src/HOL/Probability/Projective_Family.thy

 imports Finite_Product_Measure Probability_Measure
 begin
 
 lemma (in product_prob_space) distr_restrict:
   assumes "J \<noteq> {}" "J \<subseteq> K" "finite K"
-  shows "(\<Pi>\<^isub>M i\<in>J. M i) = distr (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i) (\<lambda>f. restrict f J)" (is "?P = ?D")
+  shows "(\<Pi>\<^sub>M i\<in>J. M i) = distr (\<Pi>\<^sub>M i\<in>K. M i) (\<Pi>\<^sub>M i\<in>J. M i) (\<lambda>f. restrict f J)" (is "?P = ?D")
 proof (rule measure_eqI_generator_eq)
   have "finite J" using `J \<subseteq> K` `finite K` by (auto simp add: finite_subset)
   interpret J: finite_product_prob_space M J proof qed fact
   interpret K: finite_product_prob_space M K proof qed fact
 
-  let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
-  let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. space (M k)"
-  let ?\<Omega> = "(\<Pi>\<^isub>E k\<in>J. space (M k))"
+  let ?J = "{Pi\<^sub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
+  let ?F = "\<lambda>i. \<Pi>\<^sub>E k\<in>J. space (M k)"
+  let ?\<Omega> = "(\<Pi>\<^sub>E k\<in>J. space (M k))"
   show "Int_stable ?J"
     by (rule Int_stable_PiE)
   show "range ?F \<subseteq> ?J" "(\<Union>i. ?F i) = ?\<Omega>"
     using `finite J` by (auto intro!: prod_algebraI_finite)
   { fix i show "emeasure ?P (?F i) \<noteq> \<infinity>" by simp }
   show "?J \<subseteq> Pow ?\<Omega>" by (auto simp: Pi_iff dest: sets.sets_into_space)
-  show "sets (\<Pi>\<^isub>M i\<in>J. M i) = sigma_sets ?\<Omega> ?J" "sets ?D = sigma_sets ?\<Omega> ?J"
+  show "sets (\<Pi>\<^sub>M i\<in>J. M i) = sigma_sets ?\<Omega> ?J" "sets ?D = sigma_sets ?\<Omega> ?J"
     using `finite J` by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff)
 
   fix X assume "X \<in> ?J"
-  then obtain E where [simp]: "X = Pi\<^isub>E J E" and E: "\<forall>i\<in>J. E i \<in> sets (M i)" by auto
-  with `finite J` have X: "X \<in> sets (Pi\<^isub>M J M)"
+  then obtain E where [simp]: "X = Pi\<^sub>E J E" and E: "\<forall>i\<in>J. E i \<in> sets (M i)" by auto
+  with `finite J` have X: "X \<in> sets (Pi\<^sub>M J M)"
     by simp
 
   have "emeasure ?P X = (\<Prod> i\<in>J. emeasure (M i) (E i))"
     using E by (simp add: J.measure_times)
   also have "\<dots> = (\<Prod> i\<in>J. emeasure (M i) (if i \<in> J then E i else space (M i)))"
     by simp
   also have "\<dots> = (\<Prod> i\<in>K. emeasure (M i) (if i \<in> J then E i else space (M i)))"
     using `finite K` `J \<subseteq> K`
     by (intro setprod_mono_one_left) (auto simp: M.emeasure_space_1)
-  also have "\<dots> = emeasure (Pi\<^isub>M K M) (\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i))"
+  also have "\<dots> = emeasure (Pi\<^sub>M K M) (\<Pi>\<^sub>E i\<in>K. if i \<in> J then E i else space (M i))"
     using E by (simp add: K.measure_times)
-  also have "(\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i)) = (\<lambda>f. restrict f J) -` Pi\<^isub>E J E \<inter> (\<Pi>\<^isub>E i\<in>K. space (M i))"
+  also have "(\<Pi>\<^sub>E i\<in>K. if i \<in> J then E i else space (M i)) = (\<lambda>f. restrict f J) -` Pi\<^sub>E J E \<inter> (\<Pi>\<^sub>E i\<in>K. space (M i))"
     using `J \<subseteq> K` sets.sets_into_space E by (force simp: Pi_iff PiE_def split: split_if_asm)
-  finally show "emeasure (Pi\<^isub>M J M) X = emeasure ?D X"
+  finally show "emeasure (Pi\<^sub>M J M) X = emeasure ?D X"
     using X `J \<subseteq> K` apply (subst emeasure_distr)
     by (auto intro!: measurable_restrict_subset simp: space_PiM)
 qed
 
