'((/_ in _./ _)'))\)

translations
  "\

(x in M. P)" => "CONST measure M {x \ CONST space M. P}"

print_translation \<open>
  let
    fun to_pattern (Const (\<^const_syntax>\<open>Pair\<close>, _) $ l $ r) =
      Syntax.const \<^const_syntax>\<open>Pair\<close> :: to_pattern l @ to_pattern r
    | to_pattern (t as (Const (\<^syntax_const>\<open>_bound\<close>, _)) $ _) = [t]

    fun mk_pattern ((t, n) :: xs) = mk_patterns n xs |>> curry list_comb t
    and mk_patterns 0 xs = ([], xs)
    | mk_patterns n xs =
      let
        val (t, xs') = mk_pattern xs
        val (ts, xs'') = mk_patterns (n - 1) xs'
      in
        (t :: ts, xs'')
      end

    fun unnest_tuples
      (Const (\<^syntax_const>\<open>_pattern\<close>, _) $
        t1 $
        (t as (Const (\<^syntax_const>\<open>_pattern\<close>, _) $ _ $ _)))
      = let
        val (_ $ t2 $ t3) = unnest_tuples t
      in
        Syntax.const \<^syntax_const>\<open>_pattern\<close> $
          unnest_tuples t1 $
          (Syntax.const \<^syntax_const>\<open>_patterns\<close> $ t2 $ t3)
      end
    | unnest_tuples pat = pat

    fun tr' [sig_alg, Const (\<^const_syntax>\Collect\, _) $ t] =
      let
        val bound_dummyT = Const (\<^syntax_const>\<open>_bound\<close>, dummyT)

        fun go pattern elem
          (Const (\<^const_syntax>\<open>conj\<close>, _) $
            (Const (\<^const_syntax>\<open>Set.member\<close>, _) $ elem' $ (Const (\<^const_syntax>\<open>space\<close>, _) $ sig_alg')) $
            u)
          = let
              val _ = if sig_alg aconv sig_alg' andalso to_pattern elem' = rev elem then () else raise Match;
              val (pat, rest) = mk_pattern (rev pattern);
              val _ = case rest of [] => () | _ => raise Match
            in
              Syntax.const \<^syntax_const>\<open>_prob\<close> $ unnest_tuples pat $ sig_alg $ u
            end
        | go pattern elem (Abs abs) =
            let
              val (x as (_ $ tx), t) = Syntax_Trans.atomic_abs_tr' abs
            in
              go ((x, 0) :: pattern) (bound_dummyT $ tx :: elem) t
            end
        | go pattern elem (Const (\<^const_syntax>\<open>case_prod\<close>, _) $ t) =
            go
              ((Syntax.const \<^syntax_const>\<open>_pattern\<close>, 2) :: pattern)
              (Syntax.const \<^const_syntax>\<open>Pair\<close> :: elem)
              t
      in
        go [] [] t
      end
  in
    [(\<^const_syntax>\<open>Sigma_Algebra.measure\<close>, K tr')]
  end
\<close>

definition
  "cond_prob M P Q = \

(\ in M. P \ \ Q \) / \

(\ in M. Q \)"

syntax
  "_conditional_prob" :: "pttrn \ logic \ logic \ logic \ logic" (\('\

'(_ in _. _ \/ _'))\)

translations
  "\

(x in M. P \ Q)" => "CONST cond_prob M (\x. P) (\x. Q)"

lemma (in prob_space) AE_E_prob:
  assumes ae: "AE x in M. P x"
  obtains S where "S \ {x \ space M. P x}" "S \ events" "prob S = 1"
proof -
  from ae[THEN AE_E] guess N .
  then show thesis
    by (intro that[of "space M - N"])
       (auto simp: prob_compl prob_space emeasure_eq_measure measure_nonneg)
qed

lemma (in prob_space) prob_neg: "{x\space M. P x} \ events \ \

(x in M. \ P x) = 1 - \

(x in M. P x)"
  by (auto intro!: arg_cong[where f=prob] simp add: prob_compl[symmetric])

lemma (in prob_space) prob_eq_AE:
  "(AE x in M. P x \ Q x) \ {x\space M. P x} \ events \ {x\space M. Q x} \ events \ \

(x in M. P x) = \

(x in M. Q x)"
  by (rule finite_measure_eq_AE) auto

lemma (in prob_space) prob_eq_0_AE:
  assumes not: "AE x in M. \ P x" shows "\

(x in M. P x) = 0"
proof cases
  assume "{x\space M. P x} \ events"
  with not have "\

(x in M. P x) = \

(x in M. False)"
    by (intro prob_eq_AE) auto
  then show ?thesis by simp
qed (simp add: measure_notin_sets)

lemma (in prob_space) prob_Collect_eq_0:
  "{x \ space M. P x} \ sets M \ \

(x in M. P x) = 0 \ (AE x in M. \ P x)"
  using AE_iff_measurable[OF _ refl, of M "\x. \ P x"] by (simp add: emeasure_eq_measure measure_nonneg)

lemma (in prob_space) prob_Collect_eq_1:
  "{x \ space M. P x} \ sets M \ \

(x in M. P x) = 1 \ (AE x in M. P x)"
  using AE_in_set_eq_1[of "{x\space M. P x}"] by simp

lemma (in prob_space) prob_eq_0:
  "A \ sets M \ prob A = 0 \ (AE x in M. x \ A)"
  using AE_iff_measurable[OF _ refl, of M "\x. x \ A"]
  by (auto simp add: emeasure_eq_measure Int_def[symmetric] measure_nonneg)

lemma (in prob_space) prob_eq_1:
  "A \ sets M \ prob A = 1 \ (AE x in M. x \ A)"
  using AE_in_set_eq_1[of A] by simp

lemma (in prob_space) prob_sums:
  assumes P: "\n. {x\space M. P n x} \ events"
  assumes Q: "{x\space M. Q x} \ events"
  assumes ae: "AE x in M. (\n. P n x \ Q x) \ (Q x \ (\!n. P n x))"
  shows "(\n. \

(x in M. P n x)) sums \

(x in M. Q x)"
proof -
  from ae[THEN AE_E_prob] guess S . note S = this
  then have disj: "disjoint_family (\n. {x\space M. P n x} \ S)"
    by (auto simp: disjoint_family_on_def)
  from S have ae_S:
    "AE x in M. x \ {x\space M. Q x} \ x \ (\n. {x\space M. P n x} \ S)"
    "\n. AE x in M. x \ {x\space M. P n x} \ x \ {x\space M. P n x} \ S"
    using ae by (auto dest!: AE_prob_1)
  from ae_S have *:
    "\

