diff -r 64023cf4d148 -r 05663f75964c src/HOL/Probability/Probability_Measure.thy
--- a/src/HOL/Probability/Probability_Measure.thy	Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Probability_Measure.thy	Mon Apr 23 12:14:35 2012 +0200
@@ -6,110 +6,219 @@
 header {*Probability measure*}
 
 theory Probability_Measure
-imports Lebesgue_Measure
+  imports Lebesgue_Measure Radon_Nikodym
 begin
 
+lemma funset_eq_UN_fun_upd_I:
+  assumes *: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A"
+  and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))"
+  and ***: "\<And>f x. \<lbrakk> f \<in> F A ; x \<in> G f \<rbrakk> \<Longrightarrow> f(a:=x) \<in> F (insert a A)"
+  shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` (G f))"
+proof safe
+  fix f assume f: "f \<in> F (insert a A)"
+  show "f \<in> (\<Union>f\<in>F A. fun_upd f a ` G f)"
+  proof (rule UN_I[of "f(a := d)"])
+    show "f(a := d) \<in> F A" using *[OF f] .
+    show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))"
+    proof (rule image_eqI[of _ _ "f a"])
+      show "f a \<in> G (f(a := d))" using **[OF f] .
+    qed simp
+  qed
+next
+  fix f x assume "f \<in> F A" "x \<in> G f"
+  from ***[OF this] show "f(a := x) \<in> F (insert a A)" .
+qed
+
+lemma extensional_funcset_insert_eq[simp]:
+  assumes "a \<notin> A"
+  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)"
+  apply (rule funset_eq_UN_fun_upd_I)
+  using assms
+  by (auto intro!: inj_onI dest: inj_onD split: split_if_asm simp: extensional_def)
+
+lemma finite_extensional_funcset[simp, intro]:
+  assumes "finite A" "finite B"
+  shows "finite (extensional A \<inter> (A \<rightarrow> B))"
+  using assms by induct auto
+
+lemma finite_PiE[simp, intro]:
+  assumes fin: "finite A" "\<And>i. i \<in> A \<Longrightarrow> finite (B i)"
+  shows "finite (Pi\<^isub>E A B)"
+proof -
+  have *: "(Pi\<^isub>E A B) \<subseteq> extensional A \<inter> (A \<rightarrow> (\<Union>i\<in>A. B i))" by auto
+  show ?thesis
+    using fin by (intro finite_subset[OF *] finite_extensional_funcset) auto
+qed
+
+
+lemma countably_additiveI[case_names countably]:
+  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)"
+  shows "countably_additive M \<mu>"
+  using assms unfolding countably_additive_def by auto
+
+lemma convex_le_Inf_differential:
+  fixes f :: "real \<Rightarrow> real"
+  assumes "convex_on I f"
+  assumes "x \<in> interior I" "y \<in> I"
+  shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
+    (is "_ \<ge> _ + Inf (?F x) * (y - x)")
+proof -
+  show ?thesis
+  proof (cases rule: linorder_cases)
+    assume "x < y"
+    moreover
+    have "open (interior I)" by auto
+    from openE[OF this `x \<in> interior I`] guess e . note e = this
+    moreover def t \<equiv> "min (x + e / 2) ((x + y) / 2)"
+    ultimately have "x < t" "t < y" "t \<in> ball x e"
+      by (auto simp: dist_real_def field_simps split: split_min)
+    with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
+
+    have "open (interior I)" by auto
+    from openE[OF this `x \<in> interior I`] guess e .
+    moreover def K \<equiv> "x - e / 2"
+    with `0 < e` have "K \<in> ball x e" "K < x" by (auto simp: dist_real_def)
+    ultimately have "K \<in> I" "K < x" "x \<in> I"
+      using interior_subset[of I] `x \<in> interior I` by auto
+
+    have "Inf (?F x) \<le> (f x - f y) / (x - y)"
+    proof (rule Inf_lower2)
+      show "(f x - f t) / (x - t) \<in> ?F x"
+        using `t \<in> I` `x < t` by auto
+      show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
+        using `convex_on I f` `x \<in> I` `y \<in> I` `x < t` `t < y` by (rule convex_on_diff)
+    next
+      fix y assume "y \<in> ?F x"
+      with order_trans[OF convex_on_diff[OF `convex_on I f` `K \<in> I` _ `K < x` _]]
+      show "(f K - f x) / (K - x) \<le> y" by auto
+    qed
+    then show ?thesis
+      using `x < y` by (simp add: field_simps)
+  next
+    assume "y < x"
+    moreover
+    have "open (interior I)" by auto
+    from openE[OF this `x \<in> interior I`] guess e . note e = this
+    moreover def t \<equiv> "x + e / 2"
+    ultimately have "x < t" "t \<in> ball x e"
+      by (auto simp: dist_real_def field_simps)
+    with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
+
+    have "(f x - f y) / (x - y) \<le> Inf (?F x)"
+    proof (rule Inf_greatest)
+      have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
+        using `y < x` by (auto simp: field_simps)
+      also
+      fix z  assume "z \<in> ?F x"
+      with order_trans[OF convex_on_diff[OF `convex_on I f` `y \<in> I` _ `y < x`]]
+      have "(f y - f x) / (y - x) \<le> z" by auto
+      finally show "(f x - f y) / (x - y) \<le> z" .
+    next
+      have "open (interior I)" by auto
+      from openE[OF this `x \<in> interior I`] guess e . note e = this
+      then have "x + e / 2 \<in> ball x e" by (auto simp: dist_real_def)
+      with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I" by auto
+      then show "?F x \<noteq> {}" by blast
+    qed
+    then show ?thesis
+      using `y < x` by (simp add: field_simps)
+  qed simp
+qed
+
+lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N"
+  by (rule measure_eqI) (auto simp: emeasure_distr)
+
 locale prob_space = finite_measure +
-  assumes measure_space_1: "measure M (space M) = 1"
+  assumes emeasure_space_1: "emeasure M (space M) = 1"
 
 lemma prob_spaceI[Pure.intro!]:
-  assumes "measure_space M"
-  assumes *: "measure M (space M) = 1"
+  assumes *: "emeasure M (space M) = 1"
   shows "prob_space M"
 proof -
   interpret finite_measure M
   proof
-    show "measure_space M" by fact
-    show "measure M (space M) \<noteq> \<infinity>" using * by simp 
+    show "emeasure M (space M) \<noteq> \<infinity>" using * by simp 
   qed
   show "prob_space M" by default fact
 qed
 
 abbreviation (in prob_space) "events \<equiv> sets M"
-abbreviation (in prob_space) "prob \<equiv> \<mu>'"
-abbreviation (in prob_space) "random_variable M' X \<equiv> sigma_algebra M' \<and> X \<in> measurable M M'"
+abbreviation (in prob_space) "prob \<equiv> measure M"
+abbreviation (in prob_space) "random_variable M' X \<equiv> X \<in> measurable M M'"
 abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M"
 
-definition (in prob_space)
-  "distribution X A = \<mu>' (X -` A \<inter> space M)"
-
-abbreviation (in prob_space)
-  "joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))"
-
-lemma (in prob_space) prob_space_cong:
-  assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "space N = space M" "sets N = sets M"
-  shows "prob_space N"
-proof
-  show "measure_space N" by (intro measure_space_cong assms)
-  show "measure N (space N) = 1"
-    using measure_space_1 assms by simp
+lemma (in prob_space) prob_space_distr:
+  assumes f: "f \<in> measurable M M'" shows "prob_space (distr M M' f)"
+proof (rule prob_spaceI)
+  have "f -` space M' \<inter> space M = space M" using f by (auto dest: measurable_space)
+  with f show "emeasure (distr M M' f) (space (distr M M' f)) = 1"
+    by (auto simp: emeasure_distr emeasure_space_1)
 qed
 
-lemma (in prob_space) distribution_cong:
-  assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x"
-  shows "distribution X = distribution Y"
-  unfolding distribution_def fun_eq_iff
-  using assms by (auto intro!: arg_cong[where f="\<mu>'"])
-
-lemma (in prob_space) joint_distribution_cong:
-  assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
-  assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
-  shows "joint_distribution X Y = joint_distribution X' Y'"
-  unfolding distribution_def fun_eq_iff
-  using assms by (auto intro!: arg_cong[where f="\<mu>'"])
-
-lemma (in prob_space) distribution_id[simp]:
-  "N \<in> events \<Longrightarrow> distribution (\<lambda>x. x) N = prob N"
-  by (auto simp: distribution_def intro!: arg_cong[where f=prob])
-
 lemma (in prob_space) prob_space: "prob (space M) = 1"
-  using measure_space_1 unfolding \<mu>'_def by (simp add: one_ereal_def)
+  using emeasure_space_1 unfolding measure_def by (simp add: one_ereal_def)
 
 lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1"
   using bounded_measure[of A] by (simp add: prob_space)
 
-lemma (in prob_space) distribution_positive[simp, intro]:
-  "0 \<le> distribution X A" unfolding distribution_def by auto
-
-lemma (in prob_space) not_zero_less_distribution[simp]:
-  "(\<not> 0 < distribution X A) \<longleftrightarrow> distribution X A = 0"
-  using distribution_positive[of X A] by arith
-
-lemma (in prob_space) joint_distribution_remove[simp]:
-    "joint_distribution X X {(x, x)} = distribution X {x}"
-  unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
+lemma (in prob_space) not_empty: "space M \<noteq> {}"
+  using prob_space by auto
 
-lemma (in prob_space) distribution_unit[simp]: "distribution (\<lambda>x. ()) {()} = 1"
-  unfolding distribution_def using prob_space by auto
-
-lemma (in prob_space) joint_distribution_unit[simp]: "distribution (\<lambda>x. (X x, ())) {(a, ())} = distribution X {a}"
-  unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
-
-lemma (in prob_space) not_empty: "space M \<noteq> {}"
-  using prob_space empty_measure' by auto
-
-lemma (in prob_space) measure_le_1: "X \<in> sets M \<Longrightarrow> \<mu> X \<le> 1"
-  unfolding measure_space_1[symmetric]
-  using sets_into_space
-  by (intro measure_mono) auto
+lemma (in prob_space) measure_le_1: "emeasure M X \<le> 1"
+  using emeasure_space[of M X] by (simp add: emeasure_space_1)
 
 lemma (in prob_space) AE_I_eq_1:
-  assumes "\<mu> {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M"
-  shows "AE x. P x"
+  assumes "emeasure M {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M"
+  shows "AE x in M. P x"
 proof (rule AE_I)
-  show "\<mu> (space M - {x \<in> space M. P x}) = 0"
-    using assms measure_space_1 by (simp add: measure_compl)
+  show "emeasure M (space M - {x \<in> space M. P x}) = 0"
+    using assms emeasure_space_1 by (simp add: emeasure_compl)
 qed (insert assms, auto)
 
-lemma (in prob_space) distribution_1:
-  "distribution X A \<le> 1"
-  unfolding distribution_def by simp
-
 lemma (in prob_space) prob_compl:
   assumes A: "A \<in> events"
   shows "prob (space M - A) = 1 - prob A"
   using finite_measure_compl[OF A] by (simp add: prob_space)
 
+lemma (in prob_space) AE_in_set_eq_1:
+  assumes "A \<in> events" shows "(AE x in M. x \<in> A) \<longleftrightarrow> prob A = 1"
+proof
+  assume ae: "AE x in M. x \<in> A"
+  have "{x \<in> space M. x \<in> A} = A" "{x \<in> space M. x \<notin> A} = space M - A"
+    using `A \<in> events`[THEN sets_into_space] by auto
+  with AE_E2[OF ae] `A \<in> events` have "1 - emeasure M A = 0"
+    by (simp add: emeasure_compl emeasure_space_1)
+  then show "prob A = 1"
+    using `A \<in> events` by (simp add: emeasure_eq_measure one_ereal_def)
+next
+  assume prob: "prob A = 1"
+  show "AE x in M. x \<in> A"
+  proof (rule AE_I)
+    show "{x \<in> space M. x \<notin> A} \<subseteq> space M - A" by auto
+    show "emeasure M (space M - A) = 0"
+      using `A \<in> events` prob
+      by (simp add: prob_compl emeasure_space_1 emeasure_eq_measure one_ereal_def)
+    show "space M - A \<in> events"
+      using `A \<in> events` by auto
+  qed
+qed
+
+lemma (in prob_space) AE_False: "(AE x in M. False) \<longleftrightarrow> False"
+proof
+  assume "AE x in M. False"
+  then have "AE x in M. x \<in> {}" by simp
+  then show False
+    by (subst (asm) AE_in_set_eq_1) auto
+qed simp
+
+lemma (in prob_space) AE_prob_1:
+  assumes "prob A = 1" shows "AE x in M. x \<in> A"
+proof -
+  from `prob A = 1` have "A \<in> events"
+    by (metis measure_notin_sets zero_neq_one)
+  with AE_in_set_eq_1 assms show ?thesis by simp
+qed
+
 lemma (in prob_space) prob_space_increasing: "increasing M prob"
   by (auto intro!: finite_measure_mono simp: increasing_def)
 
@@ -164,9 +273,8 @@
   shows "prob (\<Union> i :: nat. c i) = 0"
 proof (rule antisym)
   show "prob (\<Union> i :: nat. c i) \<le> 0"
-    using finite_measure_countably_subadditive[OF assms(1)]
-    by (simp add: assms(2) suminf_zero summable_zero)
-qed simp
+    using finite_measure_subadditive_countably[OF assms(1)] by (simp add: assms(2))
+qed (simp add: measure_nonneg)
 
 lemma (in prob_space) prob_equiprobable_finite_unions:
   assumes "s \<in> events"
@@ -178,7 +286,7 @@
   from someI_ex[OF this] assms
   have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast
   have "prob s = (\<Sum> x \<in> s. prob {x})"
-    using finite_measure_finite_singleton[OF s_finite] by simp
+    using finite_measure_eq_setsum_singleton[OF s_finite] by simp
   also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto
   also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}"
     using setsum_constant assms by (simp add: real_eq_of_nat)
@@ -199,96 +307,20 @@
   also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))"
   proof (rule finite_measure_finite_Union)
     show "finite s" by fact
-    show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact
+    show "(\<lambda>i. e \<inter> f i)`s \<subseteq> events" using assms(2) by auto
     show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"
       using disjoint by (auto simp: disjoint_family_on_def)
   qed
   finally show ?thesis .
 qed
 
-lemma (in prob_space) prob_space_vimage:
-  assumes S: "sigma_algebra S"
-  assumes T: "T \<in> measure_preserving M S"
-  shows "prob_space S"
-proof
-  interpret S: measure_space S
-    using S and T by (rule measure_space_vimage)
-  show "measure_space S" ..
-  
-  from T[THEN measure_preservingD2]
-  have "T -` space S \<inter> space M = space M"
-    by (auto simp: measurable_def)
-  with T[THEN measure_preservingD, of "space S", symmetric]
-  show  "measure S (space S) = 1"
-    using measure_space_1 by simp
-qed
-
-lemma prob_space_unique_Int_stable:
-  fixes E :: "('a, 'b) algebra_scheme" and A :: "nat \<Rightarrow> 'a set"
-  assumes E: "Int_stable E" "space E \<in> sets E"
-  and M: "prob_space M" "space M = space E" "sets M = sets (sigma E)"
-  and N: "prob_space N" "space N = space E" "sets N = sets (sigma E)"
-  and eq: "\<And>X. X \<in> sets E \<Longrightarrow> finite_measure.\<mu>' M X = finite_measure.\<mu>' N X"
-  assumes "X \<in> sets (sigma E)"
-  shows "finite_measure.\<mu>' M X = finite_measure.\<mu>' N X"
-proof -
-  interpret M!: prob_space M by fact
-  interpret N!: prob_space N by fact
-  have "measure M X = measure N X"
-  proof (rule measure_unique_Int_stable[OF `Int_stable E`])
-    show "range (\<lambda>i. space M) \<subseteq> sets E" "incseq (\<lambda>i. space M)" "(\<Union>i. space M) = space E"
-      using E M N by auto
-    show "\<And>i. M.\<mu> (space M) \<noteq> \<infinity>"
-      using M.measure_space_1 by simp
-    show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure_space.measure = M.\<mu>\<rparr>"
-      using E M N by (auto intro!: M.measure_space_cong)
-    show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure_space.measure = N.\<mu>\<rparr>"
-      using E M N by (auto intro!: N.measure_space_cong)
-    { fix X assume "X \<in> sets E"
-      then have "X \<in> sets (sigma E)"
-        by (auto simp: sets_sigma sigma_sets.Basic)
-      with eq[OF `X \<in> sets E`] M N show "M.\<mu> X = N.\<mu> X"
-        by (simp add: M.finite_measure_eq N.finite_measure_eq) }
-  qed fact
-  with `X \<in> sets (sigma E)` M N show ?thesis
-    by (simp add: M.finite_measure_eq N.finite_measure_eq)
-qed
-
-lemma (in prob_space) distribution_prob_space:
-  assumes X: "random_variable S X"
-  shows "prob_space (S\<lparr>measure := ereal \<circ> distribution X\<rparr>)" (is "prob_space ?S")
-proof (rule prob_space_vimage)
-  show "X \<in> measure_preserving M ?S"
-    using X
-    unfolding measure_preserving_def distribution_def [abs_def]
-    by (auto simp: finite_measure_eq measurable_sets)
-  show "sigma_algebra ?S" using X by simp
-qed
-
-lemma (in prob_space) AE_distribution:
-  assumes X: "random_variable MX X" and "AE x in MX\<lparr>measure := ereal \<circ> distribution X\<rparr>. Q x"
-  shows "AE x. Q (X x)"
-proof -
-  interpret X: prob_space "MX\<lparr>measure := ereal \<circ> distribution X\<rparr>" using X by (rule distribution_prob_space)
-  obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N"
-    using assms unfolding X.almost_everywhere_def by auto
-  from X[unfolded measurable_def] N show "AE x. Q (X x)"
-    by (intro AE_I'[where N="X -` N \<inter> space M"])
-       (auto simp: finite_measure_eq distribution_def measurable_sets)
-qed
-
-lemma (in prob_space) distribution_eq_integral:
-  "random_variable S X \<Longrightarrow> A \<in> sets S \<Longrightarrow> distribution X A = expectation (indicator (X -` A \<inter> space M))"
-  using finite_measure_eq[of "X -` A \<inter> space M"]
-  by (auto simp: measurable_sets distribution_def)
-
 lemma (in prob_space) expectation_less:
   assumes [simp]: "integrable M X"
   assumes gt: "\<forall>x\<in>space M. X x < b"
   shows "expectation X < b"
 proof -
   have "expectation X < expectation (\<lambda>x. b)"
-    using gt measure_space_1
+    using gt emeasure_space_1
     by (intro integral_less_AE_space) auto
   then show ?thesis using prob_space by simp
 qed
@@ -299,80 +331,11 @@
   shows "a < expectation X"
 proof -
   have "expectation (\<lambda>x. a) < expectation X"
-    using gt measure_space_1
+    using gt emeasure_space_1
     by (intro integral_less_AE_space) auto
   then show ?thesis using prob_space by simp
 qed
 
