src/HOL/Probability/Binary_Product_Measure.thy
 author hoelzl Fri Nov 02 14:23:40 2012 +0100 (2012-11-02) changeset 50002 ce0d316b5b44 parent 49999 dfb63b9b8908 child 50003 8c213922ed49 permissions -rw-r--r--
add measurability prover; add support for Borel sets
1 (*  Title:      HOL/Probability/Binary_Product_Measure.thy
2     Author:     Johannes Hölzl, TU München
3 *)
5 header {*Binary product measures*}
7 theory Binary_Product_Measure
8 imports Lebesgue_Integration
9 begin
11 lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
12   by auto
14 lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
15   by auto
17 lemma Pair_vimage_times[simp]: "\<And>A B x. Pair x - (A \<times> B) = (if x \<in> A then B else {})"
18   by auto
20 lemma rev_Pair_vimage_times[simp]: "\<And>A B y. (\<lambda>x. (x, y)) - (A \<times> B) = (if y \<in> B then A else {})"
21   by auto
23 lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))"
24   by (cases x) simp
26 lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
27   by (auto simp: fun_eq_iff)
29 section "Binary products"
31 definition pair_measure (infixr "\<Otimes>\<^isub>M" 80) where
32   "A \<Otimes>\<^isub>M B = measure_of (space A \<times> space B)
33       {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}
34       (\<lambda>X. \<integral>\<^isup>+x. (\<integral>\<^isup>+y. indicator X (x,y) \<partial>B) \<partial>A)"
36 lemma pair_measure_closed: "{a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} \<subseteq> Pow (space A \<times> space B)"
37   using space_closed[of A] space_closed[of B] by auto
39 lemma space_pair_measure:
40   "space (A \<Otimes>\<^isub>M B) = space A \<times> space B"
41   unfolding pair_measure_def using pair_measure_closed[of A B]
42   by (rule space_measure_of)
44 lemma sets_pair_measure:
45   "sets (A \<Otimes>\<^isub>M B) = sigma_sets (space A \<times> space B) {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
46   unfolding pair_measure_def using pair_measure_closed[of A B]
47   by (rule sets_measure_of)
49 lemma sets_pair_measure_cong[cong]:
50   "sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^isub>M M2) = sets (M1' \<Otimes>\<^isub>M M2')"
51   unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq)
53 lemma pair_measureI[intro, simp]:
54   "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^isub>M B)"
55   by (auto simp: sets_pair_measure)
57 lemma measurable_pair_measureI:
58   assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2"
59   assumes 2: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> f - (A \<times> B) \<inter> space M \<in> sets M"
60   shows "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"
61   unfolding pair_measure_def using 1 2
62   by (intro measurable_measure_of) (auto dest: sets_into_space)
64 lemma measurable_Pair:
65   assumes f: "f \<in> measurable M M1" and g: "g \<in> measurable M M2"
66   shows "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"
67 proof (rule measurable_pair_measureI)
68   show "(\<lambda>x. (f x, g x)) \<in> space M \<rightarrow> space M1 \<times> space M2"
69     using f g by (auto simp: measurable_def)
70   fix A B assume *: "A \<in> sets M1" "B \<in> sets M2"
71   have "(\<lambda>x. (f x, g x)) - (A \<times> B) \<inter> space M = (f - A \<inter> space M) \<inter> (g - B \<inter> space M)"
72     by auto
73   also have "\<dots> \<in> sets M"
74     by (rule Int) (auto intro!: measurable_sets * f g)
75   finally show "(\<lambda>x. (f x, g x)) - (A \<times> B) \<inter> space M \<in> sets M" .
76 qed
78 lemma measurable_pair:
79   assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"
80   shows "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"
81   using measurable_Pair[OF assms] by simp
83 lemma measurable_fst[intro!, simp]: "fst \<in> measurable (M1 \<Otimes>\<^isub>M M2) M1"
84   by (auto simp: fst_vimage_eq_Times space_pair_measure sets_into_space times_Int_times measurable_def)
86 lemma measurable_snd[intro!, simp]: "snd \<in> measurable (M1 \<Otimes>\<^isub>M M2) M2"
87   by (auto simp: snd_vimage_eq_Times space_pair_measure sets_into_space times_Int_times measurable_def)
89 lemma measurable_fst': "f \<in> measurable M (N \<Otimes>\<^isub>M P) \<Longrightarrow> (\<lambda>x. fst (f x)) \<in> measurable M N"
90   using measurable_comp[OF _ measurable_fst] by (auto simp: comp_def)
92 lemma measurable_snd': "f \<in> measurable M (N \<Otimes>\<^isub>M P) \<Longrightarrow> (\<lambda>x. snd (f x)) \<in> measurable M P"
93     using measurable_comp[OF _ measurable_snd] by (auto simp: comp_def)
95 lemma measurable_fst'': "f \<in> measurable M N \<Longrightarrow> (\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^isub>M P) N"
96   using measurable_comp[OF measurable_fst _] by (auto simp: comp_def)
98 lemma measurable_snd'': "f \<in> measurable M N \<Longrightarrow> (\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^isub>M M) N"
99   using measurable_comp[OF measurable_snd _] by (auto simp: comp_def)
101 lemma measurable_pair_iff:
102   "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2) \<longleftrightarrow> (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"
103   using measurable_pair[of f M M1 M2]
104   using [[simproc del: measurable]] (* the measurable method is nonterminating when using measurable_pair *)
105   by auto
107 lemma measurable_split_conv:
108   "(\<lambda>(x, y). f x y) \<in> measurable A B \<longleftrightarrow> (\<lambda>x. f (fst x) (snd x)) \<in> measurable A B"
109   by (intro arg_cong2[where f="op \<in>"]) auto
111 lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
112   by (auto intro!: measurable_Pair simp: measurable_split_conv)
114 lemma measurable_pair_swap:
115   assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^isub>M M1) M"
116   using measurable_comp[OF measurable_Pair f] by (auto simp: measurable_split_conv comp_def)
118 lemma measurable_pair_swap_iff:
119   "f \<in> measurable (M2 \<Otimes>\<^isub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) M"
120   using measurable_pair_swap[of "\<lambda>(x,y). f (y, x)"]
121   by (auto intro!: measurable_pair_swap)
123 lemma measurable_ident[intro, simp]: "(\<lambda>x. x) \<in> measurable M M"
124   unfolding measurable_def by auto
126 lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^isub>M M2)"
127   by (auto intro!: measurable_Pair)
129 lemma sets_Pair1: assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "Pair x - A \<in> sets M2"
130 proof -
131   have "Pair x - A = (if x \<in> space M1 then Pair x - A \<inter> space M2 else {})"
132     using A[THEN sets_into_space] by (auto simp: space_pair_measure)
133   also have "\<dots> \<in> sets M2"
134     using A by (auto simp add: measurable_Pair1' intro!: measurable_sets split: split_if_asm)
135   finally show ?thesis .
