src/HOL/Probability/Infinite_Product_Measure.thy
 author hoelzl Tue Apr 05 19:55:04 2011 +0200 (2011-04-05) changeset 42257 08d717c82828 parent 42166 efd229daeb2c child 42865 36353d913424 permissions -rw-r--r--
prove measurable_into_infprod_algebra and measure_infprod
```     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 lemma (in prob_space) measure_le_1: "X \<in> sets M \<Longrightarrow> \<mu> X \<le> 1"
```
```   227   unfolding measure_space_1[symmetric]
```
```   228   using sets_into_space
```
```   229   by (intro measure_mono) auto
```
```   230
```
```   231 definition (in product_prob_space)
```
```   232   "\<mu>G A =
```
```   233     (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))"
```
```   234
```
```   235 lemma (in product_prob_space) \<mu>G_spec:
```
```   236   assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)"
```
```   237   shows "\<mu>G A = measure (Pi\<^isub>M J M) X"
```
```   238   unfolding \<mu>G_def
```
```   239 proof (intro the_equality allI impI ballI)
```
```   240   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)"
```
```   241   have "measure (Pi\<^isub>M K M) Y = measure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) K Y)"
```
```   242     using K J by simp
```
```   243   also have "emb (K \<union> J) K Y = emb (K \<union> J) J X"
```
```   244     using K J by (simp add: emb_injective[of "K \<union> J" I])
```
```   245   also have "measure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) J X) = measure (Pi\<^isub>M J M) X"
```
```   246     using K J by simp
```
```   247   finally show "measure (Pi\<^isub>M J M) X = measure (Pi\<^isub>M K M) Y" ..
```
```   248 qed (insert J, force)
```
```   249
```
```   250 lemma (in product_prob_space) \<mu>G_eq:
```
```   251   "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"
```
```   252   by (intro \<mu>G_spec) auto
```
```   253
```
```   254 lemma (in product_prob_space) generator_Ex:
```
```   255   assumes *: "A \<in> sets generator"
```
```   256   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"
```
```   257 proof -
```
```   258   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)"
```
```   259     unfolding generator_def by auto
```
```   260   with \<mu>G_spec[OF this] show ?thesis by auto
```
```   261 qed
```
```   262
```
```   263 lemma (in product_prob_space) generatorE:
```
```   264   assumes A: "A \<in> sets generator"
```
```   265   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"
```
```   266 proof -
```
```   267   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"
```
```   268     "\<mu>G A = measure (Pi\<^isub>M J M) X" by auto
```
```   269   then show thesis by (intro that) auto
```
```   270 qed
```
```   271
```
```   272 lemma (in product_prob_space) merge_sets:
```
```   273   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)"
```
```   274   shows "merge J x K -` A \<inter> space (Pi\<^isub>M K M) \<in> sets (Pi\<^isub>M K M)"
```
```   275 proof -
```
```   276   interpret J: finite_product_sigma_algebra M J by default fact
```
```   277   interpret K: finite_product_sigma_algebra M K by default fact
```
```   278   interpret JK: pair_sigma_algebra J.P K.P ..
```
```   279
```
```   280   from JK.measurable_cut_fst[OF
```
```   281     measurable_merge[THEN measurable_sets, OF `J \<inter> K = {}`], OF A, of x] x
```
```   282   show ?thesis
```
```   283       by (simp add: space_pair_measure comp_def vimage_compose[symmetric])
```
```   284 qed
```
```   285
```
```   286 lemma (in product_prob_space) merge_emb:
```
```   287   assumes "K \<subseteq> I" "J \<subseteq> I" and y: "y \<in> space (Pi\<^isub>M J M)"
```
```   288   shows "(merge J y (I - J) -` emb I K X \<inter> space (Pi\<^isub>M I M)) =
```
```   289     emb I (K - J) (merge J y (K - J) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M))"
```
```   290 proof -
```
```   291   have [simp]: "\<And>x J K L. merge J y K (restrict x L) = merge J y (K \<inter> L) x"
```
```   292     by (auto simp: restrict_def merge_def)
```
```   293   have [simp]: "\<And>x J K L. restrict (merge J y K x) L = merge (J \<inter> L) y (K \<inter> L) x"
```
```   294     by (auto simp: restrict_def merge_def)
```
```   295   have [simp]: "(I - J) \<inter> K = K - J" using `K \<subseteq> I` `J \<subseteq> I` by auto
```
```   296   have [simp]: "(K - J) \<inter> (K \<union> J) = K - J" by auto
```
```   297   have [simp]: "(K - J) \<inter> K = K - J" by auto
```
```   298   from y `K \<subseteq> I` `J \<subseteq> I` show ?thesis
```
```   299     by (simp split: split_merge add: emb_def Pi_iff extensional_merge_sub set_eq_iff) auto
```
```   300 qed
```
```   301
```
```   302 definition (in product_prob_space) infprod_algebra :: "('i \<Rightarrow> 'a) measure_space" where
```
```   303   "infprod_algebra = sigma generator \<lparr> measure :=
```
```   304     (SOME \<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and>
```
```   305        measure_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>)\<rparr>"
```
```   306
```
```   307 syntax
```
```   308   "_PiP"  :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme"  ("(3PIP _:_./ _)" 10)
```
```   309
```
```   310 syntax (xsymbols)
```
```   311   "_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme"  ("(3\<Pi>\<^isub>P _\<in>_./ _)"   10)
```
```   312
```
```   313 syntax (HTML output)
```
```   314   "_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme"  ("(3\<Pi>\<^isub>P _\<in>_./ _)"   10)
```
```   315
```
```   316 abbreviation
```
