equal
deleted
inserted
replaced
704 proof - |
704 proof - |
705 interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI) |
705 interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI) |
706 have "\<And>A. measure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1" |
706 have "\<And>A. measure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1" |
707 using assms by (subst measure_times) auto |
707 using assms by (subst measure_times) auto |
708 then show ?thesis |
708 then show ?thesis |
709 unfolding positive_integral_def simple_function_def simple_integral_def_raw |
709 unfolding positive_integral_def simple_function_def simple_integral_def [abs_def] |
710 proof (simp add: P_empty, intro antisym) |
710 proof (simp add: P_empty, intro antisym) |
711 show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))" |
711 show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))" |
712 by (intro SUP_upper) (auto simp: le_fun_def split: split_max) |
712 by (intro SUP_upper) (auto simp: le_fun_def split: split_max) |
713 show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos |
713 show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos |
714 by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm) |
714 by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm) |