Theory Independent_Family

theory Independent_Family
imports Infinite_Product_Measure
(*  Title:      HOL/Probability/Independent_Family.thy
    Author:     Johannes Hölzl, TU München
    Author:     Sudeep Kanav, TU München
*)

section ‹Independent families of events, event sets, and random variables›

theory Independent_Family
  imports Infinite_Product_Measure
begin

definition (in prob_space)
  "indep_sets F I ⟷ (∀i∈I. F i ⊆ events) ∧
    (∀J⊆I. J ≠ {} ⟶ finite J ⟶ (∀A∈Pi J F. prob (⋂j∈J. A j) = (∏j∈J. prob (A j))))"

definition (in prob_space)
  "indep_set A B ⟷ indep_sets (case_bool A B) UNIV"

definition (in prob_space)
  indep_events_def_alt: "indep_events A I ⟷ indep_sets (λi. {A i}) I"

lemma (in prob_space) indep_events_def:
  "indep_events A I ⟷ (A`I ⊆ events) ∧
    (∀J⊆I. J ≠ {} ⟶ finite J ⟶ prob (⋂j∈J. A j) = (∏j∈J. prob (A j)))"
  unfolding indep_events_def_alt indep_sets_def
  apply (simp add: Ball_def Pi_iff image_subset_iff_funcset)
  apply (intro conj_cong refl arg_cong[where f=All] ext imp_cong)
  apply auto
  done

lemma (in prob_space) indep_eventsI:
  "(⋀i. i ∈ I ⟹ F i ∈ sets M) ⟹ (⋀J. J ⊆ I ⟹ finite J ⟹ J ≠ {} ⟹ prob (⋂i∈J. F i) = (∏i∈J. prob (F i))) ⟹ indep_events F I"
  by (auto simp: indep_events_def)

definition (in prob_space)
  "indep_event A B ⟷ indep_events (case_bool A B) UNIV"

lemma (in prob_space) indep_sets_cong:
  "I = J ⟹ (⋀i. i ∈ I ⟹ F i = G i) ⟹ indep_sets F I ⟷ indep_sets G J"
  by (simp add: indep_sets_def, intro conj_cong all_cong imp_cong ball_cong) blast+

lemma (in prob_space) indep_events_finite_index_events:
  "indep_events F I ⟷ (∀J⊆I. J ≠ {} ⟶ finite J ⟶ indep_events F J)"
  by (auto simp: indep_events_def)

lemma (in prob_space) indep_sets_finite_index_sets:
  "indep_sets F I ⟷ (∀J⊆I. J ≠ {} ⟶ finite J ⟶ indep_sets F J)"
proof (intro iffI allI impI)
  assume *: "∀J⊆I. J ≠ {} ⟶ finite J ⟶ indep_sets F J"
  show "indep_sets F I" unfolding indep_sets_def
  proof (intro conjI ballI allI impI)
    fix i assume "i ∈ I"
    with *[THEN spec, of "{i}"] show "F i ⊆ events"
      by (auto simp: indep_sets_def)
  qed (insert *, auto simp: indep_sets_def)
qed (auto simp: indep_sets_def)

lemma (in prob_space) indep_sets_mono_index:
  "J ⊆ I ⟹ indep_sets F I ⟹ indep_sets F J"
  unfolding indep_sets_def by auto

lemma (in prob_space) indep_sets_mono_sets:
  assumes indep: "indep_sets F I"
  assumes mono: "⋀i. i∈I ⟹ G i ⊆ F i"
  shows "indep_sets G I"
proof -
  have "(∀i∈I. F i ⊆ events) ⟹ (∀i∈I. G i ⊆ events)"
    using mono by auto
  moreover have "⋀A J. J ⊆ I ⟹ A ∈ (Π j∈J. G j) ⟹ A ∈ (Π j∈J. F j)"
    using mono by (auto simp: Pi_iff)
  ultimately show ?thesis
    using indep by (auto simp: indep_sets_def)
qed

lemma (in prob_space) indep_sets_mono:
  assumes indep: "indep_sets F I"
  assumes mono: "J ⊆ I" "⋀i. i∈J ⟹ G i ⊆ F i"
  shows "indep_sets G J"
  apply (rule indep_sets_mono_sets)
  apply (rule indep_sets_mono_index)
  apply (fact +)
  done

lemma (in prob_space) indep_setsI:
  assumes "⋀i. i ∈ I ⟹ F i ⊆ events"
    and "⋀A J. J ≠ {} ⟹ J ⊆ I ⟹ finite J ⟹ (∀j∈J. A j ∈ F j) ⟹ prob (⋂j∈J. A j) = (∏j∈J. prob (A j))"
  shows "indep_sets F I"
  using assms unfolding indep_sets_def by (auto simp: Pi_iff)

lemma (in prob_space) indep_setsD:
  assumes "indep_sets F I" and "J ⊆ I" "J ≠ {}" "finite J" "∀j∈J. A j ∈ F j"
  shows "prob (⋂j∈J. A j) = (∏j∈J. prob (A j))"
  using assms unfolding indep_sets_def by auto

lemma (in prob_space) indep_setI:
  assumes ev: "A ⊆ events" "B ⊆ events"
    and indep: "⋀a b. a ∈ A ⟹ b ∈ B ⟹ prob (a ∩ b) = prob a * prob b"
  shows "indep_set A B"
  unfolding indep_set_def
proof (rule indep_setsI)
  fix F J assume "J ≠ {}" "J ⊆ UNIV"
    and F: "∀j∈J. F j ∈ (case j of True ⇒ A | False ⇒ B)"
  have "J ∈ Pow UNIV" by auto
  with F ‹J ≠ {}› indep[of "F True" "F False"]
  show "prob (⋂j∈J. F j) = (∏j∈J. prob (F j))"
    unfolding UNIV_bool Pow_insert by (auto simp: ac_simps)
qed (auto split: bool.split simp: ev)

lemma (in prob_space) indep_setD:
  assumes indep: "indep_set A B" and ev: "a ∈ A" "b ∈ B"
  shows "prob (a ∩ b) = prob a * prob b"
  using indep[unfolded indep_set_def, THEN indep_setsD, of UNIV "case_bool a b"] ev
  by (simp add: ac_simps UNIV_bool)

lemma (in prob_space)
  assumes indep: "indep_set A B"
  shows indep_setD_ev1: "A ⊆ events"
    and indep_setD_ev2: "B ⊆ events"
  using indep unfolding indep_set_def indep_sets_def UNIV_bool by auto

