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author | hoelzl |

Wed, 17 Dec 2014 16:10:30 +0100 | |

changeset 59144 | c9b75c03de3c |

parent 59143 | 15c342a9a8e0 |

child 59145 | 0e304b1568a5 |

unfortunately, there is no general function space in the measurable spaces

--- a/src/HOL/Probability/document/root.tex Mon Dec 15 07:20:49 2014 +0100 +++ b/src/HOL/Probability/document/root.tex Wed Dec 17 16:10:30 2014 +0100 @@ -3,6 +3,7 @@ \documentclass[11pt,a4paper]{article} \usepackage{graphicx,isabelle,isabellesym,latexsym,textcomp} +\usepackage{amssymb} \usepackage[only,bigsqcap]{stmaryrd} \usepackage[utf8]{inputenc} \usepackage{pdfsetup}

--- /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 +