src/HOL/Probability/Product_Measure.thy
 author hoelzl Tue Mar 22 18:53:05 2011 +0100 (2011-03-22) changeset 42066 6db76c88907a parent 41981 cdf7693bbe08 child 42067 66c8281349ec permissions -rw-r--r--
generalized Caratheodory from algebra to ring_of_sets
1 theory Product_Measure
2 imports Lebesgue_Integration
3 begin
5 lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
6 proof
7   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
8     by induct (insert A \<subseteq> B, auto intro: sigma_sets.intros)
9 qed
11 lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
12   by auto
14 lemma Pair_vimage_times[simp]: "\<And>A B x. Pair x - (A \<times> B) = (if x \<in> A then B else {})"
15   by auto
17 lemma rev_Pair_vimage_times[simp]: "\<And>A B y. (\<lambda>x. (x, y)) - (A \<times> B) = (if y \<in> B then A else {})"
18   by auto
20 lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))"
21   by (cases x) simp
23 lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
24   by (auto simp: fun_eq_iff)
26 abbreviation
27   "Pi\<^isub>E A B \<equiv> Pi A B \<inter> extensional A"
29 syntax
30   "_PiE"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PIE _:_./ _)" 10)
32 syntax (xsymbols)
33   "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
35 syntax (HTML output)
36   "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
38 translations
39   "PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)"
41 abbreviation
42   funcset_extensional :: "['a set, 'b set] => ('a => 'b) set"
43     (infixr "->\<^isub>E" 60) where
44   "A ->\<^isub>E B \<equiv> Pi\<^isub>E A (%_. B)"
46 notation (xsymbols)
47   funcset_extensional  (infixr "\<rightarrow>\<^isub>E" 60)
49 lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}"
50   by safe (auto simp add: extensional_def fun_eq_iff)
52 lemma extensional_insert[intro, simp]:
53   assumes "a \<in> extensional (insert i I)"
54   shows "a(i := b) \<in> extensional (insert i I)"
55   using assms unfolding extensional_def by auto
57 lemma extensional_Int[simp]:
58   "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
59   unfolding extensional_def by auto
61 definition
62   "merge I x J y = (\<lambda>i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
64 lemma merge_apply[simp]:
65   "I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"
66   "I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"
67   "J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"
68   "J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"
69   "i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I x J y i = undefined"
70   unfolding merge_def by auto
72 lemma merge_commute:
73   "I \<inter> J = {} \<Longrightarrow> merge I x J y = merge J y I x"
74   by (auto simp: merge_def intro!: ext)
76 lemma Pi_cancel_merge_range[simp]:
77   "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"
78   "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"
79   "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"
80   "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"
81   by (auto simp: Pi_def)
83 lemma Pi_cancel_merge[simp]:
84   "I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
85   "J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
86   "I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
87   "J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
88   by (auto simp: Pi_def)
90 lemma extensional_merge[simp]: "merge I x J y \<in> extensional (I \<union> J)"
91   by (auto simp: extensional_def)
93 lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
94   by (auto simp: restrict_def Pi_def)
96 lemma restrict_merge[simp]:
97   "I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"
98   "I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
99   "J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"
100   "J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
101   by (auto simp: restrict_def intro!: ext)
103 lemma extensional_insert_undefined[intro, simp]:
104   assumes "a \<in> extensional (insert i I)"
105   shows "a(i := undefined) \<in> extensional I"
106   using assms unfolding extensional_def by auto
108 lemma extensional_insert_cancel[intro, simp]:
109   assumes "a \<in> extensional I"
110   shows "a \<in> extensional (insert i I)"
111   using assms unfolding extensional_def by auto
113 lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I x {i} y = restrict (x(i := y i)) (insert i I)"
114   unfolding merge_def by (auto simp: fun_eq_iff)
116 lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
117   by auto
119 lemma PiE_Int: "(Pi\<^isub>E I A) \<inter> (Pi\<^isub>E I B) = Pi\<^isub>E I (\<lambda>x. A x \<inter> B x)"
120   by auto
122 lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B"
123   by (auto simp: Pi_def)
125 lemma Pi_split_insert_domain[simp]: "x \<in> Pi (insert i I) X \<longleftrightarrow> x \<in> Pi I X \<and> x i \<in> X i"
126   by (auto simp: Pi_def)
128 lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X"
129   by (auto simp: Pi_def)
131 lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
132   by (auto simp: Pi_def)
134 lemma restrict_vimage:
135   assumes "I \<inter> J = {}"
136   shows "(\<lambda>x. (restrict x I, restrict x J)) - (Pi\<^isub>E I E \<times> Pi\<^isub>E J F) = Pi (I \<union> J) (merge I E J F)"
137   using assms by (auto simp: restrict_Pi_cancel)
139 lemma merge_vimage:
140   assumes "I \<inter> J = {}"
141   shows "(\<lambda>(x,y). merge I x J y) - Pi\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E"
142   using assms by (auto simp: restrict_Pi_cancel)
144 lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
145   by (auto simp: restrict_def intro!: ext)
147 lemma merge_restrict[simp]:
148   "merge I (restrict x I) J y = merge I x J y"
149   "merge I x J (restrict y J) = merge I x J y"
150   unfolding merge_def by (auto intro!: ext)
152 lemma merge_x_x_eq_restrict[simp]:
153   "merge I x J x = restrict x (I \<union> J)"
154   unfolding merge_def by (auto intro!: ext)
156 lemma Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I - {i}) B \<and> f i \<in> A"
157   apply auto
158   apply (drule_tac x=x in Pi_mem)
159   apply (simp_all split: split_if_asm)
160   apply (drule_tac x=i in Pi_mem)
161   apply (auto dest!: Pi_mem)
162   done
164 lemma Pi_UN:
165   fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
166   assumes "finite I" and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
167   shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
168 proof (intro set_eqI iffI)
169   fix f assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
170   then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" by auto
171   from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
172   obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
173     using finite I finite_nat_set_iff_bounded_le[of "nI"] by auto
174   have "f \<in> Pi I (A k)"
175   proof (intro Pi_I)
176     fix i assume "i \<in> I"
177     from mono[OF this, of "n i" k] k[OF this] n[OF this]
178     show "f i \<in> A k i" by auto
179   qed
180   then show "f \<in> (\<Union>n. Pi I (A n))" by auto
181 qed auto
183 lemma PiE_cong:
184   assumes "\<And>i. i\<in>I \<Longrightarrow> A i = B i"
185   shows "Pi\<^isub>E I A = Pi\<^isub>E I B"
186   using assms by (auto intro!: Pi_cong)
188 lemma restrict_upd[simp]:
189   "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
190   by (auto simp: fun_eq_iff)
192 lemma Pi_eq_subset:
193   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
194   assumes eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and "i \<in> I"
195   shows "F i \<subseteq> F' i"
196 proof
197   fix x assume "x \<in> F i"
198   with ne have "\<forall>j. \<exists>y. ((j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined))" by auto
199   from choice[OF this] guess f .. note f = this
200   then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def)
201   then have "f \<in> Pi\<^isub>E I F'" using assms by simp
202   then show "x \<in> F' i" using f i \<in> I by auto
203 qed
205 lemma Pi_eq_iff_not_empty:
206   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
207   shows "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
208 proof (intro iffI ballI)
209   fix i assume eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and i: "i \<in> I"
210   show "F i = F' i"
211     using Pi_eq_subset[of I F F', OF ne eq i]
212     using Pi_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
213     by auto
214 qed auto
216 lemma Pi_eq_empty_iff:
217   "Pi\<^isub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
218 proof
219   assume "Pi\<^isub>E I F = {}"
220   show "\<exists>i\<in>I. F i = {}"
221   proof (rule ccontr)
222     assume "\<not> ?thesis"
223     then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)" by auto
224     from choice[OF this] guess f ..
225     then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def)
226     with Pi\<^isub>E I F = {} show False by auto
227   qed
228 qed auto
230 lemma Pi_eq_iff:
231   "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i) \<or> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
232 proof (intro iffI disjCI)
233   assume eq[simp]: "Pi\<^isub>E I F = Pi\<^isub>E I F'"
234   assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
235   then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
236     using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto
237   with Pi_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i" by auto
238 next
239   assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"
240   then show "Pi\<^isub>E I F = Pi\<^isub>E I F'"
241     using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto
242 qed
244 section "Binary products"
246 definition
247   "pair_measure_generator A B =
248     \<lparr> space = space A \<times> space B,
249       sets = {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B},
250       measure = \<lambda>X. \<integral>\<^isup>+x. (\<integral>\<^isup>+y. indicator X (x,y) \<partial>B) \<partial>A \<rparr>"
252 definition pair_measure (infixr "\<Otimes>\<^isub>M" 80) where
253   "A \<Otimes>\<^isub>M B = sigma (pair_measure_generator A B)"
255 locale pair_sigma_algebra = M1: sigma_algebra M1 + M2: sigma_algebra M2
256   for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
258 abbreviation (in pair_sigma_algebra)
259   "E \<equiv> pair_measure_generator M1 M2"
261 abbreviation (in pair_sigma_algebra)
262   "P \<equiv> M1 \<Otimes>\<^isub>M M2"
264 lemma sigma_algebra_pair_measure:
265   "sets M1 \<subseteq> Pow (space M1) \<Longrightarrow> sets M2 \<subseteq> Pow (space M2) \<Longrightarrow> sigma_algebra (pair_measure M1 M2)"
266   by (force simp: pair_measure_def pair_measure_generator_def intro!: sigma_algebra_sigma)
268 sublocale pair_sigma_algebra \<subseteq> sigma_algebra P
269   using M1.space_closed M2.space_closed
270   by (rule sigma_algebra_pair_measure)
272 lemma pair_measure_generatorI[intro, simp]:
273   "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (pair_measure_generator A B)"
274   by (auto simp add: pair_measure_generator_def)
276 lemma pair_measureI[intro, simp]:
277   "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^isub>M B)"
278   by (auto simp add: pair_measure_def)
280 lemma space_pair_measure:
281   "space (A \<Otimes>\<^isub>M B) = space A \<times> space B"
282   by (simp add: pair_measure_def pair_measure_generator_def)
284 lemma sets_pair_measure_generator:
285   "sets (pair_measure_generator N M) = (\<lambda>(x, y). x \<times> y)  (sets N \<times> sets M)"
286   unfolding pair_measure_generator_def by auto
288 lemma pair_measure_generator_sets_into_space:
289   assumes "sets M \<subseteq> Pow (space M)" "sets N \<subseteq> Pow (space N)"
290   shows "sets (pair_measure_generator M N) \<subseteq> Pow (space (pair_measure_generator M N))"
291   using assms by (auto simp: pair_measure_generator_def)
293 lemma pair_measure_generator_Int_snd:
294   assumes "sets S1 \<subseteq> Pow (space S1)"
295   shows "sets (pair_measure_generator S1 (algebra.restricted_space S2 A)) =
296          sets (algebra.restricted_space (pair_measure_generator S1 S2) (space S1 \<times> A))"
297   (is "?L = ?R")
298   apply (auto simp: pair_measure_generator_def image_iff)
299   using assms
300   apply (rule_tac x="a \<times> xa" in exI)
301   apply force
302   using assms
303   apply (rule_tac x="a" in exI)
304   apply (rule_tac x="b \<inter> A" in exI)
305   apply auto
306   done
308 lemma (in pair_sigma_algebra)
309   shows measurable_fst[intro!, simp]:
310     "fst \<in> measurable P M1" (is ?fst)
311   and measurable_snd[intro!, simp]:
312     "snd \<in> measurable P M2" (is ?snd)
313 proof -
314   { fix X assume "X \<in> sets M1"
315     then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. fst - X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
316       apply - apply (rule bexI[of _ X]) apply (rule bexI[of _ "space M2"])
317       using M1.sets_into_space by force+ }
318   moreover
319   { fix X assume "X \<in> sets M2"
320     then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. snd - X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
321       apply - apply (rule bexI[of _ "space M1"]) apply (rule bexI[of _ X])
322       using M2.sets_into_space by force+ }
