src/HOL/Probability/Projective_Limit.thy
 author wenzelm Wed Nov 28 15:59:18 2012 +0100 (2012-11-28) changeset 50252 4aa34bd43228 parent 50245 dea9363887a6 child 50884 2b21b4e2d7cb permissions -rw-r--r--
eliminated slightly odd identifiers;
```     1 (*  Title:      HOL/Probability/Projective_Limit.thy
```
```     2     Author:     Fabian Immler, TU München
```
```     3 *)
```
```     4
```
```     5 header {* Projective Limit *}
```
```     6
```
```     7 theory Projective_Limit
```
```     8   imports
```
```     9     Caratheodory
```
```    10     Fin_Map
```
```    11     Regularity
```
```    12     Projective_Family
```
```    13     Infinite_Product_Measure
```
```    14     "~~/src/HOL/Library/Countable_Set"
```
```    15 begin
```
```    16
```
```    17 subsection {* Sequences of Finite Maps in Compact Sets *}
```
```    18
```
```    19 locale finmap_seqs_into_compact =
```
```    20   fixes K::"nat \<Rightarrow> (nat \<Rightarrow>\<^isub>F 'a::metric_space) set" and f::"nat \<Rightarrow> (nat \<Rightarrow>\<^isub>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)\<^isub>F t \<in> (\<lambda>k. (k)\<^isub>F t) ` K n"
```
```    26 begin
```
```    27
```
```    28 lemma proj_in_K': "(\<exists>n. \<forall>m \<ge> n. (f m)\<^isub>F t \<in> (\<lambda>k. (k)\<^isub>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
```
```    37
```
```    38 lemma proj_in_KE:
```
```    39   obtains n where "\<And>m. m \<ge> n \<Longrightarrow> (f m)\<^isub>F t \<in> (\<lambda>k. (k)\<^isub>F t) ` K n"
```
```    40   using proj_in_K' by blast
```
```    41
```
```    42 lemma compact_projset:
```
```    43   shows "compact ((\<lambda>k. (k)\<^isub>F i) ` K n)"
```
```    44   using continuous_proj compact by (rule compact_continuous_image)
```
```    45
```
```    46 end
```
```    47
```
```    48 lemma compactE':
```
```    49   assumes "compact S" "\<forall>n\<ge>m. f n \<in> S"
```
```    50   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
```
```    51 proof atomize_elim
```
```    52   have "subseq (op + m)" by (simp add: subseq_def)
```
```    53   have "\<forall>n. (f o (\<lambda>i. m + i)) n \<in> S" using assms by auto
```
```    54   from compactE[OF `compact S` this] guess l r .
```
```    55   hence "l \<in> S" "subseq ((\<lambda>i. m + i) o r) \<and> (f \<circ> ((\<lambda>i. m + i) o r)) ----> l"
```
```    56     using subseq_o[OF `subseq (op + m)` `subseq r`] by (auto simp: o_def)
```
```    57   thus "\<exists>l r. l \<in> S \<and> subseq r \<and> (f \<circ> r) ----> l" by blast
```
```    58 qed
```
```    59
```
```    60 sublocale finmap_seqs_into_compact \<subseteq> subseqs "\<lambda>n s. (\<exists>l. (\<lambda>i. ((f o s) i)\<^isub>F n) ----> l)"
```
```    61 proof
```
```    62   fix n s
```
```    63   assume "subseq s"
```
```    64   from proj_in_KE[of n] guess n0 . note n0 = this
```
```    65   have "\<forall>i \<ge> n0. ((f \<circ> s) i)\<^isub>F n \<in> (\<lambda>k. (k)\<^isub>F n) ` K n0"
```
```    66   proof safe
```
```    67     fix i assume "n0 \<le> i"
```
```    68     also have "\<dots> \<le> s i" by (rule seq_suble) fact
```
```    69     finally have "n0 \<le> s i" .
```
```    70     with n0 show "((f \<circ> s) i)\<^isub>F n \<in> (\<lambda>k. (k)\<^isub>F n) ` K n0 "
```
```    71       by auto
```
```    72   qed
```
```    73   from compactE'[OF compact_projset this] guess ls rs .
```
```    74   thus "\<exists>r'. subseq r' \<and> (\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r')) i)\<^isub>F n) ----> l)" by (auto simp: o_def)
```
```    75 qed
```
```    76
```
```    77 lemma (in finmap_seqs_into_compact) diagonal_tendsto: "\<exists>l. (\<lambda>i. (f (diagseq i))\<^isub>F n) ----> l"
```
```    78 proof -
```
```    79   have "\<And>i n0. (f o seqseq i) i = f (diagseq i)" unfolding diagseq_def by simp
```
```    80   from reducer_reduces obtain l where l: "(\<lambda>i. ((f \<circ> seqseq (Suc n)) i)\<^isub>F n) ----> l"
```
```    81     unfolding seqseq_reducer
```
```    82   by auto
```
```    83   have "(\<lambda>i. (f (diagseq (i + Suc n)))\<^isub>F n) =
```
```    84     (\<lambda>i. ((f o (diagseq o (op + (Suc n)))) i)\<^isub>F n)" by (simp add: add_commute)
```
```    85   also have "\<dots> =
```
```    86     (\<lambda>i. ((f o ((seqseq (Suc n) o (\<lambda>x. fold_reduce (Suc n) x (Suc n + x))))) i)\<^isub>F n)"
```
```    87     unfolding diagseq_seqseq by simp
```
```    88   also have "\<dots> = (\<lambda>i. ((f o ((seqseq (Suc n)))) i)\<^isub>F n) o (\<lambda>x. fold_reduce (Suc n) x (Suc n + x))"
```
```    89     by (simp add: o_def)
```
```    90   also have "\<dots> ----> l"
```
```    91   proof (rule LIMSEQ_subseq_LIMSEQ[OF _ subseq_diagonal_rest], rule tendstoI)
```
```    92     fix e::real assume "0 < e"
```
```    93     from tendstoD[OF l `0 < e`]
```
```    94     show "eventually (\<lambda>x. dist (((f \<circ> seqseq (Suc n)) x)\<^isub>F n) l < e)
```
```    95       sequentially" .