lemma (in prob_space) indep_sets_dynkin:
  assumes indep: "indep_sets F I"
  shows "indep_sets (λi. dynkin (space M) (F i)) I"
    (is "indep_sets ?F I")
proof (subst indep_sets_finite_index_sets, intro allI impI ballI)
  fix J assume "finite J" "J ⊆ I" "J ≠ {}"
  with indep have "indep_sets F J"
    by (subst (asm) indep_sets_finite_index_sets) auto
  { fix J K assume "indep_sets F K"
    let ?G = "λS i. if i ∈ S then ?F i else F i"
    assume "finite J" "J ⊆ K"
    then have "indep_sets (?G J) K"
    proof induct
      case (insert j J)
      moreover define G where "G = ?G J"
      ultimately have G: "indep_sets G K" "⋀i. i ∈ K ⟹ G i ⊆ events" and "j ∈ K"
        by (auto simp: indep_sets_def)
      let ?D = "{E∈events. indep_sets (G(j := {E})) K }"
      { fix X assume X: "X ∈ events"
        assume indep: "⋀J A. J ≠ {} ⟹ J ⊆ K ⟹ finite J ⟹ j ∉ J ⟹ (∀i∈J. A i ∈ G i)
          ⟹ prob ((⋂i∈J. A i) ∩ X) = prob X * (∏i∈J. prob (A i))"
        have "indep_sets (G(j := {X})) K"
        proof (rule indep_setsI)
          fix i assume "i ∈ K" then show "(G(j:={X})) i ⊆ events"
            using G X by auto
        next
          fix A J assume J: "J ≠ {}" "J ⊆ K" "finite J" "∀i∈J. A i ∈ (G(j := {X})) i"
          show "prob (⋂j∈J. A j) = (∏j∈J. prob (A j))"
          proof cases
            assume "j ∈ J"
            with J have "A j = X" by auto
            show ?thesis
            proof cases
              assume "J = {j}" then show ?thesis by simp
            next
              assume "J ≠ {j}"
              have "prob (⋂i∈J. A i) = prob ((⋂i∈J-{j}. A i) ∩ X)"
                using ‹j ∈ J› ‹A j = X› by (auto intro!: arg_cong[where f=prob] split: if_split_asm)
              also have "… = prob X * (∏i∈J-{j}. prob (A i))"
              proof (rule indep)
                show "J - {j} ≠ {}" "J - {j} ⊆ K" "finite (J - {j})" "j ∉ J - {j}"
                  using J ‹J ≠ {j}› ‹j ∈ J› by auto
                show "∀i∈J - {j}. A i ∈ G i"
                  using J by auto
              qed
              also have "… = prob (A j) * (∏i∈J-{j}. prob (A i))"
                using ‹A j = X› by simp
              also have "… = (∏i∈J. prob (A i))"
                unfolding prod.insert_remove[OF ‹finite J›, symmetric, of "λi. prob  (A i)"]
                using ‹j ∈ J› by (simp add: insert_absorb)
              finally show ?thesis .
            qed
          next
            assume "j ∉ J"
            with J have "∀i∈J. A i ∈ G i" by (auto split: if_split_asm)
            with J show ?thesis
              by (intro indep_setsD[OF G(1)]) auto
          qed
        qed }
      note indep_sets_insert = this
      have "dynkin_system (space M) ?D"
      proof (rule dynkin_systemI', simp_all cong del: indep_sets_cong, safe)
        show "indep_sets (G(j := {{}})) K"
          by (rule indep_sets_insert) auto
      next
        fix X assume X: "X ∈ events" and G': "indep_sets (G(j := {X})) K"
        show "indep_sets (G(j := {space M - X})) K"
        proof (rule indep_sets_insert)
          fix J A assume J: "J ≠ {}" "J ⊆ K" "finite J" "j ∉ J" and A: "∀i∈J. A i ∈ G i"
          then have A_sets: "⋀i. i∈J ⟹ A i ∈ events"
            using G by auto
          have "prob ((⋂j∈J. A j) ∩ (space M - X)) =
              prob ((⋂j∈J. A j) - (⋂i∈insert j J. (A(j := X)) i))"
            using A_sets sets.sets_into_space[of _ M] X ‹J ≠ {}›
            by (auto intro!: arg_cong[where f=prob] split: if_split_asm)
          also have "… = prob (⋂j∈J. A j) - prob (⋂i∈insert j J. (A(j := X)) i)"
            using J ‹J ≠ {}› ‹j ∉ J› A_sets X sets.sets_into_space
            by (auto intro!: finite_measure_Diff sets.finite_INT split: if_split_asm)
          finally have "prob ((⋂j∈J. A j) ∩ (space M - X)) =
              prob (⋂j∈J. A j) - prob (⋂i∈insert j J. (A(j := X)) i)" .
          moreover {
            have "prob (⋂j∈J. A j) = (∏j∈J. prob (A j))"
              using J A ‹finite J› by (intro indep_setsD[OF G(1)]) auto
            then have "prob (⋂j∈J. A j) = prob (space M) * (∏i∈J. prob (A i))"
              using prob_space by simp }
          moreover {
            have "prob (⋂i∈insert j J. (A(j := X)) i) = (∏i∈insert j J. prob ((A(j := X)) i))"
              using J A ‹j ∈ K› by (intro indep_setsD[OF G']) auto
            then have "prob (⋂i∈insert j J. (A(j := X)) i) = prob X * (∏i∈J. prob (A i))"
              using ‹finite J› ‹j ∉ J› by (auto intro!: prod.cong) }
          ultimately have "prob ((⋂j∈J. A j) ∩ (space M - X)) = (prob (space M) - prob X) * (∏i∈J. prob (A i))"
            by (simp add: field_simps)
          also have "… = prob (space M - X) * (∏i∈J. prob (A i))"
            using X A by (simp add: finite_measure_compl)
          finally show "prob ((⋂j∈J. A j) ∩ (space M - X)) = prob (space M - X) * (∏i∈J. prob (A i))" .
        qed (insert X, auto)
      next
        fix F :: "nat ⇒ 'a set" assume disj: "disjoint_family F" and "range F ⊆ ?D"
        then have F: "⋀i. F i ∈ events" "⋀i. indep_sets (G(j:={F i})) K" by auto
        show "indep_sets (G(j := {⋃k. F k})) K"
        proof (rule indep_sets_insert)
          fix J A assume J: "j ∉ J" "J ≠ {}" "J ⊆ K" "finite J" and A: "∀i∈J. A i ∈ G i"
          then have A_sets: "⋀i. i∈J ⟹ A i ∈ events"
            using G by auto
          have "prob ((⋂j∈J. A j) ∩ (⋃k. F k)) = prob (⋃k. (⋂i∈insert j J. (A(j := F k)) i))"
            using ‹J ≠ {}› ‹j ∉ J› ‹j ∈ K› by (auto intro!: arg_cong[where f=prob] split: if_split_asm)
          moreover have "(λk. prob (⋂i∈insert j J. (A(j := F k)) i)) sums prob (⋃k. (⋂i∈insert j J. (A(j := F k)) i))"
          proof (rule finite_measure_UNION)
            show "disjoint_family (λk. ⋂i∈insert j J. (A(j := F k)) i)"
              using disj by (rule disjoint_family_on_bisimulation) auto
            show "range (λk. ⋂i∈insert j J. (A(j := F k)) i) ⊆ events"
              using A_sets F ‹finite J› ‹J ≠ {}› ‹j ∉ J› by (auto intro!: sets.Int)
          qed
          moreover { fix k
            from J A ‹j ∈ K› have "prob (⋂i∈insert j J. (A(j := F k)) i) = prob (F k) * (∏i∈J. prob (A i))"
              by (subst indep_setsD[OF F(2)]) (auto intro!: prod.cong split: if_split_asm)
            also have "… = prob (F k) * prob (⋂i∈J. A i)"
              using J A ‹j ∈ K› by (subst indep_setsD[OF G(1)]) auto
            finally have "prob (⋂i∈insert j J. (A(j := F k)) i) = prob (F k) * prob (⋂i∈J. A i)" . }
          ultimately have "(λk. prob (F k) * prob (⋂i∈J. A i)) sums (prob ((⋂j∈J. A j) ∩ (⋃k. F k)))"
            by simp
          moreover
          have "(λk. prob (F k) * prob (⋂i∈J. A i)) sums (prob (⋃k. F k) * prob (⋂i∈J. A i))"
            using disj F(1) by (intro finite_measure_UNION sums_mult2) auto
          then have "(λk. prob (F k) * prob (⋂i∈J. A i)) sums (prob (⋃k. F k) * (∏i∈J. prob (A i)))"
            using J A ‹j ∈ K› by (subst indep_setsD[OF G(1), symmetric]) auto
          ultimately
          show "prob ((⋂j∈J. A j) ∩ (⋃k. F k)) = prob (⋃k. F k) * (∏j∈J. prob (A j))"
            by (auto dest!: sums_unique)
        qed (insert F, auto)
      qed (insert sets.sets_into_space, auto)
      then have mono: "dynkin (space M) (G j) ⊆ {E ∈ events. indep_sets (G(j := {E})) K}"
      proof (rule dynkin_system.dynkin_subset, safe)
        fix X assume "X ∈ G j"
        then show "X ∈ events" using G ‹j ∈ K› by auto
        from ‹indep_sets G K›
        show "indep_sets (G(j := {X})) K"
          by (rule indep_sets_mono_sets) (insert ‹X ∈ G j›, auto)
      qed
      have "indep_sets (G(j:=?D)) K"
      proof (rule indep_setsI)
        fix i assume "i ∈ K" then show "(G(j := ?D)) i ⊆ events"
          using G(2) by auto
      next
        fix A J assume J: "J≠{}" "J ⊆ K" "finite J" and A: "∀i∈J. A i ∈ (G(j := ?D)) i"
        show "prob (⋂j∈J. A j) = (∏j∈J. prob (A j))"
        proof cases
          assume "j ∈ J"
          with A have indep: "indep_sets (G(j := {A j})) K" by auto
          from J A show ?thesis
            by (intro indep_setsD[OF indep]) auto
        next
          assume "j ∉ J"
          with J A have "∀i∈J. A i ∈ G i" by (auto split: if_split_asm)
          with J show ?thesis
            by (intro indep_setsD[OF G(1)]) auto
        qed
      qed
      then have "indep_sets (G(j := dynkin (space M) (G j))) K"
        by (rule indep_sets_mono_sets) (insert mono, auto)
      then show ?case
        by (rule indep_sets_mono_sets) (insert ‹j ∈ K› ‹j ∉ J›, auto simp: G_def)
    qed (insert ‹indep_sets F K›, simp) }
  from this[OF ‹indep_sets F J› ‹finite J› subset_refl]
  show "indep_sets ?F J"
    by (rule indep_sets_mono_sets) auto
qed

lemma (in prob_space) indep_sets_sigma:
  assumes indep: "indep_sets F I"
  assumes stable: "⋀i. i ∈ I ⟹ Int_stable (F i)"
  shows "indep_sets (λi. sigma_sets (space M) (F i)) I"
proof -
  from indep_sets_dynkin[OF indep]
  show ?thesis
  proof (rule indep_sets_mono_sets, subst sigma_eq_dynkin, simp_all add: stable)
    fix i assume "i ∈ I"
    with indep have "F i ⊆ events" by (auto simp: indep_sets_def)
    with sets.sets_into_space show "F i ⊆ Pow (space M)" by auto
  qed
qed

lemma (in prob_space) indep_sets_sigma_sets_iff:
  assumes "⋀i. i ∈ I ⟹ Int_stable (F i)"
  shows "indep_sets (λi. sigma_sets (space M) (F i)) I ⟷ indep_sets F I"
proof
  assume "indep_sets F I" then show "indep_sets (λi. sigma_sets (space M) (F i)) I"
    by (rule indep_sets_sigma) fact
next
  assume "indep_sets (λi. sigma_sets (space M) (F i)) I" then show "indep_sets F I"
    by (rule indep_sets_mono_sets) (intro subsetI sigma_sets.Basic)
qed

definition (in prob_space)
  indep_vars_def2: "indep_vars M' X I ⟷
    (∀i∈I. random_variable (M' i) (X i)) ∧
    indep_sets (λi. { X i -` A ∩ space M | A. A ∈ sets (M' i)}) I"

definition (in prob_space)
  "indep_var Ma A Mb B ⟷ indep_vars (case_bool Ma Mb) (case_bool A B) UNIV"

lemma (in prob_space) indep_vars_def:
  "indep_vars M' X I ⟷
    (∀i∈I. random_variable (M' i) (X i)) ∧
    indep_sets (λi. sigma_sets (space M) { X i -` A ∩ space M | A. A ∈ sets (M' i)}) I"
  unfolding indep_vars_def2
  apply (rule conj_cong[OF refl])
  apply (rule indep_sets_sigma_sets_iff[symmetric])
  apply (auto simp: Int_stable_def)
  apply (rule_tac x="A ∩ Aa" in exI)
  apply auto
  done

lemma (in prob_space) indep_var_eq:
  "indep_var S X T Y ⟷
    (random_variable S X ∧ random_variable T Y) ∧
    indep_set
      (sigma_sets (space M) { X -` A ∩ space M | A. A ∈ sets S})
      (sigma_sets (space M) { Y -` A ∩ space M | A. A ∈ sets T})"
  unfolding indep_var_def indep_vars_def indep_set_def UNIV_bool
  by (intro arg_cong2[where f="(∧)"] arg_cong2[where f=indep_sets] ext)
     (auto split: bool.split)