-lemma convex_le_Inf_differential:
-  fixes f :: "real \<Rightarrow> real"
-  assumes "convex_on I f"
-  assumes "x \<in> interior I" "y \<in> I"
-  shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
-    (is "_ \<ge> _ + Inf (?F x) * (y - x)")
-proof -
-  show ?thesis
-  proof (cases rule: linorder_cases)
-    assume "x < y"
-    moreover
-    have "open (interior I)" by auto
-    from openE[OF this `x \<in> interior I`] guess e . note e = this
-    moreover def t \<equiv> "min (x + e / 2) ((x + y) / 2)"
-    ultimately have "x < t" "t < y" "t \<in> ball x e"
-      by (auto simp: mem_ball dist_real_def field_simps split: split_min)
-    with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
-
-    have "open (interior I)" by auto
-    from openE[OF this `x \<in> interior I`] guess e .
-    moreover def K \<equiv> "x - e / 2"
-    with `0 < e` have "K \<in> ball x e" "K < x" by (auto simp: mem_ball dist_real_def)
-    ultimately have "K \<in> I" "K < x" "x \<in> I"
-      using interior_subset[of I] `x \<in> interior I` by auto
-
-    have "Inf (?F x) \<le> (f x - f y) / (x - y)"
-    proof (rule Inf_lower2)
-      show "(f x - f t) / (x - t) \<in> ?F x"
-        using `t \<in> I` `x < t` by auto
-      show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
-        using `convex_on I f` `x \<in> I` `y \<in> I` `x < t` `t < y` by (rule convex_on_diff)
-    next
-      fix y assume "y \<in> ?F x"
-      with order_trans[OF convex_on_diff[OF `convex_on I f` `K \<in> I` _ `K < x` _]]
-      show "(f K - f x) / (K - x) \<le> y" by auto
-    qed
-    then show ?thesis
-      using `x < y` by (simp add: field_simps)
-  next
-    assume "y < x"
-    moreover
-    have "open (interior I)" by auto
-    from openE[OF this `x \<in> interior I`] guess e . note e = this
-    moreover def t \<equiv> "x + e / 2"
-    ultimately have "x < t" "t \<in> ball x e"
-      by (auto simp: mem_ball dist_real_def field_simps)
-    with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
-
-    have "(f x - f y) / (x - y) \<le> Inf (?F x)"
-    proof (rule Inf_greatest)
-      have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
-        using `y < x` by (auto simp: field_simps)
-      also
-      fix z  assume "z \<in> ?F x"
-      with order_trans[OF convex_on_diff[OF `convex_on I f` `y \<in> I` _ `y < x`]]
-      have "(f y - f x) / (y - x) \<le> z" by auto
-      finally show "(f x - f y) / (x - y) \<le> z" .
-    next
-      have "open (interior I)" by auto
-      from openE[OF this `x \<in> interior I`] guess e . note e = this
-      then have "x + e / 2 \<in> ball x e" by (auto simp: mem_ball dist_real_def)
-      with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I" by auto
-      then show "?F x \<noteq> {}" by blast
-    qed
-    then show ?thesis
-      using `y < x` by (simp add: field_simps)
-  qed simp
-qed
-
 lemma (in prob_space) jensens_inequality:
   fixes a b :: real
   assumes X: "integrable M X" "X ` space M \<subseteq> I"
@@ -410,8 +373,7 @@
     fix k assume "k \<in> (\<lambda>x. q x + ?F x * (expectation X - x)) ` I"
     then guess x .. note x = this
     have "q x + ?F x * (expectation X  - x) = expectation (\<lambda>w. q x + ?F x * (X w - x))"
-      using prob_space
-      by (simp add: integral_add integral_cmult integral_diff lebesgue_integral_const X)
+      using prob_space by (simp add: X)
     also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
       using `x \<in> I` `open I` X(2)
       by (intro integral_mono integral_add integral_cmult integral_diff
@@ -422,31 +384,6 @@
   finally show "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" .
 qed
 
-lemma (in prob_space) distribution_eq_translated_integral:
-  assumes "random_variable S X" "A \<in> sets S"
-  shows "distribution X A = integral\<^isup>P (S\<lparr>measure := ereal \<circ> distribution X\<rparr>) (indicator A)"
-proof -
-  interpret S: prob_space "S\<lparr>measure := ereal \<circ> distribution X\<rparr>"
-    using assms(1) by (rule distribution_prob_space)
-  show ?thesis
-    using S.positive_integral_indicator(1)[of A] assms by simp
-qed
-
-lemma (in prob_space) finite_expectation1:
-  assumes f: "finite (X`space M)" and rv: "random_variable borel X"
-  shows "expectation X = (\<Sum>r \<in> X ` space M. r * prob (X -` {r} \<inter> space M))" (is "_ = ?r")
-proof (subst integral_on_finite)
-  show "X \<in> borel_measurable M" "finite (X`space M)" using assms by auto
-  show "(\<Sum> r \<in> X ` space M. r * real (\<mu> (X -` {r} \<inter> space M))) = ?r"
-    "\<And>x. \<mu> (X -` {x} \<inter> space M) \<noteq> \<infinity>"
-    using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto
-qed
-
-lemma (in prob_space) finite_expectation:
-  assumes "finite (X`space M)" "random_variable borel X"
-  shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})"
-  using assms unfolding distribution_def using finite_expectation1 by auto
-
 lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0:
   assumes "{x} \<in> events"
   assumes "prob {x} = 1"
@@ -455,119 +392,25 @@
   shows "prob {y} = 0"
   using prob_one_inter[of "{y}" "{x}"] assms by auto
 
-lemma (in prob_space) distribution_empty[simp]: "distribution X {} = 0"
-  unfolding distribution_def by simp
-
-lemma (in prob_space) distribution_space[simp]: "distribution X (X ` space M) = 1"
-proof -
-  have "X -` X ` space M \<inter> space M = space M" by auto
-  thus ?thesis unfolding distribution_def by (simp add: prob_space)
-qed
-
-lemma (in prob_space) distribution_one:
-  assumes "random_variable M' X" and "A \<in> sets M'"
-  shows "distribution X A \<le> 1"
-proof -
-  have "distribution X A \<le> \<mu>' (space M)" unfolding distribution_def
-    using assms[unfolded measurable_def] by (auto intro!: finite_measure_mono)
-  thus ?thesis by (simp add: prob_space)
-qed
-
-lemma (in prob_space) distribution_x_eq_1_imp_distribution_y_eq_0:
-  assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X"
-    (is "random_variable ?S X")
-  assumes "distribution X {x} = 1"
-  assumes "y \<noteq> x"
-  shows "distribution X {y} = 0"
-proof cases
-  { fix x have "X -` {x} \<inter> space M \<in> sets M"
-    proof cases
-      assume "x \<in> X`space M" with X show ?thesis
-        by (auto simp: measurable_def image_iff)
-    next
-      assume "x \<notin> X`space M" then have "X -` {x} \<inter> space M = {}" by auto
-      then show ?thesis by auto
-    qed } note single = this
-  have "X -` {x} \<inter> space M - X -` {y} \<inter> space M = X -` {x} \<inter> space M"
-    "X -` {y} \<inter> space M \<inter> (X -` {x} \<inter> space M) = {}"
-    using `y \<noteq> x` by auto
-  with finite_measure_inter_full_set[OF single single, of x y] assms(2)
-  show ?thesis by (auto simp: distribution_def prob_space)
-next
-  assume "{y} \<notin> sets ?S"
-  then have "X -` {y} \<inter> space M = {}" by auto
-  thus "distribution X {y} = 0" unfolding distribution_def by auto
-qed
-
 lemma (in prob_space) joint_distribution_Times_le_fst:
-  assumes X: "random_variable MX X" and Y: "random_variable MY Y"
-    and A: "A \<in> sets MX" and B: "B \<in> sets MY"
-  shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
-  unfolding distribution_def
-proof (intro finite_measure_mono)
-  show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
-  show "X -` A \<inter> space M \<in> events"
-    using X A unfolding measurable_def by simp
-  have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
-    (X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
-qed
-
-lemma (in prob_space) joint_distribution_commute:
-  "joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)"
-  unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
+  "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
+    \<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"
+  by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
 
 lemma (in prob_space) joint_distribution_Times_le_snd:
-  assumes X: "random_variable MX X" and Y: "random_variable MY Y"
-    and A: "A \<in> sets MX" and B: "B \<in> sets MY"
-  shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B"
-  using assms
-  by (subst joint_distribution_commute)
-     (simp add: swap_product joint_distribution_Times_le_fst)
-
-lemma (in prob_space) random_variable_pairI:
-  assumes "random_variable MX X"
-  assumes "random_variable MY Y"
-  shows "random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))"
-proof
-  interpret MX: sigma_algebra MX using assms by simp
-  interpret MY: sigma_algebra MY using assms by simp
-  interpret P: pair_sigma_algebra MX MY by default
-  show "sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default
-  have sa: "sigma_algebra M" by default
-  show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)"
-    unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
-qed
-
-lemma (in prob_space) joint_distribution_commute_singleton:
-  "joint_distribution X Y {(x, y)} = joint_distribution Y X {(y, x)}"
-  unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
-
-lemma (in prob_space) joint_distribution_assoc_singleton:
-  "joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} =
-   joint_distribution (\<lambda>x. (X x, Y x)) Z {((x, y), z)}"
-  unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
+  "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
+    \<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"
+  by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
 
 locale pair_prob_space = pair_sigma_finite M1 M2 + M1: prob_space M1 + M2: prob_space M2 for M1 M2
 
-sublocale pair_prob_space \<subseteq> P: prob_space P
+sublocale pair_prob_space \<subseteq> P: prob_space "M1 \<Otimes>\<^isub>M M2"
 proof
-  show "measure_space P" ..
-  show "measure P (space P) = 1"
-    by (simp add: pair_measure_times M1.measure_space_1 M2.measure_space_1 space_pair_measure)
+  show "emeasure (M1 \<Otimes>\<^isub>M M2) (space (M1 \<Otimes>\<^isub>M M2)) = 1"
+    by (simp add: emeasure_pair_measure_Times M1.emeasure_space_1 M2.emeasure_space_1 space_pair_measure)
 qed
 
-lemma countably_additiveI[case_names countably]:
-  assumes "\<And>A. \<lbrakk> range A \<subseteq> sets M ; disjoint_family A ; (\<Union>i. A i) \<in> sets M\<rbrakk> \<Longrightarrow>
-    (\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
-  shows "countably_additive M \<mu>"
-  using assms unfolding countably_additive_def by auto
-
-lemma (in prob_space) joint_distribution_prob_space:
-  assumes "random_variable MX X" "random_variable MY Y"
-  shows "prob_space ((MX \<Otimes>\<^isub>M MY) \<lparr> measure := ereal \<circ> joint_distribution X Y\<rparr>)"
-  using random_variable_pairI[OF assms] by (rule distribution_prob_space)
-
-locale product_prob_space = product_sigma_finite M for M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme" +
+locale product_prob_space = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
   fixes I :: "'i set"
   assumes prob_space: "\<And>i. prob_space (M i)"
 
