src/HOL/Probability/Product_Measure.thy
 author hoelzl Wed Dec 08 16:15:14 2010 +0100 (2010-12-08) changeset 41095 c335d880ff82 parent 41026 bea75746dc9d child 41096 843c40bbc379 permissions -rw-r--r--
cleanup bijectivity btw. product spaces and pairs
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 lemma (in pair_sigma_algebra) pair_sigma_algebra_swap:
527   "sigma (pair_algebra M2 M1) = (vimage_algebra (space M2 \<times> space M1) (\<lambda>(x, y). (y, x)))"
528   unfolding vimage_algebra_def
529   apply (simp add: sets_sigma)
530   apply (subst sigma_sets_vimage[symmetric])
531   apply (fastsimp simp: pair_algebra_def)
532   using M1.sets_into_space M2.sets_into_space apply (fastsimp simp: pair_algebra_def)
533 proof -
534   have "(\<lambda>X. (\<lambda>(x, y). (y, x)) - X \<inter> space M2 \<times> space M1)  sets E
535     = sets (pair_algebra M2 M1)" (is "?S = _")
536     by (rule pair_algebra_swap)
537   then show "sigma (pair_algebra M2 M1) = \<lparr>space = space M2 \<times> space M1,
538        sets = sigma_sets (space M2 \<times> space M1) ?S\<rparr>"
539     by (simp add: pair_algebra_def sigma_def)
540 qed
542 definition (in pair_sigma_finite)
543   "pair_measure A = M1.positive_integral (\<lambda>x.
544     M2.positive_integral (\<lambda>y. indicator A (x, y)))"
546 lemma (in pair_sigma_finite) pair_measure_alt:
547   assumes "A \<in> sets P"
548   shows "pair_measure A = M1.positive_integral (\<lambda>x. \<mu>2 (Pair x - A))"
549   unfolding pair_measure_def
550 proof (rule M1.positive_integral_cong)
551   fix x assume "x \<in> space M1"
552   have *: "\<And>y. indicator A (x, y) = (indicator (Pair x - A) y :: pextreal)"
553     unfolding indicator_def by auto
554   show "M2.positive_integral (\<lambda>y. indicator A (x, y)) = \<mu>2 (Pair x - A)"
555     unfolding *
556     apply (subst M2.positive_integral_indicator)
557     apply (rule measurable_cut_fst[OF assms])
558     by simp
559 qed
561 lemma (in pair_sigma_finite) pair_measure_times:
562   assumes A: "A \<in> sets M1" and "B \<in> sets M2"
563   shows "pair_measure (A \<times> B) = \<mu>1 A * \<mu>2 B"
564 proof -
565   from assms have "pair_measure (A \<times> B) =
566       M1.positive_integral (\<lambda>x. \<mu>2 B * indicator A x)"
567     by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt)
568   with assms show ?thesis
569     by (simp add: M1.positive_integral_cmult_indicator ac_simps)
570 qed
572 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_algebra:
573   "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> F \<up> space E \<and>
574     (\<forall>i. pair_measure (F i) \<noteq> \<omega>)"
575 proof -
576   obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where
577     F1: "range F1 \<subseteq> sets M1" "F1 \<up> space M1" "\<And>i. \<mu>1 (F1 i) \<noteq> \<omega>" and
578     F2: "range F2 \<subseteq> sets M2" "F2 \<up> space M2" "\<And>i. \<mu>2 (F2 i) \<noteq> \<omega>"
579     using M1.sigma_finite_up M2.sigma_finite_up by auto
580   then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)"
581     unfolding isoton_def by auto
582   let ?F = "\<lambda>i. F1 i \<times> F2 i"
583   show ?thesis unfolding isoton_def space_pair_algebra
584   proof (intro exI[of _ ?F] conjI allI)
585     show "range ?F \<subseteq> sets E" using F1 F2
586       by (fastsimp intro!: pair_algebraI)
587   next
588     have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
589     proof (intro subsetI)
590       fix x assume "x \<in> space M1 \<times> space M2"
591       then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
592         by (auto simp: space)
593       then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
594         using F1 \<up> space M1 F2 \<up> space M2
595         by (auto simp: max_def dest: isoton_mono_le)
596       then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
597         by (intro SigmaI) (auto simp add: min_max.sup_commute)
598       then show "x \<in> (\<Union>i. ?F i)" by auto
599     qed
600     then show "(\<Union>i. ?F i) = space M1 \<times> space M2"
601       using space by (auto simp: space)
602   next
603     fix i show "F1 i \<times> F2 i \<subseteq> F1 (Suc i) \<times> F2 (Suc i)"
604       using F1 \<up> space M1 F2 \<up> space M2 unfolding isoton_def
605       by auto
606   next
607     fix i
608     from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
609     with F1 F2 show "pair_measure (F1 i \<times> F2 i) \<noteq> \<omega>"
610       by (simp add: pair_measure_times)
611   qed
612 qed
614 sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P pair_measure
615 proof
616   show "pair_measure {} = 0"
617     unfolding pair_measure_def by auto
619   show "countably_additive P pair_measure"
621   proof (intro allI impI)
622     fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"
623     assume F: "range F \<subseteq> sets P" "disjoint_family F"
624     from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto
625     moreover from F have "\<And>i. (\<lambda>x. \<mu>2 (Pair x - F i)) \<in> borel_measurable M1"
626       by (intro measure_cut_measurable_fst) auto
627     moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x - F i)"
628       by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
629     moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x - F i) \<subseteq> sets M2"
630       using F by (auto intro!: measurable_cut_fst)
631     ultimately show "(\<Sum>\<^isub>\<infinity>n. pair_measure (F n)) = pair_measure (\<Union>i. F i)"
632       by (simp add: pair_measure_alt vimage_UN M1.positive_integral_psuminf[symmetric]
634                cong: M1.positive_integral_cong)
635   qed
637   from sigma_finite_up_in_pair_algebra guess F :: "nat \<Rightarrow> ('a \<times> 'c) set" .. note F = this
638   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>)"
639   proof (rule exI[of _ F], intro conjI)
640     show "range F \<subseteq> sets P" using F by auto
641     show "(\<Union>i. F i) = space P"
642       using F by (auto simp: space_pair_algebra isoton_def)
643     show "\<forall>i. pair_measure (F i) \<noteq> \<omega>" using F by auto
644   qed
645 qed
647 lemma (in pair_sigma_finite) pair_measure_alt2:
648   assumes "A \<in> sets P"
649   shows "pair_measure A = M2.positive_integral (\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) - A))"
650     (is "_ = ?\<nu> A")
651 proof -
652   from sigma_finite_up_in_pair_algebra guess F :: "nat \<Rightarrow> ('a \<times> 'c) set" .. note F = this
653   show ?thesis
654   proof (rule measure_unique_Int_stable[where \<nu>="?\<nu>", OF Int_stable_pair_algebra],
655          simp_all add: pair_sigma_algebra_def[symmetric])
656     show "range F \<subseteq> sets E" "F \<up> space E" "\<And>i. pair_measure (F i) \<noteq> \<omega>"
657       using F by auto
658     show "measure_space P pair_measure" by default
659   next
660     show "measure_space P ?\<nu>"
661     proof
662       show "?\<nu> {} = 0" by auto
663       show "countably_additive P ?\<nu>"
665       proof (intro allI impI)
666         fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"
667         assume F: "range F \<subseteq> sets P" "disjoint_family F"
668         from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto
669         moreover from F have "\<And>i. (\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) - F i)) \<in> borel_measurable M2"
670           by (intro measure_cut_measurable_snd) auto
671         moreover have "\<And>y. disjoint_family (\<lambda>i. (\<lambda>x. (x, y)) - F i)"
672           by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
673         moreover have "\<And>y. y \<in> space M2 \<Longrightarrow> range (\<lambda>i. (\<lambda>x. (x, y)) - F i) \<subseteq> sets M1"
674           using F by (auto intro!: measurable_cut_snd)
675         ultimately show "(\<Sum>\<^isub>\<infinity>n. ?\<nu> (F n)) = ?\<nu> (\<Union>i. F i)"
676           by (simp add: vimage_UN M2.positive_integral_psuminf[symmetric]
678                    cong: M2.positive_integral_cong)
679       qed
680     qed
681   next
682     fix X assume "X \<in> sets E"
683     then obtain A B where X: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2"
684       unfolding pair_algebra_def by auto
685     show "pair_measure X = ?\<nu> X"
686     proof -
687       from AB have "?\<nu> (A \<times> B) =
688           M2.positive_integral (\<lambda>y. \<mu>1 A * indicator B y)"
689         by (auto intro!: M2.positive_integral_cong)
690       with AB show ?thesis
691         unfolding pair_measure_times[OF AB] X
692         by (simp add: M2.positive_integral_cmult_indicator ac_simps)
693     qed
694   qed fact
695 qed
697 section "Fubinis theorem"
699 lemma (in pair_sigma_finite) simple_function_cut:
700   assumes f: "simple_function f"
701   shows "(\<lambda>x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"
702     and "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))
703       = positive_integral f"
704 proof -
705   have f_borel: "f \<in> borel_measurable P"
706     using f by (rule borel_measurable_simple_function)
707   let "?F z" = "f - {z} \<inter> space P"
708   let "?F' x z" = "Pair x - ?F z"
709   { fix x assume "x \<in> space M1"
710     have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y"
711       by (auto simp: indicator_def)
712     have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space P" using x \<in> space M1
713       by (simp add: space_pair_algebra)
