equal
deleted
inserted
replaced
12 by auto |
12 by auto |
13 |
13 |
14 lemma rev_Pair_vimage_times[simp]: "(\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})" |
14 lemma rev_Pair_vimage_times[simp]: "(\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})" |
15 by auto |
15 by auto |
16 |
16 |
17 section "Binary products" |
17 subsection "Binary products" |
18 |
18 |
19 definition pair_measure (infixr "\<Otimes>\<^sub>M" 80) where |
19 definition pair_measure (infixr "\<Otimes>\<^sub>M" 80) where |
20 "A \<Otimes>\<^sub>M B = measure_of (space A \<times> space B) |
20 "A \<Otimes>\<^sub>M B = measure_of (space A \<times> space B) |
21 {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} |
21 {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} |
22 (\<lambda>X. \<integral>\<^sup>+x. (\<integral>\<^sup>+y. indicator X (x,y) \<partial>B) \<partial>A)" |
22 (\<lambda>X. \<integral>\<^sup>+x. (\<integral>\<^sup>+y. indicator X (x,y) \<partial>B) \<partial>A)" |
478 apply (subst Q.AE_pair_iff) |
478 apply (subst Q.AE_pair_iff) |
479 apply simp_all |
479 apply simp_all |
480 done |
480 done |
481 qed |
481 qed |
482 |
482 |
483 section "Fubinis theorem" |
483 subsection "Fubinis theorem" |
484 |
484 |
485 lemma measurable_compose_Pair1: |
485 lemma measurable_compose_Pair1: |
486 "x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^sub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L" |
486 "x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^sub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L" |
487 by simp |
487 by simp |
488 |
488 |
555 lemma (in pair_sigma_finite) Fubini: |
555 lemma (in pair_sigma_finite) Fubini: |
556 assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" |
556 assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" |
557 shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x, y) \<partial>M2) \<partial>M1)" |
557 shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x, y) \<partial>M2) \<partial>M1)" |
558 unfolding positive_integral_snd[OF assms] M2.positive_integral_fst[OF assms] .. |
558 unfolding positive_integral_snd[OF assms] M2.positive_integral_fst[OF assms] .. |
559 |
559 |
560 section {* Products on counting spaces, densities and distributions *} |
560 subsection {* Products on counting spaces, densities and distributions *} |
561 |
561 |
562 lemma sigma_sets_pair_measure_generator_finite: |
562 lemma sigma_sets_pair_measure_generator_finite: |
563 assumes "finite A" and "finite B" |
563 assumes "finite A" and "finite B" |
564 shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)" |
564 shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)" |
565 (is "sigma_sets ?prod ?sets = _") |
565 (is "sigma_sets ?prod ?sets = _") |