(x in M. Q x) = prob (\n. {x\space M. P n x} \ S)"
    using P Q S by (intro finite_measure_eq_AE) auto
  from ae_S have **:
    "\n. \

(x in M. P n x) = prob ({x\space M. P n x} \ S)"
    using P Q S by (intro finite_measure_eq_AE) auto
  show ?thesis
    unfolding * ** using S P disj
    by (intro finite_measure_UNION) auto
qed

lemma (in prob_space) prob_sum:
  assumes [simp, intro]: "finite I"
  assumes P: "\n. n \ I \ {x\space M. P n x} \ events"
  assumes Q: "{x\space M. Q x} \ events"
  assumes ae: "AE x in M. (\n\I. P n x \ Q x) \ (Q x \ (\!n\I. P n x))"
  shows "\

(x in M. Q x) = (\n\I. \

(x in M. P n x))"
proof -
  from ae[THEN AE_E_prob] guess S . note S = this
  then have disj: "disjoint_family_on (\n. {x\space M. P n x} \ S) I"
    by (auto simp: disjoint_family_on_def)
  from S have ae_S:
    "AE x in M. x \ {x\space M. Q x} \ x \ (\n\I. {x\space M. P n x} \ S)"
    "\n. n \ I \ AE x in M. x \ {x\space M. P n x} \ x \ {x\space M. P n x} \ S"
    using ae by (auto dest!: AE_prob_1)
  from ae_S have *:
    "\

(x in M. Q x) = prob (\n\I. {x\space M. P n x} \ S)"
    using P Q S by (intro finite_measure_eq_AE) (auto intro!: sets.Int)
  from ae_S have **:
    "\n. n \ I \ \

(x in M. P n x) = prob ({x\space M. P n x} \ S)"
    using P Q S by (intro finite_measure_eq_AE) auto
  show ?thesis
    using S P disj
    by (auto simp add: * ** simp del: UN_simps intro!: finite_measure_finite_Union)
qed

lemma (in prob_space) prob_EX_countable:
  assumes sets: "\i. i \ I \ {x\space M. P i x} \ sets M" and I: "countable I"
  assumes disj: "AE x in M. \i\I. \j\I. P i x \ P j x \ i = j"
  shows "\

(x in M. \i\I. P i x) = (\\<^sup>+i. \

(x in M. P i x) \count_space I)"
proof -
  let ?N= "\x. \!i\I. P i x"
  have "ennreal (\

(x in M. \i\I. P i x)) = \

(x in M. (\i\I. P i x \ ?N x))"
    unfolding ennreal_inj[OF measure_nonneg measure_nonneg]
  proof (rule prob_eq_AE)
    show "AE x in M. (\i\I. P i x) = (\i\I. P i x \ ?N x)"
      using disj by eventually_elim blast
  qed (auto intro!: sets.sets_Collect_countable_Ex' sets.sets_Collect_conj sets.sets_Collect_countable_Ex1' I sets)+
  also have "\

(x in M. (\i\I. P i x \ ?N x)) = emeasure M (\i\I. {x\space M. P i x \ ?N x})"
    unfolding emeasure_eq_measure by (auto intro!: arg_cong[where f=prob] simp: measure_nonneg)
  also have "\ = (\\<^sup>+i. emeasure M {x\space M. P i x \ ?N x} \count_space I)"
    by (rule emeasure_UN_countable)
       (auto intro!: sets.sets_Collect_countable_Ex' sets.sets_Collect_conj sets.sets_Collect_countable_Ex1' I sets
             simp: disjoint_family_on_def)
  also have "\ = (\\<^sup>+i. \

(x in M. P i x) \count_space I)"
    unfolding emeasure_eq_measure using disj
    by (intro nn_integral_cong ennreal_inj[THEN iffD2] prob_eq_AE)
       (auto intro!: sets.sets_Collect_countable_Ex' sets.sets_Collect_conj sets.sets_Collect_countable_Ex1' I sets measure_nonneg)+
  finally show ?thesis .
qed

lemma (in prob_space) cond_prob_eq_AE:
  assumes P: "AE x in M. Q x \ P x \ P' x" "{x\space M. P x} \ events" "{x\space M. P' x} \ events"
  assumes Q: "AE x in M. Q x \ Q' x" "{x\space M. Q x} \ events" "{x\space M. Q' x} \ events"
  shows "cond_prob M P Q = cond_prob M P' Q'"
  using P Q
  by (auto simp: cond_prob_def intro!: arg_cong2[where f="(/)"] prob_eq_AE sets.sets_Collect_conj)


lemma (in prob_space) joint_distribution_Times_le_fst:
  "random_variable MX X \ random_variable MY Y \ A \ sets MX \ B \ sets MY
    \<Longrightarrow> emeasure (distr M (MX \<Otimes>\<^sub>M MY) (\<lambda>x. (X x, Y x))) (A \<times> B) \<le> emeasure (distr M MX X) A"
  by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)

lemma (in prob_space) joint_distribution_Times_le_snd:
  "random_variable MX X \ random_variable MY Y \ A \ sets MX \ B \ sets MY
    \<Longrightarrow> emeasure (distr M (MX \<Otimes>\<^sub>M MY) (\<lambda>x. (X x, Y x))) (A \<times> B) \<le> emeasure (distr M MY Y) B"
  by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)

lemma (in prob_space) variance_eq:
  fixes X :: "'a \ real"
  assumes [simp]: "integrable M X"
  assumes [simp]: "integrable M (\x. (X x)\<^sup>2)"
  shows "variance X = expectation (\x. (X x)\<^sup>2) - (expectation X)\<^sup>2"
  by (simp add: field_simps prob_space power2_diff power2_eq_square[symmetric])

lemma (in prob_space) variance_positive: "0 \ variance (X::'a \ real)"
  by (intro integral_nonneg_AE) (auto intro!: integral_nonneg_AE)

lemma (in prob_space) variance_mean_zero:
  "expectation X = 0 \ variance X = expectation (\x. (X x)^2)"
  by simp