323   ultimately have "?fst \<and> ?snd"
324     by (fastsimp simp: measurable_def sets_sigma space_pair_measure
325                  intro!: sigma_sets.Basic)
326   then show ?fst ?snd by auto
327 qed
329 lemma (in pair_sigma_algebra) measurable_pair_iff:
330   assumes "sigma_algebra M"
331   shows "f \<in> measurable M P \<longleftrightarrow>
332     (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"
333 proof -
334   interpret M: sigma_algebra M by fact
335   from assms show ?thesis
336   proof (safe intro!: measurable_comp[where b=P])
337     assume f: "(fst \<circ> f) \<in> measurable M M1" and s: "(snd \<circ> f) \<in> measurable M M2"
338     show "f \<in> measurable M P" unfolding pair_measure_def
339     proof (rule M.measurable_sigma)
340       show "sets (pair_measure_generator M1 M2) \<subseteq> Pow (space E)"
341         unfolding pair_measure_generator_def using M1.sets_into_space M2.sets_into_space by auto
342       show "f \<in> space M \<rightarrow> space E"
343         using f s by (auto simp: mem_Times_iff measurable_def comp_def space_sigma pair_measure_generator_def)
344       fix A assume "A \<in> sets E"
345       then obtain B C where "B \<in> sets M1" "C \<in> sets M2" "A = B \<times> C"
346         unfolding pair_measure_generator_def by auto
347       moreover have "(fst \<circ> f) - B \<inter> space M \<in> sets M"
348         using f B \<in> sets M1 unfolding measurable_def by auto
349       moreover have "(snd \<circ> f) - C \<inter> space M \<in> sets M"
350         using s C \<in> sets M2 unfolding measurable_def by auto
351       moreover have "f - A \<inter> space M = ((fst \<circ> f) - B \<inter> space M) \<inter> ((snd \<circ> f) - C \<inter> space M)"
352         unfolding A = B \<times> C by (auto simp: vimage_Times)
353       ultimately show "f - A \<inter> space M \<in> sets M" by auto
354     qed
355   qed
356 qed
358 lemma (in pair_sigma_algebra) measurable_pair:
359   assumes "sigma_algebra M"
360   assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"
361   shows "f \<in> measurable M P"
362   unfolding measurable_pair_iff[OF assms(1)] using assms(2,3) by simp
364 lemma pair_measure_generatorE:
365   assumes "X \<in> sets (pair_measure_generator M1 M2)"
366   obtains A B where "X = A \<times> B" "A \<in> sets M1" "B \<in> sets M2"
367   using assms unfolding pair_measure_generator_def by auto
369 lemma (in pair_sigma_algebra) pair_measure_generator_swap:
370   "(\<lambda>X. (\<lambda>(x,y). (y,x)) - X \<inter> space M2 \<times> space M1)  sets E = sets (pair_measure_generator M2 M1)"
371 proof (safe elim!: pair_measure_generatorE)
372   fix A B assume "A \<in> sets M1" "B \<in> sets M2"
373   moreover then have "(\<lambda>(x, y). (y, x)) - (A \<times> B) \<inter> space M2 \<times> space M1 = B \<times> A"
374     using M1.sets_into_space M2.sets_into_space by auto
375   ultimately show "(\<lambda>(x, y). (y, x)) - (A \<times> B) \<inter> space M2 \<times> space M1 \<in> sets (pair_measure_generator M2 M1)"
376     by (auto intro: pair_measure_generatorI)
377 next
378   fix A B assume "A \<in> sets M1" "B \<in> sets M2"
379   then show "B \<times> A \<in> (\<lambda>X. (\<lambda>(x, y). (y, x)) - X \<inter> space M2 \<times> space M1)  sets E"
380     using M1.sets_into_space M2.sets_into_space
381     by (auto intro!: image_eqI[where x="A \<times> B"] pair_measure_generatorI)
382 qed
384 lemma (in pair_sigma_algebra) sets_pair_sigma_algebra_swap:
385   assumes Q: "Q \<in> sets P"
386   shows "(\<lambda>(x,y). (y, x)) - Q \<in> sets (M2 \<Otimes>\<^isub>M M1)" (is "_ \<in> sets ?Q")
387 proof -
388   let "?f Q" = "(\<lambda>(x,y). (y, x)) - Q \<inter> space M2 \<times> space M1"
389   have *: "(\<lambda>(x,y). (y, x)) - Q = ?f Q"
390     using sets_into_space[OF Q] by (auto simp: space_pair_measure)
391   have "sets (M2 \<Otimes>\<^isub>M M1) = sets (sigma (pair_measure_generator M2 M1))"
392     unfolding pair_measure_def ..
393   also have "\<dots> = sigma_sets (space M2 \<times> space M1) (?f  sets E)"
394     unfolding sigma_def pair_measure_generator_swap[symmetric]
395     by (simp add: pair_measure_generator_def)
396   also have "\<dots> = ?f  sigma_sets (space M1 \<times> space M2) (sets E)"
397     using M1.sets_into_space M2.sets_into_space
398     by (intro sigma_sets_vimage) (auto simp: pair_measure_generator_def)
399   also have "\<dots> = ?f  sets P"
400     unfolding pair_measure_def pair_measure_generator_def sigma_def by simp
401   finally show ?thesis
402     using Q by (subst *) auto
403 qed
405 lemma (in pair_sigma_algebra) pair_sigma_algebra_swap_measurable:
406   shows "(\<lambda>(x,y). (y, x)) \<in> measurable P (M2 \<Otimes>\<^isub>M M1)"
407     (is "?f \<in> measurable ?P ?Q")
408   unfolding measurable_def
409 proof (intro CollectI conjI Pi_I ballI)
410   fix x assume "x \<in> space ?P" then show "(case x of (x, y) \<Rightarrow> (y, x)) \<in> space ?Q"
411     unfolding pair_measure_generator_def pair_measure_def by auto
412 next
413   fix A assume "A \<in> sets (M2 \<Otimes>\<^isub>M M1)"
414   interpret Q: pair_sigma_algebra M2 M1 by default
415   with Q.sets_pair_sigma_algebra_swap[OF A \<in> sets (M2 \<Otimes>\<^isub>M M1)]
416   show "?f - A \<inter> space ?P \<in> sets ?P" by simp
417 qed
419 lemma (in pair_sigma_algebra) measurable_cut_fst[simp,intro]:
420   assumes "Q \<in> sets P" shows "Pair x - Q \<in> sets M2"
421 proof -
422   let ?Q' = "{Q. Q \<subseteq> space P \<and> Pair x - Q \<in> sets M2}"
423   let ?Q = "\<lparr> space = space P, sets = ?Q' \<rparr>"
424   interpret Q: sigma_algebra ?Q
425     proof qed (auto simp: vimage_UN vimage_Diff space_pair_measure)
426   have "sets E \<subseteq> sets ?Q"
427     using M1.sets_into_space M2.sets_into_space
428     by (auto simp: pair_measure_generator_def space_pair_measure)
429   then have "sets P \<subseteq> sets ?Q"
430     apply (subst pair_measure_def, intro Q.sets_sigma_subset)
431     by (simp add: pair_measure_def)
432   with assms show ?thesis by auto
433 qed
435 lemma (in pair_sigma_algebra) measurable_cut_snd:
436   assumes Q: "Q \<in> sets P" shows "(\<lambda>x. (x, y)) - Q \<in> sets M1" (is "?cut Q \<in> sets M1")
437 proof -
438   interpret Q: pair_sigma_algebra M2 M1 by default
439   with Q.measurable_cut_fst[OF sets_pair_sigma_algebra_swap[OF Q], of y]
440   show ?thesis by (simp add: vimage_compose[symmetric] comp_def)
441 qed
443 lemma (in pair_sigma_algebra) measurable_pair_image_snd:
444   assumes m: "f \<in> measurable P M" and "x \<in> space M1"
445   shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
446   unfolding measurable_def
447 proof (intro CollectI conjI Pi_I ballI)
448   fix y assume "y \<in> space M2" with f \<in> measurable P M x \<in> space M1
449   show "f (x, y) \<in> space M"
450     unfolding measurable_def pair_measure_generator_def pair_measure_def by auto
451 next
452   fix A assume "A \<in> sets M"
453   then have "Pair x - (f - A \<inter> space P) \<in> sets M2" (is "?C \<in> _")
454     using f \<in> measurable P M
455     by (intro measurable_cut_fst) (auto simp: measurable_def)
456   also have "?C = (\<lambda>y. f (x, y)) - A \<inter> space M2"
457     using x \<in> space M1 by (auto simp: pair_measure_generator_def pair_measure_def)
458   finally show "(\<lambda>y. f (x, y)) - A \<inter> space M2 \<in> sets M2" .
459 qed
461 lemma (in pair_sigma_algebra) measurable_pair_image_fst:
462   assumes m: "f \<in> measurable P M" and "y \<in> space M2"
463   shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
464 proof -
465   interpret Q: pair_sigma_algebra M2 M1 by default
466   from Q.measurable_pair_image_snd[OF measurable_comp y \<in> space M2,
467                                       OF Q.pair_sigma_algebra_swap_measurable m]
468   show ?thesis by simp
469 qed
471 lemma (in pair_sigma_algebra) Int_stable_pair_measure_generator: "Int_stable E"
472   unfolding Int_stable_def
473 proof (intro ballI)
474   fix A B assume "A \<in> sets E" "B \<in> sets E"
475   then obtain A1 A2 B1 B2 where "A = A1 \<times> A2" "B = B1 \<times> B2"
476     "A1 \<in> sets M1" "A2 \<in> sets M2" "B1 \<in> sets M1" "B2 \<in> sets M2"
477     unfolding pair_measure_generator_def by auto
478   then show "A \<inter> B \<in> sets E"
479     by (auto simp add: times_Int_times pair_measure_generator_def)
480 qed
482 lemma finite_measure_cut_measurable:
483   fixes M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
484   assumes "sigma_finite_measure M1" "finite_measure M2"
485   assumes "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)"
486   shows "(\<lambda>x. measure M2 (Pair x - Q)) \<in> borel_measurable M1"
487     (is "?s Q \<in> _")
488 proof -
489   interpret M1: sigma_finite_measure M1 by fact
490   interpret M2: finite_measure M2 by fact
491   interpret pair_sigma_algebra M1 M2 by default
492   have [intro]: "sigma_algebra M1" by fact
493   have [intro]: "sigma_algebra M2" by fact
494   let ?D = "\<lparr> space = space P, sets = {A\<in>sets P. ?s A \<in> borel_measurable M1}  \<rparr>"
495   note space_pair_measure[simp]
496   interpret dynkin_system ?D
497   proof (intro dynkin_systemI)
498     fix A assume "A \<in> sets ?D" then show "A \<subseteq> space ?D"
499       using sets_into_space by simp
500   next
501     from top show "space ?D \<in> sets ?D"
502       by (auto simp add: if_distrib intro!: M1.measurable_If)
503   next
504     fix A assume "A \<in> sets ?D"
505     with sets_into_space have "\<And>x. measure M2 (Pair x - (space M1 \<times> space M2 - A)) =
506         (if x \<in> space M1 then measure M2 (space M2) - ?s A x else 0)"
507       by (auto intro!: M2.measure_compl simp: vimage_Diff)
508     with A \<in> sets ?D top show "space ?D - A \<in> sets ?D"
509       by (auto intro!: Diff M1.measurable_If M1.borel_measurable_extreal_diff)
510   next
511     fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> sets ?D"
512     moreover then have "\<And>x. measure M2 (\<Union>i. Pair x - F i) = (\<Sum>i. ?s (F i) x)"
513       by (intro M2.measure_countably_additive[symmetric])
514          (auto simp: disjoint_family_on_def)
515     ultimately show "(\<Union>i. F i) \<in> sets ?D"
516       by (auto simp: vimage_UN intro!: M1.borel_measurable_psuminf)
517   qed
518   have "sets P = sets ?D" apply (subst pair_measure_def)
519   proof (intro dynkin_lemma)
520     show "Int_stable E" by (rule Int_stable_pair_measure_generator)
521     from M1.sets_into_space have "\<And>A. A \<in> sets M1 \<Longrightarrow> {x \<in> space M1. x \<in> A} = A"
522       by auto
523     then show "sets E \<subseteq> sets ?D"
524       by (auto simp: pair_measure_generator_def sets_sigma if_distrib
525                intro: sigma_sets.Basic intro!: M1.measurable_If)
526   qed (auto simp: pair_measure_def)
527   with Q \<in> sets P have "Q \<in> sets ?D" by simp
528   then show "?s Q \<in> borel_measurable M1" by simp
529 qed
531 subsection {* Binary products of $\sigma$-finite measure spaces *}
533 locale pair_sigma_finite = M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2
534   for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
536 sublocale pair_sigma_finite \<subseteq> pair_sigma_algebra M1 M2
537   by default
539 lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
540   by auto
542 lemma (in pair_sigma_finite) measure_cut_measurable_fst:
543   assumes "Q \<in> sets P" shows "(\<lambda>x. measure M2 (Pair x - Q)) \<in> borel_measurable M1" (is "?s Q \<in> _")
544 proof -
545   have [intro]: "sigma_algebra M1" and [intro]: "sigma_algebra M2" by default+
546   have M1: "sigma_finite_measure M1" by default
547   from M2.disjoint_sigma_finite guess F .. note F = this
548   then have F_sets: "\<And>i. F i \<in> sets M2" by auto
549   let "?C x i" = "F i \<inter> Pair x - Q"
550   { fix i
551     let ?R = "M2.restricted_space (F i)"
552     have [simp]: "space M1 \<times> F i \<inter> space M1 \<times> space M2 = space M1 \<times> F i"
553       using F M2.sets_into_space by auto
554     let ?R2 = "M2.restricted_space (F i)"
555     have "(\<lambda>x. measure ?R2 (Pair x - (space M1 \<times> space ?R2 \<inter> Q))) \<in> borel_measurable M1"
556     proof (intro finite_measure_cut_measurable[OF M1])
557       show "finite_measure ?R2"
558         using F by (intro M2.restricted_to_finite_measure) auto
559       have "(space M1 \<times> space ?R2) \<inter> Q \<in> (op \<inter> (space M1 \<times> F i))  sets P"
560         using Q \<in> sets P by (auto simp: image_iff)
561       also have "\<dots> = sigma_sets (space M1 \<times> F i) ((op \<inter> (space M1 \<times> F i))  sets E)"
562         unfolding pair_measure_def pair_measure_generator_def sigma_def
563         using F i \<in> sets M2 M2.sets_into_space
564         by (auto intro!: sigma_sets_Int sigma_sets.Basic)
565       also have "\<dots> \<subseteq> sets (M1 \<Otimes>\<^isub>M ?R2)"
566         using M1.sets_into_space
567         apply (auto simp: times_Int_times pair_measure_def pair_measure_generator_def sigma_def
568                     intro!: sigma_sets_subseteq)
569         apply (rule_tac x="a" in exI)
570         apply (rule_tac x="b \<inter> F i" in exI)
571         by auto
572       finally show "(space M1 \<times> space ?R2) \<inter> Q \<in> sets (M1 \<Otimes>\<^isub>M ?R2)" .
