--- a/src/HOL/Probability/Borel_Space.thy Tue Jul 19 14:37:49 2011 +0200
+++ b/src/HOL/Probability/Borel_Space.thy Tue Jul 19 14:38:29 2011 +0200
@@ -1122,7 +1122,7 @@
show "(real -` B \<inter> space borel :: ereal set) \<in> sets borel"
proof cases
assume "0 \<in> B"
- then have *: "real -` B = real -` (B - {0}) \<union> {-\<infinity>, \<infinity>, 0}"
+ then have *: "real -` B = real -` (B - {0}) \<union> {-\<infinity>, \<infinity>, 0::ereal}"
by (auto simp add: real_of_ereal_eq_0)
then show "(real -` B :: ereal set) \<inter> space borel \<in> sets borel"
using open_real by auto
@@ -1148,7 +1148,8 @@
qed auto
lemma (in sigma_algebra) borel_measurable_ereal_iff_real:
- "f \<in> borel_measurable M \<longleftrightarrow>
+ fixes f :: "'a \<Rightarrow> ereal"
+ shows "f \<in> borel_measurable M \<longleftrightarrow>
((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)"
proof safe
assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M"
@@ -1208,7 +1209,8 @@
qed auto
lemma (in sigma_algebra) borel_measurable_eq_atMost_ereal:
- "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
+ fixes f :: "'a \<Rightarrow> ereal"
+ shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
proof (intro iffI allI)
assume pos[rule_format]: "\<forall>a. f -` {..a} \<inter> space M \<in> sets M"
show "f \<in> borel_measurable M"
@@ -1226,7 +1228,7 @@
by (auto simp: not_le)
then show "f -` {\<infinity>} \<inter> space M \<in> sets M" using pos by (auto simp del: UN_simps intro!: Diff)
moreover
- have "{-\<infinity>} = {..-\<infinity>}" by auto
+ have "{-\<infinity>::ereal} = {..-\<infinity>}" by auto
then show "f -` {-\<infinity>} \<inter> space M \<in> sets M" using pos by auto
moreover have "{x\<in>space M. f x \<le> ereal a} \<in> sets M"
using pos[of "ereal a"] by (simp add: vimage_def Int_def conj_commute)