714     moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel
715       by (intro borel_measurable_vimage measurable_cut_fst)
716     ultimately have "M2.simple_function (\<lambda> y. f (x, y))"
717       apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _])
718       apply (rule simple_function_indicator_representation[OF f])
719       using x \<in> space M1 by (auto simp del: space_sigma) }
720   note M2_sf = this
721   { fix x assume x: "x \<in> space M1"
722     then have "M2.positive_integral (\<lambda> y. f (x, y)) =
723         (\<Sum>z\<in>f  space P. z * \<mu>2 (?F' x z))"
724       unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x]]
725       unfolding M2.simple_integral_def
726     proof (safe intro!: setsum_mono_zero_cong_left)
727       from f show "finite (f  space P)" by (rule simple_functionD)
728     next
729       fix y assume "y \<in> space M2" then show "f (x, y) \<in> f  space P"
730         using x \<in> space M1 by (auto simp: space_pair_algebra)
731     next
732       fix x' y assume "(x', y) \<in> space P"
733         "f (x', y) \<notin> (\<lambda>y. f (x, y))  space M2"
734       then have *: "?F' x (f (x', y)) = {}"
735         by (force simp: space_pair_algebra)
736       show  "f (x', y) * \<mu>2 (?F' x (f (x', y))) = 0"
737         unfolding * by simp
738     qed (simp add: vimage_compose[symmetric] comp_def
739                    space_pair_algebra) }
740   note eq = this
741   moreover have "\<And>z. ?F z \<in> sets P"
742     by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma)
743   moreover then have "\<And>z. (\<lambda>x. \<mu>2 (?F' x z)) \<in> borel_measurable M1"
744     by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int)
745   ultimately
746   show "(\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"
747     and "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))
748     = positive_integral f"
749     by (auto simp del: vimage_Int cong: measurable_cong
750              intro!: M1.borel_measurable_pextreal_setsum
751              simp add: M1.positive_integral_setsum simple_integral_def
752                        M1.positive_integral_cmult
753                        M1.positive_integral_cong[OF eq]
754                        positive_integral_eq_simple_integral[OF f]
755                        pair_measure_alt[symmetric])
756 qed
758 lemma (in pair_sigma_finite) positive_integral_fst_measurable:
759   assumes f: "f \<in> borel_measurable P"
760   shows "(\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"
761       (is "?C f \<in> borel_measurable M1")
762     and "M1.positive_integral (\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) =
763       positive_integral f"
764 proof -
765   from borel_measurable_implies_simple_function_sequence[OF f]
766   obtain F where F: "\<And>i. simple_function (F i)" "F \<up> f" by auto
767   then have F_borel: "\<And>i. F i \<in> borel_measurable P"
768     and F_mono: "\<And>i x. F i x \<le> F (Suc i) x"
769     and F_SUPR: "\<And>x. (SUP i. F i x) = f x"
770     unfolding isoton_def le_fun_def SUPR_fun_expand
771     by (auto intro: borel_measurable_simple_function)
772   note sf = simple_function_cut[OF F(1)]
773   then have "(SUP i. ?C (F i)) \<in> borel_measurable M1"
774     using F(1) by (auto intro!: M1.borel_measurable_SUP)
775   moreover
776   { fix x assume "x \<in> space M1"
777     have isotone: "(\<lambda> i y. F i (x, y)) \<up> (\<lambda>y. f (x, y))"
778       using F \<up> f unfolding isoton_fun_expand
779       by (auto simp: isoton_def)
780     note measurable_pair_image_snd[OF F_borelx \<in> space M1]
781     from M2.positive_integral_isoton[OF isotone this]
782     have "(SUP i. ?C (F i) x) = ?C f x"
783       by (simp add: isoton_def) }
784   note SUPR_C = this
785   ultimately show "?C f \<in> borel_measurable M1"
786     unfolding SUPR_fun_expand by (simp cong: measurable_cong)
787   have "positive_integral (\<lambda>x. SUP i. F i x) = (SUP i. positive_integral (F i))"
788     using F_borel F_mono
789     by (auto intro!: positive_integral_monotone_convergence_SUP[symmetric])
790   also have "(SUP i. positive_integral (F i)) =
791     (SUP i. M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. F i (x, y))))"
792     unfolding sf(2) by simp
793   also have "\<dots> = M1.positive_integral (\<lambda>x. SUP i. M2.positive_integral (\<lambda>y. F i (x, y)))"
794     by (auto intro!: M1.positive_integral_monotone_convergence_SUP[OF _ sf(1)]
795                      M2.positive_integral_mono F_mono)
796   also have "\<dots> = M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. SUP i. F i (x, y)))"
797     using F_borel F_mono
798     by (auto intro!: M2.positive_integral_monotone_convergence_SUP
799                      M1.positive_integral_cong measurable_pair_image_snd)
800   finally show "M1.positive_integral (\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) =
801       positive_integral f"
802     unfolding F_SUPR by simp
803 qed
805 lemma (in pair_sigma_finite) positive_integral_snd_measurable:
806   assumes f: "f \<in> borel_measurable P"
807   shows "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =
808       positive_integral f"
809 proof -
810   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
811   have s: "\<And>x y. (case (x, y) of (x, y) \<Rightarrow> f (y, x)) = f (y, x)" by simp
812   have t: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> (y, x))) = (\<lambda>(x, y). f (y, x))" by (auto simp: fun_eq_iff)
813   have bij: "bij_betw (\<lambda>(x, y). (y, x)) (space M2 \<times> space M1) (space P)"
814     by (auto intro!: inj_onI simp: space_pair_algebra bij_betw_def)
815   note pair_sigma_algebra_measurable[OF f]
816   from Q.positive_integral_fst_measurable[OF this]
817   have "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =
818     Q.positive_integral (\<lambda>(x, y). f (y, x))"
819     by simp
820   also have "\<dots> = positive_integral f"
821     unfolding positive_integral_vimage[OF bij, of f] t
822     unfolding pair_sigma_algebra_swap[symmetric]
823   proof (rule Q.positive_integral_cong_measure[symmetric])
824     fix A assume "A \<in> sets Q.P"
825     from this Q.sets_pair_sigma_algebra_swap[OF this]
826     show "pair_measure ((\<lambda>(x, y). (y, x))  A) = Q.pair_measure A"
827       by (auto intro!: M1.positive_integral_cong arg_cong[where f=\<mu>2]
828                simp: pair_measure_alt Q.pair_measure_alt2)
829   qed
830   finally show ?thesis .
831 qed
833 lemma (in pair_sigma_finite) Fubini:
834   assumes f: "f \<in> borel_measurable P"
835   shows "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =
836       M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))"
837   unfolding positive_integral_snd_measurable[OF assms]
838   unfolding positive_integral_fst_measurable[OF assms] ..
840 lemma (in pair_sigma_finite) AE_pair:
841   assumes "almost_everywhere (\<lambda>x. Q x)"
842   shows "M1.almost_everywhere (\<lambda>x. M2.almost_everywhere (\<lambda>y. Q (x, y)))"
843 proof -
844   obtain N where N: "N \<in> sets P" "pair_measure N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N"
845     using assms unfolding almost_everywhere_def by auto
846   show ?thesis
847   proof (rule M1.AE_I)
848     from N measure_cut_measurable_fst[OF N \<in> sets P]
849     show "\<mu>1 {x\<in>space M1. \<mu>2 (Pair x - N) \<noteq> 0} = 0"
850       by (simp add: M1.positive_integral_0_iff pair_measure_alt vimage_def)
851     show "{x \<in> space M1. \<mu>2 (Pair x - N) \<noteq> 0} \<in> sets M1"
852       by (intro M1.borel_measurable_pextreal_neq_const measure_cut_measurable_fst N)
853     { fix x assume "x \<in> space M1" "\<mu>2 (Pair x - N) = 0"
854       have "M2.almost_everywhere (\<lambda>y. Q (x, y))"
855       proof (rule M2.AE_I)
856         show "\<mu>2 (Pair x - N) = 0" by fact
857         show "Pair x - N \<in> sets M2" by (intro measurable_cut_fst N)
858         show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x - N"
859           using N x \<in> space M1 unfolding space_sigma space_pair_algebra by auto
860       qed }
861     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}"
862       by auto
863   qed
864 qed
866 lemma (in pair_sigma_finite) positive_integral_product_swap:
867   "measure_space.positive_integral
868     (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) f =
869   positive_integral (\<lambda>(x,y). f (y,x))"
870 proof -
871   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
872   have t: "(\<lambda>y. case case y of (y, x) \<Rightarrow> (x, y) of (x, y) \<Rightarrow> f (y, x)) = f"
873     by (auto simp: fun_eq_iff)
874   have bij: "bij_betw (\<lambda>(x, y). (y, x)) (space M2 \<times> space M1) (space P)"
875     by (auto intro!: inj_onI simp: space_pair_algebra bij_betw_def)
876   show ?thesis
877     unfolding positive_integral_vimage[OF bij, of "\<lambda>(y,x). f (x,y)"]
878     unfolding pair_sigma_algebra_swap[symmetric] t
879   proof (rule Q.positive_integral_cong_measure[symmetric])
880     fix A assume "A \<in> sets Q.P"
881     from this Q.sets_pair_sigma_algebra_swap[OF this]
882     show "pair_measure ((\<lambda>(x, y). (y, x))  A) = Q.pair_measure A"
883       by (auto intro!: M1.positive_integral_cong arg_cong[where f=\<mu>2]
884                simp: pair_measure_alt Q.pair_measure_alt2)
885   qed
886 qed
888 lemma (in pair_sigma_algebra) measurable_product_swap:
889   "f \<in> measurable (sigma (pair_algebra M2 M1)) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable P M"
890 proof -
891   interpret Q: pair_sigma_algebra M2 M1 by default
892   show ?thesis
893     using pair_sigma_algebra_measurable[of "\<lambda>(x,y). f (y, x)"]
894     by (auto intro!: pair_sigma_algebra_measurable Q.pair_sigma_algebra_measurable iffI)
895 qed
897 lemma (in pair_sigma_finite) integrable_product_swap:
898   "measure_space.integrable
899     (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) f \<longleftrightarrow>
900   integrable (\<lambda>(x,y). f (y,x))"
901 proof -
902   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
903   show ?thesis
904     unfolding Q.integrable_def integrable_def
905     unfolding positive_integral_product_swap
906     unfolding measurable_product_swap
907     by (simp add: case_prod_distrib)
908 qed
910 lemma (in pair_sigma_finite) integral_product_swap:
911   "measure_space.integral
912     (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) f =
913   integral (\<lambda>(x,y). f (y,x))"
914 proof -
915   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
916   show ?thesis
917     unfolding integral_def Q.integral_def positive_integral_product_swap
918     by (simp add: case_prod_distrib)
919 qed
921 lemma (in pair_sigma_finite) integrable_fst_measurable:
922   assumes f: "integrable f"
923   shows "M1.almost_everywhere (\<lambda>x. M2.integrable (\<lambda> y. f (x, y)))" (is "?AE")
924     and "M1.integral (\<lambda> x. M2.integral (\<lambda> y. f (x, y))) = integral f" (is "?INT")
925 proof -
926   let "?pf x" = "Real (f x)" and "?nf x" = "Real (- f x)"
927   have
928     borel: "?nf \<in> borel_measurable P""?pf \<in> borel_measurable P" and
929     int: "positive_integral ?nf \<noteq> \<omega>" "positive_integral ?pf \<noteq> \<omega>"
930     using assms by auto
931   have "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. Real (f (x, y)))) \<noteq> \<omega>"
932      "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. Real (- f (x, y)))) \<noteq> \<omega>"
933     using borel[THEN positive_integral_fst_measurable(1)] int
934     unfolding borel[THEN positive_integral_fst_measurable(2)] by simp_all
935   with borel[THEN positive_integral_fst_measurable(1)]
936   have AE: "M1.almost_everywhere (\<lambda>x. M2.positive_integral (\<lambda>y. Real (f (x, y))) \<noteq> \<omega>)"
937     "M1.almost_everywhere (\<lambda>x. M2.positive_integral (\<lambda>y. Real (- f (x, y))) \<noteq> \<omega>)"
938     by (auto intro!: M1.positive_integral_omega_AE)
939   then show ?AE
940     apply (rule M1.AE_mp[OF _ M1.AE_mp])
941     apply (rule M1.AE_cong)
942     using assms unfolding M2.integrable_def
943     by (auto intro!: measurable_pair_image_snd)
944   have "M1.integrable
945      (\<lambda>x. real (M2.positive_integral (\<lambda>xa. Real (f (x, xa)))))" (is "M1.integrable ?f")
946   proof (unfold M1.integrable_def, intro conjI)
947     show "?f \<in> borel_measurable M1"
948       using borel by (auto intro!: M1.borel_measurable_real positive_integral_fst_measurable)
949     have "M1.positive_integral (\<lambda>x. Real (?f x)) =
950         M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>xa. Real (f (x, xa))))"
951       apply (rule M1.positive_integral_cong_AE)
952       apply (rule M1.AE_mp[OF AE(1)])
953       apply (rule M1.AE_cong)
954       by (auto simp: Real_real)
955     then show "M1.positive_integral (\<lambda>x. Real (?f x)) \<noteq> \<omega>"
956       using positive_integral_fst_measurable[OF borel(2)] int by simp
957     have "M1.positive_integral (\<lambda>x. Real (- ?f x)) = M1.positive_integral (\<lambda>x. 0)"
958       by (intro M1.positive_integral_cong) simp
959     then show "M1.positive_integral (\<lambda>x. Real (- ?f x)) \<noteq> \<omega>" by simp
960   qed
961   moreover have "M1.integrable
962      (\<lambda>x. real (M2.positive_integral (\<lambda>xa. Real (- f (x, xa)))))" (is "M1.integrable ?f")
963   proof (unfold M1.integrable_def, intro conjI)
964     show "?f \<in> borel_measurable M1"
965       using borel by (auto intro!: M1.borel_measurable_real positive_integral_fst_measurable)
966     have "M1.positive_integral (\<lambda>x. Real (?f x)) =
967         M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>xa. Real (- f (x, xa))))"
968       apply (rule M1.positive_integral_cong_AE)
969       apply (rule M1.AE_mp[OF AE(2)])
970       apply (rule M1.AE_cong)
971       by (auto simp: Real_real)
972     then show "M1.positive_integral (\<lambda>x. Real (?f x)) \<noteq> \<omega>"
973       using positive_integral_fst_measurable[OF borel(1)] int by simp
974     have "M1.positive_integral (\<lambda>x. Real (- ?f x)) = M1.positive_integral (\<lambda>x. 0)"
975       by (intro M1.positive_integral_cong) simp
976     then show "M1.positive_integral (\<lambda>x. Real (- ?f x)) \<noteq> \<omega>" by simp
977   qed
978   ultimately show ?INT
979     unfolding M2.integral_def integral_def
980       borel[THEN positive_integral_fst_measurable(2), symmetric]
981     by (simp add: M1.integral_real[OF AE(1)] M1.integral_real[OF AE(2)])
982 qed
984 lemma (in pair_sigma_finite) integrable_snd_measurable:
985   assumes f: "integrable f"
986   shows "M2.almost_everywhere (\<lambda>y. M1.integrable (\<lambda>x. f (x, y)))" (is "?AE")
987     and "M2.integral (\<lambda>y. M1.integral (\<lambda>x. f (x, y))) = integral f" (is "?INT")
988 proof -
989   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
990   have Q_int: "Q.integrable (\<lambda>(x, y). f (y, x))"
991     using f unfolding integrable_product_swap by simp
992   show ?INT
993     using Q.integrable_fst_measurable(2)[OF Q_int]
994     unfolding integral_product_swap by simp
995   show ?AE
996     using Q.integrable_fst_measurable(1)[OF Q_int]
997     by simp
998 qed
1000 lemma (in pair_sigma_finite) Fubini_integral:
1001   assumes f: "integrable f"
1002   shows "M2.integral (\<lambda>y. M1.integral (\<lambda>x. f (x, y))) =
1003       M1.integral (\<lambda>x. M2.integral (\<lambda>y. f (x, y)))"
1004   unfolding integrable_snd_measurable[OF assms]
1005   unfolding integrable_fst_measurable[OF assms] ..
1007 section "Finite product spaces"
1009 section "Products"
1011 locale product_sigma_algebra =
1012   fixes M :: "'i \<Rightarrow> 'a algebra"
1013   assumes sigma_algebras: "\<And>i. sigma_algebra (M i)"
1015 locale finite_product_sigma_algebra = product_sigma_algebra M for M :: "'i \<Rightarrow> 'a algebra" +
1016   fixes I :: "'i set"
1017   assumes finite_index: "finite I"
1019 syntax
1020   "_PiE"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PIE _:_./ _)" 10)
1022 syntax (xsymbols)
1023   "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
1025 syntax (HTML output)
1026   "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
1028 translations
1029   "PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)"
1031 definition
1032   "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>"
1034 abbreviation (in finite_product_sigma_algebra) "G \<equiv> product_algebra M I"
1035 abbreviation (in finite_product_sigma_algebra) "P \<equiv> sigma G"
1037 sublocale product_sigma_algebra \<subseteq> M: sigma_algebra "M i" for i by (rule sigma_algebras)
1039 lemma (in finite_product_sigma_algebra) product_algebra_into_space:
1040   "sets G \<subseteq> Pow (space G)"
1041   using M.sets_into_space unfolding product_algebra_def
1042   by auto blast
1044 sublocale finite_product_sigma_algebra \<subseteq> sigma_algebra P
1045   using product_algebra_into_space by (rule sigma_algebra_sigma)
1047 lemma product_algebraE:
1048   assumes "A \<in> sets (product_algebra M I)"
1049   obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
1050   using assms unfolding product_algebra_def by auto
1052 lemma product_algebraI[intro]:
1053   assumes "E \<in> (\<Pi> i\<in>I. sets (M i))"
1054   shows "Pi\<^isub>E I E \<in> sets (product_algebra M I)"
1055   using assms unfolding product_algebra_def by auto
1057 lemma space_product_algebra[simp]:
1058   "space (product_algebra M I) = Pi\<^isub>E I (\<lambda>i. space (M i))"
1059   unfolding product_algebra_def by simp
1061 lemma product_algebra_sets_into_space:
1062   assumes "\<And>i. i\<in>I \<Longrightarrow> sets (M i) \<subseteq> Pow (space (M i))"
1063   shows "sets (product_algebra M I) \<subseteq> Pow (space (product_algebra M I))"
1064   using assms by (auto simp: product_algebra_def) blast
1066 lemma (in finite_product_sigma_algebra) P_empty:
1067   "I = {} \<Longrightarrow> P = \<lparr> space = {\<lambda>k. undefined}, sets = { {}, {\<lambda>k. undefined} }\<rparr>"
1068   unfolding product_algebra_def by (simp add: sigma_def image_constant)
1070 lemma (in finite_product_sigma_algebra) in_P[simp, intro]:
1071   "\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P"
1072   by (auto simp: product_algebra_def sets_sigma intro!: sigma_sets.Basic)
1074 lemma (in product_sigma_algebra) bij_inv_restrict_merge:
1075   assumes [simp]: "I \<inter> J = {}"
1076   shows "bij_inv
1077     (space (sigma (product_algebra M (I \<union> J))))
1078     (space (sigma (pair_algebra (product_algebra M I) (product_algebra M J))))
1079     (\<lambda>x. (restrict x I, restrict x J)) (\<lambda>(x, y). merge I x J y)"
1080   by (rule bij_invI) (auto simp: space_pair_algebra extensional_restrict)
1082 lemma (in product_sigma_algebra) bij_inv_singleton:
1083   "bij_inv (space (sigma (product_algebra M {i}))) (space (M i))