573     qed
574     moreover have "\<And>x. Pair x - (space M1 \<times> F i \<inter> Q) = ?C x i"
575       using Q \<in> sets P sets_into_space by (auto simp: space_pair_measure)
576     ultimately have "(\<lambda>x. measure M2 (?C x i)) \<in> borel_measurable M1"
577       by simp }
578   moreover
579   { fix x
580     have "(\<Sum>i. measure M2 (?C x i)) = measure M2 (\<Union>i. ?C x i)"
581     proof (intro M2.measure_countably_additive)
582       show "range (?C x) \<subseteq> sets M2"
583         using F Q \<in> sets P by (auto intro!: M2.Int)
584       have "disjoint_family F" using F by auto
585       show "disjoint_family (?C x)"
586         by (rule disjoint_family_on_bisimulation[OF disjoint_family F]) auto
587     qed
588     also have "(\<Union>i. ?C x i) = Pair x - Q"
589       using F sets_into_space Q \<in> sets P
590       by (auto simp: space_pair_measure)
591     finally have "measure M2 (Pair x - Q) = (\<Sum>i. measure M2 (?C x i))"
592       by simp }
593   ultimately show ?thesis using Q \<in> sets P F_sets
594     by (auto intro!: M1.borel_measurable_psuminf M2.Int)
595 qed
597 lemma (in pair_sigma_finite) measure_cut_measurable_snd:
598   assumes "Q \<in> sets P" shows "(\<lambda>y. M1.\<mu> ((\<lambda>x. (x, y)) - Q)) \<in> borel_measurable M2"
599 proof -
600   interpret Q: pair_sigma_finite M2 M1 by default
601   note sets_pair_sigma_algebra_swap[OF assms]
602   from Q.measure_cut_measurable_fst[OF this]
603   show ?thesis by (simp add: vimage_compose[symmetric] comp_def)
604 qed
606 lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable:
607   assumes "f \<in> measurable P M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^isub>M M1) M"
608 proof -
609   interpret Q: pair_sigma_algebra M2 M1 by default
610   have *: "(\<lambda>(x,y). f (y, x)) = f \<circ> (\<lambda>(x,y). (y, x))" by (simp add: fun_eq_iff)
611   show ?thesis
612     using Q.pair_sigma_algebra_swap_measurable assms
613     unfolding * by (rule measurable_comp)
614 qed
616 lemma (in pair_sigma_finite) pair_measure_alt:
617   assumes "A \<in> sets P"
618   shows "measure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+ x. measure M2 (Pair x - A) \<partial>M1)"
619   apply (simp add: pair_measure_def pair_measure_generator_def)
620 proof (rule M1.positive_integral_cong)
621   fix x assume "x \<in> space M1"
622   have *: "\<And>y. indicator A (x, y) = (indicator (Pair x - A) y :: extreal)"
623     unfolding indicator_def by auto
624   show "(\<integral>\<^isup>+ y. indicator A (x, y) \<partial>M2) = measure M2 (Pair x - A)"
625     unfolding *
626     apply (subst M2.positive_integral_indicator)
627     apply (rule measurable_cut_fst[OF assms])
628     by simp
629 qed
631 lemma (in pair_sigma_finite) pair_measure_times:
632   assumes A: "A \<in> sets M1" and "B \<in> sets M2"
633   shows "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = M1.\<mu> A * measure M2 B"
634 proof -
635   have "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = (\<integral>\<^isup>+ x. measure M2 B * indicator A x \<partial>M1)"
636     using assms by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt)
637   with assms show ?thesis
638     by (simp add: M1.positive_integral_cmult_indicator ac_simps)
639 qed
641 lemma (in measure_space) measure_not_negative[simp,intro]:
642   assumes A: "A \<in> sets M" shows "\<mu> A \<noteq> - \<infinity>"
643   using positive_measure[OF A] by auto
645 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
646   "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> incseq F \<and> (\<Union>i. F i) = space E \<and>
647     (\<forall>i. measure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
648 proof -
649   obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where
650     F1: "range F1 \<subseteq> sets M1" "incseq F1" "(\<Union>i. F1 i) = space M1" "\<And>i. M1.\<mu> (F1 i) \<noteq> \<infinity>" and
651     F2: "range F2 \<subseteq> sets M2" "incseq F2" "(\<Union>i. F2 i) = space M2" "\<And>i. M2.\<mu> (F2 i) \<noteq> \<infinity>"
652     using M1.sigma_finite_up M2.sigma_finite_up by auto
653   then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
654   let ?F = "\<lambda>i. F1 i \<times> F2 i"
655   show ?thesis unfolding space_pair_measure
656   proof (intro exI[of _ ?F] conjI allI)
657     show "range ?F \<subseteq> sets E" using F1 F2
658       by (fastsimp intro!: pair_measure_generatorI)
659   next
660     have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
661     proof (intro subsetI)
662       fix x assume "x \<in> space M1 \<times> space M2"
663       then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
664         by (auto simp: space)
665       then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
666         using incseq F1 incseq F2 unfolding incseq_def
667         by (force split: split_max)+
668       then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
669         by (intro SigmaI) (auto simp add: min_max.sup_commute)
670       then show "x \<in> (\<Union>i. ?F i)" by auto
671     qed
672     then show "(\<Union>i. ?F i) = space E"
673       using space by (auto simp: space pair_measure_generator_def)
674   next
675     fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
676       using incseq F1 incseq F2 unfolding incseq_Suc_iff by auto
677   next
678     fix i
679     from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
680     with F1 F2 M1.positive_measure[OF this(1)] M2.positive_measure[OF this(2)]
681     show "measure P (F1 i \<times> F2 i) \<noteq> \<infinity>"
682       by (simp add: pair_measure_times)
683   qed
684 qed
686 sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P
687 proof
688   show "positive P (measure P)"
689     unfolding pair_measure_def pair_measure_generator_def sigma_def positive_def
690     by (auto intro: M1.positive_integral_positive M2.positive_integral_positive)
692   show "countably_additive P (measure P)"
694   proof (intro allI impI)
695     fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"
696     assume F: "range F \<subseteq> sets P" "disjoint_family F"
697     from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto
698     moreover from F have "\<And>i. (\<lambda>x. measure M2 (Pair x - F i)) \<in> borel_measurable M1"
699       by (intro measure_cut_measurable_fst) auto
700     moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x - F i)"
701       by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
702     moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x - F i) \<subseteq> sets M2"
703       using F by auto
704     ultimately show "(\<Sum>n. measure P (F n)) = measure P (\<Union>i. F i)"
705       by (simp add: pair_measure_alt vimage_UN M1.positive_integral_suminf[symmetric]
707                cong: M1.positive_integral_cong)
708   qed
710   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
711   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>)"
712   proof (rule exI[of _ F], intro conjI)
713     show "range F \<subseteq> sets P" using F by (auto simp: pair_measure_def)
714     show "(\<Union>i. F i) = space P"
715       using F by (auto simp: pair_measure_def pair_measure_generator_def)
716     show "\<forall>i. measure P (F i) \<noteq> \<infinity>" using F by auto
717   qed
718 qed
720 lemma (in pair_sigma_algebra) sets_swap:
721   assumes "A \<in> sets P"
722   shows "(\<lambda>(x, y). (y, x)) - A \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
723     (is "_ - A \<inter> space ?Q \<in> sets ?Q")
724 proof -
725   have *: "(\<lambda>(x, y). (y, x)) - A \<inter> space ?Q = (\<lambda>(x, y). (y, x)) - A"
726     using A \<in> sets P sets_into_space by (auto simp: space_pair_measure)
727   show ?thesis
728     unfolding * using assms by (rule sets_pair_sigma_algebra_swap)
729 qed
731 lemma (in pair_sigma_finite) pair_measure_alt2:
732   assumes A: "A \<in> sets P"
733   shows "\<mu> A = (\<integral>\<^isup>+y. M1.\<mu> ((\<lambda>x. (x, y)) - A) \<partial>M2)"
734     (is "_ = ?\<nu> A")
735 proof -
736   interpret Q: pair_sigma_finite M2 M1 by default
737   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
738   have [simp]: "\<And>m. \<lparr> space = space E, sets = sets (sigma E), measure = m \<rparr> = P\<lparr> measure := m \<rparr>"
739     unfolding pair_measure_def by simp
741   have "\<mu> A = Q.\<mu> ((\<lambda>(y, x). (x, y)) - A \<inter> space Q.P)"
742   proof (rule measure_unique_Int_stable_vimage[OF Int_stable_pair_measure_generator])
743     show "measure_space P" "measure_space Q.P" by default
744     show "(\<lambda>(y, x). (x, y)) \<in> measurable Q.P P" by (rule Q.pair_sigma_algebra_swap_measurable)
745     show "sets (sigma E) = sets P" "space E = space P" "A \<in> sets (sigma E)"
746       using assms unfolding pair_measure_def by auto
747     show "range F \<subseteq> sets E" "incseq F" "(\<Union>i. F i) = space E" "\<And>i. \<mu> (F i) \<noteq> \<infinity>"
748       using F A \<in> sets P by (auto simp: pair_measure_def)
749     fix X assume "X \<in> sets E"
750     then obtain A B where X[simp]: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2"
751       unfolding pair_measure_def pair_measure_generator_def by auto
752     then have "(\<lambda>(y, x). (x, y)) - X \<inter> space Q.P = B \<times> A"
753       using M1.sets_into_space M2.sets_into_space by (auto simp: space_pair_measure)
754     then show "\<mu> X = Q.\<mu> ((\<lambda>(y, x). (x, y)) - X \<inter> space Q.P)"
755       using AB by (simp add: pair_measure_times Q.pair_measure_times ac_simps)
756   qed
757   then show ?thesis
758     using sets_into_space[OF A] Q.pair_measure_alt[OF sets_swap[OF A]]
759     by (auto simp add: Q.pair_measure_alt space_pair_measure
760              intro!: M2.positive_integral_cong arg_cong[where f="M1.\<mu>"])
761 qed
763 lemma pair_sigma_algebra_sigma:
764   assumes 1: "incseq S1" "(\<Union>i. S1 i) = space E1" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"
765   assumes 2: "decseq S2" "(\<Union>i. S2 i) = space E2" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"
766   shows "sets (sigma (pair_measure_generator (sigma E1) (sigma E2))) = sets (sigma (pair_measure_generator E1 E2))"
767     (is "sets ?S = sets ?E")
768 proof -
769   interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma)
770   interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma)
771   have P: "sets (pair_measure_generator E1 E2) \<subseteq> Pow (space E1 \<times> space E2)"
772     using E1 E2 by (auto simp add: pair_measure_generator_def)
773   interpret E: sigma_algebra ?E unfolding pair_measure_generator_def
774     using E1 E2 by (intro sigma_algebra_sigma) auto
