add restrict_space measure
Tue, 12 Nov 2013 19:28:56 +0100
changeset 54417 dbb8ecfe1337
parent 54416 7fb88ed6ff3c
child 54418 3b8e33d1a39a
add restrict_space measure
--- a/src/HOL/Library/FuncSet.thy	Tue Nov 12 19:28:55 2013 +0100
+++ b/src/HOL/Library/FuncSet.thy	Tue Nov 12 19:28:56 2013 +0100
@@ -183,18 +183,20 @@
 subsection{*Bounded Abstraction: @{term restrict}*}
-lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
+lemma restrict_in_funcset: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> (\<lambda>x\<in>A. f x) \<in> A \<rightarrow> B"
   by (simp add: Pi_def restrict_def)
-lemma restrictI[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
+lemma restrictI[intro!]: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> (\<lambda>x\<in>A. f x) \<in> Pi A B"
   by (simp add: Pi_def restrict_def)
-lemma restrict_apply [simp]:
-    "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)"
+lemma restrict_apply[simp]: "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)"
   by (simp add: restrict_def)
+lemma restrict_apply': "x \<in> A \<Longrightarrow> (\<lambda>y\<in>A. f y) x = f x"
+  by simp
 lemma restrict_ext:
-    "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
+    "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
   by (simp add: fun_eq_iff Pi_def restrict_def)
 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
@@ -364,6 +366,9 @@
 lemma PiE_empty_domain[simp]: "PiE {} T = {%x. undefined}"
   unfolding PiE_def by simp
+lemma PiE_UNIV_domain: "PiE UNIV T = Pi UNIV T"
+  unfolding PiE_def by simp
 lemma PiE_empty_range[simp]: "i \<in> I \<Longrightarrow> F i = {} \<Longrightarrow> (PIE i:I. F i) = {}"
   unfolding PiE_def by auto
--- a/src/HOL/Probability/Lebesgue_Integration.thy	Tue Nov 12 19:28:55 2013 +0100
+++ b/src/HOL/Probability/Lebesgue_Integration.thy	Tue Nov 12 19:28:56 2013 +0100
@@ -2521,6 +2521,36 @@
   "f \<in> borel_measurable (count_space A)"
   by simp
+section {* Measures with Restricted Space *}
+lemma positive_integral_restrict_space:
+  assumes \<Omega>: "\<Omega> \<in> sets M" and f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "\<And>x. x \<in> space M - \<Omega> \<Longrightarrow> f x = 0"
+  shows "positive_integral (restrict_space M \<Omega>) f = positive_integral M f"
+using f proof (induct rule: borel_measurable_induct)
+  case (cong f g) then show ?case
+    using positive_integral_cong[of M f g] positive_integral_cong[of "restrict_space M \<Omega>" f g]
+      sets.sets_into_space[OF `\<Omega> \<in> sets M`]
+    by (simp add: subset_eq space_restrict_space)
+  case (set A)
+  then have "A \<subseteq> \<Omega>"
+    unfolding indicator_eq_0_iff by (auto dest: sets.sets_into_space)
+  with set `\<Omega> \<in> sets M` sets.sets_into_space[OF `\<Omega> \<in> sets M`] show ?case
+    by (subst positive_integral_indicator')
+       (auto simp add: sets_restrict_space_iff space_restrict_space
+                  emeasure_restrict_space Int_absorb2
+                dest: sets.sets_into_space)
+  case (mult f c) then show ?case
+    by (cases "c = 0") (simp_all add: measurable_restrict_space1 \<Omega> positive_integral_cmult)
+  case (add f g) then show ?case
+    by (simp add: measurable_restrict_space1 \<Omega> positive_integral_add ereal_add_nonneg_eq_0_iff)
+  case (seq F) then show ?case
+    by (auto simp add: SUP_eq_iff measurable_restrict_space1 \<Omega> positive_integral_monotone_convergence_SUP)
 section {* Measure spaces with an associated density *}
 definition density :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
--- a/src/HOL/Probability/Measure_Space.thy	Tue Nov 12 19:28:55 2013 +0100
+++ b/src/HOL/Probability/Measure_Space.thy	Tue Nov 12 19:28:56 2013 +0100
@@ -1118,6 +1118,10 @@
     and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng"
   by (auto simp: measurable_def)
+lemma distr_cong:
+  "M = K \<Longrightarrow> sets N = sets L \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> distr M N f = distr K L g"
+  using sets_eq_imp_space_eq[of N L] by (simp add: distr_def Int_def cong: rev_conj_cong)
 lemma emeasure_distr:
   fixes f :: "'a \<Rightarrow> 'b"
   assumes f: "f \<in> measurable M N" and A: "A \<in> sets N"
@@ -1649,5 +1653,50 @@
   show "sigma_finite_measure (count_space A)" ..
