src/HOL/Probability/Binary_Product_Measure.thy
 author huffman Sun Apr 01 16:09:58 2012 +0200 (2012-04-01) changeset 47255 30a1692557b0 parent 46898 1570b30ee040 child 47694 05663f75964c permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
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
32   "pair_measure_generator A B =
33     \<lparr> space = space A \<times> space B,
34       sets = {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B},
35       measure = \<lambda>X. \<integral>\<^isup>+x. (\<integral>\<^isup>+y. indicator X (x,y) \<partial>B) \<partial>A \<rparr>"
37 definition pair_measure (infixr "\<Otimes>\<^isub>M" 80) where
38   "A \<Otimes>\<^isub>M B = sigma (pair_measure_generator A B)"
40 locale pair_sigma_algebra = M1: sigma_algebra M1 + M2: sigma_algebra M2
41   for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
43 abbreviation (in pair_sigma_algebra)
44   "E \<equiv> pair_measure_generator M1 M2"
46 abbreviation (in pair_sigma_algebra)
47   "P \<equiv> M1 \<Otimes>\<^isub>M M2"
49 lemma sigma_algebra_pair_measure:
50   "sets M1 \<subseteq> Pow (space M1) \<Longrightarrow> sets M2 \<subseteq> Pow (space M2) \<Longrightarrow> sigma_algebra (pair_measure M1 M2)"
51   by (force simp: pair_measure_def pair_measure_generator_def intro!: sigma_algebra_sigma)
53 sublocale pair_sigma_algebra \<subseteq> sigma_algebra P
54   using M1.space_closed M2.space_closed
55   by (rule sigma_algebra_pair_measure)
57 lemma pair_measure_generatorI[intro, simp]:
58   "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (pair_measure_generator A B)"
59   by (auto simp add: pair_measure_generator_def)
61 lemma pair_measureI[intro, simp]:
62   "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^isub>M B)"
63   by (auto simp add: pair_measure_def)
65 lemma space_pair_measure:
66   "space (A \<Otimes>\<^isub>M B) = space A \<times> space B"
67   by (simp add: pair_measure_def pair_measure_generator_def)
69 lemma sets_pair_measure_generator:
70   "sets (pair_measure_generator N M) = (\<lambda>(x, y). x \<times> y)  (sets N \<times> sets M)"
71   unfolding pair_measure_generator_def by auto
73 lemma pair_measure_generator_sets_into_space:
74   assumes "sets M \<subseteq> Pow (space M)" "sets N \<subseteq> Pow (space N)"
75   shows "sets (pair_measure_generator M N) \<subseteq> Pow (space (pair_measure_generator M N))"
76   using assms by (auto simp: pair_measure_generator_def)
78 lemma pair_measure_generator_Int_snd:
79   assumes "sets S1 \<subseteq> Pow (space S1)"
80   shows "sets (pair_measure_generator S1 (algebra.restricted_space S2 A)) =
81          sets (algebra.restricted_space (pair_measure_generator S1 S2) (space S1 \<times> A))"
82   (is "?L = ?R")
83   apply (auto simp: pair_measure_generator_def image_iff)
84   using assms
85   apply (rule_tac x="a \<times> xa" in exI)
86   apply force
87   using assms
88   apply (rule_tac x="a" in exI)
89   apply (rule_tac x="b \<inter> A" in exI)
90   apply auto
91   done
93 lemma (in pair_sigma_algebra)
94   shows measurable_fst[intro!, simp]:
95     "fst \<in> measurable P M1" (is ?fst)
96   and measurable_snd[intro!, simp]:
97     "snd \<in> measurable P M2" (is ?snd)
98 proof -
99   { fix X assume "X \<in> sets M1"
100     then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. fst - X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
101       apply - apply (rule bexI[of _ X]) apply (rule bexI[of _ "space M2"])
102       using M1.sets_into_space by force+ }
103   moreover
104   { fix X assume "X \<in> sets M2"
105     then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. snd - X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
106       apply - apply (rule bexI[of _ "space M1"]) apply (rule bexI[of _ X])
107       using M2.sets_into_space by force+ }
108   ultimately have "?fst \<and> ?snd"
109     by (fastforce simp: measurable_def sets_sigma space_pair_measure
110                  intro!: sigma_sets.Basic)
111   then show ?fst ?snd by auto
112 qed
114 lemma (in pair_sigma_algebra) measurable_pair_iff:
115   assumes "sigma_algebra M"
116   shows "f \<in> measurable M P \<longleftrightarrow>
117     (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"
118 proof -
119   interpret M: sigma_algebra M by fact
120   from assms show ?thesis
121   proof (safe intro!: measurable_comp[where b=P])
122     assume f: "(fst \<circ> f) \<in> measurable M M1" and s: "(snd \<circ> f) \<in> measurable M M2"
123     show "f \<in> measurable M P" unfolding pair_measure_def
124     proof (rule M.measurable_sigma)
125       show "sets (pair_measure_generator M1 M2) \<subseteq> Pow (space E)"
126         unfolding pair_measure_generator_def using M1.sets_into_space M2.sets_into_space by auto
127       show "f \<in> space M \<rightarrow> space E"
128         using f s by (auto simp: mem_Times_iff measurable_def comp_def space_sigma pair_measure_generator_def)
129       fix A assume "A \<in> sets E"
130       then obtain B C where "B \<in> sets M1" "C \<in> sets M2" "A = B \<times> C"
131         unfolding pair_measure_generator_def by auto
132       moreover have "(fst \<circ> f) - B \<inter> space M \<in> sets M"
133         using f B \<in> sets M1 unfolding measurable_def by auto
134       moreover have "(snd \<circ> f) - C \<inter> space M \<in> sets M"
135         using s C \<in> sets M2 unfolding measurable_def by auto
136       moreover have "f - A \<inter> space M = ((fst \<circ> f) - B \<inter> space M) \<inter> ((snd \<circ> f) - C \<inter> space M)"
137         unfolding A = B \<times> C by (auto simp: vimage_Times)