```
```    96   qed
```
```    97   finally show ?thesis by (intro exI) (rule LIMSEQ_offset)
```
```    98 qed
```
```    99
```
```   100 subsection {* Daniell-Kolmogorov Theorem *}
```
```   101
```
```   102 text {* Existence of Projective Limit *}
```
```   103
```
```   104 locale polish_projective = projective_family I P "\<lambda>_. borel::'a::polish_space measure"
```
```   105   for I::"'i set" and P
```
```   106 begin
```
```   107
```
```   108 abbreviation "lim\<^isub>B \<equiv> (\<lambda>J P. limP J (\<lambda>_. borel) P)"
```
```   109
```
```   110 lemma emeasure_limB_emb_not_empty:
```
```   111   assumes "I \<noteq> {}"
```
```   112   assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
```
```   113   shows "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (lim\<^isub>B J P) (Pi\<^isub>E J B)"
```
```   114 proof -
```
```   115   let ?\<Omega> = "\<Pi>\<^isub>E i\<in>I. space borel"
```
```   116   let ?G = generator
```
```   117   interpret G!: algebra ?\<Omega> generator by (intro  algebra_generator) fact
```
```   118   note mu_G_mono =
```
```   119     G.additive_increasing[OF positive_mu_G[OF `I \<noteq> {}`] additive_mu_G[OF `I \<noteq> {}`],
```
```   120       THEN increasingD]
```
```   121   write mu_G  ("\<mu>G")
```
```   122
```
```   123   have "\<exists>\<mu>. (\<forall>s\<in>?G. \<mu> s = \<mu>G s) \<and> measure_space ?\<Omega> (sigma_sets ?\<Omega> ?G) \<mu>"
```
```   124   proof (rule G.caratheodory_empty_continuous[OF positive_mu_G additive_mu_G,
```
```   125       OF `I \<noteq> {}`, OF `I \<noteq> {}`])
```
```   126     fix A assume "A \<in> ?G"
```
```   127     with generatorE guess J X . note JX = this
```
```   128     interpret prob_space "P J" using proj_prob_space[OF `finite J`] .
```
```   129     show "\<mu>G A \<noteq> \<infinity>" using JX by (simp add: limP_finite)
```
```   130   next
```
```   131     fix Z assume Z: "range Z \<subseteq> ?G" "decseq Z" "(\<Inter>i. Z i) = {}"
```
```   132     then have "decseq (\<lambda>i. \<mu>G (Z i))"
```
```   133       by (auto intro!: mu_G_mono simp: decseq_def)
```
```   134     moreover
```
```   135     have "(INF i. \<mu>G (Z i)) = 0"
```
```   136     proof (rule ccontr)
```
```   137       assume "(INF i. \<mu>G (Z i)) \<noteq> 0" (is "?a \<noteq> 0")
```
```   138       moreover have "0 \<le> ?a"
```
```   139         using Z positive_mu_G[OF `I \<noteq> {}`] by (auto intro!: INF_greatest simp: positive_def)
```
```   140       ultimately have "0 < ?a" by auto
```
```   141       hence "?a \<noteq> -\<infinity>" by auto
```
```   142       have "\<forall>n. \<exists>J B. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> B \<in> sets (Pi\<^isub>M J (\<lambda>_. borel)) \<and>
```
```   143         Z n = emb I J B \<and> \<mu>G (Z n) = emeasure (lim\<^isub>B J P) B"
```
```   144         using Z by (intro allI generator_Ex) auto
```
```   145       then obtain J' B' where J': "\<And>n. J' n \<noteq> {}" "\<And>n. finite (J' n)" "\<And>n. J' n \<subseteq> I"
```
```   146           "\<And>n. B' n \<in> sets (\<Pi>\<^isub>M i\<in>J' n. borel)"
```
```   147         and Z_emb: "\<And>n. Z n = emb I (J' n) (B' n)"
```
```   148         unfolding choice_iff by blast
```
```   149       moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)"
```
```   150       moreover def B \<equiv> "\<lambda>n. emb (J n) (J' n) (B' n)"
```
```   151       ultimately have J: "\<And>n. J n \<noteq> {}" "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I"
```
```   152         "\<And>n. B n \<in> sets (\<Pi>\<^isub>M i\<in>J n. borel)"
```
```   153         by auto
```
```   154       have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
```
```   155         unfolding J_def by force
```
```   156       have "\<forall>n. \<exists>j. j \<in> J n" using J by blast
```
```   157       then obtain j where j: "\<And>n. j n \<in> J n"
```
```   158         unfolding choice_iff by blast
```
```   159       note [simp] = `\<And>n. finite (J n)`
```
```   160       from J  Z_emb have Z_eq: "\<And>n. Z n = emb I (J n) (B n)" "\<And>n. Z n \<in> ?G"
```
```   161         unfolding J_def B_def by (subst prod_emb_trans) (insert Z, auto)
```
```   162       interpret prob_space "P (J i)" for i using proj_prob_space by simp
```
```   163       have "?a \<le> \<mu>G (Z 0)" by (auto intro: INF_lower)
```
```   164       also have "\<dots> < \<infinity>" using J by (auto simp: Z_eq mu_G_eq limP_finite proj_sets)
```
```   165       finally have "?a \<noteq> \<infinity>" by simp
```
```   166       have "\<And>n. \<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>" unfolding Z_eq using J J_mono
```
```   167         by (subst mu_G_eq) (auto simp: limP_finite proj_sets mu_G_eq)
```
```   168
```
```   169       have countable_UN_J: "countable (\<Union>n. J n)" by (simp add: countable_finite)
```
```   170       def Utn \<equiv> "to_nat_on (\<Union>n. J n)"
```
```   171       interpret function_to_finmap "J n" Utn "from_nat_into (\<Union>n. J n)" for n
```
```   172         by unfold_locales (auto simp: Utn_def intro: from_nat_into_to_nat_on[OF countable_UN_J])
```
```   173       have inj_on_Utn: "inj_on Utn (\<Union>n. J n)"
```
```   174         unfolding Utn_def using countable_UN_J by (rule inj_on_to_nat_on)
```
```   175       hence inj_on_Utn_J: "\<And>n. inj_on Utn (J n)" by (rule subset_inj_on) auto
```
```   176       def P' \<equiv> "\<lambda>n. mapmeasure n (P (J n)) (\<lambda>_. borel)"
```
```   177       let ?SUP = "\<lambda>n. SUP K : {K. K \<subseteq> fm n ` (B n) \<and> compact K}. emeasure (P' n) K"
```
```   178       {
```
```   179         fix n
```
```   180         interpret finite_measure "P (J n)" by unfold_locales
```
```   181         have "emeasure (P (J n)) (B n) = emeasure (P' n) (fm n ` (B n))"
```
```   182           using J by (auto simp: P'_def mapmeasure_PiM proj_space proj_sets)
```
```   183         also
```
```   184         have "\<dots> = ?SUP n"
```
```   185         proof (rule inner_regular)
```
```   186           show "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>"
```
```   187             unfolding P'_def
```
```   188             by (auto simp: P'_def mapmeasure_PiF fm_measurable proj_space proj_sets)
```
```   189           show "sets (P' n) = sets borel" by (simp add: borel_eq_PiF_borel P'_def)
```
```   190         next
```
```   191           show "fm n ` B n \<in> sets borel"
```
```   192             unfolding borel_eq_PiF_borel
```
```   193             by (auto simp del: J(2) simp: P'_def fm_image_measurable_finite proj_sets J)
```
```   194         qed
```
```   195         finally
```
```   196         have "emeasure (P (J n)) (B n) = ?SUP n" "?SUP n \<noteq> \<infinity>" "?SUP n \<noteq> - \<infinity>" by auto
```
```   197       } note R = this
```
```   198       have "\<forall>n. \<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> 2 powr (-n) * ?a
```
```   199         \<and> compact K \<and> K \<subseteq> fm n ` B n"
```
```   200       proof
```
```   201         fix n
```
```   202         have "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>"
```
```   203           by (simp add: mapmeasure_PiF P'_def proj_space proj_sets)
```
```   204         then interpret finite_measure "P' n" ..
```
```   205         show "\<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> ereal (2 powr - real n) * ?a \<and>
```
```   206             compact K \<and> K \<subseteq> fm n ` B n"
```
```   207           unfolding R
```
```   208         proof (rule ccontr)
```
```   209           assume H: "\<not> (\<exists>K'. ?SUP n - emeasure (P' n) K' \<le> ereal (2 powr - real n)  * ?a \<and>
```
```   210             compact K' \<and> K' \<subseteq> fm n ` B n)"
```
```   211           have "?SUP n \<le> ?SUP n - 2 powr (-n) * ?a"
```
```   212           proof (intro SUP_least)
```
```   213             fix K
```
```   214             assume "K \<in> {K. K \<subseteq> fm n ` B n \<and> compact K}"
```
```   215             with H have "\<not> ?SUP n - emeasure (P' n) K \<le> 2 powr (-n) * ?a"
```
```   216               by auto
```
```   217             hence "?SUP n - emeasure (P' n) K > 2 powr (-n) * ?a"
```
```   218               unfolding not_less[symmetric] by simp
```
```   219             hence "?SUP n - 2 powr (-n) * ?a > emeasure (P' n) K"
```
```   220               using `0 < ?a` by (auto simp add: ereal_less_minus_iff ac_simps)
```
```   221             thus "?SUP n - 2 powr (-n) * ?a \<ge> emeasure (P' n) K" by simp
```
```   222           qed
```
```   223           hence "?SUP n + 0 \<le> ?SUP n - (2 powr (-n) * ?a)" using `0 < ?a` by simp
```
```   224           hence "?SUP n + 0 \<le> ?SUP n + - (2 powr (-n) * ?a)" unfolding minus_ereal_def .
```
```   225           hence "0 \<le> - (2 powr (-n) * ?a)"
```
```   226             using `?SUP _ \<noteq> \<infinity>` `?SUP _ \<noteq> - \<infinity>`
```
```   227             by (subst (asm) ereal_add_le_add_iff) (auto simp:)
```
```   228           moreover have "ereal (2 powr - real n) * ?a > 0" using `0 < ?a`
```
```   229             by (auto simp: ereal_zero_less_0_iff)
```
```   230           ultimately show False by simp
```
```   231         qed
```
```   232       qed
```
```   233       then obtain K' where K':
```
```   234         "\<And>n. emeasure (P (J n)) (B n) - emeasure (P' n) (K' n) \<le> ereal (2 powr - real n) * ?a"
```
```   235         "\<And>n. compact (K' n)" "\<And>n. K' n \<subseteq> fm n ` B n"
```
```   236         unfolding choice_iff by blast
```
```   237       def K \<equiv> "\<lambda>n. fm n -` K' n \<inter> space (Pi\<^isub>M (J n) (\<lambda>_. borel))"
```
```   238       have K_sets: "\<And>n. K n \<in> sets (Pi\<^isub>M (J n) (\<lambda>_. borel))"
```
```   239         unfolding K_def
```
```   240         using compact_imp_closed[OF `compact (K' _)`]
```
```   241         by (intro measurable_sets[OF fm_measurable, of _ "Collect finite"])
```
```   242            (auto simp: borel_eq_PiF_borel[symmetric])
```
```   243       have K_B: "\<And>n. K n \<subseteq> B n"
```
```   244       proof
```
```   245         fix x n
```
```   246         assume "x \<in> K n" hence fm_in: "fm n x \<in> fm n ` B n"
```
```   247           using K' by (force simp: K_def)
```
```   248         show "x \<in> B n"
```
```   249           using `x \<in> K n` K_sets sets.sets_into_space J[of n]
```
```   250           by (intro inj_on_image_mem_iff[OF inj_on_fm _ fm_in, of "\<lambda>_. borel"]) auto
```
```   251       qed
```
```   252       def Z' \<equiv> "\<lambda>n. emb I (J n) (K n)"
```
```   253       have Z': "\<And>n. Z' n \<subseteq> Z n"
```
```   254         unfolding Z_eq unfolding Z'_def
```
```   255       proof (rule prod_emb_mono, safe)
```
```   256         fix n x assume "x \<in> K n"
```
```   257         hence "fm n x \<in> K' n" "x \<in> space (Pi\<^isub>M (J n) (\<lambda>_. borel))"
```
```   258           by (simp_all add: K_def proj_space)
```
```   259         note this(1)
```
```   260         also have "K' n \<subseteq> fm n ` B n" by (simp add: K')
```
```   261         finally have "fm n x \<in> fm n ` B n" .
```
```   262         thus "x \<in> B n"
```
```   263         proof safe
```
```   264           fix y assume "y \<in> B n"
```
```   265           moreover
```
```   266           hence "y \<in> space (Pi\<^isub>M (J n) (\<lambda>_. borel))" using J sets.sets_into_space[of "B n" "P (J n)"]
```
```   267             by (auto simp add: proj_space proj_sets)
```
```   268           assume "fm n x = fm n y"
```
```   269           note inj_onD[OF inj_on_fm[OF space_borel],
```
```   270             OF `fm n x = fm n y` `x \<in> space _` `y \<in> space _`]
```
```   271           ultimately show "x \<in> B n" by simp
```
```   272         qed
```
```   273       qed
```
```   274       { fix n
```
```   275         have "Z' n \<in> ?G" using K' unfolding Z'_def
```
```   276           apply (intro generatorI'[OF J(1-3)])
```
```   277           unfolding K_def proj_space
```
```   278           apply (rule measurable_sets[OF fm_measurable[of _ "Collect finite"]])
```
```   279           apply (auto simp add: P'_def borel_eq_PiF_borel[symmetric] compact_imp_closed)
```
```   280           done
```
```   281       }
```
```   282       def Y \<equiv> "\<lambda>n. \<Inter>i\<in>{1..n}. Z' i"
```
```   283       hence "\<And>n k. Y (n + k) \<subseteq> Y n" by (induct_tac k) (auto simp: Y_def)
```
```   284       hence Y_mono: "\<And>n m. n \<le> m \<Longrightarrow> Y m \<subseteq> Y n" by (auto simp: le_iff_add)
```
```   285       have Y_Z': "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z' n" by (auto simp: Y_def)
```
```   286       hence Y_Z: "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z n" using Z' by auto
```
```   287       have Y_notempty: "\<And>n. n \<ge> 1 \<Longrightarrow> (Y n) \<noteq> {}"
```
```   288       proof -
```
```   289         fix n::nat assume "n \<ge> 1" hence "Y n \<subseteq> Z n" by fact
```
```   290         have "Y n = (\<Inter> i\<in>{1..n}. emb I (J n) (emb (J n) (J i) (K i)))" using J J_mono
```
```   291           by (auto simp: Y_def Z'_def)
```
```   292         also have "\<dots> = prod_emb I (\<lambda>_. borel) (J n) (\<Inter> i\<in>{1..n}. emb (J n) (J i) (K i))"
```
```   293           using `n \<ge> 1`
```
```   294           by (subst prod_emb_INT) auto
```
```   295         finally
```
```   296         have Y_emb:
```
```   297           "Y n = prod_emb I (\<lambda>_. borel) (J n)
```
```   298             (\<Inter> i\<in>{1..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" .
