src/HOL/Probability/Finite_Product_Measure.thy
 author huffman Sun Apr 01 16:09:58 2012 +0200 (2012-04-01) changeset 47255 30a1692557b0 parent 46905 6b1c0a80a57a child 47694 05663f75964c permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
1 (*  Title:      HOL/Probability/Finite_Product_Measure.thy
2     Author:     Johannes Hölzl, TU München
3 *)
5 header {*Finite product measures*}
7 theory Finite_Product_Measure
8 imports Binary_Product_Measure
9 begin
11 lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
12   unfolding Pi_def by auto
14 abbreviation
15   "Pi\<^isub>E A B \<equiv> Pi A B \<inter> extensional A"
17 syntax
18   "_PiE"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PIE _:_./ _)" 10)
20 syntax (xsymbols)
21   "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
23 syntax (HTML output)
24   "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
26 translations
27   "PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)"
29 abbreviation
30   funcset_extensional :: "['a set, 'b set] => ('a => 'b) set"
31     (infixr "->\<^isub>E" 60) where
32   "A ->\<^isub>E B \<equiv> Pi\<^isub>E A (%_. B)"
34 notation (xsymbols)
35   funcset_extensional  (infixr "\<rightarrow>\<^isub>E" 60)
37 lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}"
38   by safe (auto simp add: extensional_def fun_eq_iff)
40 lemma extensional_insert[intro, simp]:
41   assumes "a \<in> extensional (insert i I)"
42   shows "a(i := b) \<in> extensional (insert i I)"
43   using assms unfolding extensional_def by auto
45 lemma extensional_Int[simp]:
46   "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
47   unfolding extensional_def by auto
49 definition
50   "merge I x J y = (\<lambda>i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
52 lemma merge_apply[simp]:
53   "I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"
54   "I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"
55   "J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"
56   "J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"
57   "i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I x J y i = undefined"
58   unfolding merge_def by auto
60 lemma merge_commute:
61   "I \<inter> J = {} \<Longrightarrow> merge I x J y = merge J y I x"
62   by (auto simp: merge_def intro!: ext)
64 lemma Pi_cancel_merge_range[simp]:
65   "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"
66   "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"
67   "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"
68   "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"
69   by (auto simp: Pi_def)
71 lemma Pi_cancel_merge[simp]:
72   "I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
73   "J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
74   "I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
75   "J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
76   by (auto simp: Pi_def)
78 lemma extensional_merge[simp]: "merge I x J y \<in> extensional (I \<union> J)"
79   by (auto simp: extensional_def)
81 lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
82   by (auto simp: restrict_def Pi_def)
84 lemma restrict_merge[simp]:
85   "I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"
86   "I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
87   "J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"
88   "J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
89   by (auto simp: restrict_def intro!: ext)
91 lemma extensional_insert_undefined[intro, simp]:
92   assumes "a \<in> extensional (insert i I)"
93   shows "a(i := undefined) \<in> extensional I"
94   using assms unfolding extensional_def by auto
96 lemma extensional_insert_cancel[intro, simp]:
97   assumes "a \<in> extensional I"
98   shows "a \<in> extensional (insert i I)"
99   using assms unfolding extensional_def by auto
101 lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I x {i} y = restrict (x(i := y i)) (insert i I)"
102   unfolding merge_def by (auto simp: fun_eq_iff)
104 lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
105   by auto
107 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)"
108   by auto
110 lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B"
111   by (auto simp: Pi_def)
113 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"
114   by (auto simp: Pi_def)
116 lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X"
117   by (auto simp: Pi_def)
119 lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
120   by (auto simp: Pi_def)
122 lemma restrict_vimage:
123   assumes "I \<inter> J = {}"
124   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)"
125   using assms by (auto simp: restrict_Pi_cancel)
127 lemma merge_vimage:
128   assumes "I \<inter> J = {}"
129   shows "(\<lambda>(x,y). merge I x J y) -` Pi\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E"
130   using assms by (auto simp: restrict_Pi_cancel)
132 lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
133   by (auto simp: restrict_def intro!: ext)
135 lemma merge_restrict[simp]:
136   "merge I (restrict x I) J y = merge I x J y"
137   "merge I x J (restrict y J) = merge I x J y"
138   unfolding merge_def by (auto intro!: ext)
140 lemma merge_x_x_eq_restrict[simp]:
141   "merge I x J x = restrict x (I \<union> J)"
142   unfolding merge_def by (auto intro!: ext)
144 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"
145   apply auto
146   apply (drule_tac x=x in Pi_mem)
147   apply (simp_all split: split_if_asm)
148   apply (drule_tac x=i in Pi_mem)
149   apply (auto dest!: Pi_mem)
150   done
152 lemma Pi_UN:
153   fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
154   assumes "finite I" and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
155   shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
156 proof (intro set_eqI iffI)
157   fix f assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
158   then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" by auto
159   from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
160   obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
161     using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto
162   have "f \<in> Pi I (A k)"
163   proof (intro Pi_I)
164     fix i assume "i \<in> I"
165     from mono[OF this, of "n i" k] k[OF this] n[OF this]
166     show "f i \<in> A k i" by auto
167   qed
168   then show "f \<in> (\<Union>n. Pi I (A n))" by auto
169 qed auto
171 lemma PiE_cong:
172   assumes "\<And>i. i\<in>I \<Longrightarrow> A i = B i"
173   shows "Pi\<^isub>E I A = Pi\<^isub>E I B"
174   using assms by (auto intro!: Pi_cong)
176 lemma restrict_upd[simp]:
177   "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
178   by (auto simp: fun_eq_iff)
180 lemma Pi_eq_subset:
181   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
182   assumes eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and "i \<in> I"
183   shows "F i \<subseteq> F' i"
184 proof
185   fix x assume "x \<in> F i"
186   with ne have "\<forall>j. \<exists>y. ((j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined))" by auto
187   from choice[OF this] guess f .. note f = this
188   then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def)
189   then have "f \<in> Pi\<^isub>E I F'" using assms by simp
190   then show "x \<in> F' i" using f `i \<in> I` by auto
191 qed
193 lemma Pi_eq_iff_not_empty:
194   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
195   shows "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
196 proof (intro iffI ballI)
197   fix i assume eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and i: "i \<in> I"
198   show "F i = F' i"
199     using Pi_eq_subset[of I F F', OF ne eq i]
200     using Pi_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
201     by auto
202 qed auto
204 lemma Pi_eq_empty_iff:
205   "Pi\<^isub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
206 proof
207   assume "Pi\<^isub>E I F = {}"
208   show "\<exists>i\<in>I. F i = {}"
209   proof (rule ccontr)
210     assume "\<not> ?thesis"
211     then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)" by auto
212     from choice[OF this] guess f ..