138       ultimately show "f - A \<inter> space M \<in> sets M" by auto
139     qed
140   qed
141 qed
143 lemma (in pair_sigma_algebra) measurable_pair:
144   assumes "sigma_algebra M"
145   assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"
146   shows "f \<in> measurable M P"
147   unfolding measurable_pair_iff[OF assms(1)] using assms(2,3) by simp
149 lemma pair_measure_generatorE:
150   assumes "X \<in> sets (pair_measure_generator M1 M2)"
151   obtains A B where "X = A \<times> B" "A \<in> sets M1" "B \<in> sets M2"
152   using assms unfolding pair_measure_generator_def by auto
154 lemma (in pair_sigma_algebra) pair_measure_generator_swap:
155   "(\<lambda>X. (\<lambda>(x,y). (y,x)) - X \<inter> space M2 \<times> space M1)  sets E = sets (pair_measure_generator M2 M1)"
156 proof (safe elim!: pair_measure_generatorE)
157   fix A B assume "A \<in> sets M1" "B \<in> sets M2"
158   moreover then have "(\<lambda>(x, y). (y, x)) - (A \<times> B) \<inter> space M2 \<times> space M1 = B \<times> A"
159     using M1.sets_into_space M2.sets_into_space by auto
160   ultimately show "(\<lambda>(x, y). (y, x)) - (A \<times> B) \<inter> space M2 \<times> space M1 \<in> sets (pair_measure_generator M2 M1)"
161     by (auto intro: pair_measure_generatorI)
162 next
163   fix A B assume "A \<in> sets M1" "B \<in> sets M2"
164   then show "B \<times> A \<in> (\<lambda>X. (\<lambda>(x, y). (y, x)) - X \<inter> space M2 \<times> space M1)  sets E"
165     using M1.sets_into_space M2.sets_into_space
166     by (auto intro!: image_eqI[where x="A \<times> B"] pair_measure_generatorI)
167 qed
169 lemma (in pair_sigma_algebra) sets_pair_sigma_algebra_swap:
170   assumes Q: "Q \<in> sets P"
171   shows "(\<lambda>(x,y). (y, x)) - Q \<in> sets (M2 \<Otimes>\<^isub>M M1)" (is "_ \<in> sets ?Q")
172 proof -
173   let ?f = "\<lambda>Q. (\<lambda>(x,y). (y, x)) - Q \<inter> space M2 \<times> space M1"
174   have *: "(\<lambda>(x,y). (y, x)) - Q = ?f Q"
175     using sets_into_space[OF Q] by (auto simp: space_pair_measure)
176   have "sets (M2 \<Otimes>\<^isub>M M1) = sets (sigma (pair_measure_generator M2 M1))"
177     unfolding pair_measure_def ..
178   also have "\<dots> = sigma_sets (space M2 \<times> space M1) (?f  sets E)"
179     unfolding sigma_def pair_measure_generator_swap[symmetric]
180     by (simp add: pair_measure_generator_def)
181   also have "\<dots> = ?f  sigma_sets (space M1 \<times> space M2) (sets E)"
182     using M1.sets_into_space M2.sets_into_space
183     by (intro sigma_sets_vimage) (auto simp: pair_measure_generator_def)
184   also have "\<dots> = ?f  sets P"
185     unfolding pair_measure_def pair_measure_generator_def sigma_def by simp
186   finally show ?thesis
187     using Q by (subst *) auto
188 qed
190 lemma (in pair_sigma_algebra) pair_sigma_algebra_swap_measurable:
191   shows "(\<lambda>(x,y). (y, x)) \<in> measurable P (M2 \<Otimes>\<^isub>M M1)"
192     (is "?f \<in> measurable ?P ?Q")
193   unfolding measurable_def
194 proof (intro CollectI conjI Pi_I ballI)
195   fix x assume "x \<in> space ?P" then show "(case x of (x, y) \<Rightarrow> (y, x)) \<in> space ?Q"
196     unfolding pair_measure_generator_def pair_measure_def by auto
197 next
198   fix A assume "A \<in> sets (M2 \<Otimes>\<^isub>M M1)"
199   interpret Q: pair_sigma_algebra M2 M1 by default
200   from Q.sets_pair_sigma_algebra_swap[OF A \<in> sets (M2 \<Otimes>\<^isub>M M1)]
201   show "?f - A \<inter> space ?P \<in> sets ?P" by simp
202 qed
204 lemma (in pair_sigma_algebra) measurable_cut_fst[simp,intro]:
205   assumes "Q \<in> sets P" shows "Pair x - Q \<in> sets M2"
206 proof -
207   let ?Q' = "{Q. Q \<subseteq> space P \<and> Pair x - Q \<in> sets M2}"
208   let ?Q = "\<lparr> space = space P, sets = ?Q' \<rparr>"
209   interpret Q: sigma_algebra ?Q
210     proof qed (auto simp: vimage_UN vimage_Diff space_pair_measure)
211   have "sets E \<subseteq> sets ?Q"
212     using M1.sets_into_space M2.sets_into_space
213     by (auto simp: pair_measure_generator_def space_pair_measure)
214   then have "sets P \<subseteq> sets ?Q"
215     apply (subst pair_measure_def, intro Q.sets_sigma_subset)
216     by (simp add: pair_measure_def)
217   with assms show ?thesis by auto
218 qed
220 lemma (in pair_sigma_algebra) measurable_cut_snd:
221   assumes Q: "Q \<in> sets P" shows "(\<lambda>x. (x, y)) - Q \<in> sets M1" (is "?cut Q \<in> sets M1")
222 proof -
223   interpret Q: pair_sigma_algebra M2 M1 by default
224   from Q.measurable_cut_fst[OF sets_pair_sigma_algebra_swap[OF Q], of y]
225   show ?thesis by (simp add: vimage_compose[symmetric] comp_def)
226 qed
228 lemma (in pair_sigma_algebra) measurable_pair_image_snd:
229   assumes m: "f \<in> measurable P M" and "x \<in> space M1"
230   shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
231   unfolding measurable_def
232 proof (intro CollectI conjI Pi_I ballI)
233   fix y assume "y \<in> space M2" with f \<in> measurable P M x \<in> space M1
234   show "f (x, y) \<in> space M"
235     unfolding measurable_def pair_measure_generator_def pair_measure_def by auto
236 next
237   fix A assume "A \<in> sets M"
238   then have "Pair x - (f - A \<inter> space P) \<in> sets M2" (is "?C \<in> _")
239     using f \<in> measurable P M
240     by (intro measurable_cut_fst) (auto simp: measurable_def)
241   also have "?C = (\<lambda>y. f (x, y)) - A \<inter> space M2"
242     using x \<in> space M1 by (auto simp: pair_measure_generator_def pair_measure_def)
243   finally show "(\<lambda>y. f (x, y)) - A \<inter> space M2 \<in> sets M2" .
