src/HOL/Probability/Infinite_Product_Measure.thy
 author wenzelm Tue Mar 13 16:56:56 2012 +0100 (2012-03-13) changeset 46905 6b1c0a80a57a parent 46898 1570b30ee040 child 47694 05663f75964c permissions -rw-r--r--
prefer abs_def over def_raw;
```     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 lemma (in product_prob_space) measure_preserving_restrict:
```
```    45   assumes "J \<noteq> {}" "J \<subseteq> K" "finite K"
```
```    46   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> _")
```
```    47 proof -
```
```    48   interpret K: finite_product_prob_space M K by default fact
```
```    49   have J: "J \<noteq> {}" "finite J" using assms by (auto simp add: finite_subset)
```
```    50   interpret J: finite_product_prob_space M J
```
```    51     by default (insert J, auto)
```
```    52   from J.sigma_finite_pairs guess F .. note F = this
```
```    53   then have [simp,intro]: "\<And>k i. k \<in> J \<Longrightarrow> F k i \<in> sets (M k)"
```
```    54     by auto
```
```    55   let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. F k i"
```
```    56   let ?J = "product_algebra_generator J M \<lparr> measure := measure (Pi\<^isub>M J M) \<rparr>"
```
```    57   have "?R \<in> measure_preserving (\<Pi>\<^isub>M i\<in>K. M i) (sigma ?J)"
```
```    58   proof (rule K.measure_preserving_Int_stable)
```
```    59     show "Int_stable ?J"
```
```    60       by (auto simp: Int_stable_def product_algebra_generator_def PiE_Int)
```
```    61     show "range ?F \<subseteq> sets ?J" "incseq ?F" "(\<Union>i. ?F i) = space ?J"
```
```    62       using F by auto
```
```    63     show "\<And>i. measure ?J (?F i) \<noteq> \<infinity>"
```
```    64       using F by (simp add: J.measure_times setprod_PInf)
```
```    65     have "measure_space (Pi\<^isub>M J M)" by default
```
```    66     then show "measure_space (sigma ?J)"
```
```    67       by (simp add: product_algebra_def sigma_def)
```
```    68     show "?R \<in> measure_preserving (Pi\<^isub>M K M) ?J"
```
```    69     proof (simp add: measure_preserving_def measurable_def product_algebra_generator_def del: vimage_Int,
```
```    70            safe intro!: restrict_extensional)
```
```    71       fix x k assume "k \<in> J" "x \<in> (\<Pi> i\<in>K. space (M i))"
```
```    72       then show "x k \<in> space (M k)" using `J \<subseteq> K` by auto
```
```    73     next
```
```    74       fix E assume "E \<in> (\<Pi> i\<in>J. sets (M i))"
```
```    75       then have E: "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)" by auto
```
```    76       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))"
```
```    77         (is "?X = Pi\<^isub>E K ?M")
```
```    78         using `J \<subseteq> K` sets_into_space by (auto simp: Pi_iff split: split_if_asm) blast+
```
```    79       with E show "?X \<in> sets (Pi\<^isub>M K M)"
```
```    80         by (auto intro!: product_algebra_generatorI)
```
```    81       have "measure (Pi\<^isub>M J M) (Pi\<^isub>E J E) = (\<Prod>i\<in>J. measure (M i) (?M i))"
```
```    82         using E by (simp add: J.measure_times)
```
```    83       also have "\<dots> = measure (Pi\<^isub>M K M) ?X"
```
```    84         unfolding * using E `finite K` `J \<subseteq> K`
```
```    85         by (auto simp: K.measure_times M.measure_space_1
```
```    86                  cong del: setprod_cong
```
```    87                  intro!: setprod_mono_one_left)
```
```    88       finally show "measure (Pi\<^isub>M J M) (Pi\<^isub>E J E) = measure (Pi\<^isub>M K M) ?X" .
```
```    89     qed
```
```    90   qed
```
```    91   then show ?thesis
```
```    92     by (simp add: product_algebra_def sigma_def)
```
```    93 qed
```
```    94
```
```    95 lemma (in product_prob_space) measurable_restrict:
```
```    96   assumes *: "J \<noteq> {}" "J \<subseteq> K" "finite K"
```
```    97   shows "(\<lambda>f. restrict f J) \<in> measurable (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i)"
```
```    98   using measure_preserving_restrict[OF *]
```
```    99   by (rule measure_preservingD2)
```
```   100
```
```   101 definition (in product_prob_space)
```
```   102   "emb J K X = (\<lambda>x. restrict x K) -` X \<inter> space (Pi\<^isub>M J M)"
```
```   103
```
```   104 lemma (in product_prob_space) emb_trans[simp]:
```
```   105   "J \<subseteq> K \<Longrightarrow> K \<subseteq> L \<Longrightarrow> emb L K (emb K J X) = emb L J X"
```
```   106   by (auto simp add: Int_absorb1 emb_def)
```
```   107
```
```   108 lemma (in product_prob_space) emb_empty[simp]:
```
```   109   "emb K J {} = {}"
```
```   110   by (simp add: emb_def)
```
```   111
```
```   112 lemma (in product_prob_space) emb_Pi:
```
```   113   assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
```
```   114   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))"
```
```   115   using assms space_closed
```
```   116   by (auto simp: emb_def Pi_iff split: split_if_asm) blast+
```
```   117
```
```   118 lemma (in product_prob_space) emb_injective:
```
```   119   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)"
```
```   120   assumes "emb L J X = emb L J Y"
```
```   121   shows "X = Y"
```
```   122 proof -
```
```   123   interpret J: finite_product_sigma_finite M J by default fact
```
```   124   show "X = Y"
```
```   125   proof (rule injective_vimage_restrict)
```
```   126     show "X \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" "Y \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
```
```   127       using J.sets_into_space sets by auto
```
```   128     have "\<forall>i\<in>L. \<exists>x. x \<in> space (M i)"
```
```   129       using M.not_empty by auto
```
```   130     from bchoice[OF this]
```
```   131     show "(\<Pi>\<^isub>E i\<in>L. space (M i)) \<noteq> {}" by auto
```
```   132     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))"
```
```   133       using `emb L J X = emb L J Y` by (simp add: emb_def)
```
```   134   qed fact
```
```   135 qed
```
```   136
```
```   137 lemma (in product_prob_space) emb_id:
```
```   138   "B \<subseteq> (\<Pi>\<^isub>E i\<in>L. space (M i)) \<Longrightarrow> emb L L B = B"
```
```   139   by (auto simp: emb_def Pi_iff subset_eq extensional_restrict)
```
```   140
```
```   141 lemma (in product_prob_space) emb_simps:
```
```   142   shows "emb L K (A \<union> B) = emb L K A \<union> emb L K B"
```
```   143     and "emb L K (A \<inter> B) = emb L K A \<inter> emb L K B"
```
```   144     and "emb L K (A - B) = emb L K A - emb L K B"
```
```   145   by (auto simp: emb_def)
```
```   146
```
```   147 lemma (in product_prob_space) measurable_emb[intro,simp]:
```
```   148   assumes *: "J \<noteq> {}" "J \<subseteq> L" "finite L" "X \<in> sets (Pi\<^isub>M J M)"
```
```   149   shows "emb L J X \<in> sets (Pi\<^isub>M L M)"
```
```   150   using measurable_restrict[THEN measurable_sets, OF *] by (simp add: emb_def)
```
```   151
```
```   152 lemma (in product_prob_space) measure_emb[intro,simp]:
```
```   153   assumes *: "J \<noteq> {}" "J \<subseteq> L" "finite L" "X \<in> sets (Pi\<^isub>M J M)"
```
```   154   shows "measure (Pi\<^isub>M L M) (emb L J X) = measure (Pi\<^isub>M J M) X"
```
```   155   using measure_preserving_restrict[THEN measure_preservingD, OF *]
```
```   156   by (simp add: emb_def)
```
```   157
```
```   158 definition (in product_prob_space) generator :: "('i \<Rightarrow> 'a) measure_space" where
```
```   159   "generator = \<lparr>
```
```   160     space = (\<Pi>\<^isub>E i\<in>I. space (M i)),
```
```   161     sets = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M)),
```
```   162     measure = undefined
```
```   163   \<rparr>"
```
```   164
```
```   165 lemma (in product_prob_space) generatorI:
```
```   166   "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"
```
```   167   unfolding generator_def by auto
```
```   168
```
```   169 lemma (in product_prob_space) generatorI':
```
```   170   "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"
```
```   171   unfolding generator_def by auto
```
```   172
```
```   173 lemma (in product_sigma_finite)
```
```   174   assumes "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
```
```   175   shows measure_fold_integral:
```
```   176     "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)
```
```   177     and measure_fold_measurable:
```
```   178     "(\<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)
```
```   179 proof -
```
```   180   interpret I: finite_product_sigma_finite M I by default fact
```
```   181   interpret J: finite_product_sigma_finite M J by default fact
```
```   182   interpret IJ: pair_sigma_finite I.P J.P ..
```
```   183   show ?I
```
```   184     unfolding measure_fold[OF assms]
```
```   185     apply (subst IJ.pair_measure_alt)
```
```   186     apply (intro measurable_sets[OF _ A] measurable_merge assms)
```
```   187     apply (auto simp: vimage_compose[symmetric] comp_def space_pair_measure
```
```   188       intro!: I.positive_integral_cong)
```
```   189     done
```
```   190
```
```   191   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)"
```
```   192     by (intro measurable_sets[OF _ A] measurable_merge assms)
```
```   193   from IJ.measure_cut_measurable_fst[OF this]
```
```   194   show ?B
```
```   195     apply (auto simp: vimage_compose[symmetric] comp_def space_pair_measure)
```
```   196     apply (subst (asm) measurable_cong)
```
```   197     apply auto
```
```   198     done
```
```   199 qed
```
```   200
```
```   201 definition (in product_prob_space)
```
```   202   "\<mu>G A =
```
```   203     (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))"
```
```   204
```
```   205 lemma (in product_prob_space) \<mu>G_spec:
```
```   206   assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)"
```
```   207   shows "\<mu>G A = measure (Pi\<^isub>M J M) X"
```
```   208   unfolding \<mu>G_def
```
```   209 proof (intro the_equality allI impI ballI)
```
```   210   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)"
```
```   211   have "measure (Pi\<^isub>M K M) Y = measure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) K Y)"
```
```   212     using K J by simp
```
```   213   also have "emb (K \<union> J) K Y = emb (K \<union> J) J X"
```
```   214     using K J by (simp add: emb_injective[of "K \<union> J" I])
```
```   215   also have "measure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) J X) = measure (Pi\<^isub>M J M) X"
```
```   216     using K J by simp
```
```   217   finally show "measure (Pi\<^isub>M J M) X = measure (Pi\<^isub>M K M) Y" ..
```
```   218 qed (insert J, force)
```
```   219
```
```   220 lemma (in product_prob_space) \<mu>G_eq:
```
```   221   "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"
```
```   222   by (intro \<mu>G_spec) auto
```
```   223
```
```   224 lemma (in product_prob_space) generator_Ex:
```
```   225   assumes *: "A \<in> sets generator"
```
```   226   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"
```
```   227 proof -
```
```   228   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)"
```
```   229     unfolding generator_def by auto
```
```   230   with \<mu>G_spec[OF this] show ?thesis by auto
```
```   231 qed
```
```   232
```
```   233 lemma (in product_prob_space) generatorE:
```
```   234   assumes A: "A \<in> sets generator"
```
```   235   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"
```
```   236 proof -
```
```   237   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"
```
```   238     "\<mu>G A = measure (Pi\<^isub>M J M) X" by auto
```
```   239   then show thesis by (intro that) auto
```
```   240 qed
```
```   241
```
```   242 lemma (in product_prob_space) merge_sets:
```
```   243   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)"
```
```   244   shows "merge J x K -` A \<inter> space (Pi\<^isub>M K M) \<in> sets (Pi\<^isub>M K M)"
```
```   245 proof -
```
```   246   interpret J: finite_product_sigma_algebra M J by default fact
```
```   247   interpret K: finite_product_sigma_algebra M K by default fact
```
```   248   interpret JK: pair_sigma_algebra J.P K.P ..
```
```   249
```
```   250   from JK.measurable_cut_fst[OF
```
```   251     measurable_merge[THEN measurable_sets, OF `J \<inter> K = {}`], OF A, of x] x
```
```   252   show ?thesis
```
```   253       by (simp add: space_pair_measure comp_def vimage_compose[symmetric])
```
```   254 qed
```
```   255
```
```   256 lemma (in product_prob_space) merge_emb:
```
```   257   assumes "K \<subseteq> I" "J \<subseteq> I" and y: "y \<in> space (Pi\<^isub>M J M)"
```
```   258   shows "(merge J y (I - J) -` emb I K X \<inter> space (Pi\<^isub>M I M)) =
```
```   259     emb I (K - J) (merge J y (K - J) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M))"
```
```   260 proof -
```
```   261   have [simp]: "\<And>x J K L. merge J y K (restrict x L) = merge J y (K \<inter> L) x"
```
```   262     by (auto simp: restrict_def merge_def)
```
```   263   have [simp]: "\<And>x J K L. restrict (merge J y K x) L = merge (J \<inter> L) y (K \<inter> L) x"
```
```   264     by (auto simp: restrict_def merge_def)
```
```   265   have [simp]: "(I - J) \<inter> K = K - J" using `K \<subseteq> I` `J \<subseteq> I` by auto
```
```   266   have [simp]: "(K - J) \<inter> (K \<union> J) = K - J" by auto
```
```   267   have [simp]: "(K - J) \<inter> K = K - J" by auto
```
```   268   from y `K \<subseteq> I` `J \<subseteq> I` show ?thesis
```
```   269     by (simp split: split_merge add: emb_def Pi_iff extensional_merge_sub set_eq_iff) auto
```
```   270 qed
```
```   271
```
```   272 definition (in product_prob_space) infprod_algebra :: "('i \<Rightarrow> 'a) measure_space" where
```
```   273   "infprod_algebra = sigma generator \<lparr> measure :=
```
```   274     (SOME \<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and>
```
```   275        prob_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>)\<rparr>"
```
```   276
```
```   277 syntax
```
```   278   "_PiP"  :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme"  ("(3PIP _:_./ _)" 10)
```
```   279
```
```   280 syntax (xsymbols)
```
```   281   "_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme"  ("(3\<Pi>\<^isub>P _\<in>_./ _)"   10)
```
```   282
```
```   283 syntax (HTML output)
```
```   284   "_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme"  ("(3\<Pi>\<^isub>P _\<in>_./ _)"   10)
```
```   285
```
```   286 abbreviation
```
```   287   "Pi\<^isub>P I M \<equiv> product_prob_space.infprod_algebra M I"
```
```   288
```
```   289 translations
```
```   290   "PIP x:I. M" == "CONST Pi\<^isub>P I (%x. M)"
```
```   291
```
```   292 lemma (in product_prob_space) algebra_generator:
```
```   293   assumes "I \<noteq> {}" shows "algebra generator"
```
```   294 proof
```
```   295   let ?G = generator
```
```   296   show "sets ?G \<subseteq> Pow (space ?G)"
```
```   297     by (auto simp: generator_def emb_def)
```
```   298   from `I \<noteq> {}` obtain i where "i \<in> I" by auto
```
```   299   then show "{} \<in> sets ?G"
```
```   300     by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"]
```
```   301       simp: product_algebra_def sigma_def sigma_sets.Empty generator_def emb_def)
```
```   302   from `i \<in> I` show "space ?G \<in> sets ?G"
```
```   303     by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"]
```
```   304       simp: generator_def emb_def)
```
```   305   fix A assume "A \<in> sets ?G"
```
```   306   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"
```
```   307     by (auto simp: generator_def)
```
```   308   fix B assume "B \<in> sets ?G"
```
```   309   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"
```
```   310     by (auto simp: generator_def)
```
```   311   let ?RA = "emb (JA \<union> JB) JA XA"
```
```   312   let ?RB = "emb (JA \<union> JB) JB XB"
```
```   313   interpret JAB: finite_product_sigma_algebra M "JA \<union> JB"
```
```   314     by default (insert XA XB, auto)
```
```   315   have *: "A - B = emb I (JA \<union> JB) (?RA - ?RB)" "A \<union> B = emb I (JA \<union> JB) (?RA \<union> ?RB)"
```
```   316     using XA A XB B by (auto simp: emb_simps)
```
```   317   then show "A - B \<in> sets ?G" "A \<union> B \<in> sets ?G"
```
```   318     using XA XB by (auto intro!: generatorI')
```
```   319 qed
```
```   320
```
```   321 lemma (in product_prob_space) positive_\<mu>G:
```
```   322   assumes "I \<noteq> {}"
```
```   323   shows "positive generator \<mu>G"
```
```   324 proof -
```
```   325   interpret G!: algebra generator by (rule algebra_generator) fact
```
```   326   show ?thesis
```
```   327   proof (intro positive_def[THEN iffD2] conjI ballI)
```
```   328     from generatorE[OF G.empty_sets] guess J X . note this[simp]