 lemma (in product_prob_space) emeasure_prod_emb[simp]:
-  assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" and X: "X \<in> sets (Pi\<^isub>M J M)"
-  shows "emeasure (Pi\<^isub>M L M) (prod_emb L M J X) = emeasure (Pi\<^isub>M J M) X"
+  assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" and X: "X \<in> sets (Pi\<^sub>M J M)"
+  shows "emeasure (Pi\<^sub>M L M) (prod_emb L M J X) = emeasure (Pi\<^sub>M J M) X"
   by (subst distr_restrict[OF L])
      (simp add: prod_emb_def space_PiM emeasure_distr measurable_restrict_subset L X)
 
 definition
   limP :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i set \<Rightarrow> ('i \<Rightarrow> 'a) measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
-  "limP I M P = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i))
+  "limP I M P = extend_measure (\<Pi>\<^sub>E i\<in>I. space (M i))
     {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
-    (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j))
-    (\<lambda>(J, X). emeasure (P J) (Pi\<^isub>E J X))"
-
-abbreviation "lim\<^isub>P \<equiv> limP"
+    (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^sub>E j\<in>J. X j))
+    (\<lambda>(J, X). emeasure (P J) (Pi\<^sub>E J X))"
+
+abbreviation "lim\<^sub>P \<equiv> limP"
 
 lemma space_limP[simp]: "space (limP I M P) = space (PiM I M)"
   by (auto simp add: limP_def space_PiM prod_emb_def intro!: space_extend_measure)
 
 lemma sets_limP[simp]: "sets (limP I M P) = sets (PiM I M)"
   by (auto simp add: limP_def sets_PiM prod_algebra_def prod_emb_def intro!: sets_extend_measure)
 
-lemma measurable_limP1[simp]: "measurable (limP I M P) M' = measurable (\<Pi>\<^isub>M i\<in>I. M i) M'"
+lemma measurable_limP1[simp]: "measurable (limP I M P) M' = measurable (\<Pi>\<^sub>M i\<in>I. M i) M'"
   unfolding measurable_def by auto
 
-lemma measurable_limP2[simp]: "measurable M' (limP I M P) = measurable M' (\<Pi>\<^isub>M i\<in>I. M i)"
+lemma measurable_limP2[simp]: "measurable M' (limP I M P) = measurable M' (\<Pi>\<^sub>M i\<in>I. M i)"
   unfolding measurable_def by auto
 
 locale projective_family =
   fixes I::"'i set" and P::"'i set \<Rightarrow> ('i \<Rightarrow> 'a) measure" and M::"('i \<Rightarrow> 'a measure)"
   assumes projective: "\<And>J H X. J \<noteq> {} \<Longrightarrow> J \<subseteq> H \<Longrightarrow> H \<subseteq> I \<Longrightarrow> finite H \<Longrightarrow> X \<in> sets (PiM J M) \<Longrightarrow>

 
 lemma emeasure_limP:
   assumes "finite J"
   assumes "J \<subseteq> I"
   assumes A: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> sets (M i)"
-  shows "emeasure (limP J M P) (Pi\<^isub>E J A) = emeasure (P J) (Pi\<^isub>E J A)"
-proof -
-  have "Pi\<^isub>E J (restrict A J) \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
+  shows "emeasure (limP J M P) (Pi\<^sub>E J A) = emeasure (P J) (Pi\<^sub>E J A)"
+proof -
+  have "Pi\<^sub>E J (restrict A J) \<subseteq> (\<Pi>\<^sub>E i\<in>J. space (M i))"
     using sets.sets_into_space[OF A] by (auto simp: PiE_iff) blast
-  hence "emeasure (limP J M P) (Pi\<^isub>E J A) =
-    emeasure (limP J M P) (prod_emb J M J (Pi\<^isub>E J A))"
+  hence "emeasure (limP J M P) (Pi\<^sub>E J A) =
+    emeasure (limP J M P) (prod_emb J M J (Pi\<^sub>E J A))"
     using assms(1-3) sets.sets_into_space by (auto simp add: prod_emb_id PiE_def Pi_def)
-  also have "\<dots> = emeasure (P J) (Pi\<^isub>E J A)"
+  also have "\<dots> = emeasure (P J) (Pi\<^sub>E J A)"
   proof (rule emeasure_extend_measure_Pair[OF limP_def])
     show "positive (sets (limP J M P)) (P J)" unfolding positive_def by auto
     show "countably_additive (sets (limP J M P)) (P J)" unfolding countably_additive_def
       by (auto simp: suminf_emeasure proj_sets[OF `finite J`])
     show "(J \<noteq> {} \<or> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))"
       using assms by auto
     fix K and X::"'i \<Rightarrow> 'a set"
-    show "prod_emb J M K (Pi\<^isub>E K X) \<in> Pow (\<Pi>\<^isub>E i\<in>J. space (M i))"
+    show "prod_emb J M K (Pi\<^sub>E K X) \<in> Pow (\<Pi>\<^sub>E i\<in>J. space (M i))"
       by (auto simp: prod_emb_def)
     assume JX: "(K \<noteq> {} \<or> J = {}) \<and> finite K \<and> K \<subseteq> J \<and> X \<in> (\<Pi> j\<in>K. sets (M j))"
-    thus "emeasure (P J) (prod_emb J M K (Pi\<^isub>E K X)) = emeasure (P K) (Pi\<^isub>E K X)"
+    thus "emeasure (P J) (prod_emb J M K (Pi\<^sub>E K X)) = emeasure (P K) (Pi\<^sub>E K X)"
       using assms
       apply (cases "J = {}")
       apply (simp add: prod_emb_id)
       apply (fastforce simp add: intro!: projective sets_PiM_I_finite)
       done

 lemma limP_finite:
   assumes "finite J"
   assumes "J \<subseteq> I"
   shows "limP J M P = P J" (is "?P = _")
 proof (rule measure_eqI_generator_eq)
-  let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
-  let ?\<Omega> = "(\<Pi>\<^isub>E k\<in>J. space (M k))"
+  let ?J = "{Pi\<^sub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
+  let ?\<Omega> = "(\<Pi>\<^sub>E k\<in>J. space (M k))"
   interpret prob_space "P J" using proj_prob_space `finite J` by simp
-  show "emeasure ?P (\<Pi>\<^isub>E k\<in>J. space (M k)) \<noteq> \<infinity>" using assms `finite J` by (auto simp: emeasure_limP)
+  show "emeasure ?P (\<Pi>\<^sub>E k\<in>J. space (M k)) \<noteq> \<infinity>" using assms `finite J` by (auto simp: emeasure_limP)
   show "sets (limP J M P) = sigma_sets ?\<Omega> ?J" "sets (P J) = sigma_sets ?\<Omega> ?J"
     using `finite J` proj_sets by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff)
   fix X assume "X \<in> ?J"
-  then obtain E where X: "X = Pi\<^isub>E J E" and E: "\<forall>i\<in>J. E i \<in> sets (M i)" by auto
+  then obtain E where X: "X = Pi\<^sub>E J E" and E: "\<forall>i\<in>J. E i \<in> sets (M i)" by auto
   with `finite J` have "X \<in> sets (limP J M P)" by simp
-  have emb_self: "prod_emb J M J (Pi\<^isub>E J E) = Pi\<^isub>E J E"
+  have emb_self: "prod_emb J M J (Pi\<^sub>E J E) = Pi\<^sub>E J E"
     using E sets.sets_into_space
     by (auto intro!: prod_emb_PiE_same_index)
   show "emeasure (limP J M P) X = emeasure (P J) X"
     unfolding X using E
     by (intro emeasure_limP assms) simp

 
 abbreviation
   "emb L K X \<equiv> prod_emb L M K X"
 
 lemma prod_emb_injective:
-  assumes "J \<subseteq> L" and sets: "X \<in> sets (Pi\<^isub>M J M)" "Y \<in> sets (Pi\<^isub>M J M)"
+  assumes "J \<subseteq> L" and sets: "X \<in> sets (Pi\<^sub>M J M)" "Y \<in> sets (Pi\<^sub>M J M)"
   assumes "emb L J X = emb L J Y"
   shows "X = Y"
 proof (rule injective_vimage_restrict)
-  show "X \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" "Y \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
+  show "X \<subseteq> (\<Pi>\<^sub>E i\<in>J. space (M i))" "Y \<subseteq> (\<Pi>\<^sub>E i\<in>J. space (M i))"
     using sets[THEN sets.sets_into_space] by (auto simp: space_PiM)
   have "\<forall>i\<in>L. \<exists>x. x \<in> space (M i)"
   proof
     fix i assume "i \<in> L"
     interpret prob_space "P {i}" using proj_prob_space by simp
     from not_empty show "\<exists>x. x \<in> space (M i)" by (auto simp add: proj_space space_PiM)
   qed
   from bchoice[OF this]
-  show "(\<Pi>\<^isub>E i\<in>L. space (M i)) \<noteq> {}" by (auto simp: PiE_def)
-  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))"
+  show "(\<Pi>\<^sub>E i\<in>L. space (M i)) \<noteq> {}" by (auto simp: PiE_def)
+  show "(\<lambda>x. restrict x J) -` X \<inter> (\<Pi>\<^sub>E i\<in>L. space (M i)) = (\<lambda>x. restrict x J) -` Y \<inter> (\<Pi>\<^sub>E i\<in>L. space (M i))"
     using `prod_emb L M J X = prod_emb L M J Y` by (simp add: prod_emb_def)
 qed fact
 
 definition generator :: "('i \<Rightarrow> 'a) set set" where
-  "generator = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M))"
+  "generator = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^sub>M J M))"
 
 lemma generatorI':
-  "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> emb I J X \<in> generator"
+  "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^sub>M J M) \<Longrightarrow> emb I J X \<in> generator"
   unfolding generator_def by auto
 
 lemma algebra_generator:
-  assumes "I \<noteq> {}" shows "algebra (\<Pi>\<^isub>E i\<in>I. space (M i)) generator" (is "algebra ?\<Omega> ?G")
+  assumes "I \<noteq> {}" shows "algebra (\<Pi>\<^sub>E i\<in>I. space (M i)) generator" (is "algebra ?\<Omega> ?G")
   unfolding algebra_def algebra_axioms_def ring_of_sets_iff
 proof (intro conjI ballI)
   let ?G = generator
   show "?G \<subseteq> Pow ?\<Omega>"
     by (auto simp: generator_def prod_emb_def)
   from `I \<noteq> {}` obtain i where "i \<in> I" by auto
   then show "{} \<in> ?G"
     by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"]
              simp: sigma_sets.Empty generator_def prod_emb_def)
   from `i \<in> I` show "?\<Omega> \<in> ?G"
-    by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"]
+    by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^sub>E {i} (\<lambda>i. space (M i))"]
              simp: generator_def prod_emb_def)
   fix A assume "A \<in> ?G"
-  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"
+  then obtain JA XA where XA: "JA \<noteq> {}" "finite JA" "JA \<subseteq> I" "XA \<in> sets (Pi\<^sub>M JA M)" and A: "A = emb I JA XA"
     by (auto simp: generator_def)
   fix B assume "B \<in> ?G"
-  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"
+  then obtain JB XB where XB: "JB \<noteq> {}" "finite JB" "JB \<subseteq> I" "XB \<in> sets (Pi\<^sub>M JB M)" and B: "B = emb I JB XB"
     by (auto simp: generator_def)
   let ?RA = "emb (JA \<union> JB) JA XA"
   let ?RB = "emb (JA \<union> JB) JB XB"
   have *: "A - B = emb I (JA \<union> JB) (?RA - ?RB)" "A \<union> B = emb I (JA \<union> JB) (?RA \<union> ?RB)"
     using XA A XB B by auto
   show "A - B \<in> ?G" "A \<union> B \<in> ?G"
     unfolding * using XA XB by (safe intro!: generatorI') auto
 qed
 
