src/HOL/Probability/Infinite_Product_Measure.thy
 author hoelzl Thu Jan 22 14:51:08 2015 +0100 (2015-01-22) changeset 59425 c5e79df8cc21 parent 59000 6eb0725503fc child 61169 4de9ff3ea29a permissions -rw-r--r--
import general thms from Density_Compiler
```     1 (*  Title:      HOL/Probability/Infinite_Product_Measure.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3 *)
```
```     4
```
```     5 section {*Infinite Product Measure*}
```
```     6
```
```     7 theory Infinite_Product_Measure
```
```     8   imports Probability_Measure Caratheodory Projective_Family
```
```     9 begin
```
```    10
```
```    11 lemma (in product_prob_space) emeasure_PiM_emb_not_empty:
```
```    12   assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. X i \<in> sets (M i)"
```
```    13   shows "emeasure (Pi\<^sub>M I M) (emb I J (Pi\<^sub>E J X)) = emeasure (Pi\<^sub>M J M) (Pi\<^sub>E J X)"
```
```    14 proof cases
```
```    15   assume "finite I" with X show ?thesis by simp
```
```    16 next
```
```    17   let ?\<Omega> = "\<Pi>\<^sub>E i\<in>I. space (M i)"
```
```    18   let ?G = generator
```
```    19   assume "\<not> finite I"
```
```    20   then have I_not_empty: "I \<noteq> {}" by auto
```
```    21   interpret G!: algebra ?\<Omega> generator by (rule algebra_generator) fact
```
```    22   note mu_G_mono =
```
```    23     G.additive_increasing[OF positive_mu_G[OF I_not_empty] additive_mu_G[OF I_not_empty],
```
```    24       THEN increasingD]
```
```    25   write mu_G  ("\<mu>G")
```
```    26
```
```    27   { fix Z J assume J: "J \<noteq> {}" "finite J" "J \<subseteq> I" and Z: "Z \<in> ?G"
```
```    28
```
```    29     from `infinite I` `finite J` obtain k where k: "k \<in> I" "k \<notin> J"
```
```    30       by (metis rev_finite_subset subsetI)
```
```    31     moreover from Z guess K' X' by (rule generatorE)
```
```    32     moreover def K \<equiv> "insert k K'"
```
```    33     moreover def X \<equiv> "emb K K' X'"
```
```    34     ultimately have K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "X \<in> sets (Pi\<^sub>M K M)" "Z = emb I K X"
```
```    35       "K - J \<noteq> {}" "K - J \<subseteq> I" "\<mu>G Z = emeasure (Pi\<^sub>M K M) X"
```
```    36       by (auto simp: subset_insertI)
```
```    37     let ?M = "\<lambda>y. (\<lambda>x. merge J (K - J) (y, x)) -` emb (J \<union> K) K X \<inter> space (Pi\<^sub>M (K - J) M)"
```
```    38     { fix y assume y: "y \<in> space (Pi\<^sub>M J M)"
```
```    39       note * = merge_emb[OF `K \<subseteq> I` `J \<subseteq> I` y, of X]
```
```    40       moreover
```
```    41       have **: "?M y \<in> sets (Pi\<^sub>M (K - J) M)"
```
```    42         using J K y by (intro merge_sets) auto
```
```    43       ultimately
```
```    44       have ***: "((\<lambda>x. merge J (I - J) (y, x)) -` Z \<inter> space (Pi\<^sub>M I M)) \<in> ?G"
```
```    45         using J K by (intro generatorI) auto
```
```    46       have "\<mu>G ((\<lambda>x. merge J (I - J) (y, x)) -` emb I K X \<inter> space (Pi\<^sub>M I M)) = emeasure (Pi\<^sub>M (K - J) M) (?M y)"
```
```    47         unfolding * using K J by (subst mu_G_eq[OF _ _ _ **]) auto
```
```    48       note * ** *** this }
```
```    49     note merge_in_G = this
```
```    50
```
```    51     have "finite (K - J)" using K by auto
```
```    52
```
```    53     interpret J: finite_product_prob_space M J by default fact+
```
```    54     interpret KmJ: finite_product_prob_space M "K - J" by default fact+
```
```    55
```
```    56     have "\<mu>G Z = emeasure (Pi\<^sub>M (J \<union> (K - J)) M) (emb (J \<union> (K - J)) K X)"
```
```    57       using K J by simp
```
```    58     also have "\<dots> = (\<integral>\<^sup>+ x. emeasure (Pi\<^sub>M (K - J) M) (?M x) \<partial>Pi\<^sub>M J M)"
```
```    59       using K J by (subst emeasure_fold_integral) auto
```
```    60     also have "\<dots> = (\<integral>\<^sup>+ y. \<mu>G ((\<lambda>x. merge J (I - J) (y, x)) -` Z \<inter> space (Pi\<^sub>M I M)) \<partial>Pi\<^sub>M J M)"
```
```    61       (is "_ = (\<integral>\<^sup>+x. \<mu>G (?MZ x) \<partial>Pi\<^sub>M J M)")
```
```    62     proof (intro nn_integral_cong)
```
```    63       fix x assume x: "x \<in> space (Pi\<^sub>M J M)"
```
```    64       with K merge_in_G(2)[OF this]
```
```    65       show "emeasure (Pi\<^sub>M (K - J) M) (?M x) = \<mu>G (?MZ x)"
```
```    66         unfolding `Z = emb I K X` merge_in_G(1)[OF x] by (subst mu_G_eq) auto
```
```    67     qed
```
```    68     finally have fold: "\<mu>G Z = (\<integral>\<^sup>+x. \<mu>G (?MZ x) \<partial>Pi\<^sub>M J M)" .