136 qed
138 lemma measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^isub>M M2)"
139   by (auto intro!: measurable_Pair)
141 lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "(\<lambda>x. (x, y)) - A \<in> sets M1"
142 proof -
143   have "(\<lambda>x. (x, y)) - A = (if y \<in> space M2 then (\<lambda>x. (x, y)) - A \<inter> space M1 else {})"
144     using A[THEN sets_into_space] by (auto simp: space_pair_measure)
145   also have "\<dots> \<in> sets M1"
146     using A by (auto simp add: measurable_Pair2' intro!: measurable_sets split: split_if_asm)
147   finally show ?thesis .
148 qed
150 lemma measurable_Pair2:
151   assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" and x: "x \<in> space M1"
152   shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
153   using measurable_comp[OF measurable_Pair1' f, OF x]
154   by (simp add: comp_def)
156 lemma measurable_Pair1:
157   assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" and y: "y \<in> space M2"
158   shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
159   using measurable_comp[OF measurable_Pair2' f, OF y]
160   by (simp add: comp_def)
162 lemma Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
163   unfolding Int_stable_def
164   by safe (auto simp add: times_Int_times)
166 lemma (in finite_measure) finite_measure_cut_measurable:
167   assumes "Q \<in> sets (N \<Otimes>\<^isub>M M)"
168   shows "(\<lambda>x. emeasure M (Pair x - Q)) \<in> borel_measurable N"
169     (is "?s Q \<in> _")
170   using Int_stable_pair_measure_generator pair_measure_closed assms
171   unfolding sets_pair_measure
172 proof (induct rule: sigma_sets_induct_disjoint)
173   case (compl A)
174   with sets_into_space have "\<And>x. emeasure M (Pair x - ((space N \<times> space M) - A)) =
175       (if x \<in> space N then emeasure M (space M) - ?s A x else 0)"
176     unfolding sets_pair_measure[symmetric]
177     by (auto intro!: emeasure_compl simp: vimage_Diff sets_Pair1)
178   with compl top show ?case
179     by (auto intro!: measurable_If simp: space_pair_measure)
180 next
181   case (union F)
182   moreover then have "\<And>x. emeasure M (\<Union>i. Pair x - F i) = (\<Sum>i. ?s (F i) x)"
183     unfolding sets_pair_measure[symmetric]
184     by (intro suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def sets_Pair1)
185   ultimately show ?case
186     by (auto simp: vimage_UN)
187 qed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If)
189 lemma (in sigma_finite_measure) measurable_emeasure_Pair:
190   assumes Q: "Q \<in> sets (N \<Otimes>\<^isub>M M)" shows "(\<lambda>x. emeasure M (Pair x - Q)) \<in> borel_measurable N" (is "?s Q \<in> _")
191 proof -
192   from sigma_finite_disjoint guess F . note F = this
193   then have F_sets: "\<And>i. F i \<in> sets M" by auto
194   let ?C = "\<lambda>x i. F i \<inter> Pair x - Q"
195   { fix i
196     have [simp]: "space N \<times> F i \<inter> space N \<times> space M = space N \<times> F i"
197       using F sets_into_space by auto
198     let ?R = "density M (indicator (F i))"
199     have "finite_measure ?R"
200       using F by (intro finite_measureI) (auto simp: emeasure_restricted subset_eq)
201     then have "(\<lambda>x. emeasure ?R (Pair x - (space N \<times> space ?R \<inter> Q))) \<in> borel_measurable N"
202      by (rule finite_measure.finite_measure_cut_measurable) (auto intro: Q)
203     moreover have "\<And>x. emeasure ?R (Pair x - (space N \<times> space ?R \<inter> Q))
204         = emeasure M (F i \<inter> Pair x - (space N \<times> space ?R \<inter> Q))"
205       using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1)
206     moreover have "\<And>x. F i \<inter> Pair x - (space N \<times> space ?R \<inter> Q) = ?C x i"
207       using sets_into_space[OF Q] by (auto simp: space_pair_measure)
208     ultimately have "(\<lambda>x. emeasure M (?C x i)) \<in> borel_measurable N"
209       by simp }
210   moreover
211   { fix x
212     have "(\<Sum>i. emeasure M (?C x i)) = emeasure M (\<Union>i. ?C x i)"
213     proof (intro suminf_emeasure)
214       show "range (?C x) \<subseteq> sets M"