244 qed
246 lemma (in pair_sigma_algebra) measurable_pair_image_fst:
247   assumes m: "f \<in> measurable P M" and "y \<in> space M2"
248   shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
249 proof -
250   interpret Q: pair_sigma_algebra M2 M1 by default
251   from Q.measurable_pair_image_snd[OF measurable_comp y \<in> space M2,
252                                       OF Q.pair_sigma_algebra_swap_measurable m]
253   show ?thesis by simp
254 qed
256 lemma (in pair_sigma_algebra) Int_stable_pair_measure_generator: "Int_stable E"
257   unfolding Int_stable_def
258 proof (intro ballI)
259   fix A B assume "A \<in> sets E" "B \<in> sets E"
260   then obtain A1 A2 B1 B2 where "A = A1 \<times> A2" "B = B1 \<times> B2"
261     "A1 \<in> sets M1" "A2 \<in> sets M2" "B1 \<in> sets M1" "B2 \<in> sets M2"
262     unfolding pair_measure_generator_def by auto
263   then show "A \<inter> B \<in> sets E"
264     by (auto simp add: times_Int_times pair_measure_generator_def)
265 qed
267 lemma finite_measure_cut_measurable:
268   fixes M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
269   assumes "sigma_finite_measure M1" "finite_measure M2"
270   assumes "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)"
271   shows "(\<lambda>x. measure M2 (Pair x - Q)) \<in> borel_measurable M1"
272     (is "?s Q \<in> _")
273 proof -
274   interpret M1: sigma_finite_measure M1 by fact
275   interpret M2: finite_measure M2 by fact
276   interpret pair_sigma_algebra M1 M2 by default
277   have [intro]: "sigma_algebra M1" by fact
278   have [intro]: "sigma_algebra M2" by fact
279   let ?D = "\<lparr> space = space P, sets = {A\<in>sets P. ?s A \<in> borel_measurable M1}  \<rparr>"
280   note space_pair_measure[simp]
281   interpret dynkin_system ?D
282   proof (intro dynkin_systemI)
283     fix A assume "A \<in> sets ?D" then show "A \<subseteq> space ?D"
284       using sets_into_space by simp
285   next
286     from top show "space ?D \<in> sets ?D"
287       by (auto simp add: if_distrib intro!: M1.measurable_If)
288   next
289     fix A assume "A \<in> sets ?D"
290     with sets_into_space have "\<And>x. measure M2 (Pair x - (space M1 \<times> space M2 - A)) =
291         (if x \<in> space M1 then measure M2 (space M2) - ?s A x else 0)"
292       by (auto intro!: M2.measure_compl simp: vimage_Diff)
293     with A \<in> sets ?D top show "space ?D - A \<in> sets ?D"
294       by (auto intro!: Diff M1.measurable_If M1.borel_measurable_ereal_diff)
295   next
296     fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> sets ?D"
297     moreover then have "\<And>x. measure M2 (\<Union>i. Pair x - F i) = (\<Sum>i. ?s (F i) x)"
298       by (intro M2.measure_countably_additive[symmetric])
299          (auto simp: disjoint_family_on_def)
300     ultimately show "(\<Union>i. F i) \<in> sets ?D"
301       by (auto simp: vimage_UN intro!: M1.borel_measurable_psuminf)
302   qed
303   have "sets P = sets ?D" apply (subst pair_measure_def)
304   proof (intro dynkin_lemma)
305     show "Int_stable E" by (rule Int_stable_pair_measure_generator)
306     from M1.sets_into_space have "\<And>A. A \<in> sets M1 \<Longrightarrow> {x \<in> space M1. x \<in> A} = A"
307       by auto
308     then show "sets E \<subseteq> sets ?D"
309       by (auto simp: pair_measure_generator_def sets_sigma if_distrib
310                intro: sigma_sets.Basic intro!: M1.measurable_If)
311   qed (auto simp: pair_measure_def)
312   with Q \<in> sets P have "Q \<in> sets ?D" by simp
313   then show "?s Q \<in> borel_measurable M1" by simp
314 qed
316 subsection {* Binary products of $\sigma$-finite measure spaces *}
318 locale pair_sigma_finite = pair_sigma_algebra M1 M2 + M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2
319   for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
321 lemma (in pair_sigma_finite) measure_cut_measurable_fst:
322   assumes "Q \<in> sets P" shows "(\<lambda>x. measure M2 (Pair x - Q)) \<in> borel_measurable M1" (is "?s Q \<in> _")
323 proof -
324   have [intro]: "sigma_algebra M1" and [intro]: "sigma_algebra M2" by default+
325   have M1: "sigma_finite_measure M1" by default
326   from M2.disjoint_sigma_finite guess F .. note F = this
327   then have F_sets: "\<And>i. F i \<in> sets M2" by auto
328   let ?C = "\<lambda>x i. F i \<inter> Pair x - Q"
329   { fix i
330     let ?R = "M2.restricted_space (F i)"
331     have [simp]: "space M1 \<times> F i \<inter> space M1 \<times> space M2 = space M1 \<times> F i"
332       using F M2.sets_into_space by auto
333     let ?R2 = "M2.restricted_space (F i)"
334     have "(\<lambda>x. measure ?R2 (Pair x - (space M1 \<times> space ?R2 \<inter> Q))) \<in> borel_measurable M1"
335     proof (intro finite_measure_cut_measurable[OF M1])
336       show "finite_measure ?R2"
337         using F by (intro M2.restricted_to_finite_measure) auto
338       have "(space M1 \<times> space ?R2) \<inter> Q \<in> (op \<inter> (space M1 \<times> F i))  sets P"
339         using Q \<in> sets P by (auto simp: image_iff)
340       also have "\<dots> = sigma_sets (space M1 \<times> F i) ((op \<inter> (space M1 \<times> F i))  sets E)"
341         unfolding pair_measure_def pair_measure_generator_def sigma_def
342         using F i \<in> sets M2 M2.sets_into_space
343         by (auto intro!: sigma_sets_Int sigma_sets.Basic)
344       also have "\<dots> \<subseteq> sets (M1 \<Otimes>\<^isub>M ?R2)"
345         using M1.sets_into_space
346         apply (auto simp: times_Int_times pair_measure_def pair_measure_generator_def sigma_def
347                     intro!: sigma_sets_subseteq)
348         apply (rule_tac x="a" in exI)
349         apply (rule_tac x="b \<inter> F i" in exI)
350         by auto
351       finally show "(space M1 \<times> space ?R2) \<inter> Q \<in> sets (M1 \<Otimes>\<^isub>M ?R2)" .
