src/HOL/Probability/Infinite_Product_Measure.thy
 author huffman Fri Aug 19 14:17:28 2011 -0700 (2011-08-19) changeset 44311 42c5cbf68052 parent 43920 cedb5cb948fd child 44928 7ef6505bde7f permissions -rw-r--r--
new isCont theorems;
simplify some proofs.
```     1 (*  Title:      HOL/Probability/Infinite_Product_Measure.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3 *)
```
```     4
```
```     5 header {*Infinite Product Measure*}
```
```     6
```
```     7 theory Infinite_Product_Measure
```
```     8   imports Probability_Measure
```
```     9 begin
```
```    10
```
```    11 lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
```
```    12   unfolding restrict_def extensional_def by auto
```
```    13
```
```    14 lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A \<inter> B)"
```
```    15   unfolding restrict_def by (simp add: fun_eq_iff)
```
```    16
```
```    17 lemma split_merge: "P (merge I x J y i) \<longleftrightarrow> (i \<in> I \<longrightarrow> P (x i)) \<and> (i \<in> J - I \<longrightarrow> P (y i)) \<and> (i \<notin> I \<union> J \<longrightarrow> P undefined)"
```
```    18   unfolding merge_def by auto
```
```    19
```
```    20 lemma extensional_merge_sub: "I \<union> J \<subseteq> K \<Longrightarrow> merge I x J y \<in> extensional K"
```
```    21   unfolding merge_def extensional_def by auto
```
```    22
```
```    23 lemma injective_vimage_restrict:
```
```    24   assumes J: "J \<subseteq> I"
```
```    25   and sets: "A \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" "B \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" and ne: "(\<Pi>\<^isub>E i\<in>I. S i) \<noteq> {}"
```
```    26   and eq: "(\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i) = (\<lambda>x. restrict x J) -` B \<inter> (\<Pi>\<^isub>E i\<in>I. S i)"
```
```    27   shows "A = B"
```
```    28 proof  (intro set_eqI)
```
```    29   fix x
```
```    30   from ne obtain y where y: "\<And>i. i \<in> I \<Longrightarrow> y i \<in> S i" by auto
```
```    31   have "J \<inter> (I - J) = {}" by auto
```
```    32   show "x \<in> A \<longleftrightarrow> x \<in> B"
```
```    33   proof cases
```
```    34     assume x: "x \<in> (\<Pi>\<^isub>E i\<in>J. S i)"
```
```    35     have "x \<in> A \<longleftrightarrow> merge J x (I - J) y \<in> (\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i)"
```
```    36       using y x `J \<subseteq> I` by (auto simp add: Pi_iff extensional_restrict extensional_merge_sub split: split_merge)
```
```    37     then show "x \<in> A \<longleftrightarrow> x \<in> B"
```
```    38       using y x `J \<subseteq> I` by (auto simp add: Pi_iff extensional_restrict extensional_merge_sub eq split: split_merge)
```
```    39   next
```
```    40     assume "x \<notin> (\<Pi>\<^isub>E i\<in>J. S i)" with sets show "x \<in> A \<longleftrightarrow> x \<in> B" by auto
```
```    41   qed
```
```    42 qed
```
```    43
```
```    44 locale product_prob_space =
```
```    45   fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" and I :: "'i set"
```
```    46   assumes prob_spaces: "\<And>i. prob_space (M i)"
```
```    47   and I_not_empty: "I \<noteq> {}"
```
```    48
```
```    49 locale finite_product_prob_space = product_prob_space M I
```
```    50   for M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" and I :: "'i set" +
```
```    51   assumes finite_index'[intro]: "finite I"
```
```    52
```
```    53 sublocale product_prob_space \<subseteq> M: prob_space "M i" for i
```
```    54   by (rule prob_spaces)
```
```    55
```
```    56 sublocale product_prob_space \<subseteq> product_sigma_finite
```
```    57   by default
```
```    58
```
```    59 sublocale finite_product_prob_space \<subseteq> finite_product_sigma_finite
```
```    60   by default (fact finite_index')
```
```    61
```
```    62 sublocale finite_product_prob_space \<subseteq> prob_space "Pi\<^isub>M I M"
```
```    63 proof
```
```    64   show "measure P (space P) = 1"
```
```    65     by (simp add: measure_times measure_space_1 setprod_1)
```
```    66 qed
```
```    67
```
```    68 lemma (in product_prob_space) measure_preserving_restrict:
```
```    69   assumes "J \<noteq> {}" "J \<subseteq> K" "finite K"
```
```    70   shows "(\<lambda>f. restrict f J) \<in> measure_preserving (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i)" (is "?R \<in> _")
```
```    71 proof -
```
```    72   interpret K: finite_product_prob_space M K
```
```    73     by default (insert assms, auto)
```
```    74   have J: "J \<noteq> {}" "finite J" using assms by (auto simp add: finite_subset)
```
```    75   interpret J: finite_product_prob_space M J
```
```    76     by default (insert J, auto)
```
```    77   from J.sigma_finite_pairs guess F .. note F = this
```
```    78   then have [simp,intro]: "\<And>k i. k \<in> J \<Longrightarrow> F k i \<in> sets (M k)"
```
```    79     by auto
```
```    80   let "?F i" = "\<Pi>\<^isub>E k\<in>J. F k i"
```
```    81   let ?J = "product_algebra_generator J M \<lparr> measure := measure (Pi\<^isub>M J M) \<rparr>"
```
```    82   have "?R \<in> measure_preserving (\<Pi>\<^isub>M i\<in>K. M i) (sigma ?J)"
```
```    83   proof (rule K.measure_preserving_Int_stable)
```
```    84     show "Int_stable ?J"
```
```    85       by (auto simp: Int_stable_def product_algebra_generator_def PiE_Int)
```
```    86     show "range ?F \<subseteq> sets ?J" "incseq ?F" "(\<Union>i. ?F i) = space ?J"
```
```    87       using F by auto
```
```    88     show "\<And>i. measure ?J (?F i) \<noteq> \<infinity>"
```
```    89       using F by (simp add: J.measure_times setprod_PInf)
```
```    90     have "measure_space (Pi\<^isub>M J M)" by default
```
```    91     then show "measure_space (sigma ?J)"
```
```    92       by (simp add: product_algebra_def sigma_def)
```
```    93     show "?R \<in> measure_preserving (Pi\<^isub>M K M) ?J"
```
```    94     proof (simp add: measure_preserving_def measurable_def product_algebra_generator_def del: vimage_Int,
```
```    95            safe intro!: restrict_extensional)
```
```    96       fix x k assume "k \<in> J" "x \<in> (\<Pi> i\<in>K. space (M i))"
```
```    97       then show "x k \<in> space (M k)" using `J \<subseteq> K` by auto
```
```    98     next
```
```    99       fix E assume "E \<in> (\<Pi> i\<in>J. sets (M i))"
```
```   100       then have E: "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)" by auto
```
```   101       then have *: "?R -` Pi\<^isub>E J E \<inter> (\<Pi>\<^isub>E i\<in>K. space (M i)) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i))"
```
```   102         (is "?X = Pi\<^isub>E K ?M")
```
```   103         using `J \<subseteq> K` sets_into_space by (auto simp: Pi_iff split: split_if_asm) blast+
```
```   104       with E show "?X \<in> sets (Pi\<^isub>M K M)"
```
```   105         by (auto intro!: product_algebra_generatorI)
```
```   106       have "measure (Pi\<^isub>M J M) (Pi\<^isub>E J E) = (\<Prod>i\<in>J. measure (M i) (?M i))"
```
```   107         using E by (simp add: J.measure_times)
```
```   108       also have "\<dots> = measure (Pi\<^isub>M K M) ?X"
```
```   109         unfolding * using E `finite K` `J \<subseteq> K`
```
```   110         by (auto simp: K.measure_times M.measure_space_1
```
```   111                  cong del: setprod_cong
```
```   112                  intro!: setprod_mono_one_left)
```
```   113       finally show "measure (Pi\<^isub>M J M) (Pi\<^isub>E J E) = measure (Pi\<^isub>M K M) ?X" .