215         using F Q \<in> sets (N \<Otimes>\<^isub>M M) by (auto intro!: sets_Pair1)
216       have "disjoint_family F" using F by auto
217       show "disjoint_family (?C x)"
218         by (rule disjoint_family_on_bisimulation[OF disjoint_family F]) auto
219     qed
220     also have "(\<Union>i. ?C x i) = Pair x - Q"
221       using F sets_into_space[OF Q \<in> sets (N \<Otimes>\<^isub>M M)]
222       by (auto simp: space_pair_measure)
223     finally have "emeasure M (Pair x - Q) = (\<Sum>i. emeasure M (?C x i))"
224       by simp }
225   ultimately show ?thesis using Q \<in> sets (N \<Otimes>\<^isub>M M) F_sets
226     by auto
227 qed
229 lemma (in sigma_finite_measure) emeasure_pair_measure:
230   assumes "X \<in> sets (N \<Otimes>\<^isub>M M)"
231   shows "emeasure (N \<Otimes>\<^isub>M M) X = (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. indicator X (x, y) \<partial>M \<partial>N)" (is "_ = ?\<mu> X")
232 proof (rule emeasure_measure_of[OF pair_measure_def])
233   show "positive (sets (N \<Otimes>\<^isub>M M)) ?\<mu>"
234     by (auto simp: positive_def positive_integral_positive)
235   have eq[simp]: "\<And>A x y. indicator A (x, y) = indicator (Pair x - A) y"
236     by (auto simp: indicator_def)
237   show "countably_additive (sets (N \<Otimes>\<^isub>M M)) ?\<mu>"
238   proof (rule countably_additiveI)
239     fix F :: "nat \<Rightarrow> ('b \<times> 'a) set" assume F: "range F \<subseteq> sets (N \<Otimes>\<^isub>M M)" "disjoint_family F"
240     from F have *: "\<And>i. F i \<in> sets (N \<Otimes>\<^isub>M M)" "(\<Union>i. F i) \<in> sets (N \<Otimes>\<^isub>M M)" by auto
241     moreover from F have "\<And>i. (\<lambda>x. emeasure M (Pair x - F i)) \<in> borel_measurable N"
242       by (intro measurable_emeasure_Pair) auto
243     moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x - F i)"
244       by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
245     moreover have "\<And>x. range (\<lambda>i. Pair x - F i) \<subseteq> sets M"
246       using F by (auto simp: sets_Pair1)
247     ultimately show "(\<Sum>n. ?\<mu> (F n)) = ?\<mu> (\<Union>i. F i)"
248       by (auto simp add: vimage_UN positive_integral_suminf[symmetric] suminf_emeasure subset_eq emeasure_nonneg sets_Pair1
249                intro!: positive_integral_cong positive_integral_indicator[symmetric])
250   qed
251   show "{a \<times> b |a b. a \<in> sets N \<and> b \<in> sets M} \<subseteq> Pow (space N \<times> space M)"
252     using space_closed[of N] space_closed[of M] by auto
253 qed fact
255 lemma (in sigma_finite_measure) emeasure_pair_measure_alt:
256   assumes X: "X \<in> sets (N \<Otimes>\<^isub>M M)"
257   shows "emeasure (N  \<Otimes>\<^isub>M M) X = (\<integral>\<^isup>+x. emeasure M (Pair x - X) \<partial>N)"
258 proof -
259   have [simp]: "\<And>x y. indicator X (x, y) = indicator (Pair x - X) y"
260     by (auto simp: indicator_def)
261   show ?thesis
262     using X by (auto intro!: positive_integral_cong simp: emeasure_pair_measure sets_Pair1)
263 qed
265 lemma (in sigma_finite_measure) emeasure_pair_measure_Times:
266   assumes A: "A \<in> sets N" and B: "B \<in> sets M"
267   shows "emeasure (N \<Otimes>\<^isub>M M) (A \<times> B) = emeasure N A * emeasure M B"
268 proof -
269   have "emeasure (N \<Otimes>\<^isub>M M) (A \<times> B) = (\<integral>\<^isup>+x. emeasure M B * indicator A x \<partial>N)"
270     using A B by (auto intro!: positive_integral_cong simp: emeasure_pair_measure_alt)
271   also have "\<dots> = emeasure M B * emeasure N A"
272     using A by (simp add: emeasure_nonneg positive_integral_cmult_indicator)
273   finally show ?thesis
274     by (simp add: ac_simps)
275 qed
277 subsection {* Binary products of $\sigma$-finite emeasure spaces *}
279 locale pair_sigma_finite = M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2
280   for M1 :: "'a measure" and M2 :: "'b measure"
282 lemma (in pair_sigma_finite) measurable_emeasure_Pair1:
283   "Q \<in> sets (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x - Q)) \<in> borel_measurable M1"
284   using M2.measurable_emeasure_Pair .