775   { fix A assume "A \<in> sets E1"
776     then have "fst - A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"
777       using E1 2 unfolding pair_measure_generator_def by auto
778     also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto
779     also have "\<dots> \<in> sets ?E" unfolding pair_measure_generator_def sets_sigma
780       using 2 A \<in> sets E1
781       by (intro sigma_sets.Union)
782          (force simp: image_subset_iff intro!: sigma_sets.Basic)
783     finally have "fst - A \<inter> space ?E \<in> sets ?E" . }
784   moreover
785   { fix B assume "B \<in> sets E2"
786     then have "snd - B \<inter> space ?E = (\<Union>i. S1 i) \<times> B"
787       using E2 1 unfolding pair_measure_generator_def by auto
788     also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto
789     also have "\<dots> \<in> sets ?E"
790       using 1 B \<in> sets E2 unfolding pair_measure_generator_def sets_sigma
791       by (intro sigma_sets.Union)
792          (force simp: image_subset_iff intro!: sigma_sets.Basic)
793     finally have "snd - B \<inter> space ?E \<in> sets ?E" . }
794   ultimately have proj:
795     "fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"
796     using E1 E2 by (subst (1 2) E.measurable_iff_sigma)
797                    (auto simp: pair_measure_generator_def sets_sigma)
798   { fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)"
799     with proj have "fst - A \<inter> space ?E \<in> sets ?E" "snd - B \<inter> space ?E \<in> sets ?E"
800       unfolding measurable_def by simp_all
801     moreover have "A \<times> B = (fst - A \<inter> space ?E) \<inter> (snd - B \<inter> space ?E)"
802       using A B M1.sets_into_space M2.sets_into_space
803       by (auto simp: pair_measure_generator_def)
804     ultimately have "A \<times> B \<in> sets ?E" by auto }
805   then have "sigma_sets (space ?E) (sets (pair_measure_generator (sigma E1) (sigma E2))) \<subseteq> sets ?E"
806     by (intro E.sigma_sets_subset) (auto simp add: pair_measure_generator_def sets_sigma)
807   then have subset: "sets ?S \<subseteq> sets ?E"
808     by (simp add: sets_sigma pair_measure_generator_def)
809   show "sets ?S = sets ?E"
810   proof (intro set_eqI iffI)
811     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
812       unfolding sets_sigma
813     proof induct
814       case (Basic A) then show ?case
815         by (auto simp: pair_measure_generator_def sets_sigma intro: sigma_sets.Basic)
816     qed (auto intro: sigma_sets.intros simp: pair_measure_generator_def)
817   next
818     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
819   qed
820 qed
822 section "Fubinis theorem"
824 lemma (in pair_sigma_finite) simple_function_cut:
825   assumes f: "simple_function P f" "\<And>x. 0 \<le> f x"
826   shows "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
827     and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
828 proof -
829   have f_borel: "f \<in> borel_measurable P"
830     using f(1) by (rule borel_measurable_simple_function)
831   let "?F z" = "f - {z} \<inter> space P"
832   let "?F' x z" = "Pair x - ?F z"
833   { fix x assume "x \<in> space M1"
834     have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y"
835       by (auto simp: indicator_def)
836     have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space P" using x \<in> space M1
837       by (simp add: space_pair_measure)
838     moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel
839       by (intro borel_measurable_vimage measurable_cut_fst)
840     ultimately have "simple_function M2 (\<lambda> y. f (x, y))"
841       apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _])
842       apply (rule simple_function_indicator_representation[OF f(1)])
843       using x \<in> space M1 by (auto simp del: space_sigma) }
844   note M2_sf = this
845   { fix x assume x: "x \<in> space M1"
846     then have "(\<integral>\<^isup>+y. f (x, y) \<partial>M2) = (\<Sum>z\<in>f  space P. z * M2.\<mu> (?F' x z))"
847       unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x] f(2)]
848       unfolding simple_integral_def
849     proof (safe intro!: setsum_mono_zero_cong_left)
850       from f(1) show "finite (f  space P)" by (rule simple_functionD)
851     next
852       fix y assume "y \<in> space M2" then show "f (x, y) \<in> f  space P"
853         using x \<in> space M1 by (auto simp: space_pair_measure)
854     next
855       fix x' y assume "(x', y) \<in> space P"
856         "f (x', y) \<notin> (\<lambda>y. f (x, y))  space M2"
857       then have *: "?F' x (f (x', y)) = {}"
858         by (force simp: space_pair_measure)
859       show  "f (x', y) * M2.\<mu> (?F' x (f (x', y))) = 0"
860         unfolding * by simp
861     qed (simp add: vimage_compose[symmetric] comp_def
862                    space_pair_measure) }
863   note eq = this
864   moreover have "\<And>z. ?F z \<in> sets P"
865     by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma)
866   moreover then have "\<And>z. (\<lambda>x. M2.\<mu> (?F' x z)) \<in> borel_measurable M1"
867     by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int)
868   moreover have *: "\<And>i x. 0 \<le> M2.\<mu> (Pair x - (f - {i} \<inter> space P))"
869     using f(1)[THEN simple_functionD(2)] f(2) by (intro M2.positive_measure measurable_cut_fst)
870   moreover { fix i assume "i \<in> fspace P"
871     with * have "\<And>x. 0 \<le> i * M2.\<mu> (Pair x - (f - {i} \<inter> space P))"
872       using f(2) by auto }
873   ultimately
874   show "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
875     and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f" using f(2)
876     by (auto simp del: vimage_Int cong: measurable_cong
877              intro!: M1.borel_measurable_extreal_setsum setsum_cong
878              simp add: M1.positive_integral_setsum simple_integral_def
879                        M1.positive_integral_cmult
880                        M1.positive_integral_cong[OF eq]
881                        positive_integral_eq_simple_integral[OF f]
882                        pair_measure_alt[symmetric])
883 qed
885 lemma (in pair_sigma_finite) positive_integral_fst_measurable:
886   assumes f: "f \<in> borel_measurable P"
887   shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
888       (is "?C f \<in> borel_measurable M1")
889     and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
890 proof -
891   from borel_measurable_implies_simple_function_sequence'[OF f] guess F . note F = this
892   then have F_borel: "\<And>i. F i \<in> borel_measurable P"
893     by (auto intro: borel_measurable_simple_function)
894   note sf = simple_function_cut[OF F(1,5)]
895   then have "(\<lambda>x. SUP i. ?C (F i) x) \<in> borel_measurable M1"
896     using F(1) by auto
897   moreover
898   { fix x assume "x \<in> space M1"
899     from F measurable_pair_image_snd[OF F_borelx \<in> space M1]
900     have "(\<integral>\<^isup>+y. (SUP i. F i (x, y)) \<partial>M2) = (SUP i. ?C (F i) x)"
901       by (intro M2.positive_integral_monotone_convergence_SUP)
902          (auto simp: incseq_Suc_iff le_fun_def)
903     then have "(SUP i. ?C (F i) x) = ?C f x"
904       unfolding F(4) positive_integral_max_0 by simp }
905   note SUPR_C = this
906   ultimately show "?C f \<in> borel_measurable M1"
907     by (simp cong: measurable_cong)
908   have "(\<integral>\<^isup>+x. (SUP i. F i x) \<partial>P) = (SUP i. integral\<^isup>P P (F i))"
909     using F_borel F
910     by (intro positive_integral_monotone_convergence_SUP) auto
911   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)"
912     unfolding sf(2) by simp
913   also have "\<dots> = \<integral>\<^isup>+ x. (SUP i. \<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1" using F sf(1)
914     by (intro M1.positive_integral_monotone_convergence_SUP[symmetric])
915        (auto intro!: M2.positive_integral_mono M2.positive_integral_positive
916                 simp: incseq_Suc_iff le_fun_def)
917   also have "\<dots> = \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. (SUP i. F i (x, y)) \<partial>M2) \<partial>M1"
918     using F_borel F(2,5)
919     by (auto intro!: M1.positive_integral_cong M2.positive_integral_monotone_convergence_SUP[symmetric]
920              simp: incseq_Suc_iff le_fun_def measurable_pair_image_snd)
921   finally show "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
922     using F by (simp add: positive_integral_max_0)
923 qed
925 lemma (in pair_sigma_finite) measure_preserving_swap:
926   "(\<lambda>(x,y). (y, x)) \<in> measure_preserving (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
927 proof
928   interpret Q: pair_sigma_finite M2 M1 by default
929   show *: "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
930     using pair_sigma_algebra_swap_measurable .
931   fix X assume "X \<in> sets (M2 \<Otimes>\<^isub>M M1)"
932   from measurable_sets[OF * this] this Q.sets_into_space[OF this]
933   show "measure (M1 \<Otimes>\<^isub>M M2) ((\<lambda>(x, y). (y, x)) - X \<inter> space P) = measure (M2 \<Otimes>\<^isub>M M1) X"
934     by (auto intro!: M1.positive_integral_cong arg_cong[where f="M2.\<mu>"]
935       simp: pair_measure_alt Q.pair_measure_alt2 space_pair_measure)
936 qed
938 lemma (in pair_sigma_finite) positive_integral_product_swap:
939   assumes f: "f \<in> borel_measurable P"
940   shows "(\<integral>\<^isup>+x. f (case x of (x,y)\<Rightarrow>(y,x)) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P P f"
941 proof -
942   interpret Q: pair_sigma_finite M2 M1 by default
943   have "sigma_algebra P" by default
944   with f show ?thesis
945     by (subst Q.positive_integral_vimage[OF _ Q.measure_preserving_swap]) auto
946 qed
948 lemma (in pair_sigma_finite) positive_integral_snd_measurable:
949   assumes f: "f \<in> borel_measurable P"
950   shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>P P f"
951 proof -
952   interpret Q: pair_sigma_finite M2 M1 by default
953   note pair_sigma_algebra_measurable[OF f]
954   from Q.positive_integral_fst_measurable[OF this]
955   have "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P)"
956     by simp
957   also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P) = integral\<^isup>P P f"
958     unfolding positive_integral_product_swap[OF f, symmetric]
959     by (auto intro!: Q.positive_integral_cong)
960   finally show ?thesis .
961 qed
963 lemma (in pair_sigma_finite) Fubini:
964   assumes f: "f \<in> borel_measurable P"
965   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)"
966   unfolding positive_integral_snd_measurable[OF assms]
967   unfolding positive_integral_fst_measurable[OF assms] ..
969 lemma (in pair_sigma_finite) AE_pair:
970   assumes "AE x in P. Q x"
971   shows "AE x in M1. (AE y in M2. Q (x, y))"
972 proof -
973   obtain N where N: "N \<in> sets P" "\<mu> N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N"
974     using assms unfolding almost_everywhere_def by auto
975   show ?thesis
976   proof (rule M1.AE_I)
977     from N measure_cut_measurable_fst[OF N \<in> sets P]
978     show "M1.\<mu> {x\<in>space M1. M2.\<mu> (Pair x - N) \<noteq> 0} = 0"
979       by (auto simp: pair_measure_alt M1.positive_integral_0_iff)
980     show "{x \<in> space M1. M2.\<mu> (Pair x - N) \<noteq> 0} \<in> sets M1"
981       by (intro M1.borel_measurable_extreal_neq_const measure_cut_measurable_fst N)
982     { fix x assume "x \<in> space M1" "M2.\<mu> (Pair x - N) = 0"
983       have "M2.almost_everywhere (\<lambda>y. Q (x, y))"
984       proof (rule M2.AE_I)
985         show "M2.\<mu> (Pair x - N) = 0" by fact
986         show "Pair x - N \<in> sets M2" by (intro measurable_cut_fst N)
987         show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x - N"
988           using N x \<in> space M1 unfolding space_sigma space_pair_measure by auto
989       qed }
990     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}"
991       by auto
992   qed
993 qed
995 lemma (in pair_sigma_algebra) measurable_product_swap:
996   "f \<in> measurable (M2 \<Otimes>\<^isub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable P M"
997 proof -
998   interpret Q: pair_sigma_algebra M2 M1 by default
999   show ?thesis
1000     using pair_sigma_algebra_measurable[of "\<lambda>(x,y). f (y, x)"]
1001     by (auto intro!: pair_sigma_algebra_measurable Q.pair_sigma_algebra_measurable iffI)
1002 qed
1004 lemma (in pair_sigma_finite) integrable_product_swap:
1005   assumes "integrable P f"
1006   shows "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x))"
1007 proof -
1008   interpret Q: pair_sigma_finite M2 M1 by default
1009   have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
1010   show ?thesis unfolding *
1011     using assms unfolding integrable_def
1012     apply (subst (1 2) positive_integral_product_swap)
1013     using integrable P f unfolding integrable_def
1014     by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])
1015 qed
1017 lemma (in pair_sigma_finite) integrable_product_swap_iff:
1018   "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable P f"
1019 proof -
1020   interpret Q: pair_sigma_finite M2 M1 by default
1021   from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]
1022   show ?thesis by auto
1023 qed
1025 lemma (in pair_sigma_finite) integral_product_swap:
1026   assumes "integrable P f"
1027   shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L P f"
1028 proof -
1029   interpret Q: pair_sigma_finite M2 M1 by default
1030   have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
1031   show ?thesis
1032     unfolding lebesgue_integral_def *
1033     apply (subst (1 2) positive_integral_product_swap)
1034     using integrable P f unfolding integrable_def
1035     by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])
1036 qed
1038 lemma (in pair_sigma_finite) integrable_fst_measurable:
1039   assumes f: "integrable P f"
1040   shows "M1.almost_everywhere (\<lambda>x. integrable M2 (\<lambda> y. f (x, y)))" (is "?AE")
1041     and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L P f" (is "?INT")
1042 proof -
1043   let "?pf x" = "extreal (f x)" and "?nf x" = "extreal (- f x)"
1044   have
1045     borel: "?nf \<in> borel_measurable P""?pf \<in> borel_measurable P" and
1046     int: "integral\<^isup>P P ?nf \<noteq> \<infinity>" "integral\<^isup>P P ?pf \<noteq> \<infinity>"
1047     using assms by auto
1048   have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
1049      "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
1050     using borel[THEN positive_integral_fst_measurable(1)] int
1051     unfolding borel[THEN positive_integral_fst_measurable(2)] by simp_all
1052   with borel[THEN positive_integral_fst_measurable(1)]
1053   have AE_pos: "AE x in M1. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
1054     "AE x in M1. (\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2) \<noteq> \<infinity>"
1055     by (auto intro!: M1.positive_integral_PInf_AE )
1056   then have AE: "AE x in M1. \<bar>\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
1057     "AE x in M1. \<bar>\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
1058     by (auto simp: M2.positive_integral_positive)
1059   from AE_pos show ?AE using assms
1060     by (simp add: measurable_pair_image_snd integrable_def)
1061   { fix f have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. extreal (f x y) \<partial>M2 \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
1062       using M2.positive_integral_positive
1063       by (intro M1.positive_integral_cong_pos) (auto simp: extreal_uminus_le_reorder)
1064     then have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. extreal (f x y) \<partial>M2 \<partial>M1) = 0" by simp }
1065   note this[simp]
1066   { fix f assume borel: "(\<lambda>x. extreal (f x)) \<in> borel_measurable P"
1067       and int: "integral\<^isup>P P (\<lambda>x. extreal (f x)) \<noteq> \<infinity>"
1068       and AE: "M1.almost_everywhere (\<lambda>x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<noteq> \<infinity>)"
1069     have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f")
1070     proof (intro integrable_def[THEN iffD2] conjI)
1071       show "?f \<in> borel_measurable M1"
1072         using borel by (auto intro!: M1.borel_measurable_real_of_extreal positive_integral_fst_measurable)
1073       have "(\<integral>\<^isup>+x. extreal (?f x) \<partial>M1) = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (f (x, y))  \<partial>M2) \<partial>M1)"
1074         using AE M2.positive_integral_positive
1075         by (auto intro!: M1.positive_integral_cong_AE simp: extreal_real)
1076       then show "(\<integral>\<^isup>+x. extreal (?f x) \<partial>M1) \<noteq> \<infinity>"
1077         using positive_integral_fst_measurable[OF borel] int by simp
1078       have "(\<integral>\<^isup>+x. extreal (- ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
1079         by (intro M1.positive_integral_cong_pos)
1080            (simp add: M2.positive_integral_positive real_of_extreal_pos)
1081       then show "(\<integral>\<^isup>+x. extreal (- ?f x) \<partial>M1) \<noteq> \<infinity>" by simp
1082     qed }
1083   with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)]
1084   show ?INT
1085     unfolding lebesgue_integral_def[of P] lebesgue_integral_def[of M2]
1086       borel[THEN positive_integral_fst_measurable(2), symmetric]
1087     using AE[THEN M1.integral_real]
1088     by simp
1089 qed
1091 lemma (in pair_sigma_finite) integrable_snd_measurable:
1092   assumes f: "integrable P f"
1093   shows "M2.almost_everywhere (\<lambda>y. integrable M1 (\<lambda>x. f (x, y)))" (is "?AE")
1094     and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L P f" (is "?INT")
1095 proof -
1096   interpret Q: pair_sigma_finite M2 M1 by default
1097   have Q_int: "integrable Q.P (\<lambda>(x, y). f (y, x))"
1098     using f unfolding integrable_product_swap_iff .
1099   show ?INT
1100     using Q.integrable_fst_measurable(2)[OF Q_int]
1101     using integral_product_swap[OF f] by simp
1102   show ?AE
1103     using Q.integrable_fst_measurable(1)[OF Q_int]
1104     by simp
1105 qed
1107 lemma (in pair_sigma_finite) Fubini_integral:
1108   assumes f: "integrable P f"
1109   shows "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1)"
1110   unfolding integrable_snd_measurable[OF assms]
1111   unfolding integrable_fst_measurable[OF assms] ..
1113 section "Finite product spaces"
1115 section "Products"
1117 locale product_sigma_algebra =
1118   fixes M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme"
1119   assumes sigma_algebras: "\<And>i. sigma_algebra (M i)"
1121 locale finite_product_sigma_algebra = product_sigma_algebra M
1122   for M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme" +
1123   fixes I :: "'i set"
1124   assumes finite_index: "finite I"
1126 definition
1127   "product_algebra_generator I M = \<lparr> space = (\<Pi>\<^isub>E i \<in> I. space (M i)),
1128     sets = Pi\<^isub>E I  (\<Pi> i \<in> I. sets (M i)),
1129     measure = \<lambda>A. (\<Prod>i\<in>I. measure (M i) ((SOME F. A = Pi\<^isub>E I F) i)) \<rparr>"
1131 definition product_algebra_def:
1132   "Pi\<^isub>M I M = sigma (product_algebra_generator I M)
1133     \<lparr> measure := (SOME \<mu>. sigma_finite_measure (sigma (product_algebra_generator I M) \<lparr> measure := \<mu> \<rparr>) \<and>
1134       (\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<mu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. measure (M i) (A i))))\<rparr>"
1136 syntax
1137   "_PiM"  :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
1138               ('i => 'a, 'b) measure_space_scheme"  ("(3PIM _:_./ _)" 10)
1140 syntax (xsymbols)
1141   "_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
1142              ('i => 'a, 'b) measure_space_scheme"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"   10)
1144 syntax (HTML output)
1145   "_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
1146              ('i => 'a, 'b) measure_space_scheme"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"   10)
1148 translations
1149   "PIM x:I. M" == "CONST Pi\<^isub>M I (%x. M)"
1151 abbreviation (in finite_product_sigma_algebra) "G \<equiv> product_algebra_generator I M"
1152 abbreviation (in finite_product_sigma_algebra) "P \<equiv> Pi\<^isub>M I M"
1154 sublocale product_sigma_algebra \<subseteq> M: sigma_algebra "M i" for i by (rule sigma_algebras)
1156 lemma sigma_into_space:
1157   assumes "sets M \<subseteq> Pow (space M)"
1158   shows "sets (sigma M) \<subseteq> Pow (space M)"
1159   using sigma_sets_into_sp[OF assms] unfolding sigma_def by auto
1161 lemma (in product_sigma_algebra) product_algebra_generator_into_space:
1162   "sets (product_algebra_generator I M) \<subseteq> Pow (space (product_algebra_generator I M))"
1163   using M.sets_into_space unfolding product_algebra_generator_def
1164   by auto blast
1166 lemma (in product_sigma_algebra) product_algebra_into_space:
1167   "sets (Pi\<^isub>M I M) \<subseteq> Pow (space (Pi\<^isub>M I M))"
1168   using product_algebra_generator_into_space
1169   by (auto intro!: sigma_into_space simp add: product_algebra_def)
1171 lemma (in product_sigma_algebra) sigma_algebra_product_algebra: "sigma_algebra (Pi\<^isub>M I M)"
1172   using product_algebra_generator_into_space unfolding product_algebra_def
1173   by (rule sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) simp_all
1175 sublocale finite_product_sigma_algebra \<subseteq> sigma_algebra P
1176   using sigma_algebra_product_algebra .
1178 lemma product_algebraE:
1179   assumes "A \<in> sets (product_algebra_generator I M)"
1180   obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
1181   using assms unfolding product_algebra_generator_def by auto
1183 lemma product_algebra_generatorI[intro]:
1184   assumes "E \<in> (\<Pi> i\<in>I. sets (M i))"
1185   shows "Pi\<^isub>E I E \<in> sets (product_algebra_generator I M)"
1186   using assms unfolding product_algebra_generator_def by auto
1188 lemma space_product_algebra_generator[simp]:
1189   "space (product_algebra_generator I M) = Pi\<^isub>E I (\<lambda>i. space (M i))"
1190   unfolding product_algebra_generator_def by simp
1192 lemma space_product_algebra[simp]:
1193   "space (Pi\<^isub>M I M) = (\<Pi>\<^isub>E i\<in>I. space (M i))"
1194   unfolding product_algebra_def product_algebra_generator_def by simp
1196 lemma sets_product_algebra:
1197   "sets (Pi\<^isub>M I M) = sets (sigma (product_algebra_generator I M))"
1198   unfolding product_algebra_def sigma_def by simp
1200 lemma product_algebra_generator_sets_into_space:
1201   assumes "\<And>i. i\<in>I \<Longrightarrow> sets (M i) \<subseteq> Pow (space (M i))"
1202   shows "sets (product_algebra_generator I M) \<subseteq> Pow (space (product_algebra_generator I M))"
1203   using assms by (auto simp: product_algebra_generator_def) blast
1205 lemma (in finite_product_sigma_algebra) in_P[simp, intro]:
1206   "\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P"
1207   by (auto simp: sets_product_algebra)
1209 section "Generating set generates also product algebra"
1211 lemma sigma_product_algebra_sigma_eq:
1212   assumes "finite I"
1213   assumes mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)"
1214   assumes union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (E i)"
1215   assumes sets_into: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> sets (E i)"
1216   and E: "\<And>i. sets (E i) \<subseteq> Pow (space (E i))"
1217   shows "sets (\<Pi>\<^isub>M i\<in>I. sigma (E i)) = sets (\<Pi>\<^isub>M i\<in>I. E i)"
1218     (is "sets ?S = sets ?E")
1219 proof cases
1220   assume "I = {}" then show ?thesis
1221     by (simp add: product_algebra_def product_algebra_generator_def)
1222 next
1223   assume "I \<noteq> {}"
1224   interpret E: sigma_algebra "sigma (E i)" for i
1225     using E by (rule sigma_algebra_sigma)
1226   have into_space[intro]: "\<And>i x A. A \<in> sets (E i) \<Longrightarrow> x i \<in> A \<Longrightarrow> x i \<in> space (E i)"
1227     using E by auto
1228   interpret G: sigma_algebra ?E
1229     unfolding product_algebra_def product_algebra_generator_def using E
1230     by (intro sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) (auto dest: Pi_mem)
1231   { fix A i assume "i \<in> I" and A: "A \<in> sets (E i)"
1232     then have "(\<lambda>x. x i) - A \<inter> space ?E = (\<Pi>\<^isub>E j\<in>I. if j = i then A else \<Union>n. S j n) \<inter> space ?E"
1233       using mono union unfolding incseq_Suc_iff space_product_algebra
1234       by (auto dest: Pi_mem)
1235     also have "\<dots> = (\<Union>n. (\<Pi>\<^isub>E j\<in>I. if j = i then A else S j n))"
1236       unfolding space_product_algebra
1237       apply simp
1238       apply (subst Pi_UN[OF finite I])
1239       using mono[THEN incseqD] apply simp
1240       apply (simp add: PiE_Int)
1241       apply (intro PiE_cong)
1242       using A sets_into by (auto intro!: into_space)
1243     also have "\<dots> \<in> sets ?E"
1244       using sets_into A \<in> sets (E i)
1245       unfolding sets_product_algebra sets_sigma
1246       by (intro sigma_sets.Union)
1247          (auto simp: image_subset_iff intro!: sigma_sets.Basic)
1248     finally have "(\<lambda>x. x i) - A \<inter> space ?E \<in> sets ?E" . }
1249   then have proj:
1250     "\<And>i. i\<in>I \<Longrightarrow> (\<lambda>x. x i) \<in> measurable ?E (sigma (E i))"
1251     using E by (subst G.measurable_iff_sigma)
1252                (auto simp: sets_product_algebra sets_sigma)
1253   { fix A assume A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (sigma (E i))"
1254     with proj have basic: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) - (A i) \<inter> space ?E \<in> sets ?E"
1255       unfolding measurable_def by simp
1256     have "Pi\<^isub>E I A = (\<Inter>i\<in>I. (\<lambda>x. x i) - (A i) \<inter> space ?E)"
1257       using A E.sets_into_space I \<noteq> {} unfolding product_algebra_def by auto blast
1258     then have "Pi\<^isub>E I A \<in> sets ?E"
1259       using G.finite_INT[OF finite I I \<noteq> {} basic, of "\<lambda>i. i"] by simp }
1260   then have "sigma_sets (space ?E) (sets (product_algebra_generator I (\<lambda>i. sigma (E i)))) \<subseteq> sets ?E"
1261     by (intro G.sigma_sets_subset) (auto simp add: product_algebra_generator_def)
1262   then have subset: "sets ?S \<subseteq> sets ?E"
1263     by (simp add: sets_sigma sets_product_algebra)
1264   show "sets ?S = sets ?E"
1265   proof (intro set_eqI iffI)
1266     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
1267       unfolding sets_sigma sets_product_algebra
1268     proof induct
1269       case (Basic A) then show ?case
1270         by (auto simp: sets_sigma product_algebra_generator_def intro: sigma_sets.Basic)
1271     qed (auto intro: sigma_sets.intros simp: product_algebra_generator_def)
1272   next
1273     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
1274   qed
1275 qed
1277 lemma product_algebraI[intro]:
1278     "E \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> Pi\<^isub>E I E \<in> sets (Pi\<^isub>M I M)"
1279   using assms unfolding product_algebra_def by (auto intro: product_algebra_generatorI)
1281 lemma (in product_sigma_algebra) measurable_component_update:
1282   assumes "x \<in> space (Pi\<^isub>M I M)" and "i \<notin> I"
1283   shows "(\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)" (is "?f \<in> _")
1284   unfolding product_algebra_def apply simp
1285 proof (intro measurable_sigma)
1286   let ?G = "product_algebra_generator (insert i I) M"
1287   show "sets ?G \<subseteq> Pow (space ?G)" using product_algebra_generator_into_space .
