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src/HOL/Probability/ex/Measure_Not_CCC.thy
changeset 59144 c9b75c03de3c
child 61424 c3658c18b7bc 
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/ex/Measure_Not_CCC.thy	Wed Dec 17 16:10:30 2014 +0100
@@ -0,0 +1,171 @@
+(* Author: Johannes Hölzl <hoelzl@in.tum.de> *)
+
+section \<open>The Category of Measurable Spaces is not Cartesian Closed\<close>
+
+theory Measure_Not_CCC
+  imports "~~/src/HOL/Probability/Probability" "~~/src/HOL/Library/ContNotDenum"
+begin
+
+text \<open>
+  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 "\<lambda>x y. x = y"} is $\mathbb{R}$-$\mathbb{B}^\mathbb{C}$-measurable,
+  but @{term "\<lambda>(x, y). x = y"} is not $(\mathbb{R} \times \mathbb{C})$-$\mathbb{B}$-measurable.
+\<close>
+
+definition COCOUNT :: "real measure" where
+  "COCOUNT = sigma UNIV {{x} | x. True}"
+
+abbreviation POW :: "real measure" where
+  "POW \<equiv> count_space UNIV"
+
+abbreviation BOOL :: "bool measure" where
+  "BOOL \<equiv> count_space UNIV"
+
+lemma measurable_const_iff: "(\<lambda>x. c) \<in> measurable A B \<longleftrightarrow> (space A = {} \<or> c \<in> space B)"
+  by (auto simp: measurable_def)
+
+lemma measurable_eq[measurable]: "(op = x) \<in> measurable COCOUNT BOOL"
+  unfolding pred_def by (auto simp: COCOUNT_def)
+
+lemma COCOUNT_eq: "A \<in> COCOUNT \<longleftrightarrow> countable A \<or> countable (UNIV - A)"
+proof
+  fix A assume "A \<in> COCOUNT"
+  then have "A \<in> sigma_sets UNIV {{x} | x. True}"
+    by (auto simp: COCOUNT_def)
+  then show "countable A \<or> countable (UNIV - A)"
+  proof induction
+    case (Union F)
+    moreover
+    { fix i assume "countable (UNIV - F i)"
+      then have "countable (UNIV - (\<Union>i. F i))"
+        by (rule countable_subset[rotated]) auto }
+    ultimately show "countable (\<Union>i. F i) \<or> countable (UNIV - (\<Union>i. F i))"
+      by blast
+  qed (auto simp: Diff_Diff_Int)
+next
+  assume "countable A \<or> countable (UNIV - A)"
+  moreover
+  { fix A :: "real set" assume A: "countable A"
+    have "A = (\<Union>a\<in>A. {a})"
+      by auto
+    also have "\<dots> \<in> COCOUNT"
+      by (intro sets.countable_UN' A) (auto simp: COCOUNT_def) 
+    finally have "A \<in> COCOUNT" . }
+  note A = this
+  note A[of A]
+  moreover
+  { assume "countable (UNIV - A)"
+    with A have "space COCOUNT - (UNIV - A) \<in> COCOUNT" by simp
+    then have "A \<in> COCOUNT"
+      by (auto simp: COCOUNT_def Diff_Diff_Int) }
+  ultimately show "A \<in> COCOUNT" 
+    by blast
+qed
+
+lemma pair_COCOUNT:
+  assumes A: "A \<in> sets (COCOUNT \<Otimes>\<^sub>M M)"
+  shows "\<exists>J F X. X \<in> sets M \<and> F \<in> J \<rightarrow> sets M \<and> countable J \<and> A = (UNIV - J) \<times> X \<union> (SIGMA j:J. F j)"
+  using A unfolding sets_pair_measure
+proof induction
+  case (Basic X)
+  then obtain a b where X: "X = a \<times> b" and b: "b \<in> sets M" and a: "countable a \<or> countable (UNIV - a)"
+    by (auto simp: COCOUNT_eq)
+  from a show ?case
+  proof 
+    assume "countable a" with X b show ?thesis
+      by (intro exI[of _ a] exI[of _ "\<lambda>_. b"] exI[of _ "{}"]) auto
+  next
+    assume "countable (UNIV - a)" with X b show ?thesis
+      by (intro exI[of _ "UNIV - a"] exI[of _ "\<lambda>_. {}"] exI[of _ "b"]) auto
+  qed
+next
+  case Empty then show ?case