@@ -578,648 +421,401 @@
 
 sublocale finite_product_prob_space \<subseteq> prob_space "\<Pi>\<^isub>M i\<in>I. M i"
 proof
-  show "measure_space P" ..
-  show "measure P (space P) = 1"
-    by (simp add: measure_times M.measure_space_1 setprod_1)
+  show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (space (\<Pi>\<^isub>M i\<in>I. M i)) = 1"
+    by (simp add: measure_times M.emeasure_space_1 setprod_1 space_PiM)
 qed
 
 lemma (in finite_product_prob_space) prob_times:
   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets (M i)"
   shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
 proof -
-  have "ereal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)"
-    using X by (intro finite_measure_eq[symmetric] in_P) auto
-  also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))"
+  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)"
+    using X by (simp add: emeasure_eq_measure)
+  also have "\<dots> = (\<Prod>i\<in>I. emeasure (M i) (X i))"
     using measure_times X by simp
-  also have "\<dots> = ereal (\<Prod>i\<in>I. M.\<mu>' i (X i))"
-    using X by (simp add: M.finite_measure_eq setprod_ereal)
-  finally show ?thesis by simp
-qed
-
-lemma (in prob_space) random_variable_restrict:
-  assumes I: "finite I"
-  assumes X: "\<And>i. i \<in> I \<Longrightarrow> random_variable (N i) (X i)"
-  shows "random_variable (Pi\<^isub>M I N) (\<lambda>x. \<lambda>i\<in>I. X i x)"
-proof
-  { fix i assume "i \<in> I"
-    with X interpret N: sigma_algebra "N i" by simp
-    have "sets (N i) \<subseteq> Pow (space (N i))" by (rule N.space_closed) }
-  note N_closed = this
-  then show "sigma_algebra (Pi\<^isub>M I N)"
-    by (simp add: product_algebra_def)
-       (intro sigma_algebra_sigma product_algebra_generator_sets_into_space)
-  show "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable M (Pi\<^isub>M I N)"
-    using X by (intro measurable_restrict[OF I N_closed]) auto
-qed
-
-section "Probability spaces on finite sets"
-
-locale finite_prob_space = prob_space + finite_measure_space
-
-abbreviation (in prob_space) "finite_random_variable M' X \<equiv> finite_sigma_algebra M' \<and> X \<in> measurable M M'"
-
-lemma (in prob_space) finite_random_variableD:
-  assumes "finite_random_variable M' X" shows "random_variable M' X"
-proof -
-  interpret M': finite_sigma_algebra M' using assms by simp
-  show "random_variable M' X" using assms by simp default
-qed
-
-lemma (in prob_space) distribution_finite_prob_space:
-  assumes "finite_random_variable MX X"
-  shows "finite_prob_space (MX\<lparr>measure := ereal \<circ> distribution X\<rparr>)"
-proof -
-  interpret X: prob_space "MX\<lparr>measure := ereal \<circ> distribution X\<rparr>"
-    using assms[THEN finite_random_variableD] by (rule distribution_prob_space)
-  interpret MX: finite_sigma_algebra MX
-    using assms by auto
-  show ?thesis by default (simp_all add: MX.finite_space)
-qed
-
-lemma (in prob_space) simple_function_imp_finite_random_variable[simp, intro]:
-  assumes "simple_function M X"
-  shows "finite_random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = x \<rparr> X"
-    (is "finite_random_variable ?X _")
-proof (intro conjI)
-  have [simp]: "finite (X ` space M)" using assms unfolding simple_function_def by simp
-  interpret X: sigma_algebra ?X by (rule sigma_algebra_Pow)
-  show "finite_sigma_algebra ?X"
-    by default auto
-  show "X \<in> measurable M ?X"
-  proof (unfold measurable_def, clarsimp)
-    fix A assume A: "A \<subseteq> X`space M"
-    then have "finite A" by (rule finite_subset) simp
-    then have "X -` (\<Union>a\<in>A. {a}) \<inter> space M \<in> events"
-      unfolding vimage_UN UN_extend_simps
-      apply (rule finite_UN)
-      using A assms unfolding simple_function_def by auto
-    then show "X -` A \<inter> space M \<in> events" by simp
-  qed
-qed
-
-lemma (in prob_space) simple_function_imp_random_variable[simp, intro]:
-  assumes "simple_function M X"
-  shows "random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = ext \<rparr> X"
-  using simple_function_imp_finite_random_variable[OF assms, of ext]
-  by (auto dest!: finite_random_variableD)
-
-lemma (in prob_space) sum_over_space_real_distribution:
-  "simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1"
-  unfolding distribution_def prob_space[symmetric]
-  by (subst finite_measure_finite_Union[symmetric])
-     (auto simp add: disjoint_family_on_def simple_function_def
-           intro!: arg_cong[where f=prob])
-
-lemma (in prob_space) finite_random_variable_pairI:
-  assumes "finite_random_variable MX X"
-  assumes "finite_random_variable MY Y"
-  shows "finite_random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))"
-proof
-  interpret MX: finite_sigma_algebra MX using assms by simp
-  interpret MY: finite_sigma_algebra MY using assms by simp
-  interpret P: pair_finite_sigma_algebra MX MY by default
-  show "finite_sigma_algebra (MX \<Otimes>\<^isub>M MY)" ..
-  have sa: "sigma_algebra M" by default
-  show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)"
-    unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
-qed
-
-lemma (in prob_space) finite_random_variable_imp_sets:
-  "finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX"
-  unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp
-
-lemma (in prob_space) finite_random_variable_measurable:
-  assumes X: "finite_random_variable MX X" shows "X -` A \<inter> space M \<in> events"
-proof -
-  interpret X: finite_sigma_algebra MX using X by simp
-  from X have vimage: "\<And>A. A \<subseteq> space MX \<Longrightarrow> X -` A \<inter> space M \<in> events" and
-    "X \<in> space M \<rightarrow> space MX"
-    by (auto simp: measurable_def)
-  then have *: "X -` A \<inter> space M = X -` (A \<inter> space MX) \<inter> space M"
-    by auto
-  show "X -` A \<inter> space M \<in> events"
-    unfolding * by (intro vimage) auto
-qed
-
-lemma (in prob_space) joint_distribution_finite_Times_le_fst:
-  assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
-  shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
-  unfolding distribution_def
-proof (intro finite_measure_mono)
-  show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
-  show "X -` A \<inter> space M \<in> events"
-    using finite_random_variable_measurable[OF X] .
-  have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
-    (X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
-qed
-
-lemma (in prob_space) joint_distribution_finite_Times_le_snd:
-  assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
-  shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B"
-  using assms
-  by (subst joint_distribution_commute)
-     (simp add: swap_product joint_distribution_finite_Times_le_fst)
-
-lemma (in prob_space) finite_distribution_order:
-  fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
-  assumes "finite_random_variable MX X" "finite_random_variable MY Y"
-  shows "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}"
-    and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}"
-    and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}"
-    and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}"
-    and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
-    and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
-  using joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"]
-  using joint_distribution_finite_Times_le_fst[OF assms, of "{x}" "{y}"]
-  by (auto intro: antisym)
-
-lemma (in prob_space) setsum_joint_distribution:
-  assumes X: "finite_random_variable MX X"
-  assumes Y: "random_variable MY Y" "B \<in> sets MY"
-  shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B"
-  unfolding distribution_def
-proof (subst finite_measure_finite_Union[symmetric])
-  interpret MX: finite_sigma_algebra MX using X by auto
-  show "finite (space MX)" using MX.finite_space .
-  let ?d = "\<lambda>i. (\<lambda>x. (X x, Y x)) -` ({i} \<times> B) \<inter> space M"
-  { fix i assume "i \<in> space MX"
-    moreover have "?d i = (X -` {i} \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
-    ultimately show "?d i \<in> events"
-      using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y
-      using MX.sets_eq_Pow by auto }
-  show "disjoint_family_on ?d (space MX)" by (auto simp: disjoint_family_on_def)
-  show "\<mu>' (\<Union>i\<in>space MX. ?d i) = \<mu>' (Y -` B \<inter> space M)"
-    using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>'])
-qed
-
-lemma (in prob_space) setsum_joint_distribution_singleton:
-  assumes X: "finite_random_variable MX X"
-  assumes Y: "finite_random_variable MY Y" "b \<in> space MY"
-  shows "(\<Sum>a\<in>space MX. joint_distribution X Y {(a, b)}) = distribution Y {b}"
-  using setsum_joint_distribution[OF X
-    finite_random_variableD[OF Y(1)]
-    finite_random_variable_imp_sets[OF Y]] by simp
-
-lemma (in prob_space) setsum_distribution:
-  assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
-  using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV \<rparr>" "\<lambda>x. ()" "{()}"]
-  using sigma_algebra_Pow[of "UNIV::unit set" "()"] by simp
-
-locale pair_finite_prob_space = pair_prob_space M1 M2 + pair_finite_space M1 M2 + M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2
-
-sublocale pair_finite_prob_space \<subseteq> finite_prob_space P by default
-
-lemma funset_eq_UN_fun_upd_I:
-  assumes *: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A"
-  and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))"
-  and ***: "\<And>f x. \<lbrakk> f \<in> F A ; x \<in> G f \<rbrakk> \<Longrightarrow> f(a:=x) \<in> F (insert a A)"
-  shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` (G f))"
-proof safe
-  fix f assume f: "f \<in> F (insert a A)"
-  show "f \<in> (\<Union>f\<in>F A. fun_upd f a ` G f)"
-  proof (rule UN_I[of "f(a := d)"])
-    show "f(a := d) \<in> F A" using *[OF f] .
-    show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))"
-    proof (rule image_eqI[of _ _ "f a"])
-      show "f a \<in> G (f(a := d))" using **[OF f] .
-    qed simp
-  qed
-next
-  fix f x assume "f \<in> F A" "x \<in> G f"
-  from ***[OF this] show "f(a := x) \<in> F (insert a A)" .
-qed
-
-lemma extensional_funcset_insert_eq[simp]:
-  assumes "a \<notin> A"
-  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)"
-  apply (rule funset_eq_UN_fun_upd_I)
-  using assms
-  by (auto intro!: inj_onI dest: inj_onD split: split_if_asm simp: extensional_def)
-
-lemma finite_extensional_funcset[simp, intro]:
-  assumes "finite A" "finite B"
-  shows "finite (extensional A \<inter> (A \<rightarrow> B))"
-  using assms by induct (auto simp: extensional_funcset_insert_eq)
-
-lemma finite_PiE[simp, intro]:
-  assumes fin: "finite A" "\<And>i. i \<in> A \<Longrightarrow> finite (B i)"
-  shows "finite (Pi\<^isub>E A B)"
-proof -
-  have *: "(Pi\<^isub>E A B) \<subseteq> extensional A \<inter> (A \<rightarrow> (\<Union>i\<in>A. B i))" by auto
-  show ?thesis
-    using fin by (intro finite_subset[OF *] finite_extensional_funcset) auto
-qed
-
-locale finite_product_finite_prob_space = finite_product_prob_space M I for M I +
-  assumes finite_space: "\<And>i. finite_prob_space (M i)"
-
-sublocale finite_product_finite_prob_space \<subseteq> M!: finite_prob_space "M i" using finite_space .
-
-lemma (in finite_product_finite_prob_space) singleton_eq_product:
-  assumes x: "x \<in> space P" shows "{x} = (\<Pi>\<^isub>E i\<in>I. {x i})"
-proof (safe intro!: ext[of _ x])
-  fix y i assume *: "y \<in> (\<Pi> i\<in>I. {x i})" "y \<in> extensional I"
-  with x show "y i = x i"
-    by (cases "i \<in> I") (auto simp: extensional_def)
-qed (insert x, auto)
-
-sublocale finite_product_finite_prob_space \<subseteq> finite_prob_space "Pi\<^isub>M I M"
-proof
-  show "finite (space P)"
-    using finite_index M.finite_space by auto
-
-  { fix x assume "x \<in> space P"
-    with this[THEN singleton_eq_product]
-    have "{x} \<in> sets P"
-      by (auto intro!: in_P) }
-  note x_in_P = this
-
-  have "Pow (space P) \<subseteq> sets P"
-  proof
-    fix X assume "X \<in> Pow (space P)"
-    moreover then have "finite X"
-      using `finite (space P)` by (blast intro: finite_subset)
-    ultimately have "(\<Union>x\<in>X. {x}) \<in> sets P"
-      by (intro finite_UN x_in_P) auto
-    then show "X \<in> sets P" by simp
-  qed
-  with space_closed show [simp]: "sets P = Pow (space P)" ..
-qed
-
-lemma (in finite_product_finite_prob_space) measure_finite_times:
-  "(\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)) \<Longrightarrow> \<mu> (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.\<mu> i (X i))"
-  by (rule measure_times) simp
-
-lemma (in finite_product_finite_prob_space) measure_singleton_times:
-  assumes x: "x \<in> space P" shows "\<mu> {x} = (\<Prod>i\<in>I. M.\<mu> i {x i})"
-  unfolding singleton_eq_product[OF x] using x
-  by (intro measure_finite_times) auto
-
-lemma (in finite_product_finite_prob_space) prob_finite_times:
-  assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)"
-  shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
-proof -
-  have "ereal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)"
-    using X by (intro finite_measure_eq[symmetric] in_P) auto
-  also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))"
-    using measure_finite_times X by simp
-  also have "\<dots> = ereal (\<Prod>i\<in>I. M.\<mu>' i (X i))"
-    using X by (simp add: M.finite_measure_eq setprod_ereal)
+  also have "\<dots> = ereal (\<Prod>i\<in>I. measure (M i) (X i))"
+    using X by (simp add: M.emeasure_eq_measure setprod_ereal)
   finally show ?thesis by simp
 qed
 