+section {* Measure restricted to space *}
+lemma emeasure_restrict_space:
+  assumes "\<Omega> \<in> sets M" "A \<subseteq> \<Omega>"
+  shows "emeasure (restrict_space M \<Omega>) A = emeasure M A"
+proof cases
+  assume "A \<in> sets M"
+  have "emeasure (restrict_space M \<Omega>) A = emeasure M (A \<inter> \<Omega>)"
+  proof (rule emeasure_measure_of[OF restrict_space_def])
+    show "op \<inter> \<Omega> ` sets M \<subseteq> Pow \<Omega>" "A \<in> sets (restrict_space M \<Omega>)"
+      using assms `A \<in> sets M` by (auto simp: sets_restrict_space sets.sets_into_space)
+    show "positive (sets (restrict_space M \<Omega>)) (\<lambda>A. emeasure M (A \<inter> \<Omega>))"
+      by (auto simp: positive_def emeasure_nonneg)
+    show "countably_additive (sets (restrict_space M \<Omega>)) (\<lambda>A. emeasure M (A \<inter> \<Omega>))"
+    proof (rule countably_additiveI)
+      fix A :: "nat \<Rightarrow> _" assume "range A \<subseteq> sets (restrict_space M \<Omega>)" "disjoint_family A"
+      with assms have "\<And>i. A i \<in> sets M" "\<And>i. A i \<subseteq> space M" "disjoint_family A"
+        by (auto simp: sets_restrict_space_iff subset_eq dest: sets.sets_into_space)
+      with `\<Omega> \<in> sets M` show "(\<Sum>i. emeasure M (A i \<inter> \<Omega>)) = emeasure M ((\<Union>i. A i) \<inter> \<Omega>)"
+        by (subst suminf_emeasure) (auto simp: disjoint_family_subset)
+    qed
+  qed
+  with `A \<subseteq> \<Omega>` show ?thesis
+    by (simp add: Int_absorb2)
+  assume "A \<notin> sets M"
+  moreover with assms have "A \<notin> sets (restrict_space M \<Omega>)"
+    by (simp add: sets_restrict_space_iff)
+  ultimately show ?thesis
+    by (simp add: emeasure_notin_sets)
+lemma restrict_count_space:
+  assumes "A \<subseteq> B" shows "restrict_space (count_space B) A = count_space A"
+proof (rule measure_eqI)
+  show "sets (restrict_space (count_space B) A) = sets (count_space A)"
+    using `A \<subseteq> B` by (subst sets_restrict_space) auto
+  moreover fix X assume "X \<in> sets (restrict_space (count_space B) A)"
+  moreover note `A \<subseteq> B`
+  ultimately have "X \<subseteq> A" by auto
+  with `A \<subseteq> B` show "emeasure (restrict_space (count_space B) A) X = emeasure (count_space A) X"
+    by (cases "finite X") (auto simp add: emeasure_restrict_space)
--- a/src/HOL/Probability/Sigma_Algebra.thy	Tue Nov 12 19:28:55 2013 +0100
+++ b/src/HOL/Probability/Sigma_Algebra.thy	Tue Nov 12 19:28:56 2013 +0100
@@ -1171,16 +1171,13 @@
 using assms
 by(simp_all add: sets_measure_of_conv space_measure_of_conv)
-lemma (in sigma_algebra) sets_measure_of_eq[simp]:
-  "sets (measure_of \<Omega> M \<mu>) = M"
+lemma (in sigma_algebra) sets_measure_of_eq[simp]: "sets (measure_of \<Omega> M \<mu>) = M"
   using space_closed by (auto intro!: sigma_sets_eq)
-lemma (in sigma_algebra) space_measure_of_eq[simp]:
