src/HOL/Probability/ex/Measure_Not_CCC.thy
author haftmann
Tue Oct 13 09:21:15 2015 +0200 (2015-10-13)
changeset 61424 c3658c18b7bc
parent 59144 c9b75c03de3c
child 61808 fc1556774cfe
permissions -rw-r--r--
prod_case as canonical name for product type eliminator
     1 (* Author: Johannes Hölzl <hoelzl@in.tum.de> *)
     2 
     3 section \<open>The Category of Measurable Spaces is not Cartesian Closed\<close>
     4 
     5 theory Measure_Not_CCC
     6   imports "~~/src/HOL/Probability/Probability" "~~/src/HOL/Library/ContNotDenum"
     7 begin
     8 
     9 text \<open>
    10   We show that the category of measurable spaces with measurable functions as morphisms is not a
    11   Cartesian closed category. While the category has products and terminal objects, the exponential
    12   does not exist for each pair of measurable spaces.
    13 
    14   We show that the exponential $\mathbb{B}^\mathbb{C}$ does not exist, where $\mathbb{B}$ is the
    15   discrete measurable space on boolean values, and $\mathbb{C}$ is the $\sigma$-algebra consisting
    16   of all countable and co-countable real sets. We also define $\mathbb{R}$ to be the discrete
    17   measurable space on the reals.
    18 
    19   Now, the diagonal predicate @{term "\<lambda>x y. x = y"} is $\mathbb{R}$-$\mathbb{B}^\mathbb{C}$-measurable,
    20   but @{term "\<lambda>(x, y). x = y"} is not $(\mathbb{R} \times \mathbb{C})$-$\mathbb{B}$-measurable.
    21 \<close>
    22 
    23 definition COCOUNT :: "real measure" where
    24   "COCOUNT = sigma UNIV {{x} | x. True}"
    25 
    26 abbreviation POW :: "real measure" where
    27   "POW \<equiv> count_space UNIV"
    28 
    29 abbreviation BOOL :: "bool measure" where
    30   "BOOL \<equiv> count_space UNIV"
    31 
    32 lemma measurable_const_iff: "(\<lambda>x. c) \<in> measurable A B \<longleftrightarrow> (space A = {} \<or> c \<in> space B)"
    33   by (auto simp: measurable_def)
    34 
    35 lemma measurable_eq[measurable]: "(op = x) \<in> measurable COCOUNT BOOL"
    36   unfolding pred_def by (auto simp: COCOUNT_def)
    37 
    38 lemma COCOUNT_eq: "A \<in> COCOUNT \<longleftrightarrow> countable A \<or> countable (UNIV - A)"
    39 proof
    40   fix A assume "A \<in> COCOUNT"
    41   then have "A \<in> sigma_sets UNIV {{x} | x. True}"
    42     by (auto simp: COCOUNT_def)
    43   then show "countable A \<or> countable (UNIV - A)"
    44   proof induction
    45     case (Union F)
    46     moreover
    47     { fix i assume "countable (UNIV - F i)"
    48       then have "countable (UNIV - (\<Union>i. F i))"
    49         by (rule countable_subset[rotated]) auto }
    50     ultimately show "countable (\<Union>i. F i) \<or> countable (UNIV - (\<Union>i. F i))"
    51       by blast
    52   qed (auto simp: Diff_Diff_Int)
    53 next
    54   assume "countable A \<or> countable (UNIV - A)"
    55   moreover
    56   { fix A :: "real set" assume A: "countable A"
    57     have "A = (\<Union>a\<in>A. {a})"
    58       by auto
    59     also have "\<dots> \<in> COCOUNT"
    60       by (intro sets.countable_UN' A) (auto simp: COCOUNT_def) 
    61     finally have "A \<in> COCOUNT" . }
    62   note A = this
    63   note A[of A]
    64   moreover
    65   { assume "countable (UNIV - A)"
    66     with A have "space COCOUNT - (UNIV - A) \<in> COCOUNT" by simp
    67     then have "A \<in> COCOUNT"
    68       by (auto simp: COCOUNT_def Diff_Diff_Int) }
    69   ultimately show "A \<in> COCOUNT" 
    70     by blast
    71 qed
    72 
    73 lemma pair_COCOUNT:
    74   assumes A: "A \<in> sets (COCOUNT \<Otimes>\<^sub>M M)"
    75   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)"
    76   using A unfolding sets_pair_measure
    77 proof induction
    78   case (Basic X)
    79   then obtain a b where X: "X = a \<times> b" and b: "b \<in> sets M" and a: "countable a \<or> countable (UNIV - a)"
    80     by (auto simp: COCOUNT_eq)
    81   from a show ?case
    82   proof 
    83     assume "countable a" with X b show ?thesis
    84       by (intro exI[of _ a] exI[of _ "\<lambda>_. b"] exI[of _ "{}"]) auto
    85   next
    86     assume "countable (UNIV - a)" with X b show ?thesis
    87       by (intro exI[of _ "UNIV - a"] exI[of _ "\<lambda>_. {}"] exI[of _ "b"]) auto