1084     (\<lambda>x. x i) (\<lambda>x. (\<lambda>j\<in>{i}. x))"
1085   by (rule bij_invI) (auto simp: restrict_def extensional_def fun_eq_iff)
1087 lemma (in product_sigma_algebra) bij_inv_restrict_insert:
1088   assumes [simp]: "i \<notin> I"
1089   shows "bij_inv
1090     (space (sigma (product_algebra M (insert i I))))
1091     (space (sigma (pair_algebra (product_algebra M I) (M i))))
1092     (\<lambda>x. (restrict x I, x i)) (\<lambda>(x, y). x(i := y))"
1093   by (rule bij_invI) (auto simp: space_pair_algebra extensional_restrict)
1095 lemma (in product_sigma_algebra) measurable_restrict_on_generating:
1096   assumes [simp]: "I \<inter> J = {}"
1097   shows "(\<lambda>x. (restrict x I, restrict x J)) \<in> measurable
1098     (product_algebra M (I \<union> J))
1099     (pair_algebra (product_algebra M I) (product_algebra M J))"
1100     (is "?R \<in> measurable ?IJ ?P")
1101 proof (unfold measurable_def, intro CollectI conjI ballI)
1102   show "?R \<in> space ?IJ \<rightarrow> space ?P" by (auto simp: space_pair_algebra)
1103   { fix F E assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "F \<in> (\<Pi> i\<in>J. sets (M i))"
1104     then have "Pi (I \<union> J) (merge I E J F) \<inter> (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) =
1105         Pi\<^isub>E (I \<union> J) (merge I E J F)"
1106       using M.sets_into_space by auto blast+ }
1107   note this[simp]
1108   show "\<And>A. A \<in> sets ?P \<Longrightarrow> ?R - A \<inter> space ?IJ \<in> sets ?IJ"
1109     by (force elim!: pair_algebraE product_algebraE
1110               simp del: vimage_Int simp: restrict_vimage merge_vimage space_pair_algebra)
1111   qed
1113 lemma (in product_sigma_algebra) measurable_merge_on_generating:
1114   assumes [simp]: "I \<inter> J = {}"
1115   shows "(\<lambda>(x, y). merge I x J y) \<in> measurable
1116     (pair_algebra (product_algebra M I) (product_algebra M J))
1117     (product_algebra M (I \<union> J))"
1118     (is "?M \<in> measurable ?P ?IJ")
1119 proof (unfold measurable_def, intro CollectI conjI ballI)
1120   show "?M \<in> space ?P \<rightarrow> space ?IJ" by (auto simp: space_pair_algebra)
1121   { fix E assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "E \<in> (\<Pi> i\<in>J. sets (M i))"
1122     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)) =
1123         Pi\<^isub>E I E \<times> Pi\<^isub>E J E"
1124       using M.sets_into_space by auto blast+ }
1125   note this[simp]
1126   show "\<And>A. A \<in> sets ?IJ \<Longrightarrow> ?M - A \<inter> space ?P \<in> sets ?P"
1127     by (force elim!: pair_algebraE product_algebraE
1128               simp del: vimage_Int simp: restrict_vimage merge_vimage space_pair_algebra)
1129   qed
1131 lemma (in product_sigma_algebra) measurable_singleton_on_generator:
1132   "(\<lambda>x. \<lambda>j\<in>{i}. x) \<in> measurable (M i) (product_algebra M {i})"
1133   (is "?f \<in> measurable _ ?P")
1134 proof (unfold measurable_def, intro CollectI conjI)
1135   show "?f \<in> space (M i) \<rightarrow> space ?P" by auto
1136   have "\<And>E. E i \<in> sets (M i) \<Longrightarrow> ?f - Pi\<^isub>E {i} E \<inter> space (M i) = E i"
1137     using M.sets_into_space by auto
1138   then show "\<forall>A \<in> sets ?P. ?f - A \<inter> space (M i) \<in> sets (M i)"
1139     by (auto elim!: product_algebraE)
1140 qed
1142 lemma (in product_sigma_algebra) measurable_component_on_generator:
1143   assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (product_algebra M I) (M i)"
1144   (is "?f \<in> measurable ?P _")
1145 proof (unfold measurable_def, intro CollectI conjI ballI)
1146   show "?f \<in> space ?P \<rightarrow> space (M i)" using i \<in> I by auto
1147   fix A assume "A \<in> sets (M i)"
1148   moreover then have "(\<lambda>x. x i) - A \<inter> (\<Pi>\<^isub>E i\<in>I. space (M i)) =
1149       (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))"
1150     using M.sets_into_space i \<in> I
1151     by (fastsimp dest: Pi_mem split: split_if_asm)
1152   ultimately show "?f - A \<inter> space ?P \<in> sets ?P"
1153     by (auto intro!: product_algebraI)
1154 qed
1156 lemma (in product_sigma_algebra) measurable_restrict_singleton_on_generating:
1157   assumes [simp]: "i \<notin> I"
1158   shows "(\<lambda>x. (restrict x I, x i)) \<in> measurable
1159     (product_algebra M (insert i I))
1160     (pair_algebra (product_algebra M I) (M i))"
1161     (is "?R \<in> measurable ?I ?P")
1162 proof (unfold measurable_def, intro CollectI conjI ballI)
1163   show "?R \<in> space ?I \<rightarrow> space ?P" by (auto simp: space_pair_algebra)
1164   { fix E F assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "F \<in> sets (M i)"
1165     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)) =
1166         Pi\<^isub>E (insert i I) (E(i := F))"
1167       using M.sets_into_space using i\<notin>I by (auto simp: restrict_Pi_cancel) blast+ }
1168   note this[simp]
1169   show "\<And>A. A \<in> sets ?P \<Longrightarrow> ?R - A \<inter> space ?I \<in> sets ?I"
1170     by (force elim!: pair_algebraE product_algebraE
1171               simp del: vimage_Int simp: space_pair_algebra)
1172 qed
1174 lemma (in product_sigma_algebra) measurable_merge_singleton_on_generating:
1175   assumes [simp]: "i \<notin> I"
1176   shows "(\<lambda>(x, y). x(i := y)) \<in> measurable
1177     (pair_algebra (product_algebra M I) (M i))
1178     (product_algebra M (insert i I))"
1179     (is "?M \<in> measurable ?P ?IJ")
1180 proof (unfold measurable_def, intro CollectI conjI ballI)
1181   show "?M \<in> space ?P \<rightarrow> space ?IJ" by (auto simp: space_pair_algebra)
1182   { fix E assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "E i \<in> sets (M i)"
1183     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) =
1184         Pi\<^isub>E I E \<times> E i"
1185       using M.sets_into_space by auto blast+ }
1186   note this[simp]
1187   show "\<And>A. A \<in> sets ?IJ \<Longrightarrow> ?M - A \<inter> space ?P \<in> sets ?P"
1188     by (force elim!: pair_algebraE product_algebraE
1189               simp del: vimage_Int simp: space_pair_algebra)
1190 qed
1192 section "Generating set generates also product algebra"
1194 lemma pair_sigma_algebra_sigma:
1195   assumes 1: "S1 \<up> (space E1)" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"
1196   assumes 2: "S2 \<up> (space E2)" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"
1197   shows "sigma (pair_algebra (sigma E1) (sigma E2)) = sigma (pair_algebra E1 E2)"
1198     (is "?S = ?E")
1199 proof -
1200   interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma)
1201   interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma)
1202   have P: "sets (pair_algebra E1 E2) \<subseteq> Pow (space E1 \<times> space E2)"
1203     using E1 E2 by (auto simp add: pair_algebra_def)
1204   interpret E: sigma_algebra ?E unfolding pair_algebra_def
1205     using E1 E2 by (intro sigma_algebra_sigma) auto
1206   { fix A assume "A \<in> sets E1"
1207     then have "fst - A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"
1208       using E1 2 unfolding isoton_def pair_algebra_def by auto
1209     also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto
1210     also have "\<dots> \<in> sets ?E" unfolding pair_algebra_def sets_sigma
1211       using 2 A \<in> sets E1
1212       by (intro sigma_sets.Union)
1213          (auto simp: image_subset_iff intro!: sigma_sets.Basic)
1214     finally have "fst - A \<inter> space ?E \<in> sets ?E" . }
1215   moreover
1216   { fix B assume "B \<in> sets E2"
1217     then have "snd - B \<inter> space ?E = (\<Union>i. S1 i) \<times> B"
1218       using E2 1 unfolding isoton_def pair_algebra_def by auto
1219     also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto
1220     also have "\<dots> \<in> sets ?E"
1221       using 1 B \<in> sets E2 unfolding pair_algebra_def sets_sigma
1222       by (intro sigma_sets.Union)
1223          (auto simp: image_subset_iff intro!: sigma_sets.Basic)
1224     finally have "snd - B \<inter> space ?E \<in> sets ?E" . }
1225   ultimately have proj:
1226     "fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"
1227     using E1 E2 by (subst (1 2) E.measurable_iff_sigma)
1228                    (auto simp: pair_algebra_def sets_sigma)
1229   { fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)"
1230     with proj have "fst - A \<inter> space ?E \<in> sets ?E" "snd - B \<inter> space ?E \<in> sets ?E"
1231       unfolding measurable_def by simp_all
1232     moreover have "A \<times> B = (fst - A \<inter> space ?E) \<inter> (snd - B \<inter> space ?E)"
1233       using A B M1.sets_into_space M2.sets_into_space
1234       by (auto simp: pair_algebra_def)
1235     ultimately have "A \<times> B \<in> sets ?E" by auto }
1236   then have "sigma_sets (space ?E) (sets (pair_algebra (sigma E1) (sigma E2))) \<subseteq> sets ?E"
1237     by (intro E.sigma_sets_subset) (auto simp add: pair_algebra_def sets_sigma)
1238   then have subset: "sets ?S \<subseteq> sets ?E"
1239     by (simp add: sets_sigma pair_algebra_def)
1240   have "sets ?S = sets ?E"
1241   proof (intro set_eqI iffI)
1242     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
1243       unfolding sets_sigma
1244     proof induct
1245       case (Basic A) then show ?case
1246         by (auto simp: pair_algebra_def sets_sigma intro: sigma_sets.Basic)
1247     qed (auto intro: sigma_sets.intros simp: pair_algebra_def)
1248   next
1249     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
