_integral_cong)
```
```   887     also have "\<dots> = emeasure (density S P') {a}"
```
```   888       using a A by (intro emeasure_density[symmetric]) (auto simp: S_def)
```
```   889     finally show "emeasure (distr M S X) {a} = emeasure (density S P') {a}" .
```
```   890   qed
```
```   891   show "random_variable S X"
```
```   892     using X(1) A by (auto simp: measurable_def simple_functionD S_def)
```
```   893 qed
```
```   894
```
```   895 lemma (in prob_space) simple_distributedI:
```
```   896   assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
```
```   897   shows "simple_distributed M X P"
```
```   898   unfolding simple_distributed_def
```
```   899 proof
```
```   900   have "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (if x \<in> X`space M then P x else 0))"
```
```   901     (is "?A")
```
```   902     using simple_functionD[OF X(1)] by (intro distributed_simple_function_superset[OF X]) auto
```
```   903   also have "?A \<longleftrightarrow> distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (P x))"
```
```   904     by (rule distributed_cong_density) auto
```
```   905   finally show "\<dots>" .
```
```   906 qed (rule simple_functionD[OF X(1)])
```
```   907
```
```   908 lemma simple_distributed_joint_finite:
```
```   909   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
```
```   910   shows "finite (X ` space M)" "finite (Y ` space M)"
```
```   911 proof -
```
```   912   have "finite ((\<lambda>x. (X x, Y x)) ` space M)"
```
```   913     using X by (auto simp: simple_distributed_def simple_functionD)
```
```   914   then have "finite (fst ` (\<lambda>x. (X x, Y x)) ` space M)" "finite (snd ` (\<lambda>x. (X x, Y x)) ` space M)"
```
```   915     by auto
```
```   916   then show fin: "finite (X ` space M)" "finite (Y ` space M)"
```
```   917     by (auto simp: image_image)
```
```   918 qed
```
```   919
```
```   920 lemma simple_distributed_joint2_finite:
```
```   921   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
```
```   922   shows "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
```
```   923 proof -
```
```   924   have "finite ((\<lambda>x. (X x, Y x, Z x)) ` space M)"
```
```   925     using X by (auto simp: simple_distributed_def simple_functionD)
```
```   926   then have "finite (fst ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
```
```   927     "finite ((fst \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
```
```   928     "finite ((snd \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
```
```   929     by auto
```
```   930   then show fin: "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
```
```   931     by (auto simp: image_image)
```
```   932 qed
```
```   933
```
```   934 lemma simple_distributed_simple_function:
```
```   935   "simple_distributed M X Px \<Longrightarrow> simple_function M X"
```
```   936   unfolding simple_distributed_def distributed_def
```
```   937   by (auto simp: simple_function_def measurable_count_space_eq2)
```
```   938
```
```   939 lemma simple_distributed_measure:
```
```   940   "simple_distributed M X P \<Longrightarrow> a \<in> X`space M \<Longrightarrow> P a = measure M (X -` {a} \<inter> space M)"
```
```   941   using distributed_count_space[of M "X`space M" X P a, symmetric]
```
```   942   by (auto simp: simple_distributed_def measure_def)
```
```   943
```
```   944 lemma simple_distributed_nonneg: "simple_distributed M X f \<Longrightarrow> x \<in> space M \<Longrightarrow> 0 \<le> f (X x)"
```
```   945   by (auto simp: simple_distributed_measure measure_nonneg)
```
```   946
```
```   947 lemma (in prob_space) simple_distributed_joint:
```
```   948   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
```
```   949   defines "S \<equiv> count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M)"
```
```   950   defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x))`space M then Px x else 0)"
```
```   951   shows "distributed M S (\<lambda>x. (X x, Y x)) P"
```
```   952 proof -
```
```   953   from simple_distributed_joint_finite[OF X, simp]
```
```   954   have S_eq: "S = count_space (X`space M \<times> Y`space M)"
```
```   955     by (simp add: S_def pair_measure_count_space)
```
```   956   show ?thesis
```
```   957     unfolding S_eq P_def
```
```   958   proof (rule distributed_simple_function_superset)
```
```   959     show "simple_function M (\<lambda>x. (X x, Y x))"
```
```   960       using X by (rule simple_distributed_simple_function)
```
```   961     fix x assume "x \<in> (\<lambda>x. (X x, Y x)) ` space M"
```
```   962     from simple_distributed_measure[OF X this]
```
```   963     show "Px x = prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)" .