```
```   114     qed
```
```   115   qed
```
```   116   then show ?thesis
```
```   117     by (simp add: product_algebra_def sigma_def)
```
```   118 qed
```
```   119
```
```   120 lemma (in product_prob_space) measurable_restrict:
```
```   121   assumes *: "J \<noteq> {}" "J \<subseteq> K" "finite K"
```
```   122   shows "(\<lambda>f. restrict f J) \<in> measurable (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i)"
```
```   123   using measure_preserving_restrict[OF *]
```
```   124   by (rule measure_preservingD2)
```
```   125
```
```   126 definition (in product_prob_space)
```
```   127   "emb J K X = (\<lambda>x. restrict x K) -` X \<inter> space (Pi\<^isub>M J M)"
```
```   128
```
```   129 lemma (in product_prob_space) emb_trans[simp]:
```
```   130   "J \<subseteq> K \<Longrightarrow> K \<subseteq> L \<Longrightarrow> emb L K (emb K J X) = emb L J X"
```
```   131   by (auto simp add: Int_absorb1 emb_def)
```
```   132
```
```   133 lemma (in product_prob_space) emb_empty[simp]:
```
```   134   "emb K J {} = {}"
```
```   135   by (simp add: emb_def)
```
```   136
```
```   137 lemma (in product_prob_space) emb_Pi:
```
```   138   assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
```
```   139   shows "emb K J (Pi\<^isub>E J X) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then X i else space (M i))"
```
```   140   using assms space_closed
```
```   141   by (auto simp: emb_def Pi_iff split: split_if_asm) blast+
```
```   142
```
```   143 lemma (in product_prob_space) emb_injective:
```
```   144   assumes "J \<noteq> {}" "J \<subseteq> L" "finite J" and sets: "X \<in> sets (Pi\<^isub>M J M)" "Y \<in> sets (Pi\<^isub>M J M)"
```
```   145   assumes "emb L J X = emb L J Y"
```
```   146   shows "X = Y"
```
```   147 proof -
```
```   148   interpret J: finite_product_sigma_finite M J by default fact
```
```   149   show "X = Y"
```
```   150   proof (rule injective_vimage_restrict)
```
```   151     show "X \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" "Y \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
```
```   152       using J.sets_into_space sets by auto
```
```   153     have "\<forall>i\<in>L. \<exists>x. x \<in> space (M i)"
```
```   154       using M.not_empty by auto
```
```   155     from bchoice[OF this]
```
```   156     show "(\<Pi>\<^isub>E i\<in>L. space (M i)) \<noteq> {}" by auto
```
```   157     show "(\<lambda>x. restrict x J) -` X \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i)) = (\<lambda>x. restrict x J) -` Y \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i))"
```
```   158       using `emb L J X = emb L J Y` by (simp add: emb_def)
```
```   159   qed fact
```
```   160 qed
```
```   161
```
```   162 lemma (in product_prob_space) emb_id:
```
```   163   "B \<subseteq> (\<Pi>\<^isub>E i\<in>L. space (M i)) \<Longrightarrow> emb L L B = B"
```
```   164   by (auto simp: emb_def Pi_iff subset_eq extensional_restrict)
```
```   165
```
```   166 lemma (in product_prob_space) emb_simps:
```
```   167   shows "emb L K (A \<union> B) = emb L K A \<union> emb L K B"
```
```   168     and "emb L K (A \<inter> B) = emb L K A \<inter> emb L K B"
```
```   169     and "emb L K (A - B) = emb L K A - emb L K B"
```
```   170   by (auto simp: emb_def)
```
```   171
```
```   172 lemma (in product_prob_space) measurable_emb[intro,simp]:
```
```   173   assumes *: "J \<noteq> {}" "J \<subseteq> L" "finite L" "X \<in> sets (Pi\<^isub>M J M)"
```
```   174   shows "emb L J X \<in> sets (Pi\<^isub>M L M)"
```
```   175   using measurable_restrict[THEN measurable_sets, OF *] by (simp add: emb_def)
```
```   176
```
```   177 lemma (in product_prob_space) measure_emb[intro,simp]:
```
```   178   assumes *: "J \<noteq> {}" "J \<subseteq> L" "finite L" "X \<in> sets (Pi\<^isub>M J M)"
```
```   179   shows "measure (Pi\<^isub>M L M) (emb L J X) = measure (Pi\<^isub>M J M) X"
```
```   180   using measure_preserving_restrict[THEN measure_preservingD, OF *]
```
```   181   by (simp add: emb_def)
```
```   182
```
```   183 definition (in product_prob_space) generator :: "('i \<Rightarrow> 'a) measure_space" where
```
```   184   "generator = \<lparr>
```
```   185     space = (\<Pi>\<^isub>E i\<in>I. space (M i)),
```
```   186     sets = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M)),
```
```   187     measure = undefined
```
```   188   \<rparr>"
```
```   189
```
```   190 lemma (in product_prob_space) generatorI:
```
```   191   "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> A = emb I J X \<Longrightarrow> A \<in> sets generator"
```
```   192   unfolding generator_def by auto
```
```   193
```
```   194 lemma (in product_prob_space) generatorI':
```
```   195   "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> emb I J X \<in> sets generator"
```
```   196   unfolding generator_def by auto
```
```   197
```
```   198 lemma (in product_sigma_finite)
```
```   199   assumes "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
```
```   200   shows measure_fold_integral:
```
```   201     "measure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. measure (Pi\<^isub>M J M) (merge I x J -` A \<inter> space (Pi\<^isub>M J M)) \<partial>Pi\<^isub>M I M)" (is ?I)
```
```   202     and measure_fold_measurable:
```
```   203     "(\<lambda>x. measure (Pi\<^isub>M J M) (merge I x J -` A \<inter> space (Pi\<^isub>M J M))) \<in> borel_measurable (Pi\<^isub>M I M)" (is ?B)
```
```   204 proof -
```
```   205   interpret I: finite_product_sigma_finite M I by default fact
```
```   206   interpret J: finite_product_sigma_finite M J by default fact
```
```   207   interpret IJ: pair_sigma_finite I.P J.P ..
```
```   208   show ?I
```
```   209     unfolding measure_fold[OF assms]
```
```   210     apply (subst IJ.pair_measure_alt)
```
```   211     apply (intro measurable_sets[OF _ A] measurable_merge assms)
```
```   212     apply (auto simp: vimage_compose[symmetric] comp_def space_pair_measure
```
```   213       intro!: I.positive_integral_cong)
```
```   214     done
```
```   215
```
```   216   have "(\<lambda>(x, y). merge I x J y) -` A \<inter> space (I.P \<Otimes>\<^isub>M J.P) \<in> sets (I.P \<Otimes>\<^isub>M J.P)"
```
```   217     by (intro measurable_sets[OF _ A] measurable_merge assms)
```
```   218   from IJ.measure_cut_measurable_fst[OF this]
```
```   219   show ?B
```
```   220     apply (auto simp: vimage_compose[symmetric] comp_def space_pair_measure)
```
```   221     apply (subst (asm) measurable_cong)
```
```   222     apply auto
```
```   223     done
```
```   224 qed
```
```   225
```
```   226 definition (in product_prob_space)
```
```   227   "\<mu>G A =
```
```   228     (THE x. \<forall>J. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> J \<subseteq> I \<longrightarrow> (\<forall>X\<in>sets (Pi\<^isub>M J M). A = emb I J X \<longrightarrow> x = measure (Pi\<^isub>M J M) X))"
```
```   229
```
```   230 lemma (in product_prob_space) \<mu>G_spec:
```
```   231   assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)"
```
```   232   shows "\<mu>G A = measure (Pi\<^isub>M J M) X"
```
```   233   unfolding \<mu>G_def
```
```   234 proof (intro the_equality allI impI ballI)
```
```   235   fix K Y assume K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "A = emb I K Y" "Y \<in> sets (Pi\<^isub>M K M)"
```
```   236   have "measure (Pi\<^isub>M K M) Y = measure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) K Y)"
```
```   237     using K J by simp
```
```   238   also have "emb (K \<union> J) K Y = emb (K \<union> J) J X"
```
```   239     using K J by (simp add: emb_injective[of "K \<union> J" I])
```
```   240   also have "measure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) J X) = measure (Pi\<^isub>M J M) X"
```
```   241     using K J by simp
```
```   242   finally show "measure (Pi\<^isub>M J M) X = measure (Pi\<^isub>M K M) Y" ..
```
```   243 qed (insert J, force)
```
```   244
```
```   245 lemma (in product_prob_space) \<mu>G_eq:
```
```   246   "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> \<mu>G (emb I J X) = measure (Pi\<^isub>M J M) X"
```
```   247   by (intro \<mu>G_spec) auto
```
```   248
```
```   249 lemma (in product_prob_space) generator_Ex:
```
```   250   assumes *: "A \<in> sets generator"
```
```   251   shows "\<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A = emb I J X \<and> \<mu>G A = measure (Pi\<^isub>M J M) X"
```
```   252 proof -
```
```   253   from * obtain J X where J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)"
```
```   254     unfolding generator_def by auto
```
```   255   with \<mu>G_spec[OF this] show ?thesis by auto
```
```   256 qed
```
```   257
```
```   258 lemma (in product_prob_space) generatorE:
```
```   259   assumes A: "A \<in> sets generator"
```
```   260   obtains J X where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" "\<mu>G A = measure (Pi\<^isub>M J M) X"
```
```   261 proof -
```
```   262   from generator_Ex[OF A] obtain X J where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A"
```
```   263     "\<mu>G A = measure (Pi\<^isub>M J M) X" by auto
```
```   264   then show thesis by (intro that) auto
```
```   265 qed
```
```   266
```
```   267 lemma (in product_prob_space) merge_sets:
```
```   268   assumes "finite J" "finite K" "J \<inter> K = {}" and A: "A \<in> sets (Pi\<^isub>M (J \<union> K) M)" and x: "x \<in> space (Pi\<^isub>M J M)"
```
```   269   shows "merge J x K -` A \<inter> space (Pi\<^isub>M K M) \<in> sets (Pi\<^isub>M K M)"
```
```   270 proof -
```
```   271   interpret J: finite_product_sigma_algebra M J by default fact
```
```   272   interpret K: finite_product_sigma_algebra M K by default fact
```
```   273   interpret JK: pair_sigma_algebra J.P K.P ..