1250   qed
1251   then show ?thesis
1252     by (simp add: pair_algebra_def sigma_def)
1253 qed
1255 lemma sigma_product_algebra_sigma_eq:
1256   assumes "finite I"
1257   assumes isotone: "\<And>i. i \<in> I \<Longrightarrow> (S i) \<up> (space (E i))"
1258   assumes sets_into: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> sets (E i)"
1259   and E: "\<And>i. sets (E i) \<subseteq> Pow (space (E i))"
1260   shows "sigma (product_algebra (\<lambda>i. sigma (E i)) I) = sigma (product_algebra E I)"
1261     (is "?S = ?E")
1262 proof cases
1263   assume "I = {}" then show ?thesis by (simp add: product_algebra_def)
1264 next
1265   assume "I \<noteq> {}"
1266   interpret E: sigma_algebra "sigma (E i)" for i
1267     using E by (rule sigma_algebra_sigma)
1269   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)"
1270     using E by auto
1272   interpret G: sigma_algebra ?E
1273     unfolding product_algebra_def using E
1274     by (intro sigma_algebra_sigma) (auto dest: Pi_mem)
1276   { fix A i assume "i \<in> I" and A: "A \<in> sets (E i)"
1277     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"
1278       using isotone unfolding isoton_def product_algebra_def by (auto dest: Pi_mem)
1279     also have "\<dots> = (\<Union>n. (\<Pi>\<^isub>E j\<in>I. if j = i then A else S j n))"
1280       unfolding product_algebra_def
1281       apply simp
1282       apply (subst Pi_UN[OF finite I])
1283       using isotone[THEN isoton_mono_le] apply simp
1284       apply (simp add: PiE_Int)
1285       apply (intro PiE_cong)
1286       using A sets_into by (auto intro!: into_space)
1287     also have "\<dots> \<in> sets ?E" unfolding product_algebra_def sets_sigma
1288       using sets_into A \<in> sets (E i)
1289       by (intro sigma_sets.Union)
1290          (auto simp: image_subset_iff intro!: sigma_sets.Basic)
1291     finally have "(\<lambda>x. x i) - A \<inter> space ?E \<in> sets ?E" . }
1292   then have proj:
1293     "\<And>i. i\<in>I \<Longrightarrow> (\<lambda>x. x i) \<in> measurable ?E (sigma (E i))"
1294     using E by (subst G.measurable_iff_sigma)
1295                (auto simp: product_algebra_def sets_sigma)
1297   { fix A assume A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (sigma (E i))"
1298     with proj have basic: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) - (A i) \<inter> space ?E \<in> sets ?E"
1299       unfolding measurable_def by simp
1300     have "Pi\<^isub>E I A = (\<Inter>i\<in>I. (\<lambda>x. x i) - (A i) \<inter> space ?E)"
1301       using A E.sets_into_space I \<noteq> {} unfolding product_algebra_def by auto blast
1302     then have "Pi\<^isub>E I A \<in> sets ?E"
1303       using G.finite_INT[OF finite I I \<noteq> {} basic, of "\<lambda>i. i"] by simp }
1304   then have "sigma_sets (space ?E) (sets (product_algebra (\<lambda>i. sigma (E i)) I)) \<subseteq> sets ?E"
1305     by (intro G.sigma_sets_subset) (auto simp add: sets_sigma product_algebra_def)
1306   then have subset: "sets ?S \<subseteq> sets ?E"
1307     by (simp add: sets_sigma product_algebra_def)
1309   have "sets ?S = sets ?E"
1310   proof (intro set_eqI iffI)
1311     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
1312       unfolding sets_sigma
1313     proof induct
1314       case (Basic A) then show ?case
1315         by (auto simp: sets_sigma product_algebra_def intro: sigma_sets.Basic)
1316     qed (auto intro: sigma_sets.intros simp: product_algebra_def)
1317   next
1318     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
1319   qed
1320   then show ?thesis
1321     by (simp add: product_algebra_def sigma_def)
1322 qed
1324 lemma (in product_sigma_algebra) sigma_pair_algebra_sigma_eq:
1325   "sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))) =
1326    sigma (pair_algebra (product_algebra M I) (product_algebra M J))"
1327   using M.sets_into_space
1328   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)"])
1329      (auto simp: isoton_const product_algebra_def, blast+)
1331 lemma (in product_sigma_algebra) sigma_pair_algebra_product_singleton:
1332   "sigma (pair_algebra (sigma (product_algebra M I)) (M i)) =
1333    sigma (pair_algebra (product_algebra M I) (M i))"
1334   using M.sets_into_space apply (subst M.sigma_eq[symmetric])
1335   by (intro pair_sigma_algebra_sigma[of "\<lambda>_. \<Pi>\<^isub>E i\<in>I. space (M i)" _ "\<lambda>_. space (M i)"])
1336      (auto simp: isoton_const product_algebra_def, blast+)
1338 lemma (in product_sigma_algebra) measurable_restrict:
1339   assumes [simp]: "I \<inter> J = {}"
1340   shows "(\<lambda>x. (restrict x I, restrict x J)) \<in> measurable
1341     (sigma (product_algebra M (I \<union> J)))
1342     (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))"
1343   unfolding sigma_pair_algebra_sigma_eq using M.sets_into_space
1344   by (intro measurable_sigma_sigma measurable_restrict_on_generating
1345             pair_algebra_sets_into_space product_algebra_sets_into_space)
1346      auto
1348 lemma (in product_sigma_algebra) measurable_merge:
1349   assumes [simp]: "I \<inter> J = {}"
1350   shows "(\<lambda>(x, y). merge I x J y) \<in> measurable
1351     (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))
1352     (sigma (product_algebra M (I \<union> J)))"
1353   unfolding sigma_pair_algebra_sigma_eq using M.sets_into_space
1354   by (intro measurable_sigma_sigma measurable_merge_on_generating
1355             pair_algebra_sets_into_space product_algebra_sets_into_space)
1356      auto
1358 lemma (in product_sigma_algebra) product_product_vimage_algebra:
1359   assumes [simp]: "I \<inter> J = {}"
1360   shows "sigma_algebra.vimage_algebra
1361     (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))
1362     (space (sigma (product_algebra M (I \<union> J)))) (\<lambda>x. ((\<lambda>i\<in>I. x i), (\<lambda>i\<in>J. x i))) =
1363     sigma (product_algebra M (I \<union> J))"
1364   unfolding sigma_pair_algebra_sigma_eq using sets_into_space
1365   by (intro vimage_algebra_sigma[OF bij_inv_restrict_merge]
1366             pair_algebra_sets_into_space product_algebra_sets_into_space
1367             measurable_merge_on_generating measurable_restrict_on_generating)
1368      auto
1370 lemma (in product_sigma_algebra) pair_product_product_vimage_algebra:
1371   assumes [simp]: "I \<inter> J = {}"
1372   shows "sigma_algebra.vimage_algebra (sigma (product_algebra M (I \<union> J)))
1373     (space (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))) (\<lambda>(x,y). merge I x J y) =
1374     (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))"
1375   unfolding sigma_pair_algebra_sigma_eq using sets_into_space
1376   by (intro vimage_algebra_sigma[OF bij_inv_restrict_merge[symmetric]]
1377             pair_algebra_sets_into_space product_algebra_sets_into_space
1378             measurable_merge_on_generating measurable_restrict_on_generating)
1379      auto
1381 lemma (in product_sigma_algebra) pair_product_singleton_vimage_algebra:
1382   assumes [simp]: "i \<notin> I"
1383   shows "sigma_algebra.vimage_algebra (sigma (pair_algebra (sigma (product_algebra M I)) (M i)))
1384     (space (sigma (product_algebra M (insert i I)))) (\<lambda>x. (restrict x I, x i)) =
1385     (sigma (product_algebra M (insert i I)))"
1386   unfolding sigma_pair_algebra_product_singleton using sets_into_space
1387   by (intro vimage_algebra_sigma[OF bij_inv_restrict_insert]
1388             pair_algebra_sets_into_space product_algebra_sets_into_space
1389             measurable_merge_singleton_on_generating measurable_restrict_singleton_on_generating)
1390       auto
1392 lemma (in product_sigma_algebra) singleton_vimage_algebra:
1393   "sigma_algebra.vimage_algebra (sigma (product_algebra M {i})) (space (M i)) (\<lambda>x. \<lambda>j\<in>{i}. x) = M i"
1394   using sets_into_space
1395   by (intro vimage_algebra_sigma[of "M i", unfolded M.sigma_eq, OF bij_inv_singleton[symmetric]]
1396              product_algebra_sets_into_space measurable_singleton_on_generator measurable_component_on_generator)
1397      auto
1399 lemma (in product_sigma_algebra) measurable_restrict_iff:
1400   assumes IJ[simp]: "I \<inter> J = {}"
1401   shows "f \<in> measurable (sigma (pair_algebra
1402       (sigma (product_algebra M I)) (sigma (product_algebra M J)))) M' \<longleftrightarrow>
1403     (\<lambda>x. f (restrict x I, restrict x J)) \<in> measurable (sigma (product_algebra M (I \<union> J))) M'"
1404   using M.sets_into_space
1405   apply (subst pair_product_product_vimage_algebra[OF IJ, symmetric])
1406   apply (subst sigma_pair_algebra_sigma_eq)
1407   apply (subst sigma_algebra.measurable_vimage_iff_inv[OF _
1408       bij_inv_restrict_merge[symmetric]])
1409   apply (intro sigma_algebra_sigma product_algebra_sets_into_space)
1410   by auto
1412 lemma (in product_sigma_algebra) measurable_merge_iff:
1413   assumes IJ: "I \<inter> J = {}"
1414   shows "f \<in> measurable (sigma (product_algebra M (I \<union> J))) M' \<longleftrightarrow>
1415     (\<lambda>(x, y). f (merge I x J y)) \<in>
1416       measurable (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J)))) M'"
1417   unfolding measurable_restrict_iff[OF IJ]
1418   by (rule measurable_cong) (auto intro!: arg_cong[where f=f] simp: extensional_restrict)
1420 lemma (in product_sigma_algebra) measurable_component:
1421   assumes "i \<in> I" and f: "f \<in> measurable (M i) M'"