```
```   299         hence "Y n \<in> ?G" using J J_mono K_sets `n \<ge> 1` by (intro generatorI[OF _ _ _ _ Y_emb]) auto
```
```   300         hence "\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>" unfolding Y_emb using J J_mono K_sets `n \<ge> 1`
```
```   301           by (subst mu_G_eq) (auto simp: limP_finite proj_sets mu_G_eq)
```
```   302         interpret finite_measure "(limP (J n) (\<lambda>_. borel) P)"
```
```   303         proof
```
```   304           have "emeasure (limP (J n) (\<lambda>_. borel) P) (J n \<rightarrow>\<^isub>E space borel) \<noteq> \<infinity>"
```
```   305             using J by (subst emeasure_limP) auto
```
```   306           thus  "emeasure (limP (J n) (\<lambda>_. borel) P) (space (limP (J n) (\<lambda>_. borel) P)) \<noteq> \<infinity>"
```
```   307              by (simp add: space_PiM)
```
```   308         qed
```
```   309         have "\<mu>G (Z n) = limP (J n) (\<lambda>_. borel) P (B n)"
```
```   310           unfolding Z_eq using J by (auto simp: mu_G_eq)
```
```   311         moreover have "\<mu>G (Y n) =
```
```   312           limP (J n) (\<lambda>_. borel) P (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))"
```
```   313           unfolding Y_emb using J J_mono K_sets `n \<ge> 1` by (subst mu_G_eq) auto
```
```   314         moreover have "\<mu>G (Z n - Y n) = limP (J n) (\<lambda>_. borel) P
```
```   315           (B n - (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))"
```
```   316           unfolding Z_eq Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets `n \<ge> 1`
```
```   317           by (subst mu_G_eq) (auto intro!: sets.Diff)
```
```   318         ultimately
```
```   319         have "\<mu>G (Z n) - \<mu>G (Y n) = \<mu>G (Z n - Y n)"
```
```   320           using J J_mono K_sets `n \<ge> 1`
```
```   321           by (simp only: emeasure_eq_measure)
```
```   322             (auto dest!: bspec[where x=n]
```
```   323             simp: extensional_restrict emeasure_eq_measure prod_emb_iff
```
```   324             intro!: measure_Diff[symmetric] set_mp[OF K_B])
```
```   325         also have subs: "Z n - Y n \<subseteq> (\<Union> i\<in>{1..n}. (Z i - Z' i))" using Z' Z `n \<ge> 1`
```
```   326           unfolding Y_def by (force simp: decseq_def)
```
```   327         have "Z n - Y n \<in> ?G" "(\<Union> i\<in>{1..n}. (Z i - Z' i)) \<in> ?G"
```
```   328           using `Z' _ \<in> ?G` `Z _ \<in> ?G` `Y _ \<in> ?G` by auto
```
```   329         hence "\<mu>G (Z n - Y n) \<le> \<mu>G (\<Union> i\<in>{1..n}. (Z i - Z' i))"
```
```   330           using subs G.additive_increasing[OF positive_mu_G[OF `I \<noteq> {}`] additive_mu_G[OF `I \<noteq> {}`]]
```
```   331           unfolding increasing_def by auto
```
```   332         also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. \<mu>G (Z i - Z' i))" using `Z _ \<in> ?G` `Z' _ \<in> ?G`
```
```   333           by (intro G.subadditive[OF positive_mu_G additive_mu_G, OF `I \<noteq> {}` `I \<noteq> {}`]) auto
```
```   334         also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. 2 powr -real i * ?a)"
```
```   335         proof (rule setsum_mono)
```
```   336           fix i assume "i \<in> {1..n}" hence "i \<le> n" by simp
```
```   337           have "\<mu>G (Z i - Z' i) = \<mu>G (prod_emb I (\<lambda>_. borel) (J i) (B i - K i))"
```
```   338             unfolding Z'_def Z_eq by simp
```
```   339           also have "\<dots> = P (J i) (B i - K i)"
```
```   340             apply (subst mu_G_eq) using J K_sets apply auto
```
```   341             apply (subst limP_finite) apply auto
```
```   342             done
```
```   343           also have "\<dots> = P (J i) (B i) - P (J i) (K i)"
```
```   344             apply (subst emeasure_Diff) using K_sets J `K _ \<subseteq> B _` apply (auto simp: proj_sets)
```
```   345             done
```
```   346           also have "\<dots> = P (J i) (B i) - P' i (K' i)"
```
```   347             unfolding K_def P'_def
```
```   348             by (auto simp: mapmeasure_PiF proj_space proj_sets borel_eq_PiF_borel[symmetric]
```
```   349               compact_imp_closed[OF `compact (K' _)`] space_PiM PiE_def)
```
```   350           also have "\<dots> \<le> ereal (2 powr - real i) * ?a" using K'(1)[of i] .
```
```   351           finally show "\<mu>G (Z i - Z' i) \<le> (2 powr - real i) * ?a" .
```
```   352         qed
```
```   353         also have "\<dots> = (\<Sum> i\<in>{1..n}. ereal (2 powr -real i) * ereal(real ?a))"
```
```   354           using `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` by (subst ereal_real') auto
```
```   355         also have "\<dots> = ereal (\<Sum> i\<in>{1..n}. (2 powr -real i) * (real ?a))" by simp
```
```   356         also have "\<dots> = ereal ((\<Sum> i\<in>{1..n}. (2 powr -real i)) * real ?a)"
```
```   357           by (simp add: setsum_left_distrib)
```
```   358         also have "\<dots> < ereal (1 * real ?a)" unfolding less_ereal.simps
```
```   359         proof (rule mult_strict_right_mono)
```
```   360           have "(\<Sum>i\<in>{1..n}. 2 powr - real i) = (\<Sum>i\<in>{1..<Suc n}. (1/2) ^ i)"
```
```   361             by (rule setsum_cong)
```
```   362                (auto simp: powr_realpow[symmetric] powr_minus powr_divide inverse_eq_divide)
```
```   363           also have "{1..<Suc n} = {0..<Suc n} - {0}" by auto
```
```   364           also have "setsum (op ^ (1 / 2::real)) ({0..<Suc n} - {0}) =
```
```   365             setsum (op ^ (1 / 2)) ({0..<Suc n}) - 1" by (auto simp: setsum_diff1)
```
```   366           also have "\<dots> < 1" by (subst sumr_geometric) auto
```
```   367           finally show "(\<Sum>i = 1..n. 2 powr - real i) < 1" .
```
```   368         qed (auto simp:
```
```   369           `0 < ?a` `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` ereal_less_real_iff zero_ereal_def[symmetric])
```
```   370         also have "\<dots> = ?a" using `0 < ?a` `?a \<noteq> \<infinity>` by (auto simp: ereal_real')
```
```   371         also have "\<dots> \<le> \<mu>G (Z n)" by (auto intro: INF_lower)
```
```   372         finally have "\<mu>G (Z n) - \<mu>G (Y n) < \<mu>G (Z n)" .
