equal
deleted
inserted
replaced
120 by (rule nn_integral_cong) (simp split: split_indicator) |
120 by (rule nn_integral_cong) (simp split: split_indicator) |
121 also from compl have "... = emeasure lborel ({g a..g b} - A)" using derivg_nonneg |
121 also from compl have "... = emeasure lborel ({g a..g b} - A)" using derivg_nonneg |
122 by (simp add: vimage_Compl diff_eq Int_commute[of "-A"]) |
122 by (simp add: vimage_Compl diff_eq Int_commute[of "-A"]) |
123 also have "{g a..g b} - A = {g a..g b} - A \<inter> {g a..g b}" by blast |
123 also have "{g a..g b} - A = {g a..g b} - A \<inter> {g a..g b}" by blast |
124 also have "emeasure lborel ... = g b - g a - emeasure lborel (A \<inter> {g a..g b})" |
124 also have "emeasure lborel ... = g b - g a - emeasure lborel (A \<inter> {g a..g b})" |
125 using \<open>A \<in> sets borel\<close> by (subst emeasure_Diff) (auto simp: ) |
125 using \<open>A \<in> sets borel\<close> by (subst emeasure_Diff) auto |
126 also have "emeasure lborel (A \<inter> {g a..g b}) = |
126 also have "emeasure lborel (A \<inter> {g a..g b}) = |
127 \<integral>\<^sup>+x. indicator A x * indicator {g a..g b} x \<partial>lborel" |
127 \<integral>\<^sup>+x. indicator A x * indicator {g a..g b} x \<partial>lborel" |
128 using \<open>A \<in> sets borel\<close> |
128 using \<open>A \<in> sets borel\<close> |
129 by (subst nn_integral_indicator[symmetric], simp, intro nn_integral_cong) |
129 by (subst nn_integral_indicator[symmetric], simp, intro nn_integral_cong) |
130 (simp split: split_indicator) |
130 (simp split: split_indicator) |