286 lemma (in pair_sigma_finite) measurable_emeasure_Pair2:
287   assumes Q: "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) - Q)) \<in> borel_measurable M2"
288 proof -
289   have "(\<lambda>(x, y). (y, x)) - Q \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
290     using Q measurable_pair_swap' by (auto intro: measurable_sets)
291   note M1.measurable_emeasure_Pair[OF this]
292   moreover have "\<And>y. Pair y - ((\<lambda>(x, y). (y, x)) - Q \<inter> space (M2 \<Otimes>\<^isub>M M1)) = (\<lambda>x. (x, y)) - Q"
293     using Q[THEN sets_into_space] by (auto simp: space_pair_measure)
294   ultimately show ?thesis by simp
295 qed
297 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
298   defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}"
299   shows "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> E \<and> incseq F \<and> (\<Union>i. F i) = space M1 \<times> space M2 \<and>
300     (\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
301 proof -
302   from M1.sigma_finite_incseq guess F1 . note F1 = this
303   from M2.sigma_finite_incseq guess F2 . note F2 = this
304   from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
305   let ?F = "\<lambda>i. F1 i \<times> F2 i"
306   show ?thesis
307   proof (intro exI[of _ ?F] conjI allI)
308     show "range ?F \<subseteq> E" using F1 F2 by (auto simp: E_def) (metis range_subsetD)
309   next
310     have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
311     proof (intro subsetI)
312       fix x assume "x \<in> space M1 \<times> space M2"
313       then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
314         by (auto simp: space)
315       then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
316         using incseq F1 incseq F2 unfolding incseq_def
317         by (force split: split_max)+
318       then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
319         by (intro SigmaI) (auto simp add: min_max.sup_commute)
320       then show "x \<in> (\<Union>i. ?F i)" by auto
321     qed
322     then show "(\<Union>i. ?F i) = space M1 \<times> space M2"
323       using space by (auto simp: space)
324   next
325     fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
326       using incseq F1 incseq F2 unfolding incseq_Suc_iff by auto
327   next
328     fix i
329     from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
330     with F1 F2 emeasure_nonneg[of M1 "F1 i"] emeasure_nonneg[of M2 "F2 i"]
331     show "emeasure (M1 \<Otimes>\<^isub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>"
332       by (auto simp add: emeasure_pair_measure_Times)
333   qed
334 qed
336 sublocale pair_sigma_finite \<subseteq> P: sigma_finite_measure "M1 \<Otimes>\<^isub>M M2"
337 proof
338   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
339   show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2) \<and> (\<Union>i. F i) = space (M1 \<Otimes>\<^isub>M M2) \<and> (\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
340   proof (rule exI[of _ F], intro conjI)
341     show "range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2)" using F by (auto simp: pair_measure_def)
342     show "(\<Union>i. F i) = space (M1 \<Otimes>\<^isub>M M2)"
343       using F by (auto simp: space_pair_measure)
344     show "\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>" using F by auto
345   qed
346 qed
348 lemma sigma_finite_pair_measure:
349   assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"
350   shows "sigma_finite_measure (A \<Otimes>\<^isub>M B)"
351 proof -
352   interpret A: sigma_finite_measure A by fact
353   interpret B: sigma_finite_measure B by fact
354   interpret AB: pair_sigma_finite A  B ..
355   show ?thesis ..
356 qed
358 lemma sets_pair_swap:
359   assumes "A \<in> sets (M1 \<Otimes>\<^isub>M M2)"
360   shows "(\<lambda>(x, y). (y, x)) - A \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
361   using measurable_pair_swap' assms by (rule measurable_sets)
363 lemma (in pair_sigma_finite) distr_pair_swap:
364   "M1 \<Otimes>\<^isub>M M2 = distr (M2 \<Otimes>\<^isub>M M1) (M1 \<Otimes>\<^isub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D")
365 proof -
366   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
367   let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
368   show ?thesis
369   proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
370     show "?E \<subseteq> Pow (space ?P)"
371       using space_closed[of M1] space_closed[of M2] by (auto simp: space_pair_measure)
372     show "sets ?P = sigma_sets (space ?P) ?E"
373       by (simp add: sets_pair_measure space_pair_measure)
374     then show "sets ?D = sigma_sets (space ?P) ?E"
375       by simp
376   next
377     show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
378       using F by (auto simp: space_pair_measure)
379   next
380     fix X assume "X \<in> ?E"
381     then obtain A B where X[simp]: "X = A \<times> B" and A: "A \<in> sets M1" and B: "B \<in> sets M2" by auto
382     have "(\<lambda>(y, x). (x, y)) - X \<inter> space (M2 \<Otimes>\<^isub>M M1) = B \<times> A"
383       using sets_into_space[OF A] sets_into_space[OF B] by (auto simp: space_pair_measure)
384     with A B show "emeasure (M1 \<Otimes>\<^isub>M M2) X = emeasure ?D X"
385       by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_pair_measure_Times emeasure_distr
386                     measurable_pair_swap' ac_simps)
387   qed
388 qed
390 lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:
391   assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)"
392   shows "emeasure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+y. emeasure M1 ((\<lambda>x. (x, y)) - A) \<partial>M2)"
393     (is "_ = ?\<nu> A")
394 proof -
395   have [simp]: "\<And>y. (Pair y - ((\<lambda>(x, y). (y, x)) - A \<inter> space (M2 \<Otimes>\<^isub>M M1))) = (\<lambda>x. (x, y)) - A"
396     using sets_into_space[OF A] by (auto simp: space_pair_measure)
397   show ?thesis using A
398     by (subst distr_pair_swap)
399        (simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap']
400                  M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])
401 qed
403 lemma (in pair_sigma_finite) AE_pair:
404   assumes "AE x in (M1 \<Otimes>\<^isub>M M2). Q x"
405   shows "AE x in M1. (AE y in M2. Q (x, y))"
406 proof -
407   obtain N where N: "N \<in> sets (M1 \<Otimes>\<^isub>M M2)" "emeasure (M1 \<Otimes>\<^isub>M M2) N = 0" "{x\<in>space (M1 \<Otimes>\<^isub>M M2). \<not> Q x} \<subseteq> N"
408     using assms unfolding eventually_ae_filter by auto
409   show ?thesis
410   proof (rule AE_I)
411     from N measurable_emeasure_Pair1[OF N \<in> sets (M1 \<Otimes>\<^isub>M M2)]
412     show "emeasure M1 {x\<in>space M1. emeasure M2 (Pair x - N) \<noteq> 0} = 0"
413       by (auto simp: M2.emeasure_pair_measure_alt positive_integral_0_iff emeasure_nonneg)
414     show "{x \<in> space M1. emeasure M2 (Pair x - N) \<noteq> 0} \<in> sets M1"
415       by (intro borel_measurable_ereal_neq_const measurable_emeasure_Pair1 N)
416     { fix x assume "x \<in> space M1" "emeasure M2 (Pair x - N) = 0"
