Measurable on euclidean space is equiv. to measurable components

--- a/src/HOL/Probability/Borel.thy	Fri Aug 27 13:48:10 2010 +0200
+++ b/src/HOL/Probability/Borel.thy	Fri Aug 27 14:05:03 2010 +0200
@@ -658,6 +658,30 @@
   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
   using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
 
+lemma borel_measureable_euclidean_component:
+  "(\<lambda>x::'a::euclidean_space. x $$ i) \<in> borel_measurable borel_space"
+  unfolding borel_space_def[where 'a=real]
+proof (rule borel_space.measurable_sigma)
+  fix S::"real set" assume "S \<in> open" then have "open S" unfolding mem_def .
+  from open_vimage_euclidean_component[OF this]
+  show "(\<lambda>x. x $$ i) -` S \<inter> space borel_space \<in> sets borel_space"
+    by (auto intro: borel_space_open)
+qed auto
+
+lemma (in sigma_algebra) borel_measureable_euclidean_space:
+  fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
+  shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M)"
+proof safe
+  fix i assume "f \<in> borel_measurable M"
+  then show "(\<lambda>x. f x $$ i) \<in> borel_measurable M"
+    using measurable_comp[of f _ _ "\<lambda>x. x $$ i", unfolded comp_def]
+    by (auto intro: borel_measureable_euclidean_component)
+next
+  assume f: "\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M"
+  then show "f \<in> borel_measurable M"
+    unfolding borel_measurable_iff_halfspace_le by auto
+qed
+
 subsection "Borel measurable operators"
 
 lemma (in sigma_algebra) affine_borel_measurable_vector: