src/HOL/Probability/Projective_Limit.thy
author haftmann
Sun Jun 23 21:16:07 2013 +0200 (2013-06-23)
changeset 52435 6646bb548c6b
parent 51351 dd1dd470690b
child 52681 8cc7f76b827a
permissions -rw-r--r--
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
     1 (*  Title:      HOL/Probability/Projective_Limit.thy
     2     Author:     Fabian Immler, TU M√ľnchen
     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/Diagonal_Subsequence"
    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   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
    60 
    61 sublocale finmap_seqs_into_compact \<subseteq> subseqs "\<lambda>n s. (\<exists>l. (\<lambda>i. ((f o s) i)\<^isub>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)\<^isub>F n \<in> (\<lambda>k. (k)\<^isub>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)\<^isub>F n \<in> (\<lambda>k. (k)\<^isub>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)\<^isub>F n) ----> l)" by (auto simp: o_def)
    76 qed
    77 
    78 lemma (in finmap_seqs_into_compact) diagonal_tendsto: "\<exists>l. (\<lambda>i. (f (diagseq i))\<^isub>F n) ----> l"
    79 proof -
    80   have "\<And>i n0. (f o seqseq i) i = f (diagseq i)" unfolding diagseq_def by simp
    81   from reducer_reduces obtain l where l: "(\<lambda>i. ((f \<circ> seqseq (Suc n)) i)\<^isub>F n) ----> l"
    82     unfolding seqseq_reducer
    83   by auto
    84   have "(\<lambda>i. (f (diagseq (i + Suc n)))\<^isub>F n) =
    85     (\<lambda>i. ((f o (diagseq o (op + (Suc n)))) i)\<^isub>F n)" by (simp add: add_commute)
    86   also have "\<dots> =
    87     (\<lambda>i. ((f o ((seqseq (Suc n) o (\<lambda>x. fold_reduce (Suc n) x (Suc n + x))))) i)\<^isub>F n)"
    88     unfolding diagseq_seqseq by simp
    89   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))"
    90     by (simp add: o_def)
    91   also have "\<dots> ----> l"
    92   proof (rule LIMSEQ_subseq_LIMSEQ[OF _ subseq_diagonal_rest], rule tendstoI)
    93     fix e::real assume "0 < e"
    94     from tendstoD[OF l `0 < e`]
    95     show "eventually (\<lambda>x. dist (((f \<circ> seqseq (Suc n)) x)\<^isub>F n) l < e)
    96       sequentially" .
    97   qed
    98   finally show ?thesis by (intro exI) (rule LIMSEQ_offset)
    99 qed
   100 
   101 subsection {* Daniell-Kolmogorov Theorem *}
   102 
   103 text {* Existence of Projective Limit *}
   104 
   105 locale polish_projective = projective_family I P "\<lambda>_. borel::'a::polish_space measure"
   106   for I::"'i set" and P
   107 begin
   108 
   109 abbreviation "lim\<^isub>B \<equiv> (\<lambda>J P. limP J (\<lambda>_. borel) P)"
   110 
   111 lemma emeasure_limB_emb_not_empty:
   112   assumes "I \<noteq> {}"
   113   assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
   114   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)"
   115 proof -
   116   let ?\<Omega> = "\<Pi>\<^isub>E i\<in>I. space borel"
   117   let ?G = generator
   118   interpret G!: algebra ?\<Omega> generator by (intro  algebra_generator) fact
   119   note mu_G_mono =
   120     G.additive_increasing[OF positive_mu_G[OF `I \<noteq> {}`] additive_mu_G[OF `I \<noteq> {}`],
   121       THEN increasingD]
   122   write mu_G  ("\<mu>G")
   123 
   124   have "\<exists>\<mu>. (\<forall>s\<in>?G. \<mu> s = \<mu>G s) \<and> measure_space ?\<Omega> (sigma_sets ?\<Omega> ?G) \<mu>"
   125   proof (rule G.caratheodory_empty_continuous[OF positive_mu_G additive_mu_G,
   126       OF `I \<noteq> {}`, OF `I \<noteq> {}`])
   127     fix A assume "A \<in> ?G"
   128     with generatorE guess J X . note JX = this
   129     interpret prob_space "P J" using proj_prob_space[OF `finite J`] .
   130     show "\<mu>G A \<noteq> \<infinity>" using JX by (simp add: limP_finite)
   131   next
   132     fix Z assume Z: "range Z \<subseteq> ?G" "decseq Z" "(\<Inter>i. Z i) = {}"
   133     then have "decseq (\<lambda>i. \<mu>G (Z i))"
   134       by (auto intro!: mu_G_mono simp: decseq_def)
   135     moreover
   136     have "(INF i. \<mu>G (Z i)) = 0"
   137     proof (rule ccontr)
   138       assume "(INF i. \<mu>G (Z i)) \<noteq> 0" (is "?a \<noteq> 0")
   139       moreover have "0 \<le> ?a"
   140         using Z positive_mu_G[OF `I \<noteq> {}`] by (auto intro!: INF_greatest simp: positive_def)
   141       ultimately have "0 < ?a" by auto
   142       hence "?a \<noteq> -\<infinity>" by auto
   143       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>
   144         Z n = emb I J B \<and> \<mu>G (Z n) = emeasure (lim\<^isub>B J P) B"
   145         using Z by (intro allI generator_Ex) auto
   146       then obtain J' B' where J': "\<And>n. J' n \<noteq> {}" "\<And>n. finite (J' n)" "\<And>n. J' n \<subseteq> I"
   147           "\<And>n. B' n \<in> sets (\<Pi>\<^isub>M i\<in>J' n. borel)"
   148         and Z_emb: "\<And>n. Z n = emb I (J' n) (B' n)"
   149         unfolding choice_iff by blast
   150       moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)"
   151       moreover def B \<equiv> "\<lambda>n. emb (J n) (J' n) (B' n)"
   152       ultimately have J: "\<And>n. J n \<noteq> {}" "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I"
   153         "\<And>n. B n \<in> sets (\<Pi>\<^isub>M i\<in>J n. borel)"
   154         by auto
   155       have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
   156         unfolding J_def by force
   157       have "\<forall>n. \<exists>j. j \<in> J n" using J by blast
   158       then obtain j where j: "\<And>n. j n \<in> J n"
   159         unfolding choice_iff by blast
   160       note [simp] = `\<And>n. finite (J n)`
   161       from J  Z_emb have Z_eq: "\<And>n. Z n = emb I (J n) (B n)" "\<And>n. Z n \<in> ?G"
   162         unfolding J_def B_def by (subst prod_emb_trans) (insert Z, auto)
   163       interpret prob_space "P (J i)" for i using proj_prob_space by simp
   164       have "?a \<le> \<mu>G (Z 0)" by (auto intro: INF_lower)
   165       also have "\<dots> < \<infinity>" using J by (auto simp: Z_eq mu_G_eq limP_finite proj_sets)
   166       finally have "?a \<noteq> \<infinity>" by simp
   167       have "\<And>n. \<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>" unfolding Z_eq using J J_mono
   168         by (subst mu_G_eq) (auto simp: limP_finite proj_sets mu_G_eq)
   169 
   170       have countable_UN_J: "countable (\<Union>n. J n)" by (simp add: countable_finite)
   171       def Utn \<equiv> "to_nat_on (\<Union>n. J n)"
   172       interpret function_to_finmap "J n" Utn "from_nat_into (\<Union>n. J n)" for n
   173         by unfold_locales (auto simp: Utn_def intro: from_nat_into_to_nat_on[OF countable_UN_J])
   174       have inj_on_Utn: "inj_on Utn (\<Union>n. J n)"
   175         unfolding Utn_def using countable_UN_J by (rule inj_on_to_nat_on)
   176       hence inj_on_Utn_J: "\<And>n. inj_on Utn (J n)" by (rule subset_inj_on) auto
   177       def P' \<equiv> "\<lambda>n. mapmeasure n (P (J n)) (\<lambda>_. borel)"
   178       let ?SUP = "\<lambda>n. SUP K : {K. K \<subseteq> fm n ` (B n) \<and> compact K}. emeasure (P' n) K"
   179       {
   180         fix n
   181         interpret finite_measure "P (J n)" by unfold_locales
   182         have "emeasure (P (J n)) (B n) = emeasure (P' n) (fm n ` (B n))"
   183           using J by (auto simp: P'_def mapmeasure_PiM proj_space proj_sets)
   184         also
   185         have "\<dots> = ?SUP n"
   186         proof (rule inner_regular)
   187           show "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>"
   188             unfolding P'_def
   189             by (auto simp: P'_def mapmeasure_PiF fm_measurable proj_space proj_sets)
   190           show "sets (P' n) = sets borel" by (simp add: borel_eq_PiF_borel P'_def)
   191         next
   192           show "fm n ` B n \<in> sets borel"
   193             unfolding borel_eq_PiF_borel
   194             by (auto simp del: J(2) simp: P'_def fm_image_measurable_finite proj_sets J)
   195         qed
   196         finally
   197         have "emeasure (P (J n)) (B n) = ?SUP n" "?SUP n \<noteq> \<infinity>" "?SUP n \<noteq> - \<infinity>" by auto
   198       } note R = this
   199       have "\<forall>n. \<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> 2 powr (-n) * ?a
   200         \<and> compact K \<and> K \<subseteq> fm n ` B n"
   201       proof
   202         fix n
   203         have "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>"
   204           by (simp add: mapmeasure_PiF P'_def proj_space proj_sets)
   205         then interpret finite_measure "P' n" ..
   206         show "\<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> ereal (2 powr - real n) * ?a \<and>
   207             compact K \<and> K \<subseteq> fm n ` B n"
   208           unfolding R
   209         proof (rule ccontr)
   210           assume H: "\<not> (\<exists>K'. ?SUP n - emeasure (P' n) K' \<le> ereal (2 powr - real n)  * ?a \<and>
   211             compact K' \<and> K' \<subseteq> fm n ` B n)"
   212           have "?SUP n \<le> ?SUP n - 2 powr (-n) * ?a"
   213           proof (intro SUP_least)
   214             fix K
   215             assume "K \<in> {K. K \<subseteq> fm n ` B n \<and> compact K}"
   216             with H have "\<not> ?SUP n - emeasure (P' n) K \<le> 2 powr (-n) * ?a"
   217               by auto
   218             hence "?SUP n - emeasure (P' n) K > 2 powr (-n) * ?a"
   219               unfolding not_less[symmetric] by simp
   220             hence "?SUP n - 2 powr (-n) * ?a > emeasure (P' n) K"
   221               using `0 < ?a` by (auto simp add: ereal_less_minus_iff ac_simps)
   222             thus "?SUP n - 2 powr (-n) * ?a \<ge> emeasure (P' n) K" by simp
   223           qed
   224           hence "?SUP n + 0 \<le> ?SUP n - (2 powr (-n) * ?a)" using `0 < ?a` by simp
   225           hence "?SUP n + 0 \<le> ?SUP n + - (2 powr (-n) * ?a)" unfolding minus_ereal_def .
   226           hence "0 \<le> - (2 powr (-n) * ?a)"
   227             using `?SUP _ \<noteq> \<infinity>` `?SUP _ \<noteq> - \<infinity>`
   228             by (subst (asm) ereal_add_le_add_iff) (auto simp:)
   229           moreover have "ereal (2 powr - real n) * ?a > 0" using `0 < ?a`
   230             by (auto simp: ereal_zero_less_0_iff)
   231           ultimately show False by simp
   232         qed
   233       qed
   234       then obtain K' where K':
   235         "\<And>n. emeasure (P (J n)) (B n) - emeasure (P' n) (K' n) \<le> ereal (2 powr - real n) * ?a"
   236         "\<And>n. compact (K' n)" "\<And>n. K' n \<subseteq> fm n ` B n"
   237         unfolding choice_iff by blast
   238       def K \<equiv> "\<lambda>n. fm n -` K' n \<inter> space (Pi\<^isub>M (J n) (\<lambda>_. borel))"
   239       have K_sets: "\<And>n. K n \<in> sets (Pi\<^isub>M (J n) (\<lambda>_. borel))"
   240         unfolding K_def
   241         using compact_imp_closed[OF `compact (K' _)`]
   242         by (intro measurable_sets[OF fm_measurable, of _ "Collect finite"])
   243            (auto simp: borel_eq_PiF_borel[symmetric])
   244       have K_B: "\<And>n. K n \<subseteq> B n"
   245       proof
   246         fix x n
   247         assume "x \<in> K n" hence fm_in: "fm n x \<in> fm n ` B n"
   248           using K' by (force simp: K_def)
   249         show "x \<in> B n"
   250           using `x \<in> K n` K_sets sets.sets_into_space J[of n]
   251           by (intro inj_on_image_mem_iff[OF inj_on_fm _ fm_in, of "\<lambda>_. borel"]) auto
   252       qed
   253       def Z' \<equiv> "\<lambda>n. emb I (J n) (K n)"
   254       have Z': "\<And>n. Z' n \<subseteq> Z n"
   255         unfolding Z_eq unfolding Z'_def
   256       proof (rule prod_emb_mono, safe)
   257         fix n x assume "x \<in> K n"
   258         hence "fm n x \<in> K' n" "x \<in> space (Pi\<^isub>M (J n) (\<lambda>_. borel))"
   259           by (simp_all add: K_def proj_space)
   260         note this(1)
   261         also have "K' n \<subseteq> fm n ` B n" by (simp add: K')
   262         finally have "fm n x \<in> fm n ` B n" .
   263         thus "x \<in> B n"
   264         proof safe
   265           fix y assume "y \<in> B n"
   266           moreover
   267           hence "y \<in> space (Pi\<^isub>M (J n) (\<lambda>_. borel))" using J sets.sets_into_space[of "B n" "P (J n)"]
   268             by (auto simp add: proj_space proj_sets)
   269           assume "fm n x = fm n y"
   270           note inj_onD[OF inj_on_fm[OF space_borel],
   271             OF `fm n x = fm n y` `x \<in> space _` `y \<in> space _`]
   272           ultimately show "x \<in> B n" by simp
   273         qed
   274       qed
   275       { fix n
   276         have "Z' n \<in> ?G" using K' unfolding Z'_def
   277           apply (intro generatorI'[OF J(1-3)])
   278           unfolding K_def proj_space
   279           apply (rule measurable_sets[OF fm_measurable[of _ "Collect finite"]])
   280           apply (auto simp add: P'_def borel_eq_PiF_borel[symmetric] compact_imp_closed)
   281           done
   282       }
   283       def Y \<equiv> "\<lambda>n. \<Inter>i\<in>{1..n}. Z' i"
   284       hence "\<And>n k. Y (n + k) \<subseteq> Y n" by (induct_tac k) (auto simp: Y_def)
   285       hence Y_mono: "\<And>n m. n \<le> m \<Longrightarrow> Y m \<subseteq> Y n" by (auto simp: le_iff_add)
   286       have Y_Z': "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z' n" by (auto simp: Y_def)
   287       hence Y_Z: "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z n" using Z' by auto
   288       have Y_notempty: "\<And>n. n \<ge> 1 \<Longrightarrow> (Y n) \<noteq> {}"
   289       proof -
   290         fix n::nat assume "n \<ge> 1" hence "Y n \<subseteq> Z n" by fact
   291         have "Y n = (\<Inter> i\<in>{1..n}. emb I (J n) (emb (J n) (J i) (K i)))" using J J_mono
   292           by (auto simp: Y_def Z'_def)
   293         also have "\<dots> = prod_emb I (\<lambda>_. borel) (J n) (\<Inter> i\<in>{1..n}. emb (J n) (J i) (K i))"
   294           using `n \<ge> 1`
   295           by (subst prod_emb_INT) auto
   296         finally
   297         have Y_emb:
   298           "Y n = prod_emb I (\<lambda>_. borel) (J n)
   299             (\<Inter> i\<in>{1..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" .
   300         hence "Y n \<in> ?G" using J J_mono K_sets `n \<ge> 1` by (intro generatorI[OF _ _ _ _ Y_emb]) auto
   301         hence "\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>" unfolding Y_emb using J J_mono K_sets `n \<ge> 1`
   302           by (subst mu_G_eq) (auto simp: limP_finite proj_sets mu_G_eq)
   303         interpret finite_measure "(limP (J n) (\<lambda>_. borel) P)"
   304         proof
   305           have "emeasure (limP (J n) (\<lambda>_. borel) P) (J n \<rightarrow>\<^isub>E space borel) \<noteq> \<infinity>"
   306             using J by (subst emeasure_limP) auto
   307           thus  "emeasure (limP (J n) (\<lambda>_. borel) P) (space (limP (J n) (\<lambda>_. borel) P)) \<noteq> \<infinity>"
   308              by (simp add: space_PiM)
   309         qed
   310         have "\<mu>G (Z n) = limP (J n) (\<lambda>_. borel) P (B n)"
   311           unfolding Z_eq using J by (auto simp: mu_G_eq)
   312         moreover have "\<mu>G (Y n) =
   313           limP (J n) (\<lambda>_. borel) P (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))"
   314           unfolding Y_emb using J J_mono K_sets `n \<ge> 1` by (subst mu_G_eq) auto
   315         moreover have "\<mu>G (Z n - Y n) = limP (J n) (\<lambda>_. borel) P
   316           (B n - (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))"
   317           unfolding Z_eq Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets `n \<ge> 1`
   318           by (subst mu_G_eq) (auto intro!: sets.Diff)
   319         ultimately
   320         have "\<mu>G (Z n) - \<mu>G (Y n) = \<mu>G (Z n - Y n)"
   321           using J J_mono K_sets `n \<ge> 1`
   322           by (simp only: emeasure_eq_measure)
   323             (auto dest!: bspec[where x=n]
   324             simp: extensional_restrict emeasure_eq_measure prod_emb_iff
   325             intro!: measure_Diff[symmetric] set_mp[OF K_B])
   326         also have subs: "Z n - Y n \<subseteq> (\<Union> i\<in>{1..n}. (Z i - Z' i))" using Z' Z `n \<ge> 1`
   327           unfolding Y_def by (force simp: decseq_def)
   328         have "Z n - Y n \<in> ?G" "(\<Union> i\<in>{1..n}. (Z i - Z' i)) \<in> ?G"
   329           using `Z' _ \<in> ?G` `Z _ \<in> ?G` `Y _ \<in> ?G` by auto
   330         hence "\<mu>G (Z n - Y n) \<le> \<mu>G (\<Union> i\<in>{1..n}. (Z i - Z' i))"
   331           using subs G.additive_increasing[OF positive_mu_G[OF `I \<noteq> {}`] additive_mu_G[OF `I \<noteq> {}`]]
   332           unfolding increasing_def by auto
   333         also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. \<mu>G (Z i - Z' i))" using `Z _ \<in> ?G` `Z' _ \<in> ?G`
   334           by (intro G.subadditive[OF positive_mu_G additive_mu_G, OF `I \<noteq> {}` `I \<noteq> {}`]) auto
   335         also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. 2 powr -real i * ?a)"
   336         proof (rule setsum_mono)
   337           fix i assume "i \<in> {1..n}" hence "i \<le> n" by simp
   338           have "\<mu>G (Z i - Z' i) = \<mu>G (prod_emb I (\<lambda>_. borel) (J i) (B i - K i))"
   339             unfolding Z'_def Z_eq by simp
   340           also have "\<dots> = P (J i) (B i - K i)"
   341             apply (subst mu_G_eq) using J K_sets apply auto
   342             apply (subst limP_finite) apply auto
   343             done
   344           also have "\<dots> = P (J i) (B i) - P (J i) (K i)"
   345             apply (subst emeasure_Diff) using K_sets J `K _ \<subseteq> B _` apply (auto simp: proj_sets)
   346             done
   347           also have "\<dots> = P (J i) (B i) - P' i (K' i)"
   348             unfolding K_def P'_def
   349             by (auto simp: mapmeasure_PiF proj_space proj_sets borel_eq_PiF_borel[symmetric]
   350               compact_imp_closed[OF `compact (K' _)`] space_PiM PiE_def)
   351           also have "\<dots> \<le> ereal (2 powr - real i) * ?a" using K'(1)[of i] .
   352           finally show "\<mu>G (Z i - Z' i) \<le> (2 powr - real i) * ?a" .
   353         qed
   354         also have "\<dots> = (\<Sum> i\<in>{1..n}. ereal (2 powr -real i) * ereal(real ?a))"
   355           using `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` by (subst ereal_real') auto
   356         also have "\<dots> = ereal (\<Sum> i\<in>{1..n}. (2 powr -real i) * (real ?a))" by simp
   357         also have "\<dots> = ereal ((\<Sum> i\<in>{1..n}. (2 powr -real i)) * real ?a)"
   358           by (simp add: setsum_left_distrib)
   359         also have "\<dots> < ereal (1 * real ?a)" unfolding less_ereal.simps
   360         proof (rule mult_strict_right_mono)
   361           have "(\<Sum>i\<in>{1..n}. 2 powr - real i) = (\<Sum>i\<in>{1..<Suc n}. (1/2) ^ i)"
   362             by (rule setsum_cong)
   363                (auto simp: powr_realpow[symmetric] powr_minus powr_divide inverse_eq_divide)
   364           also have "{1..<Suc n} = {0..<Suc n} - {0}" by auto
   365           also have "setsum (op ^ (1 / 2::real)) ({0..<Suc n} - {0}) =
   366             setsum (op ^ (1 / 2)) ({0..<Suc n}) - 1" by (auto simp: setsum_diff1)
   367           also have "\<dots> < 1" by (subst sumr_geometric) auto
   368           finally show "(\<Sum>i = 1..n. 2 powr - real i) < 1" .
   369         qed (auto simp:
   370           `0 < ?a` `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` ereal_less_real_iff zero_ereal_def[symmetric])
   371         also have "\<dots> = ?a" using `0 < ?a` `?a \<noteq> \<infinity>` by (auto simp: ereal_real')
   372         also have "\<dots> \<le> \<mu>G (Z n)" by (auto intro: INF_lower)
   373         finally have "\<mu>G (Z n) - \<mu>G (Y n) < \<mu>G (Z n)" .
   374         hence R: "\<mu>G (Z n) < \<mu>G (Z n) + \<mu>G (Y n)"
   375           using `\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>` by (simp add: ereal_minus_less)
   376         have "0 \<le> (- \<mu>G (Z n)) + \<mu>G (Z n)" using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
   377         also have "\<dots> < (- \<mu>G (Z n)) + (\<mu>G (Z n) + \<mu>G (Y n))"
   378           apply (rule ereal_less_add[OF _ R]) using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
   379         finally have "\<mu>G (Y n) > 0"
   380           using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by (auto simp: ac_simps zero_ereal_def[symmetric])
   381         thus "Y n \<noteq> {}" using positive_mu_G `I \<noteq> {}` by (auto simp add: positive_def)
   382       qed
   383       hence "\<forall>n\<in>{1..}. \<exists>y. y \<in> Y n" by auto
   384       then obtain y where y: "\<And>n. n \<ge> 1 \<Longrightarrow> y n \<in> Y n" unfolding bchoice_iff by force
   385       {
   386         fix t and n m::nat
   387         assume "1 \<le> n" "n \<le> m" hence "1 \<le> m" by simp
   388         from Y_mono[OF `m \<ge> n`] y[OF `1 \<le> m`] have "y m \<in> Y n" by auto
   389         also have "\<dots> \<subseteq> Z' n" using Y_Z'[OF `1 \<le> n`] .