lemma (in prob_space) indep_sets2_eq:
  "indep_set A B ⟷ A ⊆ events ∧ B ⊆ events ∧ (∀a∈A. ∀b∈B. prob (a ∩ b) = prob a * prob b)"
  unfolding indep_set_def
proof (intro iffI ballI conjI)
  assume indep: "indep_sets (case_bool A B) UNIV"
  { fix a b assume "a ∈ A" "b ∈ B"
    with indep_setsD[OF indep, of UNIV "case_bool a b"]
    show "prob (a ∩ b) = prob a * prob b"
      unfolding UNIV_bool by (simp add: ac_simps) }
  from indep show "A ⊆ events" "B ⊆ events"
    unfolding indep_sets_def UNIV_bool by auto
next
  assume *: "A ⊆ events ∧ B ⊆ events ∧ (∀a∈A. ∀b∈B. prob (a ∩ b) = prob a * prob b)"
  show "indep_sets (case_bool A B) UNIV"
  proof (rule indep_setsI)
    fix i show "(case i of True ⇒ A | False ⇒ B) ⊆ events"
      using * by (auto split: bool.split)
  next
    fix J X assume "J ≠ {}" "J ⊆ UNIV" and X: "∀j∈J. X j ∈ (case j of True ⇒ A | False ⇒ B)"
    then have "J = {True} ∨ J = {False} ∨ J = {True,False}"
      by (auto simp: UNIV_bool)
    then show "prob (⋂j∈J. X j) = (∏j∈J. prob (X j))"
      using X * by auto
  qed
qed

lemma (in prob_space) indep_set_sigma_sets:
  assumes "indep_set A B"
  assumes A: "Int_stable A" and B: "Int_stable B"
  shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)"
proof -
  have "indep_sets (λi. sigma_sets (space M) (case i of True ⇒ A | False ⇒ B)) UNIV"
  proof (rule indep_sets_sigma)
    show "indep_sets (case_bool A B) UNIV"
      by (rule ‹indep_set A B›[unfolded indep_set_def])
    fix i show "Int_stable (case i of True ⇒ A | False ⇒ B)"
      using A B by (cases i) auto
  qed
  then show ?thesis
    unfolding indep_set_def
    by (rule indep_sets_mono_sets) (auto split: bool.split)
qed

lemma (in prob_space) indep_eventsI_indep_vars:
  assumes indep: "indep_vars N X I"
  assumes P: "⋀i. i ∈ I ⟹ {x∈space (N i). P i x} ∈ sets (N i)"
  shows "indep_events (λi. {x∈space M. P i (X i x)}) I"
proof -
  have "indep_sets (λi. {X i -` A ∩ space M |A. A ∈ sets (N i)}) I"
    using indep unfolding indep_vars_def2 by auto
  then show ?thesis
    unfolding indep_events_def_alt
  proof (rule indep_sets_mono_sets)
    fix i assume "i ∈ I"
    then have "{{x ∈ space M. P i (X i x)}} = {X i -` {x∈space (N i). P i x} ∩ space M}"
      using indep by (auto simp: indep_vars_def dest: measurable_space)
    also have "… ⊆ {X i -` A ∩ space M |A. A ∈ sets (N i)}"
      using P[OF ‹i ∈ I›] by blast
    finally show "{{x ∈ space M. P i (X i x)}} ⊆ {X i -` A ∩ space M |A. A ∈ sets (N i)}" .
  qed
qed

lemma (in prob_space) indep_sets_collect_sigma:
  fixes I :: "'j ⇒ 'i set" and J :: "'j set" and E :: "'i ⇒ 'a set set"
  assumes indep: "indep_sets E (⋃j∈J. I j)"
  assumes Int_stable: "⋀i j. j ∈ J ⟹ i ∈ I j ⟹ Int_stable (E i)"
  assumes disjoint: "disjoint_family_on I J"
  shows "indep_sets (λj. sigma_sets (space M) (⋃i∈I j. E i)) J"
proof -
  let ?E = "λj. {⋂k∈K. E' k| E' K. finite K ∧ K ≠ {} ∧ K ⊆ I j ∧ (∀k∈K. E' k ∈ E k) }"

  from indep have E: "⋀j i. j ∈ J ⟹ i ∈ I j ⟹ E i ⊆ events"
    unfolding indep_sets_def by auto
  { fix j
    let ?S = "sigma_sets (space M) (⋃i∈I j. E i)"
    assume "j ∈ J"
    from E[OF this] interpret S: sigma_algebra "space M" ?S
      using sets.sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto

    have "sigma_sets (space M) (⋃i∈I j. E i) = sigma_sets (space M) (?E j)"
    proof (rule sigma_sets_eqI)
      fix A assume "A ∈ (⋃i∈I j. E i)"
      then guess i ..
      then show "A ∈ sigma_sets (space M) (?E j)"
        by (auto intro!: sigma_sets.intros(2-) exI[of _ "{i}"] exI[of _ "λi. A"])
    next
      fix A assume "A ∈ ?E j"
      then obtain E' K where "finite K" "K ≠ {}" "K ⊆ I j" "⋀k. k ∈ K ⟹ E' k ∈ E k"
        and A: "A = (⋂k∈K. E' k)"
        by auto
      then have "A ∈ ?S" unfolding A
        by (safe intro!: S.finite_INT) auto
      then show "A ∈ sigma_sets (space M) (⋃i∈I j. E i)"
        by simp
    qed }
  moreover have "indep_sets (λj. sigma_sets (space M) (?E j)) J"
  proof (rule indep_sets_sigma)
    show "indep_sets ?E J"
    proof (intro indep_setsI)
      fix j assume "j ∈ J" with E show "?E j ⊆ events" by (force  intro!: sets.finite_INT)
    next
      fix K A assume K: "K ≠ {}" "K ⊆ J" "finite K"
        and "∀j∈K. A j ∈ ?E j"
      then have "∀j∈K. ∃E' L. A j = (⋂l∈L. E' l) ∧ finite L ∧ L ≠ {} ∧ L ⊆ I j ∧ (∀l∈L. E' l ∈ E l)"
        by simp
      from bchoice[OF this] guess E' ..
      from bchoice[OF this] obtain L
        where A: "⋀j. j∈K ⟹ A j = (⋂l∈L j. E' j l)"
        and L: "⋀j. j∈K ⟹ finite (L j)" "⋀j. j∈K ⟹ L j ≠ {}" "⋀j. j∈K ⟹ L j ⊆ I j"
        and E': "⋀j l. j∈K ⟹ l ∈ L j ⟹ E' j l ∈ E l"
        by auto

      { fix k l j assume "k ∈ K" "j ∈ K" "l ∈ L j" "l ∈ L k"
        have "k = j"
        proof (rule ccontr)
          assume "k ≠ j"
          with disjoint ‹K ⊆ J› ‹k ∈ K› ‹j ∈ K› have "I k ∩ I j = {}"
            unfolding disjoint_family_on_def by auto
          with L(2,3)[OF ‹j ∈ K›] L(2,3)[OF ‹k ∈ K›]
          show False using ‹l ∈ L k› ‹l ∈ L j› by auto
        qed }
      note L_inj = this

      define k where "k l = (SOME k. k ∈ K ∧ l ∈ L k)" for l
      { fix x j l assume *: "j ∈ K" "l ∈ L j"
        have "k l = j" unfolding k_def
        proof (rule some_equality)
          fix k assume "k ∈ K ∧ l ∈ L k"
          with * L_inj show "k = j" by auto
        qed (insert *, simp) }
      note k_simp[simp] = this
      let ?E' = "λl. E' (k l) l"
      have "prob (⋂j∈K. A j) = prob (⋂l∈(⋃k∈K. L k). ?E' l)"
        by (auto simp: A intro!: arg_cong[where f=prob])
      also have "… = (∏l∈(⋃k∈K. L k). prob (?E' l))"
        using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono)
      also have "… = (∏j∈K. ∏l∈L j. prob (E' j l))"
        using K L L_inj by (subst prod.UNION_disjoint) auto
      also have "… = (∏j∈K. prob (A j))"
        using K L E' by (auto simp add: A intro!: prod.cong indep_setsD[OF indep, symmetric]) blast
      finally show "prob (⋂j∈K. A j) = (∏j∈K. prob (A j))" .
    qed
  next
    fix j assume "j ∈ J"
    show "Int_stable (?E j)"
    proof (rule Int_stableI)
      fix a assume "a ∈ ?E j" then obtain Ka Ea
        where a: "a = (⋂k∈Ka. Ea k)" "finite Ka" "Ka ≠ {}" "Ka ⊆ I j" "⋀k. k∈Ka ⟹ Ea k ∈ E k" by auto
      fix b assume "b ∈ ?E j" then obtain Kb Eb
        where b: "b = (⋂k∈Kb. Eb k)" "finite Kb" "Kb ≠ {}" "Kb ⊆ I j" "⋀k. k∈Kb ⟹ Eb k ∈ E k" by auto
      let ?f = "λk. (if k ∈ Ka ∩ Kb then Ea k ∩ Eb k else if k ∈ Kb then Eb k else if k ∈ Ka then Ea k else {})"
      have "Ka ∪ Kb = (Ka ∩ Kb) ∪ (Kb - Ka) ∪ (Ka - Kb)"
        by blast
      moreover have "(⋂x∈Ka ∩ Kb. Ea x ∩ Eb x) ∩
        (⋂x∈Kb - Ka. Eb x) ∩ (⋂x∈Ka - Kb. Ea x) = (⋂k∈Ka. Ea k) ∩ (⋂k∈Kb. Eb k)"
        by auto
      ultimately have "(⋂k∈Ka ∪ Kb. ?f k) = (⋂k∈Ka. Ea k) ∩ (⋂k∈Kb. Eb k)" (is "?lhs = ?rhs")
        by (simp only: image_Un Inter_Un_distrib) simp
      then have "a ∩ b = (⋂k∈Ka ∪ Kb. ?f k)"
        by (simp only: a(1) b(1))
      with a b ‹j ∈ J› Int_stableD[OF Int_stable] show "a ∩ b ∈ ?E j"
        by (intro CollectI exI[of _ "Ka ∪ Kb"] exI[of _ ?f]) auto
    qed
  qed
  ultimately show ?thesis
    by (simp cong: indep_sets_cong)
qed