```
```   274
```
```   275   from JK.measurable_cut_fst[OF
```
```   276     measurable_merge[THEN measurable_sets, OF `J \<inter> K = {}`], OF A, of x] x
```
```   277   show ?thesis
```
```   278       by (simp add: space_pair_measure comp_def vimage_compose[symmetric])
```
```   279 qed
```
```   280
```
```   281 lemma (in product_prob_space) merge_emb:
```
```   282   assumes "K \<subseteq> I" "J \<subseteq> I" and y: "y \<in> space (Pi\<^isub>M J M)"
```
```   283   shows "(merge J y (I - J) -` emb I K X \<inter> space (Pi\<^isub>M I M)) =
```
```   284     emb I (K - J) (merge J y (K - J) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M))"
```
```   285 proof -
```
```   286   have [simp]: "\<And>x J K L. merge J y K (restrict x L) = merge J y (K \<inter> L) x"
```
```   287     by (auto simp: restrict_def merge_def)
```
```   288   have [simp]: "\<And>x J K L. restrict (merge J y K x) L = merge (J \<inter> L) y (K \<inter> L) x"
```
```   289     by (auto simp: restrict_def merge_def)
```
```   290   have [simp]: "(I - J) \<inter> K = K - J" using `K \<subseteq> I` `J \<subseteq> I` by auto
```
```   291   have [simp]: "(K - J) \<inter> (K \<union> J) = K - J" by auto
```
```   292   have [simp]: "(K - J) \<inter> K = K - J" by auto
```
```   293   from y `K \<subseteq> I` `J \<subseteq> I` show ?thesis
```
```   294     by (simp split: split_merge add: emb_def Pi_iff extensional_merge_sub set_eq_iff) auto
```
```   295 qed
```
```   296
```
```   297 definition (in product_prob_space) infprod_algebra :: "('i \<Rightarrow> 'a) measure_space" where
```
```   298   "infprod_algebra = sigma generator \<lparr> measure :=
```
```   299     (SOME \<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and>
```
```   300        measure_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>)\<rparr>"
```
```   301
```
```   302 syntax
```
```   303   "_PiP"  :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme"  ("(3PIP _:_./ _)" 10)
```
```   304
```
```   305 syntax (xsymbols)
```
```   306   "_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme"  ("(3\<Pi>\<^isub>P _\<in>_./ _)"   10)
```
```   307
```
```   308 syntax (HTML output)
```
```   309   "_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme"  ("(3\<Pi>\<^isub>P _\<in>_./ _)"   10)
```
```   310
```
```   311 abbreviation
```
```   312   "Pi\<^isub>P I M \<equiv> product_prob_space.infprod_algebra M I"
```
```   313
```
```   314 translations
```
```   315   "PIP x:I. M" == "CONST Pi\<^isub>P I (%x. M)"
```
```   316
```
```   317 sublocale product_prob_space \<subseteq> G!: algebra generator
```
```   318 proof
```
```   319   let ?G = generator
```
```   320   show "sets ?G \<subseteq> Pow (space ?G)"
```
```   321     by (auto simp: generator_def emb_def)
```
```   322   from I_not_empty
```
```   323   obtain i where "i \<in> I" by auto
```
```   324   then show "{} \<in> sets ?G"
```
```   325     by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"]
```
```   326       simp: product_algebra_def sigma_def sigma_sets.Empty generator_def emb_def)
```
```   327   from `i \<in> I` show "space ?G \<in> sets ?G"
```
```   328     by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"]
```
```   329       simp: generator_def emb_def)
```
```   330   fix A assume "A \<in> sets ?G"
```
```   331   then obtain JA XA where XA: "JA \<noteq> {}" "finite JA" "JA \<subseteq> I" "XA \<in> sets (Pi\<^isub>M JA M)" and A: "A = emb I JA XA"
```
```   332     by (auto simp: generator_def)
```
```   333   fix B assume "B \<in> sets ?G"
```
```   334   then obtain JB XB where XB: "JB \<noteq> {}" "finite JB" "JB \<subseteq> I" "XB \<in> sets (Pi\<^isub>M JB M)" and B: "B = emb I JB XB"
```
```   335     by (auto simp: generator_def)
```
```   336   let ?RA = "emb (JA \<union> JB) JA XA"
```
```   337   let ?RB = "emb (JA \<union> JB) JB XB"
```
```   338   interpret JAB: finite_product_sigma_algebra M "JA \<union> JB"
```
```   339     by default (insert XA XB, auto)
```
```   340   have *: "A - B = emb I (JA \<union> JB) (?RA - ?RB)" "A \<union> B = emb I (JA \<union> JB) (?RA \<union> ?RB)"
```
```   341     using XA A XB B by (auto simp: emb_simps)
```
```   342   then show "A - B \<in> sets ?G" "A \<union> B \<in> sets ?G"
```
```   343     using XA XB by (auto intro!: generatorI')
```
```   344 qed
```
```   345
```
```   346 lemma (in product_prob_space) positive_\<mu>G: "positive generator \<mu>G"
```
```   347 proof (intro positive_def[THEN iffD2] conjI ballI)
```
```   348   from generatorE[OF G.empty_sets] guess J X . note this[simp]
```
```   349   interpret J: finite_product_sigma_finite M J by default fact
```
```   350   have "X = {}"
```
```   351     by (rule emb_injective[of J I]) simp_all
```
```   352   then show "\<mu>G {} = 0" by simp
```
```   353 next
```
```   354   fix A assume "A \<in> sets generator"
```
```   355   from generatorE[OF this] guess J X . note this[simp]
```
```   356   interpret J: finite_product_sigma_finite M J by default fact
```
```   357   show "0 \<le> \<mu>G A" by simp
```
```   358 qed
```
```   359
```
```   360 lemma (in product_prob_space) additive_\<mu>G: "additive generator \<mu>G"
```
```   361 proof (intro additive_def[THEN iffD2] ballI impI)
```
```   362   fix A assume "A \<in> sets generator" with generatorE guess J X . note J = this
```
```   363   fix B assume "B \<in> sets generator" with generatorE guess K Y . note K = this
```
```   364   assume "A \<inter> B = {}"
```
```   365   have JK: "J \<union> K \<noteq> {}" "J \<union> K \<subseteq> I" "finite (J \<union> K)"
```
```   366     using J K by auto
```
```   367   interpret JK: finite_product_sigma_finite M "J \<union> K" by default fact
```
```   368   have JK_disj: "emb (J \<union> K) J X \<inter> emb (J \<union> K) K Y = {}"
```
```   369     apply (rule emb_injective[of "J \<union> K" I])
```
```   370     apply (insert `A \<inter> B = {}` JK J K)
```
```   371     apply (simp_all add: JK.Int emb_simps)
```
```   372     done
```
```   373   have AB: "A = emb I (J \<union> K) (emb (J \<union> K) J X)" "B = emb I (J \<union> K) (emb (J \<union> K) K Y)"
```
```   374     using J K by simp_all
```
```   375   then have "\<mu>G (A \<union> B) = \<mu>G (emb I (J \<union> K) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y))"
```
```   376     by (simp add: emb_simps)
```
```   377   also have "\<dots> = measure (Pi\<^isub>M (J \<union> K) M) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)"
```
```   378     using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq JK.Un)
```
```   379   also have "\<dots> = \<mu>G A + \<mu>G B"
```
```   380     using J K JK_disj by (simp add: JK.measure_additive[symmetric])
```
```   381   finally show "\<mu>G (A \<union> B) = \<mu>G A + \<mu>G B" .