1422   shows "(\<lambda>x. f (x i)) \<in> measurable (sigma (product_algebra M I)) M'"
1423     (is "?f \<in> measurable ?P M'")
1424 proof -
1425   have "f \<circ> (\<lambda>x. x i) \<in> measurable ?P M'"
1426     apply (rule measurable_comp[OF _ f])
1427     using measurable_up_sigma[of "product_algebra M I" "M i"]
1428     using measurable_component_on_generator[OF i \<in> I]
1429     by auto
1430   then show "?f \<in> measurable ?P M'" by (simp add: comp_def)
1431 qed
1433 lemma (in product_sigma_algebra) measurable_component_singleton:
1434   "(\<lambda>x. f (x i)) \<in> measurable (sigma (product_algebra M {i})) M' \<longleftrightarrow>
1435     f \<in> measurable (M i) M'"
1436   using sets_into_space
1437   apply (subst singleton_vimage_algebra[symmetric])
1438   apply (subst sigma_algebra.measurable_vimage_iff_inv[OF _ bij_inv_singleton[symmetric]])
1439   by (auto intro!: sigma_algebra_sigma product_algebra_sets_into_space)
1441 lemma (in product_sigma_algebra) measurable_component_iff:
1442   assumes "i \<in> I" and not_empty: "\<forall>i\<in>I. space (M i) \<noteq> {}"
1443   shows "(\<lambda>x. f (x i)) \<in> measurable (sigma (product_algebra M I)) M' \<longleftrightarrow>
1444     f \<in> measurable (M i) M'"
1445     (is "?f \<in> measurable ?P M' \<longleftrightarrow> _")
1446 proof
1447   assume "f \<in> measurable (M i) M'" then show "?f \<in> measurable ?P M'"
1448     by (rule measurable_component[OF i \<in> I])
1449 next
1450   assume f: "?f \<in> measurable ?P M'"
1451   def t \<equiv> "\<lambda>i. SOME x. x \<in> space (M i)"
1452   have t: "\<And>i. i\<in>I \<Longrightarrow> t i \<in> space (M i)"
1453      unfolding t_def using not_empty by (rule_tac someI_ex) auto
1454   have "?f \<circ> (\<lambda>x. (\<lambda>j\<in>I. if j = i then x else t j)) \<in> measurable (M i) M'"
1455     (is "?f \<circ> ?t \<in> measurable _ _")
1456   proof (rule measurable_comp[OF _ f])
1457     have "?t \<in> measurable (M i) (product_algebra M I)"
1458     proof (unfold measurable_def, intro CollectI conjI ballI)
1459       from t show "?t \<in> space (M i) \<rightarrow> (space (product_algebra M I))" by auto
1460     next
1461       fix A assume A: "A \<in> sets (product_algebra M I)"
1462       { fix E assume "E \<in> (\<Pi> i\<in>I. sets (M i))"
1463         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 {})"
1464           using i \<in> I sets_into_space by (auto dest: Pi_mem[where B=E]) }
1465       note * = this
1466       with A i \<in> I show "?t - A \<inter> space (M i) \<in> sets (M i)"
1467         by (auto elim!: product_algebraE simp del: vimage_Int)
1468     qed
1469     also have "\<dots> \<subseteq> measurable (M i) (sigma (product_algebra M I))"
1470       using M.sets_into_space by (intro measurable_subset) (auto simp: product_algebra_def, blast)
1471     finally show "?t \<in> measurable (M i) (sigma (product_algebra M I))" .
1472   qed
1473   then show "f \<in> measurable (M i) M'" unfolding comp_def using i \<in> I by simp
1474 qed
1476 lemma (in product_sigma_algebra) measurable_singleton:
1477   shows "f \<in> measurable (sigma (product_algebra M {i})) M' \<longleftrightarrow>
1478     (\<lambda>x. f (\<lambda>j\<in>{i}. x)) \<in> measurable (M i) M'"
1479   using sets_into_space unfolding measurable_component_singleton[symmetric]
1480   by (auto intro!: measurable_cong arg_cong[where f=f] simp: fun_eq_iff extensional_def)
1482 locale product_sigma_finite =
1483   fixes M :: "'i \<Rightarrow> 'a algebra" and \<mu> :: "'i \<Rightarrow> 'a set \<Rightarrow> pextreal"
1484   assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i) (\<mu> i)"
1486 locale finite_product_sigma_finite = product_sigma_finite M \<mu> for M :: "'i \<Rightarrow> 'a algebra" and \<mu> +
1487   fixes I :: "'i set" assumes finite_index': "finite I"
1489 sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" "\<mu> i" for i
1490   by (rule sigma_finite_measures)
1492 sublocale product_sigma_finite \<subseteq> product_sigma_algebra
1493   by default
1495 sublocale finite_product_sigma_finite \<subseteq> finite_product_sigma_algebra
1496   by default (fact finite_index')
1498 lemma (in finite_product_sigma_finite) sigma_finite_pairs:
1499   "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
1500     (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
1501     (\<forall>k. \<forall>i\<in>I. \<mu> i (F i k) \<noteq> \<omega>) \<and>
1502     (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<up> space G"
1503 proof -
1504   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>)"
1505     using M.sigma_finite_up by simp
1506   from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
1507   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>"
1508     by auto
1509   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"
1510   note space_product_algebra[simp]
1511   show ?thesis
1512   proof (intro exI[of _ F] conjI allI isotoneI set_eqI iffI ballI)
1513     fix i show "range (F i) \<subseteq> sets (M i)" by fact
1514   next
1515     fix i k show "\<mu> i (F i k) \<noteq> \<omega>" by fact
1516   next
1517     fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space G"
1518       using \<And>i. range (F i) \<subseteq> sets (M i) M.sets_into_space by auto blast
1519   next
1520     fix f assume "f \<in> space G"
1521     with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"]
1522       \<And>i. F i \<up> space (M i)[THEN isotonD(2)]
1523       \<And>i. F i \<up> space (M i)[THEN isoton_mono_le]
1524     show "f \<in> (\<Union>i. ?F i)" by auto
1525   next
1526     fix i show "?F i \<subseteq> ?F (Suc i)"
1527       using \<And>i. F i \<up> space (M i)[THEN isotonD(1)] by auto
1528   qed
1529 qed
1531 lemma (in product_sigma_finite) product_measure_exists:
1532   assumes "finite I"
1533   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>
1534      sigma_finite_measure (sigma (product_algebra M I)) \<nu>"
1535 using finite I proof induct
1536   case empty then show ?case unfolding product_algebra_def
1537     by (auto intro!: exI[of _ "\<lambda>A. if A = {} then 0 else 1"] sigma_algebra_sigma
1539              simp add: positive_def additive_def sets_sigma sigma_finite_measure_def
1540                        sigma_finite_measure_axioms_def image_constant)
1541 next
1542   case (insert i I)
1543   interpret finite_product_sigma_finite M \<mu> I by default fact
1544   have "finite (insert i I)" using finite I by auto
1545   interpret I': finite_product_sigma_finite M \<mu> "insert i I" by default fact
1546   from insert obtain \<nu> where
1547     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
1548     "sigma_finite_measure P \<nu>" by auto
1549   interpret I: sigma_finite_measure P \<nu> by fact
1550   interpret P: pair_sigma_finite P \<nu> "M i" "\<mu> i" ..
1552   let ?h = "\<lambda>x. (restrict x I, x i)"
1553   let ?\<nu> = "\<lambda>A. P.pair_measure (?h  A)"
1554   interpret I': measure_space "sigma (product_algebra M (insert i I))" ?\<nu>
1555     apply (subst pair_product_singleton_vimage_algebra[OF i \<notin> I, symmetric])
1556     apply (intro P.measure_space_isomorphic bij_inv_bij_betw)
1557     unfolding sigma_pair_algebra_product_singleton
1558     by (rule bij_inv_restrict_insert[OF i \<notin> I])
1559   show ?case
1560   proof (intro exI[of _ ?\<nu>] conjI ballI)
1561     { fix A assume A: "A \<in> (\<Pi> i\<in>insert i I. sets (M i))"
1562       moreover then have "A \<in> (\<Pi> i\<in>I. sets (M i))" by auto
1563       moreover have "(\<lambda>x. (restrict x I, x i))  Pi\<^isub>E (insert i I) A = Pi\<^isub>E I A \<times> A i"
1564         using i \<notin> I
1565         apply auto
1566         apply (rule_tac x="a(i:=b)" in image_eqI)
1567         apply (auto simp: extensional_def fun_eq_iff)
1568         done
1569       ultimately show "?\<nu> (Pi\<^isub>E (insert i I) A) = (\<Prod>i\<in>insert i I. \<mu> i (A i))"
1570         apply simp
1571         apply (subst P.pair_measure_times)
1572         apply fastsimp
1573         apply fastsimp
1574         using i \<notin> I finite I prod[of A] by (auto simp: ac_simps) }
1575     note product = this
1576     show "sigma_finite_measure I'.P ?\<nu>"
1577     proof
1578       from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
1579       then have F: "\<And>j. j \<in> insert i I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
1580         "(\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) \<up> space I'.G"
1581         "\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> \<mu> j (F j k) \<noteq> \<omega>"
1582         by blast+
1583       let "?F k" = "\<Pi>\<^isub>E j \<in> insert i I. F j k"
1584       show "\<exists>F::nat \<Rightarrow> ('i \<Rightarrow> 'a) set. range F \<subseteq> sets I'.P \<and>
1585           (\<Union>i. F i) = space I'.P \<and> (\<forall>i. ?\<nu> (F i) \<noteq> \<omega>)"
1586       proof (intro exI[of _ ?F] conjI allI)
1587         show "range ?F \<subseteq> sets I'.P" using F(1) by auto
1588       next
1589         from F(2)[THEN isotonD(2)]
1590         show "(\<Union>i. ?F i) = space I'.P" by simp
1591       next
1592         fix j
1593         show "?\<nu> (?F j) \<noteq> \<omega>"
1594           using F finite I
1595           by (subst product) (auto simp: setprod_\<omega>)
1596       qed
1597     qed
1598   qed
1599 qed
1601 definition (in finite_product_sigma_finite)
1602   measure :: "('i \<Rightarrow> 'a) set \<Rightarrow> pextreal" where
1603   "measure = (SOME \<nu>.