417       have "AE y in M2. Q (x, y)"
418       proof (rule AE_I)
419         show "emeasure M2 (Pair x - N) = 0" by fact
420         show "Pair x - N \<in> sets M2" using N(1) by (rule sets_Pair1)
421         show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x - N"
422           using N x \<in> space M1 unfolding space_pair_measure by auto
423       qed }
424     then show "{x \<in> space M1. \<not> (AE y in M2. Q (x, y))} \<subseteq> {x \<in> space M1. emeasure M2 (Pair x - N) \<noteq> 0}"
425       by auto
426   qed
427 qed
429 lemma (in pair_sigma_finite) AE_pair_measure:
430   assumes "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P x} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
431   assumes ae: "AE x in M1. AE y in M2. P (x, y)"
432   shows "AE x in M1 \<Otimes>\<^isub>M M2. P x"
433 proof (subst AE_iff_measurable[OF _ refl])
434   show "{x\<in>space (M1 \<Otimes>\<^isub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
435     by (rule sets_Collect) fact
436   then have "emeasure (M1 \<Otimes>\<^isub>M M2) {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} =
437       (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. indicator {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} (x, y) \<partial>M2 \<partial>M1)"
438     by (simp add: M2.emeasure_pair_measure)
439   also have "\<dots> = (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. 0 \<partial>M2 \<partial>M1)"
440     using ae
441     apply (safe intro!: positive_integral_cong_AE)
442     apply (intro AE_I2)
443     apply (safe intro!: positive_integral_cong_AE)
444     apply auto
445     done
446   finally show "emeasure (M1 \<Otimes>\<^isub>M M2) {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} = 0" by simp
447 qed
449 lemma (in pair_sigma_finite) AE_pair_iff:
450   "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow>
451     (AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE x in (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x))"
452   using AE_pair[of "\<lambda>x. P (fst x) (snd x)"] AE_pair_measure[of "\<lambda>x. P (fst x) (snd x)"] by auto
454 lemma (in pair_sigma_finite) AE_commute:
455   assumes P: "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
456   shows "(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE y in M2. AE x in M1. P x y)"
457 proof -
458   interpret Q: pair_sigma_finite M2 M1 ..
459   have [simp]: "\<And>x. (fst (case x of (x, y) \<Rightarrow> (y, x))) = snd x" "\<And>x. (snd (case x of (x, y) \<Rightarrow> (y, x))) = fst x"
460     by auto
461   have "{x \<in> space (M2 \<Otimes>\<^isub>M M1). P (snd x) (fst x)} =
462     (\<lambda>(x, y). (y, x)) - {x \<in> space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<inter> space (M2 \<Otimes>\<^isub>M M1)"
463     by (auto simp: space_pair_measure)
464   also have "\<dots> \<in> sets (M2 \<Otimes>\<^isub>M M1)"
465     by (intro sets_pair_swap P)
466   finally show ?thesis
467     apply (subst AE_pair_iff[OF P])
468     apply (subst distr_pair_swap)
469     apply (subst AE_distr_iff[OF measurable_pair_swap' P])
470     apply (subst Q.AE_pair_iff)
471     apply simp_all
472     done
473 qed
475 section "Fubinis theorem"
477 lemma measurable_compose_Pair1:
478   "x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^isub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L"
479   by (rule measurable_compose[OF measurable_Pair]) auto
481 lemma (in sigma_finite_measure) borel_measurable_positive_integral_fst':
482   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M)" "\<And>x. 0 \<le> f x"
483   shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1"
484 using f proof induct
485   case (cong u v)
486   then have "\<And>w x. w \<in> space M1 \<Longrightarrow> x \<in> space M \<Longrightarrow> u (w, x) = v (w, x)"
487     by (auto simp: space_pair_measure)
488   show ?case
489     apply (subst measurable_cong)
490     apply (rule positive_integral_cong)
491     apply fact+
492     done
493 next
494   case (set Q)
495   have [simp]: "\<And>x y. indicator Q (x, y) = indicator (Pair x - Q) y"
496     by (auto simp: indicator_def)
497   have "\<And>x. x \<in> space M1 \<Longrightarrow> emeasure M (Pair x - Q) = \<integral>\<^isup>+ y. indicator Q (x, y) \<partial>M"
498     by (simp add: sets_Pair1[OF set])
499   from this measurable_emeasure_Pair[OF set] show ?case
500     by (rule measurable_cong[THEN iffD1])
501 qed (simp_all add: positive_integral_add positive_integral_cmult measurable_compose_Pair1
502                    positive_integral_monotone_convergence_SUP incseq_def le_fun_def
503               cong: measurable_cong)
505 lemma (in sigma_finite_measure) positive_integral_fst:
506   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M)" "\<And>x. 0 \<le> f x"
507   shows "(\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. f (x, y) \<partial>M \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M) f" (is "?I f = _")
508 using f proof induct
509   case (cong u v)
510   moreover then have "?I u = ?I v"
511     by (intro positive_integral_cong) (auto simp: space_pair_measure)
512   ultimately show ?case
513     by (simp cong: positive_integral_cong)
514 qed (simp_all add: emeasure_pair_measure positive_integral_cmult positive_integral_add
515                    positive_integral_monotone_convergence_SUP
516                    measurable_compose_Pair1 positive_integral_positive
517                    borel_measurable_positive_integral_fst' positive_integral_mono incseq_def le_fun_def
518               cong: positive_integral_cong)
520 lemma (in sigma_finite_measure) positive_integral_fst_measurable:
521   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M)"
522   shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1"
523       (is "?C f \<in> borel_measurable M1")
524     and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M) f"
525   using f
526     borel_measurable_positive_integral_fst'[of "\<lambda>x. max 0 (f x)"]
527     positive_integral_fst[of "\<lambda>x. max 0 (f x)"]
528   unfolding positive_integral_max_0 by auto
530 lemma (in sigma_finite_measure) borel_measurable_positive_integral:
531   "(\<lambda>(x, y). f x y) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M) \<Longrightarrow> (\<lambda>x. \<integral>\<^isup>+ y. f x y \<partial>M) \<in> borel_measurable M1"
532   using positive_integral_fst_measurable(1)[of "split f" M1] by simp
534 lemma (in pair_sigma_finite) positive_integral_snd_measurable:
535   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
536   shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
537 proof -
538   note measurable_pair_swap[OF f]
539   from M1.positive_integral_fst_measurable[OF this]
540   have "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^isub>M M1))"
541     by simp
542   also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
543     by (subst distr_pair_swap)
544        (auto simp: positive_integral_distr[OF measurable_pair_swap' f] intro!: positive_integral_cong)
545   finally show ?thesis .