lemma (in prob_space) indep_vars_restrict:
  assumes ind: "indep_vars M' X I" and K: "⋀j. j ∈ L ⟹ K j ⊆ I" and J: "disjoint_family_on K L"
  shows "indep_vars (λj. PiM (K j) M') (λj ω. restrict (λi. X i ω) (K j)) L"
  unfolding indep_vars_def
proof safe
  fix j assume "j ∈ L" then show "random_variable (PiM (K j) M') (λω. λi∈K j. X i ω)"
    using K ind by (auto simp: indep_vars_def intro!: measurable_restrict)
next
  have X: "⋀i. i ∈ I ⟹ X i ∈ measurable M (M' i)"
    using ind by (auto simp: indep_vars_def)
  let ?proj = "λj S. {(λω. λi∈K j. X i ω) -` A ∩ space M |A. A ∈ S}"
  let ?UN = "λj. sigma_sets (space M) (⋃i∈K j. { X i -` A ∩ space M| A. A ∈ sets (M' i) })"
  show "indep_sets (λi. sigma_sets (space M) (?proj i (sets (PiM (K i) M')))) L"
  proof (rule indep_sets_mono_sets)
    fix j assume j: "j ∈ L"
    have "sigma_sets (space M) (?proj j (sets (PiM (K j) M'))) =
      sigma_sets (space M) (sigma_sets (space M) (?proj j (prod_algebra (K j) M')))"
      using j K X[THEN measurable_space] unfolding sets_PiM
      by (subst sigma_sets_vimage_commute) (auto simp add: Pi_iff)
    also have "… = sigma_sets (space M) (?proj j (prod_algebra (K j) M'))"
      by (rule sigma_sets_sigma_sets_eq) auto
    also have "… ⊆ ?UN j"
    proof (rule sigma_sets_mono, safe del: disjE elim!: prod_algebraE)
      fix J E assume J: "finite J" "J ≠ {} ∨ K j = {}"  "J ⊆ K j" and E: "∀i. i ∈ J ⟶ E i ∈ sets (M' i)"
      show "(λω. λi∈K j. X i ω) -` prod_emb (K j) M' J (PiE J E) ∩ space M ∈ ?UN j"
      proof cases
        assume "K j = {}" with J show ?thesis
          by (auto simp add: sigma_sets_empty_eq prod_emb_def)
      next
        assume "K j ≠ {}" with J have "J ≠ {}"
          by auto
        { interpret sigma_algebra "space M" "?UN j"
            by (rule sigma_algebra_sigma_sets) auto
          have "⋀A. (⋀i. i ∈ J ⟹ A i ∈ ?UN j) ⟹ INTER J A ∈ ?UN j"
            using ‹finite J› ‹J ≠ {}› by (rule finite_INT) blast }
        note INT = this

        from ‹J ≠ {}› J K E[rule_format, THEN sets.sets_into_space] j
        have "(λω. λi∈K j. X i ω) -` prod_emb (K j) M' J (PiE J E) ∩ space M
          = (⋂i∈J. X i -` E i ∩ space M)"
          apply (subst prod_emb_PiE[OF _ ])
          apply auto []
          apply auto []
          apply (auto simp add: Pi_iff intro!: X[THEN measurable_space])
          apply (erule_tac x=i in ballE)
          apply auto
          done
        also have "… ∈ ?UN j"
          apply (rule INT)
          apply (rule sigma_sets.Basic)
          using ‹J ⊆ K j› E
          apply auto
          done
        finally show ?thesis .
      qed
    qed
    finally show "sigma_sets (space M) (?proj j (sets (PiM (K j) M'))) ⊆ ?UN j" .
  next
    show "indep_sets ?UN L"
    proof (rule indep_sets_collect_sigma)
      show "indep_sets (λi. {X i -` A ∩ space M |A. A ∈ sets (M' i)}) (⋃j∈L. K j)"
      proof (rule indep_sets_mono_index)
        show "indep_sets (λi. {X i -` A ∩ space M |A. A ∈ sets (M' i)}) I"
          using ind unfolding indep_vars_def2 by auto
        show "(⋃l∈L. K l) ⊆ I"
          using K by auto
      qed
    next
      fix l i assume "l ∈ L" "i ∈ K l"
      show "Int_stable {X i -` A ∩ space M |A. A ∈ sets (M' i)}"
        apply (auto simp: Int_stable_def)
        apply (rule_tac x="A ∩ Aa" in exI)
        apply auto
        done
    qed fact
  qed
qed

lemma (in prob_space) indep_var_restrict:
  assumes ind: "indep_vars M' X I" and AB: "A ∩ B = {}" "A ⊆ I" "B ⊆ I"
  shows "indep_var (PiM A M') (λω. restrict (λi. X i ω) A) (PiM B M') (λω. restrict (λi. X i ω) B)"
proof -
  have *:
    "case_bool (PiM A M') (PiM B M') = (λb. PiM (case_bool A B b) M')"
    "case_bool (λω. λi∈A. X i ω) (λω. λi∈B. X i ω) = (λb ω. λi∈case_bool A B b. X i ω)"
    by (simp_all add: fun_eq_iff split: bool.split)
  show ?thesis
    unfolding indep_var_def * using AB
    by (intro indep_vars_restrict[OF ind]) (auto simp: disjoint_family_on_def split: bool.split)
qed

lemma (in prob_space) indep_vars_subset:
  assumes "indep_vars M' X I" "J ⊆ I"
  shows "indep_vars M' X J"
  using assms unfolding indep_vars_def indep_sets_def
  by auto

lemma (in prob_space) indep_vars_cong:
  "I = J ⟹ (⋀i. i ∈ I ⟹ X i = Y i) ⟹ (⋀i. i ∈ I ⟹ M' i = N' i) ⟹ indep_vars M' X I ⟷ indep_vars N' Y J"
  unfolding indep_vars_def2 by (intro conj_cong indep_sets_cong) auto

definition (in prob_space) tail_events where
  "tail_events A = (⋂n. sigma_sets (space M) (UNION {n..} A))"

lemma (in prob_space) tail_events_sets:
  assumes A: "⋀i::nat. A i ⊆ events"
  shows "tail_events A ⊆ events"
proof
  fix X assume X: "X ∈ tail_events A"
  let ?A = "(⋂n. sigma_sets (space M) (UNION {n..} A))"
  from X have "⋀n::nat. X ∈ sigma_sets (space M) (UNION {n..} A)" by (auto simp: tail_events_def)
  from this[of 0] have "X ∈ sigma_sets (space M) (UNION UNIV A)" by simp
  then show "X ∈ events"
    by induct (insert A, auto)
qed

lemma (in prob_space) sigma_algebra_tail_events:
  assumes "⋀i::nat. sigma_algebra (space M) (A i)"
  shows "sigma_algebra (space M) (tail_events A)"
  unfolding tail_events_def
proof (simp add: sigma_algebra_iff2, safe)
  let ?A = "(⋂n. sigma_sets (space M) (UNION {n..} A))"
  interpret A: sigma_algebra "space M" "A i" for i by fact
  { fix X x assume "X ∈ ?A" "x ∈ X"
    then have "⋀n. X ∈ sigma_sets (space M) (UNION {n..} A)" by auto
    from this[of 0] have "X ∈ sigma_sets (space M) (UNION UNIV A)" by simp
    then have "X ⊆ space M"
      by induct (insert A.sets_into_space, auto)
    with ‹x ∈ X› show "x ∈ space M" by auto }
  { fix F :: "nat ⇒ 'a set" and n assume "range F ⊆ ?A"
    then show "(UNION UNIV F) ∈ sigma_sets (space M) (UNION {n..} A)"
      by (intro sigma_sets.Union) auto }
qed (auto intro!: sigma_sets.Compl sigma_sets.Empty)

lemma (in prob_space) kolmogorov_0_1_law:
  fixes A :: "nat ⇒ 'a set set"
  assumes "⋀i::nat. sigma_algebra (space M) (A i)"
  assumes indep: "indep_sets A UNIV"
  and X: "X ∈ tail_events A"
  shows "prob X = 0 ∨ prob X = 1"
proof -
  have A: "⋀i. A i ⊆ events"
    using indep unfolding indep_sets_def by simp

  let ?D = "{D ∈ events. prob (X ∩ D) = prob X * prob D}"
  interpret A: sigma_algebra "space M" "A i" for i by fact
  interpret T: sigma_algebra "space M" "tail_events A"
    by (rule sigma_algebra_tail_events) fact
  have "X ⊆ space M" using T.space_closed X by auto

  have X_in: "X ∈ events"
    using tail_events_sets A X by auto

  interpret D: dynkin_system "space M" ?D
  proof (rule dynkin_systemI)
    fix D assume "D ∈ ?D" then show "D ⊆ space M"
      using sets.sets_into_space by auto
  next
    show "space M ∈ ?D"
      using prob_space ‹X ⊆ space M› by (simp add: Int_absorb2)
  next
    fix A assume A: "A ∈ ?D"
    have "prob (X ∩ (space M - A)) = prob (X - (X ∩ A))"
      using ‹X ⊆ space M› by (auto intro!: arg_cong[where f=prob])
    also have "… = prob X - prob (X ∩ A)"
      using X_in A by (intro finite_measure_Diff) auto
    also have "… = prob X * prob (space M) - prob X * prob A"
      using A prob_space by auto
    also have "… = prob X * prob (space M - A)"
      using X_in A sets.sets_into_space
      by (subst finite_measure_Diff) (auto simp: field_simps)
    finally show "space M - A ∈ ?D"
      using A ‹X ⊆ space M› by auto
  next
    fix F :: "nat ⇒ 'a set" assume dis: "disjoint_family F" and "range F ⊆ ?D"
    then have F: "range F ⊆ events" "⋀i. prob (X ∩ F i) = prob X * prob (F i)"
      by auto
    have "(λi. prob (X ∩ F i)) sums prob (⋃i. X ∩ F i)"
    proof (rule finite_measure_UNION)
      show "range (λi. X ∩ F i) ⊆ events"
        using F X_in by auto
      show "disjoint_family (λi. X ∩ F i)"
        using dis by (rule disjoint_family_on_bisimulation) auto
    qed
    with F have "(λi. prob X * prob (F i)) sums prob (X ∩ (⋃i. F i))"
      by simp
    moreover have "(λi. prob X * prob (F i)) sums (prob X * prob (⋃i. F i))"
      by (intro sums_mult finite_measure_UNION F dis)
    ultimately have "prob (X ∩ (⋃i. F i)) = prob X * prob (⋃i. F i)"
      by (auto dest!: sums_unique)
    with F show "(⋃i. F i) ∈ ?D"
      by auto
  qed