213     then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def)
214     with `Pi\<^isub>E I F = {}` show False by auto
215   qed
216 qed auto
218 lemma Pi_eq_iff:
219   "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i) \<or> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
220 proof (intro iffI disjCI)
221   assume eq[simp]: "Pi\<^isub>E I F = Pi\<^isub>E I F'"
222   assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
223   then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
224     using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto
225   with Pi_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i" by auto
226 next
227   assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"
228   then show "Pi\<^isub>E I F = Pi\<^isub>E I F'"
229     using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto
230 qed
232 section "Finite product spaces"
234 section "Products"
236 locale product_sigma_algebra =
237   fixes M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme"
238   assumes sigma_algebras: "\<And>i. sigma_algebra (M i)"
240 locale finite_product_sigma_algebra = product_sigma_algebra M
241   for M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme" +
242   fixes I :: "'i set"
243   assumes finite_index[simp, intro]: "finite I"
245 definition
246   "product_algebra_generator I M = \<lparr> space = (\<Pi>\<^isub>E i \<in> I. space (M i)),
247     sets = Pi\<^isub>E I ` (\<Pi> i \<in> I. sets (M i)),
248     measure = \<lambda>A. (\<Prod>i\<in>I. measure (M i) ((SOME F. A = Pi\<^isub>E I F) i)) \<rparr>"
250 definition product_algebra_def:
251   "Pi\<^isub>M I M = sigma (product_algebra_generator I M)
252     \<lparr> measure := (SOME \<mu>. sigma_finite_measure (sigma (product_algebra_generator I M) \<lparr> measure := \<mu> \<rparr>) \<and>
253       (\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<mu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. measure (M i) (A i))))\<rparr>"
255 syntax
256   "_PiM"  :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
257               ('i => 'a, 'b) measure_space_scheme"  ("(3PIM _:_./ _)" 10)
259 syntax (xsymbols)
260   "_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
261              ('i => 'a, 'b) measure_space_scheme"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"   10)
263 syntax (HTML output)
264   "_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
265              ('i => 'a, 'b) measure_space_scheme"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"   10)
267 translations
268   "PIM x:I. M" == "CONST Pi\<^isub>M I (%x. M)"
270 abbreviation (in finite_product_sigma_algebra) "G \<equiv> product_algebra_generator I M"
271 abbreviation (in finite_product_sigma_algebra) "P \<equiv> Pi\<^isub>M I M"
273 sublocale product_sigma_algebra \<subseteq> M: sigma_algebra "M i" for i by (rule sigma_algebras)
275 lemma sigma_into_space:
276   assumes "sets M \<subseteq> Pow (space M)"
277   shows "sets (sigma M) \<subseteq> Pow (space M)"
278   using sigma_sets_into_sp[OF assms] unfolding sigma_def by auto
280 lemma (in product_sigma_algebra) product_algebra_generator_into_space:
281   "sets (product_algebra_generator I M) \<subseteq> Pow (space (product_algebra_generator I M))"
282   using M.sets_into_space unfolding product_algebra_generator_def
283   by auto blast
285 lemma (in product_sigma_algebra) product_algebra_into_space:
286   "sets (Pi\<^isub>M I M) \<subseteq> Pow (space (Pi\<^isub>M I M))"
287   using product_algebra_generator_into_space
288   by (auto intro!: sigma_into_space simp add: product_algebra_def)
290 lemma (in product_sigma_algebra) sigma_algebra_product_algebra: "sigma_algebra (Pi\<^isub>M I M)"
291   using product_algebra_generator_into_space unfolding product_algebra_def
292   by (rule sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) simp_all
294 sublocale finite_product_sigma_algebra \<subseteq> sigma_algebra P
295   using sigma_algebra_product_algebra .