```   317   "Pi\<^isub>P I M \<equiv> product_prob_space.infprod_algebra M I"
```
```   318
```
```   319 translations
```
```   320   "PIP x:I. M" == "CONST Pi\<^isub>P I (%x. M)"
```
```   321
```
```   322 sublocale product_prob_space \<subseteq> G!: algebra generator
```
```   323 proof
```
```   324   let ?G = generator
```
```   325   show "sets ?G \<subseteq> Pow (space ?G)"
```
```   326     by (auto simp: generator_def emb_def)
```
```   327   from I_not_empty
```
```   328   obtain i where "i \<in> I" by auto
```
```   329   then show "{} \<in> sets ?G"
```
```   330     by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"]
```
```   331       simp: product_algebra_def sigma_def sigma_sets.Empty generator_def emb_def)
```
```   332   from `i \<in> I` show "space ?G \<in> sets ?G"
```
```   333     by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"]
```
```   334       simp: generator_def emb_def)
```
```   335   fix A assume "A \<in> sets ?G"
```
```   336   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"
```
```   337     by (auto simp: generator_def)
```
```   338   fix B assume "B \<in> sets ?G"
```
```   339   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"
```
```   340     by (auto simp: generator_def)
```
```   341   let ?RA = "emb (JA \<union> JB) JA XA"
```
```   342   let ?RB = "emb (JA \<union> JB) JB XB"
```
```   343   interpret JAB: finite_product_sigma_algebra M "JA \<union> JB"
```
```   344     by default (insert XA XB, auto)
```
```   345   have *: "A - B = emb I (JA \<union> JB) (?RA - ?RB)" "A \<union> B = emb I (JA \<union> JB) (?RA \<union> ?RB)"
```
```   346     using XA A XB B by (auto simp: emb_simps)
```
```   347   then show "A - B \<in> sets ?G" "A \<union> B \<in> sets ?G"
```
```   348     using XA XB by (auto intro!: generatorI')
```
```   349 qed
```
```   350
```
```   351 lemma (in product_prob_space) positive_\<mu>G: "positive generator \<mu>G"
```
```   352 proof (intro positive_def[THEN iffD2] conjI ballI)
```
```   353   from generatorE[OF G.empty_sets] guess J X . note this[simp]
```
```   354   interpret J: finite_product_sigma_finite M J by default fact
```
```   355   have "X = {}"
```
```   356     by (rule emb_injective[of J I]) simp_all
```
```   357   then show "\<mu>G {} = 0" by simp
```
```   358 next
```
```   359   fix A assume "A \<in> sets generator"
```
```   360   from generatorE[OF this] guess J X . note this[simp]
```
```   361   interpret J: finite_product_sigma_finite M J by default fact
```
```   362   show "0 \<le> \<mu>G A" by simp
```
```   363 qed
```
```   364
```
```   365 lemma (in product_prob_space) additive_\<mu>G: "additive generator \<mu>G"
```
```   366 proof (intro additive_def[THEN iffD2] ballI impI)
```
```   367   fix A assume "A \<in> sets generator" with generatorE guess J X . note J = this
```
```   368   fix B assume "B \<in> sets generator" with generatorE guess K Y . note K = this
```
```   369   assume "A \<inter> B = {}"
```
```   370   have JK: "J \<union> K \<noteq> {}" "J \<union> K \<subseteq> I" "finite (J \<union> K)"
```
```   371     using J K by auto
```
```   372   interpret JK: finite_product_sigma_finite M "J \<union> K" by default fact
```
```   373   have JK_disj: "emb (J \<union> K) J X \<inter> emb (J \<union> K) K Y = {}"
```
```   374     apply (rule emb_injective[of "J \<union> K" I])
```
```   375     apply (insert `A \<inter> B = {}` JK J K)
```
```   376     apply (simp_all add: JK.Int emb_simps)
```
```   377     done
```
```   378   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)"
```
```   379     using J K by simp_all
```
```   380   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))"
```
```   381     by (simp add: emb_simps)
```
```   382   also have "\<dots> = measure (Pi\<^isub>M (J \<union> K) M) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)"
```
```   383     using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq JK.Un)
```
```   384   also have "\<dots> = \<mu>G A + \<mu>G B"
```
```   385     using J K JK_disj by (simp add: JK.measure_additive[symmetric])
```
```   386   finally show "\<mu>G (A \<union> B) = \<mu>G A + \<mu>G B" .
```
```   387 qed
```
```   388
```
```   389 lemma (in product_prob_space) finite_index_eq_finite_product:
```
```   390   assumes "finite I"
```
```   391   shows "sets (sigma generator) = sets (Pi\<^isub>M I M)"
```
```   392 proof safe
```
```   393   interpret I: finite_product_sigma_algebra M I by default fact
```
```   394   have [simp]: "space generator = space (Pi\<^isub>M I M)"
```
```   395     by (simp add: generator_def product_algebra_def)
```
```   396   { fix A assume "A \<in> sets (sigma generator)"
```
```   397     then show "A \<in> sets I.P" unfolding sets_sigma
```
```   398     proof induct
```
```   399       case (Basic A)
```
```   400       from generatorE[OF this] guess J X . note J = this
```
```   401       with `finite I` have "emb I J X \<in> sets I.P" by auto
```
```   402       with `emb I J X = A` show "A \<in> sets I.P" by simp
```
```   403     qed auto }
```
```   404   { fix A assume "A \<in> sets I.P"
```
```   405     moreover with I.sets_into_space have "emb I I A = A" by (intro emb_id) auto
```
```   406     ultimately show "A \<in> sets (sigma generator)"
```
```   407       using `finite I` I_not_empty unfolding sets_sigma
```
```   408       by (intro sigma_sets.Basic generatorI[of I A]) auto }
```
```   409 qed
```
```   410
```
```   411 lemma (in product_prob_space) extend_\<mu>G:
```
```   412   "\<exists>\<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and>