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 "M1 \<Otimes>\<^sub>M M2"
proof
  show "emeasure (M1 \\<^sub>M M2) (space (M1 \\<^sub>M M2)) = 1"
    by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_space_1 M2.emeasure_space_1 space_pair_measure)
qed

locale product_prob_space = product_sigma_finite M for M :: "'i \ 'a measure" +
  fixes I :: "'i set"
  assumes prob_space: "\i. prob_space (M i)"

sublocale product_prob_space \<subseteq> M?: prob_space "M i" for i
  by (rule prob_space)

locale finite_product_prob_space = finite_product_sigma_finite M I + product_prob_space M I for M I

sublocale finite_product_prob_space \<subseteq> prob_space "\<Pi>\<^sub>M i\<in>I. M i"
proof
  show "emeasure (\\<^sub>M i\I. M i) (space (\\<^sub>M i\I. M i)) = 1"
    by (simp add: measure_times M.emeasure_space_1 prod.neutral_const space_PiM)
qed

lemma (in finite_product_prob_space) prob_times:
  assumes X: "\i. i \ I \ X i \ sets (M i)"
  shows "prob (\\<^sub>E i\I. X i) = (\i\I. M.prob i (X i))"
proof -
  have "ennreal (measure (\\<^sub>M i\I. M i) (\\<^sub>E i\I. X i)) = emeasure (\\<^sub>M i\I. M i) (\\<^sub>E i\I. X i)"
    using X by (simp add: emeasure_eq_measure)
  also have "\ = (\i\I. emeasure (M i) (X i))"
    using measure_times X by simp
  also have "\ = ennreal (\i\I. measure (M i) (X i))"
    using X by (simp add: M.emeasure_eq_measure prod_ennreal measure_nonneg)
  finally show ?thesis by (simp add: measure_nonneg prod_nonneg)
qed

subsection \<open>Distributions\<close>

definition distributed :: "'a measure \ 'b measure \ ('a \ 'b) \ ('b \ ennreal) \ bool"
where
  "distributed M N X f \
  distr M N X = density N f \<and> f \<in> borel_measurable N \<and> X \<in> measurable M N"

lemma
  assumes "distributed M N X f"
  shows distributed_distr_eq_density: "distr M N X = density N f"
    and distributed_measurable: "X \ measurable M N"
    and distributed_borel_measurable: "f \ borel_measurable N"
  using assms by (simp_all add: distributed_def)

lemma
  assumes D: "distributed M N X f"
  shows distributed_measurable'[measurable_dest]:
      "g \ measurable L M \ (\x. X (g x)) \ measurable L N"
    and distributed_borel_measurable'[measurable_dest]:
      "h \ measurable L N \ (\x. f (h x)) \ borel_measurable L"
  using distributed_measurable[OF D] distributed_borel_measurable[OF D]
  by simp_all

lemma distributed_real_measurable:
  "(\x. x \ space N \ 0 \ f x) \ distributed M N X (\x. ennreal (f x)) \ f \ borel_measurable N"
  by (simp_all add: distributed_def)

lemma distributed_real_measurable':
  "(\x. x \ space N \ 0 \ f x) \ distributed M N X (\x. ennreal (f x)) \
    h \<in> measurable L N \<Longrightarrow> (\<lambda>x. f (h x)) \<in> borel_measurable L"
  using distributed_real_measurable[measurable] by simp

lemma joint_distributed_measurable1:
  "distributed M (S \\<^sub>M T) (\x. (X x, Y x)) f \ h1 \ measurable N M \ (\x. X (h1 x)) \ measurable N S"
  by simp

lemma joint_distributed_measurable2:
  "distributed M (S \\<^sub>M T) (\x. (X x, Y x)) f \ h2 \ measurable N M \ (\x. Y (h2 x)) \ measurable N T"
  by simp

lemma distributed_count_space:
  assumes X: "distributed M (count_space A) X P" and a: "a \ A" and A: "finite A"
  shows "P a = emeasure M (X -` {a} \ space M)"
proof -
  have "emeasure M (X -` {a} \ space M) = emeasure (distr M (count_space A) X) {a}"
    using X a A by (simp add: emeasure_distr)
  also have "\ = emeasure (density (count_space A) P) {a}"
    using X by (simp add: distributed_distr_eq_density)
  also have "\ = (\\<^sup>+x. P a * indicator {a} x \count_space A)"
    using X a by (auto simp add: emeasure_density distributed_def indicator_def intro!: nn_integral_cong)
  also have "\ = P a"
    using X a by (subst nn_integral_cmult_indicator) (auto simp: distributed_def one_ennreal_def[symmetric] AE_count_space)
  finally show ?thesis ..
qed

lemma distributed_cong_density:
  "(AE x in N. f x = g x) \ g \ borel_measurable N \ f \ borel_measurable N \
    distributed M N X f \<longleftrightarrow> distributed M N X g"
  by (auto simp: distributed_def intro!: density_cong)

lemma (in prob_space) distributed_imp_emeasure_nonzero:
  assumes X: "distributed M MX X Px"
  shows "emeasure MX {x \ space MX. Px x \ 0} \ 0"
proof
  note Px = distributed_borel_measurable[OF X]
  interpret X: prob_space "distr M MX X"
    using distributed_measurable[OF X] by (rule prob_space_distr)

  assume "emeasure MX {x \ space MX. Px x \ 0} = 0"
  with Px have "AE x in MX. Px x = 0"
    by (intro AE_I[OF subset_refl]) (auto simp: borel_measurable_ennreal_iff)
  moreover
  from X.emeasure_space_1 have "(\\<^sup>+x. Px x \MX) = 1"
    unfolding distributed_distr_eq_density[OF X] using Px
    by (subst (asm) emeasure_density)
       (auto simp: borel_measurable_ennreal_iff intro!: integral_cong cong: nn_integral_cong)
  ultimately show False
    by (simp add: nn_integral_cong_AE)
qed

lemma subdensity:
  assumes T: "T \ measurable P Q"
  assumes f: "distributed M P X f"
  assumes g: "distributed M Q Y g"
  assumes Y: "Y = T \ X"
  shows "AE x in P. g (T x) = 0 \ f x = 0"
proof -
  have "{x\space Q. g x = 0} \ null_sets (distr M Q (T \ X))"
    using g Y by (auto simp: null_sets_density_iff distributed_def)
  also have "distr M Q (T \ X) = distr (distr M P X) Q T"
    using T f[THEN distributed_measurable] by (rule distr_distr[symmetric])
  finally have "T -` {x\space Q. g x = 0} \ space P \ null_sets (distr M P X)"
    using T by (subst (asm) null_sets_distr_iff) auto
  also have "T -` {x\space Q. g x = 0} \ space P = {x\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 subdensity_real:
  fixes g :: "'a \ real" and f :: "'b \ real"
  assumes T: "T \ measurable P Q"
  assumes f: "distributed M P X f"
  assumes g: "distributed M Q Y g"
  assumes Y: "Y = T \ X"
  shows "(AE x in P. 0 \ g (T x)) \ (AE x in P. 0 \ f x) \ AE x in P. g (T x) = 0 \ f x = 0"
  using subdensity[OF T, of M X "\x. ennreal (f x)" Y "\x. ennreal (g x)"] assms
  by auto