```
```   329     interpret J: finite_product_sigma_finite M J by default fact
```
```   330     have "X = {}"
```
```   331       by (rule emb_injective[of J I]) simp_all
```
```   332     then show "\<mu>G {} = 0" by simp
```
```   333   next
```
```   334     fix A assume "A \<in> sets generator"
```
```   335     from generatorE[OF this] guess J X . note this[simp]
```
```   336     interpret J: finite_product_sigma_finite M J by default fact
```
```   337     show "0 \<le> \<mu>G A" by simp
```
```   338   qed
```
```   339 qed
```
```   340
```
```   341 lemma (in product_prob_space) additive_\<mu>G:
```
```   342   assumes "I \<noteq> {}"
```
```   343   shows "additive generator \<mu>G"
```
```   344 proof -
```
```   345   interpret G!: algebra generator by (rule algebra_generator) fact
```
```   346   show ?thesis
```
```   347   proof (intro additive_def[THEN iffD2] ballI impI)
```
```   348     fix A assume "A \<in> sets generator" with generatorE guess J X . note J = this
```
```   349     fix B assume "B \<in> sets generator" with generatorE guess K Y . note K = this
```
```   350     assume "A \<inter> B = {}"
```
```   351     have JK: "J \<union> K \<noteq> {}" "J \<union> K \<subseteq> I" "finite (J \<union> K)"
```
```   352       using J K by auto
```
```   353     interpret JK: finite_product_sigma_finite M "J \<union> K" by default fact
```
```   354     have JK_disj: "emb (J \<union> K) J X \<inter> emb (J \<union> K) K Y = {}"
```
```   355       apply (rule emb_injective[of "J \<union> K" I])
```
```   356       apply (insert `A \<inter> B = {}` JK J K)
```
```   357       apply (simp_all add: JK.Int emb_simps)
```
```   358       done
```
```   359     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)"
```
```   360       using J K by simp_all
```
```   361     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))"
```
```   362       by (simp add: emb_simps)
```
```   363     also have "\<dots> = measure (Pi\<^isub>M (J \<union> K) M) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)"
```
```   364       using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq JK.Un)
```
```   365     also have "\<dots> = \<mu>G A + \<mu>G B"
```
```   366       using J K JK_disj by (simp add: JK.measure_additive[symmetric])
```
```   367     finally show "\<mu>G (A \<union> B) = \<mu>G A + \<mu>G B" .
```
```   368   qed
```
```   369 qed
```
```   370
```
```   371 lemma (in product_prob_space) finite_index_eq_finite_product:
```
```   372   assumes "finite I"
```
```   373   shows "sets (sigma generator) = sets (Pi\<^isub>M I M)"
```
```   374 proof safe
```
```   375   interpret I: finite_product_sigma_algebra M I by default fact
```
```   376   have space_generator[simp]: "space generator = space (Pi\<^isub>M I M)"
```
```   377     by (simp add: generator_def product_algebra_def)
```
```   378   { fix A assume "A \<in> sets (sigma generator)"
```
```   379     then show "A \<in> sets I.P" unfolding sets_sigma
```
```   380     proof induct
```
```   381       case (Basic A)
```
```   382       from generatorE[OF this] guess J X . note J = this
```
```   383       with `finite I` have "emb I J X \<in> sets I.P" by auto
```
```   384       with `emb I J X = A` show "A \<in> sets I.P" by simp
```
```   385     qed auto }
```
```   386   { fix A assume A: "A \<in> sets I.P"
```
```   387     show "A \<in> sets (sigma generator)"
```
```   388     proof cases
```
```   389       assume "I = {}"
```
```   390       with I.P_empty[OF this] A
```
```   391       have "A = space generator \<or> A = {}"
```
```   392         unfolding space_generator by auto
```
```   393       then show ?thesis
```
```   394         by (auto simp: sets_sigma simp del: space_generator
```
```   395                  intro: sigma_sets.Empty sigma_sets_top)
```
```   396     next
```
```   397       assume "I \<noteq> {}"
```
```   398       note A this
```
```   399       moreover with I.sets_into_space have "emb I I A = A" by (intro emb_id) auto
```
```   400       ultimately show "A \<in> sets (sigma generator)"
```
```   401         using `finite I` unfolding sets_sigma
```
```   402         by (intro sigma_sets.Basic generatorI[of I A]) auto
```
```   403   qed }
```
```   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        prob_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>"
```
```   409 proof cases
```
```   410   assume "finite I"
```
```   411   interpret I: finite_product_prob_space 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     show "prob_space ?P"
```
```   426     proof
```
```   427       show "measure_space ?P" using `?P = I.P` by simp default
```
```   428       show "measure ?P (space ?P) = 1"
```
```   429         using I.measure_space_1 by simp
```
```   430     qed
```
```   431   qed
```
```   432 next
```
```   433   let ?G = generator
```
```   434   assume "\<not> finite I"
```
```   435   then have I_not_empty: "I \<noteq> {}" by auto
```
```   436   interpret G!: algebra generator by (rule algebra_generator) fact
```
```   437   note \<mu>G_mono =
```
```   438     G.additive_increasing[OF positive_\<mu>G[OF I_not_empty] additive_\<mu>G[OF I_not_empty], THEN increasingD]
```
```   439
```
```   440   { fix Z J assume J: "J \<noteq> {}" "finite J" "J \<subseteq> I" and Z: "Z \<in> sets ?G"
```
```   441
```
```   442     from `infinite I` `finite J` obtain k where k: "k \<in> I" "k \<notin> J"
```
```   443       by (metis rev_finite_subset subsetI)
```
```   444     moreover from Z guess K' X' by (rule generatorE)
```
```   445     moreover def K \<equiv> "insert k K'"
```
```   446     moreover def X \<equiv> "emb K K' X'"
```
```   447     ultimately have K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "X \<in> sets (Pi\<^isub>M K M)" "Z = emb I K X"
```
```   448       "K - J \<noteq> {}" "K - J \<subseteq> I" "\<mu>G Z = measure (Pi\<^isub>M K M) X"
```
```   449       by (auto simp: subset_insertI)
```
```   450
```
```   451     let ?M = "\<lambda>y. merge J y (K - J) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M)"
```
```   452     { fix y assume y: "y \<in> space (Pi\<^isub>M J M)"
```
```   453       note * = merge_emb[OF `K \<subseteq> I` `J \<subseteq> I` y, of X]
```
```   454       moreover
```
```   455       have **: "?M y \<in> sets (Pi\<^isub>M (K - J) M)"
```
```   456         using J K y by (intro merge_sets) auto
```
```   457       ultimately
```
```   458       have ***: "(merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M)) \<in> sets ?G"
```
```   459         using J K by (intro generatorI) auto
```
```   460       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)"
```
```   461         unfolding * using K J by (subst \<mu>G_eq[OF _ _ _ **]) auto
```
```   462       note * ** *** this }
```
```   463     note merge_in_G = this
```
```   464
```
```   465     have "finite (K - J)" using K by auto
```
```   466
```
```   467     interpret J: finite_product_prob_space M J by default fact+
```
```   468     interpret KmJ: finite_product_prob_space M "K - J" by default fact+
```
```   469
```
```   470     have "\<mu>G Z = measure (Pi\<^isub>M (J \<union> (K - J)) M) (emb (J \<union> (K - J)) K X)"
```
```   471       using K J by simp
```
```   472     also have "\<dots> = (\<integral>\<^isup>+ x. measure (Pi\<^isub>M (K - J) M) (?M x) \<partial>Pi\<^isub>M J M)"
```
```   473       using K J by (subst measure_fold_integral) auto
```
```   474     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)"
```
```   475       (is "_ = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)")
```
```   476     proof (intro J.positive_integral_cong)
```
```   477       fix x assume x: "x \<in> space (Pi\<^isub>M J M)"
```
```   478       with K merge_in_G(2)[OF this]
```
```   479       show "measure (Pi\<^isub>M (K - J) M) (?M x) = \<mu>G (?MZ x)"
```
```   480         unfolding `Z = emb I K X` merge_in_G(1)[OF x] by (subst \<mu>G_eq) auto
```
```   481     qed
```
```   482     finally have fold: "\<mu>G Z = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)" .