1604      (\<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>
1605      sigma_finite_measure P \<nu>)"
1607 sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure P measure
1608 proof -
1609   show "sigma_finite_measure P measure"
1610     unfolding measure_def
1611     by (rule someI2_ex[OF product_measure_exists[OF finite_index]]) auto
1612 qed
1614 lemma (in finite_product_sigma_finite) measure_times:
1615   assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)"
1616   shows "measure (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))"
1617 proof -
1618   note ex = product_measure_exists[OF finite_index]
1619   show ?thesis
1620     unfolding measure_def
1621   proof (rule someI2_ex[OF ex], elim conjE)
1622     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))"
1623     have "Pi\<^isub>E I A = Pi\<^isub>E I (\<lambda>i\<in>I. A i)" by (auto dest: Pi_mem)
1624     then have "\<nu> (Pi\<^isub>E I A) = \<nu> (Pi\<^isub>E I (\<lambda>i\<in>I. A i))" by simp
1625     also have "\<dots> = (\<Prod>i\<in>I. \<mu> i ((\<lambda>i\<in>I. A i) i))" using assms * by auto
1626     finally show "\<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))" by simp
1627   qed
1628 qed
1630 abbreviation (in product_sigma_finite)
1631   "product_measure I \<equiv> finite_product_sigma_finite.measure M \<mu> I"
1633 abbreviation (in product_sigma_finite)
1634   "product_positive_integral I \<equiv>
1635     measure_space.positive_integral (sigma (product_algebra M I)) (product_measure I)"
1637 abbreviation (in product_sigma_finite)
1638   "product_integral I \<equiv>
1639     measure_space.integral (sigma (product_algebra M I)) (product_measure I)"
1641 lemma (in product_sigma_finite) positive_integral_empty:
1642   "product_positive_integral {} f = f (\<lambda>k. undefined)"
1643 proof -
1644   interpret finite_product_sigma_finite M \<mu> "{}" by default (fact finite.emptyI)
1645   have "\<And>A. measure (Pi\<^isub>E {} A) = 1"
1646     using assms by (subst measure_times) auto
1647   then show ?thesis
1648     unfolding positive_integral_def simple_function_def simple_integral_def_raw
1649   proof (simp add: P_empty, intro antisym)
1650     show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> f}. f (\<lambda>k. undefined))"
1651       by (intro le_SUPI) auto
1652     show "(SUP f:{g. g \<le> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)"
1653       by (intro SUP_leI) (auto simp: le_fun_def)
1654   qed
1655 qed
1657 lemma (in product_sigma_finite) measure_fold:
1658   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
1659   assumes A: "A \<in> sets (sigma (product_algebra M (I \<union> J)))"
1660   shows "pair_sigma_finite.pair_measure
1661      (sigma (product_algebra M I)) (product_measure I)
1662      (sigma (product_algebra M J)) (product_measure J)
1663      ((\<lambda>x. ((\<lambda>i\<in>I. x i), (\<lambda>i\<in>J. x i)))  A) =
1664    product_measure (I \<union> J) A"
1665 proof -
1666   interpret I: finite_product_sigma_finite M \<mu> I by default fact
1667   interpret J: finite_product_sigma_finite M \<mu> J by default fact
1668   have "finite (I \<union> J)" using fin by auto
1669   interpret IJ: finite_product_sigma_finite M \<mu> "I \<union> J" by default fact
1670   interpret P: pair_sigma_finite I.P I.measure J.P J.measure by default
1671   let ?f = "\<lambda>x. ((\<lambda>i\<in>I. x i), (\<lambda>i\<in>J. x i))"
1672     from IJ.sigma_finite_pairs obtain F where
1673       F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
1674          "(\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) \<up> space IJ.G"
1675          "\<And>k. \<forall>i\<in>I\<union>J. \<mu> i (F i k) \<noteq> \<omega>"
1676       by auto
1677     let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"
1678   have split_f_image[simp]: "\<And>F. ?f  (Pi\<^isub>E (I \<union> J) F) = (Pi\<^isub>E I F) \<times> (Pi\<^isub>E J F)"
1679     apply auto apply (rule_tac x="merge I a J b" in image_eqI)
1680     by (auto dest: extensional_restrict)
1681     show "P.pair_measure (?f  A) = IJ.measure A"
1682   proof (rule measure_unique_Int_stable[OF _ _ _ _ _ _ _ _ A])
1683       show "Int_stable IJ.G" by (simp add: PiE_Int Int_stable_def product_algebra_def) auto
1684       show "range ?F \<subseteq> sets IJ.G" using F by (simp add: image_subset_iff product_algebra_def)
1685       show "?F \<up> space IJ.G " using F(2) by simp
1686       show "measure_space IJ.P (\<lambda>A. P.pair_measure (?f  A))"
1687       apply (subst product_product_vimage_algebra[OF IJ, symmetric])
1688       apply (intro P.measure_space_isomorphic bij_inv_bij_betw)
1689       unfolding sigma_pair_algebra_sigma_eq
1690       by (rule bij_inv_restrict_merge[OF I \<inter> J = {}])
1691       show "measure_space IJ.P IJ.measure" by fact
1692     next
1693       fix A assume "A \<in> sets IJ.G"
1694       then obtain F where A[simp]: "A = Pi\<^isub>E (I \<union> J) F" "F \<in> (\<Pi> i\<in>I \<union> J. sets (M i))"
1695         by (auto simp: product_algebra_def)
1696       then have F: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets (M i)" "\<And>i. i \<in> J \<Longrightarrow> F i \<in> sets (M i)"
1697         by auto
1698       have "P.pair_measure (?f  A) = (\<Prod>i\<in>I. \<mu> i (F i)) * (\<Prod>i\<in>J. \<mu> i (F i))"
1699         using finite J finite I F
1700         by (simp add: P.pair_measure_times I.measure_times J.measure_times)
1701       also have "\<dots> = (\<Prod>i\<in>I \<union> J. \<mu> i (F i))"
1702         using finite J finite I I \<inter> J = {}  by (simp add: setprod_Un_one)
1703       also have "\<dots> = IJ.measure A"
1704         using finite J finite I F unfolding A
1705         by (intro IJ.measure_times[symmetric]) auto
1706       finally show "P.pair_measure (?f  A) = IJ.measure A" .
1707     next
1708       fix k
1709       have "\<And>i. i \<in> I \<union> J \<Longrightarrow> F i k \<in> sets (M i)" using F by auto
1710       then have "P.pair_measure (?f  ?F k) = (\<Prod>i\<in>I. \<mu> i (F i k)) * (\<Prod>i\<in>J. \<mu> i (F i k))"
1711         by (simp add: P.pair_measure_times I.measure_times J.measure_times)
1712       then show "P.pair_measure (?f  ?F k) \<noteq> \<omega>"
1713         using finite I F by (simp add: setprod_\<omega>)
1714     qed simp
1715   qed
1717 lemma (in product_sigma_finite) product_positive_integral_fold:
1718   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
1719   and f: "f \<in> borel_measurable (sigma (product_algebra M (I \<union> J)))"
1720   shows "product_positive_integral (I \<union> J) f =
1721     product_positive_integral I (\<lambda>x. product_positive_integral J (\<lambda>y. f (merge I x J y)))"
1722 proof -
1723   interpret I: finite_product_sigma_finite M \<mu> I by default fact
1724   interpret J: finite_product_sigma_finite M \<mu> J by default fact
1725   have "finite (I \<union> J)" using fin by auto
1726   interpret IJ: finite_product_sigma_finite M \<mu> "I \<union> J" by default fact
1727   interpret P: pair_sigma_finite I.P I.measure J.P J.measure by default
1728   let ?f = "\<lambda>x. ((\<lambda>i\<in>I. x i), (\<lambda>i\<in>J. x i))"
1729   have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable P.P"
1730     unfolding case_prod_distrib measurable_merge_iff[OF IJ, symmetric] using f .
1731   have bij: "bij_betw ?f (space IJ.P) (space P.P)"
1732     unfolding sigma_pair_algebra_sigma_eq
1733     by (intro bij_inv_bij_betw) (rule bij_inv_restrict_merge[OF IJ])
1734   have "IJ.positive_integral f =  IJ.positive_integral (\<lambda>x. f (restrict x (I \<union> J)))"
1735     by (auto intro!: IJ.positive_integral_cong arg_cong[where f=f] dest!: extensional_restrict)
1736   also have "\<dots> = I.positive_integral (\<lambda>x. J.positive_integral (\<lambda>y. f (merge I x J y)))"
1737     unfolding P.positive_integral_fst_measurable[OF P_borel, simplified]
1738     unfolding P.positive_integral_vimage[OF bij]
1739     unfolding product_product_vimage_algebra[OF IJ]
1740     apply simp
1741     apply (rule IJ.positive_integral_cong_measure[symmetric])
1742     apply (rule measure_fold)
1743     using assms by auto
1744   finally show ?thesis .