352     qed
353     moreover have "\<And>x. Pair x - (space M1 \<times> F i \<inter> Q) = ?C x i"
354       using Q \<in> sets P sets_into_space by (auto simp: space_pair_measure)
355     ultimately have "(\<lambda>x. measure M2 (?C x i)) \<in> borel_measurable M1"
356       by simp }
357   moreover
358   { fix x
359     have "(\<Sum>i. measure M2 (?C x i)) = measure M2 (\<Union>i. ?C x i)"
360     proof (intro M2.measure_countably_additive)
361       show "range (?C x) \<subseteq> sets M2"
362         using F Q \<in> sets P by (auto intro!: M2.Int)
363       have "disjoint_family F" using F by auto
364       show "disjoint_family (?C x)"
365         by (rule disjoint_family_on_bisimulation[OF disjoint_family F]) auto
366     qed
367     also have "(\<Union>i. ?C x i) = Pair x - Q"
368       using F sets_into_space Q \<in> sets P
369       by (auto simp: space_pair_measure)
370     finally have "measure M2 (Pair x - Q) = (\<Sum>i. measure M2 (?C x i))"
371       by simp }
372   ultimately show ?thesis using Q \<in> sets P F_sets
373     by (auto intro!: M1.borel_measurable_psuminf M2.Int)
374 qed
376 lemma (in pair_sigma_finite) measure_cut_measurable_snd:
377   assumes "Q \<in> sets P" shows "(\<lambda>y. M1.\<mu> ((\<lambda>x. (x, y)) - Q)) \<in> borel_measurable M2"
378 proof -
379   interpret Q: pair_sigma_finite M2 M1 by default
380   note sets_pair_sigma_algebra_swap[OF assms]
381   from Q.measure_cut_measurable_fst[OF this]
382   show ?thesis by (simp add: vimage_compose[symmetric] comp_def)
383 qed
385 lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable:
386   assumes "f \<in> measurable P M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^isub>M M1) M"
387 proof -
388   interpret Q: pair_sigma_algebra M2 M1 by default
389   have *: "(\<lambda>(x,y). f (y, x)) = f \<circ> (\<lambda>(x,y). (y, x))" by (simp add: fun_eq_iff)
390   show ?thesis
391     using Q.pair_sigma_algebra_swap_measurable assms
392     unfolding * by (rule measurable_comp)
393 qed
395 lemma (in pair_sigma_finite) pair_measure_alt:
396   assumes "A \<in> sets P"
397   shows "measure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+ x. measure M2 (Pair x - A) \<partial>M1)"
398   apply (simp add: pair_measure_def pair_measure_generator_def)
399 proof (rule M1.positive_integral_cong)
400   fix x assume "x \<in> space M1"
401   have *: "\<And>y. indicator A (x, y) = (indicator (Pair x - A) y :: ereal)"
402     unfolding indicator_def by auto
403   show "(\<integral>\<^isup>+ y. indicator A (x, y) \<partial>M2) = measure M2 (Pair x - A)"
404     unfolding *
405     apply (subst M2.positive_integral_indicator)
406     apply (rule measurable_cut_fst[OF assms])
407     by simp
408 qed
410 lemma (in pair_sigma_finite) pair_measure_times:
411   assumes A: "A \<in> sets M1" and "B \<in> sets M2"
412   shows "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = M1.\<mu> A * measure M2 B"
413 proof -
414   have "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = (\<integral>\<^isup>+ x. measure M2 B * indicator A x \<partial>M1)"
415     using assms by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt)
416   with assms show ?thesis
417     by (simp add: M1.positive_integral_cmult_indicator ac_simps)
418 qed
420 lemma (in measure_space) measure_not_negative[simp,intro]:
421   assumes A: "A \<in> sets M" shows "\<mu> A \<noteq> - \<infinity>"
422   using positive_measure[OF A] by auto
424 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
425   "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> incseq F \<and> (\<Union>i. F i) = space E \<and>
426     (\<forall>i. measure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
427 proof -
428   obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where
429     F1: "range F1 \<subseteq> sets M1" "incseq F1" "(\<Union>i. F1 i) = space M1" "\<And>i. M1.\<mu> (F1 i) \<noteq> \<infinity>" and
430     F2: "range F2 \<subseteq> sets M2" "incseq F2" "(\<Union>i. F2 i) = space M2" "\<And>i. M2.\<mu> (F2 i) \<noteq> \<infinity>"
431     using M1.sigma_finite_up M2.sigma_finite_up by auto
432   then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
433   let ?F = "\<lambda>i. F1 i \<times> F2 i"
434   show ?thesis unfolding space_pair_measure
435   proof (intro exI[of _ ?F] conjI allI)
436     show "range ?F \<subseteq> sets E" using F1 F2
437       by (fastforce intro!: pair_measure_generatorI)
438   next
439     have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
440     proof (intro subsetI)
441       fix x assume "x \<in> space M1 \<times> space M2"
442       then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
443         by (auto simp: space)
444       then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
445         using incseq F1 incseq F2 unfolding incseq_def
446         by (force split: split_max)+
447       then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
448         by (intro SigmaI) (auto simp add: min_max.sup_commute)
449       then show "x \<in> (\<Union>i. ?F i)" by auto
450     qed
451     then show "(\<Union>i. ?F i) = space E"
452       using space by (auto simp: space pair_measure_generator_def)
453   next
454     fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
455       using incseq F1 incseq F2 unfolding incseq_Suc_iff by auto
456   next
457     fix i
458     from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
459     with F1 F2 M1.positive_measure[OF this(1)] M2.positive_measure[OF this(2)]
460     show "measure P (F1 i \<times> F2 i) \<noteq> \<infinity>"
461       by (simp add: pair_measure_times)
462   qed
463 qed
465 sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P
466 proof
467   show "positive P (measure P)"
468     unfolding pair_measure_def pair_measure_generator_def sigma_def positive_def
469     by (auto intro: M1.positive_integral_positive M2.positive_integral_positive)
471   show "countably_additive P (measure P)"
473   proof (intro allI impI)
474     fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"
475     assume F: "range F \<subseteq> sets P" "disjoint_family F"
476     from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto
477     moreover from F have "\<And>i. (\<lambda>x. measure M2 (Pair x - F i)) \<in> borel_measurable M1"
478       by (intro measure_cut_measurable_fst) auto
479     moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x - F i)"
480       by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
481     moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x - F i) \<subseteq> sets M2"