```
```   382 qed
```
```   383
```
```   384 lemma (in product_prob_space) finite_index_eq_finite_product:
```
```   385   assumes "finite I"
```
```   386   shows "sets (sigma generator) = sets (Pi\<^isub>M I M)"
```
```   387 proof safe
```
```   388   interpret I: finite_product_sigma_algebra M I by default fact
```
```   389   have [simp]: "space generator = space (Pi\<^isub>M I M)"
```
```   390     by (simp add: generator_def product_algebra_def)
```
```   391   { fix A assume "A \<in> sets (sigma generator)"
```
```   392     then show "A \<in> sets I.P" unfolding sets_sigma
```
```   393     proof induct
```
```   394       case (Basic A)
```
```   395       from generatorE[OF this] guess J X . note J = this
```
```   396       with `finite I` have "emb I J X \<in> sets I.P" by auto
```
```   397       with `emb I J X = A` show "A \<in> sets I.P" by simp
```
```   398     qed auto }
```
```   399   { fix A assume "A \<in> sets I.P"
```
```   400     moreover with I.sets_into_space have "emb I I A = A" by (intro emb_id) auto
```
```   401     ultimately show "A \<in> sets (sigma generator)"
```
```   402       using `finite I` I_not_empty unfolding sets_sigma
```
```   403       by (intro sigma_sets.Basic generatorI[of I A]) auto }
```
```   404 qed
```
```   405
```
```   406 lemma (in product_prob_space) extend_\<mu>G:
```
```   407   "\<exists>\<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and>
```
```   408        measure_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>"
```
```   409 proof cases
```
```   410   assume "finite I"
```
```   411   interpret I: finite_product_sigma_finite M I by default fact
```
```   412   show ?thesis
```
```   413   proof (intro exI[of _ "measure (Pi\<^isub>M I M)"] ballI conjI)
```
```   414     fix A assume "A \<in> sets generator"
```
```   415     from generatorE[OF this] guess J X . note J = this
```
```   416     from J(1-4) `finite I` show "measure I.P A = \<mu>G A"
```
```   417       unfolding J(6)
```
```   418       by (subst J(5)[symmetric]) (simp add: measure_emb)
```
```   419   next
```
```   420     have [simp]: "space generator = space (Pi\<^isub>M I M)"
```
```   421       by (simp add: generator_def product_algebra_def)
```
```   422     have "\<lparr>space = space generator, sets = sets (sigma generator), measure = measure I.P\<rparr>
```
```   423       = I.P" (is "?P = _")
```
```   424       by (auto intro!: measure_space.equality simp: finite_index_eq_finite_product[OF `finite I`])
```
```   425     then show "measure_space ?P" by simp default
```
```   426   qed
```
```   427 next
```
```   428   let ?G = generator
```
```   429   assume "\<not> finite I"
```
```   430   note \<mu>G_mono =
```
```   431     G.additive_increasing[OF positive_\<mu>G additive_\<mu>G, THEN increasingD]
```
```   432
```
```   433   { fix Z J assume J: "J \<noteq> {}" "finite J" "J \<subseteq> I" and Z: "Z \<in> sets ?G"
```
```   434
```
```   435     from `infinite I` `finite J` obtain k where k: "k \<in> I" "k \<notin> J"
```
```   436       by (metis rev_finite_subset subsetI)
```
```   437     moreover from Z guess K' X' by (rule generatorE)
```
```   438     moreover def K \<equiv> "insert k K'"
```
```   439     moreover def X \<equiv> "emb K K' X'"
```
```   440     ultimately have K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "X \<in> sets (Pi\<^isub>M K M)" "Z = emb I K X"
```
```   441       "K - J \<noteq> {}" "K - J \<subseteq> I" "\<mu>G Z = measure (Pi\<^isub>M K M) X"
```
```   442       by (auto simp: subset_insertI)
```
```   443
```
```   444     let "?M y" = "merge J y (K - J) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M)"
```
```   445     { fix y assume y: "y \<in> space (Pi\<^isub>M J M)"
```
```   446       note * = merge_emb[OF `K \<subseteq> I` `J \<subseteq> I` y, of X]
```
```   447       moreover
```
```   448       have **: "?M y \<in> sets (Pi\<^isub>M (K - J) M)"
```
```   449         using J K y by (intro merge_sets) auto
```
```   450       ultimately
```
```   451       have ***: "(merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M)) \<in> sets ?G"
```
```   452         using J K by (intro generatorI) auto
```
```   453       have "\<mu>G (merge J y (I - J) -` emb I K X \<inter> space (Pi\<^isub>M I M)) = measure (Pi\<^isub>M (K - J) M) (?M y)"
```
```   454         unfolding * using K J by (subst \<mu>G_eq[OF _ _ _ **]) auto
```
```   455       note * ** *** this }
```
```   456     note merge_in_G = this
```
```   457
```
```   458     have "finite (K - J)" using K by auto
```
```   459
```
```   460     interpret J: finite_product_prob_space M J by default fact+
```
```   461     interpret KmJ: finite_product_prob_space M "K - J" by default fact+
```
```   462
```
```   463     have "\<mu>G Z = measure (Pi\<^isub>M (J \<union> (K - J)) M) (emb (J \<union> (K - J)) K X)"
```
```   464       using K J by simp
```
```   465     also have "\<dots> = (\<integral>\<^isup>+ x. measure (Pi\<^isub>M (K - J) M) (?M x) \<partial>Pi\<^isub>M J M)"
```
```   466       using K J by (subst measure_fold_integral) auto
```
```   467     also have "\<dots> = (\<integral>\<^isup>+ y. \<mu>G (merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M)) \<partial>Pi\<^isub>M J M)"
```
```   468       (is "_ = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)")
```
```   469     proof (intro J.positive_integral_cong)
```
```   470       fix x assume x: "x \<in> space (Pi\<^isub>M J M)"
```
```   471       with K merge_in_G(2)[OF this]
```
```   472       show "measure (Pi\<^isub>M (K - J) M) (?M x) = \<mu>G (?MZ x)"
```
```   473         unfolding `Z = emb I K X` merge_in_G(1)[OF x] by (subst \<mu>G_eq) auto
```
```   474     qed
```
```   475     finally have fold: "\<mu>G Z = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)" .
```
```   476
```
```   477     { fix x assume x: "x \<in> space (Pi\<^isub>M J M)"
```
```   478       then have "\<mu>G (?MZ x) \<le> 1"
```
```   479         unfolding merge_in_G(4)[OF x] `Z = emb I K X`
```
```   480         by (intro KmJ.measure_le_1 merge_in_G(2)[OF x]) }
```
```   481     note le_1 = this
```
```   482
```
```   483     let "?q y" = "\<mu>G (merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M))"
```
```   484     have "?q \<in> borel_measurable (Pi\<^isub>M J M)"
```
```   485       unfolding `Z = emb I K X` using J K merge_in_G(3)
```
```   486       by (simp add: merge_in_G  \<mu>G_eq measure_fold_measurable
```
```   487                del: space_product_algebra cong: measurable_cong)
```
```   488     note this fold le_1 merge_in_G(3) }
```
```   489   note fold = this
```
```   490
```
```   491   show ?thesis
```
```   492   proof (rule G.caratheodory_empty_continuous[OF positive_\<mu>G additive_\<mu>G])
```
```   493     fix A assume "A \<in> sets ?G"
```
```   494     with generatorE guess J X . note JX = this
```
```   495     interpret JK: finite_product_prob_space M J by default fact+
```
```   496     with JX show "\<mu>G A \<noteq> \<infinity>" by simp
```
```   497   next
```
```   498     fix A assume A: "range A \<subseteq> sets ?G" "decseq A" "(\<Inter>i. A i) = {}"
```
```   499     then have "decseq (\<lambda>i. \<mu>G (A i))"
```
```   500       by (auto intro!: \<mu>G_mono simp: decseq_def)
```
```   501     moreover
```
```   502     have "(INF i. \<mu>G (A i)) = 0"
```
```   503     proof (rule ccontr)
```
```   504       assume "(INF i. \<mu>G (A i)) \<noteq> 0" (is "?a \<noteq> 0")
```
```   505       moreover have "0 \<le> ?a"
```
```   506         using A positive_\<mu>G by (auto intro!: le_INFI simp: positive_def)
```
```   507       ultimately have "0 < ?a" by auto
```
```   508
```
```   509       have "\<forall>n. \<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A n = emb I J X \<and> \<mu>G (A n) = measure (Pi\<^isub>M J M) X"
```
```   510         using A by (intro allI generator_Ex) auto
```
```   511       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\<^isub>M (J' n) M)"
```
```   512         and A': "\<And>n. A n = emb I (J' n) (X' n)"
```
```   513         unfolding choice_iff by blast
```
```   514       moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)"
```
```   515       moreover def X \<equiv> "\<lambda>n. emb (J n) (J' n) (X' n)"
```
```   516       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\<^isub>M (J n) M)"
```
```   517         by auto
```
```   518       with A' have A_eq: "\<And>n. A n = emb I (J n) (X n)" "\<And>n. A n \<in> sets ?G"
```
```   519         unfolding J_def X_def by (subst emb_trans) (insert A, auto)
```
```   520
```
```   521       have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
```
```   522         unfolding J_def by force
```
```   523
```
```   524       interpret J: finite_product_prob_space M "J i" for i by default fact+
```
```   525
```
```   526       have a_le_1: "?a \<le> 1"
```
```   527         using \<mu>G_spec[of "J 0" "A 0" "X 0"] J A_eq
```
```   528         by (auto intro!: INF_leI2[of 0] J.measure_le_1)
```
```   529
```
```   530       let "?M K Z y" = "merge K y (I - K) -` Z \<inter> space (Pi\<^isub>M I M)"
```
```   531
```
```   532       { fix Z k assume Z: "range Z \<subseteq> sets ?G" "decseq Z" "\<forall>n. ?a / 2^k \<le> \<mu>G (Z n)"
```
```   533         then have Z_sets: "\<And>n. Z n \<in> sets ?G" by auto
```
```   534         fix J' assume J': "J' \<noteq> {}" "finite J'" "J' \<subseteq> I"
```
```   535         interpret J': finite_product_prob_space M J' by default fact+
```
```   536
```
```   537         let "?q n y" = "\<mu>G (?M J' (Z n) y)"
```
```   538         let "?Q n" = "?q n -` {?a / 2^(k+1) ..} \<inter> space (Pi\<^isub>M J' M)"
```
```   539         { fix n
```
```   540           have "?q n \<in> borel_measurable (Pi\<^isub>M J' M)"
```
```   541             using Z J' by (intro fold(1)) auto
```
```   542           then have "?Q n \<in> sets (Pi\<^isub>M J' M)"
```
```   543             by (rule measurable_sets) auto }
```
```   544         note Q_sets = this
```
```   545
```
```   546         have "?a / 2^(k+1) \<le> (INF n. measure (Pi\<^isub>M J' M) (?Q n))"
```
```   547         proof (intro le_INFI)
```
```   548           fix n
```
```   549           have "?a / 2^k \<le> \<mu>G (Z n)" using Z by auto
```
```   550           also have "\<dots> \<le> (\<integral>\<^isup>+ x. indicator (?Q n) x + ?a / 2^(k+1) \<partial>Pi\<^isub>M J' M)"
```
```   551             unfolding fold(2)[OF J' `Z n \<in> sets ?G`]
```
```   552           proof (intro J'.positive_integral_mono)
```
```   553             fix x assume x: "x \<in> space (Pi\<^isub>M J' M)"
```
```   554             then have "?q n x \<le> 1 + 0"
```
```   555               using J' Z fold(3) Z_sets by auto
```
```   556             also have "\<dots> \<le> 1 + ?a / 2^(k+1)"
```
```   557               using `0 < ?a` by (intro add_mono) auto
```
```   558             finally have "?q n x \<le> 1 + ?a / 2^(k+1)" .
