equal
deleted
inserted
replaced
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) |