```
```    69
```
```    70     { fix x assume x: "x \<in> space (Pi\<^sub>M J M)"
```
```    71       then have "\<mu>G (?MZ x) \<le> 1"
```
```    72         unfolding merge_in_G(4)[OF x] `Z = emb I K X`
```
```    73         by (intro KmJ.measure_le_1 merge_in_G(2)[OF x]) }
```
```    74     note le_1 = this
```
```    75
```
```    76     let ?q = "\<lambda>y. \<mu>G ((\<lambda>x. merge J (I - J) (y,x)) -` Z \<inter> space (Pi\<^sub>M I M))"
```
```    77     have "?q \<in> borel_measurable (Pi\<^sub>M J M)"
```
```    78       unfolding `Z = emb I K X` using J K merge_in_G(3)
```
```    79       by (simp add: merge_in_G  mu_G_eq emeasure_fold_measurable cong: measurable_cong)
```
```    80     note this fold le_1 merge_in_G(3) }
```
```    81   note fold = this
```
```    82
```
```    83   have "\<exists>\<mu>. (\<forall>s\<in>?G. \<mu> s = \<mu>G s) \<and> measure_space ?\<Omega> (sigma_sets ?\<Omega> ?G) \<mu>"
```
```    84   proof (rule G.caratheodory_empty_continuous[OF positive_mu_G additive_mu_G])
```
```    85     fix A assume "A \<in> ?G"
```
```    86     with generatorE guess J X . note JX = this
```
```    87     interpret JK: finite_product_prob_space M J by default fact+
```
```    88     from JX show "\<mu>G A \<noteq> \<infinity>" by simp
```
```    89   next
```
```    90     fix A assume A: "range A \<subseteq> ?G" "decseq A" "(\<Inter>i. A i) = {}"
```
```    91     then have "decseq (\<lambda>i. \<mu>G (A i))"
```
```    92       by (auto intro!: mu_G_mono simp: decseq_def)
```
```    93     moreover
```
```    94     have "(INF i. \<mu>G (A i)) = 0"
```
```    95     proof (rule ccontr)
```
```    96       assume "(INF i. \<mu>G (A i)) \<noteq> 0" (is "?a \<noteq> 0")
```
```    97       moreover have "0 \<le> ?a"
```
```    98         using A positive_mu_G[OF I_not_empty] by (auto intro!: INF_greatest simp: positive_def)
```
```    99       ultimately have "0 < ?a" by auto
```
```   100
```
```   101       have "\<forall>n. \<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^sub>M J M) \<and> A n = emb I J X \<and> \<mu>G (A n) = emeasure (limP J M (\<lambda>J. (Pi\<^sub>M J M))) X"
```
```   102         using A by (intro allI generator_Ex) auto
```
```   103       then obtain J' X' where J': "\<And>n. J' n \<noteq> {}" "\<And>n. finite (J' n)" "\<And>n. J' n \<subseteq> I" "\<And>n. X' n \<in> sets (Pi\<^sub>M (J' n) M)"
```
```   104         and A': "\<And>n. A n = emb I (J' n) (X' n)"
```
```   105         unfolding choice_iff by blast
```
```   106       moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)"
```
```   107       moreover def X \<equiv> "\<lambda>n. emb (J n) (J' n) (X' n)"
```
```   108       ultimately have J: "\<And>n. J n \<noteq> {}" "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I" "\<And>n. X n \<in> sets (Pi\<^sub>M (J n) M)"
```
```   109         by auto
```
```   110       with A' have A_eq: "\<And>n. A n = emb I (J n) (X n)" "\<And>n. A n \<in> ?G"
```
```   111         unfolding J_def X_def by (subst prod_emb_trans) (insert A, auto)
```
```   112
```
```   113       have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
```
```   114         unfolding J_def by force
```
```   115
```
```   116       interpret J: finite_product_prob_space M "J i" for i by default fact+
```
```   117
```
```   118       have a_le_1: "?a \<le> 1"
```
```   119         using mu_G_spec[of "J 0" "A 0" "X 0"] J A_eq
```
```   120         by (auto intro!: INF_lower2[of 0] J.measure_le_1)
```
```   121
```
```   122       let ?M = "\<lambda>K Z y. (\<lambda>x. merge K (I - K) (y, x)) -` Z \<inter> space (Pi\<^sub>M I M)"
```
```   123
```
```   124       { fix Z k assume Z: "range Z \<subseteq> ?G" "decseq Z" "\<forall>n. ?a / 2^k \<le> \<mu>G (Z n)"
```
```   125         then have Z_sets: "\<And>n. Z n \<in> ?G" by auto
```
```   126         fix J' assume J': "J' \<noteq> {}" "finite J'" "J' \<subseteq> I"
```
```   127         interpret J': finite_product_prob_space M J' by default fact+
```
```   128
```
```   129         let ?q = "\<lambda>n y. \<mu>G (?M J' (Z n) y)"
```
```   130         let ?Q = "\<lambda>n. ?q n -` {?a / 2^(k+1) ..} \<inter> space (Pi\<^sub>M J' M)"
```
```   131         { fix n
```
```   132           have "?q n \<in> borel_measurable (Pi\<^sub>M J' M)"
```
```   133             using Z J' by (intro fold(1)) auto
```
```   134           then have "?Q n \<in> sets (Pi\<^sub>M J' M)"
```
```   135             by (rule measurable_sets) auto }
```
```   136         note Q_sets = this
```
```   137
```
```   138         have "?a / 2^(k+1) \<le> (INF n. emeasure (Pi\<^sub>M J' M) (?Q n))"
```
```   139         proof (intro INF_greatest)
```
```   140           fix n
```
```   141           have "?a / 2^k \<le> \<mu>G (Z n)" using Z by auto
```
```   142           also have "\<dots> \<le> (\<integral>\<^sup>+ x. indicator (?Q n) x + ?a / 2^(k+1) \<partial>Pi\<^sub>M J' M)"
```
```   143             unfolding fold(2)[OF J' `Z n \<in> ?G`]
```
```   144           proof (intro nn_integral_mono)
```
```   145             fix x assume x: "x \<in> space (Pi\<^sub>M J' M)"
```
```   146             then have "?q n x \<le> 1 + 0"
```
```   147               using J' Z fold(3) Z_sets by auto
```
```   148             also have "\<dots> \<le> 1 + ?a / 2^(k+1)"
```
```   149               using `0 < ?a` by (intro add_mono) auto
```
```   150             finally have "?q n x \<le> 1 + ?a / 2^(k+1)" .