```
```   413        measure_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>"
```
```   414 proof cases
```
```   415   assume "finite I"
```
```   416   interpret I: finite_product_sigma_finite M I by default fact
```
```   417   show ?thesis
```
```   418   proof (intro exI[of _ "measure (Pi\<^isub>M I M)"] ballI conjI)
```
```   419     fix A assume "A \<in> sets generator"
```
```   420     from generatorE[OF this] guess J X . note J = this
```
```   421     from J(1-4) `finite I` show "measure I.P A = \<mu>G A"
```
```   422       unfolding J(6)
```
```   423       by (subst J(5)[symmetric]) (simp add: measure_emb)
```
```   424   next
```
```   425     have [simp]: "space generator = space (Pi\<^isub>M I M)"
```
```   426       by (simp add: generator_def product_algebra_def)
```
```   427     have "\<lparr>space = space generator, sets = sets (sigma generator), measure = measure I.P\<rparr>
```
```   428       = I.P" (is "?P = _")
```
```   429       by (auto intro!: measure_space.equality simp: finite_index_eq_finite_product[OF `finite I`])
```
```   430     then show "measure_space ?P" by simp default
```
```   431   qed
```
```   432 next
```
```   433   let ?G = generator
```
```   434   assume "\<not> finite I"
```
```   435   note \<mu>G_mono =
```
```   436     G.additive_increasing[OF positive_\<mu>G additive_\<mu>G, THEN increasingD]
```
```   437
```
```   438   { fix Z J assume J: "J \<noteq> {}" "finite J" "J \<subseteq> I" and Z: "Z \<in> sets ?G"
```
```   439
```
```   440     from `infinite I` `finite J` obtain k where k: "k \<in> I" "k \<notin> J"
```
```   441       by (metis rev_finite_subset subsetI)
```
```   442     moreover from Z guess K' X' by (rule generatorE)
```
```   443     moreover def K \<equiv> "insert k K'"
```
```   444     moreover def X \<equiv> "emb K K' X'"
```
```   445     ultimately have K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "X \<in> sets (Pi\<^isub>M K M)" "Z = emb I K X"
```
```   446       "K - J \<noteq> {}" "K - J \<subseteq> I" "\<mu>G Z = measure (Pi\<^isub>M K M) X"
```
```   447       by (auto simp: subset_insertI)
```
```   448
```
```   449     let "?M y" = "merge J y (K - J) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M)"
```
```   450     { fix y assume y: "y \<in> space (Pi\<^isub>M J M)"
```
```   451       note * = merge_emb[OF `K \<subseteq> I` `J \<subseteq> I` y, of X]
```
```   452       moreover
```
```   453       have **: "?M y \<in> sets (Pi\<^isub>M (K - J) M)"
```
```   454         using J K y by (intro merge_sets) auto
```
```   455       ultimately
```
```   456       have ***: "(merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M)) \<in> sets ?G"
```
```   457         using J K by (intro generatorI) auto
```
```   458       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)"
```
```   459         unfolding * using K J by (subst \<mu>G_eq[OF _ _ _ **]) auto
```
```   460       note * ** *** this }
```
```   461     note merge_in_G = this
```
```   462
```
```   463     have "finite (K - J)" using K by auto
```
```   464
```
```   465     interpret J: finite_product_prob_space M J by default fact+
```
```   466     interpret KmJ: finite_product_prob_space M "K - J" by default fact+
```
```   467
```
```   468     have "\<mu>G Z = measure (Pi\<^isub>M (J \<union> (K - J)) M) (emb (J \<union> (K - J)) K X)"
```
```   469       using K J by simp
```
```   470     also have "\<dots> = (\<integral>\<^isup>+ x. measure (Pi\<^isub>M (K - J) M) (?M x) \<partial>Pi\<^isub>M J M)"
```
```   471       using K J by (subst measure_fold_integral) auto
```
```   472     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)"
```
```   473       (is "_ = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)")
```
```   474     proof (intro J.positive_integral_cong)
```
```   475       fix x assume x: "x \<in> space (Pi\<^isub>M J M)"
```
```   476       with K merge_in_G(2)[OF this]
```
```   477       show "measure (Pi\<^isub>M (K - J) M) (?M x) = \<mu>G (?MZ x)"
```
```   478         unfolding `Z = emb I K X` merge_in_G(1)[OF x] by (subst \<mu>G_eq) auto
```
```   479     qed
```
```   480     finally have fold: "\<mu>G Z = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)" .
```
```   481
```
```   482     { fix x assume x: "x \<in> space (Pi\<^isub>M J M)"
```
```   483       then have "\<mu>G (?MZ x) \<le> 1"
```
```   484         unfolding merge_in_G(4)[OF x] `Z = emb I K X`
```
```   485         by (intro KmJ.measure_le_1 merge_in_G(2)[OF x]) }
```
```   486     note le_1 = this
```
```   487
```
```   488     let "?q y" = "\<mu>G (merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M))"
```
```   489     have "?q \<in> borel_measurable (Pi\<^isub>M J M)"
```
```   490       unfolding `Z = emb I K X` using J K merge_in_G(3)
```
```   491       by (simp add: merge_in_G  \<mu>G_eq measure_fold_measurable
```
```   492                del: space_product_algebra cong: measurable_cong)
```
```   493     note this fold le_1 merge_in_G(3) }
```
```   494   note fold = this
```
```   495
```
```   496   show ?thesis
```
```   497   proof (rule G.caratheodory_empty_continuous[OF positive_\<mu>G additive_\<mu>G])
```
```   498     fix A assume "A \<in> sets ?G"
```
```   499     with generatorE guess J X . note JX = this
```
```   500     interpret JK: finite_product_prob_space M J by default fact+
```
```   501     with JX show "\<mu>G A \<noteq> \<infinity>" by simp
```
```   502   next
```
```   503     fix A assume A: "range A \<subseteq> sets ?G" "decseq A" "(\<Inter>i. A i) = {}"