-lemma (in finite_product_finite_prob_space) prob_singleton_times:
-  assumes x: "x \<in> space P"
-  shows "prob {x} = (\<Prod>i\<in>I. M.prob i {x i})"
-  unfolding singleton_eq_product[OF x] using x
-  by (intro prob_finite_times) auto
-
-lemma (in finite_product_finite_prob_space) prob_finite_product:
-  "A \<subseteq> space P \<Longrightarrow> prob A = (\<Sum>x\<in>A. \<Prod>i\<in>I. M.prob i {x i})"
-  by (auto simp add: finite_measure_singleton prob_singleton_times
-           simp del: space_product_algebra
-           intro!: setsum_cong prob_singleton_times)
+section {* Distributions *}
 
-lemma (in prob_space) joint_distribution_finite_prob_space:
-  assumes X: "finite_random_variable MX X"
-  assumes Y: "finite_random_variable MY Y"
-  shows "finite_prob_space ((MX \<Otimes>\<^isub>M MY)\<lparr> measure := ereal \<circ> joint_distribution X Y\<rparr>)"
-  by (intro distribution_finite_prob_space finite_random_variable_pairI X Y)
-
-lemma finite_prob_space_eq:
-  "finite_prob_space M \<longleftrightarrow> finite_measure_space M \<and> measure M (space M) = 1"
-  unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def
-  by auto
-
-lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1"
-  using measure_space_1 sum_over_space by simp
+definition "distributed M N X f \<longleftrightarrow> distr M N X = density N f \<and> 
+  f \<in> borel_measurable N \<and> (AE x in N. 0 \<le> f x) \<and> X \<in> measurable M N"
 
-lemma (in finite_prob_space) joint_distribution_restriction_fst:
-  "joint_distribution X Y A \<le> distribution X (fst ` A)"
-  unfolding distribution_def
-proof (safe intro!: finite_measure_mono)
-  fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A"
-  show "x \<in> X -` fst ` A"
-    by (auto intro!: image_eqI[OF _ *])
-qed (simp_all add: sets_eq_Pow)
-
-lemma (in finite_prob_space) joint_distribution_restriction_snd:
-  "joint_distribution X Y A \<le> distribution Y (snd ` A)"
-  unfolding distribution_def
-proof (safe intro!: finite_measure_mono)
-  fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A"
-  show "x \<in> Y -` snd ` A"
-    by (auto intro!: image_eqI[OF _ *])
-qed (simp_all add: sets_eq_Pow)
-
-lemma (in finite_prob_space) distribution_order:
-  shows "0 \<le> distribution X x'"
-  and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')"
-  and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}"
-  and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}"
-  and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}"
-  and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}"
-  and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
-  and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
-  using
-    joint_distribution_restriction_fst[of X Y "{(x, y)}"]
-    joint_distribution_restriction_snd[of X Y "{(x, y)}"]
-  by (auto intro: antisym)
-
-lemma (in finite_prob_space) distribution_mono:
-  assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
-  shows "distribution X x \<le> distribution Y y"
-  unfolding distribution_def
-  using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono)
+lemma
+  shows distributed_distr_eq_density: "distributed M N X f \<Longrightarrow> distr M N X = density N f"
+    and distributed_measurable: "distributed M N X f \<Longrightarrow> X \<in> measurable M N"
+    and distributed_borel_measurable: "distributed M N X f \<Longrightarrow> f \<in> borel_measurable N"
+    and distributed_AE: "distributed M N X f \<Longrightarrow> (AE x in N. 0 \<le> f x)"
+  by (simp_all add: distributed_def)
 
-lemma (in finite_prob_space) distribution_mono_gt_0:
-  assumes gt_0: "0 < distribution X x"
-  assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
-  shows "0 < distribution Y y"
-  by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *)
-
-lemma (in finite_prob_space) sum_over_space_distrib:
-  "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
-  unfolding distribution_def prob_space[symmetric] using finite_space
-  by (subst finite_measure_finite_Union[symmetric])
-     (auto simp add: disjoint_family_on_def sets_eq_Pow
-           intro!: arg_cong[where f=\<mu>'])
-
-lemma (in finite_prob_space) finite_sum_over_space_eq_1:
-  "(\<Sum>x\<in>space M. prob {x}) = 1"
-  using prob_space finite_space
-  by (subst (asm) finite_measure_finite_singleton) auto
-
-lemma (in prob_space) distribution_remove_const:
-  shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}"
-  and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}"
-  and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}"
-  and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}"
-  and "distribution (\<lambda>x. ()) {()} = 1"
-  by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric])
+lemma
+  shows distributed_real_measurable: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> f \<in> borel_measurable N"
+    and distributed_real_AE: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> (AE x in N. 0 \<le> f x)"
+  by (simp_all add: distributed_def borel_measurable_ereal_iff)
 
-lemma (in finite_prob_space) setsum_distribution_gen:
-  assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M"
-  and "inj_on f (X`space M)"
-  shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}"
-  unfolding distribution_def assms
-  using finite_space assms
-  by (subst finite_measure_finite_Union[symmetric])
-     (auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def
-      intro!: arg_cong[where f=prob])
-
-lemma (in finite_prob_space) setsum_distribution_cut:
-  "(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}"
-  "(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}"
-  "(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}"
-  "(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}"
-  "(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}"
-  by (auto intro!: inj_onI setsum_distribution_gen)
-
-lemma (in finite_prob_space) uniform_prob:
-  assumes "x \<in> space M"
-  assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}"
-  shows "prob {x} = 1 / card (space M)"
+lemma distributed_count_space:
+  assumes X: "distributed M (count_space A) X P" and a: "a \<in> A" and A: "finite A"
+  shows "P a = emeasure M (X -` {a} \<inter> space M)"
 proof -
-  have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}"
-    using assms(2)[OF _ `x \<in> space M`] by blast
-  have "1 = prob (space M)"
-    using prob_space by auto
-  also have "\<dots> = (\<Sum> x \<in> space M. prob {x})"
-    using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified]
-      sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format]
-      finite_space unfolding disjoint_family_on_def  prob_space[symmetric]
-    by (auto simp add:setsum_restrict_set)
-  also have "\<dots> = (\<Sum> y \<in> space M. prob {x})"
-    using prob_x by auto
-  also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp
-  finally have one: "1 = real (card (space M)) * prob {x}"
-    using real_eq_of_nat by auto
-  hence two: "real (card (space M)) \<noteq> 0" by fastforce
-  from one have three: "prob {x} \<noteq> 0" by fastforce
-  thus ?thesis using one two three divide_cancel_right
-    by (auto simp:field_simps)
+  have "emeasure M (X -` {a} \<inter> space M) = emeasure (distr M (count_space A) X) {a}"
+    using X a A by (simp add: distributed_measurable emeasure_distr)
+  also have "\<dots> = emeasure (density (count_space A) P) {a}"
+    using X by (simp add: distributed_distr_eq_density)
+  also have "\<dots> = (\<integral>\<^isup>+x. P a * indicator {a} x \<partial>count_space A)"
+    using X a by (auto simp add: emeasure_density distributed_def indicator_def intro!: positive_integral_cong)
+  also have "\<dots> = P a"
+    using X a by (subst positive_integral_cmult_indicator) (auto simp: distributed_def one_ereal_def[symmetric] AE_count_space)
+  finally show ?thesis ..
 qed
 