  { fix n
    have "indep_sets (λb. sigma_sets (space M) (⋃m∈case_bool {..n} {Suc n..} b. A m)) UNIV"
    proof (rule indep_sets_collect_sigma)
      have *: "(⋃b. case b of True ⇒ {..n} | False ⇒ {Suc n..}) = UNIV" (is "?U = _")
        by (simp split: bool.split add: set_eq_iff) (metis not_less_eq_eq)
      with indep show "indep_sets A ?U" by simp
      show "disjoint_family (case_bool {..n} {Suc n..})"
        unfolding disjoint_family_on_def by (auto split: bool.split)
      fix m
      show "Int_stable (A m)"
        unfolding Int_stable_def using A.Int by auto
    qed
    also have "(λb. sigma_sets (space M) (⋃m∈case_bool {..n} {Suc n..} b. A m)) =
      case_bool (sigma_sets (space M) (⋃m∈{..n}. A m)) (sigma_sets (space M) (⋃m∈{Suc n..}. A m))"
      by (auto intro!: ext split: bool.split)
    finally have indep: "indep_set (sigma_sets (space M) (⋃m∈{..n}. A m)) (sigma_sets (space M) (⋃m∈{Suc n..}. A m))"
      unfolding indep_set_def by simp

    have "sigma_sets (space M) (⋃m∈{..n}. A m) ⊆ ?D"
    proof (simp add: subset_eq, rule)
      fix D assume D: "D ∈ sigma_sets (space M) (⋃m∈{..n}. A m)"
      have "X ∈ sigma_sets (space M) (⋃m∈{Suc n..}. A m)"
        using X unfolding tail_events_def by simp
      from indep_setD[OF indep D this] indep_setD_ev1[OF indep] D
      show "D ∈ events ∧ prob (X ∩ D) = prob X * prob D"
        by (auto simp add: ac_simps)
    qed }
  then have "(⋃n. sigma_sets (space M) (⋃m∈{..n}. A m)) ⊆ ?D" (is "?A ⊆ _")
    by auto

  note ‹X ∈ tail_events A›
  also {
    have "⋀n. sigma_sets (space M) (⋃i∈{n..}. A i) ⊆ sigma_sets (space M) ?A"
      by (intro sigma_sets_subseteq UN_mono) auto
   then have "tail_events A ⊆ sigma_sets (space M) ?A"
      unfolding tail_events_def by auto }
  also have "sigma_sets (space M) ?A = dynkin (space M) ?A"
  proof (rule sigma_eq_dynkin)
    { fix B n assume "B ∈ sigma_sets (space M) (⋃m∈{..n}. A m)"
      then have "B ⊆ space M"
        by induct (insert A sets.sets_into_space[of _ M], auto) }
    then show "?A ⊆ Pow (space M)" by auto
    show "Int_stable ?A"
    proof (rule Int_stableI)
      fix a assume "a ∈ ?A" then guess n .. note a = this
      fix b assume "b ∈ ?A" then guess m .. note b = this
      interpret Amn: sigma_algebra "space M" "sigma_sets (space M) (⋃i∈{..max m n}. A i)"
        using A sets.sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
      have "sigma_sets (space M) (⋃i∈{..n}. A i) ⊆ sigma_sets (space M) (⋃i∈{..max m n}. A i)"
        by (intro sigma_sets_subseteq UN_mono) auto
      with a have "a ∈ sigma_sets (space M) (⋃i∈{..max m n}. A i)" by auto
      moreover
      have "sigma_sets (space M) (⋃i∈{..m}. A i) ⊆ sigma_sets (space M) (⋃i∈{..max m n}. A i)"
        by (intro sigma_sets_subseteq UN_mono) auto
      with b have "b ∈ sigma_sets (space M) (⋃i∈{..max m n}. A i)" by auto
      ultimately have "a ∩ b ∈ sigma_sets (space M) (⋃i∈{..max m n}. A i)"
        using Amn.Int[of a b] by simp
      then show "a ∩ b ∈ (⋃n. sigma_sets (space M) (⋃i∈{..n}. A i))" by auto
    qed
  qed
  also have "dynkin (space M) ?A ⊆ ?D"
    using ‹?A ⊆ ?D› by (auto intro!: D.dynkin_subset)
  finally show ?thesis by auto
qed

lemma (in prob_space) borel_0_1_law:
  fixes F :: "nat ⇒ 'a set"
  assumes F2: "indep_events F UNIV"
  shows "prob (⋂n. ⋃m∈{n..}. F m) = 0 ∨ prob (⋂n. ⋃m∈{n..}. F m) = 1"
proof (rule kolmogorov_0_1_law[of "λi. sigma_sets (space M) { F i }"])
  have F1: "range F ⊆ events"
    using F2 by (simp add: indep_events_def subset_eq)
  { fix i show "sigma_algebra (space M) (sigma_sets (space M) {F i})"
      using sigma_algebra_sigma_sets[of "{F i}" "space M"] F1 sets.sets_into_space
      by auto }
  show "indep_sets (λi. sigma_sets (space M) {F i}) UNIV"
  proof (rule indep_sets_sigma)
    show "indep_sets (λi. {F i}) UNIV"
      unfolding indep_events_def_alt[symmetric] by fact
    fix i show "Int_stable {F i}"
      unfolding Int_stable_def by simp
  qed
  let ?Q = "λn. ⋃i∈{n..}. F i"
  show "(⋂n. ⋃m∈{n..}. F m) ∈ tail_events (λi. sigma_sets (space M) {F i})"
    unfolding tail_events_def
  proof
    fix j
    interpret S: sigma_algebra "space M" "sigma_sets (space M) (⋃i∈{j..}. sigma_sets (space M) {F i})"
      using order_trans[OF F1 sets.space_closed]
      by (intro sigma_algebra_sigma_sets) (simp add: sigma_sets_singleton subset_eq)
    have "(⋂n. ?Q n) = (⋂n∈{j..}. ?Q n)"
      by (intro decseq_SucI INT_decseq_offset UN_mono) auto
    also have "… ∈ sigma_sets (space M) (⋃i∈{j..}. sigma_sets (space M) {F i})"
      using order_trans[OF F1 sets.space_closed]
      by (safe intro!: S.countable_INT S.countable_UN)
         (auto simp: sigma_sets_singleton intro!: sigma_sets.Basic bexI)
    finally show "(⋂n. ?Q n) ∈ sigma_sets (space M) (⋃i∈{j..}. sigma_sets (space M) {F i})"
      by simp
  qed
qed

lemma (in prob_space) borel_0_1_law_AE:
  fixes P :: "nat ⇒ 'a ⇒ bool"
  assumes "indep_events (λm. {x∈space M. P m x}) UNIV" (is "indep_events ?P _")
  shows "(AE x in M. infinite {m. P m x}) ∨ (AE x in M. finite {m. P m x})"
proof -
  have [measurable]: "⋀m. {x∈space M. P m x} ∈ sets M"
    using assms by (auto simp: indep_events_def)
  have *: "(⋂n. ⋃m∈{n..}. {x ∈ space M. P m x}) ∈ events"
    by simp
  from assms have "prob (⋂n. ⋃m∈{n..}. ?P m) = 0 ∨ prob (⋂n. ⋃m∈{n..}. ?P m) = 1"
    by (rule borel_0_1_law)
  also have "prob (⋂n. ⋃m∈{n..}. ?P m) = 1 ⟷ (AE x in M. infinite {m. P m x})"
    using * by (simp add: prob_eq_1)
      (simp add: Bex_def infinite_nat_iff_unbounded_le)
  also have "prob (⋂n. ⋃m∈{n..}. ?P m) = 0 ⟷ (AE x in M. finite {m. P m x})"
    using * by (simp add: prob_eq_0)
      (auto simp add: Ball_def finite_nat_iff_bounded not_less [symmetric])
  finally show ?thesis
    by blast
qed

lemma (in prob_space) indep_sets_finite:
  assumes I: "I ≠ {}" "finite I"
    and F: "⋀i. i ∈ I ⟹ F i ⊆ events" "⋀i. i ∈ I ⟹ space M ∈ F i"
  shows "indep_sets F I ⟷ (∀A∈Pi I F. prob (⋂j∈I. A j) = (∏j∈I. prob (A j)))"
proof
  assume *: "indep_sets F I"
  from I show "∀A∈Pi I F. prob (⋂j∈I. A j) = (∏j∈I. prob (A j))"
    by (intro indep_setsD[OF *] ballI) auto
next
  assume indep: "∀A∈Pi I F. prob (⋂j∈I. A j) = (∏j∈I. prob (A j))"
  show "indep_sets F I"
  proof (rule indep_setsI[OF F(1)])
    fix A J assume J: "J ≠ {}" "J ⊆ I" "finite J"
    assume A: "∀j∈J. A j ∈ F j"
    let ?A = "λj. if j ∈ J then A j else space M"
    have "prob (⋂j∈I. ?A j) = prob (⋂j∈J. A j)"
      using subset_trans[OF F(1) sets.space_closed] J A
      by (auto intro!: arg_cong[where f=prob] split: if_split_asm) blast
    also
    from A F have "(λj. if j ∈ J then A j else space M) ∈ Pi I F" (is "?A ∈ _")
      by (auto split: if_split_asm)
    with indep have "prob (⋂j∈I. ?A j) = (∏j∈I. prob (?A j))"
      by auto
    also have "… = (∏j∈J. prob (A j))"
      unfolding if_distrib prod.If_cases[OF ‹finite I›]
      using prob_space ‹J ⊆ I› by (simp add: Int_absorb1 prod.neutral_const)
    finally show "prob (⋂j∈J. A j) = (∏j∈J. prob (A j))" ..
  qed
qed

lemma (in prob_space) indep_vars_finite:
  fixes I :: "'i set"
  assumes I: "I ≠ {}" "finite I"
    and M': "⋀i. i ∈ I ⟹ sets (M' i) = sigma_sets (space (M' i)) (E i)"
    and rv: "⋀i. i ∈ I ⟹ random_variable (M' i) (X i)"
    and Int_stable: "⋀i. i ∈ I ⟹ Int_stable (E i)"
    and space: "⋀i. i ∈ I ⟹ space (M' i) ∈ E i" and closed: "⋀i. i ∈ I ⟹ E i ⊆ Pow (space (M' i))"
  shows "indep_vars M' X I ⟷
    (∀A∈(Π i∈I. E i). prob (⋂j∈I. X j -` A j ∩ space M) = (∏j∈I. prob (X j -` A j ∩ space M)))"
proof -
  from rv have X: "⋀i. i ∈ I ⟹ X i ∈ space M → space (M' i)"
    unfolding measurable_def by simp