```
```   483
```
```   484     { fix x assume x: "x \<in> space (Pi\<^isub>M J M)"
```
```   485       then have "\<mu>G (?MZ x) \<le> 1"
```
```   486         unfolding merge_in_G(4)[OF x] `Z = emb I K X`
```
```   487         by (intro KmJ.measure_le_1 merge_in_G(2)[OF x]) }
```
```   488     note le_1 = this
```
```   489
```
```   490     let ?q = "\<lambda>y. \<mu>G (merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M))"
```
```   491     have "?q \<in> borel_measurable (Pi\<^isub>M J M)"
```
```   492       unfolding `Z = emb I K X` using J K merge_in_G(3)
```
```   493       by (simp add: merge_in_G  \<mu>G_eq measure_fold_measurable
```
```   494                del: space_product_algebra cong: measurable_cong)
```
```   495     note this fold le_1 merge_in_G(3) }
```
```   496   note fold = this
```
```   497
```
```   498   have "\<exists>\<mu>. (\<forall>s\<in>sets ?G. \<mu> s = \<mu>G s) \<and>
```
```   499     measure_space \<lparr>space = space ?G, sets = sets (sigma ?G), measure = \<mu>\<rparr>"
```
```   500     (is "\<exists>\<mu>. _ \<and> measure_space (?ms \<mu>)")
```
```   501   proof (rule G.caratheodory_empty_continuous[OF positive_\<mu>G additive_\<mu>G])
```
```   502     fix A assume "A \<in> sets ?G"
```
```   503     with generatorE guess J X . note JX = this
```
```   504     interpret JK: finite_product_prob_space M J by default fact+
```
```   505     from JX show "\<mu>G A \<noteq> \<infinity>" by simp
```
```   506   next
```
```   507     fix A assume A: "range A \<subseteq> sets ?G" "decseq A" "(\<Inter>i. A i) = {}"
```
```   508     then have "decseq (\<lambda>i. \<mu>G (A i))"
```
```   509       by (auto intro!: \<mu>G_mono simp: decseq_def)
```
```   510     moreover
```
```   511     have "(INF i. \<mu>G (A i)) = 0"
```
```   512     proof (rule ccontr)
```
```   513       assume "(INF i. \<mu>G (A i)) \<noteq> 0" (is "?a \<noteq> 0")
```
```   514       moreover have "0 \<le> ?a"
```
```   515         using A positive_\<mu>G[OF I_not_empty] by (auto intro!: INF_greatest simp: positive_def)
```
```   516       ultimately have "0 < ?a" by auto
```
```   517
```
```   518       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"
```
```   519         using A by (intro allI generator_Ex) auto
```
```   520       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)"
```
```   521         and A': "\<And>n. A n = emb I (J' n) (X' n)"
```
```   522         unfolding choice_iff by blast
```
```   523       moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)"
```
```   524       moreover def X \<equiv> "\<lambda>n. emb (J n) (J' n) (X' n)"
```
```   525       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)"
```
```   526         by auto
```
```   527       with A' have A_eq: "\<And>n. A n = emb I (J n) (X n)" "\<And>n. A n \<in> sets ?G"
```
```   528         unfolding J_def X_def by (subst emb_trans) (insert A, auto)
```
```   529
```
```   530       have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
```
```   531         unfolding J_def by force
```
```   532
```
```   533       interpret J: finite_product_prob_space M "J i" for i by default fact+
```
```   534
```
```   535       have a_le_1: "?a \<le> 1"
```
```   536         using \<mu>G_spec[of "J 0" "A 0" "X 0"] J A_eq
```
```   537         by (auto intro!: INF_lower2[of 0] J.measure_le_1)
```
```   538
```
```   539       let ?M = "\<lambda>K Z y. merge K y (I - K) -` Z \<inter> space (Pi\<^isub>M I M)"
```
```   540
```
```   541       { fix Z k assume Z: "range Z \<subseteq> sets ?G" "decseq Z" "\<forall>n. ?a / 2^k \<le> \<mu>G (Z n)"
```
```   542         then have Z_sets: "\<And>n. Z n \<in> sets ?G" by auto
```
```   543         fix J' assume J': "J' \<noteq> {}" "finite J'" "J' \<subseteq> I"
```
```   544         interpret J': finite_product_prob_space M J' by default fact+
```
```   545
```
```   546         let ?q = "\<lambda>n y. \<mu>G (?M J' (Z n) y)"
```
```   547         let ?Q = "\<lambda>n. ?q n -` {?a / 2^(k+1) ..} \<inter> space (Pi\<^isub>M J' M)"
```
```   548         { fix n
```
```   549           have "?q n \<in> borel_measurable (Pi\<^isub>M J' M)"
```
```   550             using Z J' by (intro fold(1)) auto
```
```   551           then have "?Q n \<in> sets (Pi\<^isub>M J' M)"
```
```   552             by (rule measurable_sets) auto }
```
```   553         note Q_sets = this
```
```   554
```
```   555         have "?a / 2^(k+1) \<le> (INF n. measure (Pi\<^isub>M J' M) (?Q n))"
```
```   556         proof (intro INF_greatest)
```
```   557           fix n
```
```   558           have "?a / 2^k \<le> \<mu>G (Z n)" using Z by auto
```
```   559           also have "\<dots> \<le> (\<integral>\<^isup>+ x. indicator (?Q n) x + ?a / 2^(k+1) \<partial>Pi\<^isub>M J' M)"
```
```   560             unfolding fold(2)[OF J' `Z n \<in> sets ?G`]
```
```   561           proof (intro J'.positive_integral_mono)
```
```   562             fix x assume x: "x \<in> space (Pi\<^isub>M J' M)"
```
```   563             then have "?q n x \<le> 1 + 0"
```
```   564               using J' Z fold(3) Z_sets by auto
```
```   565             also have "\<dots> \<le> 1 + ?a / 2^(k+1)"
```
```   566               using `0 < ?a` by (intro add_mono) auto
```
```   567             finally have "?q n x \<le> 1 + ?a / 2^(k+1)" .
