--- a/src/HOL/Probability/Probability_Space.thy Fri Aug 27 15:05:07 2010 +0200
+++ b/src/HOL/Probability/Probability_Space.thy Fri Aug 27 16:23:51 2010 +0200
@@ -562,5 +562,65 @@
unfolding conditional_expectation_def by (rule someI2_ex) blast
qed
+lemma (in sigma_algebra) factorize_measurable_function:
+ fixes Z :: "'a \<Rightarrow> pinfreal" and Y :: "'a \<Rightarrow> 'c"
+ assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M"
+ shows "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)
+ \<longleftrightarrow> (\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x))"
+proof safe
+ interpret M': sigma_algebra M' by fact
+ have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
+ from M'.sigma_algebra_vimage[OF this]
+ interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
+
+ { fix g :: "'c \<Rightarrow> pinfreal" assume "g \<in> borel_measurable M'"
+ with M'.measurable_vimage_algebra[OF Y]
+ have "g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+ by (rule measurable_comp)
+ moreover assume "\<forall>x\<in>space M. Z x = g (Y x)"
+ then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow>
+ g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+ by (auto intro!: measurable_cong)
+ ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+ by simp }
+
+ assume "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+ from va.borel_measurable_implies_simple_function_sequence[OF this]
+ obtain f where f: "\<And>i. va.simple_function (f i)" and "f \<up> Z" by blast
+
+ have "\<forall>i. \<exists>g. M'.simple_function g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
+ proof
+ fix i
+ from f[of i] have "finite (f i`space M)" and B_ex:
+ "\<forall>z\<in>(f i)`space M. \<exists>B. B \<in> sets M' \<and> (f i) -` {z} \<inter> space M = Y -` B \<inter> space M"
+ unfolding va.simple_function_def by auto
+ from B_ex[THEN bchoice] guess B .. note B = this
+
+ let ?g = "\<lambda>x. \<Sum>z\<in>f i`space M. z * indicator (B z) x"
+
+ show "\<exists>g. M'.simple_function g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
+ proof (intro exI[of _ ?g] conjI ballI)
+ show "M'.simple_function ?g" using B by auto
+
+ fix x assume "x \<in> space M"
+ then have "\<And>z. z \<in> f i`space M \<Longrightarrow> indicator (B z) (Y x) = (indicator (f i -` {z} \<inter> space M) x::pinfreal)"
+ unfolding indicator_def using B by auto
+ then show "f i x = ?g (Y x)" using `x \<in> space M` f[of i]
+ by (subst va.simple_function_indicator_representation) auto
+ qed
+ qed
+ from choice[OF this] guess g .. note g = this
+
+ show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)"
+ proof (intro ballI bexI)
+ show "(SUP i. g i) \<in> borel_measurable M'"
+ using g by (auto intro: M'.borel_measurable_simple_function)
+ fix x assume "x \<in> space M"
+ have "Z x = (SUP i. f i) x" using `f \<up> Z` unfolding isoton_def by simp
+ also have "\<dots> = (SUP i. g i) (Y x)" unfolding SUPR_fun_expand
+ using g `x \<in> space M` by simp
+ finally show "Z x = (SUP i. g i) (Y x)" .
+ qed
+qed
end