    88   qed
    89 next
    90   case Empty then show ?case
    91     by (intro exI[of _ "{}"] exI[of _ "\<lambda>_. {}"] exI[of _ "{}"]) auto
    92 next
    93   case (Compl A)
    94   then obtain J F X where XFJ: "X \<in> sets M" "F \<in> J \<rightarrow> sets M" "countable J"
    95     and A: "A = (UNIV - J) \<times> X \<union> Sigma J F"
    96     by auto
    97   have *: "space COCOUNT \<times> space M - A = (UNIV - J) \<times> (space M - X) \<union> (SIGMA j:J. space M - F j)"
    98     unfolding A by (auto simp: COCOUNT_def)
    99   show ?case
   100     using XFJ unfolding *
   101     by (intro exI[of _ J] exI[of _ "space M - X"] exI[of _ "\<lambda>j. space M - F j"]) auto
   102 next
   103   case (Union A)
   104   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)"
   105     and A_eq: "A = (\<lambda>i. (UNIV - J i) \<times> X i \<union> Sigma (J i) (F i))"
   106     unfolding fun_eq_iff using Union.IH by metis
   107   show ?case
   108   proof (intro exI conjI)
   109     def G \<equiv> "\<lambda>j. (\<Union>i. if j \<in> J i then F i j else X i)"
   110     show "(\<Union>i. X i) \<in> sets M" "countable (\<Union>i. J i)" "G \<in> (\<Union>i. J i) \<rightarrow> sets M"
   111       using XFJ by (auto simp: G_def Pi_iff)
   112     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)"
   113       unfolding A_eq by (auto split: split_if_asm)
   114   qed
   115 qed
   116 
   117 context
   118   fixes EXP :: "(real \<Rightarrow> bool) measure"
   119   assumes eq: "\<And>P. case_prod P \<in> measurable (POW \<Otimes>\<^sub>M COCOUNT) BOOL \<longleftrightarrow> P \<in> measurable POW EXP"
   120 begin
   121 
   122 lemma space_EXP: "space EXP = measurable COCOUNT BOOL"
   123 proof -
   124   { fix f 
   125     have "f \<in> space EXP \<longleftrightarrow> (\<lambda>(a, b). f b) \<in> measurable (POW \<Otimes>\<^sub>M COCOUNT) BOOL"
   126       using eq[of "\<lambda>x. f"] by (simp add: measurable_const_iff)
   127     also have "\<dots> \<longleftrightarrow> f \<in> measurable COCOUNT BOOL"
   128       by auto
   129     finally have "f \<in> space EXP \<longleftrightarrow> f \<in> measurable COCOUNT BOOL" . }
   130   then show ?thesis by auto
   131 qed
   132 
   133 lemma measurable_eq_EXP: "(\<lambda>x y. x = y) \<in> measurable POW EXP"
   134   unfolding measurable_def by (auto simp: space_EXP)
   135 
   136 lemma measurable_eq_pair: "(\<lambda>(y, x). x = y) \<in> measurable (COCOUNT \<Otimes>\<^sub>M POW) BOOL"
   137   using measurable_eq_EXP unfolding eq[symmetric]
   138   by (subst measurable_pair_swap_iff) simp
   139 
   140 lemma ce: False
   141 proof -
   142   have "{(y, x) \<in> space (COCOUNT \<Otimes>\<^sub>M POW). x = y} \<in> sets (COCOUNT \<Otimes>\<^sub>M POW)"
   143     using measurable_eq_pair unfolding pred_def by (simp add: split_beta')
   144   also have "{(y, x) \<in> space (COCOUNT \<Otimes>\<^sub>M POW). x = y} = (SIGMA j:UNIV. {j})"
   145     by (auto simp: space_pair_measure COCOUNT_def)
   146   finally obtain X F J where "countable (J::real set)"
   147     and eq: "(SIGMA j:UNIV. {j}) = (UNIV - J) \<times> X \<union> (SIGMA j:J. F j)"
   148     using pair_COCOUNT[of "SIGMA j:UNIV. {j}" POW] by auto
   149   have X_single: "\<And>x. x \<notin> J \<Longrightarrow> X = {x}"
   150     using eq[unfolded set_eq_iff] by force
   151   
   152   have "uncountable (UNIV - J)"
   153     using `countable J` uncountable_UNIV_real uncountable_minus_countable by blast
   154   then have "infinite (UNIV - J)"
   155     by (auto intro: countable_finite)
   156   then have "\<exists>A. finite A \<and> card A = 2 \<and> A \<subseteq> UNIV - J"
   157     by (rule infinite_arbitrarily_large)
   158   then obtain i j where ij: "i \<in> UNIV - J" "j \<in> UNIV - J" "i \<noteq> j"
   159     by (auto simp add: card_Suc_eq numeral_2_eq_2)
   160   have "{(i, i), (j, j)} \<subseteq> (SIGMA j:UNIV. {j})" by auto
   161   with ij X_single[of i] X_single[of j] show False
   162     by auto
   163 qed
   164 
   165 end
   166 
   167 corollary "\<not> (\<exists>EXP. \<forall>P. case_prod P \<in> measurable (POW \<Otimes>\<^sub>M COCOUNT) BOOL \<longleftrightarrow> P \<in> measurable POW EXP)"
   168   using ce by blast
   169 
   170 end
   171