1288   show "?f \<in> space (M i) \<rightarrow> space ?G"
1289     using M.sets_into_space assms by auto
1290   fix A assume "A \<in> sets ?G"
1291   from product_algebraE[OF this] guess E . note E = this
1292   then have "?f - A \<inter> space (M i) = E i \<or> ?f - A \<inter> space (M i) = {}"
1293     using M.sets_into_space assms by auto
1294   then show "?f - A \<inter> space (M i) \<in> sets (M i)"
1295     using E by (auto intro!: product_algebraI)
1296 qed
1298 lemma (in product_sigma_algebra) measurable_add_dim:
1299   assumes "i \<notin> I"
1300   shows "(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M)"
1301 proof -
1302   let ?f = "(\<lambda>(f, y). f(i := y))" and ?G = "product_algebra_generator (insert i I) M"
1303   interpret Ii: pair_sigma_algebra "Pi\<^isub>M I M" "M i"
1304     unfolding pair_sigma_algebra_def
1305     by (intro sigma_algebra_product_algebra sigma_algebras conjI)
1306   have "?f \<in> measurable Ii.P (sigma ?G)"
1307   proof (rule Ii.measurable_sigma)
1308     show "sets ?G \<subseteq> Pow (space ?G)"
1309       using product_algebra_generator_into_space .
1310     show "?f \<in> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) \<rightarrow> space ?G"
1311       by (auto simp: space_pair_measure)
1312   next
1313     fix A assume "A \<in> sets ?G"
1314     then obtain F where "A = Pi\<^isub>E (insert i I) F"
1315       and F: "\<And>j. j \<in> I \<Longrightarrow> F j \<in> sets (M j)" "F i \<in> sets (M i)"
1316       by (auto elim!: product_algebraE)
1317     then have "?f - A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) = Pi\<^isub>E I F \<times> (F i)"
1318       using sets_into_space i \<notin> I
1319       by (auto simp add: space_pair_measure) blast+
1320     then show "?f - A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M M i)"
1321       using F by (auto intro!: pair_measureI)
1322   qed
1323   then show ?thesis
1324     by (simp add: product_algebra_def)
1325 qed
1327 lemma (in product_sigma_algebra) measurable_merge:
1328   assumes [simp]: "I \<inter> J = {}"
1329   shows "(\<lambda>(x, y). merge I x J y) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"
1330 proof -
1331   let ?I = "Pi\<^isub>M I M" and ?J = "Pi\<^isub>M J M"
1332   interpret P: sigma_algebra "?I \<Otimes>\<^isub>M ?J"
1333     by (intro sigma_algebra_pair_measure product_algebra_into_space)
1334   let ?f = "\<lambda>(x, y). merge I x J y"
1335   let ?G = "product_algebra_generator (I \<union> J) M"
1336   have "?f \<in> measurable (?I \<Otimes>\<^isub>M ?J) (sigma ?G)"
1337   proof (rule P.measurable_sigma)
1338     fix A assume "A \<in> sets ?G"
1339     from product_algebraE[OF this]
1340     obtain E where E: "A = Pi\<^isub>E (I \<union> J) E" "E \<in> (\<Pi> i\<in>I \<union> J. sets (M i))" .
1341     then have *: "?f - A \<inter> space (?I \<Otimes>\<^isub>M ?J) = Pi\<^isub>E I E \<times> Pi\<^isub>E J E"
1342       using sets_into_space I \<inter> J = {}
1343       by (auto simp add: space_pair_measure) fast+
1344     then show "?f - A \<inter> space (?I \<Otimes>\<^isub>M ?J) \<in> sets (?I \<Otimes>\<^isub>M ?J)"
1345       using E unfolding * by (auto intro!: pair_measureI in_sigma product_algebra_generatorI
1346         simp: product_algebra_def)
1347   qed (insert product_algebra_generator_into_space, auto simp: space_pair_measure)
1348   then show "?f \<in> measurable (?I \<Otimes>\<^isub>M ?J) (Pi\<^isub>M (I \<union> J) M)"
1349     unfolding product_algebra_def[of "I \<union> J"] by simp
1350 qed
1352 lemma (in product_sigma_algebra) measurable_component_singleton:
1353   assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)"
1354 proof (unfold measurable_def, intro CollectI conjI ballI)
1355   fix A assume "A \<in> sets (M i)"
1356   then have "(\<lambda>x. x i) - A \<inter> space (Pi\<^isub>M I M) = (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))"
1357     using M.sets_into_space i \<in> I by (fastsimp dest: Pi_mem split: split_if_asm)
1358   then show "(\<lambda>x. x i) - A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)"
1359     using A \<in> sets (M i) by (auto intro!: product_algebraI)
1360 qed (insert i \<in> I, auto)
1362 locale product_sigma_finite =
1363   fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme"
1364   assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
1366 locale finite_product_sigma_finite = product_sigma_finite M
1367   for M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" +
1368   fixes I :: "'i set" assumes finite_index'[intro]: "finite I"
1370 sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
1371   by (rule sigma_finite_measures)
1373 sublocale product_sigma_finite \<subseteq> product_sigma_algebra
1374   by default
1376 sublocale finite_product_sigma_finite \<subseteq> finite_product_sigma_algebra
1377   by default (fact finite_index')
1379 lemma setprod_extreal_0:
1380   fixes f :: "'a \<Rightarrow> extreal"
1381   shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> (finite A \<and> (\<exists>i\<in>A. f i = 0))"
1382 proof cases
1383   assume "finite A"
1384   then show ?thesis by (induct A) auto
1385 qed auto
1387 lemma setprod_extreal_pos:
1388   fixes f :: "'a \<Rightarrow> extreal" assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" shows "0 \<le> (\<Prod>i\<in>I. f i)"
1389 proof cases
1390   assume "finite I" from this pos show ?thesis by induct auto
1391 qed simp
1393 lemma setprod_PInf:
1394   assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
1395   shows "(\<Prod>i\<in>I. f i) = \<infinity> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<infinity>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)"
1396 proof cases
1397   assume "finite I" from this assms show ?thesis
1398   proof (induct I)
1399     case (insert i I)
1400     then have pos: "0 \<le> f i" "0 \<le> setprod f I" by (auto intro!: setprod_extreal_pos)
1401     from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> setprod f I * f i = \<infinity>" by auto
1402     also have "\<dots> \<longleftrightarrow> (setprod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> setprod f I \<noteq> 0"
1403       using setprod_extreal_pos[of I f] pos
1404       by (cases rule: extreal2_cases[of "f i" "setprod f I"]) auto
1405     also have "\<dots> \<longleftrightarrow> finite (insert i I) \<and> (\<exists>j\<in>insert i I. f j = \<infinity>) \<and> (\<forall>j\<in>insert i I. f j \<noteq> 0)"
1406       using insert by (auto simp: setprod_extreal_0)
1407     finally show ?case .
1408   qed simp
1409 qed simp
1411 lemma setprod_extreal: "(\<Prod>i\<in>A. extreal (f i)) = extreal (setprod f A)"
1412 proof cases
1413   assume "finite A" then show ?thesis
1414     by induct (auto simp: one_extreal_def)
1415 qed (simp add: one_extreal_def)
1417 lemma (in finite_product_sigma_finite) product_algebra_generator_measure:
1418   assumes "Pi\<^isub>E I F \<in> sets G"
1419   shows "measure G (Pi\<^isub>E I F) = (\<Prod>i\<in>I. M.\<mu> i (F i))"
1420 proof cases
1421   assume ne: "\<forall>i\<in>I. F i \<noteq> {}"
1422   have "\<forall>i\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') i = F i"
1423     by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I F'"])
1424        (insert ne, auto simp: Pi_eq_iff)
1425   then show ?thesis
1426     unfolding product_algebra_generator_def by simp
1427 next
1428   assume empty: "\<not> (\<forall>j\<in>I. F j \<noteq> {})"
1429   then have "(\<Prod>j\<in>I. M.\<mu> j (F j)) = 0"
1430     by (auto simp: setprod_extreal_0 intro!: bexI)
1431   moreover
1432   have "\<exists>j\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j = {}"
1433     by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I F'"])
1434        (insert empty, auto simp: Pi_eq_empty_iff[symmetric])
1435   then have "(\<Prod>j\<in>I. M.\<mu> j ((SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j)) = 0"
1436     by (auto simp: setprod_extreal_0 intro!: bexI)
1437   ultimately show ?thesis
1438     unfolding product_algebra_generator_def by simp
1439 qed
1441 lemma (in finite_product_sigma_finite) sigma_finite_pairs:
1442   "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
1443     (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
1444     (\<forall>k. \<forall>i\<in>I. \<mu> i (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<and>
1445     (\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space G"
1446 proof -
1447   have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. \<mu> i (F k) \<noteq> \<infinity>)"
1448     using M.sigma_finite_up by simp
1449   from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
1450   then have F: "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. incseq (F i)" "\<And>i. (\<Union>j. F i j) = space (M i)" "\<And>i k. \<mu> i (F i k) \<noteq> \<infinity>"
1451     by auto
1452   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"
1453   note space_product_algebra[simp]
1454   show ?thesis
1455   proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
1456     fix i show "range (F i) \<subseteq> sets (M i)" by fact
1457   next
1458     fix i k show "\<mu> i (F i k) \<noteq> \<infinity>" by fact
1459   next
1460     fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space G"
1461       using \<And>i. range (F i) \<subseteq> sets (M i) M.sets_into_space
1462       by (force simp: image_subset_iff)
1463   next
1464     fix f assume "f \<in> space G"
1465     with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
1466     show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def)
1467   next
1468     fix i show "?F i \<subseteq> ?F (Suc i)"
1469       using \<And>i. incseq (F i)[THEN incseq_SucD] by auto
1470   qed
1471 qed
1473 lemma sets_pair_cancel_measure[simp]:
1474   "sets (M1\<lparr>measure := m1\<rparr> \<Otimes>\<^isub>M M2) = sets (M1 \<Otimes>\<^isub>M M2)"
1475   "sets (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m2\<rparr>) = sets (M1 \<Otimes>\<^isub>M M2)"
1476   unfolding pair_measure_def pair_measure_generator_def sets_sigma
1477   by simp_all
1479 lemma measurable_pair_cancel_measure[simp]:
1480   "measurable (M1\<lparr>measure := m1\<rparr> \<Otimes>\<^isub>M M2) M = measurable (M1 \<Otimes>\<^isub>M M2) M"
1481   "measurable (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m2\<rparr>) M = measurable (M1 \<Otimes>\<^isub>M M2) M"
1482   "measurable M (M1\<lparr>measure := m3\<rparr> \<Otimes>\<^isub>M M2) = measurable M (M1 \<Otimes>\<^isub>M M2)"
1483   "measurable M (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m4\<rparr>) = measurable M (M1 \<Otimes>\<^isub>M M2)"
1484   unfolding measurable_def by (auto simp add: space_pair_measure)
1486 lemma (in product_sigma_finite) product_measure_exists:
1487   assumes "finite I"
1488   shows "\<exists>\<nu>. sigma_finite_measure (sigma (product_algebra_generator I M) \<lparr> measure := \<nu> \<rparr>) \<and>
1489     (\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i)))"
1490 using finite I proof induct
1491   case empty
1492   interpret finite_product_sigma_finite M "{}" by default simp
1493   let ?\<nu> = "(\<lambda>A. if A = {} then 0 else 1) :: 'd set \<Rightarrow> extreal"
1494   show ?case
1495   proof (intro exI conjI ballI)
1496     have "sigma_algebra (sigma G \<lparr>measure := ?\<nu>\<rparr>)"
1497       by (rule sigma_algebra_cong) (simp_all add: product_algebra_def)
1498     then have "measure_space (sigma G\<lparr>measure := ?\<nu>\<rparr>)"
1499       by (rule finite_additivity_sufficient)
1501                         product_algebra_generator_def image_constant)
1502     then show "sigma_finite_measure (sigma G\<lparr>measure := ?\<nu>\<rparr>)"
1503       by (auto intro!: exI[of _ "\<lambda>i. {\<lambda>_. undefined}"]
1504                simp: sigma_finite_measure_def sigma_finite_measure_axioms_def
1505                      product_algebra_generator_def)
1506   qed auto
1507 next
1508   case (insert i I)
1509   interpret finite_product_sigma_finite M I by default fact
1510   have "finite (insert i I)" using finite I by auto
1511   interpret I': finite_product_sigma_finite M "insert i I" by default fact
1512   from insert obtain \<nu> where
1513     prod: "\<And>A. A \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))" and
1514     "sigma_finite_measure (sigma G\<lparr> measure := \<nu> \<rparr>)" by auto
1515   then interpret I: sigma_finite_measure "P\<lparr> measure := \<nu>\<rparr>" unfolding product_algebra_def by simp
1516   interpret P: pair_sigma_finite "P\<lparr> measure := \<nu>\<rparr>" "M i" ..