```
```   504     then have "decseq (\<lambda>i. \<mu>G (A i))"
```
```   505       by (auto intro!: \<mu>G_mono simp: decseq_def)
```
```   506     moreover
```
```   507     have "(INF i. \<mu>G (A i)) = 0"
```
```   508     proof (rule ccontr)
```
```   509       assume "(INF i. \<mu>G (A i)) \<noteq> 0" (is "?a \<noteq> 0")
```
```   510       moreover have "0 \<le> ?a"
```
```   511         using A positive_\<mu>G by (auto intro!: le_INFI simp: positive_def)
```
```   512       ultimately have "0 < ?a" by auto
```
```   513
```
```   514       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"
```
```   515         using A by (intro allI generator_Ex) auto
```
```   516       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)"
```
```   517         and A': "\<And>n. A n = emb I (J' n) (X' n)"
```
```   518         unfolding choice_iff by blast
```
```   519       moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)"
```
```   520       moreover def X \<equiv> "\<lambda>n. emb (J n) (J' n) (X' n)"
```
```   521       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)"
```
```   522         by auto
```
```   523       with A' have A_eq: "\<And>n. A n = emb I (J n) (X n)" "\<And>n. A n \<in> sets ?G"
```
```   524         unfolding J_def X_def by (subst emb_trans) (insert A, auto)
```
```   525
```
```   526       have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
```
```   527         unfolding J_def by force
```
```   528
```
```   529       interpret J: finite_product_prob_space M "J i" for i by default fact+
```
```   530
```
```   531       have a_le_1: "?a \<le> 1"
```
```   532         using \<mu>G_spec[of "J 0" "A 0" "X 0"] J A_eq
```
```   533         by (auto intro!: INF_leI2[of 0] J.measure_le_1)
```
```   534
```
```   535       let "?M K Z y" = "merge K y (I - K) -` Z \<inter> space (Pi\<^isub>M I M)"
```
```   536
```
```   537       { fix Z k assume Z: "range Z \<subseteq> sets ?G" "decseq Z" "\<forall>n. ?a / 2^k \<le> \<mu>G (Z n)"
```
```   538         then have Z_sets: "\<And>n. Z n \<in> sets ?G" by auto
```
```   539         fix J' assume J': "J' \<noteq> {}" "finite J'" "J' \<subseteq> I"
```
```   540         interpret J': finite_product_prob_space M J' by default fact+
```
```   541
```
```   542         let "?q n y" = "\<mu>G (?M J' (Z n) y)"
```
```   543         let "?Q n" = "?q n -` {?a / 2^(k+1) ..} \<inter> space (Pi\<^isub>M J' M)"
```
```   544         { fix n
```
```   545           have "?q n \<in> borel_measurable (Pi\<^isub>M J' M)"
```
```   546             using Z J' by (intro fold(1)) auto
```
```   547           then have "?Q n \<in> sets (Pi\<^isub>M J' M)"
```
```   548             by (rule measurable_sets) auto }
```
```   549         note Q_sets = this
```
```   550
```
```   551         have "?a / 2^(k+1) \<le> (INF n. measure (Pi\<^isub>M J' M) (?Q n))"
```
```   552         proof (intro le_INFI)
```
```   553           fix n
```
```   554           have "?a / 2^k \<le> \<mu>G (Z n)" using Z by auto
```
```   555           also have "\<dots> \<le> (\<integral>\<^isup>+ x. indicator (?Q n) x + ?a / 2^(k+1) \<partial>Pi\<^isub>M J' M)"
```
```   556             unfolding fold(2)[OF J' `Z n \<in> sets ?G`]
```
```   557           proof (intro J'.positive_integral_mono)
```
```   558             fix x assume x: "x \<in> space (Pi\<^isub>M J' M)"
```
```   559             then have "?q n x \<le> 1 + 0"
```
```   560               using J' Z fold(3) Z_sets by auto
```
```   561             also have "\<dots> \<le> 1 + ?a / 2^(k+1)"
```
```   562               using `0 < ?a` by (intro add_mono) auto
```
```   563             finally have "?q n x \<le> 1 + ?a / 2^(k+1)" .
```
```   564             with x show "?q n x \<le> indicator (?Q n) x + ?a / 2^(k+1)"
```
```   565               by (auto split: split_indicator simp del: power_Suc)
```
```   566           qed
```
```   567           also have "\<dots> = measure (Pi\<^isub>M J' M) (?Q n) + ?a / 2^(k+1)"
```
```   568             using `0 \<le> ?a` Q_sets J'.measure_space_1
```
```   569             by (subst J'.positive_integral_add) auto
```
```   570           finally show "?a / 2^(k+1) \<le> measure (Pi\<^isub>M J' M) (?Q n)" using `?a \<le> 1`
```
```   571             by (cases rule: extreal2_cases[of ?a "measure (Pi\<^isub>M J' M) (?Q n)"])
```
```   572                (auto simp: field_simps)
```
```   573         qed
```
```   574         also have "\<dots> = measure (Pi\<^isub>M J' M) (\<Inter>n. ?Q n)"
```
```   575         proof (intro J'.continuity_from_above)
```
```   576           show "range ?Q \<subseteq> sets (Pi\<^isub>M J' M)" using Q_sets by auto
```
```   577           show "decseq ?Q"
```
```   578             unfolding decseq_def
```
```   579           proof (safe intro!: vimageI[OF refl])
```
```   580             fix m n :: nat assume "m \<le> n"
```
```   581             fix x assume x: "x \<in> space (Pi\<^isub>M J' M)"
```
```   582             assume "?a / 2^(k+1) \<le> ?q n x"
```
```   583             also have "?q n x \<le> ?q m x"
```
```   584             proof (rule \<mu>G_mono)
```
```   585               from fold(4)[OF J', OF Z_sets x]
```
```   586               show "?M J' (Z n) x \<in> sets ?G" "?M J' (Z m) x \<in> sets ?G" by auto
```
```   587               show "?M J' (Z n) x \<subseteq> ?M J' (Z m) x"
```
```   588                 using `decseq Z`[THEN decseqD, OF `m \<le> n`] by auto
```
```   589             qed
```
```   590             finally show "?a / 2^(k+1) \<le> ?q m x" .