 lemma sets_PiM_generator:
-  "sets (PiM I M) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator"
+  "sets (PiM I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) generator"
 proof cases
   assume "I = {}" then show ?thesis
     unfolding generator_def
     by (auto simp: sets_PiM_empty sigma_sets_empty_eq cong: conj_cong)
 next
   assume "I \<noteq> {}"
   show ?thesis
   proof
-    show "sets (Pi\<^isub>M I M) \<subseteq> sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator"
+    show "sets (Pi\<^sub>M I M) \<subseteq> sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) generator"
       unfolding sets_PiM
     proof (safe intro!: sigma_sets_subseteq)
       fix A assume "A \<in> prod_algebra I M" with `I \<noteq> {}` show "A \<in> generator"
         by (auto intro!: generatorI' sets_PiM_I_finite elim!: prod_algebraE)
     qed
   qed (auto simp: generator_def space_PiM[symmetric] intro!: sets.sigma_sets_subset)
 qed
 
 lemma generatorI:
-  "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> generator"
+  "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^sub>M J M) \<Longrightarrow> A = emb I J X \<Longrightarrow> A \<in> generator"
   unfolding generator_def by auto
 
 definition mu_G ("\<mu>G") where
   "\<mu>G A =
-    (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 = emeasure (limP J M P) X))"
+    (THE x. \<forall>J. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> J \<subseteq> I \<longrightarrow> (\<forall>X\<in>sets (Pi\<^sub>M J M). A = emb I J X \<longrightarrow> x = emeasure (limP J M P) X))"
 
 lemma mu_G_spec:
-  assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)"
+  assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^sub>M J M)"
   shows "\<mu>G A = emeasure (limP J M P) X"
   unfolding mu_G_def
 proof (intro the_equality allI impI ballI)
-  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)"
+  fix K Y assume K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "A = emb I K Y" "Y \<in> sets (Pi\<^sub>M K M)"
   have "emeasure (limP K M P) Y = emeasure (limP (K \<union> J) M P) (emb (K \<union> J) K Y)"
     using K J by simp
   also have "emb (K \<union> J) K Y = emb (K \<union> J) J X"
     using K J by (simp add: prod_emb_injective[of "K \<union> J" I])
   also have "emeasure (limP (K \<union> J) M P) (emb (K \<union> J) J X) = emeasure (limP J M P) X"
     using K J by simp
   finally show "emeasure (limP J M P) X = emeasure (limP K M P) Y" ..
 qed (insert J, force)
 
 lemma mu_G_eq:
-  "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) = emeasure (limP J M P) X"
+  "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^sub>M J M) \<Longrightarrow> \<mu>G (emb I J X) = emeasure (limP J M P) X"
   by (intro mu_G_spec) auto
 
 lemma generator_Ex:
   assumes *: "A \<in> generator"
-  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 = emeasure (limP J M P) X"
-proof -
-  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)"
+  shows "\<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^sub>M J M) \<and> A = emb I J X \<and> \<mu>G A = emeasure (limP J M P) X"
+proof -
+  from * obtain J X where J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^sub>M J M)"
     unfolding generator_def by auto
   with mu_G_spec[OF this] show ?thesis by auto
 qed
 
 lemma generatorE:
   assumes A: "A \<in> generator"
-  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 = emeasure (limP J M P) X"
+  obtains J X where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^sub>M J M)" "emb I J X = A" "\<mu>G A = emeasure (limP J M P) X"
   using generator_Ex[OF A] by atomize_elim auto
 
 lemma merge_sets:
-  "J \<inter> K = {} \<Longrightarrow> A \<in> sets (Pi\<^isub>M (J \<union> K) M) \<Longrightarrow> x \<in> space (Pi\<^isub>M J M) \<Longrightarrow> (\<lambda>y. merge J K (x,y)) -` A \<inter> space (Pi\<^isub>M K M) \<in> sets (Pi\<^isub>M K M)"
+  "J \<inter> K = {} \<Longrightarrow> A \<in> sets (Pi\<^sub>M (J \<union> K) M) \<Longrightarrow> x \<in> space (Pi\<^sub>M J M) \<Longrightarrow> (\<lambda>y. merge J K (x,y)) -` A \<inter> space (Pi\<^sub>M K M) \<in> sets (Pi\<^sub>M K M)"
   by simp
 
