src/HOL/Probability/Projective_Limit.thy
 author haftmann Fri Jun 19 07:53:35 2015 +0200 (2015-06-19) changeset 60517 f16e4fb20652 parent 58876 1888e3cb8048 child 60585 48fdff264eb2 permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
1 (*  Title:      HOL/Probability/Projective_Limit.thy
2     Author:     Fabian Immler, TU München
3 *)
5 section {* Projective Limit *}
7 theory Projective_Limit
8   imports
9     Caratheodory
10     Fin_Map
11     Regularity
12     Projective_Family
13     Infinite_Product_Measure
14     "~~/src/HOL/Library/Diagonal_Subsequence"
15 begin
17 subsection {* Sequences of Finite Maps in Compact Sets *}
19 locale finmap_seqs_into_compact =
20   fixes K::"nat \<Rightarrow> (nat \<Rightarrow>\<^sub>F 'a::metric_space) set" and f::"nat \<Rightarrow> (nat \<Rightarrow>\<^sub>F 'a)" and M
21   assumes compact: "\<And>n. compact (K n)"
22   assumes f_in_K: "\<And>n. K n \<noteq> {}"
23   assumes domain_K: "\<And>n. k \<in> K n \<Longrightarrow> domain k = domain (f n)"
24   assumes proj_in_K:
25     "\<And>t n m. m \<ge> n \<Longrightarrow> t \<in> domain (f n) \<Longrightarrow> (f m)\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K n"
26 begin
28 lemma proj_in_K': "(\<exists>n. \<forall>m \<ge> n. (f m)\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K n)"
29   using proj_in_K f_in_K
30 proof cases
31   obtain k where "k \<in> K (Suc 0)" using f_in_K by auto
32   assume "\<forall>n. t \<notin> domain (f n)"
33   thus ?thesis
34     by (auto intro!: exI[where x=1] image_eqI[OF _ `k \<in> K (Suc 0)`]
35       simp: domain_K[OF `k \<in> K (Suc 0)`])
36 qed blast
38 lemma proj_in_KE:
39   obtains n where "\<And>m. m \<ge> n \<Longrightarrow> (f m)\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K n"
40   using proj_in_K' by blast
42 lemma compact_projset:
43   shows "compact ((\<lambda>k. (k)\<^sub>F i) ` K n)"
44   using continuous_proj compact by (rule compact_continuous_image)
46 end
48 lemma compactE':
49   fixes S :: "'a :: metric_space set"
50   assumes "compact S" "\<forall>n\<ge>m. f n \<in> S"
51   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
52 proof atomize_elim
53   have "subseq (op + m)" by (simp add: subseq_def)
54   have "\<forall>n. (f o (\<lambda>i. m + i)) n \<in> S" using assms by auto
55   from seq_compactE[OF `compact S`[unfolded compact_eq_seq_compact_metric] this] guess l r .
56   hence "l \<in> S" "subseq ((\<lambda>i. m + i) o r) \<and> (f \<circ> ((\<lambda>i. m + i) o r)) ----> l"
57     using subseq_o[OF `subseq (op + m)` `subseq r`] by (auto simp: o_def)
58   thus "\<exists>l r. l \<in> S \<and> subseq r \<and> (f \<circ> r) ----> l" by blast
59 qed
61 sublocale finmap_seqs_into_compact \<subseteq> subseqs "\<lambda>n s. (\<exists>l. (\<lambda>i. ((f o s) i)\<^sub>F n) ----> l)"
62 proof
63   fix n s
64   assume "subseq s"
65   from proj_in_KE[of n] guess n0 . note n0 = this
66   have "\<forall>i \<ge> n0. ((f \<circ> s) i)\<^sub>F n \<in> (\<lambda>k. (k)\<^sub>F n) ` K n0"
67   proof safe
68     fix i assume "n0 \<le> i"
69     also have "\<dots> \<le> s i" by (rule seq_suble) fact
70     finally have "n0 \<le> s i" .
71     with n0 show "((f \<circ> s) i)\<^sub>F n \<in> (\<lambda>k. (k)\<^sub>F n) ` K n0 "
72       by auto
73   qed
74   from compactE'[OF compact_projset this] guess ls rs .
75   thus "\<exists>r'. subseq r' \<and> (\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r')) i)\<^sub>F n) ----> l)" by (auto simp: o_def)
76 qed
78 lemma (in finmap_seqs_into_compact) diagonal_tendsto: "\<exists>l. (\<lambda>i. (f (diagseq i))\<^sub>F n) ----> l"
79 proof -
80   obtain l where "(\<lambda>i. ((f o (diagseq o op + (Suc n))) i)\<^sub>F n) ----> l"
81   proof (atomize_elim, rule diagseq_holds)
82     fix r s n
83     assume "subseq r"
84     assume "\<exists>l. (\<lambda>i. ((f \<circ> s) i)\<^sub>F n) ----> l"
85     then obtain l where "((\<lambda>i. (f i)\<^sub>F n) o s) ----> l"
86       by (auto simp: o_def)
87     hence "((\<lambda>i. (f i)\<^sub>F n) o s o r) ----> l" using `subseq r`
88       by (rule LIMSEQ_subseq_LIMSEQ)
89     thus "\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r)) i)\<^sub>F n) ----> l" by (auto simp add: o_def)
90   qed
91   hence "(\<lambda>i. ((f (diagseq (i + Suc n))))\<^sub>F n) ----> l" by (simp add: ac_simps)
92   hence "(\<lambda>i. (f (diagseq i))\<^sub>F n) ----> l" by (rule LIMSEQ_offset)
93   thus ?thesis ..