```
```   591           qed
```
```   592         qed (intro J'.finite_measure Q_sets)
```
```   593         finally have "(\<Inter>n. ?Q n) \<noteq> {}"
```
```   594           using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq)
```
```   595         then have "\<exists>w\<in>space (Pi\<^isub>M J' M). \<forall>n. ?a / 2 ^ (k + 1) \<le> ?q n w" by auto }
```
```   596       note Ex_w = this
```
```   597
```
```   598       let "?q k n y" = "\<mu>G (?M (J k) (A n) y)"
```
```   599
```
```   600       have "\<forall>n. ?a / 2 ^ 0 \<le> \<mu>G (A n)" by (auto intro: INF_leI)
```
```   601       from Ex_w[OF A(1,2) this J(1-3), of 0] guess w0 .. note w0 = this
```
```   602
```
```   603       let "?P k wk w" =
```
```   604         "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)"
```
```   605       def w \<equiv> "nat_rec w0 (\<lambda>k wk. Eps (?P k wk))"
```
```   606
```
```   607       { fix k have w: "w k \<in> space (Pi\<^isub>M (J k) M) \<and>
```
```   608           (\<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))"
```
```   609         proof (induct k)
```
```   610           case 0 with w0 show ?case
```
```   611             unfolding w_def nat_rec_0 by auto
```
```   612         next
```
```   613           case (Suc k)
```
```   614           then have wk: "w k \<in> space (Pi\<^isub>M (J k) M)" by auto
```
```   615           have "\<exists>w'. ?P k (w k) w'"
```
```   616           proof cases
```
```   617             assume [simp]: "J k = J (Suc k)"
```
```   618             show ?thesis
```
```   619             proof (intro exI[of _ "w k"] conjI allI)
```
```   620               fix n
```
```   621               have "?a / 2 ^ (Suc k + 1) \<le> ?a / 2 ^ (k + 1)"
```
```   622                 using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: field_simps)
```
```   623               also have "\<dots> \<le> ?q k n (w k)" using Suc by auto
```
```   624               finally show "?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n (w k)" by simp
```
```   625             next
```
```   626               show "w k \<in> space (Pi\<^isub>M (J (Suc k)) M)"
```
```   627                 using Suc by simp
```
```   628               then show "restrict (w k) (J k) = w k"
```
```   629                 by (simp add: extensional_restrict)
```
```   630             qed
```
```   631           next
```
```   632             assume "J k \<noteq> J (Suc k)"
```
```   633             with J_mono[of k "Suc k"] have "J (Suc k) - J k \<noteq> {}" (is "?D \<noteq> {}") by auto
```
```   634             have "range (\<lambda>n. ?M (J k) (A n) (w k)) \<subseteq> sets ?G"
```
```   635               "decseq (\<lambda>n. ?M (J k) (A n) (w k))"
```
```   636               "\<forall>n. ?a / 2 ^ (k + 1) \<le> \<mu>G (?M (J k) (A n) (w k))"
```
```   637               using `decseq A` fold(4)[OF J(1-3) A_eq(2), of "w k" k] Suc
```
```   638               by (auto simp: decseq_def)
```
```   639             from Ex_w[OF this `?D \<noteq> {}`] J[of "Suc k"]
```
```   640             obtain w' where w': "w' \<in> space (Pi\<^isub>M ?D M)"
```
```   641               "\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> \<mu>G (?M ?D (?M (J k) (A n) (w k)) w')" by auto
```
```   642             let ?w = "merge (J k) (w k) ?D w'"
```
```   643             have [simp]: "\<And>x. merge (J k) (w k) (I - J k) (merge ?D w' (I - ?D) x) =
```
```   644               merge (J (Suc k)) ?w (I - (J (Suc k))) x"
```
```   645               using J(3)[of "Suc k"] J(3)[of k] J_mono[of k "Suc k"]
```
```   646               by (auto intro!: ext split: split_merge)
```
```   647             have *: "\<And>n. ?M ?D (?M (J k) (A n) (w k)) w' = ?M (J (Suc k)) (A n) ?w"
```
```   648               using w'(1) J(3)[of "Suc k"]
```
```   649               by (auto split: split_merge intro!: extensional_merge_sub) force+
```
```   650             show ?thesis
```
```   651               apply (rule exI[of _ ?w])
```
```   652               using w' J_mono[of k "Suc k"] wk unfolding *
```
```   653               apply (auto split: split_merge intro!: extensional_merge_sub ext)
```
```   654               apply (force simp: extensional_def)
```
```   655               done
```
```   656           qed
```
```   657           then have "?P k (w k) (w (Suc k))"
```
```   658             unfolding w_def nat_rec_Suc unfolding w_def[symmetric]
```
```   659             by (rule someI_ex)
```
```   660           then show ?case by auto
```
```   661         qed
```
```   662         moreover
```
```   663         then have "w k \<in> space (Pi\<^isub>M (J k) M)" by auto
```
```   664         moreover
```
```   665         from w have "?a / 2 ^ (k + 1) \<le> ?q k k (w k)" by auto
```
```   666         then have "?M (J k) (A k) (w k) \<noteq> {}"
```
```   667           using positive_\<mu>G[unfolded positive_def] `0 < ?a` `?a \<le> 1`
```
```   668           by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq)
```
```   669         then obtain x where "x \<in> ?M (J k) (A k) (w k)" by auto
```
```   670         then have "merge (J k) (w k) (I - J k) x \<in> A k" by auto
```
```   671         then have "\<exists>x\<in>A k. restrict x (J k) = w k"
```
```   672           using `w k \<in> space (Pi\<^isub>M (J k) M)`
```
```   673           by (intro rev_bexI) (auto intro!: ext simp: extensional_def)
```
```   674         ultimately have "w k \<in> space (Pi\<^isub>M (J k) M)"
```
```   675           "\<exists>x\<in>A k. restrict x (J k) = w k"
```
```   676           "k \<noteq> 0 \<Longrightarrow> restrict (w k) (J (k - 1)) = w (k - 1)"