 lemma merge_emb:
-  assumes "K \<subseteq> I" "J \<subseteq> I" and y: "y \<in> space (Pi\<^isub>M J M)"
-  shows "((\<lambda>x. merge J (I - J) (y, x)) -` emb I K X \<inter> space (Pi\<^isub>M I M)) =
-    emb I (K - J) ((\<lambda>x. merge J (K - J) (y, x)) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M))"
+  assumes "K \<subseteq> I" "J \<subseteq> I" and y: "y \<in> space (Pi\<^sub>M J M)"
+  shows "((\<lambda>x. merge J (I - J) (y, x)) -` emb I K X \<inter> space (Pi\<^sub>M I M)) =
+    emb I (K - J) ((\<lambda>x. merge J (K - J) (y, x)) -` emb (J \<union> K) K X \<inter> space (Pi\<^sub>M (K - J) M))"
 proof -
   have [simp]: "\<And>x J K L. merge J K (y, restrict x L) = merge J (K \<inter> L) (y, x)"
     by (auto simp: restrict_def merge_def)
   have [simp]: "\<And>x J K L. restrict (merge J K (y, x)) L = merge (J \<inter> L) (K \<inter> L) (y, x)"
     by (auto simp: restrict_def merge_def)

 
 lemma positive_mu_G:
   assumes "I \<noteq> {}"
   shows "positive generator \<mu>G"
 proof -
-  interpret G!: algebra "\<Pi>\<^isub>E i\<in>I. space (M i)" generator by (rule algebra_generator) fact
+  interpret G!: algebra "\<Pi>\<^sub>E i\<in>I. space (M i)" generator by (rule algebra_generator) fact
   show ?thesis
   proof (intro positive_def[THEN iffD2] conjI ballI)
     from generatorE[OF G.empty_sets] guess J X . note this[simp]
     have "X = {}"
       by (rule prod_emb_injective[of J I]) simp_all

 
 lemma additive_mu_G:
   assumes "I \<noteq> {}"
   shows "additive generator \<mu>G"
 proof -
-  interpret G!: algebra "\<Pi>\<^isub>E i\<in>I. space (M i)" generator by (rule algebra_generator) fact
+  interpret G!: algebra "\<Pi>\<^sub>E i\<in>I. space (M i)" generator by (rule algebra_generator) fact
   show ?thesis
   proof (intro additive_def[THEN iffD2] ballI impI)
     fix A assume "A \<in> generator" with generatorE guess J X . note J = this
     fix B assume "B \<in> generator" with generatorE guess K Y . note K = this
     assume "A \<inter> B = {}"

 
 sublocale product_prob_space \<subseteq> projective_family I "\<lambda>J. PiM J M" M
 proof
   fix J::"'i set" assume "finite J"
   interpret f: finite_product_prob_space M J proof qed fact
-  show "emeasure (Pi\<^isub>M J M) (space (Pi\<^isub>M J M)) \<noteq> \<infinity>" by simp
-  show "\<exists>A. range A \<subseteq> sets (Pi\<^isub>M J M) \<and>
-            (\<Union>i. A i) = space (Pi\<^isub>M J M) \<and>
-            (\<forall>i. emeasure (Pi\<^isub>M J M) (A i) \<noteq> \<infinity>)" using sigma_finite[OF `finite J`]
+  show "emeasure (Pi\<^sub>M J M) (space (Pi\<^sub>M J M)) \<noteq> \<infinity>" by simp
+  show "\<exists>A. range A \<subseteq> sets (Pi\<^sub>M J M) \<and>
+            (\<Union>i. A i) = space (Pi\<^sub>M J M) \<and>
+            (\<forall>i. emeasure (Pi\<^sub>M J M) (A i) \<noteq> \<infinity>)" using sigma_finite[OF `finite J`]
     by (auto simp add: sigma_finite_measure_def)
-  show "emeasure (Pi\<^isub>M J M) (space (Pi\<^isub>M J M)) = 1" by (rule f.emeasure_space_1)
+  show "emeasure (Pi\<^sub>M J M) (space (Pi\<^sub>M J M)) = 1" by (rule f.emeasure_space_1)
 qed simp_all
 
 lemma (in product_prob_space) limP_PiM_finite[simp]:
   assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" shows "limP J M (\<lambda>J. PiM J M) = PiM J M"
   using assms by (simp add: limP_finite)