```
```   151             with x show "?q n x \<le> indicator (?Q n) x + ?a / 2^(k+1)"
```
```   152               by (auto split: split_indicator simp del: power_Suc)
```
```   153           qed
```
```   154           also have "\<dots> = emeasure (Pi\<^sub>M J' M) (?Q n) + ?a / 2^(k+1)"
```
```   155             using `0 \<le> ?a` Q_sets J'.emeasure_space_1
```
```   156             by (subst nn_integral_add) auto
```
```   157           finally show "?a / 2^(k+1) \<le> emeasure (Pi\<^sub>M J' M) (?Q n)" using `?a \<le> 1`
```
```   158             by (cases rule: ereal2_cases[of ?a "emeasure (Pi\<^sub>M J' M) (?Q n)"])
```
```   159                (auto simp: field_simps)
```
```   160         qed
```
```   161         also have "\<dots> = emeasure (Pi\<^sub>M J' M) (\<Inter>n. ?Q n)"
```
```   162         proof (intro INF_emeasure_decseq)
```
```   163           show "range ?Q \<subseteq> sets (Pi\<^sub>M J' M)" using Q_sets by auto
```
```   164           show "decseq ?Q"
```
```   165             unfolding decseq_def
```
```   166           proof (safe intro!: vimageI[OF refl])
```
```   167             fix m n :: nat assume "m \<le> n"
```
```   168             fix x assume x: "x \<in> space (Pi\<^sub>M J' M)"
```
```   169             assume "?a / 2^(k+1) \<le> ?q n x"
```
```   170             also have "?q n x \<le> ?q m x"
```
```   171             proof (rule mu_G_mono)
```
```   172               from fold(4)[OF J', OF Z_sets x]
```
```   173               show "?M J' (Z n) x \<in> ?G" "?M J' (Z m) x \<in> ?G" by auto
```
```   174               show "?M J' (Z n) x \<subseteq> ?M J' (Z m) x"
```
```   175                 using `decseq Z`[THEN decseqD, OF `m \<le> n`] by auto
```
```   176             qed
```
```   177             finally show "?a / 2^(k+1) \<le> ?q m x" .
```
```   178           qed
```
```   179         qed simp
```
```   180         finally have "(\<Inter>n. ?Q n) \<noteq> {}"
```
```   181           using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq)
```
```   182         then have "\<exists>w\<in>space (Pi\<^sub>M J' M). \<forall>n. ?a / 2 ^ (k + 1) \<le> ?q n w" by auto }
```
```   183       note Ex_w = this
```
```   184
```
```   185       let ?q = "\<lambda>k n y. \<mu>G (?M (J k) (A n) y)"
```
```   186
```
```   187       let ?P = "\<lambda>w k. w \<in> space (Pi\<^sub>M (J k) M) \<and> (\<forall>n. ?a / 2 ^ (Suc k) \<le> ?q k n w)"
```
```   188       have "\<exists>w. \<forall>k. ?P (w k) k \<and> restrict (w (Suc k)) (J k) = w k"
```
```   189       proof (rule dependent_nat_choice)
```
```   190         have "\<forall>n. ?a / 2 ^ 0 \<le> \<mu>G (A n)" by (auto intro: INF_lower)
```
```   191         from Ex_w[OF A(1,2) this J(1-3), of 0] show "\<exists>w. ?P w 0" by auto
```
```   192       next
```
```   193         fix w k assume Suc: "?P w k"
```
```   194         show "\<exists>w'. ?P w' (Suc k) \<and> restrict w' (J k) = w"
```
```   195         proof cases
```
```   196           assume [simp]: "J k = J (Suc k)"
```
```   197           have "?a / 2 ^ (Suc (Suc k)) \<le> ?a / 2 ^ (k + 1)"
```
```   198             using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: field_simps)
```
```   199           with Suc show ?thesis
```
```   200             by (auto intro!: exI[of _ w] simp: extensional_restrict space_PiM intro: order_trans)
```
```   201         next
```
```   202           assume "J k \<noteq> J (Suc k)"
```
```   203           with J_mono[of k "Suc k"] have "J (Suc k) - J k \<noteq> {}" (is "?D \<noteq> {}") by auto
```
```   204           have "range (\<lambda>n. ?M (J k) (A n) w) \<subseteq> ?G" "decseq (\<lambda>n. ?M (J k) (A n) w)"
```
```   205             "\<forall>n. ?a / 2 ^ (k + 1) \<le> \<mu>G (?M (J k) (A n) w)"
```
```   206             using `decseq A` fold(4)[OF J(1-3) A_eq(2), of w k] Suc
```
```   207             by (auto simp: decseq_def)
```
```   208           from Ex_w[OF this `?D \<noteq> {}`] J[of "Suc k"]
```
```   209           obtain w' where w': "w' \<in> space (Pi\<^sub>M ?D M)"
```
```   210             "\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> \<mu>G (?M ?D (?M (J k) (A n) w) w')" by auto
```
```   211           let ?w = "merge (J k) ?D (w, w')"
```
```   212           have [simp]: "\<And>x. merge (J k) (I - J k) (w, merge ?D (I - ?D) (w', x)) =
```
```   213             merge (J (Suc k)) (I - (J (Suc k))) (?w, x)"
```
```   214             using J(3)[of "Suc k"] J(3)[of k] J_mono[of k "Suc k"]
```
```   215             by (auto intro!: ext split: split_merge)
```
```   216           have *: "\<And>n. ?M ?D (?M (J k) (A n) w) w' = ?M (J (Suc k)) (A n) ?w"
```
```   217             using w'(1) J(3)[of "Suc k"]
```
```   218             by (auto simp: space_PiM split: split_merge intro!: extensional_merge_sub) force+
```
```   219           show ?thesis
```
```   220             using Suc w' J_mono[of k "Suc k"] unfolding *
```
```   221             by (intro exI[of _ ?w])
```
```   222                (auto split: split_merge intro!: extensional_merge_sub ext simp: space_PiM PiE_iff)
```
```   223         qed
```
```   224       qed
```
