--- a/src/HOL/IsaMakefile Thu Mar 18 14:52:11 2010 +0100
+++ b/src/HOL/IsaMakefile Tue Mar 16 16:27:28 2010 +0100
@@ -1093,6 +1093,7 @@
Probability/Borel.thy \
Probability/Measure.thy \
Probability/Lebesgue.thy \
+ Probability/Product_Measure.thy \
Probability/Probability_Space.thy
@$(ISABELLE_TOOL) usedir -g true $(OUT)/HOL Probability
--- a/src/HOL/Probability/Lebesgue.thy Thu Mar 18 14:52:11 2010 +0100
+++ b/src/HOL/Probability/Lebesgue.thy Tue Mar 16 16:27:28 2010 +0100
@@ -1389,6 +1389,7 @@
qed
lemma integral_on_countable:
+ fixes enum :: "nat \<Rightarrow> real"
assumes borel: "f \<in> borel_measurable M"
and bij: "bij_betw enum S (f ` space M)"
and enum_zero: "enum ` (-S) \<subseteq> {0}"
--- a/src/HOL/Probability/Probability.thy Thu Mar 18 14:52:11 2010 +0100
+++ b/src/HOL/Probability/Probability.thy Tue Mar 16 16:27:28 2010 +0100
@@ -1,5 +1,5 @@
theory Probability
-imports Probability_Space
+imports Probability_Space Product_Measure
begin
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/Product_Measure.thy Tue Mar 16 16:27:28 2010 +0100
@@ -0,0 +1,160 @@
+theory Product_Measure
+imports "~~/src/HOL/Probability/Lebesgue"
+begin
+
+definition
+ "prod_measure M M' = (\<lambda>a. measure_space.integral M (\<lambda>s0. measure M' ((\<lambda>s1. (s0, s1)) -` a)))"
+
+definition
+ "prod_measure_space M M' \<equiv>
+ \<lparr> space = space M \<times> space M',
+ sets = sets (sigma (space M \<times> space M') (prod_sets (sets M) (sets M'))),
+ measure = prod_measure M M' \<rparr>"
+
+lemma prod_measure_times:
+ assumes "measure_space M" and "measure_space M'" and a: "a \<in> sets M"
+ shows "prod_measure M M' (a \<times> a') = measure M a * measure M' a'"
+proof -
+ interpret M: measure_space M by fact
+ interpret M': measure_space M' by fact
+
+ { fix \<omega>
+ have "(\<lambda>\<omega>'. (\<omega>, \<omega>')) -` (a \<times> a') = (if \<omega> \<in> a then a' else {})"
+ by auto
+ hence "measure M' ((\<lambda>\<omega>'. (\<omega>, \<omega>')) -` (a \<times> a')) =
+ measure M' a' * indicator_fn a \<omega>"
+ unfolding indicator_fn_def by auto }
+ note vimage_eq_indicator = this
+
+ show ?thesis
+ unfolding prod_measure_def vimage_eq_indicator
+ M.integral_cmul_indicator(1)[OF `a \<in> sets M`]
+ by simp
+qed
+
+
+
+lemma measure_space_finite_prod_measure:
+ fixes M :: "('a, 'b) measure_space_scheme"
+ and M' :: "('c, 'd) measure_space_scheme"
+ assumes "measure_space M" and "measure_space M'"
+ and finM: "finite (space M)" "Pow (space M) = sets M"
+ and finM': "finite (space M')" "Pow (space M') = sets M'"
+ shows "measure_space (prod_measure_space M M')"
+proof (rule finite_additivity_sufficient)
+ interpret M: measure_space M by fact
+ interpret M': measure_space M' by fact
+
+ have measure: "measure_space.measure (prod_measure_space M M') = prod_measure M M'"
+ unfolding prod_measure_space_def by simp
+
+ have prod_sets: "prod_sets (sets M) (sets M') \<subseteq> Pow (space M \<times> space M')"
+ using M.sets_into_space M'.sets_into_space unfolding prod_sets_def by auto
+ show sigma: "sigma_algebra (prod_measure_space M M')" unfolding prod_measure_space_def
+ by (rule sigma_algebra_sigma_sets[where a="prod_sets (sets M) (sets M')"])
+ (simp_all add: sigma_def prod_sets)
+
+ then interpret sa: sigma_algebra "prod_measure_space M M'" .