```
```   677           by auto }
```
```   678       note w = this
```
```   679
```
```   680       { fix k l i assume "k \<le> l" "i \<in> J k"
```
```   681         { fix l have "w k i = w (k + l) i"
```
```   682           proof (induct l)
```
```   683             case (Suc l)
```
```   684             from `i \<in> J k` J_mono[of k "k + l"] have "i \<in> J (k + l)" by auto
```
```   685             with w(3)[of "k + Suc l"]
```
```   686             have "w (k + l) i = w (k + Suc l) i"
```
```   687               by (auto simp: restrict_def fun_eq_iff split: split_if_asm)
```
```   688             with Suc show ?case by simp
```
```   689           qed simp }
```
```   690         from this[of "l - k"] `k \<le> l` have "w l i = w k i" by simp }
```
```   691       note w_mono = this
```
```   692
```
```   693       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"
```
```   694       { fix i k assume k: "i \<in> J k"
```
```   695         have "w k i = w (LEAST k. i \<in> J k) i"
```
```   696           by (intro w_mono Least_le k LeastI[of _ k])
```
```   697         then have "w' i = w k i"
```
```   698           unfolding w'_def using k by auto }
```
```   699       note w'_eq = this
```
```   700       have w'_simps1: "\<And>i. i \<notin> I \<Longrightarrow> w' i = undefined"
```
```   701         using J by (auto simp: w'_def)
```
```   702       have w'_simps2: "\<And>i. i \<notin> (\<Union>k. J k) \<Longrightarrow> i \<in> I \<Longrightarrow> w' i \<in> space (M i)"
```
```   703         using J by (auto simp: w'_def intro!: someI_ex[OF M.not_empty[unfolded ex_in_conv[symmetric]]])
```
```   704       { fix i assume "i \<in> I" then have "w' i \<in> space (M i)"
```
```   705           using w(1) by (cases "i \<in> (\<Union>k. J k)") (force simp: w'_simps2 w'_eq)+ }
```
```   706       note w'_simps[simp] = w'_eq w'_simps1 w'_simps2 this
```
```   707
```
```   708       have w': "w' \<in> space (Pi\<^isub>M I M)"
```
```   709         using w(1) by (auto simp add: Pi_iff extensional_def)
```
```   710
```
```   711       { fix n
```
```   712         have "restrict w' (J n) = w n" using w(1)
```
```   713           by (auto simp add: fun_eq_iff restrict_def Pi_iff extensional_def)
```
```   714         with w[of n] obtain x where "x \<in> A n" "restrict x (J n) = restrict w' (J n)" by auto
```
```   715         then have "w' \<in> A n" unfolding A_eq using w' by (auto simp: emb_def) }
```
```   716       then have "w' \<in> (\<Inter>i. A i)" by auto
```
```   717       with `(\<Inter>i. A i) = {}` show False by auto
```
```   718     qed
```
```   719     ultimately show "(\<lambda>i. \<mu>G (A i)) ----> 0"
```
```   720       using LIMSEQ_extreal_INFI[of "\<lambda>i. \<mu>G (A i)"] by simp
```
```   721   qed
```
```   722 qed
```
```   723
```
```   724 lemma (in product_prob_space) infprod_spec:
```
```   725   shows "(\<forall>s\<in>sets generator. measure (Pi\<^isub>P I M) s = \<mu>G s) \<and> measure_space (Pi\<^isub>P I M)"
```
```   726 proof -
```
```   727   let ?P = "\<lambda>\<mu>. (\<forall>A\<in>sets generator. \<mu> A = \<mu>G A) \<and>
```
```   728        measure_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>"
```
```   729   have **: "measure infprod_algebra = (SOME \<mu>. ?P \<mu>)"
```
```   730     unfolding infprod_algebra_def by simp
```
```   731   have *: "Pi\<^isub>P I M = \<lparr>space = space generator, sets = sets (sigma generator), measure = measure (Pi\<^isub>P I M)\<rparr>"
```
```   732     unfolding infprod_algebra_def by auto
```
```   733   show ?thesis
```
```   734     apply (subst (2) *)
```
```   735     apply (unfold **)
```
```   736     apply (rule someI_ex[where P="?P"])
```
```   737     apply (rule extend_\<mu>G)
```
```   738     done
```
```   739 qed
```
```   740
```
```   741 sublocale product_prob_space \<subseteq> P: measure_space "Pi\<^isub>P I M"
```
```   742   using infprod_spec by auto
```
```   743
```
```   744 lemma (in product_prob_space) measure_infprod_emb:
```
```   745   assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)"
```
```   746   shows "\<mu> (emb I J X) = measure (Pi\<^isub>M J M) X"
```
```   747 proof -
```
```   748   have "emb I J X \<in> sets generator"
```
```   749     using assms by (rule generatorI')
```
```   750   with \<mu>G_eq[OF assms] infprod_spec show ?thesis by auto
```
```   751 qed
```
```   752
```
```   753 sublocale product_prob_space \<subseteq> P: prob_space "Pi\<^isub>P I M"
```
```   754 proof
```
```   755   obtain i where "i \<in> I" using I_not_empty by auto
```
```   756   interpret i: finite_product_sigma_finite M "{i}" by default auto
```
```   757   let ?X = "\<Pi>\<^isub>E i\<in>{i}. space (M i)"
```
```   758   have "?X \<in> sets (Pi\<^isub>M {i} M)"
```
```   759     by auto
```
```   760   from measure_infprod_emb[OF _ _ _ this] `i \<in> I`
```
```   761   have "\<mu> (emb I {i} ?X) = measure (M i) (space (M i))"
```
```   762     by (simp add: i.measure_times)
```
```   763   also have "emb I {i} ?X = space (Pi\<^isub>P I M)"
```
```   764     using `i \<in> I` by (auto simp: emb_def infprod_algebra_def generator_def)
```
```   765   finally show "\<mu> (space (Pi\<^isub>P I M)) = 1"
```
```   766     using M.measure_space_1 by simp
```
```   767 qed
```
```   768
```
```   769 lemma (in product_prob_space) measurable_component:
```
```   770   assumes "i \<in> I"
```
```   771   shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>P I M) (M i)"
```
```   772 proof (unfold measurable_def, safe)
```
```   773   fix x assume "x \<in> space (Pi\<^isub>P I M)"
```