lemma distributed_emeasure:
  "distributed M N X f \ A \ sets N \ emeasure M (X -` A \ space M) = (\\<^sup>+x. f x * indicator A x \N)"
  by (auto simp: distributed_distr_eq_density[symmetric] emeasure_density[symmetric] emeasure_distr)

lemma distributed_nn_integral:
  "distributed M N X f \ g \ borel_measurable N \ (\\<^sup>+x. f x * g x \N) = (\\<^sup>+x. g (X x) \M)"
  by (auto simp: distributed_distr_eq_density[symmetric] nn_integral_density[symmetric] nn_integral_distr)

lemma distributed_integral:
  "distributed M N X f \ g \ borel_measurable N \ (\x. x \ space N \ 0 \ f x) \
    (\<integral>x. f x * g x \<partial>N) = (\<integral>x. g (X x) \<partial>M)"
  supply distributed_real_measurable[measurable]
  by (auto simp: distributed_distr_eq_density[symmetric] integral_real_density[symmetric] integral_distr)

lemma distributed_transform_integral:
  assumes Px: "distributed M N X Px" "\x. x \ space N \ 0 \ Px x"
  assumes "distributed M P Y Py" "\x. x \ space P \ 0 \ Py x"
  assumes Y: "Y = T \ X" and T: "T \ measurable N P" and f: "f \ borel_measurable P"
  shows "(\x. Py x * f x \P) = (\x. Px x * f (T x) \N)"
proof -
  have "(\x. Py x * f x \P) = (\x. f (Y x) \M)"
    by (rule distributed_integral) fact+
  also have "\ = (\x. f (T (X x)) \M)"
    using Y by simp
  also have "\ = (\x. Px x * f (T x) \N)"
    using measurable_comp[OF T f] Px by (intro distributed_integral[symmetric]) (auto simp: comp_def)
  finally show ?thesis .
qed

lemma (in prob_space) distributed_unique:
  assumes Px: "distributed M S X Px"
  assumes Py: "distributed M S X Py"
  shows "AE x in S. Px x = Py x"
proof -
  interpret X: prob_space "distr M S X"
    using Px by (intro prob_space_distr) simp
  have "sigma_finite_measure (distr M S X)" ..
  with sigma_finite_density_unique[of Px S Py ] Px Py
  show ?thesis
    by (auto simp: distributed_def)
qed

lemma (in prob_space) distributed_jointI:
  assumes "sigma_finite_measure S" "sigma_finite_measure T"
  assumes X[measurable]: "X \ measurable M S" and Y[measurable]: "Y \ measurable M T"
  assumes [measurable]: "f \ borel_measurable (S \\<^sub>M T)" and f: "AE x in S \\<^sub>M T. 0 \ f x"
  assumes eq: "\A B. A \ sets S \ B \ sets T \
    emeasure M {x \<in> space M. X x \<in> A \<and> Y x \<in> B} = (\<integral>\<^sup>+x. (\<integral>\<^sup>+y. f (x, y) * indicator B y \<partial>T) * indicator A x \<partial>S)"
  shows "distributed M (S \\<^sub>M T) (\x. (X x, Y x)) f"
  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 ..

  from ST.sigma_finite_up_in_pair_measure_generator guess F :: "nat \ ('b \ 'c) set" .. note F = this
  let ?E = "{a \ b |a b. a \ sets S \ b \ sets T}"
  let ?P = "S \\<^sub>M T"
  show "distr M ?P (\x. (X x, Y x)) = density ?P f" (is "?L = ?R")
  proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of S T]])
    show "?E \ Pow (space ?P)"
      using sets.space_closed[of S] sets.space_closed[of T] by (auto simp: space_pair_measure)
    show "sets ?L = sigma_sets (space ?P) ?E"
      by (simp add: sets_pair_measure space_pair_measure)
    then show "sets ?R = sigma_sets (space ?P) ?E"
      by simp
  next
    interpret L: prob_space ?L
      by (rule prob_space_distr) (auto intro!: measurable_Pair)
    show "range F \ ?E" "(\i. F i) = space ?P" "\i. emeasure ?L (F i) \ \"
      using F by (auto simp: space_pair_measure)
  next
    fix E assume "E \ ?E"
    then obtain A B where E[simp]: "E = A \ B"
      and A[measurable]: "A \ sets S" and B[measurable]: "B \ sets T" by auto
    have "emeasure ?L E = emeasure M {x \ space M. X x \ A \ Y x \ B}"
      by (auto intro!: arg_cong[where f="emeasure M"] simp add: emeasure_distr measurable_Pair)
    also have "\ = (\\<^sup>+x. (\\<^sup>+y. (f (x, y) * indicator B y) * indicator A x \T) \S)"
      using f by (auto simp add: eq nn_integral_multc intro!: nn_integral_cong)
    also have "\ = emeasure ?R E"
      by (auto simp add: emeasure_density T.nn_integral_fst[symmetric]
               intro!: nn_integral_cong split: split_indicator)
    finally show "emeasure ?L E = emeasure ?R E" .
  qed
qed (auto simp: f)

lemma (in prob_space) distributed_swap:
  assumes "sigma_finite_measure S" "sigma_finite_measure T"
  assumes Pxy: "distributed M (S \\<^sub>M T) (\x. (X x, Y x)) Pxy"
  shows "distributed M (T \\<^sub>M S) (\x. (Y x, X x)) (\(x, y). Pxy (y, x))"
proof -
  interpret S: sigma_finite_measure S by fact
  interpret T: sigma_finite_measure T by fact
  interpret ST: pair_sigma_finite S T ..
  interpret TS: pair_sigma_finite T S ..