-  "space (measure_of \<Omega> M \<mu>) = \<Omega>"
-  using space_closed by (auto intro!: sigma_sets_eq)
+lemma (in sigma_algebra) space_measure_of_eq[simp]: "space (measure_of \<Omega> M \<mu>) = \<Omega>"
+  by (rule space_measure_of_conv)
-lemma measure_of_subset:
-  "M \<subseteq> Pow \<Omega> \<Longrightarrow> M' \<subseteq> M \<Longrightarrow> sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')"
+lemma measure_of_subset: "M \<subseteq> Pow \<Omega> \<Longrightarrow> M' \<subseteq> M \<Longrightarrow> sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')"
   by (auto intro!: sigma_sets_subseteq)
 lemma sigma_sets_mono'':
@@ -1725,6 +1722,42 @@
   qed auto
+subsection {* Restricted Space \<sigma>-Algebra *}
+definition "restrict_space M \<Omega> = measure_of \<Omega> ((op \<inter> \<Omega>) ` sets M) (\<lambda>A. emeasure M (A \<inter> \<Omega>))"
+lemma space_restrict_space: "space (restrict_space M \<Omega>) = \<Omega>"
+  unfolding restrict_space_def by (subst space_measure_of) auto
+lemma sets_restrict_space: "\<Omega> \<subseteq> space M \<Longrightarrow> sets (restrict_space M \<Omega>) = (op \<inter> \<Omega>) ` sets M"
+  using sigma_sets_vimage[of "\<lambda>x. x" \<Omega> "space M" "sets M"]
+  unfolding restrict_space_def
+  by (subst sets_measure_of) (auto simp: sets.sigma_sets_eq Int_commute[of _ \<Omega>] sets.space_closed)
+lemma sets_restrict_space_iff:
+  "\<Omega> \<in> sets M \<Longrightarrow> A \<in> sets (restrict_space M \<Omega>) \<longleftrightarrow> (A \<subseteq> \<Omega> \<and> A \<in> sets M)"
+  by (subst sets_restrict_space) (auto dest: sets.sets_into_space)
+lemma measurable_restrict_space1:
+  assumes \<Omega>: "\<Omega> \<in> sets M" and f: "f \<in> measurable M N" shows "f \<in> measurable (restrict_space M \<Omega>) N"
+  unfolding measurable_def
+proof (intro CollectI conjI ballI)
+  show sp: "f \<in> space (restrict_space M \<Omega>) \<rightarrow> space N"
+    using measurable_space[OF f] sets.sets_into_space[OF \<Omega>] by (auto simp: space_restrict_space)
+  fix A assume "A \<in> sets N"
+  have "f -` A \<inter> space (restrict_space M \<Omega>) = (f -` A \<inter> space M) \<inter> \<Omega>"
+    using sets.sets_into_space[OF \<Omega>] by (auto simp: space_restrict_space)
+  also have "\<dots> \<in> sets (restrict_space M \<Omega>)"
+    unfolding sets_restrict_space_iff[OF \<Omega>]
+    using measurable_sets[OF f `A \<in> sets N`] \<Omega> by blast
+  finally show "f -` A \<inter> space (restrict_space M \<Omega>) \<in> sets (restrict_space M \<Omega>)" .
+lemma measurable_restrict_space2:
+  "\<Omega> \<in> sets N \<Longrightarrow> f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> f \<in> measurable M (restrict_space N \<Omega>)"
+  by (simp add: measurable_def space_restrict_space sets_restrict_space_iff)
 subsection {* A Two-Element Series *}
 definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "