src/HOL/Probability/Product_Measure.thy
 author hoelzl Mon Jan 24 22:29:50 2011 +0100 (2011-01-24) changeset 41661 baf1964bc468 parent 41659 a5d1b2df5e97 child 41689 3e39b0e730d6 permissions -rw-r--r--
use pre-image measure, instead of image
1 theory Product_Measure
2 imports Lebesgue_Integration
3 begin
5 lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
6   by auto
8 lemma Pair_vimage_times[simp]: "\<And>A B x. Pair x - (A \<times> B) = (if x \<in> A then B else {})"
9   by auto
11 lemma rev_Pair_vimage_times[simp]: "\<And>A B y. (\<lambda>x. (x, y)) - (A \<times> B) = (if y \<in> B then A else {})"
12   by auto
14 lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))"
15   by (cases x) simp
17 lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
18   by (auto simp: fun_eq_iff)
20 abbreviation
21   "Pi\<^isub>E A B \<equiv> Pi A B \<inter> extensional A"
23 abbreviation
24   funcset_extensional :: "['a set, 'b set] => ('a => 'b) set"
25     (infixr "->\<^isub>E" 60) where
26   "A ->\<^isub>E B \<equiv> Pi\<^isub>E A (%_. B)"
28 notation (xsymbols)
29   funcset_extensional  (infixr "\<rightarrow>\<^isub>E" 60)
31 lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}"
32   by safe (auto simp add: extensional_def fun_eq_iff)
34 lemma extensional_insert[intro, simp]:
35   assumes "a \<in> extensional (insert i I)"
36   shows "a(i := b) \<in> extensional (insert i I)"
37   using assms unfolding extensional_def by auto
39 lemma extensional_Int[simp]:
40   "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
41   unfolding extensional_def by auto
43 definition
44   "merge I x J y = (\<lambda>i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
46 lemma merge_apply[simp]:
47   "I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"
48   "I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"
49   "J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"
50   "J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"
51   "i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I x J y i = undefined"
52   unfolding merge_def by auto
54 lemma merge_commute:
55   "I \<inter> J = {} \<Longrightarrow> merge I x J y = merge J y I x"
56   by (auto simp: merge_def intro!: ext)
58 lemma Pi_cancel_merge_range[simp]:
59   "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"
60   "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"
61   "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"
62   "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"
63   by (auto simp: Pi_def)
65 lemma Pi_cancel_merge[simp]:
66   "I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
67   "J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
68   "I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
69   "J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
70   by (auto simp: Pi_def)
72 lemma extensional_merge[simp]: "merge I x J y \<in> extensional (I \<union> J)"
73   by (auto simp: extensional_def)
75 lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
76   by (auto simp: restrict_def Pi_def)
78 lemma restrict_merge[simp]:
79   "I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"
80   "I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
81   "J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"
82   "J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
83   by (auto simp: restrict_def intro!: ext)
85 lemma extensional_insert_undefined[intro, simp]:
86   assumes "a \<in> extensional (insert i I)"
87   shows "a(i := undefined) \<in> extensional I"
88   using assms unfolding extensional_def by auto
90 lemma extensional_insert_cancel[intro, simp]:
91   assumes "a \<in> extensional I"
92   shows "a \<in> extensional (insert i I)"
93   using assms unfolding extensional_def by auto
95 lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I x {i} y = restrict (x(i := y i)) (insert i I)"
96   unfolding merge_def by (auto simp: fun_eq_iff)
98 lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
99   by auto
101 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)"
102   by auto
104 lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B"
105   by (auto simp: Pi_def)
107 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"
108   by (auto simp: Pi_def)
110 lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X"
111   by (auto simp: Pi_def)
113 lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
114   by (auto simp: Pi_def)
116 lemma restrict_vimage:
117   assumes "I \<inter> J = {}"
118   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)"
119   using assms by (auto simp: restrict_Pi_cancel)
121 lemma merge_vimage:
122   assumes "I \<inter> J = {}"
123   shows "(\<lambda>(x,y). merge I x J y) - Pi\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E"
124   using assms by (auto simp: restrict_Pi_cancel)
126 lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
127   by (auto simp: restrict_def intro!: ext)
129 lemma merge_restrict[simp]:
130   "merge I (restrict x I) J y = merge I x J y"
131   "merge I x J (restrict y J) = merge I x J y"
132   unfolding merge_def by (auto intro!: ext)
134 lemma merge_x_x_eq_restrict[simp]:
135   "merge I x J x = restrict x (I \<union> J)"
136   unfolding merge_def by (auto intro!: ext)
138 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"
139   apply auto
140   apply (drule_tac x=x in Pi_mem)
141   apply (simp_all split: split_if_asm)
142   apply (drule_tac x=i in Pi_mem)
143   apply (auto dest!: Pi_mem)
144   done
146 lemma Pi_UN:
147   fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
148   assumes "finite I" and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
149   shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
150 proof (intro set_eqI iffI)
151   fix f assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
152   then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" by auto
153   from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
154   obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
155     using finite I finite_nat_set_iff_bounded_le[of "nI"] by auto
156   have "f \<in> Pi I (A k)"
157   proof (intro Pi_I)
158     fix i assume "i \<in> I"
159     from mono[OF this, of "n i" k] k[OF this] n[OF this]
160     show "f i \<in> A k i" by auto
161   qed
162   then show "f \<in> (\<Union>n. Pi I (A n))" by auto
163 qed auto
165 lemma PiE_cong:
166   assumes "\<And>i. i\<in>I \<Longrightarrow> A i = B i"
167   shows "Pi\<^isub>E I A = Pi\<^isub>E I B"
168   using assms by (auto intro!: Pi_cong)
170 lemma restrict_upd[simp]:
171   "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
172   by (auto simp: fun_eq_iff)
174 section "Binary products"
176 definition
177   "pair_algebra A B = \<lparr> space = space A \<times> space B,
178                            sets = {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} \<rparr>"
180 locale pair_sigma_algebra = M1: sigma_algebra M1 + M2: sigma_algebra M2
181   for M1 M2
183 abbreviation (in pair_sigma_algebra)
184   "E \<equiv> pair_algebra M1 M2"
186 abbreviation (in pair_sigma_algebra)
187   "P \<equiv> sigma E"
189 sublocale pair_sigma_algebra \<subseteq> sigma_algebra P
190   using M1.sets_into_space M2.sets_into_space
191   by (force simp: pair_algebra_def intro!: sigma_algebra_sigma)
193 lemma pair_algebraI[intro, simp]:
194   "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (pair_algebra A B)"
195   by (auto simp add: pair_algebra_def)
197 lemma space_pair_algebra:
198   "space (pair_algebra A B) = space A \<times> space B"
199   by (simp add: pair_algebra_def)
201 lemma sets_pair_algebra: "sets (pair_algebra N M) = (\<lambda>(x, y). x \<times> y)  (sets N \<times> sets M)"
202   unfolding pair_algebra_def by auto
204 lemma pair_algebra_sets_into_space:
205   assumes "sets M \<subseteq> Pow (space M)" "sets N \<subseteq> Pow (space N)"
206   shows "sets (pair_algebra M N) \<subseteq> Pow (space (pair_algebra M N))"
207   using assms by (auto simp: pair_algebra_def)
209 lemma pair_algebra_Int_snd:
210   assumes "sets S1 \<subseteq> Pow (space S1)"
211   shows "pair_algebra S1 (algebra.restricted_space S2 A) =
212          algebra.restricted_space (pair_algebra S1 S2) (space S1 \<times> A)"
213   (is "?L = ?R")
214 proof (intro algebra.equality set_eqI iffI)
215   fix X assume "X \<in> sets ?L"
216   then obtain A1 A2 where X: "X = A1 \<times> (A \<inter> A2)" and "A1 \<in> sets S1" "A2 \<in> sets S2"
217     by (auto simp: pair_algebra_def)
218   then show "X \<in> sets ?R" unfolding pair_algebra_def
219     using assms apply simp by (intro image_eqI[of _ _ "A1 \<times> A2"]) auto
220 next
221   fix X assume "X \<in> sets ?R"
222   then obtain A1 A2 where "X = space S1 \<times> A \<inter> A1 \<times> A2" "A1 \<in> sets S1" "A2 \<in> sets S2"
223     by (auto simp: pair_algebra_def)
224   moreover then have "X = A1 \<times> (A \<inter> A2)"
225     using assms by auto
226   ultimately show "X \<in> sets ?L"
227     unfolding pair_algebra_def by auto
228 qed (auto simp add: pair_algebra_def)
230 lemma (in pair_sigma_algebra)
231   shows measurable_fst[intro!, simp]:
232     "fst \<in> measurable P M1" (is ?fst)
233   and measurable_snd[intro!, simp]:
234     "snd \<in> measurable P M2" (is ?snd)
235 proof -
236   { fix X assume "X \<in> sets M1"
237     then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. fst - X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
238       apply - apply (rule bexI[of _ X]) apply (rule bexI[of _ "space M2"])
239       using M1.sets_into_space by force+ }
240   moreover
241   { fix X assume "X \<in> sets M2"
242     then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. snd - X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
243       apply - apply (rule bexI[of _ "space M1"]) apply (rule bexI[of _ X])
244       using M2.sets_into_space by force+ }
245   ultimately have "?fst \<and> ?snd"
246     by (fastsimp simp: measurable_def sets_sigma space_pair_algebra
247                  intro!: sigma_sets.Basic)
248   then show ?fst ?snd by auto
249 qed
251 lemma (in pair_sigma_algebra) measurable_pair_iff:
252   assumes "sigma_algebra M"
253   shows "f \<in> measurable M P \<longleftrightarrow>
254     (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"
255 proof -
256   interpret M: sigma_algebra M by fact
257   from assms show ?thesis
258   proof (safe intro!: measurable_comp[where b=P])
259     assume f: "(fst \<circ> f) \<in> measurable M M1" and s: "(snd \<circ> f) \<in> measurable M M2"
260     show "f \<in> measurable M P"
261     proof (rule M.measurable_sigma)
262       show "sets (pair_algebra M1 M2) \<subseteq> Pow (space E)"
263         unfolding pair_algebra_def using M1.sets_into_space M2.sets_into_space by auto
264       show "f \<in> space M \<rightarrow> space E"
265         using f s by (auto simp: mem_Times_iff measurable_def comp_def space_sigma space_pair_algebra)
266       fix A assume "A \<in> sets E"
267       then obtain B C where "B \<in> sets M1" "C \<in> sets M2" "A = B \<times> C"
268         unfolding pair_algebra_def by auto
269       moreover have "(fst \<circ> f) - B \<inter> space M \<in> sets M"
270         using f B \<in> sets M1 unfolding measurable_def by auto
271       moreover have "(snd \<circ> f) - C \<inter> space M \<in> sets M"
272         using s C \<in> sets M2 unfolding measurable_def by auto
273       moreover have "f - A \<inter> space M = ((fst \<circ> f) - B \<inter> space M) \<inter> ((snd \<circ> f) - C \<inter> space M)"
274         unfolding A = B \<times> C by (auto simp: vimage_Times)
275       ultimately show "f - A \<inter> space M \<in> sets M" by auto
276     qed
277   qed
278 qed
280 lemma (in pair_sigma_algebra) measurable_pair:
281   assumes "sigma_algebra M"
282   assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"
283   shows "f \<in> measurable M P"
284   unfolding measurable_pair_iff[OF assms(1)] using assms(2,3) by simp
286 lemma pair_algebraE:
287   assumes "X \<in> sets (pair_algebra M1 M2)"
288   obtains A B where "X = A \<times> B" "A \<in> sets M1" "B \<in> sets M2"
289   using assms unfolding pair_algebra_def by auto
291 lemma (in pair_sigma_algebra) pair_algebra_swap:
292   "(\<lambda>X. (\<lambda>(x,y). (y,x)) - X \<inter> space M2 \<times> space M1)  sets E = sets (pair_algebra M2 M1)"
293 proof (safe elim!: pair_algebraE)
294   fix A B assume "A \<in> sets M1" "B \<in> sets M2"
295   moreover then have "(\<lambda>(x, y). (y, x)) - (A \<times> B) \<inter> space M2 \<times> space M1 = B \<times> A"
296     using M1.sets_into_space M2.sets_into_space by auto
297   ultimately show "(\<lambda>(x, y). (y, x)) - (A \<times> B) \<inter> space M2 \<times> space M1 \<in> sets (pair_algebra M2 M1)"
298     by (auto intro: pair_algebraI)
299 next
300   fix A B assume "A \<in> sets M1" "B \<in> sets M2"
301   then show "B \<times> A \<in> (\<lambda>X. (\<lambda>(x, y). (y, x)) - X \<inter> space M2 \<times> space M1)  sets E"
302     using M1.sets_into_space M2.sets_into_space
303     by (auto intro!: image_eqI[where x="A \<times> B"] pair_algebraI)
304 qed
306 lemma (in pair_sigma_algebra) sets_pair_sigma_algebra_swap:
307   assumes Q: "Q \<in> sets P"
308   shows "(\<lambda>(x,y). (y, x))  Q \<in> sets (sigma (pair_algebra M2 M1))" (is "_ \<in> sets ?Q")
309 proof -
310   have *: "(\<lambda>(x,y). (y, x)) \<in> space M2 \<times> space M1 \<rightarrow> (space M1 \<times> space M2)"
311        "sets (pair_algebra M1 M2) \<subseteq> Pow (space M1 \<times> space M2)"
312     using M1.sets_into_space M2.sets_into_space by (auto elim!: pair_algebraE)
313   from Q sets_into_space show ?thesis
314     by (auto intro!: image_eqI[where x=Q]
315              simp: pair_algebra_swap[symmetric] sets_sigma
316                    sigma_sets_vimage[OF *] space_pair_algebra)
317 qed
319 lemma (in pair_sigma_algebra) pair_sigma_algebra_swap_measurable:
320   shows "(\<lambda>(x,y). (y, x)) \<in> measurable P (sigma (pair_algebra M2 M1))"
321     (is "?f \<in> measurable ?P ?Q")
322   unfolding measurable_def
323 proof (intro CollectI conjI Pi_I ballI)
324   fix x assume "x \<in> space ?P" then show "(case x of (x, y) \<Rightarrow> (y, x)) \<in> space ?Q"
325     unfolding pair_algebra_def by auto
326 next
327   fix A assume "A \<in> sets ?Q"
328   interpret Q: pair_sigma_algebra M2 M1 by default
329   have "?f - A \<inter> space ?P = (\<lambda>(x,y). (y, x))  A"
330     using Q.sets_into_space A \<in> sets ?Q by (auto simp: pair_algebra_def)
331   with Q.sets_pair_sigma_algebra_swap[OF A \<in> sets ?Q]
332   show "?f - A \<inter> space ?P \<in> sets ?P" by simp
333 qed
335 lemma (in pair_sigma_algebra) measurable_cut_fst:
336   assumes "Q \<in> sets P" shows "Pair x - Q \<in> sets M2"
337 proof -
338   let ?Q' = "{Q. Q \<subseteq> space P \<and> Pair x - Q \<in> sets M2}"
339   let ?Q = "\<lparr> space = space P, sets = ?Q' \<rparr>"
340   interpret Q: sigma_algebra ?Q
341     proof qed (auto simp: vimage_UN vimage_Diff space_pair_algebra)
342   have "sets E \<subseteq> sets ?Q"
343     using M1.sets_into_space M2.sets_into_space
344     by (auto simp: pair_algebra_def space_pair_algebra)
345   then have "sets P \<subseteq> sets ?Q"
346     by (subst pair_algebra_def, intro Q.sets_sigma_subset)
347        (simp_all add: pair_algebra_def)
348   with assms show ?thesis by auto
349 qed
351 lemma (in pair_sigma_algebra) measurable_cut_snd:
352   assumes Q: "Q \<in> sets P" shows "(\<lambda>x. (x, y)) - Q \<in> sets M1" (is "?cut Q \<in> sets M1")
353 proof -
354   interpret Q: pair_sigma_algebra M2 M1 by default
355   have "Pair y - (\<lambda>(x, y). (y, x))  Q = (\<lambda>x. (x, y)) - Q" by auto
356   with Q.measurable_cut_fst[OF sets_pair_sigma_algebra_swap[OF Q], of y]
357   show ?thesis by simp
358 qed
360 lemma (in pair_sigma_algebra) measurable_pair_image_snd:
361   assumes m: "f \<in> measurable P M" and "x \<in> space M1"
362   shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
363   unfolding measurable_def
364 proof (intro CollectI conjI Pi_I ballI)
365   fix y assume "y \<in> space M2" with f \<in> measurable P M x \<in> space M1
366   show "f (x, y) \<in> space M" unfolding measurable_def pair_algebra_def by auto
367 next
368   fix A assume "A \<in> sets M"
369   then have "Pair x - (f - A \<inter> space P) \<in> sets M2" (is "?C \<in> _")
370     using f \<in> measurable P M
371     by (intro measurable_cut_fst) (auto simp: measurable_def)
372   also have "?C = (\<lambda>y. f (x, y)) - A \<inter> space M2"
373     using x \<in> space M1 by (auto simp: pair_algebra_def)
374   finally show "(\<lambda>y. f (x, y)) - A \<inter> space M2 \<in> sets M2" .
375 qed
377 lemma (in pair_sigma_algebra) measurable_pair_image_fst:
378   assumes m: "f \<in> measurable P M" and "y \<in> space M2"
379   shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
380 proof -
381   interpret Q: pair_sigma_algebra M2 M1 by default
382   from Q.measurable_pair_image_snd[OF measurable_comp y \<in> space M2,
383                                       OF Q.pair_sigma_algebra_swap_measurable m]
384   show ?thesis by simp
385 qed
387 lemma (in pair_sigma_algebra) Int_stable_pair_algebra: "Int_stable E"
388   unfolding Int_stable_def
389 proof (intro ballI)
390   fix A B assume "A \<in> sets E" "B \<in> sets E"
391   then obtain A1 A2 B1 B2 where "A = A1 \<times> A2" "B = B1 \<times> B2"
392     "A1 \<in> sets M1" "A2 \<in> sets M2" "B1 \<in> sets M1" "B2 \<in> sets M2"
393     unfolding pair_algebra_def by auto
394   then show "A \<inter> B \<in> sets E"
395     by (auto simp add: times_Int_times pair_algebra_def)
396 qed
398 lemma finite_measure_cut_measurable:
399   fixes M1 :: "'a algebra" and M2 :: "'b algebra"
400   assumes "sigma_finite_measure M1 \<mu>1" "finite_measure M2 \<mu>2"
401   assumes "Q \<in> sets (sigma (pair_algebra M1 M2))"
402   shows "(\<lambda>x. \<mu>2 (Pair x - Q)) \<in> borel_measurable M1"
403     (is "?s Q \<in> _")
404 proof -
405   interpret M1: sigma_finite_measure M1 \<mu>1 by fact
406   interpret M2: finite_measure M2 \<mu>2 by fact
407   interpret pair_sigma_algebra M1 M2 by default
408   have [intro]: "sigma_algebra M1" by fact
409   have [intro]: "sigma_algebra M2" by fact
410   let ?D = "\<lparr> space = space P, sets = {A\<in>sets P. ?s A \<in> borel_measurable M1}  \<rparr>"
411   note space_pair_algebra[simp]
412   interpret dynkin_system ?D
413   proof (intro dynkin_systemI)
414     fix A assume "A \<in> sets ?D" then show "A \<subseteq> space ?D"
415       using sets_into_space by simp
416   next
417     from top show "space ?D \<in> sets ?D"
418       by (auto simp add: if_distrib intro!: M1.measurable_If)
419   next
420     fix A assume "A \<in> sets ?D"
421     with sets_into_space have "\<And>x. \<mu>2 (Pair x - (space M1 \<times> space M2 - A)) =
422         (if x \<in> space M1 then \<mu>2 (space M2) - ?s A x else 0)"
423       by (auto intro!: M2.finite_measure_compl measurable_cut_fst
424                simp: vimage_Diff)
425     with A \<in> sets ?D top show "space ?D - A \<in> sets ?D"
426       by (auto intro!: Diff M1.measurable_If M1.borel_measurable_pextreal_diff)
427   next
428     fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> sets ?D"
429     moreover then have "\<And>x. \<mu>2 (\<Union>i. Pair x - F i) = (\<Sum>\<^isub>\<infinity> i. ?s (F i) x)"
430       by (intro M2.measure_countably_additive[symmetric])
431          (auto intro!: measurable_cut_fst simp: disjoint_family_on_def)
432     ultimately show "(\<Union>i. F i) \<in> sets ?D"
433       by (auto simp: vimage_UN intro!: M1.borel_measurable_psuminf)
434   qed
435   have "P = ?D"
436   proof (intro dynkin_lemma)
437     show "Int_stable E" by (rule Int_stable_pair_algebra)
438     from M1.sets_into_space have "\<And>A. A \<in> sets M1 \<Longrightarrow> {x \<in> space M1. x \<in> A} = A"
439       by auto
440     then show "sets E \<subseteq> sets ?D"
441       by (auto simp: pair_algebra_def sets_sigma if_distrib
442                intro: sigma_sets.Basic intro!: M1.measurable_If)
443   qed auto
444   with Q \<in> sets P have "Q \<in> sets ?D" by simp
445   then show "?s Q \<in> borel_measurable M1" by simp
446 qed
448 subsection {* Binary products of $\sigma$-finite measure spaces *}
450 locale pair_sigma_finite = M1: sigma_finite_measure M1 \<mu>1 + M2: sigma_finite_measure M2 \<mu>2
451   for M1 \<mu>1 M2 \<mu>2
453 sublocale pair_sigma_finite \<subseteq> pair_sigma_algebra M1 M2
454   by default
456 lemma (in pair_sigma_finite) measure_cut_measurable_fst:
457   assumes "Q \<in> sets P" shows "(\<lambda>x. \<mu>2 (Pair x - Q)) \<in> borel_measurable M1" (is "?s Q \<in> _")
458 proof -
459   have [intro]: "sigma_algebra M1" and [intro]: "sigma_algebra M2" by default+
460   have M1: "sigma_finite_measure M1 \<mu>1" by default
462   from M2.disjoint_sigma_finite guess F .. note F = this
463   let "?C x i" = "F i \<inter> Pair x - Q"
464   { fix i
465     let ?R = "M2.restricted_space (F i)"
466     have [simp]: "space M1 \<times> F i \<inter> space M1 \<times> space M2 = space M1 \<times> F i"
467       using F M2.sets_into_space by auto
468     have "(\<lambda>x. \<mu>2 (Pair x - (space M1 \<times> F i \<inter> Q))) \<in> borel_measurable M1"
469     proof (intro finite_measure_cut_measurable[OF M1])
470       show "finite_measure (M2.restricted_space (F i)) \<mu>2"
471         using F by (intro M2.restricted_to_finite_measure) auto
472       have "space M1 \<times> F i \<in> sets P"
473         using M1.top F by blast
474       from sigma_sets_Int[symmetric,
475         OF this[unfolded pair_sigma_algebra_def sets_sigma]]
476       show "(space M1 \<times> F i) \<inter> Q \<in> sets (sigma (pair_algebra M1 ?R))"
477         using Q \<in> sets P
478         using pair_algebra_Int_snd[OF M1.space_closed, of "F i" M2]
479         by (auto simp: pair_algebra_def sets_sigma)
480     qed
481     moreover have "\<And>x. Pair x - (space M1 \<times> F i \<inter> Q) = ?C x i"
482       using Q \<in> sets P sets_into_space by (auto simp: space_pair_algebra)
483     ultimately have "(\<lambda>x. \<mu>2 (?C x i)) \<in> borel_measurable M1"
484       by simp }
485   moreover
486   { fix x
487     have "(\<Sum>\<^isub>\<infinity>i. \<mu>2 (?C x i)) = \<mu>2 (\<Union>i. ?C x i)"
488     proof (intro M2.measure_countably_additive)
489       show "range (?C x) \<subseteq> sets M2"
490         using F Q \<in> sets P by (auto intro!: M2.Int measurable_cut_fst)
491       have "disjoint_family F" using F by auto
492       show "disjoint_family (?C x)"
493         by (rule disjoint_family_on_bisimulation[OF disjoint_family F]) auto
494     qed
495     also have "(\<Union>i. ?C x i) = Pair x - Q"
496       using F sets_into_space Q \<in> sets P
497       by (auto simp: space_pair_algebra)
498     finally have "\<mu>2 (Pair x - Q) = (\<Sum>\<^isub>\<infinity>i. \<mu>2 (?C x i))"
499       by simp }
500   ultimately show ?thesis
501     by (auto intro!: M1.borel_measurable_psuminf)
502 qed
504 lemma (in pair_sigma_finite) measure_cut_measurable_snd:
505   assumes "Q \<in> sets P" shows "(\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) - Q)) \<in> borel_measurable M2"
506 proof -
507   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
508   have [simp]: "\<And>y. (Pair y - (\<lambda>(x, y). (y, x))  Q) = (\<lambda>x. (x, y)) - Q"
509     by auto
510   note sets_pair_sigma_algebra_swap[OF assms]
511   from Q.measure_cut_measurable_fst[OF this]
512   show ?thesis by simp
513 qed
515 lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable:
516   assumes "f \<in> measurable P M"
517   shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (sigma (pair_algebra M2 M1)) M"
518 proof -
519   interpret Q: pair_sigma_algebra M2 M1 by default
520   have *: "(\<lambda>(x,y). f (y, x)) = f \<circ> (\<lambda>(x,y). (y, x))" by (simp add: fun_eq_iff)
521   show ?thesis
522     using Q.pair_sigma_algebra_swap_measurable assms
523     unfolding * by (rule measurable_comp)
524 qed
526 definition (in pair_sigma_finite)
527   "pair_measure A = M1.positive_integral (\<lambda>x.
528     M2.positive_integral (\<lambda>y. indicator A (x, y)))"
530 lemma (in pair_sigma_finite) pair_measure_alt:
531   assumes "A \<in> sets P"
532   shows "pair_measure A = M1.positive_integral (\<lambda>x. \<mu>2 (Pair x - A))"
533   unfolding pair_measure_def
534 proof (rule M1.positive_integral_cong)
535   fix x assume "x \<in> space M1"
536   have *: "\<And>y. indicator A (x, y) = (indicator (Pair x - A) y :: pextreal)"
537     unfolding indicator_def by auto
538   show "M2.positive_integral (\<lambda>y. indicator A (x, y)) = \<mu>2 (Pair x - A)"
539     unfolding *
540     apply (subst M2.positive_integral_indicator)
541     apply (rule measurable_cut_fst[OF assms])
542     by simp
543 qed
545 lemma (in pair_sigma_finite) pair_measure_times:
546   assumes A: "A \<in> sets M1" and "B \<in> sets M2"
547   shows "pair_measure (A \<times> B) = \<mu>1 A * \<mu>2 B"
548 proof -
549   from assms have "pair_measure (A \<times> B) =
550       M1.positive_integral (\<lambda>x. \<mu>2 B * indicator A x)"
551     by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt)
552   with assms show ?thesis
553     by (simp add: M1.positive_integral_cmult_indicator ac_simps)
554 qed
556 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_algebra:
557   "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> F \<up> space E \<and>
558     (\<forall>i. pair_measure (F i) \<noteq> \<omega>)"
559 proof -
560   obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where
561     F1: "range F1 \<subseteq> sets M1" "F1 \<up> space M1" "\<And>i. \<mu>1 (F1 i) \<noteq> \<omega>" and
562     F2: "range F2 \<subseteq> sets M2" "F2 \<up> space M2" "\<And>i. \<mu>2 (F2 i) \<noteq> \<omega>"
563     using M1.sigma_finite_up M2.sigma_finite_up by auto
564   then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)"
565     unfolding isoton_def by auto
566   let ?F = "\<lambda>i. F1 i \<times> F2 i"
567   show ?thesis unfolding isoton_def space_pair_algebra
568   proof (intro exI[of _ ?F] conjI allI)
569     show "range ?F \<subseteq> sets E" using F1 F2
570       by (fastsimp intro!: pair_algebraI)
571   next
572     have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
573     proof (intro subsetI)
574       fix x assume "x \<in> space M1 \<times> space M2"
575       then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
576         by (auto simp: space)
577       then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
578         using F1 \<up> space M1 F2 \<up> space M2
579         by (auto simp: max_def dest: isoton_mono_le)
580       then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
581         by (intro SigmaI) (auto simp add: min_max.sup_commute)
582       then show "x \<in> (\<Union>i. ?F i)" by auto
583     qed
584     then show "(\<Union>i. ?F i) = space M1 \<times> space M2"
585       using space by (auto simp: space)
586   next
587     fix i show "F1 i \<times> F2 i \<subseteq> F1 (Suc i) \<times> F2 (Suc i)"
588       using F1 \<up> space M1 F2 \<up> space M2 unfolding isoton_def
589       by auto
590   next
591     fix i
592     from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
593     with F1 F2 show "pair_measure (F1 i \<times> F2 i) \<noteq> \<omega>"
594       by (simp add: pair_measure_times)
595   qed
596 qed
598 sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P pair_measure
599 proof
600   show "pair_measure {} = 0"
601     unfolding pair_measure_def by auto
603   show "countably_additive P pair_measure"
605   proof (intro allI impI)
606     fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"
607     assume F: "range F \<subseteq> sets P" "disjoint_family F"
608     from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto
609     moreover from F have "\<And>i. (\<lambda>x. \<mu>2 (Pair x - F i)) \<in> borel_measurable M1"
610       by (intro measure_cut_measurable_fst) auto
611     moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x - F i)"
612       by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
613     moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x - F i) \<subseteq> sets M2"
614       using F by (auto intro!: measurable_cut_fst)
615     ultimately show "(\<Sum>\<^isub>\<infinity>n. pair_measure (F n)) = pair_measure (\<Union>i. F i)"
616       by (simp add: pair_measure_alt vimage_UN M1.positive_integral_psuminf[symmetric]
618                cong: M1.positive_integral_cong)
619   qed
621   from sigma_finite_up_in_pair_algebra guess F :: "nat \<Rightarrow> ('a \<times> 'c) set" .. note F = this
622   show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets P \<and> (\<Union>i. F i) = space P \<and> (\<forall>i. pair_measure (F i) \<noteq> \<omega>)"
623   proof (rule exI[of _ F], intro conjI)
624     show "range F \<subseteq> sets P" using F by auto
625     show "(\<Union>i. F i) = space P"
626       using F by (auto simp: space_pair_algebra isoton_def)
627     show "\<forall>i. pair_measure (F i) \<noteq> \<omega>" using F by auto
628   qed
629 qed
631 lemma (in pair_sigma_algebra) sets_swap:
632   assumes "A \<in> sets P"
633   shows "(\<lambda>(x, y). (y, x)) - A \<inter> space (sigma (pair_algebra M2 M1)) \<in> sets (sigma (pair_algebra M2 M1))"
634     (is "_ - A \<inter> space ?Q \<in> sets ?Q")
635 proof -
636   have *: "(\<lambda>(x, y). (y, x)) - A \<inter> space ?Q = (\<lambda>(x, y). (y, x))  A"
637     using A \<in> sets P sets_into_space by (auto simp: space_pair_algebra)
638   show ?thesis
639     unfolding * using assms by (rule sets_pair_sigma_algebra_swap)
640 qed
642 lemma (in pair_sigma_finite) pair_measure_alt2:
643   assumes "A \<in> sets P"
644   shows "pair_measure A = M2.positive_integral (\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) - A))"
645     (is "_ = ?\<nu> A")
646 proof -
647   from sigma_finite_up_in_pair_algebra guess F :: "nat \<Rightarrow> ('a \<times> 'c) set" .. note F = this
648   show ?thesis
649   proof (rule measure_unique_Int_stable[where \<nu>="?\<nu>", OF Int_stable_pair_algebra],
650          simp_all add: pair_sigma_algebra_def[symmetric])
651     show "range F \<subseteq> sets E" "F \<up> space E" "\<And>i. pair_measure (F i) \<noteq> \<omega>"
652       using F by auto
653     show "measure_space P pair_measure" by default
654     interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
655     have P: "sigma_algebra P" by default
656     show "measure_space P ?\<nu>"
657       apply (rule Q.measure_space_vimage[OF P])
658       apply (rule Q.pair_sigma_algebra_swap_measurable)
659     proof -
660       fix A assume "A \<in> sets P"
661       from sets_swap[OF this]
662       show "M2.positive_integral (\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) - A)) =
663             Q.pair_measure ((\<lambda>(x, y). (y, x)) - A \<inter> space Q.P)"
664         using sets_into_space[OF A \<in> sets P]
665         by (auto simp add: Q.pair_measure_alt space_pair_algebra
666                  intro!: M2.positive_integral_cong arg_cong[where f=\<mu>1])
667     qed
668     fix X assume "X \<in> sets E"
669     then obtain A B where X: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2"
670       unfolding pair_algebra_def by auto
671     show "pair_measure X = ?\<nu> X"
672     proof -
673       from AB have "?\<nu> (A \<times> B) =
674           M2.positive_integral (\<lambda>y. \<mu>1 A * indicator B y)"
675         by (auto intro!: M2.positive_integral_cong)
676       with AB show ?thesis
677         unfolding pair_measure_times[OF AB] X
678         by (simp add: M2.positive_integral_cmult_indicator ac_simps)
679     qed
680   qed fact
681 qed
683 section "Fubinis theorem"
685 lemma (in pair_sigma_finite) simple_function_cut:
686   assumes f: "simple_function f"
687   shows "(\<lambda>x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"
688     and "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))
689       = positive_integral f"
690 proof -
691   have f_borel: "f \<in> borel_measurable P"
692     using f by (rule borel_measurable_simple_function)
693   let "?F z" = "f - {z} \<inter> space P"
694   let "?F' x z" = "Pair x - ?F z"
695   { fix x assume "x \<in> space M1"
696     have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y"
697       by (auto simp: indicator_def)
698     have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space P" using x \<in> space M1
699       by (simp add: space_pair_algebra)
700     moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel
701       by (intro borel_measurable_vimage measurable_cut_fst)
702     ultimately have "M2.simple_function (\<lambda> y. f (x, y))"
703       apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _])
704       apply (rule simple_function_indicator_representation[OF f])
705       using x \<in> space M1 by (auto simp del: space_sigma) }
706   note M2_sf = this
707   { fix x assume x: "x \<in> space M1"
708     then have "M2.positive_integral (\<lambda> y. f (x, y)) =
709         (\<Sum>z\<in>f  space P. z * \<mu>2 (?F' x z))"
710       unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x]]
711       unfolding M2.simple_integral_def
712     proof (safe intro!: setsum_mono_zero_cong_left)
713       from f show "finite (f  space P)" by (rule simple_functionD)
714     next
715       fix y assume "y \<in> space M2" then show "f (x, y) \<in> f  space P"
716         using x \<in> space M1 by (auto simp: space_pair_algebra)
717     next
718       fix x' y assume "(x', y) \<in> space P"
719         "f (x', y) \<notin> (\<lambda>y. f (x, y))  space M2"
720       then have *: "?F' x (f (x', y)) = {}"
721         by (force simp: space_pair_algebra)
722       show  "f (x', y) * \<mu>2 (?F' x (f (x', y))) = 0"
723         unfolding * by simp
724     qed (simp add: vimage_compose[symmetric] comp_def
725                    space_pair_algebra) }
726   note eq = this
727   moreover have "\<And>z. ?F z \<in> sets P"
728     by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma)
729   moreover then have "\<And>z. (\<lambda>x. \<mu>2 (?F' x z)) \<in> borel_measurable M1"
730     by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int)
731   ultimately
732   show "(\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"
733     and "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))
734     = positive_integral f"
735     by (auto simp del: vimage_Int cong: measurable_cong
736              intro!: M1.borel_measurable_pextreal_setsum
737              simp add: M1.positive_integral_setsum simple_integral_def
738                        M1.positive_integral_cmult
739                        M1.positive_integral_cong[OF eq]
740                        positive_integral_eq_simple_integral[OF f]
741                        pair_measure_alt[symmetric])
742 qed
744 lemma (in pair_sigma_finite) positive_integral_fst_measurable:
745   assumes f: "f \<in> borel_measurable P"
746   shows "(\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"
747       (is "?C f \<in> borel_measurable M1")
748     and "M1.positive_integral (\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) =
749       positive_integral f"
750 proof -
751   from borel_measurable_implies_simple_function_sequence[OF f]
752   obtain F where F: "\<And>i. simple_function (F i)" "F \<up> f" by auto
753   then have F_borel: "\<And>i. F i \<in> borel_measurable P"
754     and F_mono: "\<And>i x. F i x \<le> F (Suc i) x"
755     and F_SUPR: "\<And>x. (SUP i. F i x) = f x"
756     unfolding isoton_fun_expand unfolding isoton_def le_fun_def
757     by (auto intro: borel_measurable_simple_function)
758   note sf = simple_function_cut[OF F(1)]
759   then have "(\<lambda>x. SUP i. ?C (F i) x) \<in> borel_measurable M1"
760     using F(1) by auto
761   moreover
762   { fix x assume "x \<in> space M1"
763     have isotone: "(\<lambda> i y. F i (x, y)) \<up> (\<lambda>y. f (x, y))"
764       using F \<up> f unfolding isoton_fun_expand
765       by (auto simp: isoton_def)
766     note measurable_pair_image_snd[OF F_borelx \<in> space M1]
767     from M2.positive_integral_isoton[OF isotone this]
768     have "(SUP i. ?C (F i) x) = ?C f x"
769       by (simp add: isoton_def) }
770   note SUPR_C = this
771   ultimately show "?C f \<in> borel_measurable M1"
772     by (simp cong: measurable_cong)
773   have "positive_integral (\<lambda>x. (SUP i. F i x)) = (SUP i. positive_integral (F i))"
774     using F_borel F_mono
775     by (auto intro!: positive_integral_monotone_convergence_SUP[symmetric])
776   also have "(SUP i. positive_integral (F i)) =
777     (SUP i. M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. F i (x, y))))"
778     unfolding sf(2) by simp
779   also have "\<dots> = M1.positive_integral (\<lambda>x. SUP i. M2.positive_integral (\<lambda>y. F i (x, y)))"
780     by (auto intro!: M1.positive_integral_monotone_convergence_SUP[OF _ sf(1)]
781                      M2.positive_integral_mono F_mono)
782   also have "\<dots> = M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. SUP i. F i (x, y)))"
783     using F_borel F_mono
784     by (auto intro!: M2.positive_integral_monotone_convergence_SUP
785                      M1.positive_integral_cong measurable_pair_image_snd)
786   finally show "M1.positive_integral (\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) =
787       positive_integral f"
788     unfolding F_SUPR by simp
789 qed
791 lemma (in pair_sigma_finite) positive_integral_product_swap:
792   assumes f: "f \<in> borel_measurable P"
793   shows "measure_space.positive_integral
794     (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x))) =
795   positive_integral f"
796 proof -
797   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
798   have P: "sigma_algebra P" by default
799   show ?thesis
800     unfolding Q.positive_integral_vimage[OF P Q.pair_sigma_algebra_swap_measurable f, symmetric]
801   proof (rule positive_integral_cong_measure)
802     fix A
803     assume A: "A \<in> sets P"
804     from Q.pair_sigma_algebra_swap_measurable[THEN measurable_sets, OF this] this sets_into_space[OF this]
805     show "Q.pair_measure ((\<lambda>(x, y). (y, x)) - A \<inter> space Q.P) = pair_measure A"
806       by (auto intro!: M1.positive_integral_cong arg_cong[where f=\<mu>2]
807                simp: pair_measure_alt Q.pair_measure_alt2 space_pair_algebra)
808   qed
809 qed
811 lemma (in pair_sigma_finite) positive_integral_snd_measurable:
812   assumes f: "f \<in> borel_measurable P"
813   shows "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =
814       positive_integral f"
815 proof -
816   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
817   note pair_sigma_algebra_measurable[OF f]
818   from Q.positive_integral_fst_measurable[OF this]
819   have "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =
820     Q.positive_integral (\<lambda>(x, y). f (y, x))"
821     by simp
822   also have "Q.positive_integral (\<lambda>(x, y). f (y, x)) = positive_integral f"
823     unfolding positive_integral_product_swap[OF f, symmetric]
824     by (auto intro!: Q.positive_integral_cong)
825   finally show ?thesis .
826 qed
828 lemma (in pair_sigma_finite) Fubini:
829   assumes f: "f \<in> borel_measurable P"
830   shows "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =
831       M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))"
832   unfolding positive_integral_snd_measurable[OF assms]
833   unfolding positive_integral_fst_measurable[OF assms] ..
835 lemma (in pair_sigma_finite) AE_pair:
836   assumes "almost_everywhere (\<lambda>x. Q x)"
837   shows "M1.almost_everywhere (\<lambda>x. M2.almost_everywhere (\<lambda>y. Q (x, y)))"
838 proof -
839   obtain N where N: "N \<in> sets P" "pair_measure N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N"
840     using assms unfolding almost_everywhere_def by auto
841   show ?thesis
842   proof (rule M1.AE_I)
843     from N measure_cut_measurable_fst[OF N \<in> sets P]
844     show "\<mu>1 {x\<in>space M1. \<mu>2 (Pair x - N) \<noteq> 0} = 0"
845       by (simp add: M1.positive_integral_0_iff pair_measure_alt vimage_def)
846     show "{x \<in> space M1. \<mu>2 (Pair x - N) \<noteq> 0} \<in> sets M1"
847       by (intro M1.borel_measurable_pextreal_neq_const measure_cut_measurable_fst N)
848     { fix x assume "x \<in> space M1" "\<mu>2 (Pair x - N) = 0"
849       have "M2.almost_everywhere (\<lambda>y. Q (x, y))"
850       proof (rule M2.AE_I)
851         show "\<mu>2 (Pair x - N) = 0" by fact
852         show "Pair x - N \<in> sets M2" by (intro measurable_cut_fst N)
853         show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x - N"
854           using N x \<in> space M1 unfolding space_sigma space_pair_algebra by auto
855       qed }
856     then show "{x \<in> space M1. \<not> M2.almost_everywhere (\<lambda>y. Q (x, y))} \<subseteq> {x \<in> space M1. \<mu>2 (Pair x - N) \<noteq> 0}"
857       by auto
858   qed
859 qed
861 lemma (in pair_sigma_algebra) measurable_product_swap:
862   "f \<in> measurable (sigma (pair_algebra M2 M1)) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable P M"
863 proof -
864   interpret Q: pair_sigma_algebra M2 M1 by default
865   show ?thesis
866     using pair_sigma_algebra_measurable[of "\<lambda>(x,y). f (y, x)"]
867     by (auto intro!: pair_sigma_algebra_measurable Q.pair_sigma_algebra_measurable iffI)
868 qed
870 lemma (in pair_sigma_finite) integrable_product_swap:
871   assumes "integrable f"
872   shows "measure_space.integrable
873     (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) (\<lambda>(x,y). f (y,x))"
874 proof -
875   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
876   have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
877   show ?thesis unfolding *
878     using assms unfolding Q.integrable_def integrable_def
879     apply (subst (1 2) positive_integral_product_swap)
880     using integrable f unfolding integrable_def
881     by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])
882 qed
884 lemma (in pair_sigma_finite) integrable_product_swap_iff:
885   "measure_space.integrable
886     (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow>
887   integrable f"
888 proof -
889   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
890   from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]
891   show ?thesis by auto
892 qed
894 lemma (in pair_sigma_finite) integral_product_swap:
895   assumes "integrable f"
896   shows "measure_space.integral
897     (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) (\<lambda>(x,y). f (y,x)) =
898   integral f"
899 proof -
900   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
901   have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
902   show ?thesis
903     unfolding integral_def Q.integral_def *
904     apply (subst (1 2) positive_integral_product_swap)
905     using integrable f unfolding integrable_def
906     by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])
907 qed
909 lemma (in pair_sigma_finite) integrable_fst_measurable:
910   assumes f: "integrable f"
911   shows "M1.almost_everywhere (\<lambda>x. M2.integrable (\<lambda> y. f (x, y)))" (is "?AE")
912     and "M1.integral (\<lambda> x. M2.integral (\<lambda> y. f (x, y))) = integral f" (is "?INT")
913 proof -
914   let "?pf x" = "Real (f x)" and "?nf x" = "Real (- f x)"
915   have
916     borel: "?nf \<in> borel_measurable P""?pf \<in> borel_measurable P" and
917     int: "positive_integral ?nf \<noteq> \<omega>" "positive_integral ?pf \<noteq> \<omega>"
918     using assms by auto
919   have "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. Real (f (x, y)))) \<noteq> \<omega>"
920      "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. Real (- f (x, y)))) \<noteq> \<omega>"
921     using borel[THEN positive_integral_fst_measurable(1)] int
922     unfolding borel[THEN positive_integral_fst_measurable(2)] by simp_all
923   with borel[THEN positive_integral_fst_measurable(1)]
924   have AE: "M1.almost_everywhere (\<lambda>x. M2.positive_integral (\<lambda>y. Real (f (x, y))) \<noteq> \<omega>)"
925     "M1.almost_everywhere (\<lambda>x. M2.positive_integral (\<lambda>y. Real (- f (x, y))) \<noteq> \<omega>)"
926     by (auto intro!: M1.positive_integral_omega_AE)
927   then show ?AE
928     apply (rule M1.AE_mp[OF _ M1.AE_mp])
929     apply (rule M1.AE_cong)
930     using assms unfolding M2.integrable_def
931     by (auto intro!: measurable_pair_image_snd)
932   have "M1.integrable
933      (\<lambda>x. real (M2.positive_integral (\<lambda>xa. Real (f (x, xa)))))" (is "M1.integrable ?f")
934   proof (unfold M1.integrable_def, intro conjI)
935     show "?f \<in> borel_measurable M1"
936       using borel by (auto intro!: M1.borel_measurable_real positive_integral_fst_measurable)
937     have "M1.positive_integral (\<lambda>x. Real (?f x)) =
938         M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>xa. Real (f (x, xa))))"
939       apply (rule M1.positive_integral_cong_AE)
940       apply (rule M1.AE_mp[OF AE(1)])
941       apply (rule M1.AE_cong)
942       by (auto simp: Real_real)
943     then show "M1.positive_integral (\<lambda>x. Real (?f x)) \<noteq> \<omega>"
944       using positive_integral_fst_measurable[OF borel(2)] int by simp
945     have "M1.positive_integral (\<lambda>x. Real (- ?f x)) = M1.positive_integral (\<lambda>x. 0)"
946       by (intro M1.positive_integral_cong) simp
947     then show "M1.positive_integral (\<lambda>x. Real (- ?f x)) \<noteq> \<omega>" by simp
948   qed
949   moreover have "M1.integrable
950      (\<lambda>x. real (M2.positive_integral (\<lambda>xa. Real (- f (x, xa)))))" (is "M1.integrable ?f")
951   proof (unfold M1.integrable_def, intro conjI)
952     show "?f \<in> borel_measurable M1"
953       using borel by (auto intro!: M1.borel_measurable_real positive_integral_fst_measurable)
954     have "M1.positive_integral (\<lambda>x. Real (?f x)) =
955         M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>xa. Real (- f (x, xa))))"
956       apply (rule M1.positive_integral_cong_AE)
957       apply (rule M1.AE_mp[OF AE(2)])
958       apply (rule M1.AE_cong)
959       by (auto simp: Real_real)
960     then show "M1.positive_integral (\<lambda>x. Real (?f x)) \<noteq> \<omega>"
961       using positive_integral_fst_measurable[OF borel(1)] int by simp
962     have "M1.positive_integral (\<lambda>x. Real (- ?f x)) = M1.positive_integral (\<lambda>x. 0)"
963       by (intro M1.positive_integral_cong) simp
964     then show "M1.positive_integral (\<lambda>x. Real (- ?f x)) \<noteq> \<omega>" by simp
965   qed
966   ultimately show ?INT
967     unfolding M2.integral_def integral_def
968       borel[THEN positive_integral_fst_measurable(2), symmetric]
969     by (simp add: M1.integral_real[OF AE(1)] M1.integral_real[OF AE(2)])
970 qed
972 lemma (in pair_sigma_finite) integrable_snd_measurable:
973   assumes f: "integrable f"
974   shows "M2.almost_everywhere (\<lambda>y. M1.integrable (\<lambda>x. f (x, y)))" (is "?AE")
975     and "M2.integral (\<lambda>y. M1.integral (\<lambda>x. f (x, y))) = integral f" (is "?INT")
976 proof -
977   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
978   have Q_int: "Q.integrable (\<lambda>(x, y). f (y, x))"
979     using f unfolding integrable_product_swap_iff .
980   show ?INT
981     using Q.integrable_fst_measurable(2)[OF Q_int]
982     using integral_product_swap[OF f] by simp
983   show ?AE
984     using Q.integrable_fst_measurable(1)[OF Q_int]
985     by simp
986 qed
988 lemma (in pair_sigma_finite) Fubini_integral:
989   assumes f: "integrable f"
990   shows "M2.integral (\<lambda>y. M1.integral (\<lambda>x. f (x, y))) =
991       M1.integral (\<lambda>x. M2.integral (\<lambda>y. f (x, y)))"
992   unfolding integrable_snd_measurable[OF assms]
993   unfolding integrable_fst_measurable[OF assms] ..
995 section "Finite product spaces"
997 section "Products"
999 locale product_sigma_algebra =
1000   fixes M :: "'i \<Rightarrow> 'a algebra"
1001   assumes sigma_algebras: "\<And>i. sigma_algebra (M i)"
1003 locale finite_product_sigma_algebra = product_sigma_algebra M for M :: "'i \<Rightarrow> 'a algebra" +
1004   fixes I :: "'i set"
1005   assumes finite_index: "finite I"
1007 syntax
1008   "_PiE"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PIE _:_./ _)" 10)
1010 syntax (xsymbols)
1011   "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
1013 syntax (HTML output)
1014   "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
1016 translations
1017   "PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)"
1019 definition
1020   "product_algebra M I = \<lparr> space = (\<Pi>\<^isub>E i\<in>I. space (M i)), sets = Pi\<^isub>E I  (\<Pi> i \<in> I. sets (M i)) \<rparr>"
1022 abbreviation (in finite_product_sigma_algebra) "G \<equiv> product_algebra M I"
1023 abbreviation (in finite_product_sigma_algebra) "P \<equiv> sigma G"
1025 sublocale product_sigma_algebra \<subseteq> M: sigma_algebra "M i" for i by (rule sigma_algebras)
1027 lemma (in finite_product_sigma_algebra) product_algebra_into_space:
1028   "sets G \<subseteq> Pow (space G)"
1029   using M.sets_into_space unfolding product_algebra_def
1030   by auto blast
1032 sublocale finite_product_sigma_algebra \<subseteq> sigma_algebra P
1033   using product_algebra_into_space by (rule sigma_algebra_sigma)
1035 lemma product_algebraE:
1036   assumes "A \<in> sets (product_algebra M I)"
1037   obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
1038   using assms unfolding product_algebra_def by auto
1040 lemma product_algebraI[intro]:
1041   assumes "E \<in> (\<Pi> i\<in>I. sets (M i))"
1042   shows "Pi\<^isub>E I E \<in> sets (product_algebra M I)"
1043   using assms unfolding product_algebra_def by auto
1045 lemma space_product_algebra[simp]:
1046   "space (product_algebra M I) = Pi\<^isub>E I (\<lambda>i. space (M i))"
1047   unfolding product_algebra_def by simp
1049 lemma product_algebra_sets_into_space:
1050   assumes "\<And>i. i\<in>I \<Longrightarrow> sets (M i) \<subseteq> Pow (space (M i))"
1051   shows "sets (product_algebra M I) \<subseteq> Pow (space (product_algebra M I))"
1052   using assms by (auto simp: product_algebra_def) blast
1054 lemma (in finite_product_sigma_algebra) P_empty:
1055   "I = {} \<Longrightarrow> P = \<lparr> space = {\<lambda>k. undefined}, sets = { {}, {\<lambda>k. undefined} }\<rparr>"
1056   unfolding product_algebra_def by (simp add: sigma_def image_constant)
1058 lemma (in finite_product_sigma_algebra) in_P[simp, intro]:
1059   "\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P"
1060   by (auto simp: product_algebra_def sets_sigma intro!: sigma_sets.Basic)
1062 lemma (in product_sigma_algebra) bij_inv_restrict_merge:
1063   assumes [simp]: "I \<inter> J = {}"
1064   shows "bij_inv
1065     (space (sigma (product_algebra M (I \<union> J))))
1066     (space (sigma (pair_algebra (product_algebra M I) (product_algebra M J))))
1067     (\<lambda>x. (restrict x I, restrict x J)) (\<lambda>(x, y). merge I x J y)"
1068   by (rule bij_invI) (auto simp: space_pair_algebra extensional_restrict)
1070 lemma (in product_sigma_algebra) bij_inv_singleton:
1071   "bij_inv (space (sigma (product_algebra M {i}))) (space (M i))
1072     (\<lambda>x. x i) (\<lambda>x. (\<lambda>j\<in>{i}. x))"
1073   by (rule bij_invI) (auto simp: restrict_def extensional_def fun_eq_iff)
1075 lemma (in product_sigma_algebra) bij_inv_restrict_insert:
1076   assumes [simp]: "i \<notin> I"
1077   shows "bij_inv
1078     (space (sigma (product_algebra M (insert i I))))
1079     (space (sigma (pair_algebra (product_algebra M I) (M i))))
1080     (\<lambda>x. (restrict x I, x i)) (\<lambda>(x, y). x(i := y))"
1081   by (rule bij_invI) (auto simp: space_pair_algebra extensional_restrict)
1083 lemma (in product_sigma_algebra) measurable_restrict_on_generating:
1084   assumes [simp]: "I \<inter> J = {}"
1085   shows "(\<lambda>x. (restrict x I, restrict x J)) \<in> measurable
1086     (product_algebra M (I \<union> J))
1087     (pair_algebra (product_algebra M I) (product_algebra M J))"
1088     (is "?R \<in> measurable ?IJ ?P")
1089 proof (unfold measurable_def, intro CollectI conjI ballI)
1090   show "?R \<in> space ?IJ \<rightarrow> space ?P" by (auto simp: space_pair_algebra)
1091   { fix F E assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "F \<in> (\<Pi> i\<in>J. sets (M i))"
1092     then have "Pi (I \<union> J) (merge I E J F) \<inter> (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) =
1093         Pi\<^isub>E (I \<union> J) (merge I E J F)"
1094       using M.sets_into_space by auto blast+ }
1095   note this[simp]
1096   show "\<And>A. A \<in> sets ?P \<Longrightarrow> ?R - A \<inter> space ?IJ \<in> sets ?IJ"
1097     by (force elim!: pair_algebraE product_algebraE
1098               simp del: vimage_Int simp: restrict_vimage merge_vimage space_pair_algebra)
1099   qed
1101 lemma (in product_sigma_algebra) measurable_merge_on_generating:
1102   assumes [simp]: "I \<inter> J = {}"
1103   shows "(\<lambda>(x, y). merge I x J y) \<in> measurable
1104     (pair_algebra (product_algebra M I) (product_algebra M J))
1105     (product_algebra M (I \<union> J))"
1106     (is "?M \<in> measurable ?P ?IJ")
1107 proof (unfold measurable_def, intro CollectI conjI ballI)
1108   show "?M \<in> space ?P \<rightarrow> space ?IJ" by (auto simp: space_pair_algebra)
1109   { fix E assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "E \<in> (\<Pi> i\<in>J. sets (M i))"
1110     then have "Pi I E \<times> Pi J E \<inter> (\<Pi>\<^isub>E i\<in>I. space (M i)) \<times> (\<Pi>\<^isub>E i\<in>J. space (M i)) =
1111         Pi\<^isub>E I E \<times> Pi\<^isub>E J E"
1112       using M.sets_into_space by auto blast+ }
1113   note this[simp]
1114   show "\<And>A. A \<in> sets ?IJ \<Longrightarrow> ?M - A \<inter> space ?P \<in> sets ?P"
1115     by (force elim!: pair_algebraE product_algebraE
1116               simp del: vimage_Int simp: restrict_vimage merge_vimage space_pair_algebra)
1117   qed
1119 lemma (in product_sigma_algebra) measurable_singleton_on_generator:
1120   "(\<lambda>x. \<lambda>j\<in>{i}. x) \<in> measurable (M i) (product_algebra M {i})"
1121   (is "?f \<in> measurable _ ?P")
1122 proof (unfold measurable_def, intro CollectI conjI)
1123   show "?f \<in> space (M i) \<rightarrow> space ?P" by auto
1124   have "\<And>E. E i \<in> sets (M i) \<Longrightarrow> ?f - Pi\<^isub>E {i} E \<inter> space (M i) = E i"
1125     using M.sets_into_space by auto
1126   then show "\<forall>A \<in> sets ?P. ?f - A \<inter> space (M i) \<in> sets (M i)"
1127     by (auto elim!: product_algebraE)
1128 qed
1130 lemma (in product_sigma_algebra) measurable_component_on_generator:
1131   assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (product_algebra M I) (M i)"
1132   (is "?f \<in> measurable ?P _")
1133 proof (unfold measurable_def, intro CollectI conjI ballI)
1134   show "?f \<in> space ?P \<rightarrow> space (M i)" using i \<in> I by auto
1135   fix A assume "A \<in> sets (M i)"
1136   moreover then have "(\<lambda>x. x i) - A \<inter> (\<Pi>\<^isub>E i\<in>I. space (M i)) =
1137       (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))"
1138     using M.sets_into_space i \<in> I
1139     by (fastsimp dest: Pi_mem split: split_if_asm)
1140   ultimately show "?f - A \<inter> space ?P \<in> sets ?P"
1141     by (auto intro!: product_algebraI)
1142 qed
1144 lemma (in product_sigma_algebra) measurable_restrict_singleton_on_generating:
1145   assumes [simp]: "i \<notin> I"
1146   shows "(\<lambda>x. (restrict x I, x i)) \<in> measurable
1147     (product_algebra M (insert i I))
1148     (pair_algebra (product_algebra M I) (M i))"
1149     (is "?R \<in> measurable ?I ?P")
1150 proof (unfold measurable_def, intro CollectI conjI ballI)
1151   show "?R \<in> space ?I \<rightarrow> space ?P" by (auto simp: space_pair_algebra)
1152   { fix E F assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "F \<in> sets (M i)"
1153     then have "(\<lambda>x. (restrict x I, x i)) - (Pi\<^isub>E I E \<times> F) \<inter> (\<Pi>\<^isub>E i\<in>insert i I. space (M i)) =
1154         Pi\<^isub>E (insert i I) (E(i := F))"
1155       using M.sets_into_space using i\<notin>I by (auto simp: restrict_Pi_cancel) blast+ }
1156   note this[simp]
1157   show "\<And>A. A \<in> sets ?P \<Longrightarrow> ?R - A \<inter> space ?I \<in> sets ?I"
1158     by (force elim!: pair_algebraE product_algebraE
1159               simp del: vimage_Int simp: space_pair_algebra)
1160 qed
1162 lemma (in product_sigma_algebra) measurable_merge_singleton_on_generating:
1163   assumes [simp]: "i \<notin> I"
1164   shows "(\<lambda>(x, y). x(i := y)) \<in> measurable
1165     (pair_algebra (product_algebra M I) (M i))
1166     (product_algebra M (insert i I))"
1167     (is "?M \<in> measurable ?P ?IJ")
1168 proof (unfold measurable_def, intro CollectI conjI ballI)
1169   show "?M \<in> space ?P \<rightarrow> space ?IJ" by (auto simp: space_pair_algebra)
1170   { fix E assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "E i \<in> sets (M i)"
1171     then have "(\<lambda>(x, y). x(i := y)) - Pi\<^isub>E (insert i I) E \<inter> (\<Pi>\<^isub>E i\<in>I. space (M i)) \<times> space (M i) =
1172         Pi\<^isub>E I E \<times> E i"
1173       using M.sets_into_space by auto blast+ }
1174   note this[simp]
1175   show "\<And>A. A \<in> sets ?IJ \<Longrightarrow> ?M - A \<inter> space ?P \<in> sets ?P"
1176     by (force elim!: pair_algebraE product_algebraE
1177               simp del: vimage_Int simp: space_pair_algebra)
1178 qed
1180 section "Generating set generates also product algebra"
1182 lemma pair_sigma_algebra_sigma:
1183   assumes 1: "S1 \<up> (space E1)" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"
1184   assumes 2: "S2 \<up> (space E2)" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"
1185   shows "sigma (pair_algebra (sigma E1) (sigma E2)) = sigma (pair_algebra E1 E2)"
1186     (is "?S = ?E")
1187 proof -
1188   interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma)
1189   interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma)
1190   have P: "sets (pair_algebra E1 E2) \<subseteq> Pow (space E1 \<times> space E2)"
1191     using E1 E2 by (auto simp add: pair_algebra_def)
1192   interpret E: sigma_algebra ?E unfolding pair_algebra_def
1193     using E1 E2 by (intro sigma_algebra_sigma) auto
1194   { fix A assume "A \<in> sets E1"
1195     then have "fst - A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"
1196       using E1 2 unfolding isoton_def pair_algebra_def by auto
1197     also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto
1198     also have "\<dots> \<in> sets ?E" unfolding pair_algebra_def sets_sigma
1199       using 2 A \<in> sets E1
1200       by (intro sigma_sets.Union)
1201          (auto simp: image_subset_iff intro!: sigma_sets.Basic)
1202     finally have "fst - A \<inter> space ?E \<in> sets ?E" . }
1203   moreover
1204   { fix B assume "B \<in> sets E2"
1205     then have "snd - B \<inter> space ?E = (\<Union>i. S1 i) \<times> B"
1206       using E2 1 unfolding isoton_def pair_algebra_def by auto
1207     also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto
1208     also have "\<dots> \<in> sets ?E"
1209       using 1 B \<in> sets E2 unfolding pair_algebra_def sets_sigma
1210       by (intro sigma_sets.Union)
1211          (auto simp: image_subset_iff intro!: sigma_sets.Basic)
1212     finally have "snd - B \<inter> space ?E \<in> sets ?E" . }
1213   ultimately have proj:
1214     "fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"
1215     using E1 E2 by (subst (1 2) E.measurable_iff_sigma)
1216                    (auto simp: pair_algebra_def sets_sigma)
1217   { fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)"
1218     with proj have "fst - A \<inter> space ?E \<in> sets ?E" "snd - B \<inter> space ?E \<in> sets ?E"
1219       unfolding measurable_def by simp_all
1220     moreover have "A \<times> B = (fst - A \<inter> space ?E) \<inter> (snd - B \<inter> space ?E)"
1221       using A B M1.sets_into_space M2.sets_into_space
1222       by (auto simp: pair_algebra_def)
1223     ultimately have "A \<times> B \<in> sets ?E" by auto }
1224   then have "sigma_sets (space ?E) (sets (pair_algebra (sigma E1) (sigma E2))) \<subseteq> sets ?E"
1225     by (intro E.sigma_sets_subset) (auto simp add: pair_algebra_def sets_sigma)
1226   then have subset: "sets ?S \<subseteq> sets ?E"
1227     by (simp add: sets_sigma pair_algebra_def)
1228   have "sets ?S = sets ?E"
1229   proof (intro set_eqI iffI)
1230     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
1231       unfolding sets_sigma
1232     proof induct
1233       case (Basic A) then show ?case
1234         by (auto simp: pair_algebra_def sets_sigma intro: sigma_sets.Basic)
1235     qed (auto intro: sigma_sets.intros simp: pair_algebra_def)
1236   next
1237     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
1238   qed
1239   then show ?thesis
1240     by (simp add: pair_algebra_def sigma_def)
1241 qed
1243 lemma sigma_product_algebra_sigma_eq:
1244   assumes "finite I"
1245   assumes isotone: "\<And>i. i \<in> I \<Longrightarrow> (S i) \<up> (space (E i))"
1246   assumes sets_into: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> sets (E i)"
1247   and E: "\<And>i. sets (E i) \<subseteq> Pow (space (E i))"
1248   shows "sigma (product_algebra (\<lambda>i. sigma (E i)) I) = sigma (product_algebra E I)"
1249     (is "?S = ?E")
1250 proof cases
1251   assume "I = {}" then show ?thesis by (simp add: product_algebra_def)
1252 next
1253   assume "I \<noteq> {}"
1254   interpret E: sigma_algebra "sigma (E i)" for i
1255     using E by (rule sigma_algebra_sigma)
1257   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)"
1258     using E by auto
1260   interpret G: sigma_algebra ?E
1261     unfolding product_algebra_def using E
1262     by (intro sigma_algebra_sigma) (auto dest: Pi_mem)
1264   { fix A i assume "i \<in> I" and A: "A \<in> sets (E i)"
1265     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"
1266       using isotone unfolding isoton_def product_algebra_def by (auto dest: Pi_mem)
1267     also have "\<dots> = (\<Union>n. (\<Pi>\<^isub>E j\<in>I. if j = i then A else S j n))"
1268       unfolding product_algebra_def
1269       apply simp
1270       apply (subst Pi_UN[OF finite I])
1271       using isotone[THEN isoton_mono_le] apply simp
1272       apply (simp add: PiE_Int)
1273       apply (intro PiE_cong)
1274       using A sets_into by (auto intro!: into_space)
1275     also have "\<dots> \<in> sets ?E" unfolding product_algebra_def sets_sigma
1276       using sets_into A \<in> sets (E i)
1277       by (intro sigma_sets.Union)
1278          (auto simp: image_subset_iff intro!: sigma_sets.Basic)
1279     finally have "(\<lambda>x. x i) - A \<inter> space ?E \<in> sets ?E" . }
1280   then have proj:
1281     "\<And>i. i\<in>I \<Longrightarrow> (\<lambda>x. x i) \<in> measurable ?E (sigma (E i))"
1282     using E by (subst G.measurable_iff_sigma)
1283                (auto simp: product_algebra_def sets_sigma)
1285   { fix A assume A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (sigma (E i))"
1286     with proj have basic: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) - (A i) \<inter> space ?E \<in> sets ?E"
1287       unfolding measurable_def by simp
1288     have "Pi\<^isub>E I A = (\<Inter>i\<in>I. (\<lambda>x. x i) - (A i) \<inter> space ?E)"
1289       using A E.sets_into_space I \<noteq> {} unfolding product_algebra_def by auto blast
1290     then have "Pi\<^isub>E I A \<in> sets ?E"
1291       using G.finite_INT[OF finite I I \<noteq> {} basic, of "\<lambda>i. i"] by simp }
1292   then have "sigma_sets (space ?E) (sets (product_algebra (\<lambda>i. sigma (E i)) I)) \<subseteq> sets ?E"
1293     by (intro G.sigma_sets_subset) (auto simp add: sets_sigma product_algebra_def)
1294   then have subset: "sets ?S \<subseteq> sets ?E"
1295     by (simp add: sets_sigma product_algebra_def)
1297   have "sets ?S = sets ?E"
1298   proof (intro set_eqI iffI)
1299     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
1300       unfolding sets_sigma
1301     proof induct
1302       case (Basic A) then show ?case
1303         by (auto simp: sets_sigma product_algebra_def intro: sigma_sets.Basic)
1304     qed (auto intro: sigma_sets.intros simp: product_algebra_def)
1305   next
1306     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
1307   qed
1308   then show ?thesis
1309     by (simp add: product_algebra_def sigma_def)
1310 qed
1312 lemma (in product_sigma_algebra) sigma_pair_algebra_sigma_eq:
1313   "sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))) =
1314    sigma (pair_algebra (product_algebra M I) (product_algebra M J))"
1315   using M.sets_into_space
1316   by (intro pair_sigma_algebra_sigma[of "\<lambda>_. \<Pi>\<^isub>E i\<in>I. space (M i)", of _ "\<lambda>_. \<Pi>\<^isub>E i\<in>J. space (M i)"])
1317      (auto simp: isoton_const product_algebra_def, blast+)
1319 lemma (in product_sigma_algebra) sigma_pair_algebra_product_singleton:
1320   "sigma (pair_algebra (sigma (product_algebra M I)) (M i)) =
1321    sigma (pair_algebra (product_algebra M I) (M i))"
1322   using M.sets_into_space apply (subst M.sigma_eq[symmetric])
1323   by (intro pair_sigma_algebra_sigma[of "\<lambda>_. \<Pi>\<^isub>E i\<in>I. space (M i)" _ "\<lambda>_. space (M i)"])
1324      (auto simp: isoton_const product_algebra_def, blast+)
1326 lemma (in product_sigma_algebra) measurable_restrict:
1327   assumes [simp]: "I \<inter> J = {}"
1328   shows "(\<lambda>x. (restrict x I, restrict x J)) \<in> measurable
1329     (sigma (product_algebra M (I \<union> J)))
1330     (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))"
1331   unfolding sigma_pair_algebra_sigma_eq using M.sets_into_space
1332   by (intro measurable_sigma_sigma measurable_restrict_on_generating
1333             pair_algebra_sets_into_space product_algebra_sets_into_space)
1334      auto
1336 lemma (in product_sigma_algebra) measurable_merge:
1337   assumes [simp]: "I \<inter> J = {}"
1338   shows "(\<lambda>(x, y). merge I x J y) \<in> measurable
1339     (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))
1340     (sigma (product_algebra M (I \<union> J)))"
1341   unfolding sigma_pair_algebra_sigma_eq using M.sets_into_space
1342   by (intro measurable_sigma_sigma measurable_merge_on_generating
1343             pair_algebra_sets_into_space product_algebra_sets_into_space)
1344      auto
1346 lemma (in product_sigma_algebra) pair_product_product_vimage_algebra:
1347   assumes [simp]: "I \<inter> J = {}"
1348   shows "sigma_algebra.vimage_algebra (sigma (product_algebra M (I \<union> J)))
1349     (space (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))) (\<lambda>(x,y). merge I x J y) =
1350     (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))"
1351   unfolding sigma_pair_algebra_sigma_eq using sets_into_space
1352   by (intro vimage_algebra_sigma[OF bij_inv_restrict_merge[symmetric]]
1353             pair_algebra_sets_into_space product_algebra_sets_into_space
1354             measurable_merge_on_generating measurable_restrict_on_generating)
1355      auto
1357 lemma (in product_sigma_algebra) measurable_restrict_iff:
1358   assumes IJ[simp]: "I \<inter> J = {}"
1359   shows "f \<in> measurable (sigma (pair_algebra
1360       (sigma (product_algebra M I)) (sigma (product_algebra M J)))) M' \<longleftrightarrow>
1361     (\<lambda>x. f (restrict x I, restrict x J)) \<in> measurable (sigma (product_algebra M (I \<union> J))) M'"
1362   using M.sets_into_space
1363   apply (subst pair_product_product_vimage_algebra[OF IJ, symmetric])
1364   apply (subst sigma_pair_algebra_sigma_eq)
1365   apply (subst sigma_algebra.measurable_vimage_iff_inv[OF _
1366       bij_inv_restrict_merge[symmetric]])
1367   apply (intro sigma_algebra_sigma product_algebra_sets_into_space)
1368   by auto
1370 lemma (in product_sigma_algebra) measurable_merge_iff:
1371   assumes IJ: "I \<inter> J = {}"
1372   shows "f \<in> measurable (sigma (product_algebra M (I \<union> J))) M' \<longleftrightarrow>
1373     (\<lambda>(x, y). f (merge I x J y)) \<in>
1374       measurable (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J)))) M'"
1375   unfolding measurable_restrict_iff[OF IJ]
1376   by (rule measurable_cong) (auto intro!: arg_cong[where f=f] simp: extensional_restrict)
1378 lemma (in product_sigma_algebra) measurable_component:
1379   assumes "i \<in> I" and f: "f \<in> measurable (M i) M'"
1380   shows "(\<lambda>x. f (x i)) \<in> measurable (sigma (product_algebra M I)) M'"
1381     (is "?f \<in> measurable ?P M'")
1382 proof -
1383   have "f \<circ> (\<lambda>x. x i) \<in> measurable ?P M'"
1384     apply (rule measurable_comp[OF _ f])
1385     using measurable_up_sigma[of "product_algebra M I" "M i"]
1386     using measurable_component_on_generator[OF i \<in> I]
1387     by auto
1388   then show "?f \<in> measurable ?P M'" by (simp add: comp_def)
1389 qed
1391 lemma (in product_sigma_algebra) singleton_vimage_algebra:
1392   "sigma_algebra.vimage_algebra (sigma (product_algebra M {i})) (space (M i)) (\<lambda>x. \<lambda>j\<in>{i}. x) = M i"
1393   using sets_into_space
1394   by (intro vimage_algebra_sigma[of "M i", unfolded M.sigma_eq, OF bij_inv_singleton[symmetric]]
1395             product_algebra_sets_into_space measurable_singleton_on_generator measurable_component_on_generator)
1396      auto
1398 lemma (in product_sigma_algebra) measurable_component_singleton:
1399   "(\<lambda>x. f (x i)) \<in> measurable (sigma (product_algebra M {i})) M' \<longleftrightarrow>
1400     f \<in> measurable (M i) M'"
1401   using sets_into_space
1402   apply (subst singleton_vimage_algebra[symmetric])
1403   apply (subst sigma_algebra.measurable_vimage_iff_inv[OF _ bij_inv_singleton[symmetric]])
1404   by (auto intro!: sigma_algebra_sigma product_algebra_sets_into_space)
1406 lemma (in product_sigma_algebra) measurable_component_iff:
1407   assumes "i \<in> I" and not_empty: "\<forall>i\<in>I. space (M i) \<noteq> {}"
1408   shows "(\<lambda>x. f (x i)) \<in> measurable (sigma (product_algebra M I)) M' \<longleftrightarrow>
1409     f \<in> measurable (M i) M'"
1410     (is "?f \<in> measurable ?P M' \<longleftrightarrow> _")
1411 proof
1412   assume "f \<in> measurable (M i) M'" then show "?f \<in> measurable ?P M'"
1413     by (rule measurable_component[OF i \<in> I])
1414 next
1415   assume f: "?f \<in> measurable ?P M'"
1416   def t \<equiv> "\<lambda>i. SOME x. x \<in> space (M i)"
1417   have t: "\<And>i. i\<in>I \<Longrightarrow> t i \<in> space (M i)"
1418      unfolding t_def using not_empty by (rule_tac someI_ex) auto
1419   have "?f \<circ> (\<lambda>x. (\<lambda>j\<in>I. if j = i then x else t j)) \<in> measurable (M i) M'"
1420     (is "?f \<circ> ?t \<in> measurable _ _")
1421   proof (rule measurable_comp[OF _ f])
1422     have "?t \<in> measurable (M i) (product_algebra M I)"
1423     proof (unfold measurable_def, intro CollectI conjI ballI)
1424       from t show "?t \<in> space (M i) \<rightarrow> (space (product_algebra M I))" by auto
1425     next
1426       fix A assume A: "A \<in> sets (product_algebra M I)"
1427       { fix E assume "E \<in> (\<Pi> i\<in>I. sets (M i))"
1428         then have "?t - Pi\<^isub>E I E \<inter> space (M i) = (if (\<forall>j\<in>I-{i}. t j \<in> E j) then E i else {})"
1429           using i \<in> I sets_into_space by (auto dest: Pi_mem[where B=E]) }
1430       note * = this
1431       with A i \<in> I show "?t - A \<inter> space (M i) \<in> sets (M i)"
1432         by (auto elim!: product_algebraE simp del: vimage_Int)
1433     qed
1434     also have "\<dots> \<subseteq> measurable (M i) (sigma (product_algebra M I))"
1435       using M.sets_into_space by (intro measurable_subset) (auto simp: product_algebra_def, blast)
1436     finally show "?t \<in> measurable (M i) (sigma (product_algebra M I))" .
1437   qed
1438   then show "f \<in> measurable (M i) M'" unfolding comp_def using i \<in> I by simp
1439 qed
1441 lemma (in product_sigma_algebra) measurable_singleton:
1442   shows "f \<in> measurable (sigma (product_algebra M {i})) M' \<longleftrightarrow>
1443     (\<lambda>x. f (\<lambda>j\<in>{i}. x)) \<in> measurable (M i) M'"
1444   using sets_into_space unfolding measurable_component_singleton[symmetric]
1445   by (auto intro!: measurable_cong arg_cong[where f=f] simp: fun_eq_iff extensional_def)
1448 lemma (in pair_sigma_algebra) measurable_pair_split:
1449   assumes "sigma_algebra M1'" "sigma_algebra M2'"
1450   assumes f: "f \<in> measurable M1 M1'" and g: "g \<in> measurable M2 M2'"
1451   shows "(\<lambda>(x, y). (f x, g y)) \<in> measurable P (sigma (pair_algebra M1' M2'))"
1452 proof (rule measurable_sigma)
1453   interpret M1': sigma_algebra M1' by fact
1454   interpret M2': sigma_algebra M2' by fact
1455   interpret Q: pair_sigma_algebra M1' M2' by default
1456   show "sets Q.E \<subseteq> Pow (space Q.E)"
1457     using M1'.sets_into_space M2'.sets_into_space by (auto simp: pair_algebra_def)
1458   show "(\<lambda>(x, y). (f x, g y)) \<in> space P \<rightarrow> space Q.E"
1459     using f g unfolding measurable_def pair_algebra_def by auto
1460   fix A assume "A \<in> sets Q.E"
1461   then obtain X Y where A: "A = X \<times> Y" "X \<in> sets M1'" "Y \<in> sets M2'"
1462     unfolding pair_algebra_def by auto
1463   then have *: "(\<lambda>(x, y). (f x, g y)) - A \<inter> space P =
1464       (f - X \<inter> space M1) \<times> (g - Y \<inter> space M2)"
1465     by (auto simp: pair_algebra_def)
1466   then show "(\<lambda>(x, y). (f x, g y)) - A \<inter> space P \<in> sets P"
1467     using f g A unfolding measurable_def *
1468     by (auto intro!: pair_algebraI in_sigma)
1469 qed
1471 lemma (in product_sigma_algebra) measurable_add_dim:
1472   assumes "i \<notin> I" "finite I"
1473   shows "(\<lambda>(f, y). f(i := y)) \<in> measurable (sigma (pair_algebra (sigma (product_algebra M I)) (M i)))
1474                          (sigma (product_algebra M (insert i I)))"
1475 proof (subst measurable_cong)
1476   interpret I: finite_product_sigma_algebra M I by default fact
1477   interpret i: finite_product_sigma_algebra M "{i}" by default auto
1478   interpret Ii: pair_sigma_algebra I.P "M i" by default
1479   interpret Ii': pair_sigma_algebra I.P i.P by default
1480   have disj: "I \<inter> {i} = {}" using i \<notin> I by auto
1481   have "(\<lambda>(x, y). (id x, \<lambda>_\<in>{i}. y)) \<in> measurable Ii.P Ii'.P"
1482   proof (intro Ii.measurable_pair_split I.measurable_ident)
1483     show "(\<lambda>y. \<lambda>_\<in>{i}. y) \<in> measurable (M i) i.P"
1484       apply (rule measurable_singleton[THEN iffD1])
1485       using i.measurable_ident unfolding id_def .
1486   qed default
1487   from measurable_comp[OF this measurable_merge[OF disj]]
1488   show "(\<lambda>(x, y). merge I x {i} y) \<circ> (\<lambda>(x, y). (id x, \<lambda>_\<in>{i}. y))
1489     \<in> measurable (sigma (pair_algebra I.P (M i))) (sigma (product_algebra M (insert i I)))"
1490     (is "?f \<in> _") by simp
1491   fix x assume "x \<in> space Ii.P"
1492   with assms show "(\<lambda>(f, y). f(i := y)) x = ?f x"
1493     by (cases x) (simp add: merge_def fun_eq_iff pair_algebra_def extensional_def)
1494 qed
1496 locale product_sigma_finite =
1497   fixes M :: "'i \<Rightarrow> 'a algebra" and \<mu> :: "'i \<Rightarrow> 'a set \<Rightarrow> pextreal"
1498   assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i) (\<mu> i)"
1500 locale finite_product_sigma_finite = product_sigma_finite M \<mu> for M :: "'i \<Rightarrow> 'a algebra" and \<mu> +
1501   fixes I :: "'i set" assumes finite_index': "finite I"
1503 sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" "\<mu> i" for i
1504   by (rule sigma_finite_measures)
1506 sublocale product_sigma_finite \<subseteq> product_sigma_algebra
1507   by default
1509 sublocale finite_product_sigma_finite \<subseteq> finite_product_sigma_algebra
1510   by default (fact finite_index')
1512 lemma (in finite_product_sigma_finite) sigma_finite_pairs:
1513   "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
1514     (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
1515     (\<forall>k. \<forall>i\<in>I. \<mu> i (F i k) \<noteq> \<omega>) \<and>
1516     (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<up> space G"
1517 proof -
1518   have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> F \<up> space (M i) \<and> (\<forall>k. \<mu> i (F k) \<noteq> \<omega>)"
1519     using M.sigma_finite_up by simp
1520   from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
1521   then have "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. F i \<up> space (M i)" "\<And>i k. \<mu> i (F i k) \<noteq> \<omega>"
1522     by auto
1523   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"
1524   note space_product_algebra[simp]
1525   show ?thesis
1526   proof (intro exI[of _ F] conjI allI isotoneI set_eqI iffI ballI)
1527     fix i show "range (F i) \<subseteq> sets (M i)" by fact
1528   next
1529     fix i k show "\<mu> i (F i k) \<noteq> \<omega>" by fact
1530   next
1531     fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space G"
1532       using \<And>i. range (F i) \<subseteq> sets (M i) M.sets_into_space by auto blast
1533   next
1534     fix f assume "f \<in> space G"
1535     with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"]
1536       \<And>i. F i \<up> space (M i)[THEN isotonD(2)]
1537       \<And>i. F i \<up> space (M i)[THEN isoton_mono_le]
1538     show "f \<in> (\<Union>i. ?F i)" by auto
1539   next
1540     fix i show "?F i \<subseteq> ?F (Suc i)"
1541       using \<And>i. F i \<up> space (M i)[THEN isotonD(1)] by auto
1542   qed
1543 qed
1545 lemma (in product_sigma_finite) product_measure_exists:
1546   assumes "finite I"
1547   shows "\<exists>\<nu>. (\<forall>A\<in>(\<Pi> i\<in>I. sets (M i)). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))) \<and>
1548      sigma_finite_measure (sigma (product_algebra M I)) \<nu>"
1549 using finite I proof induct
1550   case empty then show ?case unfolding product_algebra_def
1551     by (auto intro!: exI[of _ "\<lambda>A. if A = {} then 0 else 1"] sigma_algebra_sigma
1553              simp add: positive_def additive_def sets_sigma sigma_finite_measure_def
1554                        sigma_finite_measure_axioms_def image_constant)
1555 next
1556   case (insert i I)
1557   interpret finite_product_sigma_finite M \<mu> I by default fact
1558   have "finite (insert i I)" using finite I by auto
1559   interpret I': finite_product_sigma_finite M \<mu> "insert i I" by default fact
1560   from insert obtain \<nu> where
1561     prod: "\<And>A. A \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))" and
1562     "sigma_finite_measure P \<nu>" by auto
1563   interpret I: sigma_finite_measure P \<nu> by fact
1564   interpret P: pair_sigma_finite P \<nu> "M i" "\<mu> i" ..
1566   let ?h = "(\<lambda>(f, y). f(i := y))"
1567   let ?\<nu> = "\<lambda>A. P.pair_measure (?h - A \<inter> space P.P)"
1568   have I': "sigma_algebra I'.P" by default
1569   interpret I': measure_space "sigma (product_algebra M (insert i I))" ?\<nu>
1570     apply (rule P.measure_space_vimage[OF I'])
1571     apply (rule measurable_add_dim[OF i \<notin> I finite I])
1572     by simp
1573   show ?case
1574   proof (intro exI[of _ ?\<nu>] conjI ballI)
1575     { fix A assume A: "A \<in> (\<Pi> i\<in>insert i I. sets (M i))"
1576       then have *: "?h - Pi\<^isub>E (insert i I) A \<inter> space P.P = Pi\<^isub>E I A \<times> A i"
1577         using i \<notin> I M.sets_into_space by (auto simp: pair_algebra_def) blast
1578       show "?\<nu> (Pi\<^isub>E (insert i I) A) = (\<Prod>i\<in>insert i I. \<mu> i (A i))"
1579         unfolding * using A
1580         apply (subst P.pair_measure_times)
1581         using A apply fastsimp
1582         using A apply fastsimp
1583         using i \<notin> I finite I prod[of A] A by (auto simp: ac_simps) }
1584     note product = this
1585     show "sigma_finite_measure I'.P ?\<nu>"
1586     proof
1587       from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
1588       then have F: "\<And>j. j \<in> insert i I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
1589         "(\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) \<up> space I'.G"
1590         "\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> \<mu> j (F j k) \<noteq> \<omega>"
1591         by blast+
1592       let "?F k" = "\<Pi>\<^isub>E j \<in> insert i I. F j k"
1593       show "\<exists>F::nat \<Rightarrow> ('i \<Rightarrow> 'a) set. range F \<subseteq> sets I'.P \<and>
1594           (\<Union>i. F i) = space I'.P \<and> (\<forall>i. ?\<nu> (F i) \<noteq> \<omega>)"
1595       proof (intro exI[of _ ?F] conjI allI)
1596         show "range ?F \<subseteq> sets I'.P" using F(1) by auto
1597       next
1598         from F(2)[THEN isotonD(2)]
1599         show "(\<Union>i. ?F i) = space I'.P" by simp
1600       next
1601         fix j
1602         show "?\<nu> (?F j) \<noteq> \<omega>"
1603           using F finite I
1604           by (subst product) (auto simp: setprod_\<omega>)
1605       qed
1606     qed
1607   qed
1608 qed
1610 definition (in finite_product_sigma_finite)
1611   measure :: "('i \<Rightarrow> 'a) set \<Rightarrow> pextreal" where
1612   "measure = (SOME \<nu>.
1613      (\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))) \<and>
1614      sigma_finite_measure P \<nu>)"
1616 sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure P measure
1617 proof -
1618   show "sigma_finite_measure P measure"
1619     unfolding measure_def
1620     by (rule someI2_ex[OF product_measure_exists[OF finite_index]]) auto
1621 qed
1623 lemma (in finite_product_sigma_finite) measure_times:
1624   assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)"
1625   shows "measure (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))"
1626 proof -
1627   note ex = product_measure_exists[OF finite_index]
1628   show ?thesis
1629     unfolding measure_def
1630   proof (rule someI2_ex[OF ex], elim conjE)
1631     fix \<nu> assume *: "\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))"
1632     have "Pi\<^isub>E I A = Pi\<^isub>E I (\<lambda>i\<in>I. A i)" by (auto dest: Pi_mem)
1633     then have "\<nu> (Pi\<^isub>E I A) = \<nu> (Pi\<^isub>E I (\<lambda>i\<in>I. A i))" by simp
1634     also have "\<dots> = (\<Prod>i\<in>I. \<mu> i ((\<lambda>i\<in>I. A i) i))" using assms * by auto
1635     finally show "\<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))" by simp
1636   qed
1637 qed
1639 abbreviation (in product_sigma_finite)
1640   "product_measure I \<equiv> finite_product_sigma_finite.measure M \<mu> I"
1642 abbreviation (in product_sigma_finite)
1643   "product_positive_integral I \<equiv>
1644     measure_space.positive_integral (sigma (product_algebra M I)) (product_measure I)"
1646 abbreviation (in product_sigma_finite)
1647   "product_integral I \<equiv>
1648     measure_space.integral (sigma (product_algebra M I)) (product_measure I)"
1650 abbreviation (in product_sigma_finite)
1651   "product_integrable I \<equiv>
1652     measure_space.integrable (sigma (product_algebra M I)) (product_measure I)"
1654 lemma (in product_sigma_finite) product_measure_empty[simp]:
1655   "product_measure {} {\<lambda>x. undefined} = 1"
1656 proof -
1657   interpret finite_product_sigma_finite M \<mu> "{}" by default auto
1658   from measure_times[of "\<lambda>x. {}"] show ?thesis by simp
1659 qed
1661 lemma (in product_sigma_finite) positive_integral_empty:
1662   "product_positive_integral {} f = f (\<lambda>k. undefined)"
1663 proof -
1664   interpret finite_product_sigma_finite M \<mu> "{}" by default (fact finite.emptyI)
1665   have "\<And>A. measure (Pi\<^isub>E {} A) = 1"
1666     using assms by (subst measure_times) auto
1667   then show ?thesis
1668     unfolding positive_integral_def simple_function_def simple_integral_def_raw
1669   proof (simp add: P_empty, intro antisym)
1670     show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> f}. f (\<lambda>k. undefined))"
1671       by (intro le_SUPI) auto
1672     show "(SUP f:{g. g \<le> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)"
1673       by (intro SUP_leI) (auto simp: le_fun_def)
1674   qed
1675 qed
1677 lemma (in product_sigma_finite) measure_fold:
1678   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
1679   assumes A: "A \<in> sets (sigma (product_algebra M (I \<union> J)))"
1680   shows "pair_sigma_finite.pair_measure
1681      (sigma (product_algebra M I)) (product_measure I)
1682      (sigma (product_algebra M J)) (product_measure J)
1683      ((\<lambda>(x,y). merge I x J y) - A \<inter> space (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))) =
1684    product_measure (I \<union> J) A"
1685 proof -
1686   interpret I: finite_product_sigma_finite M \<mu> I by default fact
1687   interpret J: finite_product_sigma_finite M \<mu> J by default fact
1688   have "finite (I \<union> J)" using fin by auto
1689   interpret IJ: finite_product_sigma_finite M \<mu> "I \<union> J" by default fact
1690   interpret P: pair_sigma_finite I.P I.measure J.P J.measure by default
1691   let ?g = "\<lambda>(x,y). merge I x J y"
1692   let "?X S" = "?g - S \<inter> space P.P"
1693   from IJ.sigma_finite_pairs obtain F where
1694     F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
1695        "(\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) \<up> space IJ.G"
1696        "\<And>k. \<forall>i\<in>I\<union>J. \<mu> i (F i k) \<noteq> \<omega>"
1697     by auto
1698   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"
1699   show "P.pair_measure (?X A) = IJ.measure A"
1700   proof (rule measure_unique_Int_stable[OF _ _ _ _ _ _ _ _ A])
1701     show "Int_stable IJ.G" by (simp add: PiE_Int Int_stable_def product_algebra_def) auto
1702     show "range ?F \<subseteq> sets IJ.G" using F by (simp add: image_subset_iff product_algebra_def)
1703     show "?F \<up> space IJ.G " using F(2) by simp
1704     have "sigma_algebra IJ.P" by default
1705     then show "measure_space IJ.P (\<lambda>A. P.pair_measure (?X A))"
1706       apply (rule P.measure_space_vimage)
1707       apply (rule measurable_merge[OF I \<inter> J = {}])
1708       by simp
1709     show "measure_space IJ.P IJ.measure" by fact
1710   next
1711     fix A assume "A \<in> sets IJ.G"
1712     then obtain F where A[simp]: "A = Pi\<^isub>E (I \<union> J) F"
1713       and F: "\<And>i. i \<in> I \<union> J \<Longrightarrow> F i \<in> sets (M i)"
1714       by (auto simp: product_algebra_def)
1715     then have "?X A = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"
1716       using sets_into_space by (auto simp: space_pair_algebra) blast+
1717     then have "P.pair_measure (?X A) = (\<Prod>i\<in>I. \<mu> i (F i)) * (\<Prod>i\<in>J. \<mu> i (F i))"
1718       using finite J finite I F
1719       by (simp add: P.pair_measure_times I.measure_times J.measure_times)
1720     also have "\<dots> = (\<Prod>i\<in>I \<union> J. \<mu> i (F i))"
1721       using finite J finite I I \<inter> J = {}  by (simp add: setprod_Un_one)
1722     also have "\<dots> = IJ.measure A"
1723       using finite J finite I F unfolding A
1724       by (intro IJ.measure_times[symmetric]) auto
1725     finally show "P.pair_measure (?X A) = IJ.measure A" .
1726   next
1727     fix k
1728     have k: "\<And>i. i \<in> I \<union> J \<Longrightarrow> F i k \<in> sets (M i)" using F by auto
1729     then have "?X (?F k) = (\<Pi>\<^isub>E i\<in>I. F i k) \<times> (\<Pi>\<^isub>E i\<in>J. F i k)"
1730       using sets_into_space by (auto simp: space_pair_algebra) blast+
1731     with k have "P.pair_measure (?X (?F k)) = (\<Prod>i\<in>I. \<mu> i (F i k)) * (\<Prod>i\<in>J. \<mu> i (F i k))"
1732      by (simp add: P.pair_measure_times I.measure_times J.measure_times)
1733     then show "P.pair_measure (?X (?F k)) \<noteq> \<omega>"
1734       using finite I F by (simp add: setprod_\<omega>)
1735   qed simp
1736 qed
1738 lemma (in product_sigma_finite) product_positive_integral_fold:
1739   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
1740   and f: "f \<in> borel_measurable (sigma (product_algebra M (I \<union> J)))"
1741   shows "product_positive_integral (I \<union> J) f =
1742     product_positive_integral I (\<lambda>x. product_positive_integral J (\<lambda>y. f (merge I x J y)))"
1743 proof -
1744   interpret I: finite_product_sigma_finite M \<mu> I by default fact
1745   interpret J: finite_product_sigma_finite M \<mu> J by default fact
1746   have "finite (I \<union> J)" using fin by auto
1747   interpret IJ: finite_product_sigma_finite M \<mu> "I \<union> J" by default fact
1748   interpret P: pair_sigma_finite I.P I.measure J.P J.measure by default
1749   have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable P.P"
1750     unfolding case_prod_distrib measurable_merge_iff[OF IJ, symmetric] using f .
1751   show ?thesis
1752     unfolding P.positive_integral_fst_measurable[OF P_borel, simplified]
1753     apply (subst IJ.positive_integral_cong_measure[symmetric])
1754     apply (rule measure_fold[OF IJ fin])
1755     apply assumption
1756   proof (rule P.positive_integral_vimage)
1757     show "sigma_algebra IJ.P" by default
1758     show "(\<lambda>(x, y). merge I x J y) \<in> measurable P.P IJ.P" by (rule measurable_merge[OF IJ])
1759     show "f \<in> borel_measurable IJ.P" using f .
1760   qed
1761 qed
1763 lemma (in product_sigma_finite) product_positive_integral_singleton:
1764   assumes f: "f \<in> borel_measurable (M i)"
1765   shows "product_positive_integral {i} (\<lambda>x. f (x i)) = M.positive_integral i f"
1766 proof -
1767   interpret I: finite_product_sigma_finite M \<mu> "{i}" by default simp
1768   have T: "(\<lambda>x. x i) \<in> measurable (sigma (product_algebra M {i})) (M i)"
1769     using measurable_component_singleton[of "\<lambda>x. x" i]
1770           measurable_ident[unfolded id_def] by auto
1771   show "I.positive_integral (\<lambda>x. f (x i)) = M.positive_integral i f"
1772     unfolding I.positive_integral_vimage[OF sigma_algebras T f, symmetric]
1773   proof (rule positive_integral_cong_measure)
1774     fix A let ?P = "(\<lambda>x. x i) - A \<inter> space (sigma (product_algebra M {i}))"
1775     assume A: "A \<in> sets (M i)"
1776     then have *: "?P = {i} \<rightarrow>\<^isub>E A" using sets_into_space by auto
1777     show "product_measure {i} ?P = \<mu> i A" unfolding *
1778       using A I.measure_times[of "\<lambda>_. A"] by auto
1779   qed
1780 qed
1782 lemma (in product_sigma_finite) product_positive_integral_insert:
1783   assumes [simp]: "finite I" "i \<notin> I"
1784     and f: "f \<in> borel_measurable (sigma (product_algebra M (insert i I)))"
1785   shows "product_positive_integral (insert i I) f
1786     = product_positive_integral I (\<lambda>x. M.positive_integral i (\<lambda>y. f (x(i:=y))))"
1787 proof -
1788   interpret I: finite_product_sigma_finite M \<mu> I by default auto
1789   interpret i: finite_product_sigma_finite M \<mu> "{i}" by default auto
1790   interpret P: pair_sigma_algebra I.P i.P ..
1791   have IJ: "I \<inter> {i} = {}" by auto
1792   show ?thesis
1793     unfolding product_positive_integral_fold[OF IJ, simplified, OF f]
1794   proof (rule I.positive_integral_cong, subst product_positive_integral_singleton)
1795     fix x assume x: "x \<in> space I.P"
1796     let "?f y" = "f (restrict (x(i := y)) (insert i I))"
1797     have f'_eq: "\<And>y. ?f y = f (x(i := y))"
1798       using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)
1799     note fP = f[unfolded measurable_merge_iff[OF IJ, simplified]]
1800     show "?f \<in> borel_measurable (M i)"
1801       using P.measurable_pair_image_snd[OF fP x]
1802       unfolding measurable_singleton f'_eq by (simp add: f'_eq)
1803     show "M.positive_integral i ?f = M.positive_integral i (\<lambda>y. f (x(i := y)))"
1804       unfolding f'_eq by simp
1805   qed
1806 qed
1808 lemma (in product_sigma_finite) product_positive_integral_setprod:
1809   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> pextreal"
1810   assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
1811   shows "product_positive_integral I (\<lambda>x. (\<Prod>i\<in>I. f i (x i))) =
1812     (\<Prod>i\<in>I. M.positive_integral i (f i))"
1813 using assms proof induct
1814   case empty
1815   interpret finite_product_sigma_finite M \<mu> "{}" by default auto
1816   then show ?case by simp
1817 next
1818   case (insert i I)
1819   note finite I[intro, simp]
1820   interpret I: finite_product_sigma_finite M \<mu> I by default auto
1821   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))"
1822     using insert by (auto intro!: setprod_cong)
1823   have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
1824     (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (sigma (product_algebra M J))"
1825     using sets_into_space insert
1826     by (intro sigma_algebra.borel_measurable_pextreal_setprod
1827               sigma_algebra_sigma product_algebra_sets_into_space
1828               measurable_component)
1829        auto
1830   show ?case
1831     by (simp add: product_positive_integral_insert[OF insert(1,2) prod])
1832        (simp add: insert I.positive_integral_cmult M.positive_integral_multc * prod subset_insertI)
1833 qed
1835 lemma (in product_sigma_finite) product_integral_singleton:
1836   assumes f: "f \<in> borel_measurable (M i)"
1837   shows "product_integral {i} (\<lambda>x. f (x i)) = M.integral i f"
1838 proof -
1839   interpret I: finite_product_sigma_finite M \<mu> "{i}" by default simp
1840   have *: "(\<lambda>x. Real (f x)) \<in> borel_measurable (M i)"
1841     "(\<lambda>x. Real (- f x)) \<in> borel_measurable (M i)"
1842     using assms by auto
1843   show ?thesis
1844     unfolding I.integral_def integral_def
1845     unfolding *[THEN product_positive_integral_singleton] ..
1846 qed
1848 lemma (in product_sigma_finite) product_integral_fold:
1849   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
1850   and f: "measure_space.integrable (sigma (product_algebra M (I \<union> J))) (product_measure (I \<union> J)) f"
1851   shows "product_integral (I \<union> J) f =
1852     product_integral I (\<lambda>x. product_integral J (\<lambda>y. f (merge I x J y)))"
1853 proof -
1854   interpret I: finite_product_sigma_finite M \<mu> I by default fact
1855   interpret J: finite_product_sigma_finite M \<mu> J by default fact
1856   have "finite (I \<union> J)" using fin by auto
1857   interpret IJ: finite_product_sigma_finite M \<mu> "I \<union> J" by default fact
1858   interpret P: pair_sigma_finite I.P I.measure J.P J.measure by default
1859   let ?f = "\<lambda>(x,y). f (merge I x J y)"
1860   have f_borel: "f \<in> borel_measurable IJ.P"
1861      "(\<lambda>x. Real (f x)) \<in> borel_measurable IJ.P"
1862      "(\<lambda>x. Real (- f x)) \<in> borel_measurable IJ.P"
1863     using f unfolding integrable_def by auto
1864   have f_restrict: "(\<lambda>x. f (restrict x (I \<union> J))) \<in> borel_measurable IJ.P"
1865     by (rule measurable_cong[THEN iffD2, OF _ f_borel(1)])
1866        (auto intro!: arg_cong[where f=f] simp: extensional_restrict)
1867   then have f'_borel:
1868     "(\<lambda>x. Real (?f x)) \<in> borel_measurable P.P"
1869     "(\<lambda>x. Real (- ?f x)) \<in> borel_measurable P.P"
1870     unfolding measurable_restrict_iff[OF IJ]
1871     by simp_all
1872   have PI:
1873     "P.positive_integral (\<lambda>x. Real (?f x)) = IJ.positive_integral (\<lambda>x. Real (f x))"
1874     "P.positive_integral (\<lambda>x. Real (- ?f x)) = IJ.positive_integral (\<lambda>x. Real (- f x))"
1875     using f'_borel[THEN P.positive_integral_fst_measurable(2)]
1876     using f_borel(2,3)[THEN product_positive_integral_fold[OF assms(1-3)]]
1877     by simp_all
1878   have "P.integrable ?f" using IJ.integrable f
1879     unfolding P.integrable_def IJ.integrable_def
1880     unfolding measurable_restrict_iff[OF IJ]
1881     using f_restrict PI by simp_all
1882   show ?thesis
1883     unfolding P.integrable_fst_measurable(2)[OF P.integrable ?f, simplified]
1884     unfolding IJ.integral_def P.integral_def
1885     unfolding PI by simp
1886 qed
1888 lemma (in product_sigma_finite) product_integral_insert:
1889   assumes [simp]: "finite I" "i \<notin> I"
1890     and f: "measure_space.integrable (sigma (product_algebra M (insert i I))) (product_measure (insert i I)) f"
1891   shows "product_integral (insert i I) f
1892     = product_integral I (\<lambda>x. M.integral i (\<lambda>y. f (x(i:=y))))"
1893 proof -
1894   interpret I: finite_product_sigma_finite M \<mu> I by default auto
1895   interpret I': finite_product_sigma_finite M \<mu> "insert i I" by default auto
1896   interpret i: finite_product_sigma_finite M \<mu> "{i}" by default auto
1897   interpret P: pair_sigma_algebra I.P i.P ..
1898   have IJ: "I \<inter> {i} = {}" by auto
1899   show ?thesis
1900     unfolding product_integral_fold[OF IJ, simplified, OF f]
1901   proof (rule I.integral_cong, subst product_integral_singleton)
1902     fix x assume x: "x \<in> space I.P"
1903     let "?f y" = "f (restrict (x(i := y)) (insert i I))"
1904     have f'_eq: "\<And>y. ?f y = f (x(i := y))"
1905       using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)
1906     have "f \<in> borel_measurable I'.P" using f unfolding I'.integrable_def by auto
1907     note fP = this[unfolded measurable_merge_iff[OF IJ, simplified]]
1908     show "?f \<in> borel_measurable (M i)"
1909       using P.measurable_pair_image_snd[OF fP x]
1910       unfolding measurable_singleton f'_eq by (simp add: f'_eq)
1911     show "M.integral i ?f = M.integral i (\<lambda>y. f (x(i := y)))"
1912       unfolding f'_eq by simp
1913   qed
1914 qed
1916 lemma (in product_sigma_finite) product_integrable_setprod:
1917   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
1918   assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> M.integrable i (f i)"
1919   shows "product_integrable I (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "product_integrable I ?f")
1920 proof -
1921   interpret finite_product_sigma_finite M \<mu> I by default fact
1922   have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
1923     using integrable unfolding M.integrable_def by auto
1924   then have borel: "?f \<in> borel_measurable P"
1925     by (intro borel_measurable_setprod measurable_component) auto
1926   moreover have "integrable (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"
1927   proof (unfold integrable_def, intro conjI)
1928     show "(\<lambda>x. abs (?f x)) \<in> borel_measurable P"
1929       using borel by auto
1930     have "positive_integral (\<lambda>x. Real (abs (?f x))) = positive_integral (\<lambda>x. \<Prod>i\<in>I. Real (abs (f i (x i))))"
1931       by (simp add: Real_setprod abs_setprod)
1932     also have "\<dots> = (\<Prod>i\<in>I. M.positive_integral i (\<lambda>x. Real (abs (f i x))))"
1933       using f by (subst product_positive_integral_setprod) auto
1934     also have "\<dots> < \<omega>"
1935       using integrable[THEN M.integrable_abs]
1936       unfolding pextreal_less_\<omega> setprod_\<omega> M.integrable_def by simp
1937     finally show "positive_integral (\<lambda>x. Real (abs (?f x))) \<noteq> \<omega>" by auto
1938     show "positive_integral (\<lambda>x. Real (- abs (?f x))) \<noteq> \<omega>" by simp
1939   qed
1940   ultimately show ?thesis
1941     by (rule integrable_abs_iff[THEN iffD1])
1942 qed
1944 lemma (in product_sigma_finite) product_integral_setprod:
1945   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
1946   assumes "finite I" "I \<noteq> {}" and integrable: "\<And>i. i \<in> I \<Longrightarrow> M.integrable i (f i)"
1947   shows "product_integral I (\<lambda>x. (\<Prod>i\<in>I. f i (x i))) = (\<Prod>i\<in>I. M.integral i (f i))"
1948 using assms proof (induct rule: finite_ne_induct)
1949   case (singleton i)
1950   then show ?case by (simp add: product_integral_singleton integrable_def)
1951 next
1952   case (insert i I)
1953   then have iI: "finite (insert i I)" by auto
1954   then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
1955     product_integrable J (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
1956     by (intro product_integrable_setprod insert(5)) (auto intro: finite_subset)
1957   interpret I: finite_product_sigma_finite M \<mu> I by default fact
1958   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))"
1959     using i \<notin> I by (auto intro!: setprod_cong)
1960   show ?case
1961     unfolding product_integral_insert[OF insert(1,3) prod[OF subset_refl]]
1962     by (simp add: * insert integral_multc I.integral_cmult[OF prod] subset_insertI)
1963 qed
1965 section "Products on finite spaces"
1967 lemma sigma_sets_pair_algebra_finite:
1968   assumes "finite A" and "finite B"
1969   shows "sigma_sets (A \<times> B) ((\<lambda>(x,y). x \<times> y) ` (Pow A \<times> Pow B)) = Pow (A \<times> B)"
1970   (is "sigma_sets ?prod ?sets = _")
1971 proof safe
1972   have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
1973   fix x assume subset: "x \<subseteq> A \<times> B"
1974   hence "finite x" using fin by (rule finite_subset)
1975   from this subset show "x \<in> sigma_sets ?prod ?sets"
1976   proof (induct x)
1977     case empty show ?case by (rule sigma_sets.Empty)
1978   next
1979     case (insert a x)
1980     hence "{a} \<in> sigma_sets ?prod ?sets"
1981       by (auto simp: pair_algebra_def intro!: sigma_sets.Basic)
1982     moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
1983     ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
1984   qed
1985 next
1986   fix x a b
1987   assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
1988   from sigma_sets_into_sp[OF _ this(1)] this(2)
1989   show "a \<in> A" and "b \<in> B" by auto
1990 qed
1992 locale pair_finite_sigma_algebra = M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2 for M1 M2
1994 sublocale pair_finite_sigma_algebra \<subseteq> pair_sigma_algebra by default
1996 lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra[simp]:
1997   shows "P = (| space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2) |)"
1998 proof -
1999   show ?thesis using M1.finite_space M2.finite_space
2000     by (simp add: sigma_def space_pair_algebra sets_pair_algebra
2001                   sigma_sets_pair_algebra_finite M1.sets_eq_Pow M2.sets_eq_Pow)
2002 qed
2004 sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P
2005 proof
2006   show "finite (space P)" "sets P = Pow (space P)"
2007     using M1.finite_space M2.finite_space by auto
2008 qed
2010 locale pair_finite_space = M1: finite_measure_space M1 \<mu>1 + M2: finite_measure_space M2 \<mu>2
2011   for M1 \<mu>1 M2 \<mu>2
2013 sublocale pair_finite_space \<subseteq> pair_finite_sigma_algebra
2014   by default
2016 sublocale pair_finite_space \<subseteq> pair_sigma_finite
2017   by default
2019 lemma (in pair_finite_space) finite_pair_sigma_algebra[simp]:
2020   shows "P = (| space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2) |)"
2021 proof -
2022   show ?thesis using M1.finite_space M2.finite_space
2023     by (simp add: sigma_def space_pair_algebra sets_pair_algebra
2024                   sigma_sets_pair_algebra_finite M1.sets_eq_Pow M2.sets_eq_Pow)
2025 qed
2027 lemma (in pair_finite_space) pair_measure_Pair[simp]:
2028   assumes "a \<in> space M1" "b \<in> space M2"
2029   shows "pair_measure {(a, b)} = \<mu>1 {a} * \<mu>2 {b}"
2030 proof -
2031   have "pair_measure ({a}\<times>{b}) = \<mu>1 {a} * \<mu>2 {b}"
2032     using M1.sets_eq_Pow M2.sets_eq_Pow assms
2033     by (subst pair_measure_times) auto
2034   then show ?thesis by simp
2035 qed
2037 lemma (in pair_finite_space) pair_measure_singleton[simp]:
2038   assumes "x \<in> space M1 \<times> space M2"
2039   shows "pair_measure {x} = \<mu>1 {fst x} * \<mu>2 {snd x}"
2040   using pair_measure_Pair assms by (cases x) auto
2042 sublocale pair_finite_space \<subseteq> finite_measure_space P pair_measure
2043   by default auto
2045 lemma (in pair_finite_space) finite_measure_space_finite_prod_measure_alterantive:
2046   "finite_measure_space \<lparr>space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2)\<rparr> pair_measure"
2047   unfolding finite_pair_sigma_algebra[symmetric]
2048   by default
2050 end