
diff -r 614ef6d7a6b6 -r 7582b39f51ed src/HOL/Probability/Probability_Mass_Function.thy
--- a/src/HOL/Probability/Probability_Mass_Function.thy	Wed Jan 06 13:04:31 2016 +0100
+++ b/src/HOL/Probability/Probability_Mass_Function.thy	Wed Jan 06 12:18:53 2016 +0100
@@ -32,48 +32,6 @@
 lemma ereal_divide': "b \<noteq> 0 \<Longrightarrow> ereal (a / b) = ereal a / ereal b"
   using ereal_divide[of a b] by simp
 
-lemma (in finite_measure) countable_support:
-  "countable {x. measure M {x} \<noteq> 0}"
-proof cases
-  assume "measure M (space M) = 0"
-  with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"
-    by auto
-  then show ?thesis
-    by simp
-next
-  let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"
-  assume "?M \<noteq> 0"
-  then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"
-    using reals_Archimedean[of "?m x / ?M" for x]
-    by (auto simp: field_simps not_le[symmetric] measure_nonneg divide_le_0_iff measure_le_0_iff)
-  have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"
-  proof (rule ccontr)
-    fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")
-    then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
-      by (metis infinite_arbitrarily_large)
-    from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"
-      by auto
-    { fix x assume "x \<in> X"
-      from \<open>?M \<noteq> 0\<close> *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)
-      then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
-    note singleton_sets = this
-    have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
-      using \<open>?M \<noteq> 0\<close>
-      by (simp add: \<open>card X = Suc (Suc n)\<close> of_nat_Suc field_simps less_le measure_nonneg)
-    also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
-      by (rule setsum_mono) fact
-    also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
-      using singleton_sets \<open>finite X\<close>
-      by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
-    finally have "?M < measure M (\<Union>x\<in>X. {x})" .
-    moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"
-      using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto
-    ultimately show False by simp
-  qed
-  show ?thesis
-    unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
-qed
-
 lemma (in finite_measure) AE_support_countable:
   assumes [simp]: "sets M = UNIV"
   shows "(AE x in M. measure M {x} \<noteq> 0) \<longleftrightarrow> (\<exists>S. countable S \<and> (AE x in M. x \<in> S))"