546 qed
548 lemma (in pair_sigma_finite) Fubini:
549   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
550   shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1)"
551   unfolding positive_integral_snd_measurable[OF assms]
552   unfolding M2.positive_integral_fst_measurable[OF assms] ..
554 lemma (in pair_sigma_finite) integrable_product_swap:
555   assumes "integrable (M1 \<Otimes>\<^isub>M M2) f"
556   shows "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x))"
557 proof -
558   interpret Q: pair_sigma_finite M2 M1 by default
559   have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
560   show ?thesis unfolding *
561     by (rule integrable_distr[OF measurable_pair_swap'])
562        (simp add: distr_pair_swap[symmetric] assms)
563 qed
565 lemma (in pair_sigma_finite) integrable_product_swap_iff:
566   "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable (M1 \<Otimes>\<^isub>M M2) f"
567 proof -
568   interpret Q: pair_sigma_finite M2 M1 by default
569   from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]
570   show ?thesis by auto
571 qed
573 lemma (in pair_sigma_finite) integral_product_swap:
574   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
575   shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f"
576 proof -
577   have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
578   show ?thesis unfolding *
579     by (simp add: integral_distr[symmetric, OF measurable_pair_swap' f] distr_pair_swap[symmetric])
580 qed
582 lemma (in pair_sigma_finite) integrable_fst_measurable:
583   assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
584   shows "AE x in M1. integrable M2 (\<lambda> y. f (x, y))" (is "?AE")
585     and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f" (is "?INT")
586 proof -
587   have f_borel: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
588     using f by auto
589   let ?pf = "\<lambda>x. ereal (f x)" and ?nf = "\<lambda>x. ereal (- f x)"
590   have
591     borel: "?nf \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)""?pf \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)" and
592     int: "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) ?nf \<noteq> \<infinity>" "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) ?pf \<noteq> \<infinity>"
593     using assms by auto
594   have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
595      "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
596     using borel[THEN M2.positive_integral_fst_measurable(1)] int
597     unfolding borel[THEN M2.positive_integral_fst_measurable(2)] by simp_all
598   with borel[THEN M2.positive_integral_fst_measurable(1)]
599   have AE_pos: "AE x in M1. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
600     "AE x in M1. (\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2) \<noteq> \<infinity>"
601     by (auto intro!: positive_integral_PInf_AE )
602   then have AE: "AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
603     "AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
604     by (auto simp: positive_integral_positive)
605   from AE_pos show ?AE using assms
606     by (simp add: measurable_Pair2[OF f_borel] integrable_def)
607   { fix f have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
608       using positive_integral_positive
609       by (intro positive_integral_cong_pos) (auto simp: ereal_uminus_le_reorder)
610     then have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = 0" by simp }
611   note this[simp]
612   { fix f assume borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
613       and int: "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) (\<lambda>x. ereal (f x)) \<noteq> \<infinity>"
614       and AE: "AE x in M1. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
615     have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f")
616     proof (intro integrable_def[THEN iffD2] conjI)
617       show "?f \<in> borel_measurable M1"
618         using borel by (auto intro!: M2.positive_integral_fst_measurable)
619       have "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y))  \<partial>M2) \<partial>M1)"
620         using AE positive_integral_positive[of M2]
621         by (auto intro!: positive_integral_cong_AE simp: ereal_real)
622       then show "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) \<noteq> \<infinity>"
623         using M2.positive_integral_fst_measurable[OF borel] int by simp
624       have "(\<integral>\<^isup>+x. ereal (- ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
625         by (intro positive_integral_cong_pos)
626            (simp add: positive_integral_positive real_of_ereal_pos)
627       then show "(\<integral>\<^isup>+x. ereal (- ?f x) \<partial>M1) \<noteq> \<infinity>" by simp
628     qed }
629   with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)]
630   show ?INT
631     unfolding lebesgue_integral_def[of "M1 \<Otimes>\<^isub>M M2"] lebesgue_integral_def[of M2]
632       borel[THEN M2.positive_integral_fst_measurable(2), symmetric]
633     using AE[THEN integral_real]
634     by simp
635 qed
637 lemma (in pair_sigma_finite) integrable_snd_measurable:
638   assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
639   shows "AE y in M2. integrable M1 (\<lambda>x. f (x, y))" (is "?AE")
640     and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f" (is "?INT")
641 proof -
642   interpret Q: pair_sigma_finite M2 M1 by default
643   have Q_int: "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x, y). f (y, x))"
644     using f unfolding integrable_product_swap_iff .