```   774   then show "x i \<in> space (M i)"
```
```   775     using `i \<in> I` by (auto simp: infprod_algebra_def generator_def)
```
```   776 next
```
```   777   fix A assume "A \<in> sets (M i)"
```
```   778   with `i \<in> I` have
```
```   779     "(\<Pi>\<^isub>E x \<in> {i}. A) \<in> sets (Pi\<^isub>M {i} M)"
```
```   780     "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) = emb I {i} (\<Pi>\<^isub>E x \<in> {i}. A)"
```
```   781     by (auto simp: infprod_algebra_def generator_def emb_def)
```
```   782   from generatorI[OF _ _ _ this] `i \<in> I`
```
```   783   show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) \<in> sets (Pi\<^isub>P I M)"
```
```   784     unfolding infprod_algebra_def by auto
```
```   785 qed
```
```   786
```
```   787 lemma (in product_prob_space) emb_in_infprod_algebra[intro]:
```
```   788   fixes J assumes J: "finite J" "J \<subseteq> I" and X: "X \<in> sets (Pi\<^isub>M J M)"
```
```   789   shows "emb I J X \<in> sets (\<Pi>\<^isub>P i\<in>I. M i)"
```
```   790 proof cases
```
```   791   assume "J = {}"
```
```   792   with X have "emb I J X = space (\<Pi>\<^isub>P i\<in>I. M i) \<or> emb I J X = {}"
```
```   793     by (auto simp: emb_def infprod_algebra_def generator_def
```
```   794                    product_algebra_def product_algebra_generator_def image_constant sigma_def)
```
```   795   then show ?thesis by auto
```
```   796 next
```
```   797   assume "J \<noteq> {}"
```
```   798   show ?thesis unfolding infprod_algebra_def
```
```   799     by simp (intro in_sigma generatorI'  `J \<noteq> {}` J X)
```
```   800 qed
```
```   801
```
```   802 lemma (in product_prob_space) finite_measure_infprod_emb:
```
```   803   assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)"
```
```   804   shows "\<mu>' (emb I J X) = finite_measure.\<mu>' (Pi\<^isub>M J M) X"
```
```   805 proof -
```
```   806   interpret J: finite_product_prob_space M J by default fact+
```
```   807   from assms have "emb I J X \<in> sets (Pi\<^isub>P I M)" by auto
```
```   808   with assms show "\<mu>' (emb I J X) = J.\<mu>' X"
```
```   809     unfolding \<mu>'_def J.\<mu>'_def
```
```   810     unfolding measure_infprod_emb[OF assms]
```
```   811     by auto
```
```   812 qed
```
```   813
```
```   814 lemma (in finite_product_prob_space) finite_measure_times:
```
```   815   assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)"
```
```   816   shows "\<mu>' (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu>' i (A i))"
```
```   817   using assms
```
```   818   unfolding \<mu>'_def M.\<mu>'_def
```
```   819   by (subst measure_times[OF assms])
```
```   820      (auto simp: finite_measure_eq M.finite_measure_eq setprod_extreal)
```
```   821
```
```   822 lemma (in product_prob_space) finite_measure_infprod_emb_Pi:
```
```   823   assumes J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> X j \<in> sets (M j)"
```
```   824   shows "\<mu>' (emb I J (Pi\<^isub>E J X)) = (\<Prod>j\<in>J. M.\<mu>' j (X j))"
```
```   825 proof cases
```
```   826   assume "J = {}"
```
```   827   then have "emb I J (Pi\<^isub>E J X) = space infprod_algebra"
```
```   828     by (auto simp: infprod_algebra_def generator_def sigma_def emb_def)
```
```   829   then show ?thesis using `J = {}` prob_space by simp
```
```   830 next
```
```   831   assume "J \<noteq> {}"
```
```   832   interpret J: finite_product_prob_space M J by default fact+
```
```   833   have "(\<Prod>i\<in>J. M.\<mu>' i (X i)) = J.\<mu>' (Pi\<^isub>E J X)"
```
```   834     using J `J \<noteq> {}` by (subst J.finite_measure_times) auto
```
```   835   also have "\<dots> = \<mu>' (emb I J (Pi\<^isub>E J X))"
```
```   836     using J `J \<noteq> {}` by (intro finite_measure_infprod_emb[symmetric]) auto
```
```   837   finally show ?thesis by simp
```
```   838 qed
```
```   839
```
```   840 lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
```
```   841 proof
```
```   842   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
```
```   843     by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros)
```
```   844 qed
```
```   845
```
```   846 lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
```
```   847 proof
```
```   848   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
```
```   849     by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros)
```
```   850 qed
```
```   851
```
```   852 lemma sigma_sets_subseteq: "A \<subseteq> sigma_sets X A"
```
```   853   by (auto intro: sigma_sets.Basic)
```
```   854
```
```   855 lemma (in product_prob_space) infprod_algebra_alt:
```
```   856   "Pi\<^isub>P I M = sigma \<lparr> space = space (Pi\<^isub>P I M),
```
```   857     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))),
```
```   858     measure = measure (Pi\<^isub>P I M) \<rparr>"
```
```   859   (is "_ = sigma \<lparr> space = ?O, sets = ?M, measure = ?m \<rparr>")
```
```   860 proof (rule measure_space.equality)
```
```   861   let ?G = "\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M)"
```
```   862   have "sigma_sets ?O ?M = sigma_sets ?O ?G"
```
```   863   proof (intro equalityI sigma_sets_mono UN_least)
```
```   864     fix J assume J: "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}"
```
```   865     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
```
```   866     also have "\<dots> \<subseteq> ?G" using J by (rule UN_upper)
```
```   867     also have "\<dots> \<subseteq> sigma_sets ?O ?G" by (rule sigma_sets_subseteq)
```
```   868     finally show "emb I J ` Pi\<^isub>E J ` (\<Pi> i\<in>J. sets (M i)) \<subseteq> sigma_sets ?O ?G" .