```
```   559             with x show "?q n x \<le> indicator (?Q n) x + ?a / 2^(k+1)"
```
```   560               by (auto split: split_indicator simp del: power_Suc)
```
```   561           qed
```
```   562           also have "\<dots> = measure (Pi\<^isub>M J' M) (?Q n) + ?a / 2^(k+1)"
```
```   563             using `0 \<le> ?a` Q_sets J'.measure_space_1
```
```   564             by (subst J'.positive_integral_add) auto
```
```   565           finally show "?a / 2^(k+1) \<le> measure (Pi\<^isub>M J' M) (?Q n)" using `?a \<le> 1`
```
```   566             by (cases rule: ereal2_cases[of ?a "measure (Pi\<^isub>M J' M) (?Q n)"])
```
```   567                (auto simp: field_simps)
```
```   568         qed
```
```   569         also have "\<dots> = measure (Pi\<^isub>M J' M) (\<Inter>n. ?Q n)"
```
```   570         proof (intro J'.continuity_from_above)
```
```   571           show "range ?Q \<subseteq> sets (Pi\<^isub>M J' M)" using Q_sets by auto
```
```   572           show "decseq ?Q"
```
```   573             unfolding decseq_def
```
```   574           proof (safe intro!: vimageI[OF refl])
```
```   575             fix m n :: nat assume "m \<le> n"
```
```   576             fix x assume x: "x \<in> space (Pi\<^isub>M J' M)"
```
```   577             assume "?a / 2^(k+1) \<le> ?q n x"
```
```   578             also have "?q n x \<le> ?q m x"
```
```   579             proof (rule \<mu>G_mono)
```
```   580               from fold(4)[OF J', OF Z_sets x]
```
```   581               show "?M J' (Z n) x \<in> sets ?G" "?M J' (Z m) x \<in> sets ?G" by auto
```
```   582               show "?M J' (Z n) x \<subseteq> ?M J' (Z m) x"
```
```   583                 using `decseq Z`[THEN decseqD, OF `m \<le> n`] by auto
```
```   584             qed
```
```   585             finally show "?a / 2^(k+1) \<le> ?q m x" .
```
```   586           qed
```
```   587         qed (intro J'.finite_measure Q_sets)
```
```   588         finally have "(\<Inter>n. ?Q n) \<noteq> {}"
```
```   589           using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq)
```
```   590         then have "\<exists>w\<in>space (Pi\<^isub>M J' M). \<forall>n. ?a / 2 ^ (k + 1) \<le> ?q n w" by auto }
```
```   591       note Ex_w = this
```
```   592
```
```   593       let "?q k n y" = "\<mu>G (?M (J k) (A n) y)"
```
```   594
```
```   595       have "\<forall>n. ?a / 2 ^ 0 \<le> \<mu>G (A n)" by (auto intro: INF_leI)
```
```   596       from Ex_w[OF A(1,2) this J(1-3), of 0] guess w0 .. note w0 = this
```
```   597
```
```   598       let "?P k wk w" =
```
```   599         "w \<in> space (Pi\<^isub>M (J (Suc k)) M) \<and> restrict w (J k) = wk \<and> (\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n w)"
```
```   600       def w \<equiv> "nat_rec w0 (\<lambda>k wk. Eps (?P k wk))"
```
```   601
```
```   602       { fix k have w: "w k \<in> space (Pi\<^isub>M (J k) M) \<and>
```
```   603           (\<forall>n. ?a / 2 ^ (k + 1) \<le> ?q k n (w k)) \<and> (k \<noteq> 0 \<longrightarrow> restrict (w k) (J (k - 1)) = w (k - 1))"
```
```   604         proof (induct k)
```
```   605           case 0 with w0 show ?case
```
```   606             unfolding w_def nat_rec_0 by auto
```
```   607         next
```
```   608           case (Suc k)
```
```   609           then have wk: "w k \<in> space (Pi\<^isub>M (J k) M)" by auto
```
```   610           have "\<exists>w'. ?P k (w k) w'"
```
```   611           proof cases
```
```   612             assume [simp]: "J k = J (Suc k)"
```
```   613             show ?thesis
```
```   614             proof (intro exI[of _ "w k"] conjI allI)
```
```   615               fix n
```
```   616               have "?a / 2 ^ (Suc k + 1) \<le> ?a / 2 ^ (k + 1)"
```
```   617                 using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: field_simps)
```
```   618               also have "\<dots> \<le> ?q k n (w k)" using Suc by auto
```
```   619               finally show "?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n (w k)" by simp
```
```   620             next
```
```   621               show "w k \<in> space (Pi\<^isub>M (J (Suc k)) M)"
```
```   622                 using Suc by simp
```
```   623               then show "restrict (w k) (J k) = w k"
```
```   624                 by (simp add: extensional_restrict)
```
```   625             qed
```
```   626           next
```
```   627             assume "J k \<noteq> J (Suc k)"
```
```   628             with J_mono[of k "Suc k"] have "J (Suc k) - J k \<noteq> {}" (is "?D \<noteq> {}") by auto
```
```   629             have "range (\<lambda>n. ?M (J k) (A n) (w k)) \<subseteq> sets ?G"
```
```   630               "decseq (\<lambda>n. ?M (J k) (A n) (w k))"
```
```   631               "\<forall>n. ?a / 2 ^ (k + 1) \<le> \<mu>G (?M (J k) (A n) (w k))"
```
```   632               using `decseq A` fold(4)[OF J(1-3) A_eq(2), of "w k" k] Suc
```
```   633               by (auto simp: decseq_def)
```
```   634             from Ex_w[OF this `?D \<noteq> {}`] J[of "Suc k"]
```
```   635             obtain w' where w': "w' \<in> space (Pi\<^isub>M ?D M)"
```
```   636               "\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> \<mu>G (?M ?D (?M (J k) (A n) (w k)) w')" by auto
```
```   637             let ?w = "merge (J k) (w k) ?D w'"
```
```   638             have [simp]: "\<And>x. merge (J k) (w k) (I - J k) (merge ?D w' (I - ?D) x) =
```
```   639               merge (J (Suc k)) ?w (I - (J (Suc k))) x"
```
```   640               using J(3)[of "Suc k"] J(3)[of k] J_mono[of k "Suc k"]
```
```   641               by (auto intro!: ext split: split_merge)
```
```   642             have *: "\<And>n. ?M ?D (?M (J k) (A n) (w k)) w' = ?M (J (Suc k)) (A n) ?w"
```
```   643               using w'(1) J(3)[of "Suc k"]
```
```   644               by (auto split: split_merge intro!: extensional_merge_sub) force+
```
```   645             show ?thesis
```
```   646               apply (rule exI[of _ ?w])
```
```   647               using w' J_mono[of k "Suc k"] wk unfolding *
```
```   648               apply (auto split: split_merge intro!: extensional_merge_sub ext)
```
```   649               apply (force simp: extensional_def)
```
```   650               done
```
```   651           qed
```
```   652           then have "?P k (w k) (w (Suc k))"
```
```   653             unfolding w_def nat_rec_Suc unfolding w_def[symmetric]
```
```   654             by (rule someI_ex)
```
```   655           then show ?case by auto
```
```   656         qed
```
```   657         moreover
```
```   658         then have "w k \<in> space (Pi\<^isub>M (J k) M)" by auto
```
```   659         moreover
```
```   660         from w have "?a / 2 ^ (k + 1) \<le> ?q k k (w k)" by auto
```
```   661         then have "?M (J k) (A k) (w k) \<noteq> {}"
```
```   662           using positive_\<mu>G[unfolded positive_def] `0 < ?a` `?a \<le> 1`
```
```   663           by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq)
```
```   664         then obtain x where "x \<in> ?M (J k) (A k) (w k)" by auto
```
```   665         then have "merge (J k) (w k) (I - J k) x \<in> A k" by auto
```
```   666         then have "\<exists>x\<in>A k. restrict x (J k) = w k"
```
```   667           using `w k \<in> space (Pi\<^isub>M (J k) M)`
```
```   668           by (intro rev_bexI) (auto intro!: ext simp: extensional_def)
```
```   669         ultimately have "w k \<in> space (Pi\<^isub>M (J k) M)"
```
```   670           "\<exists>x\<in>A k. restrict x (J k) = w k"
```