482       using F by auto
483     ultimately show "(\<Sum>n. measure P (F n)) = measure P (\<Union>i. F i)"
484       by (simp add: pair_measure_alt vimage_UN M1.positive_integral_suminf[symmetric]
486                cong: M1.positive_integral_cong)
487   qed
489   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
490   show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets P \<and> (\<Union>i. F i) = space P \<and> (\<forall>i. measure P (F i) \<noteq> \<infinity>)"
491   proof (rule exI[of _ F], intro conjI)
492     show "range F \<subseteq> sets P" using F by (auto simp: pair_measure_def)
493     show "(\<Union>i. F i) = space P"
494       using F by (auto simp: pair_measure_def pair_measure_generator_def)
495     show "\<forall>i. measure P (F i) \<noteq> \<infinity>" using F by auto
496   qed
497 qed
499 lemma (in pair_sigma_algebra) sets_swap:
500   assumes "A \<in> sets P"
501   shows "(\<lambda>(x, y). (y, x)) - A \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
502     (is "_ - A \<inter> space ?Q \<in> sets ?Q")
503 proof -
504   have *: "(\<lambda>(x, y). (y, x)) - A \<inter> space ?Q = (\<lambda>(x, y). (y, x)) - A"
505     using A \<in> sets P sets_into_space by (auto simp: space_pair_measure)
506   show ?thesis
507     unfolding * using assms by (rule sets_pair_sigma_algebra_swap)
508 qed
510 lemma (in pair_sigma_finite) pair_measure_alt2:
511   assumes A: "A \<in> sets P"
512   shows "\<mu> A = (\<integral>\<^isup>+y. M1.\<mu> ((\<lambda>x. (x, y)) - A) \<partial>M2)"
513     (is "_ = ?\<nu> A")
514 proof -
515   interpret Q: pair_sigma_finite M2 M1 by default
516   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
517   have [simp]: "\<And>m. \<lparr> space = space E, sets = sets (sigma E), measure = m \<rparr> = P\<lparr> measure := m \<rparr>"
518     unfolding pair_measure_def by simp
520   have "\<mu> A = Q.\<mu> ((\<lambda>(y, x). (x, y)) - A \<inter> space Q.P)"
521   proof (rule measure_unique_Int_stable_vimage[OF Int_stable_pair_measure_generator])
522     show "measure_space P" "measure_space Q.P" by default
523     show "(\<lambda>(y, x). (x, y)) \<in> measurable Q.P P" by (rule Q.pair_sigma_algebra_swap_measurable)
524     show "sets (sigma E) = sets P" "space E = space P" "A \<in> sets (sigma E)"
525       using assms unfolding pair_measure_def by auto
526     show "range F \<subseteq> sets E" "incseq F" "(\<Union>i. F i) = space E" "\<And>i. \<mu> (F i) \<noteq> \<infinity>"
527       using F A \<in> sets P by (auto simp: pair_measure_def)
528     fix X assume "X \<in> sets E"
529     then obtain A B where X[simp]: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2"
530       unfolding pair_measure_def pair_measure_generator_def by auto
531     then have "(\<lambda>(y, x). (x, y)) - X \<inter> space Q.P = B \<times> A"
532       using M1.sets_into_space M2.sets_into_space by (auto simp: space_pair_measure)
533     then show "\<mu> X = Q.\<mu> ((\<lambda>(y, x). (x, y)) - X \<inter> space Q.P)"
534       using AB by (simp add: pair_measure_times Q.pair_measure_times ac_simps)
535   qed
536   then show ?thesis
537     using sets_into_space[OF A] Q.pair_measure_alt[OF sets_swap[OF A]]
538     by (auto simp add: Q.pair_measure_alt space_pair_measure
539              intro!: M2.positive_integral_cong arg_cong[where f="M1.\<mu>"])
540 qed
542 lemma pair_sigma_algebra_sigma:
543   assumes 1: "incseq S1" "(\<Union>i. S1 i) = space E1" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"
544   assumes 2: "decseq S2" "(\<Union>i. S2 i) = space E2" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"
545   shows "sets (sigma (pair_measure_generator (sigma E1) (sigma E2))) = sets (sigma (pair_measure_generator E1 E2))"
546     (is "sets ?S = sets ?E")
547 proof -
548   interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma)
549   interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma)
550   have P: "sets (pair_measure_generator E1 E2) \<subseteq> Pow (space E1 \<times> space E2)"
551     using E1 E2 by (auto simp add: pair_measure_generator_def)
552   interpret E: sigma_algebra ?E unfolding pair_measure_generator_def
553     using E1 E2 by (intro sigma_algebra_sigma) auto
554   { fix A assume "A \<in> sets E1"
555     then have "fst - A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"
556       using E1 2 unfolding pair_measure_generator_def by auto
557     also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto
558     also have "\<dots> \<in> sets ?E" unfolding pair_measure_generator_def sets_sigma
559       using 2 A \<in> sets E1
560       by (intro sigma_sets.Union)
561          (force simp: image_subset_iff intro!: sigma_sets.Basic)
562     finally have "fst - A \<inter> space ?E \<in> sets ?E" . }
563   moreover
564   { fix B assume "B \<in> sets E2"
565     then have "snd - B \<inter> space ?E = (\<Union>i. S1 i) \<times> B"
566       using E2 1 unfolding pair_measure_generator_def by auto
567     also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto
568     also have "\<dots> \<in> sets ?E"
569       using 1 B \<in> sets E2 unfolding pair_measure_generator_def sets_sigma
570       by (intro sigma_sets.Union)
571          (force simp: image_subset_iff intro!: sigma_sets.Basic)
572     finally have "snd - B \<inter> space ?E \<in> sets ?E" . }
573   ultimately have proj:
574     "fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"
575     using E1 E2 by (subst (1 2) E.measurable_iff_sigma)
576                    (auto simp: pair_measure_generator_def sets_sigma)
577   { fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)"
578     with proj have "fst - A \<inter> space ?E \<in> sets ?E" "snd - B \<inter> space ?E \<in> sets ?E"
579       unfolding measurable_def by simp_all
580     moreover have "A \<times> B = (fst - A \<inter> space ?E) \<inter> (snd - B \<inter> space ?E)"
581       using A B M1.sets_into_space M2.sets_into_space
582       by (auto simp: pair_measure_generator_def)
583     ultimately have "A \<times> B \<in> sets ?E" by auto }
584   then have "sigma_sets (space ?E) (sets (pair_measure_generator (sigma E1) (sigma E2))) \<subseteq> sets ?E"
585     by (intro E.sigma_sets_subset) (auto simp add: pair_measure_generator_def sets_sigma)
586   then have subset: "sets ?S \<subseteq> sets ?E"
587     by (simp add: sets_sigma pair_measure_generator_def)
588   show "sets ?S = sets ?E"
589   proof (intro set_eqI iffI)
590     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
591       unfolding sets_sigma
592     proof induct