1745 qed
1747 lemma (in product_sigma_finite) product_positive_integral_singleton:
1748   assumes f: "f \<in> borel_measurable (M i)"
1749   shows "product_positive_integral {i} (\<lambda>x. f (x i)) = M.positive_integral i f"
1750 proof -
1751   interpret I: finite_product_sigma_finite M \<mu> "{i}" by default simp
1752   have bij: "bij_betw (\<lambda>x. \<lambda>j\<in>{i}. x) (space (M i)) (space I.P)"
1753     by (auto intro!: bij_betwI ext simp: extensional_def)
1754   have *: "(\<lambda>x. (\<lambda>x. \<lambda>j\<in>{i}. x) - Pi\<^isub>E {i} x \<inter> space (M i))  (\<Pi> i\<in>{i}. sets (M i)) = sets (M i)"
1755   proof (subst image_cong, rule refl)
1756     fix x assume "x \<in> (\<Pi> i\<in>{i}. sets (M i))"
1757     then show "(\<lambda>x. \<lambda>j\<in>{i}. x) - Pi\<^isub>E {i} x \<inter> space (M i) = x i"
1758       using sets_into_space by auto
1759   qed auto
1760   have vimage: "I.vimage_algebra (space (M i)) (\<lambda>x. \<lambda>j\<in>{i}. x) = M i"
1761     unfolding I.vimage_algebra_def
1762     unfolding product_sigma_algebra_def sets_sigma
1763     apply (subst sigma_sets_vimage[symmetric])
1764     apply (simp_all add: image_image sigma_sets_eq product_algebra_def * del: vimage_Int)
1765     using sets_into_space by blast
1766   show "I.positive_integral (\<lambda>x. f (x i)) = M.positive_integral i f"
1767     unfolding I.positive_integral_vimage[OF bij]
1768     unfolding vimage
1769     apply (subst positive_integral_cong_measure)
1770   proof -
1771     fix A assume A: "A \<in> sets (M i)"
1772     have "(\<lambda>x. \<lambda>j\<in>{i}. x)  A = (\<Pi>\<^isub>E i\<in>{i}. A)"
1773       by (auto intro!: image_eqI ext[where 'b='a] simp: extensional_def)
1774     with A show "product_measure {i} ((\<lambda>x. \<lambda>j\<in>{i}. x)  A) = \<mu> i A"
1775       using I.measure_times[of "\<lambda>i. A"] by simp
1776   qed simp
1777 qed
1779 lemma (in product_sigma_finite) product_integral_singleton:
1780   assumes f: "f \<in> borel_measurable (M i)"
1781   shows "product_integral {i} (\<lambda>x. f (x i)) = M.integral i f"
1782 proof -
1783   interpret I: finite_product_sigma_finite M \<mu> "{i}" by default simp
1784   have *: "(\<lambda>x. Real (f x)) \<in> borel_measurable (M i)"
1785     "(\<lambda>x. Real (- f x)) \<in> borel_measurable (M i)"
1786     using assms by auto
1787   show ?thesis
1788     unfolding I.integral_def integral_def
1789     unfolding *[THEN product_positive_integral_singleton] ..
1790 qed
1792 lemma (in product_sigma_finite) product_integral_fold:
1793   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
1794   and f: "measure_space.integrable (sigma (product_algebra M (I \<union> J))) (product_measure (I \<union> J)) f"
1795   shows "product_integral (I \<union> J) f =
1796     product_integral I (\<lambda>x. product_integral J (\<lambda>y. f (merge I x J y)))"
1797 proof -
1798   interpret I: finite_product_sigma_finite M \<mu> I by default fact
1799   interpret J: finite_product_sigma_finite M \<mu> J by default fact
1800   have "finite (I \<union> J)" using fin by auto
1801   interpret IJ: finite_product_sigma_finite M \<mu> "I \<union> J" by default fact
1802   interpret P: pair_sigma_finite I.P I.measure J.P J.measure by default
1803   let ?f = "\<lambda>(x,y). f (merge I x J y)"
1804   have f_borel: "f \<in> borel_measurable IJ.P"
1805      "(\<lambda>x. Real (f x)) \<in> borel_measurable IJ.P"
1806      "(\<lambda>x. Real (- f x)) \<in> borel_measurable IJ.P"
1807     using f unfolding integrable_def by auto
1808   have f_restrict: "(\<lambda>x. f (restrict x (I \<union> J))) \<in> borel_measurable IJ.P"
1809     by (rule measurable_cong[THEN iffD2, OF _ f_borel(1)])
1810        (auto intro!: arg_cong[where f=f] simp: extensional_restrict)
1811   then have f'_borel:
1812     "(\<lambda>x. Real (?f x)) \<in> borel_measurable P.P"
1813     "(\<lambda>x. Real (- ?f x)) \<in> borel_measurable P.P"
1814     unfolding measurable_restrict_iff[OF IJ]
1815     by simp_all
1816   have PI:
1817     "P.positive_integral (\<lambda>x. Real (?f x)) = IJ.positive_integral (\<lambda>x. Real (f x))"
1818     "P.positive_integral (\<lambda>x. Real (- ?f x)) = IJ.positive_integral (\<lambda>x. Real (- f x))"
1819     using f'_borel[THEN P.positive_integral_fst_measurable(2)]
1820     using f_borel(2,3)[THEN product_positive_integral_fold[OF assms(1-3)]]
1821     by simp_all
1822   have "P.integrable ?f" using IJ.integrable f
1823     unfolding P.integrable_def IJ.integrable_def
1824     unfolding measurable_restrict_iff[OF IJ]
1825     using f_restrict PI by simp_all
1826   show ?thesis
1827     unfolding P.integrable_fst_measurable(2)[OF P.integrable ?f, simplified]
1828     unfolding IJ.integral_def P.integral_def
1829     unfolding PI by simp
1830 qed
1832 section "Products on finite spaces"
1834 lemma sigma_sets_pair_algebra_finite:
1835   assumes "finite A" and "finite B"
1836   shows "sigma_sets (A \<times> B) ((\<lambda>(x,y). x \<times> y) ` (Pow A \<times> Pow B)) = Pow (A \<times> B)"
1837   (is "sigma_sets ?prod ?sets = _")
1838 proof safe
1839   have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
1840   fix x assume subset: "x \<subseteq> A \<times> B"
1841   hence "finite x" using fin by (rule finite_subset)
1842   from this subset show "x \<in> sigma_sets ?prod ?sets"
1843   proof (induct x)
1844     case empty show ?case by (rule sigma_sets.Empty)
1845   next
1846     case (insert a x)
1847     hence "{a} \<in> sigma_sets ?prod ?sets"
1848       by (auto simp: pair_algebra_def intro!: sigma_sets.Basic)
1849     moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
1850     ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
1851   qed
1852 next
1853   fix x a b
1854   assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
1855   from sigma_sets_into_sp[OF _ this(1)] this(2)
1856   show "a \<in> A" and "b \<in> B" by auto
1857 qed
1859 locale pair_finite_sigma_algebra = M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2 for M1 M2
1861 sublocale pair_finite_sigma_algebra \<subseteq> pair_sigma_algebra by default
1863 lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra[simp]:
1864   shows "P = (| space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2) |)"
1865 proof -
1866   show ?thesis using M1.finite_space M2.finite_space
1867     by (simp add: sigma_def space_pair_algebra sets_pair_algebra
1868                   sigma_sets_pair_algebra_finite M1.sets_eq_Pow M2.sets_eq_Pow)
1869 qed
1871 sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P
1872 proof
1873   show "finite (space P)" "sets P = Pow (space P)"
1874     using M1.finite_space M2.finite_space by auto
1875 qed
1877 locale pair_finite_space = M1: finite_measure_space M1 \<mu>1 + M2: finite_measure_space M2 \<mu>2
1878   for M1 \<mu>1 M2 \<mu>2
1880 sublocale pair_finite_space \<subseteq> pair_finite_sigma_algebra
1881   by default
1883 sublocale pair_finite_space \<subseteq> pair_sigma_finite
1884   by default
1886 lemma (in pair_finite_space) finite_pair_sigma_algebra[simp]:
1887   shows "P = (| space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2) |)"
1888 proof -
1889   show ?thesis using M1.finite_space M2.finite_space
1890     by (simp add: sigma_def space_pair_algebra sets_pair_algebra
1891                   sigma_sets_pair_algebra_finite M1.sets_eq_Pow M2.sets_eq_Pow)
1892 qed
1894 lemma (in pair_finite_space) pair_measure_Pair[simp]:
1895   assumes "a \<in> space M1" "b \<in> space M2"
1896   shows "pair_measure {(a, b)} = \<mu>1 {a} * \<mu>2 {b}"
1897 proof -
1898   have "pair_measure ({a}\<times>{b}) = \<mu>1 {a} * \<mu>2 {b}"
1899     using M1.sets_eq_Pow M2.sets_eq_Pow assms
1900     by (subst pair_measure_times) auto
1901   then show ?thesis by simp
1902 qed
1904 lemma (in pair_finite_space) pair_measure_singleton[simp]:
1905   assumes "x \<in> space M1 \<times> space M2"
1906   shows "pair_measure {x} = \<mu>1 {fst x} * \<mu>2 {snd x}"
1907   using pair_measure_Pair assms by (cases x) auto
1909 sublocale pair_finite_space \<subseteq> finite_measure_space P pair_measure
1910   by default auto
1912 lemma (in pair_finite_space) finite_measure_space_finite_prod_measure_alterantive:
1913   "finite_measure_space \<lparr>space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2)\<rparr> pair_measure"
1914   unfolding finite_pair_sigma_algebra[symmetric]
1915   by default
1917 end