-lemma (in prob_space) prob_space_subalgebra:
-  assumes "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M"
-    and "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
-  shows "prob_space N"
-proof
-  interpret N: measure_space N
-    by (rule measure_space_subalgebra[OF assms])
-  show "measure_space N" ..
-  show "measure N (space N) = 1"
-    using assms(4)[OF N.top] by (simp add: assms measure_space_1)
+lemma distributed_cong_density:
+  "(AE x in N. f x = g x) \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable N \<Longrightarrow>
+    distributed M N X f \<longleftrightarrow> distributed M N X g"
+  by (auto simp: distributed_def intro!: density_cong)
+
+lemma subdensity:
+  assumes T: "T \<in> measurable P Q"
+  assumes f: "distributed M P X f"
+  assumes g: "distributed M Q Y g"
+  assumes Y: "Y = T \<circ> X"
+  shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
+proof -
+  have "{x\<in>space Q. g x = 0} \<in> null_sets (distr M Q (T \<circ> X))"
+    using g Y by (auto simp: null_sets_density_iff distributed_def)
+  also have "distr M Q (T \<circ> X) = distr (distr M P X) Q T"
+    using T f[THEN distributed_measurable] by (rule distr_distr[symmetric])
+  finally have "T -` {x\<in>space Q. g x = 0} \<inter> space P \<in> null_sets (distr M P X)"
+    using T by (subst (asm) null_sets_distr_iff) auto
+  also have "T -` {x\<in>space Q. g x = 0} \<inter> space P = {x\<in>space P. g (T x) = 0}"
+    using T by (auto dest: measurable_space)
+  finally show ?thesis
+    using f g by (auto simp add: null_sets_density_iff distributed_def)
 qed
 
-lemma (in prob_space) prob_space_of_restricted_space:
-  assumes "\<mu> A \<noteq> 0" "A \<in> sets M"
-  shows "prob_space (restricted_space A \<lparr>measure := \<lambda>S. \<mu> S / \<mu> A\<rparr>)"
-    (is "prob_space ?P")
+lemma subdensity_real:
+  fixes g :: "'a \<Rightarrow> real" and f :: "'b \<Rightarrow> real"
+  assumes T: "T \<in> measurable P Q"
+  assumes f: "distributed M P X f"
+  assumes g: "distributed M Q Y g"
+  assumes Y: "Y = T \<circ> X"
+  shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
+  using subdensity[OF T, of M X "\<lambda>x. ereal (f x)" Y "\<lambda>x. ereal (g x)"] assms by auto
+
+lemma distributed_integral:
+  "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)"
+  by (auto simp: distributed_real_measurable distributed_real_AE distributed_measurable
+                 distributed_distr_eq_density[symmetric] integral_density[symmetric] integral_distr)
+  
+lemma distributed_transform_integral:
+  assumes Px: "distributed M N X Px"
+  assumes "distributed M P Y Py"
+  assumes Y: "Y = T \<circ> X" and T: "T \<in> measurable N P" and f: "f \<in> borel_measurable P"
+  shows "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. Px x * f (T x) \<partial>N)"
 proof -
-  interpret A: measure_space "restricted_space A"
-    using `A \<in> sets M` by (rule restricted_measure_space)
-  interpret A': sigma_algebra ?P
-    by (rule A.sigma_algebra_cong) auto
-  show "prob_space ?P"
-  proof
-    show "measure_space ?P"
-    proof
-      show "positive ?P (measure ?P)"
-      proof (simp add: positive_def, safe)
-        fix B assume "B \<in> events"
-        with real_measure[of "A \<inter> B"] real_measure[OF `A \<in> events`] `A \<in> sets M`
-        show "0 \<le> \<mu> (A \<inter> B) / \<mu> A" by (auto simp: Int)
-      qed
-      show "countably_additive ?P (measure ?P)"
-      proof (simp add: countably_additive_def, safe)
-        fix B and F :: "nat \<Rightarrow> 'a set"
-        assume F: "range F \<subseteq> op \<inter> A ` events" "disjoint_family F"
-        { fix i
-          from F have "F i \<in> op \<inter> A ` events" by auto
-          with `A \<in> events` have "F i \<in> events" by auto }
-        moreover then have "range F \<subseteq> events" by auto
-        moreover have "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S"
-          by (simp add: mult_commute divide_ereal_def)
-        moreover have "0 \<le> inverse (\<mu> A)"
-          using real_measure[OF `A \<in> events`] by auto
-        ultimately show "(\<Sum>i. \<mu> (F i) / \<mu> A) = \<mu> (\<Union>i. F i) / \<mu> A"
-          using measure_countably_additive[of F] F
-          by (auto simp: suminf_cmult_ereal)
-      qed
-    qed
-    show "measure ?P (space ?P) = 1"
-      using real_measure[OF `A \<in> events`] `\<mu> A \<noteq> 0` by auto
-  qed
-qed
-
-lemma finite_prob_spaceI:
-  assumes "finite (space M)" "sets M = Pow(space M)"
-    and 1: "measure M (space M) = 1" and "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> measure M {x}"
-    and add: "\<And>A B. A \<subseteq> space M \<Longrightarrow> measure M A = (\<Sum>x\<in>A. measure M {x})"
-  shows "finite_prob_space M"
-proof -
-  interpret finite_measure_space M
-  proof
-    show "measure M (space M) \<noteq> \<infinity>" using 1 by simp
-  qed fact+
-  show ?thesis by default fact
-qed
-
-lemma (in finite_prob_space) distribution_eq_setsum:
-  "distribution X A = (\<Sum>x\<in>A \<inter> X ` space M. distribution X {x})"
-proof -
-  have *: "X -` A \<inter> space M = (\<Union>x\<in>A \<inter> X ` space M. X -` {x} \<inter> space M)"
-    by auto
-  then show "distribution X A = (\<Sum>x\<in>A \<inter> X ` space M. distribution X {x})"
-    using finite_space unfolding distribution_def *
-    by (intro finite_measure_finite_Union)
-       (auto simp: disjoint_family_on_def)
-qed
-
-lemma (in finite_prob_space) distribution_eq_setsum_finite:
-  assumes "finite A"
-  shows "distribution X A = (\<Sum>x\<in>A. distribution X {x})"
-proof -
-  note distribution_eq_setsum[of X A]
-  also have "(\<Sum>x\<in>A \<inter> X ` space M. distribution X {x}) = (\<Sum>x\<in>A. distribution X {x})"
-  proof (intro setsum_mono_zero_cong_left ballI)
-    fix i assume "i\<in>A - A \<inter> X ` space M"
-    then have "X -` {i} \<inter> space M = {}" by auto
-    then show "distribution X {i} = 0"
-      by (simp add: distribution_def)
-  next
-    show "finite A" by fact
-  qed simp_all
+  have "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. f (Y x) \<partial>M)"
+    by (rule distributed_integral) fact+
+  also have "\<dots> = (\<integral>x. f (T (X x)) \<partial>M)"
+    using Y by simp
+  also have "\<dots> = (\<integral>x. Px x * f (T x) \<partial>N)"
+    using measurable_comp[OF T f] Px by (intro distributed_integral[symmetric]) (auto simp: comp_def)
   finally show ?thesis .
 qed
 
-lemma (in finite_prob_space) finite_measure_space:
-  fixes X :: "'a \<Rightarrow> 'x"
-  shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M), measure = ereal \<circ> distribution X\<rparr>"
-    (is "finite_measure_space ?S")
-proof (rule finite_measure_spaceI, simp_all)
-  show "finite (X ` space M)" using finite_space by simp
-next
-  fix A assume "A \<subseteq> X ` space M"
-  then show "distribution X A = (\<Sum>x\<in>A. distribution X {x})"
-    by (subst distribution_eq_setsum) (simp add: Int_absorb2)
+lemma distributed_marginal_eq_joint:
+  assumes T: "sigma_finite_measure T"
+  assumes S: "sigma_finite_measure S"
+  assumes Px: "distributed M S X Px"
+  assumes Py: "distributed M T Y Py"
+  assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
+  shows "AE y in T. Py y = (\<integral>\<^isup>+x. Pxy (x, y) \<partial>S)"
+proof (rule sigma_finite_measure.density_unique[OF T])
+  interpret ST: pair_sigma_finite S T using S T unfolding pair_sigma_finite_def by simp
+  show "Py \<in> borel_measurable T" "AE y in T. 0 \<le> Py y"
+    "(\<lambda>x. \<integral>\<^isup>+ xa. Pxy (xa, x) \<partial>S) \<in> borel_measurable T" "AE y in T. 0 \<le> \<integral>\<^isup>+ x. Pxy (x, y) \<partial>S"
+    using Pxy[THEN distributed_borel_measurable]
+    by (auto intro!: Py[THEN distributed_borel_measurable] Py[THEN distributed_AE]
+                     ST.positive_integral_snd_measurable' positive_integral_positive)
+
+  show "density T Py = density T (\<lambda>x. \<integral>\<^isup>+ xa. Pxy (xa, x) \<partial>S)"
+  proof (rule measure_eqI)
+    fix A assume A: "A \<in> sets (density T Py)"
+    have *: "\<And>x y. x \<in> space S \<Longrightarrow> indicator (space S \<times> A) (x, y) = indicator A y"
+      by (auto simp: indicator_def)
+    have "emeasure (density T Py) A = emeasure (distr M T Y) A"
+      unfolding Py[THEN distributed_distr_eq_density] ..
+    also have "\<dots> = emeasure (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) (space S \<times> A)"
+      using A Px Py Pxy
+      by (subst (1 2) emeasure_distr)
+         (auto dest: measurable_space distributed_measurable intro!: arg_cong[where f="emeasure M"])
+    also have "\<dots> = emeasure (density (S \<Otimes>\<^isub>M T) Pxy) (space S \<times> A)"
+      unfolding Pxy[THEN distributed_distr_eq_density] ..
+    also have "\<dots> = (\<integral>\<^isup>+ x. Pxy x * indicator (space S \<times> A) x \<partial>(S \<Otimes>\<^isub>M T))"
+      using A Pxy by (simp add: emeasure_density distributed_borel_measurable)
+    also have "\<dots> = (\<integral>\<^isup>+y. \<integral>\<^isup>+x. Pxy (x, y) * indicator (space S \<times> A) (x, y) \<partial>S \<partial>T)"
+      using A Pxy
+      by (subst ST.positive_integral_snd_measurable) (simp_all add: emeasure_density distributed_borel_measurable)
+    also have "\<dots> = (\<integral>\<^isup>+y. (\<integral>\<^isup>+x. Pxy (x, y) \<partial>S) * indicator A y \<partial>T)"
+      using measurable_comp[OF measurable_Pair1[OF measurable_identity] distributed_borel_measurable[OF Pxy]]
+      using distributed_borel_measurable[OF Pxy] distributed_AE[OF Pxy, THEN ST.AE_pair]
+      by (subst (asm) ST.AE_commute) (auto intro!: positive_integral_cong_AE positive_integral_multc cong: positive_integral_cong simp: * comp_def)
+    also have "\<dots> = emeasure (density T (\<lambda>x. \<integral>\<^isup>+ xa. Pxy (xa, x) \<partial>S)) A"
+      using A by (intro emeasure_density[symmetric])  (auto intro!: ST.positive_integral_snd_measurable' Pxy[THEN distributed_borel_measurable])
+    finally show "emeasure (density T Py) A = emeasure (density T (\<lambda>x. \<integral>\<^isup>+ xa. Pxy (xa, x) \<partial>S)) A" .
+  qed simp
 qed
 