593       case (Basic A) then show ?case
594         by (auto simp: pair_measure_generator_def sets_sigma intro: sigma_sets.Basic)
595     qed (auto intro: sigma_sets.intros simp: pair_measure_generator_def)
596   next
597     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
598   qed
599 qed
601 section "Fubinis theorem"
603 lemma (in pair_sigma_finite) simple_function_cut:
604   assumes f: "simple_function P f" "\<And>x. 0 \<le> f x"
605   shows "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
606     and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
607 proof -
608   have f_borel: "f \<in> borel_measurable P"
609     using f(1) by (rule borel_measurable_simple_function)
610   let ?F = "\<lambda>z. f - {z} \<inter> space P"
611   let ?F' = "\<lambda>x z. Pair x - ?F z"
612   { fix x assume "x \<in> space M1"
613     have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y"
614       by (auto simp: indicator_def)
615     have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space P" using x \<in> space M1
616       by (simp add: space_pair_measure)
617     moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel
618       by (intro borel_measurable_vimage measurable_cut_fst)
619     ultimately have "simple_function M2 (\<lambda> y. f (x, y))"
620       apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _])
621       apply (rule simple_function_indicator_representation[OF f(1)])
622       using x \<in> space M1 by (auto simp del: space_sigma) }
623   note M2_sf = this
624   { fix x assume x: "x \<in> space M1"
625     then have "(\<integral>\<^isup>+y. f (x, y) \<partial>M2) = (\<Sum>z\<in>f  space P. z * M2.\<mu> (?F' x z))"
626       unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x] f(2)]
627       unfolding simple_integral_def
628     proof (safe intro!: setsum_mono_zero_cong_left)
629       from f(1) show "finite (f  space P)" by (rule simple_functionD)
630     next
631       fix y assume "y \<in> space M2" then show "f (x, y) \<in> f  space P"
632         using x \<in> space M1 by (auto simp: space_pair_measure)
633     next
634       fix x' y assume "(x', y) \<in> space P"
635         "f (x', y) \<notin> (\<lambda>y. f (x, y))  space M2"
636       then have *: "?F' x (f (x', y)) = {}"
637         by (force simp: space_pair_measure)
638       show  "f (x', y) * M2.\<mu> (?F' x (f (x', y))) = 0"
639         unfolding * by simp
640     qed (simp add: vimage_compose[symmetric] comp_def
641                    space_pair_measure) }
642   note eq = this
643   moreover have "\<And>z. ?F z \<in> sets P"
644     by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma)
645   moreover then have "\<And>z. (\<lambda>x. M2.\<mu> (?F' x z)) \<in> borel_measurable M1"
646     by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int)
647   moreover have *: "\<And>i x. 0 \<le> M2.\<mu> (Pair x - (f - {i} \<inter> space P))"
648     using f(1)[THEN simple_functionD(2)] f(2) by (intro M2.positive_measure measurable_cut_fst)
649   moreover { fix i assume "i \<in> fspace P"
650     with * have "\<And>x. 0 \<le> i * M2.\<mu> (Pair x - (f - {i} \<inter> space P))"
651       using f(2) by auto }
652   ultimately
653   show "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
654     and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f" using f(2)
655     by (auto simp del: vimage_Int cong: measurable_cong
656              intro!: M1.borel_measurable_ereal_setsum setsum_cong
657              simp add: M1.positive_integral_setsum simple_integral_def
658                        M1.positive_integral_cmult
659                        M1.positive_integral_cong[OF eq]
660                        positive_integral_eq_simple_integral[OF f]
661                        pair_measure_alt[symmetric])
662 qed
664 lemma (in pair_sigma_finite) positive_integral_fst_measurable:
665   assumes f: "f \<in> borel_measurable P"
666   shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
667       (is "?C f \<in> borel_measurable M1")
668     and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
669 proof -
670   from borel_measurable_implies_simple_function_sequence'[OF f] guess F . note F = this
671   then have F_borel: "\<And>i. F i \<in> borel_measurable P"
672     by (auto intro: borel_measurable_simple_function)
673   note sf = simple_function_cut[OF F(1,5)]
674   then have "(\<lambda>x. SUP i. ?C (F i) x) \<in> borel_measurable M1"
675     using F(1) by auto
676   moreover
677   { fix x assume "x \<in> space M1"
678     from F measurable_pair_image_snd[OF F_borelx \<in> space M1]
679     have "(\<integral>\<^isup>+y. (SUP i. F i (x, y)) \<partial>M2) = (SUP i. ?C (F i) x)"
680       by (intro M2.positive_integral_monotone_convergence_SUP)
681          (auto simp: incseq_Suc_iff le_fun_def)
682     then have "(SUP i. ?C (F i) x) = ?C f x"
683       unfolding F(4) positive_integral_max_0 by simp }
684   note SUPR_C = this
685   ultimately show "?C f \<in> borel_measurable M1"
686     by (simp cong: measurable_cong)
687   have "(\<integral>\<^isup>+x. (SUP i. F i x) \<partial>P) = (SUP i. integral\<^isup>P P (F i))"
688     using F_borel F
689     by (intro positive_integral_monotone_convergence_SUP) auto
690   also have "(SUP i. integral\<^isup>P P (F i)) = (SUP i. \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1)"
691     unfolding sf(2) by simp
692   also have "\<dots> = \<integral>\<^isup>+ x. (SUP i. \<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1" using F sf(1)
693     by (intro M1.positive_integral_monotone_convergence_SUP[symmetric])
694        (auto intro!: M2.positive_integral_mono M2.positive_integral_positive
695                 simp: incseq_Suc_iff le_fun_def)
696   also have "\<dots> = \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. (SUP i. F i (x, y)) \<partial>M2) \<partial>M1"
697     using F_borel F(2,5)
698     by (auto intro!: M1.positive_integral_cong M2.positive_integral_monotone_convergence_SUP[symmetric]
699              simp: incseq_Suc_iff le_fun_def measurable_pair_image_snd)
700   finally show "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
701     using F by (simp add: positive_integral_max_0)
702 qed
704 lemma (in pair_sigma_finite) measure_preserving_swap:
705   "(\<lambda>(x,y). (y, x)) \<in> measure_preserving (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
706 proof
707   interpret Q: pair_sigma_finite M2 M1 by default
708   show *: "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
709     using pair_sigma_algebra_swap_measurable .