+
+ { fix x y assume "y \<in> sets (prod_measure_space M M')" and "x \<in> space M"
+ hence "y \<subseteq> space M \<times> space M'"
+ using sa.sets_into_space unfolding prod_measure_space_def by simp
+ hence "Pair x -` y \<in> sets M'"
+ using `x \<in> space M` unfolding finM'(2)[symmetric] by auto }
+ note Pair_in_sets = this
+
+ show "additive (prod_measure_space M M') (measure (prod_measure_space M M'))"
+ unfolding measure additive_def
+ proof safe
+ fix x y assume x: "x \<in> sets (prod_measure_space M M')" and y: "y \<in> sets (prod_measure_space M M')"
+ and disj_x_y: "x \<inter> y = {}"
+ { fix z have "Pair z -` x \<inter> Pair z -` y = {}" using disj_x_y by auto }
+ note Pair_disj = this
+
+ from M'.measure_additive[OF Pair_in_sets[OF x] Pair_in_sets[OF y] Pair_disj, symmetric]
+ show "prod_measure M M' (x \<union> y) = prod_measure M M' x + prod_measure M M' y"
+ unfolding prod_measure_def
+ apply (subst (1 2 3) M.integral_finite_singleton[OF finM])
+ by (simp_all add: setsum_addf[symmetric] field_simps)
+ qed
+
+ show "finite (space (prod_measure_space M M'))"
+ unfolding prod_measure_space_def using finM finM' by simp
+
+ have singletonM: "\<And>x. x \<in> space M \<Longrightarrow> {x} \<in> sets M"
+ unfolding finM(2)[symmetric] by simp
+
+ show "positive (prod_measure_space M M') (measure (prod_measure_space M M'))"
+ unfolding positive_def
+ proof (safe, simp add: M.integral_zero prod_measure_space_def prod_measure_def)
+ fix Q assume "Q \<in> sets (prod_measure_space M M')"
+ from Pair_in_sets[OF this]
+ show "0 \<le> measure (prod_measure_space M M') Q"
+ unfolding prod_measure_space_def prod_measure_def
+ apply (subst M.integral_finite_singleton[OF finM])
+ using M.positive M'.positive singletonM
+ by (auto intro!: setsum_nonneg mult_nonneg_nonneg)
+ qed
+qed
+
+lemma measure_space_finite_prod_measure_alterantive:
+ assumes "measure_space M" and "measure_space M'"
+ and finM: "finite (space M)" "Pow (space M) = sets M"
+ and finM': "finite (space M')" "Pow (space M') = sets M'"
+ shows "measure_space \<lparr> space = space M \<times> space M',
+ sets = Pow (space M \<times> space M'),
+ measure = prod_measure M M' \<rparr>"
+ (is "measure_space ?space")
+proof (rule finite_additivity_sufficient)
+ interpret M: measure_space M by fact
+ interpret M': measure_space M' by fact
+
+ show "sigma_algebra ?space"
+ using sigma_algebra.sigma_algebra_extend[where M="\<lparr> space = space M \<times> space M', sets = Pow (space M \<times> space M') \<rparr>"]
+ by (auto intro!: sigma_algebra_Pow)
+ then interpret sa: sigma_algebra ?space .
+
+ have measure: "measure_space.measure (prod_measure_space M M') = prod_measure M M'"
+ unfolding prod_measure_space_def by simp
+
+ { fix x y assume "y \<in> sets ?space" and "x \<in> space M"
+ hence "y \<subseteq> space M \<times> space M'"
+ using sa.sets_into_space by simp
+ hence "Pair x -` y \<in> sets M'"
+ using `x \<in> space M` unfolding finM'(2)[symmetric] by auto }
+ note Pair_in_sets = this
+
+ show "additive ?space (measure ?space)"
+ unfolding measure additive_def
+ proof safe
+ fix x y assume x: "x \<in> sets ?space" and y: "y \<in> sets ?space"
+ and disj_x_y: "x \<inter> y = {}"
+ { fix z have "Pair z -` x \<inter> Pair z -` y = {}" using disj_x_y by auto }
+ note Pair_disj = this
+
+ from M'.measure_additive[OF Pair_in_sets[OF x] Pair_in_sets[OF y] Pair_disj, symmetric]
+ show "measure ?space (x \<union> y) = measure ?space x + measure ?space y"
+ apply (simp add: prod_measure_def)
+ apply (subst (1 2 3) M.integral_finite_singleton[OF finM])
+ by (simp_all add: setsum_addf[symmetric] field_simps)
+ qed
+
+ show "finite (space ?space)" using finM finM' by simp
+
+ have singletonM: "\<And>x. x \<in> space M \<Longrightarrow> {x} \<in> sets M"
+ unfolding finM(2)[symmetric] by simp
+
+ show "positive ?space (measure ?space)"
+ unfolding positive_def
+ proof (safe, simp add: M.integral_zero prod_measure_def)
+ fix Q assume "Q \<in> sets ?space"
+ from Pair_in_sets[OF this]
+ show "0 \<le> measure ?space Q"
+ unfolding prod_measure_space_def prod_measure_def
+ apply (subst M.integral_finite_singleton[OF finM])
+ using M.positive M'.positive singletonM
+ by (auto intro!: setsum_nonneg mult_nonneg_nonneg)
+ qed
+qed
+
+end
\ No newline at end of file