   390         finally
   391         have "fm n (restrict (y m) (J n)) \<in> K' n"
   392           unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
   393         moreover have "finmap_of (J n) (restrict (y m) (J n)) = finmap_of (J n) (y m)"
   394           using J by (simp add: fm_def)
   395         ultimately have "fm n (y m) \<in> K' n" by simp
   396       } note fm_in_K' = this
   397       interpret finmap_seqs_into_compact "\<lambda>n. K' (Suc n)" "\<lambda>k. fm (Suc k) (y (Suc k))" borel
   398       proof
   399         fix n show "compact (K' n)" by fact
   400       next
   401         fix n
   402         from Y_mono[of n "Suc n"] y[of "Suc n"] have "y (Suc n) \<in> Y (Suc n)" by auto
   403         also have "\<dots> \<subseteq> Z' (Suc n)" using Y_Z' by auto
   404         finally
   405         have "fm (Suc n) (restrict (y (Suc n)) (J (Suc n))) \<in> K' (Suc n)"
   406           unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
   407         thus "K' (Suc n) \<noteq> {}" by auto
   408         fix k
   409         assume "k \<in> K' (Suc n)"
   410         with K'[of "Suc n"] sets.sets_into_space have "k \<in> fm (Suc n) ` B (Suc n)" by auto
   411         then obtain b where "k = fm (Suc n) b" by auto
   412         thus "domain k = domain (fm (Suc n) (y (Suc n)))"
   413           by (simp_all add: fm_def)
   414       next
   415         fix t and n m::nat
   416         assume "n \<le> m" hence "Suc n \<le> Suc m" by simp
   417         assume "t \<in> domain (fm (Suc n) (y (Suc n)))"
   418         then obtain j where j: "t = Utn j" "j \<in> J (Suc n)" by auto
   419         hence "j \<in> J (Suc m)" using J_mono[OF `Suc n \<le> Suc m`] by auto
   420         have img: "fm (Suc n) (y (Suc m)) \<in> K' (Suc n)" using `n \<le> m`
   421           by (intro fm_in_K') simp_all
   422         show "(fm (Suc m) (y (Suc m)))\<^isub>F t \<in> (\<lambda>k. (k)\<^isub>F t) ` K' (Suc n)"
   423           apply (rule image_eqI[OF _ img])
   424           using `j \<in> J (Suc n)` `j \<in> J (Suc m)`
   425           unfolding j by (subst proj_fm, auto)+
   426       qed
   427       have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) ----> z"
   428         using diagonal_tendsto ..
   429       then obtain z where z:
   430         "\<And>t. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) ----> z t"
   431         unfolding choice_iff by blast
   432       {
   433         fix n :: nat assume "n \<ge> 1"
   434         have "\<And>i. domain (fm n (y (Suc (diagseq i)))) = domain (finmap_of (Utn ` J n) z)"
   435           by simp
   436         moreover
   437         {
   438           fix t
   439           assume t: "t \<in> domain (finmap_of (Utn ` J n) z)"
   440           hence "t \<in> Utn ` J n" by simp
   441           then obtain j where j: "t = Utn j" "j \<in> J n" by auto
   442           have "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^isub>F t) ----> z t"
   443             apply (subst (2) tendsto_iff, subst eventually_sequentially)
   444           proof safe
   445             fix e :: real assume "0 < e"
   446             { fix i x assume "i \<ge> n" "t \<in> domain (fm n x)"
   447               moreover
   448               hence "t \<in> domain (fm i x)" using J_mono[OF `i \<ge> n`] by auto
   449               ultimately have "(fm i x)\<^isub>F t = (fm n x)\<^isub>F t"
   450                 using j by (auto simp: proj_fm dest!: inj_onD[OF inj_on_Utn])
   451             } note index_shift = this
   452             have I: "\<And>i. i \<ge> n \<Longrightarrow> Suc (diagseq i) \<ge> n"
   453               apply (rule le_SucI)
   454               apply (rule order_trans) apply simp
   455               apply (rule seq_suble[OF subseq_diagseq])
   456               done
   457             from z
   458             have "\<exists>N. \<forall>i\<ge>N. dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) (z t) < e"
   459               unfolding tendsto_iff eventually_sequentially using `0 < e` by auto
   460             then obtain N where N: "\<And>i. i \<ge> N \<Longrightarrow>
   461               dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) (z t) < e" by auto
   462             show "\<exists>N. \<forall>na\<ge>N. dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) < e "
   463             proof (rule exI[where x="max N n"], safe)
   464               fix na assume "max N n \<le> na"
   465               hence  "dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) =
   466                       dist ((fm (Suc (diagseq na)) (y (Suc (diagseq na))))\<^isub>F t) (z t)" using t
   467                 by (subst index_shift[OF I]) auto
   468               also have "\<dots> < e" using `max N n \<le> na` by (intro N) simp
   469               finally show "dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) < e" .