```   671           "k \<noteq> 0 \<Longrightarrow> restrict (w k) (J (k - 1)) = w (k - 1)"
```
```   672           by auto }
```
```   673       note w = this
```
```   674
```
```   675       { fix k l i assume "k \<le> l" "i \<in> J k"
```
```   676         { fix l have "w k i = w (k + l) i"
```
```   677           proof (induct l)
```
```   678             case (Suc l)
```
```   679             from `i \<in> J k` J_mono[of k "k + l"] have "i \<in> J (k + l)" by auto
```
```   680             with w(3)[of "k + Suc l"]
```
```   681             have "w (k + l) i = w (k + Suc l) i"
```
```   682               by (auto simp: restrict_def fun_eq_iff split: split_if_asm)
```
```   683             with Suc show ?case by simp
```
```   684           qed simp }
```
```   685         from this[of "l - k"] `k \<le> l` have "w l i = w k i" by simp }
```
```   686       note w_mono = this
```
```   687
```
```   688       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"
```
```   689       { fix i k assume k: "i \<in> J k"
```
```   690         have "w k i = w (LEAST k. i \<in> J k) i"
```
```   691           by (intro w_mono Least_le k LeastI[of _ k])
```
```   692         then have "w' i = w k i"
```
```   693           unfolding w'_def using k by auto }
```
```   694       note w'_eq = this
```
```   695       have w'_simps1: "\<And>i. i \<notin> I \<Longrightarrow> w' i = undefined"
```
```   696         using J by (auto simp: w'_def)
```
```   697       have w'_simps2: "\<And>i. i \<notin> (\<Union>k. J k) \<Longrightarrow> i \<in> I \<Longrightarrow> w' i \<in> space (M i)"
```
```   698         using J by (auto simp: w'_def intro!: someI_ex[OF M.not_empty[unfolded ex_in_conv[symmetric]]])
```
```   699       { fix i assume "i \<in> I" then have "w' i \<in> space (M i)"
```
```   700           using w(1) by (cases "i \<in> (\<Union>k. J k)") (force simp: w'_simps2 w'_eq)+ }
```
```   701       note w'_simps[simp] = w'_eq w'_simps1 w'_simps2 this
```
```   702
```
```   703       have w': "w' \<in> space (Pi\<^isub>M I M)"
```
```   704         using w(1) by (auto simp add: Pi_iff extensional_def)
```
```   705
```
```   706       { fix n
```
```   707         have "restrict w' (J n) = w n" using w(1)
```
```   708           by (auto simp add: fun_eq_iff restrict_def Pi_iff extensional_def)
```
```   709         with w[of n] obtain x where "x \<in> A n" "restrict x (J n) = restrict w' (J n)" by auto
```
```   710         then have "w' \<in> A n" unfolding A_eq using w' by (auto simp: emb_def) }
```
```   711       then have "w' \<in> (\<Inter>i. A i)" by auto
```
```   712       with `(\<Inter>i. A i) = {}` show False by auto
```
```   713     qed
```
```   714     ultimately show "(\<lambda>i. \<mu>G (A i)) ----> 0"
```
```   715       using LIMSEQ_ereal_INFI[of "\<lambda>i. \<mu>G (A i)"] by simp
```
```   716   qed
```
```   717 qed
```
```   718
```
```   719 lemma (in product_prob_space) infprod_spec:
```
```   720   shows "(\<forall>s\<in>sets generator. measure (Pi\<^isub>P I M) s = \<mu>G s) \<and> measure_space (Pi\<^isub>P I M)"
```
```   721 proof -
```
```   722   let ?P = "\<lambda>\<mu>. (\<forall>A\<in>sets generator. \<mu> A = \<mu>G A) \<and>
```
```   723        measure_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>"
```
```   724   have **: "measure infprod_algebra = (SOME \<mu>. ?P \<mu>)"
```
```   725     unfolding infprod_algebra_def by simp
```
```   726   have *: "Pi\<^isub>P I M = \<lparr>space = space generator, sets = sets (sigma generator), measure = measure (Pi\<^isub>P I M)\<rparr>"
```
```   727     unfolding infprod_algebra_def by auto
```
```   728   show ?thesis
```
```   729     apply (subst (2) *)
```
```   730     apply (unfold **)
```
```   731     apply (rule someI_ex[where P="?P"])
```
```   732     apply (rule extend_\<mu>G)
```
```   733     done
```
```   734 qed
```
```   735
```
```   736 sublocale product_prob_space \<subseteq> P: measure_space "Pi\<^isub>P I M"
```
```   737   using infprod_spec by auto
```
```   738
```
```   739 lemma (in product_prob_space) measure_infprod_emb:
```
```   740   assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)"
```
```   741   shows "\<mu> (emb I J X) = measure (Pi\<^isub>M J M) X"
```
```   742 proof -
```
```   743   have "emb I J X \<in> sets generator"
```
```   744     using assms by (rule generatorI')
```
```   745   with \<mu>G_eq[OF assms] infprod_spec show ?thesis by auto
```
```   746 qed
```
```   747
```
```   748 sublocale product_prob_space \<subseteq> P: prob_space "Pi\<^isub>P I M"
```
```   749 proof
```
```   750   obtain i where "i \<in> I" using I_not_empty by auto
```
```   751   interpret i: finite_product_sigma_finite M "{i}" by default auto
```
```   752   let ?X = "\<Pi>\<^isub>E i\<in>{i}. space (M i)"
```
```   753   have "?X \<in> sets (Pi\<^isub>M {i} M)"
```
```   754     by auto
```
```   755   from measure_infprod_emb[OF _ _ _ this] `i \<in> I`
```
```   756   have "\<mu> (emb I {i} ?X) = measure (M i) (space (M i))"
```
```   757     by (simp add: i.measure_times)
```
```   758   also have "emb I {i} ?X = space (Pi\<^isub>P I M)"
```
```   759     using `i \<in> I` by (auto simp: emb_def infprod_algebra_def generator_def)
```
```   760   finally show "\<mu> (space (Pi\<^isub>P I M)) = 1"
```
```   761     using M.measure_space_1 by simp
```
```   762 qed
```
```   763
```
```   764 lemma (in product_prob_space) measurable_component:
```
```   765   assumes "i \<in> I"
```
```   766   shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>P I M) (M i)"
```
```   767 proof (unfold measurable_def, safe)
```
```   768   fix x assume "x \<in> space (Pi\<^isub>P I M)"
```
```   769   then show "x i \<in> space (M i)"
```
```   770     using `i \<in> I` by (auto simp: infprod_algebra_def generator_def)
```
```   771 next
```
```   772   fix A assume "A \<in> sets (M i)"
```
```   773   with `i \<in> I` have
```
```   774     "(\<Pi>\<^isub>E x \<in> {i}. A) \<in> sets (Pi\<^isub>M {i} M)"
```
```   775     "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) = emb I {i} (\<Pi>\<^isub>E x \<in> {i}. A)"
```
```   776     by (auto simp: infprod_algebra_def generator_def emb_def)
```
```   777   from generatorI[OF _ _ _ this] `i \<in> I`
```
```   778   show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) \<in> sets (Pi\<^isub>P I M)"
```
```   779     unfolding infprod_algebra_def by auto
```
```   780 qed
```
```   781
```
```   782 lemma (in product_prob_space) emb_in_infprod_algebra[intro]:
```
```   783   fixes J assumes J: "finite J" "J \<subseteq> I" and X: "X \<in> sets (Pi\<^isub>M J M)"
```
```   784   shows "emb I J X \<in> sets (\<Pi>\<^isub>P i\<in>I. M i)"
```
```   785 proof cases
```
```   786   assume "J = {}"
```
```   787   with X have "emb I J X = space (\<Pi>\<^isub>P i\<in>I. M i) \<or> emb I J X = {}"
```
```   788     by (auto simp: emb_def infprod_algebra_def generator_def
```
```   789                    product_algebra_def product_algebra_generator_def image_constant sigma_def)
```
```   790   then show ?thesis by auto
```
```   791 next
```
```   792   assume "J \<noteq> {}"
```
```   793   show ?thesis unfolding infprod_algebra_def
```
```   794     by simp (intro in_sigma generatorI'  `J \<noteq> {}` J X)
```
```   795 qed
```
```   796
```
```   797 lemma (in product_prob_space) finite_measure_infprod_emb:
```
```   798   assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)"
```
```   799   shows "\<mu>' (emb I J X) = finite_measure.\<mu>' (Pi\<^isub>M J M) X"
```
```   800 proof -