645   show ?INT
646     using Q.integrable_fst_measurable(2)[OF Q_int]
647     using integral_product_swap[of f] f by auto
648   show ?AE
649     using Q.integrable_fst_measurable(1)[OF Q_int]
650     by simp
651 qed
653 lemma (in pair_sigma_finite) positive_integral_fst_measurable':
654   assumes f: "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
655   shows "(\<lambda>x. \<integral>\<^isup>+ y. f x y \<partial>M2) \<in> borel_measurable M1"
656   using M2.positive_integral_fst_measurable(1)[OF f] by simp
658 lemma (in pair_sigma_finite) integral_fst_measurable:
659   "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>x. \<integral> y. f x y \<partial>M2) \<in> borel_measurable M1"
660   by (auto simp: lebesgue_integral_def intro!: borel_measurable_diff positive_integral_fst_measurable')
662 lemma (in pair_sigma_finite) positive_integral_snd_measurable':
663   assumes f: "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
664   shows "(\<lambda>y. \<integral>\<^isup>+ x. f x y \<partial>M1) \<in> borel_measurable M2"
665 proof -
666   interpret Q: pair_sigma_finite M2 M1 ..
667   show ?thesis
668     using measurable_pair_swap[OF f]
669     by (intro Q.positive_integral_fst_measurable') (simp add: split_beta')
670 qed
672 lemma (in pair_sigma_finite) integral_snd_measurable:
673   "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>y. \<integral> x. f x y \<partial>M1) \<in> borel_measurable M2"
674   by (auto simp: lebesgue_integral_def intro!: borel_measurable_diff positive_integral_snd_measurable')
676 lemma (in pair_sigma_finite) Fubini_integral:
677   assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
678   shows "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1)"
679   unfolding integrable_snd_measurable[OF assms]
680   unfolding integrable_fst_measurable[OF assms] ..
682 section {* Products on counting spaces, densities and distributions *}
684 lemma sigma_sets_pair_measure_generator_finite:
685   assumes "finite A" and "finite B"
686   shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)"
687   (is "sigma_sets ?prod ?sets = _")
688 proof safe
689   have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
690   fix x assume subset: "x \<subseteq> A \<times> B"
691   hence "finite x" using fin by (rule finite_subset)
692   from this subset show "x \<in> sigma_sets ?prod ?sets"
693   proof (induct x)
694     case empty show ?case by (rule sigma_sets.Empty)
695   next
696     case (insert a x)
697     hence "{a} \<in> sigma_sets ?prod ?sets" by auto
698     moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
699     ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
700   qed
701 next
702   fix x a b
703   assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
704   from sigma_sets_into_sp[OF _ this(1)] this(2)
705   show "a \<in> A" and "b \<in> B" by auto
706 qed
708 lemma pair_measure_count_space:
709   assumes A: "finite A" and B: "finite B"
710   shows "count_space A \<Otimes>\<^isub>M count_space B = count_space (A \<times> B)" (is "?P = ?C")
711 proof (rule measure_eqI)
712   interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact
713   interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact
714   interpret P: pair_sigma_finite "count_space A" "count_space B" by default
715   show eq: "sets ?P = sets ?C"
716     by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)
717   fix X assume X: "X \<in> sets ?P"
718   with eq have X_subset: "X \<subseteq> A \<times> B" by simp
719   with A B have fin_Pair: "\<And>x. finite (Pair x - X)"
720     by (intro finite_subset[OF _ B]) auto
721   have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B)
722   show "emeasure ?P X = emeasure ?C X"
723     apply (subst B.emeasure_pair_measure_alt[OF X])
724     apply (subst emeasure_count_space)
725     using X_subset apply auto []
726     apply (simp add: fin_Pair emeasure_count_space X_subset fin_X)
727     apply (subst positive_integral_count_space)
728     using A apply simp
729     apply (simp del: real_of_nat_setsum add: real_of_nat_setsum[symmetric])
730     apply (subst card_gt_0_iff)
731     apply (simp add: fin_Pair)
732     apply (subst card_SigmaI[symmetric])
733     using A apply simp
734     using fin_Pair apply simp
735     using X_subset apply (auto intro!: arg_cong[where f=card])
736     done
737 qed
739 lemma pair_measure_density:
740   assumes f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x"
741   assumes g: "g \<in> borel_measurable M2" "AE x in M2. 0 \<le> g x"
742   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
743   assumes "sigma_finite_measure (density M1 f)" "sigma_finite_measure (density M2 g)"
744   shows "density M1 f \<Otimes>\<^isub>M density M2 g = density (M1 \<Otimes>\<^isub>M M2) (\<lambda>(x,y). f x * g y)" (is "?L = ?R")
745 proof (rule measure_eqI)
746   interpret M1: sigma_finite_measure M1 by fact
747   interpret M2: sigma_finite_measure M2 by fact
748   interpret D1: sigma_finite_measure "density M1 f" by fact
749   interpret D2: sigma_finite_measure "density M2 g" by fact
750   interpret L: pair_sigma_finite "density M1 f" "density M2 g" ..
751   interpret R: pair_sigma_finite M1 M2 ..