  { fix i assume "i∈I"
    from closed[OF ‹i ∈ I›]
    have "sigma_sets (space M) {X i -` A ∩ space M |A. A ∈ sets (M' i)}
      = sigma_sets (space M) {X i -` A ∩ space M |A. A ∈ E i}"
      unfolding sigma_sets_vimage_commute[OF X, OF ‹i ∈ I›, symmetric] M'[OF ‹i ∈ I›]
      by (subst sigma_sets_sigma_sets_eq) auto }
  note sigma_sets_X = this

  { fix i assume "i∈I"
    have "Int_stable {X i -` A ∩ space M |A. A ∈ E i}"
    proof (rule Int_stableI)
      fix a assume "a ∈ {X i -` A ∩ space M |A. A ∈ E i}"
      then obtain A where "a = X i -` A ∩ space M" "A ∈ E i" by auto
      moreover
      fix b assume "b ∈ {X i -` A ∩ space M |A. A ∈ E i}"
      then obtain B where "b = X i -` B ∩ space M" "B ∈ E i" by auto
      moreover
      have "(X i -` A ∩ space M) ∩ (X i -` B ∩ space M) = X i -` (A ∩ B) ∩ space M" by auto
      moreover note Int_stable[OF ‹i ∈ I›]
      ultimately
      show "a ∩ b ∈ {X i -` A ∩ space M |A. A ∈ E i}"
        by (auto simp del: vimage_Int intro!: exI[of _ "A ∩ B"] dest: Int_stableD)
    qed }
  note indep_sets_X = indep_sets_sigma_sets_iff[OF this]

  { fix i assume "i ∈ I"
    { fix A assume "A ∈ E i"
      with M'[OF ‹i ∈ I›] have "A ∈ sets (M' i)" by auto
      moreover
      from rv[OF ‹i∈I›] have "X i ∈ measurable M (M' i)" by auto
      ultimately
      have "X i -` A ∩ space M ∈ sets M" by (auto intro: measurable_sets) }
    with X[OF ‹i∈I›] space[OF ‹i∈I›]
    have "{X i -` A ∩ space M |A. A ∈ E i} ⊆ events"
      "space M ∈ {X i -` A ∩ space M |A. A ∈ E i}"
      by (auto intro!: exI[of _ "space (M' i)"]) }
  note indep_sets_finite_X = indep_sets_finite[OF I this]

  have "(∀A∈Π i∈I. {X i -` A ∩ space M |A. A ∈ E i}. prob (INTER I A) = (∏j∈I. prob (A j))) =
    (∀A∈Π i∈I. E i. prob ((⋂j∈I. X j -` A j) ∩ space M) = (∏x∈I. prob (X x -` A x ∩ space M)))"
    (is "?L = ?R")
  proof safe
    fix A assume ?L and A: "A ∈ (Π i∈I. E i)"
    from ‹?L›[THEN bspec, of "λi. X i -` A i ∩ space M"] A ‹I ≠ {}›
    show "prob ((⋂j∈I. X j -` A j) ∩ space M) = (∏x∈I. prob (X x -` A x ∩ space M))"
      by (auto simp add: Pi_iff)
  next
    fix A assume ?R and A: "A ∈ (Π i∈I. {X i -` A ∩ space M |A. A ∈ E i})"
    from A have "∀i∈I. ∃B. A i = X i -` B ∩ space M ∧ B ∈ E i" by auto
    from bchoice[OF this] obtain B where B: "∀i∈I. A i = X i -` B i ∩ space M"
      "B ∈ (Π i∈I. E i)" by auto
    from ‹?R›[THEN bspec, OF B(2)] B(1) ‹I ≠ {}›
    show "prob (INTER I A) = (∏j∈I. prob (A j))"
      by simp
  qed
  then show ?thesis using ‹I ≠ {}›
    by (simp add: rv indep_vars_def indep_sets_X sigma_sets_X indep_sets_finite_X cong: indep_sets_cong)
qed

lemma (in prob_space) indep_vars_compose:
  assumes "indep_vars M' X I"
  assumes rv: "⋀i. i ∈ I ⟹ Y i ∈ measurable (M' i) (N i)"
  shows "indep_vars N (λi. Y i ∘ X i) I"
  unfolding indep_vars_def
proof
  from rv ‹indep_vars M' X I›
  show "∀i∈I. random_variable (N i) (Y i ∘ X i)"
    by (auto simp: indep_vars_def)

  have "indep_sets (λi. sigma_sets (space M) {X i -` A ∩ space M |A. A ∈ sets (M' i)}) I"
    using ‹indep_vars M' X I› by (simp add: indep_vars_def)
  then show "indep_sets (λi. sigma_sets (space M) {(Y i ∘ X i) -` A ∩ space M |A. A ∈ sets (N i)}) I"
  proof (rule indep_sets_mono_sets)
    fix i assume "i ∈ I"
    with ‹indep_vars M' X I› have X: "X i ∈ space M → space (M' i)"
      unfolding indep_vars_def measurable_def by auto
    { fix A assume "A ∈ sets (N i)"
      then have "∃B. (Y i ∘ X i) -` A ∩ space M = X i -` B ∩ space M ∧ B ∈ sets (M' i)"
        by (intro exI[of _ "Y i -` A ∩ space (M' i)"])
           (auto simp: vimage_comp intro!: measurable_sets rv ‹i ∈ I› funcset_mem[OF X]) }
    then show "sigma_sets (space M) {(Y i ∘ X i) -` A ∩ space M |A. A ∈ sets (N i)} ⊆
      sigma_sets (space M) {X i -` A ∩ space M |A. A ∈ sets (M' i)}"
      by (intro sigma_sets_subseteq) (auto simp: vimage_comp)
  qed
qed

lemma (in prob_space) indep_vars_compose2:
  assumes "indep_vars M' X I"
  assumes rv: "⋀i. i ∈ I ⟹ Y i ∈ measurable (M' i) (N i)"
  shows "indep_vars N (λi x. Y i (X i x)) I"
  using indep_vars_compose [OF assms] by (simp add: comp_def)

lemma (in prob_space) indep_var_compose:
  assumes "indep_var M1 X1 M2 X2" "Y1 ∈ measurable M1 N1" "Y2 ∈ measurable M2 N2"
  shows "indep_var N1 (Y1 ∘ X1) N2 (Y2 ∘ X2)"
proof -
  have "indep_vars (case_bool N1 N2) (λb. case_bool Y1 Y2 b ∘ case_bool X1 X2 b) UNIV"
    using assms
    by (intro indep_vars_compose[where M'="case_bool M1 M2"])
       (auto simp: indep_var_def split: bool.split)
  also have "(λb. case_bool Y1 Y2 b ∘ case_bool X1 X2 b) = case_bool (Y1 ∘ X1) (Y2 ∘ X2)"
    by (simp add: fun_eq_iff split: bool.split)
  finally show ?thesis
    unfolding indep_var_def .
qed

lemma (in prob_space) indep_vars_Min:
  fixes X :: "'i ⇒ 'a ⇒ real"
  assumes I: "finite I" "i ∉ I" and indep: "indep_vars (λ_. borel) X (insert i I)"
  shows "indep_var borel (X i) borel (λω. Min ((λi. X i ω)`I))"
proof -
  have "indep_var
    borel ((λf. f i) ∘ (λω. restrict (λi. X i ω) {i}))
    borel ((λf. Min (f`I)) ∘ (λω. restrict (λi. X i ω) I))"
    using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] borel_measurable_Min) auto
  also have "((λf. f i) ∘ (λω. restrict (λi. X i ω) {i})) = X i"
    by auto
  also have "((λf. Min (f`I)) ∘ (λω. restrict (λi. X i ω) I)) = (λω. Min ((λi. X i ω)`I))"
    by (auto cong: rev_conj_cong)
  finally show ?thesis
    unfolding indep_var_def .
qed

lemma (in prob_space) indep_vars_sum:
  fixes X :: "'i ⇒ 'a ⇒ real"
  assumes I: "finite I" "i ∉ I" and indep: "indep_vars (λ_. borel) X (insert i I)"
  shows "indep_var borel (X i) borel (λω. ∑i∈I. X i ω)"
proof -
  have "indep_var
    borel ((λf. f i) ∘ (λω. restrict (λi. X i ω) {i}))
    borel ((λf. ∑i∈I. f i) ∘ (λω. restrict (λi. X i ω) I))"
    using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] ) auto
  also have "((λf. f i) ∘ (λω. restrict (λi. X i ω) {i})) = X i"
    by auto
  also have "((λf. ∑i∈I. f i) ∘ (λω. restrict (λi. X i ω) I)) = (λω. ∑i∈I. X i ω)"
    by (auto cong: rev_conj_cong)
  finally show ?thesis .
qed

lemma (in prob_space) indep_vars_prod:
  fixes X :: "'i ⇒ 'a ⇒ real"
  assumes I: "finite I" "i ∉ I" and indep: "indep_vars (λ_. borel) X (insert i I)"
  shows "indep_var borel (X i) borel (λω. ∏i∈I. X i ω)"
proof -
  have "indep_var
    borel ((λf. f i) ∘ (λω. restrict (λi. X i ω) {i}))
    borel ((λf. ∏i∈I. f i) ∘ (λω. restrict (λi. X i ω) I))"
    using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] ) auto
  also have "((λf. f i) ∘ (λω. restrict (λi. X i ω) {i})) = X i"
    by auto
  also have "((λf. ∏i∈I. f i) ∘ (λω. restrict (λi. X i ω) I)) = (λω. ∏i∈I. X i ω)"
    by (auto cong: rev_conj_cong)
  finally show ?thesis .
qed

lemma (in prob_space) indep_varsD_finite:
  assumes X: "indep_vars M' X I"
  assumes I: "I ≠ {}" "finite I" "⋀i. i ∈ I ⟹ A i ∈ sets (M' i)"
  shows "prob (⋂i∈I. X i -` A i ∩ space M) = (∏i∈I. prob (X i -` A i ∩ space M))"
proof (rule indep_setsD)
  show "indep_sets (λi. sigma_sets (space M) {X i -` A ∩ space M |A. A ∈ sets (M' i)}) I"
    using X by (auto simp: indep_vars_def)
  show "I ⊆ I" "I ≠ {}" "finite I" using I by auto
  show "∀i∈I. X i -` A i ∩ space M ∈ sigma_sets (space M) {X i -` A ∩ space M |A. A ∈ sets (M' i)}"
    using I by auto
qed