```
```   373         hence R: "\<mu>G (Z n) < \<mu>G (Z n) + \<mu>G (Y n)"
```
```   374           using `\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>` by (simp add: ereal_minus_less)
```
```   375         have "0 \<le> (- \<mu>G (Z n)) + \<mu>G (Z n)" using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
```
```   376         also have "\<dots> < (- \<mu>G (Z n)) + (\<mu>G (Z n) + \<mu>G (Y n))"
```
```   377           apply (rule ereal_less_add[OF _ R]) using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
```
```   378         finally have "\<mu>G (Y n) > 0"
```
```   379           using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by (auto simp: ac_simps zero_ereal_def[symmetric])
```
```   380         thus "Y n \<noteq> {}" using positive_mu_G `I \<noteq> {}` by (auto simp add: positive_def)
```
```   381       qed
```
```   382       hence "\<forall>n\<in>{1..}. \<exists>y. y \<in> Y n" by auto
```
```   383       then obtain y where y: "\<And>n. n \<ge> 1 \<Longrightarrow> y n \<in> Y n" unfolding bchoice_iff by force
```
```   384       {
```
```   385         fix t and n m::nat
```
```   386         assume "1 \<le> n" "n \<le> m" hence "1 \<le> m" by simp
```
```   387         from Y_mono[OF `m \<ge> n`] y[OF `1 \<le> m`] have "y m \<in> Y n" by auto
```
```   388         also have "\<dots> \<subseteq> Z' n" using Y_Z'[OF `1 \<le> n`] .
```
```   389         finally
```
```   390         have "fm n (restrict (y m) (J n)) \<in> K' n"
```
```   391           unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
```
```   392         moreover have "finmap_of (J n) (restrict (y m) (J n)) = finmap_of (J n) (y m)"
```
```   393           using J by (simp add: fm_def)
```
```   394         ultimately have "fm n (y m) \<in> K' n" by simp
```
```   395       } note fm_in_K' = this
```
```   396       interpret finmap_seqs_into_compact "\<lambda>n. K' (Suc n)" "\<lambda>k. fm (Suc k) (y (Suc k))" borel
```
```   397       proof
```
```   398         fix n show "compact (K' n)" by fact
```
```   399       next
```
```   400         fix n
```
```   401         from Y_mono[of n "Suc n"] y[of "Suc n"] have "y (Suc n) \<in> Y (Suc n)" by auto
```
```   402         also have "\<dots> \<subseteq> Z' (Suc n)" using Y_Z' by auto
```
```   403         finally
```
```   404         have "fm (Suc n) (restrict (y (Suc n)) (J (Suc n))) \<in> K' (Suc n)"
```
```   405           unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
```
```   406         thus "K' (Suc n) \<noteq> {}" by auto
```
```   407         fix k
```
```   408         assume "k \<in> K' (Suc n)"
```
```   409         with K'[of "Suc n"] sets.sets_into_space have "k \<in> fm (Suc n) ` B (Suc n)" by auto
```
```   410         then obtain b where "k = fm (Suc n) b" by auto
```
```   411         thus "domain k = domain (fm (Suc n) (y (Suc n)))"
```
```   412           by (simp_all add: fm_def)
```
```   413       next
```
```   414         fix t and n m::nat
```
```   415         assume "n \<le> m" hence "Suc n \<le> Suc m" by simp
```
```   416         assume "t \<in> domain (fm (Suc n) (y (Suc n)))"
```
```   417         then obtain j where j: "t = Utn j" "j \<in> J (Suc n)" by auto
```
```   418         hence "j \<in> J (Suc m)" using J_mono[OF `Suc n \<le> Suc m`] by auto
```
```   419         have img: "fm (Suc n) (y (Suc m)) \<in> K' (Suc n)" using `n \<le> m`
```
```   420           by (intro fm_in_K') simp_all
```
```   421         show "(fm (Suc m) (y (Suc m)))\<^isub>F t \<in> (\<lambda>k. (k)\<^isub>F t) ` K' (Suc n)"
```
```   422           apply (rule image_eqI[OF _ img])
```
```   423           using `j \<in> J (Suc n)` `j \<in> J (Suc m)`
```
```   424           unfolding j by (subst proj_fm, auto)+
```
```   425       qed
```
```   426       have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) ----> z"
```
```   427         using diagonal_tendsto ..