```
```   869     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)))"
```
```   870       by (simp add: sets_sigma product_algebra_generator_def product_algebra_def)
```
```   871     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)))"
```
```   872       using J M.sets_into_space
```
```   873       by (auto simp: emb_def_raw intro!: sigma_sets_vimage[symmetric]) blast
```
```   874     also have "\<dots> \<subseteq> sigma_sets (space (Pi\<^isub>M I M)) ?M"
```
```   875       using J by (intro sigma_sets_mono') auto
```
```   876     finally show "emb I J ` sets (Pi\<^isub>M J M) \<subseteq> sigma_sets ?O ?M"
```
```   877       by (simp add: infprod_algebra_def generator_def)
```
```   878   qed
```
```   879   then show "sets (Pi\<^isub>P I M) = sets (sigma \<lparr> space = ?O, sets = ?M, measure = ?m \<rparr>)"
```
```   880     by (simp_all add: infprod_algebra_def generator_def sets_sigma)
```
```   881 qed simp_all
```
```   882
```
```   883 lemma (in product_prob_space) infprod_algebra_alt2:
```
```   884   "Pi\<^isub>P I M = sigma \<lparr> space = space (Pi\<^isub>P I M),
```
```   885     sets = (\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (M i)),
```
```   886     measure = measure (Pi\<^isub>P I M) \<rparr>"
```
```   887   (is "_ = ?S")
```
```   888 proof (rule measure_space.equality)
```
```   889   let "sigma \<lparr> space = ?O, sets = ?A, \<dots> = _ \<rparr>" = ?S
```
```   890   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)))"
```
```   891   have "sets (Pi\<^isub>P I M) = sigma_sets ?O ?G"
```
```   892     by (subst infprod_algebra_alt) (simp add: sets_sigma)
```
```   893   also have "\<dots> = sigma_sets ?O ?A"
```
```   894   proof (intro equalityI sigma_sets_mono subsetI)
```
```   895     interpret A: sigma_algebra ?S
```
```   896       by (rule sigma_algebra_sigma) auto
```
```   897     fix A assume "A \<in> ?G"
```
```   898     then obtain J B where "finite J" "J \<noteq> {}" "J \<subseteq> I" "A = emb I J (Pi\<^isub>E J B)"
```
```   899         and B: "\<And>i. i \<in> J \<Longrightarrow> B i \<in> sets (M i)"
```
```   900       by auto
```
```   901     then have A: "A = (\<Inter>j\<in>J. (\<lambda>x. x j) -` (B j) \<inter> space (Pi\<^isub>P I M))"
```
```   902       by (auto simp: emb_def infprod_algebra_def generator_def Pi_iff)
```
```   903     { fix j assume "j\<in>J"
```
```   904       with `J \<subseteq> I` have "j \<in> I" by auto
```
```   905       with `j \<in> J` B have "(\<lambda>x. x j) -` (B j) \<inter> space (Pi\<^isub>P I M) \<in> sets ?S"
```
```   906         by (auto simp: sets_sigma intro: sigma_sets.Basic) }
```
```   907     with `finite J` `J \<noteq> {}` have "A \<in> sets ?S"
```
```   908       unfolding A by (intro A.finite_INT) auto
```
```   909     then show "A \<in> sigma_sets ?O ?A" by (simp add: sets_sigma)
```
```   910   next
```
```   911     fix A assume "A \<in> ?A"
```
```   912     then obtain i B where i: "i \<in> I" "B \<in> sets (M i)"
```
```   913         and "A = (\<lambda>x. x i) -` B \<inter> space (Pi\<^isub>P I M)"
```
```   914       by auto
```
```   915     then have "A = emb I {i} (Pi\<^isub>E {i} (\<lambda>_. B))"
```
```   916       by (auto simp: emb_def infprod_algebra_def generator_def Pi_iff)
```
```   917     with i show "A \<in> sigma_sets ?O ?G"
```
```   918       by (intro sigma_sets.Basic UN_I[where a="{i}"]) auto
```
```   919   qed
```
```   920   finally show "sets (Pi\<^isub>P I M) = sets ?S"
```
```   921     by (simp add: sets_sigma)
```
```   922 qed simp_all
```
```   923
```
```   924 lemma (in product_prob_space) measurable_into_infprod_algebra:
```
```   925   assumes "sigma_algebra N"
```
```   926   assumes f: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)"
```
```   927   assumes ext: "\<And>x. x \<in> space N \<Longrightarrow> f x \<in> extensional I"
```
```   928   shows "f \<in> measurable N (Pi\<^isub>P I M)"
```
```   929 proof -
```
```   930   interpret N: sigma_algebra N by fact
```
```   931   have f_in: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f x i) \<in> space N \<rightarrow> space (M i)"
```
```   932     using f by (auto simp: measurable_def)
```
```   933   { fix i A assume i: "i \<in> I" "A \<in> sets (M i)"
```
```   934     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"
```
```   935       using f_in ext by (auto simp: infprod_algebra_def generator_def)
```
```   936     also have "\<dots> \<in> sets N"
```
```   937       by (rule measurable_sets f i)+
```
```   938     finally have "f -` (\<lambda>x. x i) -` A \<inter> f -` space infprod_algebra \<inter> space N \<in> sets N" . }
```
```   939   with f_in ext show ?thesis
```
```   940     by (subst infprod_algebra_alt2)
```
```   941        (auto intro!: N.measurable_sigma simp: Pi_iff infprod_algebra_def generator_def)
```
```   942 qed
```
```   943
```
```   944 subsection {* Sequence space *}
```
```   945
```
```   946 locale sequence_space = product_prob_space M "UNIV :: nat set" for M
```
```   947
```
```   948 lemma (in sequence_space) infprod_in_sets[intro]:
```
```   949   fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
```
```   950   shows "Pi UNIV E \<in> sets (Pi\<^isub>P UNIV M)"
```
```   951 proof -
```
```   952   have "Pi UNIV E = (\<Inter>i. emb UNIV {..i} (\<Pi>\<^isub>E j\<in>{..i}. E j))"
```
```   953     using E E[THEN M.sets_into_space]
```
```   954     by (auto simp: emb_def Pi_iff extensional_def) blast
```
```   955   with E show ?thesis
```
```   956     by (auto intro: emb_in_infprod_algebra)
```
```   957 qed
```
```   958
```
```   959 lemma (in sequence_space) measure_infprod:
```
```   960   fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
```
```   961   shows "(\<lambda>n. \<Prod>i\<le>n. M.\<mu>' i (E i)) ----> \<mu>' (Pi UNIV E)"
```
```   962 proof -
```
```   963   let "?E n" = "emb UNIV {..n} (Pi\<^isub>E {.. n} E)"
```
```   964   { fix n :: nat
```
```   965     interpret n: finite_product_prob_space M "{..n}" by default auto
```
```   966     have "(\<Prod>i\<le>n. M.\<mu>' i (E i)) = n.\<mu>' (Pi\<^isub>E {.. n} E)"
```
```   967       using E by (subst n.finite_measure_times) auto
```
```   968     also have "\<dots> = \<mu>' (?E n)"
```
```   969       using E by (intro finite_measure_infprod_emb[symmetric]) auto
```
```   970     finally have "(\<Prod>i\<le>n. M.\<mu>' i (E i)) = \<mu>' (?E n)" . }
```
```   971   moreover have "Pi UNIV E = (\<Inter>n. ?E n)"
```
```   972     using E E[THEN M.sets_into_space]
```
```   973     by (auto simp: emb_def extensional_def Pi_iff) blast
```
```   974   moreover have "range ?E \<subseteq> sets (Pi\<^isub>P UNIV M)"
```
```   975     using E by auto
```
```   976   moreover have "decseq ?E"
```
```   977     by (auto simp: emb_def Pi_iff decseq_def)
```
```   978   ultimately show ?thesis
```
```   979     by (simp add: finite_continuity_from_above)
```
```   980 qed
```
```   981
```
`   982 end`