94 qed
96 subsection {* Daniell-Kolmogorov Theorem *}
98 text {* Existence of Projective Limit *}
100 locale polish_projective = projective_family I P "\<lambda>_. borel::'a::polish_space measure"
101   for I::"'i set" and P
102 begin
104 abbreviation "lim\<^sub>B \<equiv> (\<lambda>J P. limP J (\<lambda>_. borel) P)"
106 lemma emeasure_limB_emb_not_empty:
107   assumes "I \<noteq> {}"
108   assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
109   shows "emeasure (lim\<^sub>B I P) (emb I J (Pi\<^sub>E J B)) = emeasure (lim\<^sub>B J P) (Pi\<^sub>E J B)"
110 proof -
111   let ?\<Omega> = "\<Pi>\<^sub>E i\<in>I. space borel"
112   let ?G = generator
113   interpret G!: algebra ?\<Omega> generator by (intro  algebra_generator) fact
114   note mu_G_mono =
116       THEN increasingD]
117   write mu_G  ("\<mu>G")
119   have "\<exists>\<mu>. (\<forall>s\<in>?G. \<mu> s = \<mu>G s) \<and> measure_space ?\<Omega> (sigma_sets ?\<Omega> ?G) \<mu>"
120   proof (rule G.caratheodory_empty_continuous[OF positive_mu_G additive_mu_G,
121       OF `I \<noteq> {}`, OF `I \<noteq> {}`])
122     fix A assume "A \<in> ?G"
123     with generatorE guess J X . note JX = this
124     interpret prob_space "P J" using proj_prob_space[OF `finite J`] .
125     show "\<mu>G A \<noteq> \<infinity>" using JX by (simp add: limP_finite)
126   next
127     fix Z assume Z: "range Z \<subseteq> ?G" "decseq Z" "(\<Inter>i. Z i) = {}"
128     then have "decseq (\<lambda>i. \<mu>G (Z i))"
129       by (auto intro!: mu_G_mono simp: decseq_def)
130     moreover
131     have "(INF i. \<mu>G (Z i)) = 0"
132     proof (rule ccontr)
133       assume "(INF i. \<mu>G (Z i)) \<noteq> 0" (is "?a \<noteq> 0")
134       moreover have "0 \<le> ?a"
135         using Z positive_mu_G[OF `I \<noteq> {}`] by (auto intro!: INF_greatest simp: positive_def)
136       ultimately have "0 < ?a" by auto
137       hence "?a \<noteq> -\<infinity>" by auto
138       have "\<forall>n. \<exists>J B. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> B \<in> sets (Pi\<^sub>M J (\<lambda>_. borel)) \<and>
139         Z n = emb I J B \<and> \<mu>G (Z n) = emeasure (lim\<^sub>B J P) B"
140         using Z by (intro allI generator_Ex) auto
141       then obtain J' B' where J': "\<And>n. J' n \<noteq> {}" "\<And>n. finite (J' n)" "\<And>n. J' n \<subseteq> I"
142           "\<And>n. B' n \<in> sets (\<Pi>\<^sub>M i\<in>J' n. borel)"
143         and Z_emb: "\<And>n. Z n = emb I (J' n) (B' n)"
144         unfolding choice_iff by blast
145       moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)"
146       moreover def B \<equiv> "\<lambda>n. emb (J n) (J' n) (B' n)"
147       ultimately have J: "\<And>n. J n \<noteq> {}" "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I"
148         "\<And>n. B n \<in> sets (\<Pi>\<^sub>M i\<in>J n. borel)"
149         by auto
150       have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
151         unfolding J_def by force
152       have "\<forall>n. \<exists>j. j \<in> J n" using J by blast
153       then obtain j where j: "\<And>n. j n \<in> J n"
154         unfolding choice_iff by blast
155       note [simp] = `\<And>n. finite (J n)`
156       from J  Z_emb have Z_eq: "\<And>n. Z n = emb I (J n) (B n)" "\<And>n. Z n \<in> ?G"
157         unfolding J_def B_def by (subst prod_emb_trans) (insert Z, auto)
158       interpret prob_space "P (J i)" for i using proj_prob_space by simp
159       have "?a \<le> \<mu>G (Z 0)" by (auto intro: INF_lower)
160       also have "\<dots> < \<infinity>" using J by (auto simp: Z_eq mu_G_eq limP_finite proj_sets)
161       finally have "?a \<noteq> \<infinity>" by simp