```
```   568             with x show "?q n x \<le> indicator (?Q n) x + ?a / 2^(k+1)"
```
```   569               by (auto split: split_indicator simp del: power_Suc)
```
```   570           qed
```
```   571           also have "\<dots> = measure (Pi\<^isub>M J' M) (?Q n) + ?a / 2^(k+1)"
```
```   572             using `0 \<le> ?a` Q_sets J'.measure_space_1
```
```   573             by (subst J'.positive_integral_add) auto
```
```   574           finally show "?a / 2^(k+1) \<le> measure (Pi\<^isub>M J' M) (?Q n)" using `?a \<le> 1`
```
```   575             by (cases rule: ereal2_cases[of ?a "measure (Pi\<^isub>M J' M) (?Q n)"])
```
```   576                (auto simp: field_simps)
```
```   577         qed
```
```   578         also have "\<dots> = measure (Pi\<^isub>M J' M) (\<Inter>n. ?Q n)"
```
```   579         proof (intro J'.continuity_from_above)
```
```   580           show "range ?Q \<subseteq> sets (Pi\<^isub>M J' M)" using Q_sets by auto
```
```   581           show "decseq ?Q"
```
```   582             unfolding decseq_def
```
```   583           proof (safe intro!: vimageI[OF refl])
```
```   584             fix m n :: nat assume "m \<le> n"
```
```   585             fix x assume x: "x \<in> space (Pi\<^isub>M J' M)"
```
```   586             assume "?a / 2^(k+1) \<le> ?q n x"
```
```   587             also have "?q n x \<le> ?q m x"
```
```   588             proof (rule \<mu>G_mono)
```
```   589               from fold(4)[OF J', OF Z_sets x]
```
```   590               show "?M J' (Z n) x \<in> sets ?G" "?M J' (Z m) x \<in> sets ?G" by auto
```
```   591               show "?M J' (Z n) x \<subseteq> ?M J' (Z m) x"
```
```   592                 using `decseq Z`[THEN decseqD, OF `m \<le> n`] by auto
```
```   593             qed
```
```   594             finally show "?a / 2^(k+1) \<le> ?q m x" .
```
```   595           qed
```
```   596         qed (intro J'.finite_measure Q_sets)
```
```   597         finally have "(\<Inter>n. ?Q n) \<noteq> {}"
```
```   598           using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq)
```
```   599         then have "\<exists>w\<in>space (Pi\<^isub>M J' M). \<forall>n. ?a / 2 ^ (k + 1) \<le> ?q n w" by auto }
```
```   600       note Ex_w = this
```
```   601
```
```   602       let ?q = "\<lambda>k n y. \<mu>G (?M (J k) (A n) y)"
```
```   603
```
```   604       have "\<forall>n. ?a / 2 ^ 0 \<le> \<mu>G (A n)" by (auto intro: INF_lower)
```
```   605       from Ex_w[OF A(1,2) this J(1-3), of 0] guess w0 .. note w0 = this
```
```   606
```
```   607       let ?P =
```
```   608         "\<lambda>k wk w. w \<in> space (Pi\<^isub>M (J (Suc k)) M) \<and> restrict w (J k) = wk \<and>
```
```   609           (\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n w)"
```
```   610       def w \<equiv> "nat_rec w0 (\<lambda>k wk. Eps (?P k wk))"
```
```   611
```
```   612       { fix k have w: "w k \<in> space (Pi\<^isub>M (J k) M) \<and>
```
```   613           (\<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))"
```
```   614         proof (induct k)
```
```   615           case 0 with w0 show ?case
```
```   616             unfolding w_def nat_rec_0 by auto
```
```   617         next
```
```   618           case (Suc k)
```
```   619           then have wk: "w k \<in> space (Pi\<^isub>M (J k) M)" by auto
```
```   620           have "\<exists>w'. ?P k (w k) w'"
```
```   621           proof cases
```
```   622             assume [simp]: "J k = J (Suc k)"
```
```   623             show ?thesis
```
```   624             proof (intro exI[of _ "w k"] conjI allI)
```
```   625               fix n
```
```   626               have "?a / 2 ^ (Suc k + 1) \<le> ?a / 2 ^ (k + 1)"
```
```   627                 using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: field_simps)
```
```   628               also have "\<dots> \<le> ?q k n (w k)" using Suc by auto
```
```   629               finally show "?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n (w k)" by simp
```
```   630             next
```
```   631               show "w k \<in> space (Pi\<^isub>M (J (Suc k)) M)"
```
```   632                 using Suc by simp
```
```   633               then show "restrict (w k) (J k) = w k"
```
```   634                 by (simp add: extensional_restrict)
```
```   635             qed
```
```   636           next
```
```   637             assume "J k \<noteq> J (Suc k)"
```
```   638             with J_mono[of k "Suc k"] have "J (Suc k) - J k \<noteq> {}" (is "?D \<noteq> {}") by auto
```
```   639             have "range (\<lambda>n. ?M (J k) (A n) (w k)) \<subseteq> sets ?G"
```
```   640               "decseq (\<lambda>n. ?M (J k) (A n) (w k))"
```
```   641               "\<forall>n. ?a / 2 ^ (k + 1) \<le> \<mu>G (?M (J k) (A n) (w k))"
```
```   642               using `decseq A` fold(4)[OF J(1-3) A_eq(2), of "w k" k] Suc
```
```   643               by (auto simp: decseq_def)
```
```   644             from Ex_w[OF this `?D \<noteq> {}`] J[of "Suc k"]
```
```   645             obtain w' where w': "w' \<in> space (Pi\<^isub>M ?D M)"
```
```   646               "\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> \<mu>G (?M ?D (?M (J k) (A n) (w k)) w')" by auto
```
```   647             let ?w = "merge (J k) (w k) ?D w'"
```
```   648             have [simp]: "\<And>x. merge (J k) (w k) (I - J k) (merge ?D w' (I - ?D) x) =
```
```   649               merge (J (Suc k)) ?w (I - (J (Suc k))) x"
```
```   650               using J(3)[of "Suc k"] J(3)[of k] J_mono[of k "Suc k"]
```
```   651               by (auto intro!: ext split: split_merge)
```
```   652             have *: "\<And>n. ?M ?D (?M (J k) (A n) (w k)) w' = ?M (J (Suc k)) (A n) ?w"
```
```   653               using w'(1) J(3)[of "Suc k"]
```
```   654               by (auto split: split_merge intro!: extensional_merge_sub) force+
```
```   655             show ?thesis
```
```   656               apply (rule exI[of _ ?w])
```
```   657               using w' J_mono[of k "Suc k"] wk unfolding *
```
```   658               apply (auto split: split_merge intro!: extensional_merge_sub ext)
```
```   659               apply (force simp: extensional_def)
```
```   660               done
```
```   661           qed
```
```   662           then have "?P k (w k) (w (Suc k))"
```
```   663             unfolding w_def nat_rec_Suc unfolding w_def[symmetric]
```
```   664             by (rule someI_ex)
```
```   665           then show ?case by auto
```
```   666         qed
```
```   667         moreover
```
```   668         then have "w k \<in> space (Pi\<^isub>M (J k) M)" by auto
```
```   669         moreover
```
```   670         from w have "?a / 2 ^ (k + 1) \<le> ?q k k (w k)" by auto
```
```   671         then have "?M (J k) (A k) (w k) \<noteq> {}"
```
```   672           using positive_\<mu>G[OF I_not_empty, unfolded positive_def] `0 < ?a` `?a \<le> 1`
```
```   673           by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq)
```
```   674         then obtain x where "x \<in> ?M (J k) (A k) (w k)" by auto
```
```   675         then have "merge (J k) (w k) (I - J k) x \<in> A k" by auto
```
```   676         then have "\<exists>x\<in>A k. restrict x (J k) = w k"
```
```   677           using `w k \<in> space (Pi\<^isub>M (J k) M)`
```
```   678           by (intro rev_bexI) (auto intro!: ext simp: extensional_def)
```
```   679         ultimately have "w k \<in> space (Pi\<^isub>M (J k) M)"
```