```
```   801   interpret J: finite_product_prob_space M J by default fact+
```
```   802   from assms have "emb I J X \<in> sets (Pi\<^isub>P I M)" by auto
```
```   803   with assms show "\<mu>' (emb I J X) = J.\<mu>' X"
```
```   804     unfolding \<mu>'_def J.\<mu>'_def
```
```   805     unfolding measure_infprod_emb[OF assms]
```
```   806     by auto
```
```   807 qed
```
```   808
```
```   809 lemma (in finite_product_prob_space) finite_measure_times:
```
```   810   assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)"
```
```   811   shows "\<mu>' (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu>' i (A i))"
```
```   812   using assms
```
```   813   unfolding \<mu>'_def M.\<mu>'_def
```
```   814   by (subst measure_times[OF assms])
```
```   815      (auto simp: finite_measure_eq M.finite_measure_eq setprod_ereal)
```
```   816
```
```   817 lemma (in product_prob_space) finite_measure_infprod_emb_Pi:
```
```   818   assumes J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> X j \<in> sets (M j)"
```
```   819   shows "\<mu>' (emb I J (Pi\<^isub>E J X)) = (\<Prod>j\<in>J. M.\<mu>' j (X j))"
```
```   820 proof cases
```
```   821   assume "J = {}"
```
```   822   then have "emb I J (Pi\<^isub>E J X) = space infprod_algebra"
```
```   823     by (auto simp: infprod_algebra_def generator_def sigma_def emb_def)
```
```   824   then show ?thesis using `J = {}` prob_space by simp
```
```   825 next
```
```   826   assume "J \<noteq> {}"
```
```   827   interpret J: finite_product_prob_space M J by default fact+
```
```   828   have "(\<Prod>i\<in>J. M.\<mu>' i (X i)) = J.\<mu>' (Pi\<^isub>E J X)"
```
```   829     using J `J \<noteq> {}` by (subst J.finite_measure_times) auto
```
```   830   also have "\<dots> = \<mu>' (emb I J (Pi\<^isub>E J X))"
```
```   831     using J `J \<noteq> {}` by (intro finite_measure_infprod_emb[symmetric]) auto
```
```   832   finally show ?thesis by simp
```
```   833 qed
```
```   834
```
```   835 lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
```
```   836 proof
```
```   837   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
```
```   838     by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros)
```
```   839 qed
```
```   840
```
```   841 lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
```
```   842 proof
```
```   843   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
```
```   844     by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros)
```
```   845 qed
```
```   846
```
```   847 lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"
```
```   848   by (auto intro: sigma_sets.Basic)
```
```   849
```
```   850 lemma (in product_prob_space) infprod_algebra_alt:
```
```   851   "Pi\<^isub>P I M = sigma \<lparr> space = space (Pi\<^isub>P I M),
```
```   852     sets = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i))),
```
```   853     measure = measure (Pi\<^isub>P I M) \<rparr>"
```
```   854   (is "_ = sigma \<lparr> space = ?O, sets = ?M, measure = ?m \<rparr>")
```
```   855 proof (rule measure_space.equality)
```
```   856   let ?G = "\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M)"
```
```   857   have "sigma_sets ?O ?M = sigma_sets ?O ?G"
```
```   858   proof (intro equalityI sigma_sets_mono UN_least)
```
```   859     fix J assume J: "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}"
```
```   860     have "emb I J ` Pi\<^isub>E J ` (\<Pi> i\<in>J. sets (M i)) \<subseteq> emb I J ` sets (Pi\<^isub>M J M)" by auto
```
```   861     also have "\<dots> \<subseteq> ?G" using J by (rule UN_upper)
```
```   862     also have "\<dots> \<subseteq> sigma_sets ?O ?G" by (rule sigma_sets_superset_generator)
```
```   863     finally show "emb I J ` Pi\<^isub>E J ` (\<Pi> i\<in>J. sets (M i)) \<subseteq> sigma_sets ?O ?G" .
```
```   864     have "emb I J ` sets (Pi\<^isub>M J M) = emb I J ` sigma_sets (space (Pi\<^isub>M J M)) (Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))"
```
```   865       by (simp add: sets_sigma product_algebra_generator_def product_algebra_def)
```
```   866     also have "\<dots> = sigma_sets (space (Pi\<^isub>M I M)) (emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))"
```
```   867       using J M.sets_into_space
```
```   868       by (auto simp: emb_def_raw intro!: sigma_sets_vimage[symmetric]) blast
```
```   869     also have "\<dots> \<subseteq> sigma_sets (space (Pi\<^isub>M I M)) ?M"
```
```   870       using J by (intro sigma_sets_mono') auto
```
```   871     finally show "emb I J ` sets (Pi\<^isub>M J M) \<subseteq> sigma_sets ?O ?M"
```
```   872       by (simp add: infprod_algebra_def generator_def)
```
```   873   qed
```
```   874   then show "sets (Pi\<^isub>P I M) = sets (sigma \<lparr> space = ?O, sets = ?M, measure = ?m \<rparr>)"
```
```   875     by (simp_all add: infprod_algebra_def generator_def sets_sigma)
```
```   876 qed simp_all
```
```   877
```
```   878 lemma (in product_prob_space) infprod_algebra_alt2:
```
```   879   "Pi\<^isub>P I M = sigma \<lparr> space = space (Pi\<^isub>P I M),
```
```   880     sets = (\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (M i)),
```
```   881     measure = measure (Pi\<^isub>P I M) \<rparr>"
```
```   882   (is "_ = ?S")
```
```   883 proof (rule measure_space.equality)
```
```   884   let "sigma \<lparr> space = ?O, sets = ?A, \<dots> = _ \<rparr>" = ?S
```
```   885   let ?G = "(\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))"
```
```   886   have "sets (Pi\<^isub>P I M) = sigma_sets ?O ?G"
```
```   887     by (subst infprod_algebra_alt) (simp add: sets_sigma)
```
```   888   also have "\<dots> = sigma_sets ?O ?A"
```
```   889   proof (intro equalityI sigma_sets_mono subsetI)
```
```   890     interpret A: sigma_algebra ?S
```
```   891       by (rule sigma_algebra_sigma) auto
```
```   892     fix A assume "A \<in> ?G"
```
```   893     then obtain J B where "finite J" "J \<noteq> {}" "J \<subseteq> I" "A = emb I J (Pi\<^isub>E J B)"
```
```   894         and B: "\<And>i. i \<in> J \<Longrightarrow> B i \<in> sets (M i)"
```
```   895       by auto
```
```   896     then have A: "A = (\<Inter>j\<in>J. (\<lambda>x. x j) -` (B j) \<inter> space (Pi\<^isub>P I M))"
```
```   897       by (auto simp: emb_def infprod_algebra_def generator_def Pi_iff)
```
```   898     { fix j assume "j\<in>J"
```
```   899       with `J \<subseteq> I` have "j \<in> I" by auto
```
```   900       with `j \<in> J` B have "(\<lambda>x. x j) -` (B j) \<inter> space (Pi\<^isub>P I M) \<in> sets ?S"
```
```   901         by (auto simp: sets_sigma intro: sigma_sets.Basic) }
```
```   902     with `finite J` `J \<noteq> {}` have "A \<in> sets ?S"
```
```   903       unfolding A by (intro A.finite_INT) auto
```
```   904     then show "A \<in> sigma_sets ?O ?A" by (simp add: sets_sigma)
```
```   905   next
```
```   906     fix A assume "A \<in> ?A"
```
```   907     then obtain i B where i: "i \<in> I" "B \<in> sets (M i)"
```
```   908         and "A = (\<lambda>x. x i) -` B \<inter> space (Pi\<^isub>P I M)"
```
```   909       by auto
```
```   910     then have "A = emb I {i} (Pi\<^isub>E {i} (\<lambda>_. B))"
```
```   911       by (auto simp: emb_def infprod_algebra_def generator_def Pi_iff)
```
```   912     with i show "A \<in> sigma_sets ?O ?G"
```
```   913       by (intro sigma_sets.Basic UN_I[where a="{i}"]) auto
```
```   914   qed
```
```   915   also have "\<dots> = sets ?S"
```
```   916     by (simp add: sets_sigma)
```
```   917   finally show "sets (Pi\<^isub>P I M) = sets ?S" .