710   fix X assume "X \<in> sets (M2 \<Otimes>\<^isub>M M1)"
711   from measurable_sets[OF * this] this Q.sets_into_space[OF this]
712   show "measure (M1 \<Otimes>\<^isub>M M2) ((\<lambda>(x, y). (y, x)) - X \<inter> space P) = measure (M2 \<Otimes>\<^isub>M M1) X"
713     by (auto intro!: M1.positive_integral_cong arg_cong[where f="M2.\<mu>"]
714       simp: pair_measure_alt Q.pair_measure_alt2 space_pair_measure)
715 qed
717 lemma (in pair_sigma_finite) positive_integral_product_swap:
718   assumes f: "f \<in> borel_measurable P"
719   shows "(\<integral>\<^isup>+x. f (case x of (x,y)\<Rightarrow>(y,x)) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P P f"
720 proof -
721   interpret Q: pair_sigma_finite M2 M1 by default
722   have "sigma_algebra P" by default
723   with f show ?thesis
724     by (subst Q.positive_integral_vimage[OF _ Q.measure_preserving_swap]) auto
725 qed
727 lemma (in pair_sigma_finite) positive_integral_snd_measurable:
728   assumes f: "f \<in> borel_measurable P"
729   shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>P P f"
730 proof -
731   interpret Q: pair_sigma_finite M2 M1 by default
732   note pair_sigma_algebra_measurable[OF f]
733   from Q.positive_integral_fst_measurable[OF this]
734   have "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P)"
735     by simp
736   also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P) = integral\<^isup>P P f"
737     unfolding positive_integral_product_swap[OF f, symmetric]
738     by (auto intro!: Q.positive_integral_cong)
739   finally show ?thesis .
740 qed
742 lemma (in pair_sigma_finite) Fubini:
743   assumes f: "f \<in> borel_measurable P"
744   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)"
745   unfolding positive_integral_snd_measurable[OF assms]
746   unfolding positive_integral_fst_measurable[OF assms] ..
748 lemma (in pair_sigma_finite) AE_pair:
749   assumes "AE x in P. Q x"
750   shows "AE x in M1. (AE y in M2. Q (x, y))"
751 proof -
752   obtain N where N: "N \<in> sets P" "\<mu> N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N"
753     using assms unfolding almost_everywhere_def by auto
754   show ?thesis
755   proof (rule M1.AE_I)
756     from N measure_cut_measurable_fst[OF N \<in> sets P]
757     show "M1.\<mu> {x\<in>space M1. M2.\<mu> (Pair x - N) \<noteq> 0} = 0"
758       by (auto simp: pair_measure_alt M1.positive_integral_0_iff)
759     show "{x \<in> space M1. M2.\<mu> (Pair x - N) \<noteq> 0} \<in> sets M1"
760       by (intro M1.borel_measurable_ereal_neq_const measure_cut_measurable_fst N)
761     { fix x assume "x \<in> space M1" "M2.\<mu> (Pair x - N) = 0"
762       have "M2.almost_everywhere (\<lambda>y. Q (x, y))"
763       proof (rule M2.AE_I)
764         show "M2.\<mu> (Pair x - N) = 0" by fact
765         show "Pair x - N \<in> sets M2" by (intro measurable_cut_fst N)
766         show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x - N"
767           using N x \<in> space M1 unfolding space_sigma space_pair_measure by auto
768       qed }
769     then show "{x \<in> space M1. \<not> M2.almost_everywhere (\<lambda>y. Q (x, y))} \<subseteq> {x \<in> space M1. M2.\<mu> (Pair x - N) \<noteq> 0}"
770       by auto
771   qed
772 qed
774 lemma (in pair_sigma_algebra) measurable_product_swap:
775   "f \<in> measurable (M2 \<Otimes>\<^isub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable P M"
776 proof -
777   interpret Q: pair_sigma_algebra M2 M1 by default
778   show ?thesis
779     using pair_sigma_algebra_measurable[of "\<lambda>(x,y). f (y, x)"]
780     by (auto intro!: pair_sigma_algebra_measurable Q.pair_sigma_algebra_measurable iffI)
781 qed
783 lemma (in pair_sigma_finite) integrable_product_swap:
784   assumes "integrable P f"
785   shows "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x))"
786 proof -
787   interpret Q: pair_sigma_finite M2 M1 by default
788   have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
789   show ?thesis unfolding *
790     using assms unfolding integrable_def
791     apply (subst (1 2) positive_integral_product_swap)
792     using integrable P f unfolding integrable_def
793     by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])
794 qed
796 lemma (in pair_sigma_finite) integrable_product_swap_iff:
797   "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable P f"
798 proof -
799   interpret Q: pair_sigma_finite M2 M1 by default
800   from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]
801   show ?thesis by auto
802 qed
804 lemma (in pair_sigma_finite) integral_product_swap:
805   assumes "integrable P f"
806   shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L P f"
807 proof -
808   interpret Q: pair_sigma_finite M2 M1 by default
809   have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
810   show ?thesis
811     unfolding lebesgue_integral_def *
812     apply (subst (1 2) positive_integral_product_swap)
813     using integrable P f unfolding integrable_def
814     by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])
815 qed
817 lemma (in pair_sigma_finite) integrable_fst_measurable:
818   assumes f: "integrable P f"
819   shows "M1.almost_everywhere (\<lambda>x. integrable M2 (\<lambda> y. f (x, y)))" (is "?AE")
820     and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L P f" (is "?INT")
821 proof -
822   let ?pf = "\<lambda>x. ereal (f x)" and ?nf = "\<lambda>x. ereal (- f x)"
823   have
824     borel: "?nf \<in> borel_measurable P""?pf \<in> borel_measurable P" and
825     int: "integral\<^isup>P P ?nf \<noteq> \<infinity>" "integral\<^isup>P P ?pf \<noteq> \<infinity>"
826     using assms by auto
827   have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
828      "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
829     using borel[THEN positive_integral_fst_measurable(1)] int
830     unfolding borel[THEN positive_integral_fst_measurable(2)] by simp_all
831   with borel[THEN positive_integral_fst_measurable(1)]
832   have AE_pos: "AE x in M1. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
833     "AE x in M1. (\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2) \<noteq> \<infinity>"