162       have "\<And>n. \<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>" unfolding Z_eq using J J_mono
163         by (subst mu_G_eq) (auto simp: limP_finite proj_sets mu_G_eq)
165       have countable_UN_J: "countable (\<Union>n. J n)" by (simp add: countable_finite)
166       def Utn \<equiv> "to_nat_on (\<Union>n. J n)"
167       interpret function_to_finmap "J n" Utn "from_nat_into (\<Union>n. J n)" for n
168         by unfold_locales (auto simp: Utn_def intro: from_nat_into_to_nat_on[OF countable_UN_J])
169       have inj_on_Utn: "inj_on Utn (\<Union>n. J n)"
170         unfolding Utn_def using countable_UN_J by (rule inj_on_to_nat_on)
171       hence inj_on_Utn_J: "\<And>n. inj_on Utn (J n)" by (rule subset_inj_on) auto
172       def P' \<equiv> "\<lambda>n. mapmeasure n (P (J n)) (\<lambda>_. borel)"
173       let ?SUP = "\<lambda>n. SUP K : {K. K \<subseteq> fm n ` (B n) \<and> compact K}. emeasure (P' n) K"
174       {
175         fix n
176         interpret finite_measure "P (J n)" by unfold_locales
177         have "emeasure (P (J n)) (B n) = emeasure (P' n) (fm n ` (B n))"
178           using J by (auto simp: P'_def mapmeasure_PiM proj_space proj_sets)
179         also
180         have "\<dots> = ?SUP n"
181         proof (rule inner_regular)
182           show "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>"
183             unfolding P'_def
184             by (auto simp: P'_def mapmeasure_PiF fm_measurable proj_space proj_sets)
185           show "sets (P' n) = sets borel" by (simp add: borel_eq_PiF_borel P'_def)
186         next
187           show "fm n ` B n \<in> sets borel"
188             unfolding borel_eq_PiF_borel
189             by (auto simp del: J(2) simp: P'_def fm_image_measurable_finite proj_sets J)
190         qed
191         finally
192         have "emeasure (P (J n)) (B n) = ?SUP n" "?SUP n \<noteq> \<infinity>" "?SUP n \<noteq> - \<infinity>" by auto
193       } note R = this
194       have "\<forall>n. \<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> 2 powr (-n) * ?a
195         \<and> compact K \<and> K \<subseteq> fm n ` B n"
196       proof
197         fix n
198         have "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>"
199           by (simp add: mapmeasure_PiF P'_def proj_space proj_sets)
200         then interpret finite_measure "P' n" ..
201         show "\<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> ereal (2 powr - real n) * ?a \<and>
202             compact K \<and> K \<subseteq> fm n ` B n"
203           unfolding R
204         proof (rule ccontr)
205           assume H: "\<not> (\<exists>K'. ?SUP n - emeasure (P' n) K' \<le> ereal (2 powr - real n)  * ?a \<and>
206             compact K' \<and> K' \<subseteq> fm n ` B n)"
207           have "?SUP n \<le> ?SUP n - 2 powr (-n) * ?a"
208           proof (intro SUP_least)
209             fix K
210             assume "K \<in> {K. K \<subseteq> fm n ` B n \<and> compact K}"
211             with H have "\<not> ?SUP n - emeasure (P' n) K \<le> 2 powr (-n) * ?a"
212               by auto
213             hence "?SUP n - emeasure (P' n) K > 2 powr (-n) * ?a"
214               unfolding not_less[symmetric] by simp
215             hence "?SUP n - 2 powr (-n) * ?a > emeasure (P' n) K"
216               using `0 < ?a` by (auto simp add: ereal_less_minus_iff ac_simps)
217             thus "?SUP n - 2 powr (-n) * ?a \<ge> emeasure (P' n) K" by simp
218           qed
219           hence "?SUP n + 0 \<le> ?SUP n - (2 powr (-n) * ?a)" using `0 < ?a` by simp
220           hence "?SUP n + 0 \<le> ?SUP n + - (2 powr (-n) * ?a)" unfolding minus_ereal_def .