```
```   918 qed simp_all
```
```   919
```
```   920 lemma (in product_prob_space) measurable_into_infprod_algebra:
```
```   921   assumes "sigma_algebra N"
```
```   922   assumes f: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)"
```
```   923   assumes ext: "\<And>x. x \<in> space N \<Longrightarrow> f x \<in> extensional I"
```
```   924   shows "f \<in> measurable N (Pi\<^isub>P I M)"
```
```   925 proof -
```
```   926   interpret N: sigma_algebra N by fact
```
```   927   have f_in: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f x i) \<in> space N \<rightarrow> space (M i)"
```
```   928     using f by (auto simp: measurable_def)
```
```   929   { fix i A assume i: "i \<in> I" "A \<in> sets (M i)"
```
```   930     then have "f -` (\<lambda>x. x i) -` A \<inter> f -` space infprod_algebra \<inter> space N = (\<lambda>x. f x i) -` A \<inter> space N"
```
```   931       using f_in ext by (auto simp: infprod_algebra_def generator_def)
```
```   932     also have "\<dots> \<in> sets N"
```
```   933       by (rule measurable_sets f i)+
```
```   934     finally have "f -` (\<lambda>x. x i) -` A \<inter> f -` space infprod_algebra \<inter> space N \<in> sets N" . }
```
```   935   with f_in ext show ?thesis
```
```   936     by (subst infprod_algebra_alt2)
```
```   937        (auto intro!: N.measurable_sigma simp: Pi_iff infprod_algebra_def generator_def)
```
```   938 qed
```
```   939
```
```   940 lemma (in product_prob_space) measurable_singleton_infprod:
```
```   941   assumes "i \<in> I"
```
```   942   shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>P I M) (M i)"
```
```   943 proof (unfold measurable_def, intro CollectI conjI ballI)
```
```   944   show "(\<lambda>x. x i) \<in> space (Pi\<^isub>P I M) \<rightarrow> space (M i)"
```
```   945     using M.sets_into_space `i \<in> I`
```
```   946     by (auto simp: infprod_algebra_def generator_def)
```
```   947   fix A assume "A \<in> sets (M i)"
```
```   948   have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) = emb I {i} (\<Pi>\<^isub>E _\<in>{i}. A)"
```
```   949     by (auto simp: infprod_algebra_def generator_def emb_def)
```
```   950   also have "\<dots> \<in> sets (Pi\<^isub>P I M)"
```
```   951     using `i \<in> I` `A \<in> sets (M i)`
```
```   952     by (intro emb_in_infprod_algebra product_algebraI) auto
```
```   953   finally show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) \<in> sets (Pi\<^isub>P I M)" .
```
```   954 qed
```
```   955
```
```   956 lemma (in product_prob_space) sigma_product_algebra_sigma_eq:
```
```   957   assumes M: "\<And>i. i \<in> I \<Longrightarrow> M i = sigma (E i)"
```
```   958   shows "sets (Pi\<^isub>P I M) = sigma_sets (space (Pi\<^isub>P I M)) (\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (E i))"
```
```   959 proof -
```
```   960   let ?E = "(\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (E i))"
```
```   961   let ?M = "(\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (M i))"
```
```   962   { fix i A assume "i\<in>I" "A \<in> sets (E i)"
```
```   963     then have "A \<in> sets (M i)" using M by auto
```
```   964     then have "A \<in> Pow (space (M i))" using M.sets_into_space by auto
```
```   965     then have "A \<in> Pow (space (E i))" using M[OF `i \<in> I`] by auto }
```
```   966   moreover
```
```   967   have "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) \<in> space infprod_algebra \<rightarrow> space (E i)"
```
```   968     by (auto simp: M infprod_algebra_def generator_def Pi_iff)
```
```   969   ultimately have "sigma_sets (space (Pi\<^isub>P I M)) ?M \<subseteq> sigma_sets (space (Pi\<^isub>P I M)) ?E"
```
```   970     apply (intro sigma_sets_mono UN_least)
```
```   971     apply (simp add: sets_sigma M)
```
```   972     apply (subst sigma_sets_vimage[symmetric])
```
```   973     apply (auto intro!: sigma_sets_mono')
```
```   974     done
```
```   975   moreover have "sigma_sets (space (Pi\<^isub>P I M)) ?E \<subseteq> sigma_sets (space (Pi\<^isub>P I M)) ?M"
```
```   976     by (intro sigma_sets_mono') (auto simp: M)
```
```   977   ultimately show ?thesis
```
```   978     by (subst infprod_algebra_alt2) (auto simp: sets_sigma)
```
```   979 qed
```
```   980
```
```   981 lemma (in product_prob_space) Int_proj_eq_emb:
```
```   982   assumes "J \<noteq> {}" "J \<subseteq> I"
```
```   983   shows "(\<Inter>i\<in>J. (\<lambda>x. x i) -` A i \<inter> space (Pi\<^isub>P I M)) = emb I J (Pi\<^isub>E J A)"
```
```   984   using assms by (auto simp: infprod_algebra_def generator_def emb_def Pi_iff)
```
```   985
```
```   986 lemma (in product_prob_space) emb_insert:
```
```   987   "i \<notin> J \<Longrightarrow> emb I J (Pi\<^isub>E J f) \<inter> ((\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) =
```
```   988     emb I (insert i J) (Pi\<^isub>E (insert i J) (f(i := A)))"
```
```   989   by (auto simp: emb_def Pi_iff infprod_algebra_def generator_def split: split_if_asm)
```
```   990
```
```   991 subsection {* Sequence space *}
```
```   992
```
```   993 locale sequence_space = product_prob_space M "UNIV :: nat set" for M
```
```   994
```
```   995 lemma (in sequence_space) infprod_in_sets[intro]:
```
```   996   fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
```
```   997   shows "Pi UNIV E \<in> sets (Pi\<^isub>P UNIV M)"
```
```   998 proof -
```
```   999   have "Pi UNIV E = (\<Inter>i. emb UNIV {..i} (\<Pi>\<^isub>E j\<in>{..i}. E j))"
```
```  1000     using E E[THEN M.sets_into_space]
```
```  1001     by (auto simp: emb_def Pi_iff extensional_def) blast
```
```  1002   with E show ?thesis
```
```  1003     by (auto intro: emb_in_infprod_algebra)
```
```  1004 qed
```
```  1005
```
```  1006 lemma (in sequence_space) measure_infprod:
```
```  1007   fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
```
```  1008   shows "(\<lambda>n. \<Prod>i\<le>n. M.\<mu>' i (E i)) ----> \<mu>' (Pi UNIV E)"
```
```  1009 proof -
```
```  1010   let "?E n" = "emb UNIV {..n} (Pi\<^isub>E {.. n} E)"
```
```  1011   { fix n :: nat
```
```  1012     interpret n: finite_product_prob_space M "{..n}" by default auto
```
```  1013     have "(\<Prod>i\<le>n. M.\<mu>' i (E i)) = n.\<mu>' (Pi\<^isub>E {.. n} E)"
```
```  1014       using E by (subst n.finite_measure_times) auto
```
```  1015     also have "\<dots> = \<mu>' (?E n)"
```
```  1016       using E by (intro finite_measure_infprod_emb[symmetric]) auto
```
```  1017     finally have "(\<Prod>i\<le>n. M.\<mu>' i (E i)) = \<mu>' (?E n)" . }
```
```  1018   moreover have "Pi UNIV E = (\<Inter>n. ?E n)"
```
```  1019     using E E[THEN M.sets_into_space]
```
```  1020     by (auto simp: emb_def extensional_def Pi_iff) blast
```
```  1021   moreover have "range ?E \<subseteq> sets (Pi\<^isub>P UNIV M)"
```
```  1022     using E by auto
```
```  1023   moreover have "decseq ?E"
```
```  1024     by (auto simp: emb_def Pi_iff decseq_def)
```
```  1025   ultimately show ?thesis
```
```  1026     by (simp add: finite_continuity_from_above)
```
```  1027 qed
```
```  1028
```
`  1029 end`