lemma (in prob_space) indep_varsD:
  assumes X: "indep_vars M' X I"
  assumes I: "J ≠ {}" "finite J" "J ⊆ I" "⋀i. i ∈ J ⟹ A i ∈ sets (M' i)"
  shows "prob (⋂i∈J. X i -` A i ∩ space M) = (∏i∈J. prob (X i -` A i ∩ space M))"
proof (rule indep_setsD)
  show "indep_sets (λi. sigma_sets (space M) {X i -` A ∩ space M |A. A ∈ sets (M' i)}) I"
    using X by (auto simp: indep_vars_def)
  show "∀i∈J. X i -` A i ∩ space M ∈ sigma_sets (space M) {X i -` A ∩ space M |A. A ∈ sets (M' i)}"
    using I by auto
qed fact+

lemma (in prob_space) indep_vars_iff_distr_eq_PiM:
  fixes I :: "'i set" and X :: "'i ⇒ 'a ⇒ 'b"
  assumes "I ≠ {}"
  assumes rv: "⋀i. random_variable (M' i) (X i)"
  shows "indep_vars M' X I ⟷
    distr M (ΠM i∈I. M' i) (λx. λi∈I. X i x) = (ΠM i∈I. distr M (M' i) (X i))"
proof -
  let ?P = M i∈I. M' i"
  let ?X = "λx. λi∈I. X i x"
  let ?D = "distr M ?P ?X"
  have X: "random_variable ?P ?X" by (intro measurable_restrict rv)
  interpret D: prob_space ?D by (intro prob_space_distr X)

  let ?D' = "λi. distr M (M' i) (X i)"
  let ?P' = M i∈I. distr M (M' i) (X i)"
  interpret D': prob_space "?D' i" for i by (intro prob_space_distr rv)
  interpret P: product_prob_space ?D' I ..

  show ?thesis
  proof
    assume "indep_vars M' X I"
    show "?D = ?P'"
    proof (rule measure_eqI_generator_eq)
      show "Int_stable (prod_algebra I M')"
        by (rule Int_stable_prod_algebra)
      show "prod_algebra I M' ⊆ Pow (space ?P)"
        using prod_algebra_sets_into_space by (simp add: space_PiM)
      show "sets ?D = sigma_sets (space ?P) (prod_algebra I M')"
        by (simp add: sets_PiM space_PiM)
      show "sets ?P' = sigma_sets (space ?P) (prod_algebra I M')"
        by (simp add: sets_PiM space_PiM cong: prod_algebra_cong)
      let ?A = "λi. ΠE i∈I. space (M' i)"
      show "range ?A ⊆ prod_algebra I M'" "(⋃i. ?A i) = space (PiM I M')"
        by (auto simp: space_PiM intro!: space_in_prod_algebra cong: prod_algebra_cong)
      { fix i show "emeasure ?D (ΠE i∈I. space (M' i)) ≠ ∞" by auto }
    next
      fix E assume E: "E ∈ prod_algebra I M'"
      from prod_algebraE[OF E] guess J Y . note J = this

      from E have "E ∈ sets ?P" by (auto simp: sets_PiM)
      then have "emeasure ?D E = emeasure M (?X -` E ∩ space M)"
        by (simp add: emeasure_distr X)
      also have "?X -` E ∩ space M = (⋂i∈J. X i -` Y i ∩ space M)"
        using J ‹I ≠ {}› measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: if_split_asm)
      also have "emeasure M (⋂i∈J. X i -` Y i ∩ space M) = (∏ i∈J. emeasure M (X i -` Y i ∩ space M))"
        using ‹indep_vars M' X I› J ‹I ≠ {}› using indep_varsD[of M' X I J]
        by (auto simp: emeasure_eq_measure prod_ennreal measure_nonneg prod_nonneg)
      also have "… = (∏ i∈J. emeasure (?D' i) (Y i))"
        using rv J by (simp add: emeasure_distr)
      also have "… = emeasure ?P' E"
        using P.emeasure_PiM_emb[of J Y] J by (simp add: prod_emb_def)
      finally show "emeasure ?D E = emeasure ?P' E" .
    qed
  next
    assume "?D = ?P'"
    show "indep_vars M' X I" unfolding indep_vars_def
    proof (intro conjI indep_setsI ballI rv)
      fix i show "sigma_sets (space M) {X i -` A ∩ space M |A. A ∈ sets (M' i)} ⊆ events"
        by (auto intro!: sets.sigma_sets_subset measurable_sets rv)
    next
      fix J Y' assume J: "J ≠ {}" "J ⊆ I" "finite J"
      assume Y': "∀j∈J. Y' j ∈ sigma_sets (space M) {X j -` A ∩ space M |A. A ∈ sets (M' j)}"
      have "∀j∈J. ∃Y. Y' j = X j -` Y ∩ space M ∧ Y ∈ sets (M' j)"
      proof
        fix j assume "j ∈ J"
        from Y'[rule_format, OF this] rv[of j]
        show "∃Y. Y' j = X j -` Y ∩ space M ∧ Y ∈ sets (M' j)"
          by (subst (asm) sigma_sets_vimage_commute[symmetric, of _ _ "space (M' j)"])
             (auto dest: measurable_space simp: sets.sigma_sets_eq)
      qed
      from bchoice[OF this] obtain Y where
        Y: "⋀j. j ∈ J ⟹ Y' j = X j -` Y j ∩ space M" "⋀j. j ∈ J ⟹ Y j ∈ sets (M' j)" by auto
      let ?E = "prod_emb I M' J (PiE J Y)"
      from Y have "(⋂j∈J. Y' j) = ?X -` ?E ∩ space M"
        using J ‹I ≠ {}› measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: if_split_asm)
      then have "emeasure M (⋂j∈J. Y' j) = emeasure M (?X -` ?E ∩ space M)"
        by simp
      also have "… = emeasure ?D ?E"
        using Y  J by (intro emeasure_distr[symmetric] X sets_PiM_I) auto
      also have "… = emeasure ?P' ?E"
        using ‹?D = ?P'› by simp
      also have "… = (∏ i∈J. emeasure (?D' i) (Y i))"
        using P.emeasure_PiM_emb[of J Y] J Y by (simp add: prod_emb_def)
      also have "… = (∏ i∈J. emeasure M (Y' i))"
        using rv J Y by (simp add: emeasure_distr)
      finally have "emeasure M (⋂j∈J. Y' j) = (∏ i∈J. emeasure M (Y' i))" .
      then show "prob (⋂j∈J. Y' j) = (∏ i∈J. prob (Y' i))"
        by (auto simp: emeasure_eq_measure prod_ennreal measure_nonneg prod_nonneg)
    qed
  qed
qed

lemma (in prob_space) indep_varD:
  assumes indep: "indep_var Ma A Mb B"
  assumes sets: "Xa ∈ sets Ma" "Xb ∈ sets Mb"
  shows "prob ((λx. (A x, B x)) -` (Xa × Xb) ∩ space M) =
    prob (A -` Xa ∩ space M) * prob (B -` Xb ∩ space M)"
proof -
  have "prob ((λx. (A x, B x)) -` (Xa × Xb) ∩ space M) =
    prob (⋂i∈UNIV. (case_bool A B i -` case_bool Xa Xb i ∩ space M))"
    by (auto intro!: arg_cong[where f=prob] simp: UNIV_bool)
  also have "… = (∏i∈UNIV. prob (case_bool A B i -` case_bool Xa Xb i ∩ space M))"
    using indep unfolding indep_var_def
    by (rule indep_varsD) (auto split: bool.split intro: sets)
  also have "… = prob (A -` Xa ∩ space M) * prob (B -` Xb ∩ space M)"
    unfolding UNIV_bool by simp
  finally show ?thesis .
qed

lemma (in prob_space) prob_indep_random_variable:
  assumes ind[simp]: "indep_var N X N Y"
  assumes [simp]: "A ∈ sets N" "B ∈ sets N"
  shows "𝒫(x in M. X x ∈ A ∧ Y x ∈ B) = 𝒫(x in M. X x ∈ A) * 𝒫(x in M. Y x ∈ B)"
proof-
  have  " 𝒫(x in M. (X x)∈A ∧  (Y x)∈ B ) = prob ((λx. (X x, Y x)) -` (A × B) ∩ space M)"
    by (auto intro!: arg_cong[where f= prob])
  also have "...=  prob (X -` A ∩ space M) * prob (Y -` B ∩ space M)"
    by (auto intro!: indep_varD[where Ma=N and Mb=N])
  also have "... = 𝒫(x in M. X x ∈ A) * 𝒫(x in M. Y x ∈ B)"
    by (auto intro!: arg_cong2[where f= "( * )"] arg_cong[where f= prob])
  finally show ?thesis .
qed

lemma (in prob_space)
  assumes "indep_var S X T Y"
  shows indep_var_rv1: "random_variable S X"
    and indep_var_rv2: "random_variable T Y"
proof -
  have "∀i∈UNIV. random_variable (case_bool S T i) (case_bool X Y i)"
    using assms unfolding indep_var_def indep_vars_def by auto
  then show "random_variable S X" "random_variable T Y"
    unfolding UNIV_bool by auto
qed

lemma (in prob_space) indep_var_distribution_eq:
  "indep_var S X T Y ⟷ random_variable S X ∧ random_variable T Y ∧
    distr M S X ⨂M distr M T Y = distr M (S ⨂M T) (λx. (X x, Y x))" (is "_ ⟷ _ ∧ _ ∧ ?S ⨂M ?T = ?J")
proof safe
  assume "indep_var S X T Y"
  then show rvs: "random_variable S X" "random_variable T Y"
    by (blast dest: indep_var_rv1 indep_var_rv2)+
  then have XY: "random_variable (S ⨂M T) (λx. (X x, Y x))"
    by (rule measurable_Pair)

  interpret X: prob_space ?S by (rule prob_space_distr) fact
  interpret Y: prob_space ?T by (rule prob_space_distr) fact
  interpret XY: pair_prob_space ?S ?T ..
  show "?S ⨂M ?T = ?J"
  proof (rule pair_measure_eqI)
    show "sigma_finite_measure ?S" ..
    show "sigma_finite_measure ?T" ..