```   225       then obtain w where w:
```
```   226         "\<And>k. w k \<in> space (Pi\<^sub>M (J k) M)"
```
```   227         "\<And>k n. ?a / 2 ^ (Suc k) \<le> ?q k n (w k)"
```
```   228         "\<And>k. restrict (w (Suc k)) (J k) = w k"
```
```   229         by metis
```
```   230
```
```   231       { fix k
```
```   232         from w have "?a / 2 ^ (k + 1) \<le> ?q k k (w k)" by auto
```
```   233         then have "?M (J k) (A k) (w k) \<noteq> {}"
```
```   234           using positive_mu_G[OF I_not_empty, unfolded positive_def] `0 < ?a` `?a \<le> 1`
```
```   235           by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq)
```
```   236         then obtain x where "x \<in> ?M (J k) (A k) (w k)" by auto
```
```   237         then have "merge (J k) (I - J k) (w k, x) \<in> A k" by auto
```
```   238         then have "\<exists>x\<in>A k. restrict x (J k) = w k"
```
```   239           using `w k \<in> space (Pi\<^sub>M (J k) M)`
```
```   240           by (intro rev_bexI) (auto intro!: ext simp: extensional_def space_PiM) }
```
```   241       note w_ext = this
```
```   242
```
```   243       { fix k l i assume "k \<le> l" "i \<in> J k"
```
```   244         { fix l have "w k i = w (k + l) i"
```
```   245           proof (induct l)
```
```   246             case (Suc l)
```
```   247             from `i \<in> J k` J_mono[of k "k + l"] have "i \<in> J (k + l)" by auto
```
```   248             with w(3)[of "k + l"]
```
```   249             have "w (k + l) i = w (k + Suc l) i"
```
```   250               by (auto simp: restrict_def fun_eq_iff split: split_if_asm)
```
```   251             with Suc show ?case by simp
```
```   252           qed simp }
```
```   253         from this[of "l - k"] `k \<le> l` have "w l i = w k i" by simp }
```
```   254       note w_mono = this
```
```   255
```
```   256       def w' \<equiv> "\<lambda>i. if i \<in> (\<Union>k. J k) then w (LEAST k. i \<in> J k) i else if i \<in> I then (SOME x. x \<in> space (M i)) else undefined"
```
```   257       { fix i k assume k: "i \<in> J k"
```
```   258         have "w k i = w (LEAST k. i \<in> J k) i"
```
```   259           by (intro w_mono Least_le k LeastI[of _ k])
```
```   260         then have "w' i = w k i"
```
```   261           unfolding w'_def using k by auto }
```
```   262       note w'_eq = this
```
```   263       have w'_simps1: "\<And>i. i \<notin> I \<Longrightarrow> w' i = undefined"
```
```   264         using J by (auto simp: w'_def)
```
```   265       have w'_simps2: "\<And>i. i \<notin> (\<Union>k. J k) \<Longrightarrow> i \<in> I \<Longrightarrow> w' i \<in> space (M i)"
```
```   266         using J by (auto simp: w'_def intro!: someI_ex[OF M.not_empty[unfolded ex_in_conv[symmetric]]])
```
```   267       { fix i assume "i \<in> I" then have "w' i \<in> space (M i)"
```
```   268           using w(1) by (cases "i \<in> (\<Union>k. J k)") (force simp: w'_simps2 w'_eq space_PiM)+ }
```
```   269       note w'_simps[simp] = w'_eq w'_simps1 w'_simps2 this
```
```   270
```
```   271       have w': "w' \<in> space (Pi\<^sub>M I M)"
```
```   272         using w(1) by (auto simp add: Pi_iff extensional_def space_PiM)
```
```   273
```
```   274       { fix n
```
```   275         have "restrict w' (J n) = w n" using w(1)[of n]
```
```   276           by (auto simp add: fun_eq_iff space_PiM)
```
```   277         with w_ext[of n] obtain x where "x \<in> A n" "restrict x (J n) = restrict w' (J n)"
```
```   278           by auto
```
```   279         then have "w' \<in> A n" unfolding A_eq using w' by (auto simp: prod_emb_def space_PiM) }
```
```   280       then have "w' \<in> (\<Inter>i. A i)" by auto
```
```   281       with `(\<Inter>i. A i) = {}` show False by auto
```
```   282     qed
```
```   283     ultimately show "(\<lambda>i. \<mu>G (A i)) ----> 0"
```
```   284       using LIMSEQ_INF[of "\<lambda>i. \<mu>G (A i)"] by simp
```
```   285   qed fact+
```
```   286   then guess \<mu> .. note \<mu> = this
```
```   287   show ?thesis
```
```   288   proof (subst emeasure_extend_measure_Pair[OF PiM_def, of I M \<mu> J X])
```
```   289     from assms show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))"
```
```   290       by (simp add: Pi_iff)
```
```   291   next
```
```   292     fix J X assume J: "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))"
```
```   293     then show "emb I J (Pi\<^sub>E J X) \<in> Pow (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   294       by (auto simp: Pi_iff prod_emb_def dest: sets.sets_into_space)
```
```   295     have "emb I J (Pi\<^sub>E J X) \<in> generator"
```
```   296       using J `I \<noteq> {}` by (intro generatorI') (auto simp: Pi_iff)
```
```   297     then have "\<mu> (emb I J (Pi\<^sub>E J X)) = \<mu>G (emb I J (Pi\<^sub>E J X))"
```
```   298       using \<mu> by simp
```
```   299     also have "\<dots> = (\<Prod> j\<in>J. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))"
```
```   300       using J  `I \<noteq> {}` by (subst mu_G_spec[OF _ _ _ refl]) (auto simp: emeasure_PiM Pi_iff)
```
```   301     also have "\<dots> = (\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}.
