src/HOL/Probability/Giry_Monad.thy
changeset 67399 eab6ce8368fa
parent 67226 ec32cdaab97b
child 67649 1e1782c1aedf
equal deleted inserted replaced
67398:5eb932e604a2 67399:eab6ce8368fa
   347     fix A assume A: "A \<in> sets N"
   347     fix A assume A: "A \<in> sets N"
   348     then have "(\<lambda>M'. emeasure (distr M' N f) A) \<in> borel_measurable (subprob_algebra M) \<longleftrightarrow>
   348     then have "(\<lambda>M'. emeasure (distr M' N f) A) \<in> borel_measurable (subprob_algebra M) \<longleftrightarrow>
   349       (\<lambda>M'. emeasure M' (f -` A \<inter> space M)) \<in> borel_measurable (subprob_algebra M)"
   349       (\<lambda>M'. emeasure M' (f -` A \<inter> space M)) \<in> borel_measurable (subprob_algebra M)"
   350       by (intro measurable_cong)
   350       by (intro measurable_cong)
   351          (auto simp: emeasure_distr space_subprob_algebra
   351          (auto simp: emeasure_distr space_subprob_algebra
   352                intro!: arg_cong2[where f=emeasure] sets_eq_imp_space_eq arg_cong2[where f="op \<inter>"])
   352                intro!: arg_cong2[where f=emeasure] sets_eq_imp_space_eq arg_cong2[where f="(\<inter>)"])
   353     also have "\<dots>"
   353     also have "\<dots>"
   354       using A by (intro measurable_emeasure_subprob_algebra) simp
   354       using A by (intro measurable_emeasure_subprob_algebra) simp
   355     finally show "(\<lambda>M'. emeasure (distr M' N f) A) \<in> borel_measurable (subprob_algebra M)" .
   355     finally show "(\<lambda>M'. emeasure (distr M' N f) A) \<in> borel_measurable (subprob_algebra M)" .
   356   qed (auto intro!: subprob_space.subprob_space_distr simp: space_subprob_algebra not_empty cong: measurable_cong_sets)
   356   qed (auto intro!: subprob_space.subprob_space_distr simp: space_subprob_algebra not_empty cong: measurable_cong_sets)
   357 qed (insert assms, auto simp: measurable_empty_iff space_subprob_algebra_empty_iff)
   357 qed (insert assms, auto simp: measurable_empty_iff space_subprob_algebra_empty_iff)