src/HOL/Analysis/Lebesgue_Integral_Substitution.thy
changeset 75455 91c16c5ad3e9
parent 69861 62e47f06d22c
child 75462 7448423e5dba
equal deleted inserted replaced
75454:295e1c9d2994 75455:91c16c5ad3e9
   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)