```
```   302       if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))"
```
```   303       using J `I \<noteq> {}` by (intro setprod.mono_neutral_right) (auto simp: M.emeasure_space_1)
```
```   304     finally show "\<mu> (emb I J (Pi\<^sub>E J X)) = \<dots>" .
```
```   305   next
```
```   306     let ?F = "\<lambda>j. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j))"
```
```   307     have "(\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. ?F j) = (\<Prod>j\<in>J. ?F j)"
```
```   308       using X `I \<noteq> {}` by (intro setprod.mono_neutral_right) (auto simp: M.emeasure_space_1)
```
```   309     then show "(\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. ?F j) =
```
```   310       emeasure (Pi\<^sub>M J M) (Pi\<^sub>E J X)"
```
```   311       using X by (auto simp add: emeasure_PiM)
```
```   312   next
```
```   313     show "positive (sets (Pi\<^sub>M I M)) \<mu>" "countably_additive (sets (Pi\<^sub>M I M)) \<mu>"
```
```   314       using \<mu> unfolding sets_PiM_generator by (auto simp: measure_space_def)
```
```   315   qed
```
```   316 qed
```
```   317
```
```   318 sublocale product_prob_space \<subseteq> P: prob_space "Pi\<^sub>M I M"
```
```   319 proof
```
```   320   show "emeasure (Pi\<^sub>M I M) (space (Pi\<^sub>M I M)) = 1"
```
```   321   proof cases
```
```   322     assume "I = {}" then show ?thesis by (simp add: space_PiM_empty)
```
```   323   next
```
```   324     assume "I \<noteq> {}"
```
```   325     then obtain i where i: "i \<in> I" by auto
```
```   326     then have "emb I {i} (\<Pi>\<^sub>E i\<in>{i}. space (M i)) = (space (Pi\<^sub>M I M))"
```
```   327       by (auto simp: prod_emb_def space_PiM)
```
```   328     with i show ?thesis
```
```   329       using emeasure_PiM_emb_not_empty[of "{i}" "\<lambda>i. space (M i)"]
```
```   330       by (simp add: emeasure_PiM emeasure_space_1)
```
```   331   qed
```
```   332 qed
```
```   333
```
```   334 lemma (in product_prob_space) emeasure_PiM_emb:
```
```   335   assumes X: "J \<subseteq> I" "finite J" "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)"
```
```   336   shows "emeasure (Pi\<^sub>M I M) (emb I J (Pi\<^sub>E J X)) = (\<Prod> i\<in>J. emeasure (M i) (X i))"
```
```   337 proof cases
```
```   338   assume "J = {}"
```
```   339   moreover have "emb I {} {\<lambda>x. undefined} = space (Pi\<^sub>M I M)"
```
```   340     by (auto simp: space_PiM prod_emb_def)
```
```   341   ultimately show ?thesis
```
```   342     by (simp add: space_PiM_empty P.emeasure_space_1)
```
```   343 next
```
```   344   assume "J \<noteq> {}" with X show ?thesis
```
```   345     by (subst emeasure_PiM_emb_not_empty) (auto simp: emeasure_PiM)
```
```   346 qed
```
```   347
```
```   348 lemma (in product_prob_space) emeasure_PiM_Collect:
```
```   349   assumes X: "J \<subseteq> I" "finite J" "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)"
```
```   350   shows "emeasure (Pi\<^sub>M I M) {x\<in>space (Pi\<^sub>M I M). \<forall>i\<in>J. x i \<in> X i} = (\<Prod> i\<in>J. emeasure (M i) (X i))"
```
```   351 proof -
```
```   352   have "{x\<in>space (Pi\<^sub>M I M). \<forall>i\<in>J. x i \<in> X i} = emb I J (Pi\<^sub>E J X)"
```
```   353     unfolding prod_emb_def using assms by (auto simp: space_PiM Pi_iff)
```
```   354   with emeasure_PiM_emb[OF assms] show ?thesis by simp
```
```   355 qed
```
```   356
```
```   357 lemma (in product_prob_space) emeasure_PiM_Collect_single:
```
```   358   assumes X: "i \<in> I" "A \<in> sets (M i)"
```
```   359   shows "emeasure (Pi\<^sub>M I M) {x\<in>space (Pi\<^sub>M I M). x i \<in> A} = emeasure (M i) A"
```
```   360   using emeasure_PiM_Collect[of "{i}" "\<lambda>i. A"] assms
```
```   361   by simp
```
```   362
```
```   363 lemma (in product_prob_space) measure_PiM_emb:
```
```   364   assumes "J \<subseteq> I" "finite J" "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)"
```
```   365   shows "measure (PiM I M) (emb I J (Pi\<^sub>E J X)) = (\<Prod> i\<in>J. measure (M i) (X i))"
```
```   366   using emeasure_PiM_emb[OF assms]
```
```   367   unfolding emeasure_eq_measure M.emeasure_eq_measure by (simp add: setprod_ereal)
```
```   368
```
```   369 lemma sets_Collect_single':
```
```   370   "i \<in> I \<Longrightarrow> {x\<in>space (M i). P x} \<in> sets (M i) \<Longrightarrow> {x\<in>space (PiM I M). P (x i)} \<in> sets (PiM I M)"
```
```   371   using sets_Collect_single[of i I "{x\<in>space (M i). P x}" M]
```
```   372   by (simp add: space_PiM PiE_iff cong: conj_cong)
```
```   373
```
```   374 lemma (in finite_product_prob_space) finite_measure_PiM_emb:
```
```   375   "(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> measure (PiM I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. measure (M i) (A i))"
```
```   376   using measure_PiM_emb[of I A] finite_index prod_emb_PiE_same_index[OF sets.sets_into_space, of I A M]