    fix A B assume A: "A ∈ sets ?S" and B: "B ∈ sets ?T"
    have "emeasure ?J (A × B) = emeasure M ((λx. (X x, Y x)) -` (A × B) ∩ space M)"
      using A B by (intro emeasure_distr[OF XY]) auto
    also have "… = emeasure M (X -` A ∩ space M) * emeasure M (Y -` B ∩ space M)"
      using indep_varD[OF ‹indep_var S X T Y›, of A B] A B
      by (simp add: emeasure_eq_measure measure_nonneg ennreal_mult)
    also have "… = emeasure ?S A * emeasure ?T B"
      using rvs A B by (simp add: emeasure_distr)
    finally show "emeasure ?S A * emeasure ?T B = emeasure ?J (A × B)" by simp
  qed simp
next
  assume rvs: "random_variable S X" "random_variable T Y"
  then have XY: "random_variable (S ⨂M T) (λx. (X x, Y x))"
    by (rule measurable_Pair)

  let ?S = "distr M S X" and ?T = "distr M T Y"
  interpret X: prob_space ?S by (rule prob_space_distr) fact
  interpret Y: prob_space ?T by (rule prob_space_distr) fact
  interpret XY: pair_prob_space ?S ?T ..

  assume "?S ⨂M ?T = ?J"

  { fix S and X
    have "Int_stable {X -` A ∩ space M |A. A ∈ sets S}"
    proof (safe intro!: Int_stableI)
      fix A B assume "A ∈ sets S" "B ∈ sets S"
      then show "∃C. (X -` A ∩ space M) ∩ (X -` B ∩ space M) = (X -` C ∩ space M) ∧ C ∈ sets S"
        by (intro exI[of _ "A ∩ B"]) auto
    qed }
  note Int_stable = this

  show "indep_var S X T Y" unfolding indep_var_eq
  proof (intro conjI indep_set_sigma_sets Int_stable rvs)
    show "indep_set {X -` A ∩ space M |A. A ∈ sets S} {Y -` A ∩ space M |A. A ∈ sets T}"
    proof (safe intro!: indep_setI)
      { fix A assume "A ∈ sets S" then show "X -` A ∩ space M ∈ sets M"
        using ‹X ∈ measurable M S› by (auto intro: measurable_sets) }
      { fix A assume "A ∈ sets T" then show "Y -` A ∩ space M ∈ sets M"
        using ‹Y ∈ measurable M T› by (auto intro: measurable_sets) }
    next
      fix A B assume ab: "A ∈ sets S" "B ∈ sets T"
      then have "prob ((X -` A ∩ space M) ∩ (Y -` B ∩ space M)) = emeasure ?J (A × B)"
        using XY by (auto simp add: emeasure_distr emeasure_eq_measure measure_nonneg intro!: arg_cong[where f="prob"])
      also have "… = emeasure (?S ⨂M ?T) (A × B)"
        unfolding ‹?S ⨂M ?T = ?J› ..
      also have "… = emeasure ?S A * emeasure ?T B"
        using ab by (simp add: Y.emeasure_pair_measure_Times)
      finally show "prob ((X -` A ∩ space M) ∩ (Y -` B ∩ space M)) =
        prob (X -` A ∩ space M) * prob (Y -` B ∩ space M)"
        using rvs ab by (simp add: emeasure_eq_measure emeasure_distr measure_nonneg ennreal_mult[symmetric])
    qed
  qed
qed

lemma (in prob_space) distributed_joint_indep:
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
  assumes X: "distributed M S X Px" and Y: "distributed M T Y Py"
  assumes indep: "indep_var S X T Y"
  shows "distributed M (S ⨂M T) (λx. (X x, Y x)) (λ(x, y). Px x * Py y)"
  using indep_var_distribution_eq[of S X T Y] indep
  by (intro distributed_joint_indep'[OF S T X Y]) auto

lemma (in prob_space) indep_vars_nn_integral:
  assumes I: "finite I" "indep_vars (λ_. borel) X I" "⋀i ω. i ∈ I ⟹ 0 ≤ X i ω"
  shows "(∫+ω. (∏i∈I. X i ω) ∂M) = (∏i∈I. ∫+ω. X i ω ∂M)"
proof cases
  assume "I ≠ {}"
  define Y where [abs_def]: "Y i ω = (if i ∈ I then X i ω else 0)" for i ω
  { fix i have "i ∈ I ⟹ random_variable borel (X i)"
    using I(2) by (cases "i∈I") (auto simp: indep_vars_def) }
  note rv_X = this

  { fix i have "random_variable borel (Y i)"
    using I(2) by (cases "i∈I") (auto simp: Y_def rv_X) }
  note rv_Y = this[measurable]

  interpret Y: prob_space "distr M borel (Y i)" for i
    using I(2) by (cases "i ∈ I") (auto intro!: prob_space_distr simp: indep_vars_def prob_space_return)
  interpret product_sigma_finite "λi. distr M borel (Y i)"
    ..

  have indep_Y: "indep_vars (λi. borel) Y I"
    by (rule indep_vars_cong[THEN iffD1, OF _ _ _ I(2)]) (auto simp: Y_def)

  have "(∫+ω. (∏i∈I. X i ω) ∂M) = (∫+ω. (∏i∈I. Y i ω) ∂M)"
    using I(3) by (auto intro!: nn_integral_cong prod.cong simp add: Y_def max_def)
  also have "… = (∫+ω. (∏i∈I. ω i) ∂distr M (PiM I (λi. borel)) (λx. λi∈I. Y i x))"
    by (subst nn_integral_distr) auto
  also have "… = (∫+ω. (∏i∈I. ω i) ∂PiM I (λi. distr M borel (Y i)))"
    unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF ‹I ≠ {}› rv_Y indep_Y] ..
  also have "… = (∏i∈I. (∫+ω. ω ∂distr M borel (Y i)))"
    by (rule product_nn_integral_prod) (auto intro: ‹finite I›)
  also have "… = (∏i∈I. ∫+ω. X i ω ∂M)"
    by (intro prod.cong nn_integral_cong) (auto simp: nn_integral_distr Y_def rv_X)
  finally show ?thesis .
qed (simp add: emeasure_space_1)

lemma (in prob_space)
  fixes X :: "'i ⇒ 'a ⇒ 'b::{real_normed_field, banach, second_countable_topology}"
  assumes I: "finite I" "indep_vars (λ_. borel) X I" "⋀i. i ∈ I ⟹ integrable M (X i)"
  shows indep_vars_lebesgue_integral: "(∫ω. (∏i∈I. X i ω) ∂M) = (∏i∈I. ∫ω. X i ω ∂M)" (is ?eq)
    and indep_vars_integrable: "integrable M (λω. (∏i∈I. X i ω))" (is ?int)
proof (induct rule: case_split)
  assume "I ≠ {}"
  define Y where [abs_def]: "Y i ω = (if i ∈ I then X i ω else 0)" for i ω
  { fix i have "i ∈ I ⟹ random_variable borel (X i)"
    using I(2) by (cases "i∈I") (auto simp: indep_vars_def) }
  note rv_X = this[measurable]

  { fix i have "random_variable borel (Y i)"
    using I(2) by (cases "i∈I") (auto simp: Y_def rv_X) }
  note rv_Y = this[measurable]

  { fix i have "integrable M (Y i)"
    using I(3) by (cases "i∈I") (auto simp: Y_def) }
  note int_Y = this

  interpret Y: prob_space "distr M borel (Y i)" for i
    using I(2) by (cases "i ∈ I") (auto intro!: prob_space_distr simp: indep_vars_def prob_space_return)
  interpret product_sigma_finite "λi. distr M borel (Y i)"
    ..

  have indep_Y: "indep_vars (λi. borel) Y I"
    by (rule indep_vars_cong[THEN iffD1, OF _ _ _ I(2)]) (auto simp: Y_def)

  have "(∫ω. (∏i∈I. X i ω) ∂M) = (∫ω. (∏i∈I. Y i ω) ∂M)"
    using I(3) by (simp add: Y_def)
  also have "… = (∫ω. (∏i∈I. ω i) ∂distr M (PiM I (λi. borel)) (λx. λi∈I. Y i x))"
    by (subst integral_distr) auto
  also have "… = (∫ω. (∏i∈I. ω i) ∂PiM I (λi. distr M borel (Y i)))"
    unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF ‹I ≠ {}› rv_Y indep_Y] ..
  also have "… = (∏i∈I. (∫ω. ω ∂distr M borel (Y i)))"
    by (rule product_integral_prod) (auto intro: ‹finite I› simp: integrable_distr_eq int_Y)
  also have "… = (∏i∈I. ∫ω. X i ω ∂M)"
    by (intro prod.cong integral_cong)
       (auto simp: integral_distr Y_def rv_X)
  finally show ?eq .

  have "integrable (distr M (PiM I (λi. borel)) (λx. λi∈I. Y i x)) (λω. (∏i∈I. ω i))"
    unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF ‹I ≠ {}› rv_Y indep_Y]
    by (intro product_integrable_prod[OF ‹finite I›])
       (simp add: integrable_distr_eq int_Y)
  then show ?int
    by (simp add: integrable_distr_eq Y_def)
qed (simp_all add: prob_space)

lemma (in prob_space)
  fixes X1 X2 :: "'a ⇒ 'b::{real_normed_field, banach, second_countable_topology}"
  assumes "indep_var borel X1 borel X2" "integrable M X1" "integrable M X2"
  shows indep_var_lebesgue_integral: "(∫ω. X1 ω * X2 ω ∂M) = (∫ω. X1 ω ∂M) * (∫ω. X2 ω ∂M)" (is ?eq)
    and indep_var_integrable: "integrable M (λω. X1 ω * X2 ω)" (is ?int)
unfolding indep_var_def
proof -
  have *: "(λω. X1 ω * X2 ω) = (λω. ∏i∈UNIV. (case_bool X1 X2 i ω))"
    by (simp add: UNIV_bool mult.commute)
  have **: "(λ _. borel) = case_bool borel borel"
    by (rule ext, metis (full_types) bool.simps(3) bool.simps(4))
  show ?eq
    apply (subst *)
    apply (subst indep_vars_lebesgue_integral)
    apply (auto)
    apply (subst **, subst indep_var_def [symmetric], rule assms)
    apply (simp split: bool.split add: assms)
    by (simp add: UNIV_bool mult.commute)
  show ?int
    apply (subst *)
    apply (rule indep_vars_integrable)
    apply auto
    apply (subst **, subst indep_var_def [symmetric], rule assms)
    by (simp split: bool.split add: assms)
qed

end