1517   let ?h = "(\<lambda>(f, y). f(i := y))"
1518   let ?\<nu> = "\<lambda>A. P.\<mu> (?h - A \<inter> space P.P)"
1519   have I': "sigma_algebra (I'.P\<lparr> measure := ?\<nu> \<rparr>)"
1520     by (rule I'.sigma_algebra_cong) simp_all
1521   interpret I'': measure_space "I'.P\<lparr> measure := ?\<nu> \<rparr>"
1522     using measurable_add_dim[OF i \<notin> I]
1523     by (intro P.measure_space_vimage[OF I']) (auto simp add: measure_preserving_def)
1524   show ?case
1525   proof (intro exI[of _ ?\<nu>] conjI ballI)
1526     let "?m A" = "measure (Pi\<^isub>M I M\<lparr>measure := \<nu>\<rparr> \<Otimes>\<^isub>M M i) (?h - A \<inter> space P.P)"
1527     { fix A assume A: "A \<in> (\<Pi> i\<in>insert i I. sets (M i))"
1528       then have *: "?h - Pi\<^isub>E (insert i I) A \<inter> space P.P = Pi\<^isub>E I A \<times> A i"
1529         using i \<notin> I M.sets_into_space by (auto simp: space_pair_measure space_product_algebra) blast
1530       show "?m (Pi\<^isub>E (insert i I) A) = (\<Prod>i\<in>insert i I. M.\<mu> i (A i))"
1531         unfolding * using A
1532         apply (subst P.pair_measure_times)
1533         using A apply fastsimp
1534         using A apply fastsimp
1535         using i \<notin> I finite I prod[of A] A by (auto simp: ac_simps) }
1536     note product = this
1537     have *: "sigma I'.G\<lparr> measure := ?\<nu> \<rparr> = I'.P\<lparr> measure := ?\<nu> \<rparr>"
1538       by (simp add: product_algebra_def)
1539     show "sigma_finite_measure (sigma I'.G\<lparr> measure := ?\<nu> \<rparr>)"
1540     proof (unfold *, default, simp)
1541       from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
1542       then have F: "\<And>j. j \<in> insert i I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
1543         "incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k)"
1544         "(\<Union>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) = space I'.G"
1545         "\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> \<mu> j (F j k) \<noteq> \<infinity>"
1546         by blast+
1547       let "?F k" = "\<Pi>\<^isub>E j \<in> insert i I. F j k"
1548       show "\<exists>F::nat \<Rightarrow> ('i \<Rightarrow> 'a) set. range F \<subseteq> sets I'.P \<and>
1549           (\<Union>i. F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i)) \<and> (\<forall>i. ?m (F i) \<noteq> \<infinity>)"
1550       proof (intro exI[of _ ?F] conjI allI)
1551         show "range ?F \<subseteq> sets I'.P" using F(1) by auto
1552       next
1553         from F(3) show "(\<Union>i. ?F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i))" by simp
1554       next
1555         fix j
1556         have "\<And>k. k \<in> insert i I \<Longrightarrow> 0 \<le> measure (M k) (F k j)"
1557           using F(1) by auto
1558         with F finite I setprod_PInf[of "insert i I", OF this] show "?\<nu> (?F j) \<noteq> \<infinity>"
1559           by (subst product) auto
1560       qed
1561     qed
1562   qed
1563 qed
1565 sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure P
1566   unfolding product_algebra_def
1567   using product_measure_exists[OF finite_index]
1568   by (rule someI2_ex) auto
1570 lemma (in finite_product_sigma_finite) measure_times:
1571   assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)"
1572   shows "\<mu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
1573   unfolding product_algebra_def
1574   using product_measure_exists[OF finite_index]
1575   proof (rule someI2_ex, elim conjE)
1576     fix \<nu> assume *: "\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
1577     have "Pi\<^isub>E I A = Pi\<^isub>E I (\<lambda>i\<in>I. A i)" by (auto dest: Pi_mem)
1578     then have "\<nu> (Pi\<^isub>E I A) = \<nu> (Pi\<^isub>E I (\<lambda>i\<in>I. A i))" by simp
1579     also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i ((\<lambda>i\<in>I. A i) i))" using assms * by auto
1580     finally show "measure (sigma G\<lparr>measure := \<nu>\<rparr>) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
1581       by (simp add: sigma_def)
1582   qed
1584 lemma (in product_sigma_finite) product_measure_empty[simp]:
1585   "measure (Pi\<^isub>M {} M) {\<lambda>x. undefined} = 1"
1586 proof -
1587   interpret finite_product_sigma_finite M "{}" by default auto
1588   from measure_times[of "\<lambda>x. {}"] show ?thesis by simp
1589 qed
1591 lemma (in finite_product_sigma_algebra) P_empty:
1592   assumes "I = {}"
1593   shows "space P = {\<lambda>k. undefined}" "sets P = { {}, {\<lambda>k. undefined} }"
1594   unfolding product_algebra_def product_algebra_generator_def I = {}
1595   by (simp_all add: sigma_def image_constant)
1597 lemma (in product_sigma_finite) positive_integral_empty:
1598   assumes pos: "0 \<le> f (\<lambda>k. undefined)"
1599   shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"
1600 proof -
1601   interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
1602   have "\<And>A. measure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"
1603     using assms by (subst measure_times) auto
1604   then show ?thesis
1605     unfolding positive_integral_def simple_function_def simple_integral_def_raw
1606   proof (simp add: P_empty, intro antisym)
1607     show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
1608       by (intro le_SUPI) (auto simp: le_fun_def split: split_max)
1609     show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
1610       by (intro SUP_leI) (auto simp: le_fun_def simp: max_def split: split_if_asm)
1611   qed
1612 qed
1614 lemma (in product_sigma_finite) measure_fold:
1615   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
1616   assumes A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
1617   shows "measure (Pi\<^isub>M (I \<union> J) M) A =
1618     measure (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ((\<lambda>(x,y). merge I x J y) - A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M))"
1619 proof -
1620   interpret I: finite_product_sigma_finite M I by default fact
1621   interpret J: finite_product_sigma_finite M J by default fact
1622   have "finite (I \<union> J)" using fin by auto
1623   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
1624   interpret P: pair_sigma_finite I.P J.P by default
1625   let ?g = "\<lambda>(x,y). merge I x J y"
1626   let "?X S" = "?g - S \<inter> space P.P"
1627   from IJ.sigma_finite_pairs obtain F where
1628     F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
1629        "incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)"
1630        "(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space IJ.G"
1631        "\<And>k. \<forall>i\<in>I\<union>J. \<mu> i (F i k) \<noteq> \<infinity>"
1632     by auto
1633   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"
1634   show "IJ.\<mu> A = P.\<mu> (?X A)"
1635   proof (rule measure_unique_Int_stable_vimage)
1636     show "measure_space IJ.P" "measure_space P.P" by default
1637     show "sets (sigma IJ.G) = sets IJ.P" "space IJ.G = space IJ.P" "A \<in> sets (sigma IJ.G)"
1638       using A unfolding product_algebra_def by auto
1639   next
1640     show "Int_stable IJ.G"
1641       by (simp add: PiE_Int Int_stable_def product_algebra_def
1642                     product_algebra_generator_def)
1643           auto
1644     show "range ?F \<subseteq> sets IJ.G" using F
1645       by (simp add: image_subset_iff product_algebra_def
1646                     product_algebra_generator_def)
1647     show "incseq ?F" "(\<Union>i. ?F i) = space IJ.G " by fact+
1648   next
1649     fix k
1650     have "\<And>j. j \<in> I \<union> J \<Longrightarrow> 0 \<le> measure (M j) (F j k)"
1651       using F(1) by auto
1652     with F finite I setprod_PInf[of "I \<union> J", OF this]
1653     show "IJ.\<mu> (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
1654   next
1655     fix A assume "A \<in> sets IJ.G"
1656     then obtain F where A: "A = Pi\<^isub>E (I \<union> J) F"
1657       and F: "\<And>i. i \<in> I \<union> J \<Longrightarrow> F i \<in> sets (M i)"
1658       by (auto simp: product_algebra_generator_def)
1659     then have X: "?X A = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"
1660       using sets_into_space by (auto simp: space_pair_measure) blast+
1661     then have "P.\<mu> (?X A) = (\<Prod>i\<in>I. \<mu> i (F i)) * (\<Prod>i\<in>J. \<mu> i (F i))"
1662       using finite J finite I F
1663       by (simp add: P.pair_measure_times I.measure_times J.measure_times)
1664     also have "\<dots> = (\<Prod>i\<in>I \<union> J. \<mu> i (F i))"
1665       using finite J finite I I \<inter> J = {}  by (simp add: setprod_Un_one)
1666     also have "\<dots> = IJ.\<mu> A"
1667       using finite J finite I F unfolding A
1668       by (intro IJ.measure_times[symmetric]) auto
1669     finally show "IJ.\<mu> A = P.\<mu> (?X A)" by (rule sym)
1670   qed (rule measurable_merge[OF IJ])
1671 qed
1673 lemma (in product_sigma_finite) measure_preserving_merge:
1674   assumes IJ: "I \<inter> J = {}" and "finite I" "finite J"
1675   shows "(\<lambda>(x,y). merge I x J y) \<in> measure_preserving (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"
1676   by (intro measure_preservingI measurable_merge[OF IJ] measure_fold[symmetric] assms)
1678 lemma (in product_sigma_finite) product_positive_integral_fold:
1679   assumes IJ[simp]: "I \<inter> J = {}" "finite I" "finite J"
1680   and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
1681   shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f =
1682     (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I x J y) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))"
1683 proof -
1684   interpret I: finite_product_sigma_finite M I by default fact
1685   interpret J: finite_product_sigma_finite M J by default fact
1686   interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
1687   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default simp
1688   have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable P.P"
1689     using measurable_comp[OF measurable_merge[OF IJ(1)] f] by (simp add: comp_def)
1690   show ?thesis
1691     unfolding P.positive_integral_fst_measurable[OF P_borel, simplified]
1692   proof (rule P.positive_integral_vimage)
1693     show "sigma_algebra IJ.P" by default
1694     show "(\<lambda>(x, y). merge I x J y) \<in> measure_preserving P.P IJ.P"
1695       using IJ by (rule measure_preserving_merge)
1696     show "f \<in> borel_measurable IJ.P" using f by simp
1697   qed
1698 qed
1700 lemma (in product_sigma_finite) measure_preserving_component_singelton:
1701   "(\<lambda>x. x i) \<in> measure_preserving (Pi\<^isub>M {i} M) (M i)"
1702 proof (intro measure_preservingI measurable_component_singleton)
1703   interpret I: finite_product_sigma_finite M "{i}" by default simp
1704   fix A let ?P = "(\<lambda>x. x i) - A \<inter> space I.P"
1705   assume A: "A \<in> sets (M i)"
1706   then have *: "?P = {i} \<rightarrow>\<^isub>E A" using sets_into_space by auto
1707   show "I.\<mu> ?P = M.\<mu> i A" unfolding *
1708     using A I.measure_times[of "\<lambda>_. A"] by auto
1709 qed auto
1711 lemma (in product_sigma_finite) product_positive_integral_singleton:
1712   assumes f: "f \<in> borel_measurable (M i)"
1713   shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f"
1714 proof -
1715   interpret I: finite_product_sigma_finite M "{i}" by default simp
1716   show ?thesis
1717   proof (rule I.positive_integral_vimage[symmetric])
1718     show "sigma_algebra (M i)" by (rule sigma_algebras)
1719     show "(\<lambda>x. x i) \<in> measure_preserving (Pi\<^isub>M {i} M) (M i)"
1720       by (rule measure_preserving_component_singelton)
1721     show "f \<in> borel_measurable (M i)" by fact
1722   qed
1723 qed
1725 lemma (in product_sigma_finite) product_positive_integral_insert:
1726   assumes [simp]: "finite I" "i \<notin> I"
1727     and f: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
1728   shows "integral\<^isup>P (Pi\<^isub>M (insert i I) M) f = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^isub>M I M))"
1729 proof -
1730   interpret I: finite_product_sigma_finite M I by default auto
1731   interpret i: finite_product_sigma_finite M "{i}" by default auto
1732   interpret P: pair_sigma_algebra I.P i.P ..