  note Pxy[measurable]
  show ?thesis
    apply (subst TS.distr_pair_swap)
    unfolding distributed_def
  proof safe
    let ?D = "distr (S \\<^sub>M T) (T \\<^sub>M S) (\(x, y). (y, x))"
    show 1: "(\(x, y). Pxy (y, x)) \ borel_measurable ?D"
      by auto
    show 2: "random_variable (distr (S \\<^sub>M T) (T \\<^sub>M S) (\(x, y). (y, x))) (\x. (Y x, X x))"
      using Pxy by auto
    { fix A assume A: "A \ sets (T \\<^sub>M S)"
      let ?B = "(\(x, y). (y, x)) -` A \ space (S \\<^sub>M T)"
      from sets.sets_into_space[OF A]
      have "emeasure M ((\x. (Y x, X x)) -` A \ space M) =
        emeasure M ((\<lambda>x. (X x, Y x)) -` ?B \<inter> space M)"
        by (auto intro!: arg_cong2[where f=emeasure] simp: space_pair_measure)
      also have "\ = (\\<^sup>+ x. Pxy x * indicator ?B x \(S \\<^sub>M T))"
        using Pxy A by (intro distributed_emeasure) auto
      finally have "emeasure M ((\x. (Y x, X x)) -` A \ space M) =
        (\<integral>\<^sup>+ x. Pxy x * indicator A (snd x, fst x) \<partial>(S \<Otimes>\<^sub>M T))"
        by (auto intro!: nn_integral_cong split: split_indicator) }
    note * = this
    show "distr M ?D (\x. (Y x, X x)) = density ?D (\(x, y). Pxy (y, x))"
      apply (intro measure_eqI)
      apply (simp_all add: emeasure_distr[OF 2] emeasure_density[OF 1])
      apply (subst nn_integral_distr)
      apply (auto intro!: * simp: comp_def split_beta)
      done
  qed
qed

lemma (in prob_space) distr_marginal1:
  assumes "sigma_finite_measure S" "sigma_finite_measure T"
  assumes Pxy: "distributed M (S \\<^sub>M T) (\x. (X x, Y x)) Pxy"
  defines "Px \ \x. (\\<^sup>+z. Pxy (x, z) \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 ..

  note Pxy[measurable]
  show X: "X \ measurable M S" by simp

  show borel: "Px \ borel_measurable S"
    by (auto intro!: T.nn_integral_fst simp: Px_def)

  interpret Pxy: prob_space "distr M (S \\<^sub>M T) (\x. (X x, Y x))"
    by (intro prob_space_distr) simp

  show "distr M S X = density S Px"
  proof (rule measure_eqI)
    fix A assume A: "A \ sets (distr M S X)"
    with X measurable_space[of Y M T]
    have "emeasure (distr M S X) A = emeasure (distr M (S \\<^sub>M T) (\x. (X x, Y x))) (A \ space T)"
      by (auto simp add: emeasure_distr intro!: arg_cong[where f="emeasure M"])
    also have "\ = emeasure (density (S \\<^sub>M T) Pxy) (A \ space T)"
      using Pxy by (simp add: distributed_def)
    also have "\ = \\<^sup>+ x. \\<^sup>+ y. Pxy (x, y) * indicator (A \ space T) (x, y) \T \S"
      using A borel Pxy
      by (simp add: emeasure_density T.nn_integral_fst[symmetric])
    also have "\ = \\<^sup>+ x. Px x * indicator A x \S"
    proof (rule nn_integral_cong)
      fix x assume "x \ space S"
      moreover have eq: "\y. y \ space T \ indicator (A \ space T) (x, y) = indicator A x"
        by (auto simp: indicator_def)
      ultimately have "(\\<^sup>+ y. Pxy (x, y) * indicator (A \ space T) (x, y) \T) = (\\<^sup>+ y. Pxy (x, y) \T) * indicator A x"
        by (simp add: eq nn_integral_multc cong: nn_integral_cong)
      also have "(\\<^sup>+ y. Pxy (x, y) \T) = Px x"
        by (simp add: Px_def)
      finally show "(\\<^sup>+ y. Pxy (x, y) * indicator (A \ space T) (x, y) \T) = 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
qed

lemma (in prob_space) distr_marginal2:
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
  assumes Pxy: "distributed M (S \\<^sub>M T) (\x. (X x, Y x)) Pxy"
  shows "distributed M T Y (\y. (\\<^sup>+x. Pxy (x, y) \S))"
  using distr_marginal1[OF T S distributed_swap[OF S T]] Pxy by simp

lemma (in prob_space) distributed_marginal_eq_joint1:
  assumes T: "sigma_finite_measure T"
  assumes S: "sigma_finite_measure S"
  assumes Px: "distributed M S X Px"
  assumes Pxy: "distributed M (S \\<^sub>M T) (\x. (X x, Y x)) Pxy"
  shows "AE x in S. Px x = (\\<^sup>+y. Pxy (x, y) \T)"
  using Px distr_marginal1[OF S T Pxy] by (rule distributed_unique)

lemma (in prob_space) distributed_marginal_eq_joint2:
  assumes T: "sigma_finite_measure T"
  assumes S: "sigma_finite_measure S"
  assumes Py: "distributed M T Y Py"
  assumes Pxy: "distributed M (S \\<^sub>M T) (\x. (X x, Y x)) Pxy"
  shows "AE y in T. Py y = (\\<^sup>+x. Pxy (x, y) \S)"
  using Py distr_marginal2[OF S T Pxy] by (rule distributed_unique)

lemma (in prob_space) distributed_joint_indep':
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
  assumes X[measurable]: "distributed M S X Px" and Y[measurable]: "distributed M T Y Py"
  assumes indep: "distr M S X \\<^sub>M distr M T Y = distr M (S \\<^sub>M T) (\x. (X x, Y x))"
  shows "distributed M (S \\<^sub>M T) (\x. (X x, Y x)) (\(x, y). Px x * Py y)"
  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 ..

  interpret X: prob_space "density S Px"
    unfolding distributed_distr_eq_density[OF X, symmetric]
    by (rule prob_space_distr) simp
  have sf_X: "sigma_finite_measure (density S Px)" ..

  interpret Y: prob_space "density T Py"
    unfolding distributed_distr_eq_density[OF Y, symmetric]
    by (rule prob_space_distr) simp
  have sf_Y: "sigma_finite_measure (density T Py)" ..

  show "distr M (S \\<^sub>M T) (\x. (X x, Y x)) = density (S \\<^sub>M T) (\(x, y). Px x * Py y)"
    unfolding indep[symmetric] distributed_distr_eq_density[OF X] distributed_distr_eq_density[OF Y]
    using distributed_borel_measurable[OF X]
    using distributed_borel_measurable[OF Y]
    by (rule pair_measure_density[OF _ _ T sf_Y])

  show "random_variable (S \\<^sub>M T) (\x. (X x, Y x))" by auto

  show Pxy: "(\(x, y). Px x * Py y) \ borel_measurable (S \\<^sub>M T)" by auto
qed

lemma distributed_integrable:
  "distributed M N X f \ g \ borel_measurable N \ (\x. x \ space N \ 0 \ f x) \
    integrable N (\<lambda>x. f x * g x) \<longleftrightarrow> integrable M (\<lambda>x. g (X x))"
  supply distributed_real_measurable[measurable]
  by (auto simp: distributed_distr_eq_density[symmetric] integrable_real_density[symmetric] integrable_distr_eq)