   470             qed
   471           qed
   472           hence "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^isub>F t) ----> (finmap_of (Utn ` J n) z)\<^isub>F t"
   473             by (simp add: tendsto_intros)
   474         } ultimately
   475         have "(\<lambda>i. fm n (y (Suc (diagseq i)))) ----> finmap_of (Utn ` J n) z"
   476           by (rule tendsto_finmap)
   477         hence "((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) ----> finmap_of (Utn ` J n) z"
   478           by (intro lim_subseq) (simp add: subseq_def)
   479         moreover
   480         have "(\<forall>i. ((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) i \<in> K' n)"
   481           apply (auto simp add: o_def intro!: fm_in_K' `1 \<le> n` le_SucI)
   482           apply (rule le_trans)
   483           apply (rule le_add2)
   484           using seq_suble[OF subseq_diagseq]
   485           apply auto
   486           done
   487         moreover
   488         from `compact (K' n)` have "closed (K' n)" by (rule compact_imp_closed)
   489         ultimately
   490         have "finmap_of (Utn ` J n) z \<in> K' n"
   491           unfolding closed_sequential_limits by blast
   492         also have "finmap_of (Utn ` J n) z  = fm n (\<lambda>i. z (Utn i))"
   493           unfolding finmap_eq_iff
   494         proof clarsimp
   495           fix i assume "i \<in> J n"
   496           moreover hence "from_nat_into (\<Union>n. J n) (Utn i) = i"
   497             unfolding Utn_def
   498             by (subst from_nat_into_to_nat_on[OF countable_UN_J]) auto
   499           ultimately show "z (Utn i) = (fm n (\<lambda>i. z (Utn i)))\<^isub>F (Utn i)"
   500             by (simp add: finmap_eq_iff fm_def compose_def)
   501         qed
   502         finally have "fm n (\<lambda>i. z (Utn i)) \<in> K' n" .
   503         moreover
   504         let ?J = "\<Union>n. J n"
   505         have "(?J \<inter> J n) = J n" by auto
   506         ultimately have "restrict (\<lambda>i. z (Utn i)) (?J \<inter> J n) \<in> K n"
   507           unfolding K_def by (auto simp: proj_space space_PiM)
   508         hence "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z' n" unfolding Z'_def
   509           using J by (auto simp: prod_emb_def PiE_def extensional_def)
   510         also have "\<dots> \<subseteq> Z n" using Z' by simp
   511         finally have "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z n" .
   512       } note in_Z = this
   513       hence "(\<Inter>i\<in>{1..}. Z i) \<noteq> {}" by auto
   514       hence "(\<Inter>i. Z i) \<noteq> {}" using Z INT_decseq_offset[OF `decseq Z`] by simp
   515       thus False using Z by simp
   516     qed
   517     ultimately show "(\<lambda>i. \<mu>G (Z i)) ----> 0"
   518       using LIMSEQ_INF[of "\<lambda>i. \<mu>G (Z i)"] by simp
   519   qed
   520   then guess \<mu> .. note \<mu> = this
   521   def f \<equiv> "finmap_of J B"
   522   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)"
   523   proof (subst emeasure_extend_measure_Pair[OF limP_def, of I "\<lambda>_. borel" \<mu>])
   524     show "positive (sets (lim\<^isub>B I P)) \<mu>" "countably_additive (sets (lim\<^isub>B I P)) \<mu>"
   525       using \<mu> unfolding sets_limP sets_PiM_generator by (auto simp: measure_space_def)
   526   next
   527     show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> B \<in> J \<rightarrow> sets borel"
   528       using assms by (auto simp: f_def)
   529   next
   530     fix J and X::"'i \<Rightarrow> 'a set"
   531     show "prod_emb I (\<lambda>_. borel) J (Pi\<^isub>E J X) \<in> Pow (I \<rightarrow>\<^isub>E space borel)"
   532       by (auto simp: prod_emb_def)
   533     assume JX: "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> J \<rightarrow> sets borel"
   534     hence "emb I J (Pi\<^isub>E J X) \<in> generator" using assms
   535       by (intro generatorI[where J=J and X="Pi\<^isub>E J X"]) (auto intro: sets_PiM_I_finite)
   536     hence "\<mu> (emb I J (Pi\<^isub>E J X)) = \<mu>G (emb I J (Pi\<^isub>E J X))" using \<mu> by simp
   537     also have "\<dots> = emeasure (P J) (Pi\<^isub>E J X)"
   538       using JX assms proj_sets
   539       by (subst mu_G_eq) (auto simp: mu_G_eq limP_finite intro: sets_PiM_I_finite)
   540     finally show "\<mu> (emb I J (Pi\<^isub>E J X)) = emeasure (P J) (Pi\<^isub>E J X)" .