297 lemma product_algebraE:
298   assumes "A \<in> sets (product_algebra_generator I M)"
299   obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
300   using assms unfolding product_algebra_generator_def by auto
302 lemma product_algebra_generatorI[intro]:
303   assumes "E \<in> (\<Pi> i\<in>I. sets (M i))"
304   shows "Pi\<^isub>E I E \<in> sets (product_algebra_generator I M)"
305   using assms unfolding product_algebra_generator_def by auto
307 lemma space_product_algebra_generator[simp]:
308   "space (product_algebra_generator I M) = Pi\<^isub>E I (\<lambda>i. space (M i))"
309   unfolding product_algebra_generator_def by simp
311 lemma space_product_algebra[simp]:
312   "space (Pi\<^isub>M I M) = (\<Pi>\<^isub>E i\<in>I. space (M i))"
313   unfolding product_algebra_def product_algebra_generator_def by simp
315 lemma sets_product_algebra:
316   "sets (Pi\<^isub>M I M) = sets (sigma (product_algebra_generator I M))"
317   unfolding product_algebra_def sigma_def by simp
319 lemma product_algebra_generator_sets_into_space:
320   assumes "\<And>i. i\<in>I \<Longrightarrow> sets (M i) \<subseteq> Pow (space (M i))"
321   shows "sets (product_algebra_generator I M) \<subseteq> Pow (space (product_algebra_generator I M))"
322   using assms by (auto simp: product_algebra_generator_def) blast
324 lemma (in finite_product_sigma_algebra) in_P[simp, intro]:
325   "\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P"
326   by (auto simp: sets_product_algebra)
328 lemma Int_stable_product_algebra_generator:
329   "(\<And>i. i \<in> I \<Longrightarrow> Int_stable (M i)) \<Longrightarrow> Int_stable (product_algebra_generator I M)"
330   by (auto simp add: product_algebra_generator_def Int_stable_def PiE_Int Pi_iff)
332 section "Generating set generates also product algebra"
334 lemma sigma_product_algebra_sigma_eq:
335   assumes "finite I"
336   assumes mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)"
337   assumes union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (E i)"
338   assumes sets_into: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> sets (E i)"
339   and E: "\<And>i. sets (E i) \<subseteq> Pow (space (E i))"
340   shows "sets (\<Pi>\<^isub>M i\<in>I. sigma (E i)) = sets (\<Pi>\<^isub>M i\<in>I. E i)"
341     (is "sets ?S = sets ?E")
342 proof cases
343   assume "I = {}" then show ?thesis
344     by (simp add: product_algebra_def product_algebra_generator_def)
345 next
346   assume "I \<noteq> {}"
347   interpret E: sigma_algebra "sigma (E i)" for i
348     using E by (rule sigma_algebra_sigma)
349   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)"
350     using E by auto
351   interpret G: sigma_algebra ?E
352     unfolding product_algebra_def product_algebra_generator_def using E
353     by (intro sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) (auto dest: Pi_mem)
354   { fix A i assume "i \<in> I" and A: "A \<in> sets (E i)"
355     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"
356       using mono union unfolding incseq_Suc_iff space_product_algebra
357       by (auto dest: Pi_mem)
358     also have "\<dots> = (\<Union>n. (\<Pi>\<^isub>E j\<in>I. if j = i then A else S j n))"
359       unfolding space_product_algebra
360       apply simp
361       apply (subst Pi_UN[OF `finite I`])
362       using mono[THEN incseqD] apply simp
363       apply (simp add: PiE_Int)
364       apply (intro PiE_cong)
365       using A sets_into by (auto intro!: into_space)
366     also have "\<dots> \<in> sets ?E"
367       using sets_into `A \<in> sets (E i)`
368       unfolding sets_product_algebra sets_sigma
369       by (intro sigma_sets.Union)
370          (auto simp: image_subset_iff intro!: sigma_sets.Basic)
371     finally have "(\<lambda>x. x i) -` A \<inter> space ?E \<in> sets ?E" . }
372   then have proj:
373     "\<And>i. i\<in>I \<Longrightarrow> (\<lambda>x. x i) \<in> measurable ?E (sigma (E i))"
374     using E by (subst G.measurable_iff_sigma)
375                (auto simp: sets_product_algebra sets_sigma)
376   { fix A assume A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (sigma (E i))"
377     with proj have basic: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) -` (A i) \<inter> space ?E \<in> sets ?E"
378       unfolding measurable_def by simp
379     have "Pi\<^isub>E I A = (\<Inter>i\<in>I. (\<lambda>x. x i) -` (A i) \<inter> space ?E)"
380       using A E.sets_into_space `I \<noteq> {}` unfolding product_algebra_def by auto blast
381     then have "Pi\<^isub>E I A \<in> sets ?E"
382       using G.finite_INT[OF `finite I` `I \<noteq> {}` basic, of "\<lambda>i. i"] by simp }
383   then have "sigma_sets (space ?E) (sets (product_algebra_generator I (\<lambda>i. sigma (E i)))) \<subseteq> sets ?E"
384     by (intro G.sigma_sets_subset) (auto simp add: product_algebra_generator_def)
385   then have subset: "sets ?S \<subseteq> sets ?E"
386     by (simp add: sets_sigma sets_product_algebra)
387   show "sets ?S = sets ?E"
388   proof (intro set_eqI iffI)
389     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
390       unfolding sets_sigma sets_product_algebra
391     proof induct
392       case (Basic A) then show ?case
393         by (auto simp: sets_sigma product_algebra_generator_def intro: sigma_sets.Basic)
394     qed (auto intro: sigma_sets.intros simp: product_algebra_generator_def)
395   next
396     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
397   qed
398 qed
400 lemma product_algebraI[intro]:
401     "E \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> Pi\<^isub>E I E \<in> sets (Pi\<^isub>M I M)"
402   using assms unfolding product_algebra_def by (auto intro: product_algebra_generatorI)
404 lemma (in product_sigma_algebra) measurable_component_update:
405   assumes "x \<in> space (Pi\<^isub>M I M)" and "i \<notin> I"
406   shows "(\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)" (is "?f \<in> _")
407   unfolding product_algebra_def apply simp
408 proof (intro measurable_sigma)
409   let ?G = "product_algebra_generator (insert i I) M"
410   show "sets ?G \<subseteq> Pow (space ?G)" using product_algebra_generator_into_space .
411   show "?f \<in> space (M i) \<rightarrow> space ?G"
412     using M.sets_into_space assms by auto
413   fix A assume "A \<in> sets ?G"
414   from product_algebraE[OF this] guess E . note E = this
415   then have "?f -` A \<inter> space (M i) = E i \<or> ?f -` A \<inter> space (M i) = {}"
416     using M.sets_into_space assms by auto
417   then show "?f -` A \<inter> space (M i) \<in> sets (M i)"
418     using E by (auto intro!: product_algebraI)
419 qed
421 lemma (in product_sigma_algebra) measurable_add_dim:
422   assumes "i \<notin> I"
423   shows "(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M)"
424 proof -
425   let ?f = "(\<lambda>(f, y). f(i := y))" and ?G = "product_algebra_generator (insert i I) M"
426   interpret Ii: pair_sigma_algebra "Pi\<^isub>M I M" "M i"
427     unfolding pair_sigma_algebra_def
428     by (intro sigma_algebra_product_algebra sigma_algebras conjI)
429   have "?f \<in> measurable Ii.P (sigma ?G)"
430   proof (rule Ii.measurable_sigma)
431     show "sets ?G \<subseteq> Pow (space ?G)"
432       using product_algebra_generator_into_space .
433     show "?f \<in> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) \<rightarrow> space ?G"
434       by (auto simp: space_pair_measure)
435   next
436     fix A assume "A \<in> sets ?G"
437     then obtain F where "A = Pi\<^isub>E (insert i I) F"
438       and F: "\<And>j. j \<in> I \<Longrightarrow> F j \<in> sets (M j)" "F i \<in> sets (M i)"
439       by (auto elim!: product_algebraE)
440     then have "?f -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) = Pi\<^isub>E I F \<times> (F i)"
441       using sets_into_space `i \<notin> I`
442       by (auto simp add: space_pair_measure) blast+
443     then show "?f -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M M i)"
444       using F by (auto intro!: pair_measureI)
445   qed
446   then show ?thesis
447     by (simp add: product_algebra_def)
448 qed
450 lemma (in product_sigma_algebra) measurable_merge:
451   assumes [simp]: "I \<inter> J = {}"
452   shows "(\<lambda>(x, y). merge I x J y) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"
453 proof -
454   let ?I = "Pi\<^isub>M I M" and ?J = "Pi\<^isub>M J M"
455   interpret P: sigma_algebra "?I \<Otimes>\<^isub>M ?J"
456     by (intro sigma_algebra_pair_measure product_algebra_into_space)
457   let ?f = "\<lambda>(x, y). merge I x J y"
458   let ?G = "product_algebra_generator (I \<union> J) M"
459   have "?f \<in> measurable (?I \<Otimes>\<^isub>M ?J) (sigma ?G)"
460   proof (rule P.measurable_sigma)
461     fix A assume "A \<in> sets ?G"
462     from product_algebraE[OF this]
463     obtain E where E: "A = Pi\<^isub>E (I \<union> J) E" "E \<in> (\<Pi> i\<in>I \<union> J. sets (M i))" .
464     then have *: "?f -` A \<inter> space (?I \<Otimes>\<^isub>M ?J) = Pi\<^isub>E I E \<times> Pi\<^isub>E J E"
465       using sets_into_space `I \<inter> J = {}`
466       by (auto simp add: space_pair_measure) fast+
467     then show "?f -` A \<inter> space (?I \<Otimes>\<^isub>M ?J) \<in> sets (?I \<Otimes>\<^isub>M ?J)"
468       using E unfolding * by (auto intro!: pair_measureI in_sigma product_algebra_generatorI
469         simp: product_algebra_def)
470   qed (insert product_algebra_generator_into_space, auto simp: space_pair_measure)
471   then show "?f \<in> measurable (?I \<Otimes>\<^isub>M ?J) (Pi\<^isub>M (I \<union> J) M)"
472     unfolding product_algebra_def[of "I \<union> J"] by simp
473 qed
475 lemma (in product_sigma_algebra) measurable_component_singleton:
476   assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)"
477 proof (unfold measurable_def, intro CollectI conjI ballI)
478   fix A assume "A \<in> sets (M i)"
479   then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) = (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))"
480     using M.sets_into_space `i \<in> I` by (fastforce dest: Pi_mem split: split_if_asm)
481   then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)"
482     using `A \<in> sets (M i)` by (auto intro!: product_algebraI)
483 qed (insert `i \<in> I`, auto)
485 lemma (in sigma_algebra) measurable_restrict:
486   assumes I: "finite I"
487   assumes "\<And>i. i \<in> I \<Longrightarrow> sets (N i) \<subseteq> Pow (space (N i))"
488   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable M (N i)"
489   shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable M (Pi\<^isub>M I N)"
490   unfolding product_algebra_def
491 proof (simp, rule measurable_sigma)
492   show "sets (product_algebra_generator I N) \<subseteq> Pow (space (product_algebra_generator I N))"
493     by (rule product_algebra_generator_sets_into_space) fact
494   show "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> space M \<rightarrow> space (product_algebra_generator I N)"
495     using X by (auto simp: measurable_def)
496   fix E assume "E \<in> sets (product_algebra_generator I N)"
497   then obtain F where "E = Pi\<^isub>E I F" and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets (N i)"
498     by (auto simp: product_algebra_generator_def)
499   then have "(\<lambda>x. \<lambda>i\<in>I. X i x) -` E \<inter> space M = (\<Inter>i\<in>I. X i -` F i \<inter> space M) \<inter> space M"
500     by (auto simp: Pi_iff)
501   also have "\<dots> \<in> sets M"
502   proof cases
503     assume "I = {}" then show ?thesis by simp
504   next
505     assume "I \<noteq> {}" with X F I show ?thesis
506       by (intro finite_INT measurable_sets Int) auto
507   qed
508   finally show "(\<lambda>x. \<lambda>i\<in>I. X i x) -` E \<inter> space M \<in> sets M" .
509 qed
511 locale product_sigma_finite = product_sigma_algebra M
512   for M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" +
513   assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
515 sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
516   by (rule sigma_finite_measures)
518 locale finite_product_sigma_finite = finite_product_sigma_algebra M I + product_sigma_finite M
519   for M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" and I :: "'i set"
521 lemma (in finite_product_sigma_finite) product_algebra_generator_measure:
522   assumes "Pi\<^isub>E I F \<in> sets G"
523   shows "measure G (Pi\<^isub>E I F) = (\<Prod>i\<in>I. M.\<mu> i (F i))"
524 proof cases
525   assume ne: "\<forall>i\<in>I. F i \<noteq> {}"
526   have "\<forall>i\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') i = F i"
527     by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I F'"])
528        (insert ne, auto simp: Pi_eq_iff)
529   then show ?thesis
530     unfolding product_algebra_generator_def by simp
531 next
532   assume empty: "\<not> (\<forall>j\<in>I. F j \<noteq> {})"
533   then have "(\<Prod>j\<in>I. M.\<mu> j (F j)) = 0"
534     by (auto simp: setprod_ereal_0 intro!: bexI)
535   moreover
536   have "\<exists>j\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j = {}"
537     by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I F'"])
538        (insert empty, auto simp: Pi_eq_empty_iff[symmetric])
539   then have "(\<Prod>j\<in>I. M.\<mu> j ((SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j)) = 0"
540     by (auto simp: setprod_ereal_0 intro!: bexI)
541   ultimately show ?thesis
542     unfolding product_algebra_generator_def by simp
543 qed
545 lemma (in finite_product_sigma_finite) sigma_finite_pairs:
546   "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
547     (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
548     (\<forall>k. \<forall>i\<in>I. \<mu> i (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<and>
549     (\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space G"
550 proof -
551   have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. \<mu> i (F k) \<noteq> \<infinity>)"
552     using M.sigma_finite_up by simp
553   from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
554   then have F: "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. incseq (F i)" "\<And>i. (\<Union>j. F i j) = space (M i)" "\<And>i k. \<mu> i (F i k) \<noteq> \<infinity>"
555     by auto
556   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"
557   note space_product_algebra[simp]
558   show ?thesis
559   proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
560     fix i show "range (F i) \<subseteq> sets (M i)" by fact
561   next
562     fix i k show "\<mu> i (F i k) \<noteq> \<infinity>" by fact
563   next
564     fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space G"
565       using `\<And>i. range (F i) \<subseteq> sets (M i)` M.sets_into_space
566       by (force simp: image_subset_iff)
567   next
568     fix f assume "f \<in> space G"
569     with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
570     show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def)
571   next
572     fix i show "?F i \<subseteq> ?F (Suc i)"
573       using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
574   qed
575 qed
577 lemma sets_pair_cancel_measure[simp]:
578   "sets (M1\<lparr>measure := m1\<rparr> \<Otimes>\<^isub>M M2) = sets (M1 \<Otimes>\<^isub>M M2)"
579   "sets (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m2\<rparr>) = sets (M1 \<Otimes>\<^isub>M M2)"
580   unfolding pair_measure_def pair_measure_generator_def sets_sigma
581   by simp_all
583 lemma measurable_pair_cancel_measure[simp]:
584   "measurable (M1\<lparr>measure := m1\<rparr> \<Otimes>\<^isub>M M2) M = measurable (M1 \<Otimes>\<^isub>M M2) M"
585   "measurable (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m2\<rparr>) M = measurable (M1 \<Otimes>\<^isub>M M2) M"
586   "measurable M (M1\<lparr>measure := m3\<rparr> \<Otimes>\<^isub>M M2) = measurable M (M1 \<Otimes>\<^isub>M M2)"
587   "measurable M (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m4\<rparr>) = measurable M (M1 \<Otimes>\<^isub>M M2)"
588   unfolding measurable_def by (auto simp add: space_pair_measure)
590 lemma (in product_sigma_finite) product_measure_exists:
591   assumes "finite I"
592   shows "\<exists>\<nu>. sigma_finite_measure (sigma (product_algebra_generator I M) \<lparr> measure := \<nu> \<rparr>) \<and>
593     (\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i)))"
594 using `finite I` proof induct
595   case empty
596   interpret finite_product_sigma_finite M "{}" by default simp
597   let ?\<nu> = "(\<lambda>A. if A = {} then 0 else 1) :: 'd set \<Rightarrow> ereal"
598   show ?case
599   proof (intro exI conjI ballI)
600     have "sigma_algebra (sigma G \<lparr>measure := ?\<nu>\<rparr>)"
601       by (rule sigma_algebra_cong) (simp_all add: product_algebra_def)
602     then have "measure_space (sigma G\<lparr>measure := ?\<nu>\<rparr>)"
603       by (rule finite_additivity_sufficient)
605                         product_algebra_generator_def image_constant)
606     then show "sigma_finite_measure (sigma G\<lparr>measure := ?\<nu>\<rparr>)"
607       by (auto intro!: exI[of _ "\<lambda>i. {\<lambda>_. undefined}"]
608                simp: sigma_finite_measure_def sigma_finite_measure_axioms_def
609                      product_algebra_generator_def)
610   qed auto
611 next
612   case (insert i I)
613   interpret finite_product_sigma_finite M I by default fact
614   have "finite (insert i I)" using `finite I` by auto
615   interpret I': finite_product_sigma_finite M "insert i I" by default fact
616   from insert obtain \<nu> where
617     prod: "\<And>A. A \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))" and
618     "sigma_finite_measure (sigma G\<lparr> measure := \<nu> \<rparr>)" by auto
619   then interpret I: sigma_finite_measure "P\<lparr> measure := \<nu>\<rparr>" unfolding product_algebra_def by simp
620   interpret P: pair_sigma_finite "P\<lparr> measure := \<nu>\<rparr>" "M i" ..
621   let ?h = "(\<lambda>(f, y). f(i := y))"
622   let ?\<nu> = "\<lambda>A. P.\<mu> (?h -` A \<inter> space P.P)"
623   have I': "sigma_algebra (I'.P\<lparr> measure := ?\<nu> \<rparr>)"
624     by (rule I'.sigma_algebra_cong) simp_all
625   interpret I'': measure_space "I'.P\<lparr> measure := ?\<nu> \<rparr>"
626     using measurable_add_dim[OF `i \<notin> I`]
627     by (intro P.measure_space_vimage[OF I']) (auto simp add: measure_preserving_def)
628   show ?case
629   proof (intro exI[of _ ?\<nu>] conjI ballI)
630     let ?m = "\<lambda>A. measure (Pi\<^isub>M I M\<lparr>measure := \<nu>\<rparr> \<Otimes>\<^isub>M M i) (?h -` A \<inter> space P.P)"
631     { fix A assume A: "A \<in> (\<Pi> i\<in>insert i I. sets (M i))"
632       then have *: "?h -` Pi\<^isub>E (insert i I) A \<inter> space P.P = Pi\<^isub>E I A \<times> A i"
633         using `i \<notin> I` M.sets_into_space by (auto simp: space_pair_measure space_product_algebra) blast
634       show "?m (Pi\<^isub>E (insert i I) A) = (\<Prod>i\<in>insert i I. M.\<mu> i (A i))"
635         unfolding * using A
636         apply (subst P.pair_measure_times)
637         using A apply fastforce
638         using A apply fastforce
639         using `i \<notin> I` `finite I` prod[of A] A by (auto simp: ac_simps) }
640     note product = this
641     have *: "sigma I'.G\<lparr> measure := ?\<nu> \<rparr> = I'.P\<lparr> measure := ?\<nu> \<rparr>"
642       by (simp add: product_algebra_def)
643     show "sigma_finite_measure (sigma I'.G\<lparr> measure := ?\<nu> \<rparr>)"
644     proof (unfold *, default, simp)
645       from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
646       then have F: "\<And>j. j \<in> insert i I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
647         "incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k)"
648         "(\<Union>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) = space I'.G"
649         "\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> \<mu> j (F j k) \<noteq> \<infinity>"
650         by blast+
651       let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k"
652       show "\<exists>F::nat \<Rightarrow> ('i \<Rightarrow> 'a) set. range F \<subseteq> sets I'.P \<and>
653           (\<Union>i. F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i)) \<and> (\<forall>i. ?m (F i) \<noteq> \<infinity>)"
654       proof (intro exI[of _ ?F] conjI allI)
655         show "range ?F \<subseteq> sets I'.P" using F(1) by auto
656       next
657         from F(3) show "(\<Union>i. ?F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i))" by simp
658       next
659         fix j
660         have "\<And>k. k \<in> insert i I \<Longrightarrow> 0 \<le> measure (M k) (F k j)"
661           using F(1) by auto
662         with F `finite I` setprod_PInf[of "insert i I", OF this] show "?\<nu> (?F j) \<noteq> \<infinity>"
663           by (subst product) auto
664       qed
665     qed
666   qed
667 qed
669 sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure P
670   unfolding product_algebra_def
671   using product_measure_exists[OF finite_index]
672   by (rule someI2_ex) auto
674 lemma (in finite_product_sigma_finite) measure_times:
675   assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)"
676   shows "\<mu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
677   unfolding product_algebra_def
678   using product_measure_exists[OF finite_index]
679   proof (rule someI2_ex, elim conjE)
680     fix \<nu> assume *: "\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
681     have "Pi\<^isub>E I A = Pi\<^isub>E I (\<lambda>i\<in>I. A i)" by (auto dest: Pi_mem)
682     then have "\<nu> (Pi\<^isub>E I A) = \<nu> (Pi\<^isub>E I (\<lambda>i\<in>I. A i))" by simp
683     also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i ((\<lambda>i\<in>I. A i) i))" using assms * by auto
684     finally show "measure (sigma G\<lparr>measure := \<nu>\<rparr>) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
685       by (simp add: sigma_def)
686   qed
688 lemma (in product_sigma_finite) product_measure_empty[simp]:
689   "measure (Pi\<^isub>M {} M) {\<lambda>x. undefined} = 1"
690 proof -
691   interpret finite_product_sigma_finite M "{}" by default auto
692   from measure_times[of "\<lambda>x. {}"] show ?thesis by simp
693 qed
695 lemma (in finite_product_sigma_algebra) P_empty:
696   assumes "I = {}"
697   shows "space P = {\<lambda>k. undefined}" "sets P = { {}, {\<lambda>k. undefined} }"
698   unfolding product_algebra_def product_algebra_generator_def `I = {}`
699   by (simp_all add: sigma_def image_constant)
701 lemma (in product_sigma_finite) positive_integral_empty:
702   assumes pos: "0 \<le> f (\<lambda>k. undefined)"
703   shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"
704 proof -
705   interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
706   have "\<And>A. measure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"
707     using assms by (subst measure_times) auto
708   then show ?thesis
709     unfolding positive_integral_def simple_function_def simple_integral_def [abs_def]
710   proof (simp add: P_empty, intro antisym)
711     show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
712       by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
713     show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
714       by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm)
715   qed
716 qed
718 lemma (in product_sigma_finite) measure_fold:
719   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
720   assumes A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
721   shows "measure (Pi\<^isub>M (I \<union> J) M) A =
722     measure (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ((\<lambda>(x,y). merge I x J y) -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M))"
723 proof -
724   interpret I: finite_product_sigma_finite M I by default fact
725   interpret J: finite_product_sigma_finite M J by default fact
726   have "finite (I \<union> J)" using fin by auto
727   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
728   interpret P: pair_sigma_finite I.P J.P by default
729   let ?g = "\<lambda>(x,y). merge I x J y"
730   let ?X = "\<lambda>S. ?g -` S \<inter> space P.P"
731   from IJ.sigma_finite_pairs obtain F where
732     F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
733        "incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)"
734        "(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space IJ.G"
735        "\<And>k. \<forall>i\<in>I\<union>J. \<mu> i (F i k) \<noteq> \<infinity>"
736     by auto
737   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"
738   show "IJ.\<mu> A = P.\<mu> (?X A)"
739   proof (rule measure_unique_Int_stable_vimage)
740     show "measure_space IJ.P" "measure_space P.P" by default
741     show "sets (sigma IJ.G) = sets IJ.P" "space IJ.G = space IJ.P" "A \<in> sets (sigma IJ.G)"
742       using A unfolding product_algebra_def by auto
743   next
744     show "Int_stable IJ.G"
745       by (rule Int_stable_product_algebra_generator)
746          (auto simp: Int_stable_def)
747     show "range ?F \<subseteq> sets IJ.G" using F
748       by (simp add: image_subset_iff product_algebra_def
749                     product_algebra_generator_def)
750     show "incseq ?F" "(\<Union>i. ?F i) = space IJ.G " by fact+
751   next
752     fix k
753     have "\<And>j. j \<in> I \<union> J \<Longrightarrow> 0 \<le> measure (M j) (F j k)"
754       using F(1) by auto
755     with F `finite I` setprod_PInf[of "I \<union> J", OF this]
756     show "IJ.\<mu> (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
757   next
758     fix A assume "A \<in> sets IJ.G"
759     then obtain F where A: "A = Pi\<^isub>E (I \<union> J) F"
760       and F: "\<And>i. i \<in> I \<union> J \<Longrightarrow> F i \<in> sets (M i)"
761       by (auto simp: product_algebra_generator_def)
762     then have X: "?X A = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"
763       using sets_into_space by (auto simp: space_pair_measure) blast+
764     then have "P.\<mu> (?X A) = (\<Prod>i\<in>I. \<mu> i (F i)) * (\<Prod>i\<in>J. \<mu> i (F i))"
765       using `finite J` `finite I` F
766       by (simp add: P.pair_measure_times I.measure_times J.measure_times)
767     also have "\<dots> = (\<Prod>i\<in>I \<union> J. \<mu> i (F i))"
768       using `finite J` `finite I` `I \<inter> J = {}`  by (simp add: setprod_Un_one)
769     also have "\<dots> = IJ.\<mu> A"
770       using `finite J` `finite I` F unfolding A
771       by (intro IJ.measure_times[symmetric]) auto
772     finally show "IJ.\<mu> A = P.\<mu> (?X A)" by (rule sym)
773   qed (rule measurable_merge[OF IJ])
774 qed
776 lemma (in product_sigma_finite) measure_preserving_merge:
777   assumes IJ: "I \<inter> J = {}" and "finite I" "finite J"
778   shows "(\<lambda>(x,y). merge I x J y) \<in> measure_preserving (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"
779   by (intro measure_preservingI measurable_merge[OF IJ] measure_fold[symmetric] assms)
781 lemma (in product_sigma_finite) product_positive_integral_fold:
782   assumes IJ[simp]: "I \<inter> J = {}" "finite I" "finite J"
783   and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
784   shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f =
785     (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I x J y) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))"
786 proof -
787   interpret I: finite_product_sigma_finite M I by default fact
788   interpret J: finite_product_sigma_finite M J by default fact
789   interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
790   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default simp
791   have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable P.P"
792     using measurable_comp[OF measurable_merge[OF IJ(1)] f] by (simp add: comp_def)
793   show ?thesis
794     unfolding P.positive_integral_fst_measurable[OF P_borel, simplified]
795   proof (rule P.positive_integral_vimage)
796     show "sigma_algebra IJ.P" by default
797     show "(\<lambda>(x, y). merge I x J y) \<in> measure_preserving P.P IJ.P"
798       using IJ by (rule measure_preserving_merge)
799     show "f \<in> borel_measurable IJ.P" using f by simp
800   qed
801 qed
803 lemma (in product_sigma_finite) measure_preserving_component_singelton:
804   "(\<lambda>x. x i) \<in> measure_preserving (Pi\<^isub>M {i} M) (M i)"
805 proof (intro measure_preservingI measurable_component_singleton)
806   interpret I: finite_product_sigma_finite M "{i}" by default simp
807   fix A let ?P = "(\<lambda>x. x i) -` A \<inter> space I.P"
808   assume A: "A \<in> sets (M i)"
809   then have *: "?P = {i} \<rightarrow>\<^isub>E A" using sets_into_space by auto
810   show "I.\<mu> ?P = M.\<mu> i A" unfolding *
811     using A I.measure_times[of "\<lambda>_. A"] by auto
812 qed auto
814 lemma (in product_sigma_finite) product_positive_integral_singleton:
815   assumes f: "f \<in> borel_measurable (M i)"
816   shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f"
817 proof -
818   interpret I: finite_product_sigma_finite M "{i}" by default simp
819   show ?thesis
820   proof (rule I.positive_integral_vimage[symmetric])
821     show "sigma_algebra (M i)" by (rule sigma_algebras)
822     show "(\<lambda>x. x i) \<in> measure_preserving (Pi\<^isub>M {i} M) (M i)"
823       by (rule measure_preserving_component_singelton)
824     show "f \<in> borel_measurable (M i)" by fact
825   qed
826 qed
828 lemma (in product_sigma_finite) product_positive_integral_insert:
829   assumes [simp]: "finite I" "i \<notin> I"
830     and f: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
831   shows "integral\<^isup>P (Pi\<^isub>M (insert i I) M) f = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^isub>M I M))"
832 proof -
833   interpret I: finite_product_sigma_finite M I by default auto
834   interpret i: finite_product_sigma_finite M "{i}" by default auto
835   interpret P: pair_sigma_algebra I.P i.P ..
836   have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
837     using f by auto
838   show ?thesis
839     unfolding product_positive_integral_fold[OF IJ, unfolded insert, simplified, OF f]
840   proof (rule I.positive_integral_cong, subst product_positive_integral_singleton)
841     fix x assume x: "x \<in> space I.P"
842     let ?f = "\<lambda>y. f (restrict (x(i := y)) (insert i I))"
843     have f'_eq: "\<And>y. ?f y = f (x(i := y))"
844       using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)
845     show "?f \<in> borel_measurable (M i)" unfolding f'_eq
846       using measurable_comp[OF measurable_component_update f] x `i \<notin> I`
847       by (simp add: comp_def)
848     show "integral\<^isup>P (M i) ?f = \<integral>\<^isup>+ y. f (x(i:=y)) \<partial>M i"
849       unfolding f'_eq by simp
850   qed
851 qed
853 lemma (in product_sigma_finite) product_positive_integral_setprod:
854   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
855   assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
856   and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
857   shows "(\<integral>\<^isup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>P (M i) (f i))"
858 using assms proof induct
859   case empty
860   interpret finite_product_sigma_finite M "{}" by default auto
861   show ?case by simp
862 next
863   case (insert i I)
864   note `finite I`[intro, simp]
865   interpret I: finite_product_sigma_finite M I by default auto
866   have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))"
867     using insert by (auto intro!: setprod_cong)
868   have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^isub>M J M)"
869     using sets_into_space insert
870     by (intro sigma_algebra.borel_measurable_ereal_setprod sigma_algebra_product_algebra
871               measurable_comp[OF measurable_component_singleton, unfolded comp_def])
872        auto
873   then show ?case
874     apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
875     apply (simp add: insert * pos borel setprod_ereal_pos M.positive_integral_multc)
876     apply (subst I.positive_integral_cmult)
877     apply (auto simp add: pos borel insert setprod_ereal_pos positive_integral_positive)
878     done
879 qed
881 lemma (in product_sigma_finite) product_integral_singleton:
882   assumes f: "f \<in> borel_measurable (M i)"
883   shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f"
884 proof -
885   interpret I: finite_product_sigma_finite M "{i}" by default simp
886   have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)"
887     "(\<lambda>x. ereal (- f x)) \<in> borel_measurable (M i)"
888     using assms by auto
889   show ?thesis
890     unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] ..
891 qed
893 lemma (in product_sigma_finite) product_integral_fold:
894   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
895   and f: "integrable (Pi\<^isub>M (I \<union> J) M) f"
896   shows "integral\<^isup>L (Pi\<^isub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I x J y) \<partial>Pi\<^isub>M J M) \<partial>Pi\<^isub>M I M)"
897 proof -
898   interpret I: finite_product_sigma_finite M I by default fact
899   interpret J: finite_product_sigma_finite M J by default fact
900   have "finite (I \<union> J)" using fin by auto
901   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
902   interpret P: pair_sigma_finite I.P J.P by default
903   let ?M = "\<lambda>(x, y). merge I x J y"
904   let ?f = "\<lambda>x. f (?M x)"
905   show ?thesis
906   proof (subst P.integrable_fst_measurable(2)[of ?f, simplified])
907     have 1: "sigma_algebra IJ.P" by default
908     have 2: "?M \<in> measure_preserving P.P IJ.P" using measure_preserving_merge[OF assms(1,2,3)] .
909     have 3: "integrable (Pi\<^isub>M (I \<union> J) M) f" by fact
910     then have 4: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
911       by (simp add: integrable_def)
912     show "integrable P.P ?f"
913       by (rule P.integrable_vimage[where f=f, OF 1 2 3])
914     show "integral\<^isup>L IJ.P f = integral\<^isup>L P.P ?f"
915       by (rule P.integral_vimage[where f=f, OF 1 2 4])
916   qed
917 qed
919 lemma (in product_sigma_finite) product_integral_insert:
920   assumes [simp]: "finite I" "i \<notin> I"
921     and f: "integrable (Pi\<^isub>M (insert i I) M) f"
922   shows "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = (\<integral>x. (\<integral>y. f (x(i:=y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
923 proof -
924   interpret I: finite_product_sigma_finite M I by default auto
925   interpret I': finite_product_sigma_finite M "insert i I" by default auto
926   interpret i: finite_product_sigma_finite M "{i}" by default auto
927   interpret P: pair_sigma_finite I.P i.P ..
928   have IJ: "I \<inter> {i} = {}" by auto
929   show ?thesis
930     unfolding product_integral_fold[OF IJ, simplified, OF f]
931   proof (rule I.integral_cong, subst product_integral_singleton)
932     fix x assume x: "x \<in> space I.P"
933     let ?f = "\<lambda>y. f (restrict (x(i := y)) (insert i I))"
934     have f'_eq: "\<And>y. ?f y = f (x(i := y))"
935       using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)
936     have f: "f \<in> borel_measurable I'.P" using f unfolding integrable_def by auto
937     show "?f \<in> borel_measurable (M i)"
938       unfolding measurable_cong[OF f'_eq]
939       using measurable_comp[OF measurable_component_update f] x `i \<notin> I`
940       by (simp add: comp_def)
941     show "integral\<^isup>L (M i) ?f = integral\<^isup>L (M i) (\<lambda>y. f (x(i := y)))"
942       unfolding f'_eq by simp
943   qed
944 qed
946 lemma (in product_sigma_finite) product_integrable_setprod:
947   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
948   assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
949   shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
950 proof -
951   interpret finite_product_sigma_finite M I by default fact
952   have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
953     using integrable unfolding integrable_def by auto
954   then have borel: "?f \<in> borel_measurable P"
955     using measurable_comp[OF measurable_component_singleton f]
956     by (auto intro!: borel_measurable_setprod simp: comp_def)
957   moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"
958   proof (unfold integrable_def, intro conjI)
959     show "(\<lambda>x. abs (?f x)) \<in> borel_measurable P"
960       using borel by auto
961     have "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. ereal (abs (f i (x i)))) \<partial>P)"
962       by (simp add: setprod_ereal abs_setprod)
963     also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))"
964       using f by (subst product_positive_integral_setprod) auto
965     also have "\<dots> < \<infinity>"
966       using integrable[THEN M.integrable_abs]
967       by (simp add: setprod_PInf integrable_def M.positive_integral_positive)
968     finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>P) \<noteq> \<infinity>" by auto
969     have "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. 0 \<partial>P)"
970       by (intro positive_integral_cong_pos) auto
971     then show "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>P) \<noteq> \<infinity>" by simp
972   qed
973   ultimately show ?thesis
974     by (rule integrable_abs_iff[THEN iffD1])
975 qed
977 lemma (in product_sigma_finite) product_integral_setprod:
978   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
979   assumes "finite I" "I \<noteq> {}" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
980   shows "(\<integral>x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>L (M i) (f i))"
981 using assms proof (induct rule: finite_ne_induct)
982   case (singleton i)
983   then show ?case by (simp add: product_integral_singleton integrable_def)
984 next
985   case (insert i I)
986   then have iI: "finite (insert i I)" by auto
987   then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
988     integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
989     by (intro product_integrable_setprod insert(5)) (auto intro: finite_subset)
990   interpret I: finite_product_sigma_finite M I by default fact
991   have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))"
992     using `i \<notin> I` by (auto intro!: setprod_cong)
993   show ?case
994     unfolding product_integral_insert[OF insert(1,3) prod[OF subset_refl]]
995     by (simp add: * insert integral_multc I.integral_cmult[OF prod] subset_insertI)
996 qed
998 end