src/HOL/Analysis/Bochner_Integration.thy
changeset 69064 5840724b1d71
parent 68833 fde093888c16
child 69144 f13b82281715
equal deleted inserted replaced
69045:8c240fdeffcb 69064:5840724b1d71
  1854   then have s'_eq_s: "\<And>i x. x \<in> space M \<Longrightarrow> s' i x = s i x"
  1854   then have s'_eq_s: "\<And>i x. x \<in> space M \<Longrightarrow> s' i x = s i x"
  1855     by simp
  1855     by simp
  1856 
  1856 
  1857   have sf[measurable]: "\<And>i. simple_function M (s' i)"
  1857   have sf[measurable]: "\<And>i. simple_function M (s' i)"
  1858     unfolding s'_def using s(1)
  1858     unfolding s'_def using s(1)
  1859     by (intro simple_function_compose2[where h="( *\<^sub>R)"] simple_function_indicator) auto
  1859     by (intro simple_function_compose2[where h="(*\<^sub>R)"] simple_function_indicator) auto
  1860 
  1860 
  1861   { fix i
  1861   { fix i
  1862     have "\<And>z. {y. s' i z = y \<and> y \<in> s' i ` space M \<and> y \<noteq> 0 \<and> z \<in> space M} =
  1862     have "\<And>z. {y. s' i z = y \<and> y \<in> s' i ` space M \<and> y \<noteq> 0 \<and> z \<in> space M} =
  1863         (if z \<in> space M \<and> s' i z \<noteq> 0 then {s' i z} else {})"
  1863         (if z \<in> space M \<and> s' i z \<noteq> 0 then {s' i z} else {})"
  1864       by (auto simp add: s'_def split: split_indicator)
  1864       by (auto simp add: s'_def split: split_indicator)