Theory Measure_Not_CCC

theory Measure_Not_CCC
imports Probability
(* Author: Johannes Hölzl <> *)

section ‹The Category of Measurable Spaces is not Cartesian Closed›

theory Measure_Not_CCC
  imports Probability

text ‹
  We show that the category of measurable spaces with measurable functions as morphisms is not a
  Cartesian closed category. While the category has products and terminal objects, the exponential
  does not exist for each pair of measurable spaces.

  We show that the exponential $\mathbb{B}^\mathbb{C}$ does not exist, where $\mathbb{B}$ is the
  discrete measurable space on boolean values, and $\mathbb{C}$ is the $\sigma$-algebra consisting
  of all countable and co-countable real sets. We also define $\mathbb{R}$ to be the discrete
  measurable space on the reals.

  Now, the diagonal predicate @{term "λx y. x = y"} is $\mathbb{R}$-$\mathbb{B}^\mathbb{C}$-measurable,
  but @{term "λ(x, y). x = y"} is not $(\mathbb{R} \times \mathbb{C})$-$\mathbb{B}$-measurable.

definition COCOUNT :: "real measure" where
  "COCOUNT = sigma UNIV {{x} | x. True}"

abbreviation POW :: "real measure" where
  "POW ≡ count_space UNIV"

abbreviation BOOL :: "bool measure" where
  "BOOL ≡ count_space UNIV"

lemma measurable_const_iff: "(λx. c) ∈ measurable A B ⟷ (space A = {} ∨ c ∈ space B)"
  by (auto simp: measurable_def)

lemma measurable_eq[measurable]: "(op = x) ∈ measurable COCOUNT BOOL"
  unfolding pred_def by (auto simp: COCOUNT_def)

lemma COCOUNT_eq: "A ∈ COCOUNT ⟷ countable A ∨ countable (UNIV - A)"
  fix A assume "A ∈ COCOUNT"
  then have "A ∈ sigma_sets UNIV {{x} | x. True}"
    by (auto simp: COCOUNT_def)
  then show "countable A ∨ countable (UNIV - A)"
  proof induction
    case (Union F)
    { fix i assume "countable (UNIV - F i)"
      then have "countable (UNIV - (⋃i. F i))"
        by (rule countable_subset[rotated]) auto }
    ultimately show "countable (⋃i. F i) ∨ countable (UNIV - (⋃i. F i))"
      by blast
  qed (auto simp: Diff_Diff_Int)
  assume "countable A ∨ countable (UNIV - A)"
  { fix A :: "real set" assume A: "countable A"
    have "A = (⋃a∈A. {a})"
      by auto
    also have "… ∈ COCOUNT"
      by (intro sets.countable_UN' A) (auto simp: COCOUNT_def) 
    finally have "A ∈ COCOUNT" . }
  note A = this
  note A[of A]
  { assume "countable (UNIV - A)"
    with A have "space COCOUNT - (UNIV - A) ∈ COCOUNT" by simp
    then have "A ∈ COCOUNT"
      by (auto simp: COCOUNT_def Diff_Diff_Int) }
  ultimately show "A ∈ COCOUNT" 
    by blast

