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theory Product_Measure
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imports "~~/src/HOL/Probability/Lebesgue"
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begin
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definition
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"prod_measure M M' = (\<lambda>a. measure_space.integral M (\<lambda>s0. measure M' ((\<lambda>s1. (s0, s1)) -` a)))"
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definition
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"prod_measure_space M M' \<equiv>
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\<lparr> space = space M \<times> space M',
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sets = sets (sigma (space M \<times> space M') (prod_sets (sets M) (sets M'))),
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measure = prod_measure M M' \<rparr>"
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lemma prod_measure_times:
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assumes "measure_space M" and "measure_space M'" and a: "a \<in> sets M"
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shows "prod_measure M M' (a \<times> a') = measure M a * measure M' a'"
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proof -
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interpret M: measure_space M by fact
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interpret M': measure_space M' by fact
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{ fix \<omega>
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have "(\<lambda>\<omega>'. (\<omega>, \<omega>')) -` (a \<times> a') = (if \<omega> \<in> a then a' else {})"
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by auto
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hence "measure M' ((\<lambda>\<omega>'. (\<omega>, \<omega>')) -` (a \<times> a')) =
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measure M' a' * indicator_fn a \<omega>"
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unfolding indicator_fn_def by auto }
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note vimage_eq_indicator = this
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show ?thesis
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unfolding prod_measure_def vimage_eq_indicator
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M.integral_cmul_indicator(1)[OF `a \<in> sets M`]
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by simp
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qed
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lemma finite_prod_measure_space:
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assumes "finite_measure_space M" and "finite_measure_space M'"
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shows "prod_measure_space M M' =
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\<lparr> space = space M \<times> space M',
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sets = Pow (space M \<times> space M'),
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measure = prod_measure M M' \<rparr>"
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proof -
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interpret M: finite_measure_space M by fact
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interpret M': finite_measure_space M' by fact
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show ?thesis using M.finite_space M'.finite_space
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by (simp add: sigma_prod_sets_finite M.sets_eq_Pow M'.sets_eq_Pow
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prod_measure_space_def)
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qed
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lemma finite_measure_space_finite_prod_measure:
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assumes "finite_measure_space M" and "finite_measure_space M'"
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shows "finite_measure_space (prod_measure_space M M')"
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proof (rule finite_Pow_additivity_sufficient)
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interpret M: finite_measure_space M by fact
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interpret M': finite_measure_space M' by fact
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from M.finite_space M'.finite_space
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show "finite (space (prod_measure_space M M'))" and
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"sets (prod_measure_space M M') = Pow (space (prod_measure_space M M'))"
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by (simp_all add: finite_prod_measure_space[OF assms])
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show "additive (prod_measure_space M M') (measure (prod_measure_space M M'))"
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unfolding additive_def finite_prod_measure_space[OF assms]
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proof (simp, safe)
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fix x y assume subs: "x \<subseteq> space M \<times> space M'" "y \<subseteq> space M \<times> space M'"
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and disj_x_y: "x \<inter> y = {}"
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have "\<And>z. measure M' (Pair z -` x \<union> Pair z -` y) =
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measure M' (Pair z -` x) + measure M' (Pair z -` y)"
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using disj_x_y subs by (subst M'.measure_additive) (auto simp: M'.sets_eq_Pow)
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thus "prod_measure M M' (x \<union> y) = prod_measure M M' x + prod_measure M M' y"
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unfolding prod_measure_def M.integral_finite_singleton
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by (simp_all add: setsum_addf[symmetric] field_simps)
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qed
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show "positive (prod_measure_space M M') (measure (prod_measure_space M M'))"
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unfolding positive_def
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by (auto intro!: setsum_nonneg mult_nonneg_nonneg M'.positive M.positive
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simp add: M.integral_zero finite_prod_measure_space[OF assms]
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prod_measure_def M.integral_finite_singleton
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M.sets_eq_Pow M'.sets_eq_Pow)
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qed
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lemma finite_measure_space_finite_prod_measure_alterantive:
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assumes M: "finite_measure_space M" and M': "finite_measure_space M'"
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shows "finite_measure_space \<lparr> space = space M \<times> space M', sets = Pow (space M \<times> space M'), measure = prod_measure M M' \<rparr>"
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(is "finite_measure_space ?M")
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proof -
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interpret M: finite_measure_space M by fact
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interpret M': finite_measure_space M' by fact
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show ?thesis
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using finite_measure_space_finite_prod_measure[OF assms]
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unfolding prod_measure_space_def M.sets_eq_Pow M'.sets_eq_Pow
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using M.finite_space M'.finite_space
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by (simp add: sigma_prod_sets_finite)
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qed
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end |