```   680           "\<exists>x\<in>A k. restrict x (J k) = w k"
```
```   681           "k \<noteq> 0 \<Longrightarrow> restrict (w k) (J (k - 1)) = w (k - 1)"
```
```   682           by auto }
```
```   683       note w = this
```
```   684
```
```   685       { fix k l i assume "k \<le> l" "i \<in> J k"
```
```   686         { fix l have "w k i = w (k + l) i"
```
```   687           proof (induct l)
```
```   688             case (Suc l)
```
```   689             from `i \<in> J k` J_mono[of k "k + l"] have "i \<in> J (k + l)" by auto
```
```   690             with w(3)[of "k + Suc l"]
```
```   691             have "w (k + l) i = w (k + Suc l) i"
```
```   692               by (auto simp: restrict_def fun_eq_iff split: split_if_asm)
```
```   693             with Suc show ?case by simp
```
```   694           qed simp }
```
```   695         from this[of "l - k"] `k \<le> l` have "w l i = w k i" by simp }
```
```   696       note w_mono = this
```
```   697
```
```   698       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"
```
```   699       { fix i k assume k: "i \<in> J k"
```
```   700         have "w k i = w (LEAST k. i \<in> J k) i"
```
```   701           by (intro w_mono Least_le k LeastI[of _ k])
```
```   702         then have "w' i = w k i"
```
```   703           unfolding w'_def using k by auto }
```
```   704       note w'_eq = this
```
```   705       have w'_simps1: "\<And>i. i \<notin> I \<Longrightarrow> w' i = undefined"
```
```   706         using J by (auto simp: w'_def)
```
```   707       have w'_simps2: "\<And>i. i \<notin> (\<Union>k. J k) \<Longrightarrow> i \<in> I \<Longrightarrow> w' i \<in> space (M i)"
```
```   708         using J by (auto simp: w'_def intro!: someI_ex[OF M.not_empty[unfolded ex_in_conv[symmetric]]])
```
```   709       { fix i assume "i \<in> I" then have "w' i \<in> space (M i)"
```
```   710           using w(1) by (cases "i \<in> (\<Union>k. J k)") (force simp: w'_simps2 w'_eq)+ }
```
```   711       note w'_simps[simp] = w'_eq w'_simps1 w'_simps2 this
```
```   712
```
```   713       have w': "w' \<in> space (Pi\<^isub>M I M)"
```
```   714         using w(1) by (auto simp add: Pi_iff extensional_def)
```
```   715
```
```   716       { fix n
```
```   717         have "restrict w' (J n) = w n" using w(1)
```
```   718           by (auto simp add: fun_eq_iff restrict_def Pi_iff extensional_def)
```
```   719         with w[of n] obtain x where "x \<in> A n" "restrict x (J n) = restrict w' (J n)" by auto
```
```   720         then have "w' \<in> A n" unfolding A_eq using w' by (auto simp: emb_def) }
```
```   721       then have "w' \<in> (\<Inter>i. A i)" by auto
```
```   722       with `(\<Inter>i. A i) = {}` show False by auto
```
```   723     qed
```
```   724     ultimately show "(\<lambda>i. \<mu>G (A i)) ----> 0"
```
```   725       using LIMSEQ_ereal_INFI[of "\<lambda>i. \<mu>G (A i)"] by simp
```
```   726   qed fact+
```
```   727   then guess \<mu> .. note \<mu> = this
```
```   728   show ?thesis
```
```   729   proof (intro exI[of _ \<mu>] conjI)
```
```   730     show "\<forall>S\<in>sets ?G. \<mu> S = \<mu>G S" using \<mu> by simp
```
```   731     show "prob_space (?ms \<mu>)"
```
```   732     proof
```
```   733       show "measure_space (?ms \<mu>)" using \<mu> by simp
```
```   734       obtain i where "i \<in> I" using I_not_empty by auto
```
```   735       interpret i: finite_product_sigma_finite M "{i}" by default auto
```
```   736       let ?X = "\<Pi>\<^isub>E i\<in>{i}. space (M i)"
```
```   737       have X: "?X \<in> sets (Pi\<^isub>M {i} M)"
```
```   738         by auto
```
```   739       with `i \<in> I` have "emb I {i} ?X \<in> sets generator"
```
```   740         by (intro generatorI') auto
```
```   741       with \<mu> have "\<mu> (emb I {i} ?X) = \<mu>G (emb I {i} ?X)" by auto
```
```   742       with \<mu>G_eq[OF _ _ _ X] `i \<in> I`
```
```   743       have "\<mu> (emb I {i} ?X) = measure (M i) (space (M i))"
```
```   744         by (simp add: i.measure_times)
```
```   745       also have "emb I {i} ?X = space (Pi\<^isub>P I M)"
```
```   746         using `i \<in> I` by (auto simp: emb_def infprod_algebra_def generator_def)
```
```   747       finally show "measure (?ms \<mu>) (space (?ms \<mu>)) = 1"
```
```   748         using M.measure_space_1 by (simp add: infprod_algebra_def)
```
```   749     qed
```
```   750   qed
```
```   751 qed
```
```   752
```
```   753 lemma (in product_prob_space) infprod_spec:
```
```   754   "(\<forall>s\<in>sets generator. measure (Pi\<^isub>P I M) s = \<mu>G s) \<and> prob_space (Pi\<^isub>P I M)"
```
```   755   (is "?Q infprod_algebra")
```
```   756   unfolding infprod_algebra_def
```
```   757   by (rule someI2_ex[OF extend_\<mu>G])
```
```   758      (auto simp: sigma_def generator_def)
```
```   759
```
```   760 sublocale product_prob_space \<subseteq> P: prob_space "Pi\<^isub>P I M"
```
```   761   using infprod_spec by simp
```
```   762
```
```   763 lemma (in product_prob_space) measure_infprod_emb:
```
```   764   assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)"
```
```   765   shows "\<mu> (emb I J X) = measure (Pi\<^isub>M J M) X"
```
```   766 proof -
```
```   767   have "emb I J X \<in> sets generator"
```
```   768     using assms by (rule generatorI')
```
```   769   with \<mu>G_eq[OF assms] infprod_spec show ?thesis by auto
```
```   770 qed
```
```   771
```
```   772 lemma (in product_prob_space) measurable_component:
```
```   773   assumes "i \<in> I"
```
```   774   shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>P I M) (M i)"
```
```   775 proof (unfold measurable_def, safe)
```
```   776   fix x assume "x \<in> space (Pi\<^isub>P I M)"
```
```   777   then show "x i \<in> space (M i)"
```
```   778     using `i \<in> I` by (auto simp: infprod_algebra_def generator_def)
```
```   779 next
```
```   780   fix A assume "A \<in> sets (M i)"
```
```   781   with `i \<in> I` have
```
```   782     "(\<Pi>\<^isub>E x \<in> {i}. A) \<in> sets (Pi\<^isub>M {i} M)"
```
```   783     "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) = emb I {i} (\<Pi>\<^isub>E x \<in> {i}. A)"
```
```   784     by (auto simp: infprod_algebra_def generator_def emb_def)
```
```   785   from generatorI[OF _ _ _ this] `i \<in> I`
```
```   786   show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) \<in> sets (Pi\<^isub>P I M)"
```
```   787     unfolding infprod_algebra_def by auto
```
```   788 qed
```
```   789
```
```   790 lemma (in product_prob_space) emb_in_infprod_algebra[intro]:
```
```   791   fixes J assumes J: "finite J" "J \<subseteq> I" and X: "X \<in> sets (Pi\<^isub>M J M)"
```
```   792   shows "emb I J X \<in> sets (\<Pi>\<^isub>P i\<in>I. M i)"
```
```   793 proof cases
```
```   794   assume "J = {}"
```
```   795   with X have "emb I J X = space (\<Pi>\<^isub>P i\<in>I. M i) \<or> emb I J X = {}"
```
```   796     by (auto simp: emb_def infprod_algebra_def generator_def
```
```   797                    product_algebra_def product_algebra_generator_def image_constant sigma_def)
```
```   798   then show ?thesis by auto
```
```   799 next
```
```   800   assume "J \<noteq> {}"
```
```   801   show ?thesis unfolding infprod_algebra_def
```
```   802     by simp (intro in_sigma generatorI'  `J \<noteq> {}` J X)
```
```   803 qed
```
```   804
```
```   805 lemma (in product_prob_space) finite_measure_infprod_emb:
```
```   806   assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)"