lemma pair_COCOUNT:
  assumes A: "A ∈ sets (COCOUNT ⨂M M)"
  shows "∃J F X. X ∈ sets M ∧ F ∈ J → sets M ∧ countable J ∧ A = (UNIV - J) × X ∪ (SIGMA j:J. F j)"
  using A unfolding sets_pair_measure
proof induction
  case (Basic X)
  then obtain a b where X: "X = a × b" and b: "b ∈ sets M" and a: "countable a ∨ countable (UNIV - a)"
    by (auto simp: COCOUNT_eq)
  from a show ?case
    assume "countable a" with X b show ?thesis
      by (intro exI[of _ a] exI[of _ "λ_. b"] exI[of _ "{}"]) auto
    assume "countable (UNIV - a)" with X b show ?thesis
      by (intro exI[of _ "UNIV - a"] exI[of _ "λ_. {}"] exI[of _ "b"]) auto
  case Empty then show ?case
    by (intro exI[of _ "{}"] exI[of _ "λ_. {}"] exI[of _ "{}"]) auto
  case (Compl A)
  then obtain J F X where XFJ: "X ∈ sets M" "F ∈ J → sets M" "countable J"
    and A: "A = (UNIV - J) × X ∪ Sigma J F"
    by auto
  have *: "space COCOUNT × space M - A = (UNIV - J) × (space M - X) ∪ (SIGMA j:J. space M - F j)"
    unfolding A by (auto simp: COCOUNT_def)
  show ?case
    using XFJ unfolding *
    by (intro exI[of _ J] exI[of _ "space M - X"] exI[of _ "λj. space M - F j"]) auto
  case (Union A)
  obtain J F X where XFJ: "⋀i. X i ∈ sets M" "⋀i. F i ∈ J i → sets M" "⋀i. countable (J i)"
    and A_eq: "A = (λi. (UNIV - J i) × X i ∪ Sigma (J i) (F i))"
    unfolding fun_eq_iff using Union.IH by metis
  show ?case
  proof (intro exI conjI)
    define G where "G j = (⋃i. if j ∈ J i then F i j else X i)" for j
    show "(⋃i. X i) ∈ sets M" "countable (⋃i. J i)" "G ∈ (⋃i. J i) → sets M"
      using XFJ by (auto simp: G_def Pi_iff)
    show "UNION UNIV A = (UNIV - (⋃i. J i)) × (⋃i. X i) ∪ (SIGMA j:⋃i. J i. ⋃i. if j ∈ J i then F i j else X i)"
      unfolding A_eq by (auto split: if_split_asm)

  fixes EXP :: "(real ⇒ bool) measure"
  assumes eq: "⋀P. case_prod P ∈ measurable (POW ⨂M COCOUNT) BOOL ⟷ P ∈ measurable POW EXP"

lemma space_EXP: "space EXP = measurable COCOUNT BOOL"
proof -
  { fix f 
    have "f ∈ space EXP ⟷ (λ(a, b). f b) ∈ measurable (POW ⨂M COCOUNT) BOOL"
      using eq[of "λx. f"] by (simp add: measurable_const_iff)
    also have "… ⟷ f ∈ measurable COCOUNT BOOL"
      by auto
    finally have "f ∈ space EXP ⟷ f ∈ measurable COCOUNT BOOL" . }
  then show ?thesis by auto

lemma measurable_eq_EXP: "(λx y. x = y) ∈ measurable POW EXP"
  unfolding measurable_def by (auto simp: space_EXP)

lemma measurable_eq_pair: "(λ(y, x). x = y) ∈ measurable (COCOUNT ⨂M POW) BOOL"
  using measurable_eq_EXP unfolding eq[symmetric]
  by (subst measurable_pair_swap_iff) simp

lemma ce: False
proof -
  have "{(y, x) ∈ space (COCOUNT ⨂M POW). x = y} ∈ sets (COCOUNT ⨂M POW)"
    using measurable_eq_pair unfolding pred_def by (simp add: split_beta')
  also have "{(y, x) ∈ space (COCOUNT ⨂M POW). x = y} = (SIGMA j:UNIV. {j})"
    by (auto simp: space_pair_measure COCOUNT_def)
  finally obtain X F J where "countable (J::real set)"
    and eq: "(SIGMA j:UNIV. {j}) = (UNIV - J) × X ∪ (SIGMA j:J. F j)"
    using pair_COCOUNT[of "SIGMA j:UNIV. {j}" POW] by auto
  have X_single: "⋀x. x ∉ J ⟹ X = {x}"
    using eq[unfolded set_eq_iff] by force
  have "uncountable (UNIV - J)"
    using ‹countable J› uncountable_UNIV_real uncountable_minus_countable by blast
  then have "infinite (UNIV - J)"
    by (auto intro: countable_finite)
  then have "∃A. finite A ∧ card A = 2 ∧ A ⊆ UNIV - J"
    by (rule infinite_arbitrarily_large)
  then obtain i j where ij: "i ∈ UNIV - J" "j ∈ UNIV - J" "i ≠ j"
    by (auto simp add: card_Suc_eq numeral_2_eq_2)
  have "{(i, i), (j, j)} ⊆ (SIGMA j:UNIV. {j})" by auto
  with ij X_single[of i] X_single[of j] show False
    by auto


corollary "¬ (∃EXP. ∀P. case_prod P ∈ measurable (POW ⨂M COCOUNT) BOOL ⟷ P ∈ measurable POW EXP)"
  using ce by blast