equal
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inserted
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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 |