```
```   428       then obtain z where z:
```
```   429         "\<And>t. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) ----> z t"
```
```   430         unfolding choice_iff by blast
```
```   431       {
```
```   432         fix n :: nat assume "n \<ge> 1"
```
```   433         have "\<And>i. domain (fm n (y (Suc (diagseq i)))) = domain (finmap_of (Utn ` J n) z)"
```
```   434           by simp
```
```   435         moreover
```
```   436         {
```
```   437           fix t
```
```   438           assume t: "t \<in> domain (finmap_of (Utn ` J n) z)"
```
```   439           hence "t \<in> Utn ` J n" by simp
```
```   440           then obtain j where j: "t = Utn j" "j \<in> J n" by auto
```
```   441           have "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^isub>F t) ----> z t"
```
```   442             apply (subst (2) tendsto_iff, subst eventually_sequentially)
```
```   443           proof safe
```
```   444             fix e :: real assume "0 < e"
```
```   445             { fix i x assume "i \<ge> n" "t \<in> domain (fm n x)"
```
```   446               moreover
```
```   447               hence "t \<in> domain (fm i x)" using J_mono[OF `i \<ge> n`] by auto
```
```   448               ultimately have "(fm i x)\<^isub>F t = (fm n x)\<^isub>F t"
```
```   449                 using j by (auto simp: proj_fm dest!: inj_onD[OF inj_on_Utn])
```
```   450             } note index_shift = this
```
```   451             have I: "\<And>i. i \<ge> n \<Longrightarrow> Suc (diagseq i) \<ge> n"
```
```   452               apply (rule le_SucI)
```
```   453               apply (rule order_trans) apply simp
```
```   454               apply (rule seq_suble[OF subseq_diagseq])
```
```   455               done
```
```   456             from z
```
```   457             have "\<exists>N. \<forall>i\<ge>N. dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) (z t) < e"
```
```   458               unfolding tendsto_iff eventually_sequentially using `0 < e` by auto
```
```   459             then obtain N where N: "\<And>i. i \<ge> N \<Longrightarrow>
```
```   460               dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) (z t) < e" by auto
```
```   461             show "\<exists>N. \<forall>na\<ge>N. dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) < e "
```
```   462             proof (rule exI[where x="max N n"], safe)
```
```   463               fix na assume "max N n \<le> na"
```
```   464               hence  "dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) =
```
```   465                       dist ((fm (Suc (diagseq na)) (y (Suc (diagseq na))))\<^isub>F t) (z t)" using t
```
```   466                 by (subst index_shift[OF I]) auto
```
```   467               also have "\<dots> < e" using `max N n \<le> na` by (intro N) simp
```
```   468               finally show "dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) < e" .
```
```   469             qed
```
```   470           qed
```
```   471           hence "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^isub>F t) ----> (finmap_of (Utn ` J n) z)\<^isub>F t"
```
```   472             by (simp add: tendsto_intros)
```
```   473         } ultimately
```
```   474         have "(\<lambda>i. fm n (y (Suc (diagseq i)))) ----> finmap_of (Utn ` J n) z"
```
```   475           by (rule tendsto_finmap)
```
```   476         hence "((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) ----> finmap_of (Utn ` J n) z"
```
```   477           by (intro lim_subseq) (simp add: subseq_def)
```
```   478         moreover
```
```   479         have "(\<forall>i. ((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) i \<in> K' n)"
```
```   480           apply (auto simp add: o_def intro!: fm_in_K' `1 \<le> n` le_SucI)
```
```   481           apply (rule le_trans)
```
```   482           apply (rule le_add2)
```
```   483           using seq_suble[OF subseq_diagseq]
```
```   484           apply auto
```
```   485           done
```
```   486         moreover
```
```   487         from `compact (K' n)` have "closed (K' n)" by (rule compact_imp_closed)
```
```   488         ultimately
```
```   489         have "finmap_of (Utn ` J n) z \<in> K' n"
```
```   490           unfolding closed_sequential_limits by blast
```
```   491         also have "finmap_of (Utn ` J n) z  = fm n (\<lambda>i. z (Utn i))"
```
```   492           unfolding finmap_eq_iff
```
```   493         proof clarsimp
```
```   494           fix i assume "i \<in> J n"
```
```   495           moreover hence "from_nat_into (\<Union>n. J n) (Utn i) = i"
```
```   496             unfolding Utn_def
```
```   497             by (subst from_nat_into_to_nat_on[OF countable_UN_J]) auto
```
```   498           ultimately show "z (Utn i) = (fm n (\<lambda>i. z (Utn i)))\<^isub>F (Utn i)"
```
```   499             by (simp add: finmap_eq_iff fm_def compose_def)
```
```   500         qed
```
```   501         finally have "fm n (\<lambda>i. z (Utn i)) \<in> K' n" .
```
```   502         moreover
```
```   503         let ?J = "\<Union>n. J n"
```
```   504         have "(?J \<inter> J n) = J n" by auto
```
```   505         ultimately have "restrict (\<lambda>i. z (Utn i)) (?J \<inter> J n) \<in> K n"
```
```   506           unfolding K_def by (auto simp: proj_space space_PiM)
```
```   507         hence "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z' n" unfolding Z'_def
```
```   508           using J by (auto simp: prod_emb_def PiE_def extensional_def)
```
```   509         also have "\<dots> \<subseteq> Z n" using Z' by simp
```
```   510         finally have "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z n" .
```
```   511       } note in_Z = this
```
```   512       hence "(\<Inter>i\<in>{1..}. Z i) \<noteq> {}" by auto
```
```   513       hence "(\<Inter>i. Z i) \<noteq> {}" using Z INT_decseq_offset[OF `decseq Z`] by simp
```
```   514       thus False using Z by simp
```
```   515     qed
```
```   516     ultimately show "(\<lambda>i. \<mu>G (Z i)) ----> 0"
```
```   517       using LIMSEQ_ereal_INFI[of "\<lambda>i. \<mu>G (Z i)"] by simp
```
```   518   qed
```
```   519   then guess \<mu> .. note \<mu> = this
```
```   520   def f \<equiv> "finmap_of J B"
```
```   521   show "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (lim\<^isub>B J P) (Pi\<^isub>E J B)"
```
```   522   proof (subst emeasure_extend_measure_Pair[OF limP_def, of I "\<lambda>_. borel" \<mu>])
```
```   523     show "positive (sets (lim\<^isub>B I P)) \<mu>" "countably_additive (sets (lim\<^isub>B I P)) \<mu>"
```
```   524       using \<mu> unfolding sets_limP sets_PiM_generator by (auto simp: measure_space_def)
```
```   525   next
```
```   526     show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> B \<in> J \<rightarrow> sets borel"
```
```   527       using assms by (auto simp: f_def)
```
```   528   next
```
```   529     fix J and X::"'i \<Rightarrow> 'a set"
```
```   530     show "prod_emb I (\<lambda>_. borel) J (Pi\<^isub>E J X) \<in> Pow (I \<rightarrow>\<^isub>E space borel)"
```
```   531       by (auto simp: prod_emb_def)
```
```   532     assume JX: "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> J \<rightarrow> sets borel"
```
```   533     hence "emb I J (Pi\<^isub>E J X) \<in> generator" using assms
```
```   534       by (intro generatorI[where J=J and X="Pi\<^isub>E J X"]) (auto intro: sets_PiM_I_finite)
```
```   535     hence "\<mu> (emb I J (Pi\<^isub>E J X)) = \<mu>G (emb I J (Pi\<^isub>E J X))" using \<mu> by simp
```
```   536     also have "\<dots> = emeasure (P J) (Pi\<^isub>E J X)"
```
```   537       using JX assms proj_sets
```
```   538       by (subst mu_G_eq) (auto simp: mu_G_eq limP_finite intro: sets_PiM_I_finite)
```
```   539     finally show "\<mu> (emb I J (Pi\<^isub>E J X)) = emeasure (P J) (Pi\<^isub>E J X)" .