   541   next
   542     show "emeasure (P J) (Pi\<^isub>E J B) = emeasure (limP J (\<lambda>_. borel) P) (Pi\<^isub>E J B)"
   543       using assms by (simp add: f_def limP_finite Pi_def)
   544   qed
   545 qed
   546 
   547 end
   548 
   549 hide_const (open) PiF
   550 hide_const (open) Pi\<^isub>F
   551 hide_const (open) Pi'
   552 hide_const (open) Abs_finmap
   553 hide_const (open) Rep_finmap
   554 hide_const (open) finmap_of
   555 hide_const (open) proj
   556 hide_const (open) domain
   557 hide_const (open) basis_finmap
   558 
   559 sublocale polish_projective \<subseteq> P!: prob_space "(lim\<^isub>B I P)"
   560 proof
   561   show "emeasure (lim\<^isub>B I P) (space (lim\<^isub>B I P)) = 1"
   562   proof cases
   563     assume "I = {}"
   564     interpret prob_space "P {}" using proj_prob_space by simp
   565     show ?thesis
   566       by (simp add: space_PiM_empty limP_finite emeasure_space_1 `I = {}`)
   567   next
   568     assume "I \<noteq> {}"
   569     then obtain i where "i \<in> I" by auto
   570     interpret prob_space "P {i}" using proj_prob_space by simp
   571     have R: "(space (lim\<^isub>B I P)) = (emb I {i} (Pi\<^isub>E {i} (\<lambda>_. space borel)))"
   572       by (auto simp: prod_emb_def space_PiM)
   573     moreover have "extensional {i} = space (P {i})" by (simp add: proj_space space_PiM PiE_def)
   574     ultimately show ?thesis using `i \<in> I`
   575       apply (subst R)
   576       apply (subst emeasure_limB_emb_not_empty)
   577       apply (auto simp: limP_finite emeasure_space_1 PiE_def)
   578       done
   579   qed
   580 qed
   581 
   582 context polish_projective begin
   583 
   584 lemma emeasure_limB_emb:
   585   assumes X: "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
   586   shows "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (P J) (Pi\<^isub>E J B)"
   587 proof cases
   588   interpret prob_space "P {}" using proj_prob_space by simp
   589   assume "J = {}"
   590   moreover have "emb I {} {\<lambda>x. undefined} = space (lim\<^isub>B I P)"
   591     by (auto simp: space_PiM prod_emb_def)
   592   moreover have "{\<lambda>x. undefined} = space (lim\<^isub>B {} P)"
   593     by (auto simp: space_PiM prod_emb_def)
   594   ultimately show ?thesis
   595     by (simp add: P.emeasure_space_1 limP_finite emeasure_space_1 del: space_limP)
   596 next
   597   assume "J \<noteq> {}" with X show ?thesis
   598     by (subst emeasure_limB_emb_not_empty) (auto simp: limP_finite)
   599 qed
   600 
   601 lemma measure_limB_emb:
   602   assumes "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
   603   shows "measure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = measure (P J) (Pi\<^isub>E J B)"
   604 proof -
   605   interpret prob_space "P J" using proj_prob_space assms by simp
   606   show ?thesis
   607     using emeasure_limB_emb[OF assms]
   608     unfolding emeasure_eq_measure limP_finite[OF `finite J` `J \<subseteq> I`] P.emeasure_eq_measure
   609     by simp
   610 qed
   611 
   612 end
   613 
   614 locale polish_product_prob_space =
   615   product_prob_space "\<lambda>_. borel::('a::polish_space) measure" I for I::"'i set"
   616 
   617 sublocale polish_product_prob_space \<subseteq> P: polish_projective I "\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)"
   618 proof qed
   619 
   620 lemma (in polish_product_prob_space) limP_eq_PiM:
   621   "I \<noteq> {} \<Longrightarrow> lim\<^isub>P I (\<lambda>_. borel) (\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)) =
   622     PiM I (\<lambda>_. borel)"
   623   by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_limB_emb)
   624 
   625 end