lemma distributed_transform_integrable:
  assumes Px: "distributed M N X Px" "\x. x \ space N \ 0 \ Px x"
  assumes "distributed M P Y Py" "\x. x \ space P \ 0 \ Py x"
  assumes Y: "Y = (\x. T (X x))" and T: "T \ measurable N P" and f: "f \ borel_measurable P"
  shows "integrable P (\x. Py x * f x) \ integrable N (\x. Px x * f (T x))"
proof -
  have "integrable P (\x. Py x * f x) \ integrable M (\x. f (Y x))"
    by (rule distributed_integrable) fact+
  also have "\ \ integrable M (\x. f (T (X x)))"
    using Y by simp
  also have "\ \ integrable N (\x. Px x * f (T x))"
    using measurable_comp[OF T f] Px by (intro distributed_integrable[symmetric]) (auto simp: comp_def)
  finally show ?thesis .
qed

lemma distributed_integrable_var:
  fixes X :: "'a \ real"
  shows "distributed M lborel X (\x. ennreal (f x)) \ (\x. 0 \ f x) \
    integrable lborel (\<lambda>x. f x * x) \<Longrightarrow> integrable M X"
  using distributed_integrable[of M lborel X f "\x. x"] by simp

lemma (in prob_space) distributed_variance:
  fixes f::"real \ real"
  assumes D: "distributed M lborel X f" and [simp]: "\x. 0 \ f x"
  shows "variance X = (\x. x\<^sup>2 * f (x + expectation X) \lborel)"
proof (subst distributed_integral[OF D, symmetric])
  show "(\ x. f x * (x - expectation X)\<^sup>2 \lborel) = (\ x. x\<^sup>2 * f (x + expectation X) \lborel)"
    by (subst lborel_integral_real_affine[where c=1 and t="expectation X"])  (auto simp: ac_simps)
qed simp_all

lemma (in prob_space) variance_affine:
  fixes f::"real \ real"
  assumes [arith]: "b \ 0"
  assumes D[intro]: "distributed M lborel X f"
  assumes [simp]: "prob_space (density lborel f)"
  assumes I[simp]: "integrable M X"
  assumes I2[simp]: "integrable M (\x. (X x)\<^sup>2)"
  shows "variance (\x. a + b * X x) = b\<^sup>2 * variance X"
  by (subst variance_eq)
     (auto simp: power2_sum power_mult_distrib prob_space variance_eq right_diff_distrib)

definition
  "simple_distributed M X f \
    (\<forall>x. 0 \<le> f x) \<and>
    distributed M (count_space (X`space M)) X (\<lambda>x. ennreal (f x)) \<and>
    finite (X`space M)"

lemma simple_distributed_nonneg[dest]: "simple_distributed M X f \ 0 \ f x"
  by (auto simp: simple_distributed_def)

lemma simple_distributed:
  "simple_distributed M X Px \ distributed M (count_space (X`space M)) X Px"
  unfolding simple_distributed_def by auto

lemma simple_distributed_finite[dest]: "simple_distributed M X P \ finite (X`space M)"
  by (simp add: simple_distributed_def)

lemma (in prob_space) distributed_simple_function_superset:
  assumes X: "simple_function M X" "\x. x \ X ` space M \ P x = measure M (X -` {x} \ space M)"
  assumes A: "X`space M \ A" "finite A"
  defines "S \ count_space A" and "P' \ (\x. if x \ X`space M then P x else 0)"
  shows "distributed M S X P'"
  unfolding distributed_def
proof safe
  show "(\x. ennreal (P' x)) \ borel_measurable S" unfolding S_def by simp
  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 \ A"
    then have "a \ X`space M \ X -` {a} \ 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 measurable_count_space_eq2
               intro!: arg_cong[where f=prob])
    also have "\ = (\\<^sup>+x. ennreal (P' a) * indicator {a} x \S)"
      using A X a
      by (subst nn_integral_cmult_indicator)
         (auto simp: S_def P'_def simple_distributed_def simple_functionD measure_nonneg)
    also have "\ = (\\<^sup>+x. ennreal (P' x) * indicator {a} x \S)"
      by (auto simp: indicator_def intro!: nn_integral_cong)
    also have "\ = 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 (in prob_space) simple_distributedI:
  assumes X: "simple_function M X"
    "\x. 0 \ P x"
    "\x. x \ X ` space M \ P x = measure M (X -` {x} \ space M)"
  shows "simple_distributed M X P"
  unfolding simple_distributed_def
proof (safe intro!: X)
  have "distributed M (count_space (X ` space M)) X (\x. ennreal (if x \ X`space M then P x else 0))"
    (is "?A")
    using simple_functionD[OF X(1)] by (intro distributed_simple_function_superset[OF X(1,3)]) auto
  also have "?A \ distributed M (count_space (X ` space M)) X (\x. ennreal (P x))"
    by (rule distributed_cong_density) auto
  finally show "\" .
qed (rule simple_functionD[OF X(1)])

lemma simple_distributed_joint_finite:
  assumes X: "simple_distributed M (\x. (X x, Y x)) Px"
  shows "finite (X ` space M)" "finite (Y ` space M)"
proof -
  have "finite ((\x. (X x, Y x)) ` space M)"
    using X by (auto simp: simple_distributed_def simple_functionD)
  then have "finite (fst ` (\x. (X x, Y x)) ` space M)" "finite (snd ` (\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 (\x. (X x, Y x, Z x)) Px"
  shows "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
proof -
  have "finite ((\x. (X x, Y x, Z x)) ` space M)"
    using X by (auto simp: simple_distributed_def simple_functionD)
  then have "finite (fst ` (\x. (X x, Y x, Z x)) ` space M)"
    "finite ((fst \ snd) ` (\x. (X x, Y x, Z x)) ` space M)"
    "finite ((snd \ snd) ` (\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

lemma simple_distributed_simple_function:
  "simple_distributed M X Px \ simple_function M X"
  unfolding simple_distributed_def distributed_def
  by (auto simp: simple_function_def measurable_count_space_eq2)

lemma simple_distributed_measure:
  "simple_distributed M X P \ a \ X`space M \ P a = measure M (X -` {a} \ space M)"
  using distributed_count_space[of M "X`space M" X P a, symmetric]
  by (auto simp: simple_distributed_def measure_def)