-lemma (in finite_prob_space) finite_prob_space_of_images:
-  "finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = ereal \<circ> distribution X \<rparr>"
-  by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_ereal_def)
+lemma (in prob_space) distr_marginal1:
+  fixes Pxy :: "('b \<times> 'c) \<Rightarrow> real"
+  assumes "sigma_finite_measure S" "sigma_finite_measure T"
+  assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
+  defines "Px \<equiv> \<lambda>x. real (\<integral>\<^isup>+z. Pxy (x, z) \<partial>T)"
+  shows "distributed M S X Px"
+  unfolding distributed_def
+proof safe
+  interpret S: sigma_finite_measure S by fact
+  interpret T: sigma_finite_measure T by fact
+  interpret ST: pair_sigma_finite S T by default
+
+  have XY: "(\<lambda>x. (X x, Y x)) \<in> measurable M (S \<Otimes>\<^isub>M T)"
+    using Pxy by (rule distributed_measurable)
+  then show X: "X \<in> measurable M S"
+    unfolding measurable_pair_iff by (simp add: comp_def)
+  from XY have Y: "Y \<in> measurable M T"
+    unfolding measurable_pair_iff by (simp add: comp_def)
+
+  from Pxy show borel: "(\<lambda>x. ereal (Px x)) \<in> borel_measurable S"
+    by (auto intro!: ST.positive_integral_fst_measurable borel_measurable_real_of_ereal dest!: distributed_real_measurable simp: Px_def)
 
-lemma (in finite_prob_space) finite_product_measure_space:
-  fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y"
-  assumes "finite s1" "finite s2"
-  shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2), measure = ereal \<circ> joint_distribution X Y\<rparr>"
-    (is "finite_measure_space ?M")
-proof (rule finite_measure_spaceI, simp_all)
-  show "finite (s1 \<times> s2)"
-    using assms by auto
-next
-  fix A assume "A \<subseteq> (s1 \<times> s2)"
-  with assms show "joint_distribution X Y A = (\<Sum>x\<in>A. joint_distribution X Y {x})"
-    by (intro distribution_eq_setsum_finite) (auto dest: finite_subset)
-qed
-
-lemma (in finite_prob_space) finite_product_measure_space_of_images:
-  shows "finite_measure_space \<lparr> space = X ` space M \<times> Y ` space M,
-                                sets = Pow (X ` space M \<times> Y ` space M),
-                                measure = ereal \<circ> joint_distribution X Y \<rparr>"
-  using finite_space by (auto intro!: finite_product_measure_space)
-
-lemma (in finite_prob_space) finite_product_prob_space_of_images:
-  "finite_prob_space \<lparr> space = X ` space M \<times> Y ` space M, sets = Pow (X ` space M \<times> Y ` space M),
-                       measure = ereal \<circ> joint_distribution X Y \<rparr>"
-  (is "finite_prob_space ?S")
-proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_ereal_def)
-  have "X -` X ` space M \<inter> Y -` Y ` space M \<inter> space M = space M" by auto
-  thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1"
-    by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1)
+  interpret Pxy: prob_space "distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
+    using XY by (rule prob_space_distr)
+  have "(\<integral>\<^isup>+ x. max 0 (ereal (- Pxy x)) \<partial>(S \<Otimes>\<^isub>M T)) = (\<integral>\<^isup>+ x. 0 \<partial>(S \<Otimes>\<^isub>M T))"
+    using Pxy
+    by (intro positive_integral_cong_AE) (auto simp: max_def dest: distributed_real_measurable distributed_real_AE)
+  then have Pxy_integrable: "integrable (S \<Otimes>\<^isub>M T) Pxy"
+    using Pxy Pxy.emeasure_space_1
+    by (simp add: integrable_def emeasure_density positive_integral_max_0 distributed_def borel_measurable_ereal_iff cong: positive_integral_cong)
+    
+  show "distr M S X = density S Px"
+  proof (rule measure_eqI)
+    fix A assume A: "A \<in> sets (distr M S X)"
+    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)"
+      by (auto simp add: emeasure_distr
+               intro!: arg_cong[where f="emeasure M"] dest: measurable_space)
+    also have "\<dots> = emeasure (density (S \<Otimes>\<^isub>M T) Pxy) (A \<times> space T)"
+      using Pxy by (simp add: distributed_def)
+    also have "\<dots> = \<integral>\<^isup>+ x. \<integral>\<^isup>+ y. ereal (Pxy (x, y)) * indicator (A \<times> space T) (x, y) \<partial>T \<partial>S"
+      using A borel Pxy
+      by (simp add: emeasure_density ST.positive_integral_fst_measurable(2)[symmetric] distributed_def)
+    also have "\<dots> = \<integral>\<^isup>+ x. ereal (Px x) * indicator A x \<partial>S"
+      apply (rule positive_integral_cong_AE)
+      using Pxy[THEN distributed_real_AE, THEN ST.AE_pair] ST.integrable_fst_measurable(1)[OF Pxy_integrable] AE_space
+    proof eventually_elim
+      fix x assume "x \<in> space S" "AE y in T. 0 \<le> Pxy (x, y)" and i: "integrable T (\<lambda>y. Pxy (x, y))"
+      moreover have eq: "\<And>y. y \<in> space T \<Longrightarrow> indicator (A \<times> space T) (x, y) = indicator A x"
+        by (auto simp: indicator_def)
+      ultimately have "(\<integral>\<^isup>+ y. ereal (Pxy (x, y)) * indicator (A \<times> space T) (x, y) \<partial>T) =
+          (\<integral>\<^isup>+ y. ereal (Pxy (x, y)) \<partial>T) * indicator A x"
+        using Pxy[THEN distributed_real_measurable] by (simp add: eq positive_integral_multc measurable_Pair2 cong: positive_integral_cong)
+      also have "(\<integral>\<^isup>+ y. ereal (Pxy (x, y)) \<partial>T) = Px x"
+        using i by (simp add: Px_def ereal_real integrable_def positive_integral_positive)
+      finally show "(\<integral>\<^isup>+ y. ereal (Pxy (x, y)) * indicator (A \<times> space T) (x, y) \<partial>T) = ereal (Px x) * indicator A x" .
+    qed
+    finally show "emeasure (distr M S X) A = emeasure (density S Px) A"
+      using A borel Pxy by (simp add: emeasure_density)
+  qed simp
+  
+  show "AE x in S. 0 \<le> ereal (Px x)"
+    by (simp add: Px_def positive_integral_positive real_of_ereal_pos)
 qed
 
-subsection "Borel Measure on {0 ..< 1}"
-
-definition pborel :: "real measure_space" where
-  "pborel = lborel.restricted_space {0 ..< 1}"
-
-lemma space_pborel[simp]:
-  "space pborel = {0 ..< 1}"
-  unfolding pborel_def by auto
-
-lemma sets_pborel:
-  "A \<in> sets pborel \<longleftrightarrow> A \<in> sets borel \<and> A \<subseteq> { 0 ..< 1}"
-  unfolding pborel_def by auto
+definition
+  "simple_distributed M X f \<longleftrightarrow> distributed M (count_space (X`space M)) X (\<lambda>x. ereal (f x)) \<and>
+    finite (X`space M)"
 
-lemma in_pborel[intro, simp]:
-  "A \<subseteq> {0 ..< 1} \<Longrightarrow> A \<in> sets borel \<Longrightarrow> A \<in> sets pborel"
-  unfolding pborel_def by auto
-
-interpretation pborel: measure_space pborel
-  using lborel.restricted_measure_space[of "{0 ..< 1}"]
-  by (simp add: pborel_def)
+lemma simple_distributed:
+  "simple_distributed M X Px \<Longrightarrow> distributed M (count_space (X`space M)) X Px"
+  unfolding simple_distributed_def by auto
 
-interpretation pborel: prob_space pborel
-proof
-  show "measure pborel (space pborel) = 1"
-    by (simp add: one_ereal_def pborel_def)
-qed default
-
-lemma pborel_prob: "pborel.prob A = (if A \<in> sets borel \<and> A \<subseteq> {0 ..< 1} then real (lborel.\<mu> A) else 0)"
-  unfolding pborel.\<mu>'_def by (auto simp: pborel_def)
+lemma simple_distributed_finite[dest]: "simple_distributed M X P \<Longrightarrow> finite (X`space M)"
+  by (simp add: simple_distributed_def)
 
-lemma pborel_singelton[simp]: "pborel.prob {a} = 0"
-  by (auto simp: pborel_prob)
-
-lemma
-  shows pborel_atLeastAtMost[simp]: "pborel.\<mu>' {a .. b} = (if 0 \<le> a \<and> a \<le> b \<and> b < 1 then b - a else 0)"
-    and pborel_atLeastLessThan[simp]: "pborel.\<mu>' {a ..< b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)"
-    and pborel_greaterThanAtMost[simp]: "pborel.\<mu>' {a <.. b} = (if 0 \<le> a \<and> a \<le> b \<and> b < 1 then b - a else 0)"
-    and pborel_greaterThanLessThan[simp]: "pborel.\<mu>' {a <..< b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)"
-  unfolding pborel_prob
-  by (auto simp: atLeastAtMost_subseteq_atLeastLessThan_iff
-    greaterThanAtMost_subseteq_atLeastLessThan_iff greaterThanLessThan_subseteq_atLeastLessThan_iff)
-
-lemma pborel_lebesgue_measure:
-  "A \<in> sets pborel \<Longrightarrow> pborel.prob A = real (measure lebesgue A)"
-  by (simp add: sets_pborel pborel_prob)
+lemma (in prob_space) distributed_simple_function_superset:
+  assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
+  assumes A: "X`space M \<subseteq> A" "finite A"
+  defines "S \<equiv> count_space A" and "P' \<equiv> (\<lambda>x. if x \<in> X`space M then P x else 0)"
+  shows "distributed M S X P'"
+  unfolding distributed_def
+proof safe
+  show "(\<lambda>x. ereal (P' x)) \<in> borel_measurable S" unfolding S_def by simp
+  show "AE x in S. 0 \<le> ereal (P' x)"
+    using X by (auto simp: S_def P'_def simple_distributed_def intro!: measure_nonneg)
+  show "distr M S X = density S P'"
+  proof (rule measure_eqI_finite)
+    show "sets (distr M S X) = Pow A" "sets (density S P') = Pow A"
+      using A unfolding S_def by auto
+    show "finite A" by fact
+    fix a assume a: "a \<in> A"
+    then have "a \<notin> X`space M \<Longrightarrow> X -` {a} \<inter> space M = {}" by auto
+    with A a X have "emeasure (distr M S X) {a} = P' a"
+      by (subst emeasure_distr)
+         (auto simp add: S_def P'_def simple_functionD emeasure_eq_measure
+               intro!: arg_cong[where f=prob])
+    also have "\<dots> = (\<integral>\<^isup>+x. ereal (P' a) * indicator {a} x \<partial>S)"
+      using A X a
+      by (subst positive_integral_cmult_indicator)
+         (auto simp: S_def P'_def simple_distributed_def simple_functionD measure_nonneg)
+    also have "\<dots> = (\<integral>\<^isup>+x. ereal (P' x) * indicator {a} x \<partial>S)"
+      by (auto simp: indicator_def intro!: positive_integral_cong)
+    also have "\<dots> = emeasure (density S P') {a}"
+      using a A by (intro emeasure_density[symmetric]) (auto simp: S_def)
+    finally show "emeasure (distr M S X) {a} = emeasure (density S P') {a}" .
+  qed
+  show "random_variable S X"
+    using X(1) A by (auto simp: measurable_def simple_functionD S_def)
+qed
 