```
```   377   by auto
```
```   378
```
```   379 lemma (in product_prob_space) PiM_component:
```
```   380   assumes "i \<in> I"
```
```   381   shows "distr (PiM I M) (M i) (\<lambda>\<omega>. \<omega> i) = M i"
```
```   382 proof (rule measure_eqI[symmetric])
```
```   383   fix A assume "A \<in> sets (M i)"
```
```   384   moreover have "((\<lambda>\<omega>. \<omega> i) -` A \<inter> space (PiM I M)) = {x\<in>space (PiM I M). x i \<in> A}"
```
```   385     by auto
```
```   386   ultimately show "emeasure (M i) A = emeasure (distr (PiM I M) (M i) (\<lambda>\<omega>. \<omega> i)) A"
```
```   387     by (auto simp: `i\<in>I` emeasure_distr measurable_component_singleton emeasure_PiM_Collect_single)
```
```   388 qed simp
```
```   389
```
```   390 lemma (in product_prob_space) PiM_eq:
```
```   391   assumes "I \<noteq> {}"
```
```   392   assumes "sets M' = sets (PiM I M)"
```
```   393   assumes eq: "\<And>J F. finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>j. j \<in> J \<Longrightarrow> F j \<in> sets (M j)) \<Longrightarrow>
```
```   394     emeasure M' (prod_emb I M J (\<Pi>\<^sub>E j\<in>J. F j)) = (\<Prod>j\<in>J. emeasure (M j) (F j))"
```
```   395   shows "M' = (PiM I M)"
```
```   396 proof (rule measure_eqI_generator_eq[symmetric, OF Int_stable_prod_algebra prod_algebra_sets_into_space])
```
```   397   show "sets (PiM I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) (prod_algebra I M)"
```
```   398     by (rule sets_PiM)
```
```   399   then show "sets M' = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) (prod_algebra I M)"
```
```   400     unfolding `sets M' = sets (PiM I M)` by simp
```
```   401
```
```   402   def i \<equiv> "SOME i. i \<in> I"
```
```   403   with `I \<noteq> {}` have i: "i \<in> I"
```
```   404     by (auto intro: someI_ex)
```
```   405
```
```   406   def A \<equiv> "\<lambda>n::nat. prod_emb I M {i} (\<Pi>\<^sub>E j\<in>{i}. space (M i))"
```
```   407   then show "range A \<subseteq> prod_algebra I M"
```
```   408     by (auto intro!: prod_algebraI i)
```
```   409
```
```   410   have A_eq: "\<And>i. A i = space (PiM I M)"
```
```   411     by (auto simp: prod_emb_def space_PiM Pi_iff A_def i)
```
```   412   show "(\<Union>i. A i) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   413     unfolding A_eq by (auto simp: space_PiM)
```
```   414   show "\<And>i. emeasure (PiM I M) (A i) \<noteq> \<infinity>"
```
```   415     unfolding A_eq P.emeasure_space_1 by simp
```
```   416 next
```
```   417   fix X assume X: "X \<in> prod_algebra I M"
```
```   418   then obtain J E where X: "X = prod_emb I M J (PIE j:J. E j)"
```
```   419     and J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)"
```
```   420     by (force elim!: prod_algebraE)
```
```   421   from eq[OF J] have "emeasure M' X = (\<Prod>j\<in>J. emeasure (M j) (E j))"
```
```   422     by (simp add: X)
```
```   423   also have "\<dots> = emeasure (PiM I M) X"
```
```   424     unfolding X using J by (intro emeasure_PiM_emb[symmetric]) auto
```
```   425   finally show "emeasure (PiM I M) X = emeasure M' X" ..
```
```   426 qed
```
```   427
```
```   428 lemma (in product_prob_space) AE_component: "i \<in> I \<Longrightarrow> AE x in M i. P x \<Longrightarrow> AE x in PiM I M. P (x i)"
```
```   429   apply (rule AE_distrD[of "\<lambda>\<omega>. \<omega> i" "PiM I M" "M i" P])
```
```   430   apply simp
```
```   431   apply (subst PiM_component)
```
```   432   apply simp_all
```
```   433   done
```
```   434
```
```   435 subsection {* Sequence space *}
```
```   436
```
```   437 definition comb_seq :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a)" where
```
```   438   "comb_seq i \<omega> \<omega>' j = (if j < i then \<omega> j else \<omega>' (j - i))"
```
```   439
```
```   440 lemma split_comb_seq: "P (comb_seq i \<omega> \<omega>' j) \<longleftrightarrow> (j < i \<longrightarrow> P (\<omega> j)) \<and> (\<forall>k. j = i + k \<longrightarrow> P (\<omega>' k))"
```
```   441   by (auto simp: comb_seq_def not_less)
```
```   442
```
```   443 lemma split_comb_seq_asm: "P (comb_seq i \<omega> \<omega>' j) \<longleftrightarrow> \<not> ((j < i \<and> \<not> P (\<omega> j)) \<or> (\<exists>k. j = i + k \<and> \<not> P (\<omega>' k)))"
```
```   444   by (auto simp: comb_seq_def)
```
```   445
```
```   446 lemma measurable_comb_seq:
```
```   447   "(\<lambda>(\<omega>, \<omega>'). comb_seq i \<omega> \<omega>') \<in> measurable ((\<Pi>\<^sub>M i\<in>UNIV. M) \<Otimes>\<^sub>M (\<Pi>\<^sub>M i\<in>UNIV. M)) (\<Pi>\<^sub>M i\<in>UNIV. M)"
```
```   448 proof (rule measurable_PiM_single)
```
```   449   show "(\<lambda>(\<omega>, \<omega>'). comb_seq i \<omega> \<omega>') \<in> space ((\<Pi>\<^sub>M i\<in>UNIV. M) \<Otimes>\<^sub>M (\<Pi>\<^sub>M i\<in>UNIV. M)) \<rightarrow> (UNIV \<rightarrow>\<^sub>E space M)"
```
```   450     by (auto simp: space_pair_measure space_PiM PiE_iff split: split_comb_seq)
```
```   451   fix j :: nat and A assume A: "A \<in> sets M"
```
```   452   then have *: "{\<omega> \<in> space ((\<Pi>\<^sub>M i\<in>UNIV. M) \<Otimes>\<^sub>M (\<Pi>\<^sub>M i\<in>UNIV. M)). case_prod (comb_seq i) \<omega> j \<in> A} =
```
```   453     (if j < i then {\<omega> \<in> space (\<Pi>\<^sub>M i\<in>UNIV. M). \<omega> j \<in> A} \<times> space (\<Pi>\<^sub>M i\<in>UNIV. M)
```
```   454               else space (\<Pi>\<^sub>M i\<in>UNIV. M) \<times> {\<omega> \<in> space (\<Pi>\<^sub>M i\<in>UNIV. M). \<omega> (j - i) \<in> A})"
```
```   455     by (auto simp: space_PiM space_pair_measure comb_seq_def dest: sets.sets_into_space)
```
```   456   show "{\<omega> \<in> space ((\<Pi>\<^sub>M i\<in>UNIV. M) \<Otimes>\<^sub>M (\<Pi>\<^sub>M i\<in>UNIV. M)). case_prod (comb_seq i) \<omega> j \<in> A} \<in> sets ((\<Pi>\<^sub>M i\<in>UNIV. M) \<Otimes>\<^sub>M (\<Pi>\<^sub>M i\<in>UNIV. M))"
```
```   457     unfolding * by (auto simp: A intro!: sets_Collect_single)
```
```   458 qed
```
```   459
```
```   460 lemma measurable_comb_seq'[measurable (raw)]:
```
```   461   assumes f: "f \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)" and g: "g \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
```
```   462   shows "(\<lambda>x. comb_seq i (f x) (g x)) \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
```
```   463   using measurable_compose[OF measurable_Pair[OF f g] measurable_comb_seq] by simp
```
```   464
```
```   465 lemma comb_seq_0: "comb_seq 0 \<omega> \<omega>' = \<omega>'"
```
```   466   by (auto simp add: comb_seq_def)
```
```   467
```
```   468 lemma comb_seq_Suc: "comb_seq (Suc n) \<omega> \<omega>' = comb_seq n \<omega> (case_nat (\<omega> n) \<omega>')"
```
```   469   by (auto simp add: comb_seq_def not_less less_Suc_eq le_imp_diff_is_add intro!: ext split: nat.split)
```
```   470
```
```   471 lemma comb_seq_Suc_0[simp]: "comb_seq (Suc 0) \<omega> = case_nat (\<omega> 0)"
```
```   472   by (intro ext) (simp add: comb_seq_Suc comb_seq_0)
```
```   473
```
```   474 lemma comb_seq_less: "i < n \<Longrightarrow> comb_seq n \<omega> \<omega>' i = \<omega> i"
```
```   475   by (auto split: split_comb_seq)
```
```   476
```
```   477 lemma comb_seq_add: "comb_seq n \<omega> \<omega>' (i + n) = \<omega>' i"
```
```   478   by (auto split: nat.split split_comb_seq)
```
```   479
```
```   480 lemma case_nat_comb_seq: "case_nat s' (comb_seq n \<omega> \<omega>') (i + n) = case_nat (case_nat s' \<omega> n) \<omega>' i"
```
```   481   by (auto split: nat.split split_comb_seq)
```
```   482
```
```   483 lemma case_nat_comb_seq':
```
```   484   "case_nat s (comb_seq i \<omega> \<omega>') = comb_seq (Suc i) (case_nat s \<omega>) \<omega>'"
```
```   485   by (auto split: split_comb_seq nat.split)
```
```   486
```
```   487 locale sequence_space = product_prob_space "\<lambda>i. M" "UNIV :: nat set" for M
```
```   488 begin
```
```   489
```
```   490 abbreviation "S \<equiv> \<Pi>\<^sub>M i\<in>UNIV::nat set. M"
```
```   491
```
```   492 lemma infprod_in_sets[intro]:
```
```   493   fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets M"
```
```   494   shows "Pi UNIV E \<in> sets S"
```
```   495 proof -
```
```   496   have "Pi UNIV E = (\<Inter>i. emb UNIV {..i} (\<Pi>\<^sub>E j\<in>{..i}. E j))"
```
```   497     using E E[THEN sets.sets_into_space]
```
```   498     by (auto simp: prod_emb_def Pi_iff extensional_def) blast
```
```   499   with E show ?thesis by auto
```
```   500 qed
```
```   501
```
```   502 lemma measure_PiM_countable:
```
```   503   fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets M"
```
```   504   shows "(\<lambda>n. \<Prod>i\<le>n. measure M (E i)) ----> measure S (Pi UNIV E)"
```
```   505 proof -
```
```   506   let ?E = "\<lambda>n. emb UNIV {..n} (Pi\<^sub>E {.. n} E)"
```
```   507   have "\<And>n. (\<Prod>i\<le>n. measure M (E i)) = measure S (?E n)"
```
```   508     using E by (simp add: measure_PiM_emb)
```
```   509   moreover have "Pi UNIV E = (\<Inter>n. ?E n)"
```
```   510     using E E[THEN sets.sets_into_space]
```
```   511     by (auto simp: prod_emb_def extensional_def Pi_iff) blast
```
```   512   moreover have "range ?E \<subseteq> sets S"
```
```   513     using E by auto
```
```   514   moreover have "decseq ?E"
```
```   515     by (auto simp: prod_emb_def Pi_iff decseq_def)
```
```   516   ultimately show ?thesis
```
```   517     by (simp add: finite_Lim_measure_decseq)
```
```   518 qed
```
```   519
```
```   520 lemma nat_eq_diff_eq:
```
```   521   fixes a b c :: nat
```
```   522   shows "c \<le> b \<Longrightarrow> a = b - c \<longleftrightarrow> a + c = b"
```
```   523   by auto
```
```   524
```
```   525 lemma PiM_comb_seq:
```
```   526   "distr (S \<Otimes>\<^sub>M S) S (\<lambda>(\<omega>, \<omega>'). comb_seq i \<omega> \<omega>') = S" (is "?D = _")
```
```   527 proof (rule PiM_eq)
```
```   528   let ?I = "UNIV::nat set" and ?M = "\<lambda>n. M"
```
```   529   let "distr _ _ ?f" = "?D"
```
```   530
```
```   531   fix J E assume J: "finite J" "J \<subseteq> ?I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets M"
```
```   532   let ?X = "prod_emb ?I ?M J (\<Pi>\<^sub>E j\<in>J. E j)"
```
```   533   have "\<And>j x. j \<in> J \<Longrightarrow> x \<in> E j \<Longrightarrow> x \<in> space M"
```
```   534     using J(3)[THEN sets.sets_into_space] by (auto simp: space_PiM Pi_iff subset_eq)
```
```   535   with J have "?f -` ?X \<inter> space (S \<Otimes>\<^sub>M S) =
```
```   536     (prod_emb ?I ?M (J \<inter> {..<i}) (PIE j:J \<inter> {..<i}. E j)) \<times>
```
```   537     (prod_emb ?I ?M ((op + i) -` J) (PIE j:(op + i) -` J. E (i + j)))" (is "_ = ?E \<times> ?F")
```
```   538    by (auto simp: space_pair_measure space_PiM prod_emb_def all_conj_distrib PiE_iff
```
```   539                split: split_comb_seq split_comb_seq_asm)
```
```   540   then have "emeasure ?D ?X = emeasure (S \<Otimes>\<^sub>M S) (?E \<times> ?F)"
```
```   541     by (subst emeasure_distr[OF measurable_comb_seq])
```
```   542        (auto intro!: sets_PiM_I simp: split_beta' J)
```
```   543   also have "\<dots> = emeasure S ?E * emeasure S ?F"
```
```   544     using J by (intro P.emeasure_pair_measure_Times)  (auto intro!: sets_PiM_I finite_vimageI simp: inj_on_def)
```
```   545   also have "emeasure S ?F = (\<Prod>j\<in>(op + i) -` J. emeasure M (E (i + j)))"
```
```   546     using J by (intro emeasure_PiM_emb) (simp_all add: finite_vimageI inj_on_def)
```
```   547   also have "\<dots> = (\<Prod>j\<in>J - (J \<inter> {..<i}). emeasure M (E j))"
```
```   548     by (rule setprod.reindex_cong [of "\<lambda>x. x - i"])
```
```   549        (auto simp: image_iff Bex_def not_less nat_eq_diff_eq ac_simps cong: conj_cong intro!: inj_onI)
```
```   550   also have "emeasure S ?E = (\<Prod>j\<in>J \<inter> {..<i}. emeasure M (E j))"
```
```   551     using J by (intro emeasure_PiM_emb) simp_all
```
```   552   also have "(\<Prod>j\<in>J \<inter> {..<i}. emeasure M (E j)) * (\<Prod>j\<in>J - (J \<inter> {..<i}). emeasure M (E j)) = (\<Prod>j\<in>J. emeasure M (E j))"
```
```   553     by (subst mult.commute) (auto simp: J setprod.subset_diff[symmetric])
```
```   554   finally show "emeasure ?D ?X = (\<Prod>j\<in>J. emeasure M (E j))" .
```
```   555 qed simp_all
```
```   556
```
```   557 lemma PiM_iter:
```
```   558   "distr (M \<Otimes>\<^sub>M S) S (\<lambda>(s, \<omega>). case_nat s \<omega>) = S" (is "?D = _")
```
```   559 proof (rule PiM_eq)
```
```   560   let ?I = "UNIV::nat set" and ?M = "\<lambda>n. M"
```
```   561   let "distr _ _ ?f" = "?D"
```
```   562
```
```   563   fix J E assume J: "finite J" "J \<subseteq> ?I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets M"
```
```   564   let ?X = "prod_emb ?I ?M J (PIE j:J. E j)"
```
```   565   have "\<And>j x. j \<in> J \<Longrightarrow> x \<in> E j \<Longrightarrow> x \<in> space M"
```
```   566     using J(3)[THEN sets.sets_into_space] by (auto simp: space_PiM Pi_iff subset_eq)
```
```   567   with J have "?f -` ?X \<inter> space (M \<Otimes>\<^sub>M S) = (if 0 \<in> J then E 0 else space M) \<times>
```
```   568     (prod_emb ?I ?M (Suc -` J) (PIE j:Suc -` J. E (Suc j)))" (is "_ = ?E \<times> ?F")
```
```   569    by (auto simp: space_pair_measure space_PiM PiE_iff prod_emb_def all_conj_distrib
```
```   570       split: nat.split nat.split_asm)
```
```   571   then have "emeasure ?D ?X = emeasure (M \<Otimes>\<^sub>M S) (?E \<times> ?F)"
```
```   572     by (subst emeasure_distr)
```
```   573        (auto intro!: sets_PiM_I simp: split_beta' J)
```
```   574   also have "\<dots> = emeasure M ?E * emeasure S ?F"
```
```   575     using J by (intro P.emeasure_pair_measure_Times) (auto intro!: sets_PiM_I finite_vimageI)
```
```   576   also have "emeasure S ?F = (\<Prod>j\<in>Suc -` J. emeasure M (E (Suc j)))"
```
```   577     using J by (intro emeasure_PiM_emb) (simp_all add: finite_vimageI)
```
```   578   also have "\<dots> = (\<Prod>j\<in>J - {0}. emeasure M (E j))"
```
```   579     by (rule setprod.reindex_cong [of "\<lambda>x. x - 1"])
```
```   580        (auto simp: image_iff Bex_def not_less nat_eq_diff_eq ac_simps cong: conj_cong intro!: inj_onI)
```
```   581   also have "emeasure M ?E * (\<Prod>j\<in>J - {0}. emeasure M (E j)) = (\<Prod>j\<in>J. emeasure M (E j))"
```
```   582     by (auto simp: M.emeasure_space_1 setprod.remove J)
```
```   583   finally show "emeasure ?D ?X = (\<Prod>j\<in>J. emeasure M (E j))" .
```
```   584 qed simp_all
```
```   585
```
```   586 end
```
```   587
```
`   588 end`