834     by (auto intro!: M1.positive_integral_PInf_AE )
835   then have AE: "AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
836     "AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
837     by (auto simp: M2.positive_integral_positive)
838   from AE_pos show ?AE using assms
839     by (simp add: measurable_pair_image_snd integrable_def)
840   { fix f have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
841       using M2.positive_integral_positive
842       by (intro M1.positive_integral_cong_pos) (auto simp: ereal_uminus_le_reorder)
843     then have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = 0" by simp }
844   note this[simp]
845   { fix f assume borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable P"
846       and int: "integral\<^isup>P P (\<lambda>x. ereal (f x)) \<noteq> \<infinity>"
847       and AE: "M1.almost_everywhere (\<lambda>x. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>)"
848     have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f")
849     proof (intro integrable_def[THEN iffD2] conjI)
850       show "?f \<in> borel_measurable M1"
851         using borel by (auto intro!: M1.borel_measurable_real_of_ereal positive_integral_fst_measurable)
852       have "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y))  \<partial>M2) \<partial>M1)"
853         using AE M2.positive_integral_positive
854         by (auto intro!: M1.positive_integral_cong_AE simp: ereal_real)
855       then show "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) \<noteq> \<infinity>"
856         using positive_integral_fst_measurable[OF borel] int by simp
857       have "(\<integral>\<^isup>+x. ereal (- ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
858         by (intro M1.positive_integral_cong_pos)
859            (simp add: M2.positive_integral_positive real_of_ereal_pos)
860       then show "(\<integral>\<^isup>+x. ereal (- ?f x) \<partial>M1) \<noteq> \<infinity>" by simp
861     qed }
862   with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)]
863   show ?INT
864     unfolding lebesgue_integral_def[of P] lebesgue_integral_def[of M2]
865       borel[THEN positive_integral_fst_measurable(2), symmetric]
866     using AE[THEN M1.integral_real]
867     by simp
868 qed
870 lemma (in pair_sigma_finite) integrable_snd_measurable:
871   assumes f: "integrable P f"
872   shows "M2.almost_everywhere (\<lambda>y. integrable M1 (\<lambda>x. f (x, y)))" (is "?AE")
873     and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L P f" (is "?INT")
874 proof -
875   interpret Q: pair_sigma_finite M2 M1 by default
876   have Q_int: "integrable Q.P (\<lambda>(x, y). f (y, x))"
877     using f unfolding integrable_product_swap_iff .
878   show ?INT
879     using Q.integrable_fst_measurable(2)[OF Q_int]
880     using integral_product_swap[OF f] by simp
881   show ?AE
882     using Q.integrable_fst_measurable(1)[OF Q_int]
883     by simp
884 qed
886 lemma (in pair_sigma_finite) Fubini_integral:
887   assumes f: "integrable P f"
888   shows "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1)"
889   unfolding integrable_snd_measurable[OF assms]
890   unfolding integrable_fst_measurable[OF assms] ..
892 section "Products on finite spaces"
894 lemma sigma_sets_pair_measure_generator_finite:
895   assumes "finite A" and "finite B"
896   shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<in> Pow A \<and> b \<in> Pow B} = Pow (A \<times> B)"
897   (is "sigma_sets ?prod ?sets = _")
898 proof safe
899   have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
900   fix x assume subset: "x \<subseteq> A \<times> B"
901   hence "finite x" using fin by (rule finite_subset)
902   from this subset show "x \<in> sigma_sets ?prod ?sets"
903   proof (induct x)
904     case empty show ?case by (rule sigma_sets.Empty)
905   next
906     case (insert a x)
907     hence "{a} \<in> sigma_sets ?prod ?sets"
908       by (auto simp: pair_measure_generator_def intro!: sigma_sets.Basic)
909     moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
910     ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
911   qed
912 next
913   fix x a b
914   assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
915   from sigma_sets_into_sp[OF _ this(1)] this(2)
916   show "a \<in> A" and "b \<in> B" by auto
917 qed
919 locale pair_finite_sigma_algebra = pair_sigma_algebra M1 M2 + M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2 for M1 M2
921 lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra:
922   shows "P = \<lparr> space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2), \<dots> = algebra.more P \<rparr>"
923 proof -
924   show ?thesis
925     using sigma_sets_pair_measure_generator_finite[OF M1.finite_space M2.finite_space]
926     by (intro algebra.equality) (simp_all add: pair_measure_def pair_measure_generator_def sigma_def)
927 qed
929 sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P
930 proof
931   show "finite (space P)"
932     using M1.finite_space M2.finite_space
933     by (subst finite_pair_sigma_algebra) simp
934   show "sets P = Pow (space P)"
935     by (subst (1 2) finite_pair_sigma_algebra) simp
936 qed
938 locale pair_finite_space = pair_sigma_finite M1 M2 + pair_finite_sigma_algebra M1 M2 +
939   M1: finite_measure_space M1 + M2: finite_measure_space M2 for M1 M2
941 lemma (in pair_finite_space) pair_measure_Pair[simp]:
942   assumes "a \<in> space M1" "b \<in> space M2"
943   shows "\<mu> {(a, b)} = M1.\<mu> {a} * M2.\<mu> {b}"
944 proof -
945   have "\<mu> ({a}\<times>{b}) = M1.\<mu> {a} * M2.\<mu> {b}"
946     using M1.sets_eq_Pow M2.sets_eq_Pow assms
947     by (subst pair_measure_times) auto
948   then show ?thesis by simp
949 qed
951 lemma (in pair_finite_space) pair_measure_singleton[simp]:
952   assumes "x \<in> space M1 \<times> space M2"
953   shows "\<mu> {x} = M1.\<mu> {fst x} * M2.\<mu> {snd x}"
954   using pair_measure_Pair assms by (cases x) auto
956 sublocale pair_finite_space \<subseteq> finite_measure_space P
957 proof unfold_locales
958   show "measure P (space P) \<noteq> \<infinity>"
959     by (subst (2) finite_pair_sigma_algebra)
960        (simp add: pair_measure_times)
961 qed
963 end