lemma (in prob_space) simple_distributed_joint:
  assumes X: "simple_distributed M (\x. (X x, Y x)) Px"
  defines "S \ count_space (X`space M) \\<^sub>M count_space (Y`space M)"
  defines "P \ (\x. if x \ (\x. (X x, Y x))`space M then Px x else 0)"
  shows "distributed M S (\x. (X x, Y x)) P"
proof -
  from simple_distributed_joint_finite[OF X, simp]
  have S_eq: "S = count_space (X`space M \ 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 (\x. (X x, Y x))"
      using X by (rule simple_distributed_simple_function)
    fix x assume "x \ (\x. (X x, Y x)) ` space M"
    from simple_distributed_measure[OF X this]
    show "Px x = prob ((\x. (X x, Y x)) -` {x} \ space M)" .
  qed auto
qed

lemma (in prob_space) simple_distributed_joint2:
  assumes X: "simple_distributed M (\x. (X x, Y x, Z x)) Px"
  defines "S \ count_space (X`space M) \\<^sub>M count_space (Y`space M) \\<^sub>M count_space (Z`space M)"
  defines "P \ (\x. if x \ (\x. (X x, Y x, Z x))`space M then Px x else 0)"
  shows "distributed M S (\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 \ Y`space M \ 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 (\x. (X x, Y x, Z x))"
      using X by (rule simple_distributed_simple_function)
    fix x assume "x \ (\x. (X x, Y x, Z x)) ` space M"
    from simple_distributed_measure[OF X this]
    show "Px x = prob ((\x. (X x, Y x, Z x)) -` {x} \ space M)" .
  qed auto
qed

lemma (in prob_space) simple_distributed_sum_space:
  assumes X: "simple_distributed M X f"
  shows "sum f (X`space M) = 1"
proof -
  from X have "sum f (X`space M) = prob (\i\X`space M. X -` {i} \ 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!: sum.cong arg_cong[where f="prob"])
  also have "\ = prob (space M)"
    by (auto intro!: arg_cong[where f=prob])
  finally show ?thesis
    using emeasure_space_1 by (simp add: emeasure_eq_measure)
qed

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 (\x. (X x, Y x)) Pxy"
  assumes y: "y \ Y`space M"
  shows "Py y = (\x\X`space M. if (x, y) \ (\x. (X x, Y x)) ` space M then Pxy (x, y) else 0)"
proof -
  note Px = simple_distributedI[OF Px measure_nonneg refl]
  have "AE y in count_space (Y ` space M). ennreal (Py y) =
       \<integral>\<^sup>+ x. ennreal (if (x, y) \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy (x, y) else 0) \<partial>count_space (X ` space M)"
    using sigma_finite_measure_count_space_finite sigma_finite_measure_count_space_finite
      simple_distributed[OF Py] simple_distributed_joint[OF Pxy]
    by (rule distributed_marginal_eq_joint2)
       (auto intro: Py Px simple_distributed_finite)
  then have "ennreal (Py y) =
    (\<Sum>x\<in>X`space M. ennreal (if (x, y) \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy (x, y) else 0))"
    using y Px[THEN simple_distributed_finite]
    by (auto simp: AE_count_space nn_integral_count_space_finite)
  also have "\ = (\x\X`space M. if (x, y) \ (\x. (X x, Y x)) ` space M then Pxy (x, y) else 0)"
    using Pxy by (intro sum_ennreal) auto
  finally show ?thesis
    using simple_distributed_nonneg[OF Py] simple_distributed_nonneg[OF Pxy]
    by (subst (asm) ennreal_inj) (auto intro!: sum_nonneg)
qed

lemma distributedI_real:
  fixes f :: "'a \ real"
  assumes gen: "sets M1 = sigma_sets (space M1) E" and "Int_stable E"
    and A: "range A \ E" "(\i::nat. A i) = space M1" "\i. emeasure (distr M M1 X) (A i) \ \"
    and X: "X \ measurable M M1"
    and f: "f \ borel_measurable M1" "AE x in M1. 0 \ f x"
    and eq: "\A. A \ E \ emeasure M (X -` A \ space M) = (\\<^sup>+ x. f x * indicator A x \M1)"
  shows "distributed M M1 X f"
  unfolding distributed_def
proof (intro conjI)
  show "distr M M1 X = density M1 f"
  proof (rule measure_eqI_generator_eq[where A=A])
    { fix A assume A: "A \ E"
      then have "A \ sigma_sets (space M1) E" by auto
      then have "A \ sets M1"
        using gen by simp
      with f A eq[of A] X show "emeasure (distr M M1 X) A = emeasure (density M1 f) A"
        by (auto simp add: emeasure_distr emeasure_density ennreal_indicator
                 intro!: nn_integral_cong split: split_indicator) }
    note eq_E = this
    show "Int_stable E" by fact
    { fix e assume "e \ E"
      then have "e \ sigma_sets (space M1) E" by auto
      then have "e \ sets M1" unfolding gen .
      then have "e \ space M1" by (rule sets.sets_into_space) }
    then show "E \ Pow (space M1)" by auto
    show "sets (distr M M1 X) = sigma_sets (space M1) E"
      "sets (density M1 (\x. ennreal (f x))) = sigma_sets (space M1) E"
      unfolding gen[symmetric] by auto
  qed fact+
qed (insert X f, auto)

lemma distributedI_borel_atMost:
  fixes f :: "real \ real"
  assumes [measurable]: "X \ borel_measurable M"
    and [measurable]: "f \ borel_measurable borel" and f[simp]: "AE x in lborel. 0 \ f x"
    and g_eq: "\a. (\\<^sup>+x. f x * indicator {..a} x \lborel) = ennreal (g a)"
    and M_eq: "\a. emeasure M {x\space M. X x \ a} = ennreal (g a)"
  shows "distributed M lborel X f"
proof (rule distributedI_real)
  show "sets (lborel::real measure) = sigma_sets (space lborel) (range atMost)"
    by (simp add: borel_eq_atMost)
  show "Int_stable (range atMost :: real set set)"
    by (auto simp: Int_stable_def)
  have vimage_eq: "\a. (X -` {..a} \ space M) = {x\space M. X x \ a}" by auto
  define A where "A i = {.. real i}" for i :: nat
  then show "range A \ range atMost" "(\i. A i) = space lborel"
    "\i. emeasure (distr M lborel X) (A i) \ \"
    by (auto simp: real_arch_simple emeasure_distr vimage_eq M_eq)

  fix A :: "real set" assume "A \ range atMost"
  then obtain a where A: "A = {..a}" by auto
  show "emeasure M (X -` A \ space M) = (\\<^sup>+x. f x * indicator A x \lborel)"
    unfolding vimage_eq A M_eq g_eq ..
qed auto

