--- 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)" ..
qed
+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)
+next
+ 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)
+qed
+
+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)
+qed
+
end