221           hence "0 \<le> - (2 powr (-n) * ?a)"
222             using `?SUP _ \<noteq> \<infinity>` `?SUP _ \<noteq> - \<infinity>`
224           moreover have "ereal (2 powr - real n) * ?a > 0" using `0 < ?a`
225             by (auto simp: ereal_zero_less_0_iff)
226           ultimately show False by simp
227         qed
228       qed
229       then obtain K' where K':
230         "\<And>n. emeasure (P (J n)) (B n) - emeasure (P' n) (K' n) \<le> ereal (2 powr - real n) * ?a"
231         "\<And>n. compact (K' n)" "\<And>n. K' n \<subseteq> fm n ` B n"
232         unfolding choice_iff by blast
233       def K \<equiv> "\<lambda>n. fm n -` K' n \<inter> space (Pi\<^sub>M (J n) (\<lambda>_. borel))"
234       have K_sets: "\<And>n. K n \<in> sets (Pi\<^sub>M (J n) (\<lambda>_. borel))"
235         unfolding K_def
236         using compact_imp_closed[OF `compact (K' _)`]
237         by (intro measurable_sets[OF fm_measurable, of _ "Collect finite"])
238            (auto simp: borel_eq_PiF_borel[symmetric])
239       have K_B: "\<And>n. K n \<subseteq> B n"
240       proof
241         fix x n
242         assume "x \<in> K n" hence fm_in: "fm n x \<in> fm n ` B n"
243           using K' by (force simp: K_def)
244         show "x \<in> B n"
245           using `x \<in> K n` K_sets sets.sets_into_space J[of n]
246           by (intro inj_on_image_mem_iff[OF inj_on_fm _ fm_in, of "\<lambda>_. borel"]) auto
247       qed
248       def Z' \<equiv> "\<lambda>n. emb I (J n) (K n)"
249       have Z': "\<And>n. Z' n \<subseteq> Z n"
250         unfolding Z_eq unfolding Z'_def
251       proof (rule prod_emb_mono, safe)
252         fix n x assume "x \<in> K n"
253         hence "fm n x \<in> K' n" "x \<in> space (Pi\<^sub>M (J n) (\<lambda>_. borel))"
254           by (simp_all add: K_def proj_space)
255         note this(1)
256         also have "K' n \<subseteq> fm n ` B n" by (simp add: K')
257         finally have "fm n x \<in> fm n ` B n" .
258         thus "x \<in> B n"
259         proof safe
260           fix y assume y: "y \<in> B n"
261           hence "y \<in> space (Pi\<^sub>M (J n) (\<lambda>_. borel))" using J sets.sets_into_space[of "B n" "P (J n)"]
262             by (auto simp add: proj_space proj_sets)
263           assume "fm n x = fm n y"
264           note inj_onD[OF inj_on_fm[OF space_borel],
265             OF `fm n x = fm n y` `x \<in> space _` `y \<in> space _`]
266           with y show "x \<in> B n" by simp
267         qed
268       qed
269       { fix n
270         have "Z' n \<in> ?G" using K' unfolding Z'_def
271           apply (intro generatorI'[OF J(1-3)])
272           unfolding K_def proj_space
273           apply (rule measurable_sets[OF fm_measurable[of _ "Collect finite"]])
274           apply (auto simp add: P'_def borel_eq_PiF_borel[symmetric] compact_imp_closed)
275           done
276       }
277       def Y \<equiv> "\<lambda>n. \<Inter>i\<in>{1..n}. Z' i"
278       hence "\<And>n k. Y (n + k) \<subseteq> Y n" by (induct_tac k) (auto simp: Y_def)
279       hence Y_mono: "\<And>n m. n \<le> m \<Longrightarrow> Y m \<subseteq> Y n" by (auto simp: le_iff_add)
280       have Y_Z': "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z' n" by (auto simp: Y_def)
281       hence Y_Z: "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z n" using Z' by auto
282       have Y_notempty: "\<And>n. n \<ge> 1 \<Longrightarrow> (Y n) \<noteq> {}"
283       proof -
284         fix n::nat assume "n \<ge> 1" hence "Y n \<subseteq> Z n" by fact
285         have "Y n = (\<Inter> i\<in>{1..n}. emb I (J n) (emb (J n) (J i) (K i)))" using J J_mono
286           by (auto simp: Y_def Z'_def)
287         also have "\<dots> = prod_emb I (\<lambda>_. borel) (J n) (\<Inter> i\<in>{1..n}. emb (J n) (J i) (K i))"
288           using `n \<ge> 1`
289           by (subst prod_emb_INT) auto
290         finally
291         have Y_emb:
292           "Y n = prod_emb I (\<lambda>_. borel) (J n)
293             (\<Inter> i\<in>{1..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" .
294         hence "Y n \<in> ?G" using J J_mono K_sets `n \<ge> 1` by (intro generatorI[OF _ _ _ _ Y_emb]) auto
295         hence "\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>" unfolding Y_emb using J J_mono K_sets `n \<ge> 1`
296           by (subst mu_G_eq) (auto simp: limP_finite proj_sets mu_G_eq)
297         interpret finite_measure "(limP (J n) (\<lambda>_. borel) P)"
298         proof
299           have "emeasure (limP (J n) (\<lambda>_. borel) P) (J n \<rightarrow>\<^sub>E space borel) \<noteq> \<infinity>"
300             using J by (subst emeasure_limP) auto
301           thus  "emeasure (limP (J n) (\<lambda>_. borel) P) (space (limP (J n) (\<lambda>_. borel) P)) \<noteq> \<infinity>"
303         qed
304         have "\<mu>G (Z n) = limP (J n) (\<lambda>_. borel) P (B n)"
305           unfolding Z_eq using J by (auto simp: mu_G_eq)
306         moreover have "\<mu>G (Y n) =
307           limP (J n) (\<lambda>_. borel) P (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))"
308           unfolding Y_emb using J J_mono K_sets `n \<ge> 1` by (subst mu_G_eq) auto
309         moreover have "\<mu>G (Z n - Y n) = limP (J n) (\<lambda>_. borel) P
310           (B n - (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))"
311           unfolding Z_eq Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets `n \<ge> 1`
312           by (subst mu_G_eq) (auto intro!: sets.Diff)
313         ultimately
314         have "\<mu>G (Z n) - \<mu>G (Y n) = \<mu>G (Z n - Y n)"
315           using J J_mono K_sets `n \<ge> 1`
316           by (simp only: emeasure_eq_measure)
317             (auto dest!: bspec[where x=n]
318             simp: extensional_restrict emeasure_eq_measure prod_emb_iff simp del: limP_finite
319             intro!: measure_Diff[symmetric] set_mp[OF K_B])
320         also have subs: "Z n - Y n \<subseteq> (\<Union> i\<in>{1..n}. (Z i - Z' i))" using Z' Z `n \<ge> 1`
321           unfolding Y_def by (force simp: decseq_def)
322         have "Z n - Y n \<in> ?G" "(\<Union> i\<in>{1..n}. (Z i - Z' i)) \<in> ?G"
323           using `Z' _ \<in> ?G` `Z _ \<in> ?G` `Y _ \<in> ?G` by auto
324         hence "\<mu>G (Z n - Y n) \<le> \<mu>G (\<Union> i\<in>{1..n}. (Z i - Z' i))"
325           using subs G.additive_increasing[OF positive_mu_G[OF `I \<noteq> {}`] additive_mu_G[OF `I \<noteq> {}`]]
326           unfolding increasing_def by auto
327         also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. \<mu>G (Z i - Z' i))" using `Z _ \<in> ?G` `Z' _ \<in> ?G`
328           by (intro G.subadditive[OF positive_mu_G additive_mu_G, OF `I \<noteq> {}` `I \<noteq> {}`]) auto
329         also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. 2 powr -real i * ?a)"
330         proof (rule setsum_mono)
331           fix i assume "i \<in> {1..n}" hence "i \<le> n" by simp
332           have "\<mu>G (Z i - Z' i) = \<mu>G (prod_emb I (\<lambda>_. borel) (J i) (B i - K i))"
333             unfolding Z'_def Z_eq by simp
334           also have "\<dots> = P (J i) (B i - K i)"
335             using J K_sets by (subst mu_G_eq) auto
336           also have "\<dots> = P (J i) (B i) - P (J i) (K i)"
337             apply (subst emeasure_Diff) using K_sets J `K _ \<subseteq> B _` apply (auto simp: proj_sets)
338             done
339           also have "\<dots> = P (J i) (B i) - P' i (K' i)"
340             unfolding K_def P'_def
341             by (auto simp: mapmeasure_PiF proj_space proj_sets borel_eq_PiF_borel[symmetric]
342               compact_imp_closed[OF `compact (K' _)`] space_PiM PiE_def)
343           also have "\<dots> \<le> ereal (2 powr - real i) * ?a" using K'(1)[of i] .
344           finally show "\<mu>G (Z i - Z' i) \<le> (2 powr - real i) * ?a" .
345         qed
346         also have "\<dots> = (\<Sum> i\<in>{1..n}. ereal (2 powr -real i) * ereal(real ?a))"
347           using `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` by (subst ereal_real') auto
348         also have "\<dots> = ereal (\<Sum> i\<in>{1..n}. (2 powr -real i) * (real ?a))" by simp
349         also have "\<dots> = ereal ((\<Sum> i\<in>{1..n}. (2 powr -real i)) * real ?a)"
351         also have "\<dots> < ereal (1 * real ?a)" unfolding less_ereal.simps
352         proof (rule mult_strict_right_mono)
353           have "(\<Sum>i\<in>{1..n}. 2 powr - real i) = (\<Sum>i\<in>{1..<Suc n}. (1/2) ^ i)"
354             by (rule setsum.cong)
355                (auto simp: powr_realpow[symmetric] powr_minus powr_divide inverse_eq_divide)
356           also have "{1..<Suc n} = {..<Suc n} - {0}" by auto
357           also have "setsum (op ^ (1 / 2::real)) ({..<Suc n} - {0}) =
358             setsum (op ^ (1 / 2)) ({..<Suc n}) - 1" by (auto simp: setsum_diff1)
359           also have "\<dots> < 1" by (subst geometric_sum) auto
360           finally show "(\<Sum>i = 1..n. 2 powr - real i) < 1" .
361         qed (auto simp:
362           `0 < ?a` `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` ereal_less_real_iff zero_ereal_def[symmetric])
363         also have "\<dots> = ?a" using `0 < ?a` `?a \<noteq> \<infinity>` by (auto simp: ereal_real')
364         also have "\<dots> \<le> \<mu>G (Z n)" by (auto intro: INF_lower)
365         finally have "\<mu>G (Z n) - \<mu>G (Y n) < \<mu>G (Z n)" .
366         hence R: "\<mu>G (Z n) < \<mu>G (Z n) + \<mu>G (Y n)"
367           using `\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>` by (simp add: ereal_minus_less)
368         have "0 \<le> (- \<mu>G (Z n)) + \<mu>G (Z n)" using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
369         also have "\<dots> < (- \<mu>G (Z n)) + (\<mu>G (Z n) + \<mu>G (Y n))"
370           apply (rule ereal_less_add[OF _ R]) using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
371         finally have "\<mu>G (Y n) > 0"
372           using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by (auto simp: ac_simps zero_ereal_def[symmetric])
373         thus "Y n \<noteq> {}" using positive_mu_G `I \<noteq> {}` by (auto simp add: positive_def)
374       qed
375       hence "\<forall>n\<in>{1..}. \<exists>y. y \<in> Y n" by auto
376       then obtain y where y: "\<And>n. n \<ge> 1 \<Longrightarrow> y n \<in> Y n" unfolding bchoice_iff by force
377       {
378         fix t and n m::nat
379         assume "1 \<le> n" "n \<le> m" hence "1 \<le> m" by simp
380         from Y_mono[OF `m \<ge> n`] y[OF `1 \<le> m`] have "y m \<in> Y n" by auto
381         also have "\<dots> \<subseteq> Z' n" using Y_Z'[OF `1 \<le> n`] .
382         finally
383         have "fm n (restrict (y m) (J n)) \<in> K' n"
384           unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
385         moreover have "finmap_of (J n) (restrict (y m) (J n)) = finmap_of (J n) (y m)"
386           using J by (simp add: fm_def)
387         ultimately have "fm n (y m) \<in> K' n" by simp
388       } note fm_in_K' = this
389       interpret finmap_seqs_into_compact "\<lambda>n. K' (Suc n)" "\<lambda>k. fm (Suc k) (y (Suc k))" borel
390       proof
391         fix n show "compact (K' n)" by fact
392       next
393         fix n
394         from Y_mono[of n "Suc n"] y[of "Suc n"] have "y (Suc n) \<in> Y (Suc n)" by auto
395         also have "\<dots> \<subseteq> Z' (Suc n)" using Y_Z' by auto
396         finally
397         have "fm (Suc n) (restrict (y (Suc n)) (J (Suc n))) \<in> K' (Suc n)"
398           unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
399         thus "K' (Suc n) \<noteq> {}" by auto
400         fix k
401         assume "k \<in> K' (Suc n)"
402         with K'[of "Suc n"] sets.sets_into_space have "k \<in> fm (Suc n) ` B (Suc n)" by auto
403         then obtain b where "k = fm (Suc n) b" by auto
404         thus "domain k = domain (fm (Suc n) (y (Suc n)))"
406       next
407         fix t and n m::nat
408         assume "n \<le> m" hence "Suc n \<le> Suc m" by simp
409         assume "t \<in> domain (fm (Suc n) (y (Suc n)))"
410         then obtain j where j: "t = Utn j" "j \<in> J (Suc n)" by auto
411         hence "j \<in> J (Suc m)" using J_mono[OF `Suc n \<le> Suc m`] by auto
412         have img: "fm (Suc n) (y (Suc m)) \<in> K' (Suc n)" using `n \<le> m`
413           by (intro fm_in_K') simp_all
414         show "(fm (Suc m) (y (Suc m)))\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K' (Suc n)"
415           apply (rule image_eqI[OF _ img])
416           using `j \<in> J (Suc n)` `j \<in> J (Suc m)`
417           unfolding j by (subst proj_fm, auto)+
418       qed
419       have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) ----> z"
420         using diagonal_tendsto ..
421       then obtain z where z:
422         "\<And>t. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) ----> z t"
423         unfolding choice_iff by blast
424       {
425         fix n :: nat assume "n \<ge> 1"
426         have "\<And>i. domain (fm n (y (Suc (diagseq i)))) = domain (finmap_of (Utn ` J n) z)"
427           by simp
428         moreover
429         {
430           fix t
431           assume t: "t \<in> domain (finmap_of (Utn ` J n) z)"
432           hence "t \<in> Utn ` J n" by simp
433           then obtain j where j: "t = Utn j" "j \<in> J n" by auto
434           have "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) ----> z t"
435             apply (subst (2) tendsto_iff, subst eventually_sequentially)
436           proof safe
437             fix e :: real assume "0 < e"
438             { fix i x
439               assume i: "i \<ge> n"
440               assume "t \<in> domain (fm n x)"
441               hence "t \<in> domain (fm i x)" using J_mono[OF `i \<ge> n`] by auto
442               with i have "(fm i x)\<^sub>F t = (fm n x)\<^sub>F t"
443                 using j by (auto simp: proj_fm dest!: inj_onD[OF inj_on_Utn])
444             } note index_shift = this
445             have I: "\<And>i. i \<ge> n \<Longrightarrow> Suc (diagseq i) \<ge> n"
446               apply (rule le_SucI)
447               apply (rule order_trans) apply simp
448               apply (rule seq_suble[OF subseq_diagseq])
449               done
450             from z
451             have "\<exists>N. \<forall>i\<ge>N. dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e"
452               unfolding tendsto_iff eventually_sequentially using `0 < e` by auto
453             then obtain N where N: "\<And>i. i \<ge> N \<Longrightarrow>
454               dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e" by auto
455             show "\<exists>N. \<forall>na\<ge>N. dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e "
456             proof (rule exI[where x="max N n"], safe)
457               fix na assume "max N n \<le> na"
458               hence  "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) =
459                       dist ((fm (Suc (diagseq na)) (y (Suc (diagseq na))))\<^sub>F t) (z t)" using t
460                 by (subst index_shift[OF I]) auto
461               also have "\<dots> < e" using `max N n \<le> na` by (intro N) simp
462               finally show "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e" .
463             qed
464           qed
465           hence "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) ----> (finmap_of (Utn ` J n) z)\<^sub>F t"
467         } ultimately
468         have "(\<lambda>i. fm n (y (Suc (diagseq i)))) ----> finmap_of (Utn ` J n) z"
469           by (rule tendsto_finmap)
470         hence "((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) ----> finmap_of (Utn ` J n) z"
471           by (intro lim_subseq) (simp add: subseq_def)
472         moreover
473         have "(\<forall>i. ((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) i \<in> K' n)"
474           apply (auto simp add: o_def intro!: fm_in_K' `1 \<le> n` le_SucI)
475           apply (rule le_trans)
477           using seq_suble[OF subseq_diagseq]
478           apply auto
479           done
480         moreover
481         from `compact (K' n)` have "closed (K' n)" by (rule compact_imp_closed)
482         ultimately
483         have "finmap_of (Utn ` J n) z \<in> K' n"
484           unfolding closed_sequential_limits by blast
485         also have "finmap_of (Utn ` J n) z  = fm n (\<lambda>i. z (Utn i))"
486           unfolding finmap_eq_iff
487         proof clarsimp
488           fix i assume i: "i \<in> J n"
489           hence "from_nat_into (\<Union>n. J n) (Utn i) = i"
490             unfolding Utn_def
491             by (subst from_nat_into_to_nat_on[OF countable_UN_J]) auto
492           with i show "z (Utn i) = (fm n (\<lambda>i. z (Utn i)))\<^sub>F (Utn i)"
493             by (simp add: finmap_eq_iff fm_def compose_def)
494         qed
495         finally have "fm n (\<lambda>i. z (Utn i)) \<in> K' n" .
496         moreover
497         let ?J = "\<Union>n. J n"
498         have "(?J \<inter> J n) = J n" by auto
499         ultimately have "restrict (\<lambda>i. z (Utn i)) (?J \<inter> J n) \<in> K n"
500           unfolding K_def by (auto simp: proj_space space_PiM)
501         hence "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z' n" unfolding Z'_def
502           using J by (auto simp: prod_emb_def PiE_def extensional_def)
503         also have "\<dots> \<subseteq> Z n" using Z' by simp
504         finally have "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z n" .
505       } note in_Z = this
506       hence "(\<Inter>i\<in>{1..}. Z i) \<noteq> {}" by auto
507       hence "(\<Inter>i. Z i) \<noteq> {}" using Z INT_decseq_offset[OF `decseq Z`] by simp
508       thus False using Z by simp
509     qed
510     ultimately show "(\<lambda>i. \<mu>G (Z i)) ----> 0"
511       using LIMSEQ_INF[of "\<lambda>i. \<mu>G (Z i)"] by simp
512   qed
513   then guess \<mu> .. note \<mu> = this
514   def f \<equiv> "finmap_of J B"
515   show "emeasure (lim\<^sub>B I P) (emb I J (Pi\<^sub>E J B)) = emeasure (lim\<^sub>B J P) (Pi\<^sub>E J B)"
516   proof (subst emeasure_extend_measure_Pair[OF limP_def, of I "\<lambda>_. borel" \<mu>])
517     show "positive (sets (lim\<^sub>B I P)) \<mu>" "countably_additive (sets (lim\<^sub>B I P)) \<mu>"
518       using \<mu> unfolding sets_limP sets_PiM_generator by (auto simp: measure_space_def)
519   next
520     show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> B \<in> J \<rightarrow> sets borel"
521       using assms by (auto simp: f_def)
522   next
523     fix J and X::"'i \<Rightarrow> 'a set"
524     show "prod_emb I (\<lambda>_. borel) J (Pi\<^sub>E J X) \<in> Pow (I \<rightarrow>\<^sub>E space borel)"
525       by (auto simp: prod_emb_def)
526     assume JX: "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> J \<rightarrow> sets borel"
527     hence "emb I J (Pi\<^sub>E J X) \<in> generator" using assms
528       by (intro generatorI[where J=J and X="Pi\<^sub>E J X"]) (auto intro: sets_PiM_I_finite)
529     hence "\<mu> (emb I J (Pi\<^sub>E J X)) = \<mu>G (emb I J (Pi\<^sub>E J X))" using \<mu> by simp
530     also have "\<dots> = emeasure (P J) (Pi\<^sub>E J X)"
531       using JX assms proj_sets
532       by (subst mu_G_eq) (auto simp: mu_G_eq limP_finite intro: sets_PiM_I_finite)
533     finally show "\<mu> (emb I J (Pi\<^sub>E J X)) = emeasure (P J) (Pi\<^sub>E J X)" .
534   next
535     show "emeasure (P J) (Pi\<^sub>E J B) = emeasure (limP J (\<lambda>_. borel) P) (Pi\<^sub>E J B)"
536       using assms by (simp add: f_def limP_finite Pi_def)
537   qed
538 qed
540 end
542 hide_const (open) PiF
543 hide_const (open) Pi\<^sub>F
544 hide_const (open) Pi'
545 hide_const (open) Abs_finmap
546 hide_const (open) Rep_finmap
547 hide_const (open) finmap_of
548 hide_const (open) proj
549 hide_const (open) domain
550 hide_const (open) basis_finmap
552 sublocale polish_projective \<subseteq> P!: prob_space "(lim\<^sub>B I P)"
553 proof
554   show "emeasure (lim\<^sub>B I P) (space (lim\<^sub>B I P)) = 1"
555   proof cases
556     assume "I = {}"
557     interpret prob_space "P {}" using proj_prob_space by simp
558     show ?thesis
559       by (simp add: space_PiM_empty limP_finite emeasure_space_1 `I = {}`)
560   next
561     assume "I \<noteq> {}"
562     then obtain i where "i \<in> I" by auto
563     interpret prob_space "P {i}" using proj_prob_space by simp
564     have R: "(space (lim\<^sub>B I P)) = (emb I {i} (Pi\<^sub>E {i} (\<lambda>_. space borel)))"
565       by (auto simp: prod_emb_def space_PiM)
566     moreover have "extensional {i} = space (P {i})" by (simp add: proj_space space_PiM PiE_def)
567     ultimately show ?thesis using `i \<in> I`
568       apply (subst R)
569       apply (subst emeasure_limB_emb_not_empty)
570       apply (auto simp: limP_finite emeasure_space_1 PiE_def)
571       done
572   qed
573 qed
575 context polish_projective begin
577 lemma emeasure_limB_emb:
578   assumes X: "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
579   shows "emeasure (lim\<^sub>B I P) (emb I J (Pi\<^sub>E J B)) = emeasure (P J) (Pi\<^sub>E J B)"
580 proof cases
581   interpret prob_space "P {}" using proj_prob_space by simp
582   assume "J = {}"
583   moreover have "emb I {} {\<lambda>x. undefined} = space (lim\<^sub>B I P)"
584     by (auto simp: space_PiM prod_emb_def)
585   moreover have "{\<lambda>x. undefined} = space (lim\<^sub>B {} P)"
586     by (auto simp: space_PiM prod_emb_def simp del: limP_finite)
587   ultimately show ?thesis
588     by (simp add: P.emeasure_space_1 limP_finite emeasure_space_1 del: space_limP)
589 next
590   assume "J \<noteq> {}" with X show ?thesis
591     by (subst emeasure_limB_emb_not_empty) (auto simp: limP_finite)
592 qed
594 lemma measure_limB_emb:
595   assumes "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
596   shows "measure (lim\<^sub>B I P) (emb I J (Pi\<^sub>E J B)) = measure (P J) (Pi\<^sub>E J B)"
597 proof -
598   interpret prob_space "P J" using proj_prob_space assms by simp
599   show ?thesis
600     using emeasure_limB_emb[OF assms]
601     unfolding emeasure_eq_measure limP_finite[OF `finite J` `J \<subseteq> I`] P.emeasure_eq_measure
602     by simp
603 qed
605 end
607 locale polish_product_prob_space =
608   product_prob_space "\<lambda>_. borel::('a::polish_space) measure" I for I::"'i set"
610 sublocale polish_product_prob_space \<subseteq> P: polish_projective I "\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)"
611 proof qed
613 lemma (in polish_product_prob_space) limP_eq_PiM:
614   "I \<noteq> {} \<Longrightarrow> lim\<^sub>P I (\<lambda>_. borel) (\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)) =
615     PiM I (\<lambda>_. borel)"
616   by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_limB_emb)
618 end