```
```   807   shows "\<mu>' (emb I J X) = finite_measure.\<mu>' (Pi\<^isub>M J M) X"
```
```   808 proof -
```
```   809   interpret J: finite_product_prob_space M J by default fact+
```
```   810   from assms have "emb I J X \<in> sets (Pi\<^isub>P I M)" by auto
```
```   811   with assms show "\<mu>' (emb I J X) = J.\<mu>' X"
```
```   812     unfolding \<mu>'_def J.\<mu>'_def
```
```   813     unfolding measure_infprod_emb[OF assms]
```
```   814     by auto
```
```   815 qed
```
```   816
```
```   817 lemma (in finite_product_prob_space) finite_measure_times:
```
```   818   assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)"
```
```   819   shows "\<mu>' (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu>' i (A i))"
```
```   820   using assms
```
```   821   unfolding \<mu>'_def M.\<mu>'_def
```
```   822   by (subst measure_times[OF assms])
```
```   823      (auto simp: finite_measure_eq M.finite_measure_eq setprod_ereal)
```
```   824
```
```   825 lemma (in product_prob_space) finite_measure_infprod_emb_Pi:
```
```   826   assumes J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> X j \<in> sets (M j)"
```
```   827   shows "\<mu>' (emb I J (Pi\<^isub>E J X)) = (\<Prod>j\<in>J. M.\<mu>' j (X j))"
```
```   828 proof cases
```
```   829   assume "J = {}"
```
```   830   then have "emb I J (Pi\<^isub>E J X) = space infprod_algebra"
```
```   831     by (auto simp: infprod_algebra_def generator_def sigma_def emb_def)
```
```   832   then show ?thesis using `J = {}` P.prob_space
```
```   833     by simp
```
```   834 next
```
```   835   assume "J \<noteq> {}"
```
```   836   interpret J: finite_product_prob_space M J by default fact+
```
```   837   have "(\<Prod>i\<in>J. M.\<mu>' i (X i)) = J.\<mu>' (Pi\<^isub>E J X)"
```
```   838     using J `J \<noteq> {}` by (subst J.finite_measure_times) auto
```
```   839   also have "\<dots> = \<mu>' (emb I J (Pi\<^isub>E J X))"
```
```   840     using J `J \<noteq> {}` by (intro finite_measure_infprod_emb[symmetric]) auto
```
```   841   finally show ?thesis by simp
```
```   842 qed
```
```   843
```
```   844 lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
```
```   845 proof
```
```   846   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
```
```   847     by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros)
```
```   848 qed
```
```   849
```
```   850 lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
```
```   851 proof
```
```   852   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
```
```   853     by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros)
```
```   854 qed
```
```   855
```
```   856 lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"
```
```   857   by (auto intro: sigma_sets.Basic)
```
```   858
```
```   859 lemma (in product_prob_space) infprod_algebra_alt:
```
```   860   "Pi\<^isub>P I M = sigma \<lparr> space = space (Pi\<^isub>P I M),
```
```   861     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))),
```
```   862     measure = measure (Pi\<^isub>P I M) \<rparr>"
```
```   863   (is "_ = sigma \<lparr> space = ?O, sets = ?M, measure = ?m \<rparr>")
```
```   864 proof (rule measure_space.equality)
```
```   865   let ?G = "\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M)"
```
```   866   have "sigma_sets ?O ?M = sigma_sets ?O ?G"
```
```   867   proof (intro equalityI sigma_sets_mono UN_least)
```
```   868     fix J assume J: "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}"
```
```   869     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
```
```   870     also have "\<dots> \<subseteq> ?G" using J by (rule UN_upper)
```
```   871     also have "\<dots> \<subseteq> sigma_sets ?O ?G" by (rule sigma_sets_superset_generator)
```
```   872     finally show "emb I J ` Pi\<^isub>E J ` (\<Pi> i\<in>J. sets (M i)) \<subseteq> sigma_sets ?O ?G" .
```
```   873     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)))"
```
```   874       by (simp add: sets_sigma product_algebra_generator_def product_algebra_def)
```
```   875     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)))"
```
```   876       using J M.sets_into_space
```
```   877       by (auto simp: emb_def [abs_def] intro!: sigma_sets_vimage[symmetric]) blast
```
```   878     also have "\<dots> \<subseteq> sigma_sets (space (Pi\<^isub>M I M)) ?M"
```
```   879       using J by (intro sigma_sets_mono') auto
```
```   880     finally show "emb I J ` sets (Pi\<^isub>M J M) \<subseteq> sigma_sets ?O ?M"
```
```   881       by (simp add: infprod_algebra_def generator_def)
```
```   882   qed
```
```   883   then show "sets (Pi\<^isub>P I M) = sets (sigma \<lparr> space = ?O, sets = ?M, measure = ?m \<rparr>)"
```
```   884     by (simp_all add: infprod_algebra_def generator_def sets_sigma)
```
```   885 qed simp_all
```
```   886
```
```   887 lemma (in product_prob_space) infprod_algebra_alt2:
```
```   888   "Pi\<^isub>P I M = sigma \<lparr> space = space (Pi\<^isub>P I M),
```
```   889     sets = (\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (M i)),
```
```   890     measure = measure (Pi\<^isub>P I M) \<rparr>"
```
```   891   (is "_ = ?S")
```
```   892 proof (rule measure_space.equality)
```
```   893   let "sigma \<lparr> space = ?O, sets = ?A, \<dots> = _ \<rparr>" = ?S
```
```   894   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)))"
```
```   895   have "sets (Pi\<^isub>P I M) = sigma_sets ?O ?G"
```
```   896     by (subst infprod_algebra_alt) (simp add: sets_sigma)
```
```   897   also have "\<dots> = sigma_sets ?O ?A"
```
```   898   proof (intro equalityI sigma_sets_mono subsetI)
```
```   899     interpret A: sigma_algebra ?S
```
```   900       by (rule sigma_algebra_sigma) auto
```
```   901     fix A assume "A \<in> ?G"
```
```   902     then obtain J B where "finite J" "J \<noteq> {}" "J \<subseteq> I" "A = emb I J (Pi\<^isub>E J B)"
```
```   903         and B: "\<And>i. i \<in> J \<Longrightarrow> B i \<in> sets (M i)"
```
```   904       by auto
```
```   905     then have A: "A = (\<Inter>j\<in>J. (\<lambda>x. x j) -` (B j) \<inter> space (Pi\<^isub>P I M))"
```
```   906       by (auto simp: emb_def infprod_algebra_def generator_def Pi_iff)
```
```   907     { fix j assume "j\<in>J"
```
```   908       with `J \<subseteq> I` have "j \<in> I" by auto
```
```   909       with `j \<in> J` B have "(\<lambda>x. x j) -` (B j) \<inter> space (Pi\<^isub>P I M) \<in> sets ?S"
```
```   910         by (auto simp: sets_sigma intro: sigma_sets.Basic) }
```
```   911     with `finite J` `J \<noteq> {}` have "A \<in> sets ?S"
```
```   912       unfolding A by (intro A.finite_INT) auto
```
```   913     then show "A \<in> sigma_sets ?O ?A" by (simp add: sets_sigma)
```
```   914   next
```
```   915     fix A assume "A \<in> ?A"
```
```   916     then obtain i B where i: "i \<in> I" "B \<in> sets (M i)"
```
```   917         and "A = (\<lambda>x. x i) -` B \<inter> space (Pi\<^isub>P I M)"
```
```   918       by auto
```
```   919     then have "A = emb I {i} (Pi\<^isub>E {i} (\<lambda>_. B))"
```
```   920       by (auto simp: emb_def infprod_algebra_def generator_def Pi_iff)
```
```   921     with i show "A \<in> sigma_sets ?O ?G"
```
```   922       by (intro sigma_sets.Basic UN_I[where a="{i}"]) auto
```
```   923   qed
```
```   924   also have "\<dots> = sets ?S"
```
```   925     by (simp add: sets_sigma)
```
```   926   finally show "sets (Pi\<^isub>P I M) = sets ?S" .
