equal
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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) |