753   fix A assume A: "A \<in> sets ?L"
754   then have indicator_eq: "\<And>x y. indicator A (x, y) = indicator (Pair x - A) y"
755    and Pair_A: "\<And>x. Pair x - A \<in> sets M2"
756     by (auto simp: indicator_def sets_Pair1)
757   have f_fst: "(\<lambda>p. f (fst p)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
758     using measurable_comp[OF measurable_fst f(1)] by (simp add: comp_def)
759   have g_snd: "(\<lambda>p. g (snd p)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
760     using measurable_comp[OF measurable_snd g(1)] by (simp add: comp_def)
761   have "(\<lambda>x. \<integral>\<^isup>+ y. g (snd (x, y)) * indicator A (x, y) \<partial>M2) \<in> borel_measurable M1"
762     using g_snd Pair_A A by (intro M2.positive_integral_fst_measurable) auto
763   then have int_g: "(\<lambda>x. \<integral>\<^isup>+ y. g y * indicator A (x, y) \<partial>M2) \<in> borel_measurable M1"
764     by simp
766   show "emeasure ?L A = emeasure ?R A"
767     apply (subst D2.emeasure_pair_measure[OF A])
768     apply (subst emeasure_density)
769         using f_fst g_snd apply (simp add: split_beta')
770       using A apply simp
771     apply (subst positive_integral_density[OF g])
772       apply (simp add: indicator_eq Pair_A)
773     apply (subst positive_integral_density[OF f])
774       apply (rule int_g)
775     apply (subst M2.positive_integral_fst_measurable(2)[symmetric])
776       using f g A Pair_A f_fst g_snd
777       apply (auto intro!: positive_integral_cong_AE R.measurable_emeasure_Pair1
778                   simp: positive_integral_cmult indicator_eq split_beta')
779     apply (intro AE_I2 impI)
780     apply (subst mult_assoc)
781     apply (subst positive_integral_cmult)
782           apply auto
783     done
784 qed simp
786 lemma sigma_finite_measure_distr:
787   assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N"
788   shows "sigma_finite_measure M"
789 proof -
790   interpret sigma_finite_measure "distr M N f" by fact
791   from sigma_finite_disjoint guess A . note A = this
792   show ?thesis
793   proof (unfold_locales, intro conjI exI allI)
794     show "range (\<lambda>i. f - A i \<inter> space M) \<subseteq> sets M"
795       using A f by (auto intro!: measurable_sets)
796     show "(\<Union>i. f - A i \<inter> space M) = space M"
797       using A(1) A(2)[symmetric] f by (auto simp: measurable_def Pi_def)
798     fix i show "emeasure M (f - A i \<inter> space M) \<noteq> \<infinity>"
799       using f A(1,2) A(3)[of i] by (simp add: emeasure_distr subset_eq)
800   qed
801 qed
803 lemma measurable_cong':
804   assumes sets: "sets M = sets M'" "sets N = sets N'"
805   shows "measurable M N = measurable M' N'"
806   using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)
808 lemma pair_measure_distr:
809   assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T"
810   assumes "sigma_finite_measure (distr M S f)" "sigma_finite_measure (distr N T g)"
811   shows "distr M S f \<Otimes>\<^isub>M distr N T g = distr (M \<Otimes>\<^isub>M N) (S \<Otimes>\<^isub>M T) (\<lambda>(x, y). (f x, g y))" (is "?P = ?D")
812 proof (rule measure_eqI)
813   show "sets ?P = sets ?D"
814     by simp
815   interpret S: sigma_finite_measure "distr M S f" by fact
816   interpret T: sigma_finite_measure "distr N T g" by fact
817   interpret ST: pair_sigma_finite "distr M S f"  "distr N T g" ..
818   interpret M: sigma_finite_measure M by (rule sigma_finite_measure_distr) fact+
819   interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+
820   interpret MN: pair_sigma_finite M N ..
821   interpret SN: pair_sigma_finite "distr M S f" N ..
822   have [simp]:
823     "\<And>f g. fst \<circ> (\<lambda>(x, y). (f x, g y)) = f \<circ> fst" "\<And>f g. snd \<circ> (\<lambda>(x, y). (f x, g y)) = g \<circ> snd"
824     by auto
825   then have fg: "(\<lambda>(x, y). (f x, g y)) \<in> measurable (M \<Otimes>\<^isub>M N) (S \<Otimes>\<^isub>M T)"
826     using measurable_comp[OF measurable_fst f] measurable_comp[OF measurable_snd g]
827     by (auto simp: measurable_pair_iff)
828   fix A assume A: "A \<in> sets ?P"
829   then have "emeasure ?P A = (\<integral>\<^isup>+x. emeasure (distr N T g) (Pair x - A) \<partial>distr M S f)"
830     by (rule T.emeasure_pair_measure_alt)
831   also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (g - (Pair x - A) \<inter> space N) \<partial>distr M S f)"
832     using g A by (simp add: sets_Pair1 emeasure_distr)
833   also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (g - (Pair (f x) - A) \<inter> space N) \<partial>M)"
834     using f g A ST.measurable_emeasure_Pair1[OF A]
835     by (intro positive_integral_distr) (auto simp add: sets_Pair1 emeasure_distr)
836   also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (Pair x - ((\<lambda>(x, y). (f x, g y)) - A \<inter> space (M \<Otimes>\<^isub>M N))) \<partial>M)"
837     by (intro positive_integral_cong arg_cong2[where f=emeasure]) (auto simp: space_pair_measure)
838   also have "\<dots> = emeasure (M \<Otimes>\<^isub>M N) ((\<lambda>(x, y). (f x, g y)) - A \<inter> space (M \<Otimes>\<^isub>M N))"
839     using fg by (intro N.emeasure_pair_measure_alt[symmetric] measurable_sets[OF _ A])
840                 (auto cong: measurable_cong')
841   also have "\<dots> = emeasure ?D A"
842     using fg A by (subst emeasure_distr) auto
843   finally show "emeasure ?P A = emeasure ?D A" .
844 qed
846 end