```
```   927 qed simp_all
```
```   928
```
```   929 lemma (in product_prob_space) measurable_into_infprod_algebra:
```
```   930   assumes "sigma_algebra N"
```
```   931   assumes f: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)"
```
```   932   assumes ext: "\<And>x. x \<in> space N \<Longrightarrow> f x \<in> extensional I"
```
```   933   shows "f \<in> measurable N (Pi\<^isub>P I M)"
```
```   934 proof -
```
```   935   interpret N: sigma_algebra N by fact
```
```   936   have f_in: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f x i) \<in> space N \<rightarrow> space (M i)"
```
```   937     using f by (auto simp: measurable_def)
```
```   938   { fix i A assume i: "i \<in> I" "A \<in> sets (M i)"
```
```   939     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"
```
```   940       using f_in ext by (auto simp: infprod_algebra_def generator_def)
```
```   941     also have "\<dots> \<in> sets N"
```
```   942       by (rule measurable_sets f i)+
```
```   943     finally have "f -` (\<lambda>x. x i) -` A \<inter> f -` space infprod_algebra \<inter> space N \<in> sets N" . }
```
```   944   with f_in ext show ?thesis
```
```   945     by (subst infprod_algebra_alt2)
```
```   946        (auto intro!: N.measurable_sigma simp: Pi_iff infprod_algebra_def generator_def)
```
```   947 qed
```
```   948
```
```   949 lemma (in product_prob_space) measurable_singleton_infprod:
```
```   950   assumes "i \<in> I"
```
```   951   shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>P I M) (M i)"
```
```   952 proof (unfold measurable_def, intro CollectI conjI ballI)
```
```   953   show "(\<lambda>x. x i) \<in> space (Pi\<^isub>P I M) \<rightarrow> space (M i)"
```
```   954     using M.sets_into_space `i \<in> I`
```
```   955     by (auto simp: infprod_algebra_def generator_def)
```
```   956   fix A assume "A \<in> sets (M i)"
```
```   957   have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) = emb I {i} (\<Pi>\<^isub>E _\<in>{i}. A)"
```
```   958     by (auto simp: infprod_algebra_def generator_def emb_def)
```
```   959   also have "\<dots> \<in> sets (Pi\<^isub>P I M)"
```
```   960     using `i \<in> I` `A \<in> sets (M i)`
```
```   961     by (intro emb_in_infprod_algebra product_algebraI) auto
```
```   962   finally show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) \<in> sets (Pi\<^isub>P I M)" .
```
```   963 qed
```
```   964
```
```   965 lemma (in product_prob_space) sigma_product_algebra_sigma_eq:
```
```   966   assumes M: "\<And>i. i \<in> I \<Longrightarrow> M i = sigma (E i)"
```
```   967   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))"
```
```   968 proof -
```
```   969   let ?E = "(\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (E i))"
```
```   970   let ?M = "(\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (M i))"
```
```   971   { fix i A assume "i\<in>I" "A \<in> sets (E i)"
```
```   972     then have "A \<in> sets (M i)" using M by auto
```
```   973     then have "A \<in> Pow (space (M i))" using M.sets_into_space by auto
```
```   974     then have "A \<in> Pow (space (E i))" using M[OF `i \<in> I`] by auto }
```
```   975   moreover
```
```   976   have "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) \<in> space infprod_algebra \<rightarrow> space (E i)"
```
```   977     by (auto simp: M infprod_algebra_def generator_def Pi_iff)
```
```   978   ultimately have "sigma_sets (space (Pi\<^isub>P I M)) ?M \<subseteq> sigma_sets (space (Pi\<^isub>P I M)) ?E"
```
```   979     apply (intro sigma_sets_mono UN_least)
```
```   980     apply (simp add: sets_sigma M)
```
```   981     apply (subst sigma_sets_vimage[symmetric])
```
```   982     apply (auto intro!: sigma_sets_mono')
```
```   983     done
```
```   984   moreover have "sigma_sets (space (Pi\<^isub>P I M)) ?E \<subseteq> sigma_sets (space (Pi\<^isub>P I M)) ?M"
```
```   985     by (intro sigma_sets_mono') (auto simp: M)
```
```   986   ultimately show ?thesis
```
```   987     by (subst infprod_algebra_alt2) (auto simp: sets_sigma)
```
```   988 qed
```
```   989
```
```   990 lemma (in product_prob_space) Int_proj_eq_emb:
```
```   991   assumes "J \<noteq> {}" "J \<subseteq> I"
```
```   992   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)"
```
```   993   using assms by (auto simp: infprod_algebra_def generator_def emb_def Pi_iff)
```
```   994
```
```   995 lemma (in product_prob_space) emb_insert:
```
```   996   "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)) =
```
```   997     emb I (insert i J) (Pi\<^isub>E (insert i J) (f(i := A)))"
```
```   998   by (auto simp: emb_def Pi_iff infprod_algebra_def generator_def split: split_if_asm)
```
```   999
```
```  1000 subsection {* Sequence space *}
```
```  1001
```
```  1002 locale sequence_space = product_prob_space M "UNIV :: nat set" for M
```
```  1003
```
```  1004 lemma (in sequence_space) infprod_in_sets[intro]:
```
```  1005   fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
```
```  1006   shows "Pi UNIV E \<in> sets (Pi\<^isub>P UNIV M)"
```
```  1007 proof -
```
```  1008   have "Pi UNIV E = (\<Inter>i. emb UNIV {..i} (\<Pi>\<^isub>E j\<in>{..i}. E j))"
```
```  1009     using E E[THEN M.sets_into_space]
```
```  1010     by (auto simp: emb_def Pi_iff extensional_def) blast
```
```  1011   with E show ?thesis
```
```  1012     by (auto intro: emb_in_infprod_algebra)
```
```  1013 qed
```
```  1014
```
```  1015 lemma (in sequence_space) measure_infprod:
```
```  1016   fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
```
```  1017   shows "(\<lambda>n. \<Prod>i\<le>n. M.\<mu>' i (E i)) ----> \<mu>' (Pi UNIV E)"
```
```  1018 proof -
```
```  1019   let ?E = "\<lambda>n. emb UNIV {..n} (Pi\<^isub>E {.. n} E)"
```
```  1020   { fix n :: nat
```
```  1021     interpret n: finite_product_prob_space M "{..n}" by default auto
```
```  1022     have "(\<Prod>i\<le>n. M.\<mu>' i (E i)) = n.\<mu>' (Pi\<^isub>E {.. n} E)"
```
```  1023       using E by (subst n.finite_measure_times) auto
```
```  1024     also have "\<dots> = \<mu>' (?E n)"
```
```  1025       using E by (intro finite_measure_infprod_emb[symmetric]) auto
```
```  1026     finally have "(\<Prod>i\<le>n. M.\<mu>' i (E i)) = \<mu>' (?E n)" . }
```
```  1027   moreover have "Pi UNIV E = (\<Inter>n. ?E n)"
```
```  1028     using E E[THEN M.sets_into_space]
```
```  1029     by (auto simp: emb_def extensional_def Pi_iff) blast
```
```  1030   moreover have "range ?E \<subseteq> sets (Pi\<^isub>P UNIV M)"
```
```  1031     using E by auto
```
```  1032   moreover have "decseq ?E"
```
```  1033     by (auto simp: emb_def Pi_iff decseq_def)
```
```  1034   ultimately show ?thesis
```
```  1035     by (simp add: finite_continuity_from_above)
```
```  1036 qed
```
```  1037
```
`  1038 end`