1733   have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
1734     using f by auto
1735   show ?thesis
1736     unfolding product_positive_integral_fold[OF IJ, unfolded insert, simplified, OF f]
1737   proof (rule I.positive_integral_cong, subst product_positive_integral_singleton)
1738     fix x assume x: "x \<in> space I.P"
1739     let "?f y" = "f (restrict (x(i := y)) (insert i I))"
1740     have f'_eq: "\<And>y. ?f y = f (x(i := y))"
1741       using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)
1742     show "?f \<in> borel_measurable (M i)" unfolding f'_eq
1743       using measurable_comp[OF measurable_component_update f] x i \<notin> I
1744       by (simp add: comp_def)
1745     show "integral\<^isup>P (M i) ?f = \<integral>\<^isup>+ y. f (x(i:=y)) \<partial>M i"
1746       unfolding f'_eq by simp
1747   qed
1748 qed
1750 lemma (in product_sigma_finite) product_positive_integral_setprod:
1751   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> extreal"
1752   assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
1753   and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
1754   shows "(\<integral>\<^isup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>P (M i) (f i))"
1755 using assms proof induct
1756   case empty
1757   interpret finite_product_sigma_finite M "{}" by default auto
1758   then show ?case by simp
1759 next
1760   case (insert i I)
1761   note finite I[intro, simp]
1762   interpret I: finite_product_sigma_finite M I by default auto
1763   have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))"
1764     using insert by (auto intro!: setprod_cong)
1765   have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^isub>M J M)"
1766     using sets_into_space insert
1767     by (intro sigma_algebra.borel_measurable_extreal_setprod sigma_algebra_product_algebra
1768               measurable_comp[OF measurable_component_singleton, unfolded comp_def])
1769        auto
1770   then show ?case
1771     apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
1772     apply (simp add: insert * pos borel setprod_extreal_pos M.positive_integral_multc)
1773     apply (subst I.positive_integral_cmult)
1774     apply (auto simp add: pos borel insert setprod_extreal_pos positive_integral_positive)
1775     done
1776 qed
1778 lemma (in product_sigma_finite) product_integral_singleton:
1779   assumes f: "f \<in> borel_measurable (M i)"
1780   shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f"
1781 proof -
1782   interpret I: finite_product_sigma_finite M "{i}" by default simp
1783   have *: "(\<lambda>x. extreal (f x)) \<in> borel_measurable (M i)"
1784     "(\<lambda>x. extreal (- f x)) \<in> borel_measurable (M i)"
1785     using assms by auto
1786   show ?thesis
1787     unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] ..
1788 qed
1790 lemma (in product_sigma_finite) product_integral_fold:
1791   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
1792   and f: "integrable (Pi\<^isub>M (I \<union> J) M) f"
1793   shows "integral\<^isup>L (Pi\<^isub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I x J y) \<partial>Pi\<^isub>M J M) \<partial>Pi\<^isub>M I M)"
1794 proof -
1795   interpret I: finite_product_sigma_finite M I by default fact
1796   interpret J: finite_product_sigma_finite M J by default fact
1797   have "finite (I \<union> J)" using fin by auto
1798   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
1799   interpret P: pair_sigma_finite I.P J.P by default
1800   let ?M = "\<lambda>(x, y). merge I x J y"
1801   let ?f = "\<lambda>x. f (?M x)"
1802   show ?thesis
1803   proof (subst P.integrable_fst_measurable(2)[of ?f, simplified])
1804     have 1: "sigma_algebra IJ.P" by default
1805     have 2: "?M \<in> measure_preserving P.P IJ.P" using measure_preserving_merge[OF assms(1,2,3)] .
1806     have 3: "integrable (Pi\<^isub>M (I \<union> J) M) f" by fact
1807     then have 4: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
1808       by (simp add: integrable_def)
1809     show "integrable P.P ?f"
1810       by (rule P.integrable_vimage[where f=f, OF 1 2 3])
1811     show "integral\<^isup>L IJ.P f = integral\<^isup>L P.P ?f"
1812       by (rule P.integral_vimage[where f=f, OF 1 2 4])
1813   qed
1814 qed
1816 lemma (in product_sigma_finite) product_integral_insert:
1817   assumes [simp]: "finite I" "i \<notin> I"
1818     and f: "integrable (Pi\<^isub>M (insert i I) M) f"
1819   shows "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = (\<integral>x. (\<integral>y. f (x(i:=y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
1820 proof -
1821   interpret I: finite_product_sigma_finite M I by default auto
1822   interpret I': finite_product_sigma_finite M "insert i I" by default auto
1823   interpret i: finite_product_sigma_finite M "{i}" by default auto
1824   interpret P: pair_sigma_finite I.P i.P ..
1825   have IJ: "I \<inter> {i} = {}" by auto
1826   show ?thesis
1827     unfolding product_integral_fold[OF IJ, simplified, OF f]
1828   proof (rule I.integral_cong, subst product_integral_singleton)
1829     fix x assume x: "x \<in> space I.P"
1830     let "?f y" = "f (restrict (x(i := y)) (insert i I))"
1831     have f'_eq: "\<And>y. ?f y = f (x(i := y))"
1832       using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)
1833     have f: "f \<in> borel_measurable I'.P" using f unfolding integrable_def by auto
1834     show "?f \<in> borel_measurable (M i)"
1835       unfolding measurable_cong[OF f'_eq]
1836       using measurable_comp[OF measurable_component_update f] x i \<notin> I
1837       by (simp add: comp_def)
1838     show "integral\<^isup>L (M i) ?f = integral\<^isup>L (M i) (\<lambda>y. f (x(i := y)))"
1839       unfolding f'_eq by simp
1840   qed
1841 qed
1843 lemma (in product_sigma_finite) product_integrable_setprod:
1844   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
1845   assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
1846   shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
1847 proof -
1848   interpret finite_product_sigma_finite M I by default fact
1849   have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
1850     using integrable unfolding integrable_def by auto
1851   then have borel: "?f \<in> borel_measurable P"
1852     using measurable_comp[OF measurable_component_singleton f]
1853     by (auto intro!: borel_measurable_setprod simp: comp_def)
1854   moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"
1855   proof (unfold integrable_def, intro conjI)
1856     show "(\<lambda>x. abs (?f x)) \<in> borel_measurable P"
1857       using borel by auto
1858     have "(\<integral>\<^isup>+x. extreal (abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. extreal (abs (f i (x i)))) \<partial>P)"
1859       by (simp add: setprod_extreal abs_setprod)
1860     also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. extreal (abs (f i x)) \<partial>M i))"
1861       using f by (subst product_positive_integral_setprod) auto
1862     also have "\<dots> < \<infinity>"
1863       using integrable[THEN M.integrable_abs]
1864       by (simp add: setprod_PInf integrable_def M.positive_integral_positive)
1865     finally show "(\<integral>\<^isup>+x. extreal (abs (?f x)) \<partial>P) \<noteq> \<infinity>" by auto
1866     have "(\<integral>\<^isup>+x. extreal (- abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. 0 \<partial>P)"
1867       by (intro positive_integral_cong_pos) auto
1868     then show "(\<integral>\<^isup>+x. extreal (- abs (?f x)) \<partial>P) \<noteq> \<infinity>" by simp
1869   qed
1870   ultimately show ?thesis
1871     by (rule integrable_abs_iff[THEN iffD1])
1872 qed
1874 lemma (in product_sigma_finite) product_integral_setprod:
1875   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
1876   assumes "finite I" "I \<noteq> {}" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
1877   shows "(\<integral>x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>L (M i) (f i))"
1878 using assms proof (induct rule: finite_ne_induct)
1879   case (singleton i)
1880   then show ?case by (simp add: product_integral_singleton integrable_def)
1881 next
1882   case (insert i I)
1883   then have iI: "finite (insert i I)" by auto
1884   then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
1885     integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
1886     by (intro product_integrable_setprod insert(5)) (auto intro: finite_subset)
1887   interpret I: finite_product_sigma_finite M I by default fact
1888   have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))"
1889     using i \<notin> I` by (auto intro!: setprod_cong)
1890   show ?case
1891     unfolding product_integral_insert[OF insert(1,3) prod[OF subset_refl]]
1892     by (simp add: * insert integral_multc I.integral_cmult[OF prod] subset_insertI)
1893 qed
1895 section "Products on finite spaces"
1897 lemma sigma_sets_pair_measure_generator_finite:
1898   assumes "finite A" and "finite B"
1899   shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<in> Pow A \<and> b \<in> Pow B} = Pow (A \<times> B)"
1900   (is "sigma_sets ?prod ?sets = _")
1901 proof safe
1902   have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
1903   fix x assume subset: "x \<subseteq> A \<times> B"
1904   hence "finite x" using fin by (rule finite_subset)
1905   from this subset show "x \<in> sigma_sets ?prod ?sets"
1906   proof (induct x)
1907     case empty show ?case by (rule sigma_sets.Empty)
1908   next
1909     case (insert a x)
1910     hence "{a} \<in> sigma_sets ?prod ?sets"
1911       by (auto simp: pair_measure_generator_def intro!: sigma_sets.Basic)
1912     moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
1913     ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
1914   qed
1915 next
1916   fix x a b
1917   assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
1918   from sigma_sets_into_sp[OF _ this(1)] this(2)
1919   show "a \<in> A" and "b \<in> B" by auto
1920 qed
1922 locale pair_finite_sigma_algebra = M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2
1923   for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
1925 sublocale pair_finite_sigma_algebra \<subseteq> pair_sigma_algebra by default
1927 lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra:
1928   shows "P = \<lparr> space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2), \<dots> = algebra.more P \<rparr>"
1929 proof -
1930   show ?thesis
1931     using sigma_sets_pair_measure_generator_finite[OF M1.finite_space M2.finite_space]
1932     by (intro algebra.equality) (simp_all add: pair_measure_def pair_measure_generator_def sigma_def)
1933 qed
1935 sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P
1936   apply default
1937   using M1.finite_space M2.finite_space
1938   apply (subst finite_pair_sigma_algebra) apply simp
1939   apply (subst (1 2) finite_pair_sigma_algebra) apply simp
1940   done
1942 locale pair_finite_space = M1: finite_measure_space M1 + M2: finite_measure_space M2
1943   for M1 M2
1945 sublocale pair_finite_space \<subseteq> pair_finite_sigma_algebra
1946   by default
1948 sublocale pair_finite_space \<subseteq> pair_sigma_finite
1949   by default
1951 lemma (in pair_finite_space) pair_measure_Pair[simp]:
1952   assumes "a \<in> space M1" "b \<in> space M2"
1953   shows "\<mu> {(a, b)} = M1.\<mu> {a} * M2.\<mu> {b}"
1954 proof -
1955   have "\<mu> ({a}\<times>{b}) = M1.\<mu> {a} * M2.\<mu> {b}"
1956     using M1.sets_eq_Pow M2.sets_eq_Pow assms
1957     by (subst pair_measure_times) auto
1958   then show ?thesis by simp
1959 qed
1961 lemma (in pair_finite_space) pair_measure_singleton[simp]:
1962   assumes "x \<in> space M1 \<times> space M2"
1963   shows "\<mu> {x} = M1.\<mu> {fst x} * M2.\<mu> {snd x}"
1964   using pair_measure_Pair assms by (cases x) auto
1966 sublocale pair_finite_space \<subseteq> finite_measure_space P
1967   by default (auto simp: space_pair_measure)
1969 end