+    by (intro exI[of _ "{}"] exI[of _ "\<lambda>_. {}"] exI[of _ "{}"]) auto
+next
+  case (Compl A)
+  then obtain J F X where XFJ: "X \<in> sets M" "F \<in> J \<rightarrow> sets M" "countable J"
+    and A: "A = (UNIV - J) \<times> X \<union> Sigma J F"
+    by auto
+  have *: "space COCOUNT \<times> space M - A = (UNIV - J) \<times> (space M - X) \<union> (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 _ "\<lambda>j. space M - F j"]) auto
+next
+  case (Union A)
+  obtain J F X where XFJ: "\<And>i. X i \<in> sets M" "\<And>i. F i \<in> J i \<rightarrow> sets M" "\<And>i. countable (J i)"
+    and A_eq: "A = (\<lambda>i. (UNIV - J i) \<times> X i \<union> Sigma (J i) (F i))"
+    unfolding fun_eq_iff using Union.IH by metis
+  show ?case
+  proof (intro exI conjI)
+    def G \<equiv> "\<lambda>j. (\<Union>i. if j \<in> J i then F i j else X i)"
+    show "(\<Union>i. X i) \<in> sets M" "countable (\<Union>i. J i)" "G \<in> (\<Union>i. J i) \<rightarrow> sets M"
+      using XFJ by (auto simp: G_def Pi_iff)
+    show "UNION UNIV A = (UNIV - (\<Union>i. J i)) \<times> (\<Union>i. X i) \<union> (SIGMA j:\<Union>i. J i. \<Union>i. if j \<in> J i then F i j else X i)"
+      unfolding A_eq by (auto split: split_if_asm)
+  qed
+qed
+
+context
+  fixes EXP :: "(real \<Rightarrow> bool) measure"
+  assumes eq: "\<And>P. split P \<in> measurable (POW \<Otimes>\<^sub>M COCOUNT) BOOL \<longleftrightarrow> P \<in> measurable POW EXP"
+begin
+
+lemma space_EXP: "space EXP = measurable COCOUNT BOOL"
+proof -
+  { fix f 
+    have "f \<in> space EXP \<longleftrightarrow> (\<lambda>(a, b). f b) \<in> measurable (POW \<Otimes>\<^sub>M COCOUNT) BOOL"
+      using eq[of "\<lambda>x. f"] by (simp add: measurable_const_iff)
+    also have "\<dots> \<longleftrightarrow> f \<in> measurable COCOUNT BOOL"
+      by auto
+    finally have "f \<in> space EXP \<longleftrightarrow> f \<in> measurable COCOUNT BOOL" . }
+  then show ?thesis by auto
+qed
+
+lemma measurable_eq_EXP: "(\<lambda>x y. x = y) \<in> measurable POW EXP"
+  unfolding measurable_def by (auto simp: space_EXP)
+
+lemma measurable_eq_pair: "(\<lambda>(y, x). x = y) \<in> measurable (COCOUNT \<Otimes>\<^sub>M POW) BOOL"
+  using measurable_eq_EXP unfolding eq[symmetric]
+  by (subst measurable_pair_swap_iff) simp
+
+lemma ce: False
+proof -
+  have "{(y, x) \<in> space (COCOUNT \<Otimes>\<^sub>M POW). x = y} \<in> sets (COCOUNT \<Otimes>\<^sub>M POW)"
+    using measurable_eq_pair unfolding pred_def by (simp add: split_beta')
+  also have "{(y, x) \<in> space (COCOUNT \<Otimes>\<^sub>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) \<times> X \<union> (SIGMA j:J. F j)"
+    using pair_COCOUNT[of "SIGMA j:UNIV. {j}" POW] by auto
+  have X_single: "\<And>x. x \<notin> J \<Longrightarrow> 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 "\<exists>A. finite A \<and> card A = 2 \<and> A \<subseteq> UNIV - J"
+    by (rule infinite_arbitrarily_large)
+  then obtain i j where ij: "i \<in> UNIV - J" "j \<in> UNIV - J" "i \<noteq> j"
+    by (auto simp add: card_Suc_eq numeral_2_eq_2)
+  have "{(i, i), (j, j)} \<subseteq> (SIGMA j:UNIV. {j})" by auto
+  with ij X_single[of i] X_single[of j] show False
+    by auto
+qed
+
+end
+
+corollary "\<not> (\<exists>EXP. \<forall>P. split P \<in> measurable (POW \<Otimes>\<^sub>M COCOUNT) BOOL \<longleftrightarrow> P \<in> measurable POW EXP)"
+  using ce by blast
+
+end
+
isabelle