```
```   964   qed auto
```
```   965 qed
```
```   966
```
```   967 lemma (in prob_space) simple_distributed_joint2:
```
```   968   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
```
```   969   defines "S \<equiv> count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M) \<Otimes>\<^isub>M count_space (Z`space M)"
```
```   970   defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x))`space M then Px x else 0)"
```
```   971   shows "distributed M S (\<lambda>x. (X x, Y x, Z x)) P"
```
```   972 proof -
```
```   973   from simple_distributed_joint2_finite[OF X, simp]
```
```   974   have S_eq: "S = count_space (X`space M \<times> Y`space M \<times> Z`space M)"
```
```   975     by (simp add: S_def pair_measure_count_space)
```
```   976   show ?thesis
```
```   977     unfolding S_eq P_def
```
```   978   proof (rule distributed_simple_function_superset)
```
```   979     show "simple_function M (\<lambda>x. (X x, Y x, Z x))"
```
```   980       using X by (rule simple_distributed_simple_function)
```
```   981     fix x assume "x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M"
```
```   982     from simple_distributed_measure[OF X this]
```
```   983     show "Px x = prob ((\<lambda>x. (X x, Y x, Z x)) -` {x} \<inter> space M)" .
```
```   984   qed auto
```
```   985 qed
```
```   986
```
```   987 lemma (in prob_space) simple_distributed_setsum_space:
```
```   988   assumes X: "simple_distributed M X f"
```
```   989   shows "setsum f (X`space M) = 1"
```
```   990 proof -
```
```   991   from X have "setsum f (X`space M) = prob (\<Union>i\<in>X`space M. X -` {i} \<inter> space M)"
```
```   992     by (subst finite_measure_finite_Union)
```
```   993        (auto simp add: disjoint_family_on_def simple_distributed_measure simple_distributed_simple_function simple_functionD
```
```   994              intro!: setsum_cong arg_cong[where f="prob"])
```
```   995   also have "\<dots> = prob (space M)"
```
```   996     by (auto intro!: arg_cong[where f=prob])
```
```   997   finally show ?thesis
```
```   998     using emeasure_space_1 by (simp add: emeasure_eq_measure one_ereal_def)
```
```   999 qed
```
```  1000
```
```  1001 lemma (in prob_space) distributed_marginal_eq_joint_simple:
```
```  1002   assumes Px: "simple_function M X"
```
```  1003   assumes Py: "simple_distributed M Y Py"
```
```  1004   assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
```
```  1005   assumes y: "y \<in> Y`space M"
```
```  1006   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)"
```
```  1007 proof -
```
```  1008   note Px = simple_distributedI[OF Px refl]
```
```  1009   have *: "\<And>f A. setsum (\<lambda>x. max 0 (ereal (f x))) A = ereal (setsum (\<lambda>x. max 0 (f x)) A)"
```
```  1010     by (simp add: setsum_ereal[symmetric] zero_ereal_def)
```
```  1011   from distributed_marginal_eq_joint2[OF
```
```  1012     sigma_finite_measure_count_space_finite
```
```  1013     sigma_finite_measure_count_space_finite
```
```  1014     simple_distributed[OF Py] simple_distributed_joint[OF Pxy],
```
```  1015     OF Py[THEN simple_distributed_finite] Px[THEN simple_distributed_finite]]
```
```  1016     y
```
```  1017     Px[THEN simple_distributed_finite]
```
```  1018     Py[THEN simple_distributed_finite]
```
```  1019     Pxy[THEN simple_distributed, THEN distributed_real_AE]
```
```  1020   show ?thesis
```
```  1021     unfolding AE_count_space
```
```  1022     apply (auto simp add: positive_integral_count_space_finite * intro!: setsum_cong split: split_max)
```
```  1023     done
```
```  1024 qed
```
```  1025
```
```  1026 lemma prob_space_uniform_measure:
```
```  1027   assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>"
```
```  1028   shows "prob_space (uniform_measure M A)"
```
```  1029 proof
```
```  1030   show "emeasure (uniform_measure M A) (space (uniform_measure M A)) = 1"
```
```  1031     using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)], of "space M"]
```
```  1032     using sets_into_space[OF emeasure_neq_0_sets[OF A(1)]] A
```
```  1033     by (simp add: Int_absorb2 emeasure_nonneg)
```
```  1034 qed
```
```  1035
```
```  1036 lemma prob_space_uniform_count_measure: "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> prob_space (uniform_count_measure A)"
```
```  1037   by default (auto simp: emeasure_uniform_count_measure space_uniform_count_measure one_ereal_def)
```
```  1038
```
```  1039 end
```