-lemma pborel_alt:
-  "pborel = sigma \<lparr>
-    space = {0..<1},
-    sets = range (\<lambda>(x,y). {x..<y} \<inter> {0..<1}),
-    measure = measure lborel \<rparr>" (is "_ = ?R")
+lemma (in prob_space) simple_distributedI:
+  assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
+  shows "simple_distributed M X P"
+  unfolding simple_distributed_def
+proof
+  have "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (if x \<in> X`space M then P x else 0))"
+    (is "?A")
+    using simple_functionD[OF X(1)] by (intro distributed_simple_function_superset[OF X]) auto
+  also have "?A \<longleftrightarrow> distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (P x))"
+    by (rule distributed_cong_density) auto
+  finally show "\<dots>" .
+qed (rule simple_functionD[OF X(1)])
+
+lemma simple_distributed_joint_finite:
+  assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
+  shows "finite (X ` space M)" "finite (Y ` space M)"
 proof -
-  have *: "{0..<1::real} \<in> sets borel" by auto
-  have **: "op \<inter> {0..<1::real} ` range (\<lambda>(x, y). {x..<y}) = range (\<lambda>(x,y). {x..<y} \<inter> {0..<1})"
-    unfolding image_image by (intro arg_cong[where f=range]) auto
-  have "pborel = algebra.restricted_space (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a, b). {a ..< b :: real}),
-    measure = measure pborel\<rparr>) {0 ..< 1}"
-    by (simp add: sigma_def lebesgue_def pborel_def borel_eq_atLeastLessThan lborel_def)
-  also have "\<dots> = ?R"
-    by (subst restricted_sigma)
-       (simp_all add: sets_sigma sigma_sets.Basic ** pborel_def image_eqI[of _ _ "(0,1)"])
-  finally show ?thesis .
+  have "finite ((\<lambda>x. (X x, Y x)) ` space M)"
+    using X by (auto simp: simple_distributed_def simple_functionD)
+  then have "finite (fst ` (\<lambda>x. (X x, Y x)) ` space M)" "finite (snd ` (\<lambda>x. (X x, Y x)) ` space M)"
+    by auto
+  then show fin: "finite (X ` space M)" "finite (Y ` space M)"
+    by (auto simp: image_image)
+qed
+
+lemma simple_distributed_joint2_finite:
+  assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
+  shows "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
+proof -
+  have "finite ((\<lambda>x. (X x, Y x, Z x)) ` space M)"
+    using X by (auto simp: simple_distributed_def simple_functionD)
+  then have "finite (fst ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
+    "finite ((fst \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
+    "finite ((snd \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
+    by auto
+  then show fin: "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
+    by (auto simp: image_image)
 qed
 
-subsection "Bernoulli space"
+lemma simple_distributed_simple_function:
+  "simple_distributed M X Px \<Longrightarrow> simple_function M X"
+  unfolding simple_distributed_def distributed_def
+  by (auto simp: simple_function_def)
+
+lemma simple_distributed_measure:
+  "simple_distributed M X P \<Longrightarrow> a \<in> X`space M \<Longrightarrow> P a = measure M (X -` {a} \<inter> space M)"
+  using distributed_count_space[of M "X`space M" X P a, symmetric]
+  by (auto simp: simple_distributed_def measure_def)
+
+lemma simple_distributed_nonneg: "simple_distributed M X f \<Longrightarrow> x \<in> space M \<Longrightarrow> 0 \<le> f (X x)"
+  by (auto simp: simple_distributed_measure measure_nonneg)
 
-definition "bernoulli_space p = \<lparr> space = UNIV, sets = UNIV,
-  measure = ereal \<circ> setsum (\<lambda>b. if b then min 1 (max 0 p) else 1 - min 1 (max 0 p)) \<rparr>"
+lemma (in prob_space) simple_distributed_joint:
+  assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
+  defines "S \<equiv> count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M)"
+  defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x))`space M then Px x else 0)"
+  shows "distributed M S (\<lambda>x. (X x, Y x)) P"
+proof -
+  from simple_distributed_joint_finite[OF X, simp]
+  have S_eq: "S = count_space (X`space M \<times> Y`space M)"
+    by (simp add: S_def pair_measure_count_space)
+  show ?thesis
+    unfolding S_eq P_def
+  proof (rule distributed_simple_function_superset)
+    show "simple_function M (\<lambda>x. (X x, Y x))"
+      using X by (rule simple_distributed_simple_function)
+    fix x assume "x \<in> (\<lambda>x. (X x, Y x)) ` space M"
+    from simple_distributed_measure[OF X this]
+    show "Px x = prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)" .
+  qed auto
+qed
 
-interpretation bernoulli: finite_prob_space "bernoulli_space p" for p
-  by (rule finite_prob_spaceI)
-     (auto simp: bernoulli_space_def UNIV_bool one_ereal_def setsum_Un_disjoint intro!: setsum_nonneg)
+lemma (in prob_space) simple_distributed_joint2:
+  assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
+  defines "S \<equiv> count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M) \<Otimes>\<^isub>M count_space (Z`space M)"
+  defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x))`space M then Px x else 0)"
+  shows "distributed M S (\<lambda>x. (X x, Y x, Z x)) P"
+proof -
+  from simple_distributed_joint2_finite[OF X, simp]
+  have S_eq: "S = count_space (X`space M \<times> Y`space M \<times> Z`space M)"
+    by (simp add: S_def pair_measure_count_space)
+  show ?thesis
+    unfolding S_eq P_def
+  proof (rule distributed_simple_function_superset)
+    show "simple_function M (\<lambda>x. (X x, Y x, Z x))"
+      using X by (rule simple_distributed_simple_function)
+    fix x assume "x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M"
+    from simple_distributed_measure[OF X this]
+    show "Px x = prob ((\<lambda>x. (X x, Y x, Z x)) -` {x} \<inter> space M)" .
+  qed auto
+qed
+
+lemma (in prob_space) simple_distributed_setsum_space:
+  assumes X: "simple_distributed M X f"
+  shows "setsum f (X`space M) = 1"
+proof -
+  from X have "setsum f (X`space M) = prob (\<Union>i\<in>X`space M. X -` {i} \<inter> space M)"
+    by (subst finite_measure_finite_Union)
+       (auto simp add: disjoint_family_on_def simple_distributed_measure simple_distributed_simple_function simple_functionD
+             intro!: setsum_cong arg_cong[where f="prob"])
+  also have "\<dots> = prob (space M)"
+    by (auto intro!: arg_cong[where f=prob])
+  finally show ?thesis
+    using emeasure_space_1 by (simp add: emeasure_eq_measure one_ereal_def)
+qed
 
-lemma bernoulli_measure:
-  "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p B = (\<Sum>b\<in>B. if b then p else 1 - p)"
-  unfolding bernoulli.\<mu>'_def unfolding bernoulli_space_def by (auto intro!: setsum_cong)
+lemma (in prob_space) distributed_marginal_eq_joint_simple:
+  assumes Px: "simple_function M X"
+  assumes Py: "simple_distributed M Y Py"
+  assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
+  assumes y: "y \<in> Y`space M"
+  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)"
+proof -
+  note Px = simple_distributedI[OF Px refl]
+  have *: "\<And>f A. setsum (\<lambda>x. max 0 (ereal (f x))) A = ereal (setsum (\<lambda>x. max 0 (f x)) A)"
+    by (simp add: setsum_ereal[symmetric] zero_ereal_def)
+  from distributed_marginal_eq_joint[OF sigma_finite_measure_count_space_finite sigma_finite_measure_count_space_finite
+    simple_distributed[OF Px] simple_distributed[OF Py] simple_distributed_joint[OF Pxy],
+    OF Py[THEN simple_distributed_finite] Px[THEN simple_distributed_finite]]
+    y Px[THEN simple_distributed_finite] Py[THEN simple_distributed_finite]
+    Pxy[THEN simple_distributed, THEN distributed_real_AE]
+  show ?thesis
+    unfolding AE_count_space
+    apply (elim ballE[where x=y])
+    apply (auto simp add: positive_integral_count_space_finite * intro!: setsum_cong split: split_max)
+    done
+qed
 
-lemma bernoulli_measure_True: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {True} = p"
-  and bernoulli_measure_False: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {False} = 1 - p"
-  unfolding bernoulli_measure by simp_all
+
+lemma prob_space_uniform_measure:
+  assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>"
+  shows "prob_space (uniform_measure M A)"
+proof
+  show "emeasure (uniform_measure M A) (space (uniform_measure M A)) = 1"
+    using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)], of "space M"]
+    using sets_into_space[OF emeasure_neq_0_sets[OF A(1)]] A
+    by (simp add: Int_absorb2 emeasure_nonneg)
+qed
+
+lemma prob_space_uniform_count_measure: "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> prob_space (uniform_count_measure A)"
+  by default (auto simp: emeasure_uniform_count_measure space_uniform_count_measure one_ereal_def)
 
 end