```
```   540   next
```
```   541     show "emeasure (P J) (Pi\<^isub>E J B) = emeasure (limP J (\<lambda>_. borel) P) (Pi\<^isub>E J B)"
```
```   542       using assms by (simp add: f_def limP_finite Pi_def)
```
```   543   qed
```
```   544 qed
```
```   545
```
```   546 end
```
```   547
```
```   548 hide_const (open) PiF
```
```   549 hide_const (open) Pi\<^isub>F
```
```   550 hide_const (open) Pi'
```
```   551 hide_const (open) Abs_finmap
```
```   552 hide_const (open) Rep_finmap
```
```   553 hide_const (open) finmap_of
```
```   554 hide_const (open) proj
```
```   555 hide_const (open) domain
```
```   556 hide_const (open) basis_finmap
```
```   557
```
```   558 sublocale polish_projective \<subseteq> P!: prob_space "(lim\<^isub>B I P)"
```
```   559 proof
```
```   560   show "emeasure (lim\<^isub>B I P) (space (lim\<^isub>B I P)) = 1"
```
```   561   proof cases
```
```   562     assume "I = {}"
```
```   563     interpret prob_space "P {}" using proj_prob_space by simp
```
```   564     show ?thesis
```
```   565       by (simp add: space_PiM_empty limP_finite emeasure_space_1 `I = {}`)
```
```   566   next
```
```   567     assume "I \<noteq> {}"
```
```   568     then obtain i where "i \<in> I" by auto
```
```   569     interpret prob_space "P {i}" using proj_prob_space by simp
```
```   570     have R: "(space (lim\<^isub>B I P)) = (emb I {i} (Pi\<^isub>E {i} (\<lambda>_. space borel)))"
```
```   571       by (auto simp: prod_emb_def space_PiM)
```
```   572     moreover have "extensional {i} = space (P {i})" by (simp add: proj_space space_PiM PiE_def)
```
```   573     ultimately show ?thesis using `i \<in> I`
```
```   574       apply (subst R)
```
```   575       apply (subst emeasure_limB_emb_not_empty)
```
```   576       apply (auto simp: limP_finite emeasure_space_1 PiE_def)
```
```   577       done
```
```   578   qed
```
```   579 qed
```
```   580
```
```   581 context polish_projective begin
```
```   582
```
```   583 lemma emeasure_limB_emb:
```
```   584   assumes X: "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
```
```   585   shows "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (P J) (Pi\<^isub>E J B)"
```
```   586 proof cases
```
```   587   interpret prob_space "P {}" using proj_prob_space by simp
```
```   588   assume "J = {}"
```
```   589   moreover have "emb I {} {\<lambda>x. undefined} = space (lim\<^isub>B I P)"
```
```   590     by (auto simp: space_PiM prod_emb_def)
```
```   591   moreover have "{\<lambda>x. undefined} = space (lim\<^isub>B {} P)"
```
```   592     by (auto simp: space_PiM prod_emb_def)
```
```   593   ultimately show ?thesis
```
```   594     by (simp add: P.emeasure_space_1 limP_finite emeasure_space_1 del: space_limP)
```
```   595 next
```
```   596   assume "J \<noteq> {}" with X show ?thesis
```
```   597     by (subst emeasure_limB_emb_not_empty) (auto simp: limP_finite)
```
```   598 qed
```
```   599
```
```   600 lemma measure_limB_emb:
```
```   601   assumes "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
```
```   602   shows "measure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = measure (P J) (Pi\<^isub>E J B)"
```
```   603 proof -
```
```   604   interpret prob_space "P J" using proj_prob_space assms by simp
```
```   605   show ?thesis
```
```   606     using emeasure_limB_emb[OF assms]
```
```   607     unfolding emeasure_eq_measure limP_finite[OF `finite J` `J \<subseteq> I`] P.emeasure_eq_measure
```
```   608     by simp
```
```   609 qed
```
```   610
```
```   611 end
```
```   612
```
```   613 locale polish_product_prob_space =
```
```   614   product_prob_space "\<lambda>_. borel::('a::polish_space) measure" I for I::"'i set"
```
```   615
```
```   616 sublocale polish_product_prob_space \<subseteq> P: polish_projective I "\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)"
```
```   617 proof qed
```
```   618
```
```   619 lemma (in polish_product_prob_space) limP_eq_PiM:
```
```   620   "I \<noteq> {} \<Longrightarrow> lim\<^isub>P I (\<lambda>_. borel) (\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)) =
```
```   621     PiM I (\<lambda>_. borel)"
```
```   622   by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_limB_emb)
```
```   623
```
```   624 end
```