src/HOL/Probability/Probability_Measure.thy
changeset 61810 3c5040d5694a
parent 61808 fc1556774cfe
child 62083 7582b39f51ed
equal deleted inserted replaced
61809:81d34cf268d8 61810:3c5040d5694a
   219       using prob_space by (simp add: X)
   219       using prob_space by (simp add: X)
   220     also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
   220     also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
   221       using \<open>x \<in> I\<close> \<open>open I\<close> X(2)
   221       using \<open>x \<in> I\<close> \<open>open I\<close> X(2)
   222       apply (intro integral_mono_AE integrable_add integrable_mult_right integrable_diff
   222       apply (intro integral_mono_AE integrable_add integrable_mult_right integrable_diff
   223                 integrable_const X q)
   223                 integrable_const X q)
   224       apply (elim eventually_elim1)
   224       apply (elim eventually_mono)
   225       apply (intro convex_le_Inf_differential)
   225       apply (intro convex_le_Inf_differential)
   226       apply (auto simp: interior_open q)
   226       apply (auto simp: interior_open q)
   227       done
   227       done
   228     finally show "k \<le> expectation (\<lambda>w. q (X w))" using x by auto
   228     finally show "k \<le> expectation (\<lambda>w. q (X w))" using x by auto
   229   qed
   229   qed