lemma (in prob_space) uniform_distributed_params:
  assumes X: "distributed M MX X (\x. indicator A x / measure MX A)"
  shows "A \ sets MX" "measure MX A \ 0"
proof -
  interpret X: prob_space "distr M MX X"
    using distributed_measurable[OF X] by (rule prob_space_distr)

  show "measure MX A \ 0"
  proof
    assume "measure MX A = 0"
    with X.emeasure_space_1 X.prob_space distributed_distr_eq_density[OF X]
    show False
      by (simp add: emeasure_density zero_ennreal_def[symmetric])
  qed
  with measure_notin_sets[of A MX] show "A \ sets MX"
    by blast
qed

lemma prob_space_uniform_measure:
  assumes A: "emeasure M A \ 0" "emeasure M A \ \"
  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.sets_into_space[OF emeasure_neq_0_sets[OF A(1)]] A
    by (simp add: Int_absorb2 less_top)
qed

lemma prob_space_uniform_count_measure: "finite A \ A \ {} \ prob_space (uniform_count_measure A)"
  by standard (auto simp: emeasure_uniform_count_measure space_uniform_count_measure one_ennreal_def)

lemma (in prob_space) measure_uniform_measure_eq_cond_prob:
  assumes [measurable]: "Measurable.pred M P" "Measurable.pred M Q"
  shows "\

(x in uniform_measure M {x\space M. Q x}. P x) = \

(x in M. P x \ Q x)"
proof cases
  assume Q: "measure M {x\space M. Q x} = 0"
  then have *: "AE x in M. \ Q x"
    by (simp add: prob_eq_0)
  then have "density M (\x. indicator {x \ space M. Q x} x / emeasure M {x \ space M. Q x}) = density M (\x. 0)"
    by (intro density_cong) auto
  with * show ?thesis
    unfolding uniform_measure_def
    by (simp add: emeasure_density measure_def cond_prob_def emeasure_eq_0_AE)
next
  assume Q: "measure M {x\space M. Q x} \ 0"
  then show "\

(x in uniform_measure M {x \ space M. Q x}. P x) = cond_prob M P Q"
    by (subst measure_uniform_measure)
       (auto simp: emeasure_eq_measure cond_prob_def measure_nonneg intro!: arg_cong[where f=prob])
qed

lemma prob_space_point_measure:
  "finite S \ (\s. s \ S \ 0 \ p s) \ (\s\S. p s) = 1 \ prob_space (point_measure S p)"
  by (rule prob_spaceI) (simp add: space_point_measure emeasure_point_measure_finite)

lemma (in prob_space) distr_pair_fst: "distr (N \\<^sub>M M) N fst = N"
proof (intro measure_eqI)
  fix A assume A: "A \ sets (distr (N \\<^sub>M M) N fst)"
  from A have "emeasure (distr (N \\<^sub>M M) N fst) A = emeasure (N \\<^sub>M M) (A \ space M)"
    by (auto simp add: emeasure_distr space_pair_measure dest: sets.sets_into_space intro!: arg_cong2[where f=emeasure])
  with A show "emeasure (distr (N \\<^sub>M M) N fst) A = emeasure N A"
    by (simp add: emeasure_pair_measure_Times emeasure_space_1)
qed simp

lemma (in product_prob_space) distr_reorder:
  assumes "inj_on t J" "t \ J \ K" "finite K"
  shows "distr (PiM K M) (Pi\<^sub>M J (\x. M (t x))) (\\. \n\J. \ (t n)) = PiM J (\x. M (t x))"
proof (rule product_sigma_finite.PiM_eqI)
  show "product_sigma_finite (\x. M (t x))" ..
  have "t`J \ K" using assms by auto
  then show [simp]: "finite J"
    by (rule finite_imageD[OF finite_subset]) fact+
  fix A assume A: "\i. i \ J \ A i \ sets (M (t i))"
  moreover have "((\\. \n\J. \ (t n)) -` Pi\<^sub>E J A \ space (Pi\<^sub>M K M)) =
    (\<Pi>\<^sub>E i\<in>K. if i \<in> t`J then A (the_inv_into J t i) else space (M i))"
    using A A[THEN sets.sets_into_space] \<open>t \<in> J \<rightarrow> K\<close> \<open>inj_on t J\<close>
    by (subst prod_emb_Pi[symmetric]) (auto simp: space_PiM PiE_iff the_inv_into_f_f prod_emb_def)
  ultimately show "distr (Pi\<^sub>M K M) (Pi\<^sub>M J (\x. M (t x))) (\\. \n\J. \ (t n)) (Pi\<^sub>E J A) = (\i\J. M (t i) (A i))"
    using assms
    apply (subst emeasure_distr)
    apply (auto intro!: sets_PiM_I_finite simp: Pi_iff)
    apply (subst emeasure_PiM)
    apply (auto simp: the_inv_into_f_f \<open>inj_on t J\<close> prod.reindex[OF \<open>inj_on t J\<close>]
      if_distrib[where f="emeasure (M _)"] prod.If_cases emeasure_space_1 Int_absorb1 \<open>t`J \<subseteq> K\<close>)
    done
qed simp

lemma (in product_prob_space) distr_restrict:
  "J \ K \ finite K \ (\\<^sub>M i\J. M i) = distr (\\<^sub>M i\K. M i) (\\<^sub>M i\J. M i) (\f. restrict f J)"
  using distr_reorder[of "\x. x" J K] by (simp add: Pi_iff subset_eq)

lemma (in product_prob_space) emeasure_prod_emb[simp]:
  assumes L: "J \ L" "finite L" and X: "X \ sets (Pi\<^sub>M J M)"
  shows "emeasure (Pi\<^sub>M L M) (prod_emb L M J X) = emeasure (Pi\<^sub>M J M) X"
  by (subst distr_restrict[OF L])
     (simp add: prod_emb_def space_PiM emeasure_distr measurable_restrict_subset L X)

lemma emeasure_distr_restrict:
  assumes "I \ K" and Q[measurable_cong]: "sets Q = sets (PiM K M)" and A[measurable]: "A \ sets (PiM I M)"
  shows "emeasure (distr Q (PiM I M) (\\. restrict \ I)) A = emeasure Q (prod_emb K M I A)"
  using \<open>I\<subseteq>K\<close> sets_eq_imp_space_eq[OF Q]
  by (subst emeasure_distr)
     (auto simp: measurable_cong_sets[OF Q] prod_emb_def space_PiM[symmetric] intro!: measurable_restrict)

lemma (in prob_space) prob_space_completion: "prob_space (completion M)"
  by (rule prob_spaceI) (simp add: emeasure_space_1)

end

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