author immler Fri, 16 Nov 2012 11:34:34 +0100 changeset 50095 94d7dfa9f404 parent 50094 84ddcf5364b4 child 50096 7c9c5b1b6cd7 child 50102 5e01e32dadbe
renamed to more appropriate lim_P for projective limit
```--- a/src/HOL/Probability/Infinite_Product_Measure.thy	Fri Nov 16 11:22:22 2012 +0100
+++ b/src/HOL/Probability/Infinite_Product_Measure.thy	Fri Nov 16 11:34:34 2012 +0100
@@ -96,7 +96,7 @@
using A positive_\<mu>G[OF I_not_empty] by (auto intro!: INF_greatest simp: positive_def)
ultimately have "0 < ?a" by auto

-      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) = emeasure (PiP J M (\<lambda>J. (Pi\<^isub>M J M))) X"
+      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) = emeasure (limP J M (\<lambda>J. (Pi\<^isub>M J M))) X"
using A by (intro allI generator_Ex) auto
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)"
and A': "\<And>n. A n = emb I (J' n) (X' n)"```
```--- a/src/HOL/Probability/Projective_Family.thy	Fri Nov 16 11:22:22 2012 +0100
+++ b/src/HOL/Probability/Projective_Family.thy	Fri Nov 16 11:34:34 2012 +0100
@@ -57,22 +57,24 @@
(simp add: prod_emb_def space_PiM emeasure_distr measurable_restrict_subset L X)

definition
-  PiP :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i set \<Rightarrow> ('i \<Rightarrow> 'a) measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
-  "PiP I M P = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i))
+  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))
{(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))"

-lemma space_PiP[simp]: "space (PiP I M P) = space (PiM I M)"
-  by (auto simp add: PiP_def space_PiM prod_emb_def intro!: space_extend_measure)
+abbreviation "lim\<^isub>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_PiP[simp]: "sets (PiP I M P) = sets (PiM I M)"
-  by (auto simp add: PiP_def sets_PiM prod_algebra_def prod_emb_def intro!: sets_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_PiP1[simp]: "measurable (PiP 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>\<^isub>M i\<in>I. M i) M'"
unfolding measurable_def by auto

-lemma measurable_PiP2[simp]: "measurable M' (PiP 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>\<^isub>M i\<in>I. M i)"
unfolding measurable_def by auto

locale projective_family =
@@ -84,11 +86,11 @@
assumes proj_sets: "\<And>J. finite J \<Longrightarrow> sets (P J) = sets (PiM J M)"
begin

-lemma emeasure_PiP:
+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 (PiP J M P) (Pi\<^isub>E J A) = emeasure (P J) (Pi\<^isub>E J A)"
+  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))"
proof safe
@@ -97,13 +99,13 @@
also have "\<dots> \<subseteq> space (M j)" using sets_into_space A `j \<in> J` by auto
finally show "x j \<in> space (M j)" .
qed
-  hence "emeasure (PiP J M P) (Pi\<^isub>E J A) =
-    emeasure (PiP J M P) (prod_emb J M J (Pi\<^isub>E J A))"
+  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))"
using assms(1-3) sets_into_space by (auto simp add: prod_emb_id Pi_def)
also have "\<dots> = emeasure (P J) (Pi\<^isub>E J A)"
-  proof (rule emeasure_extend_measure_Pair[OF PiP_def])
-    show "positive (sets (PiP J M P)) (P J)" unfolding positive_def by auto
-    show "countably_additive (sets (PiP J M P)) (P J)" unfolding countably_additive_def
+  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
@@ -121,10 +123,10 @@
finally show ?thesis .
qed

-lemma PiP_finite:
+lemma limP_finite:
assumes "finite J"
assumes "J \<subseteq> I"
-  shows "PiP J M P = P J" (is "?P = _")
+  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 ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. space (M k)"
@@ -132,26 +134,26 @@
show "Int_stable ?J"
by (rule Int_stable_PiE)
interpret prob_space "P J" using prob_space `finite J` by simp
-  show "emeasure ?P (?F _) \<noteq> \<infinity>" using assms `finite J` by (auto simp: emeasure_PiP)
+  show "emeasure ?P (?F _) \<noteq> \<infinity>" using assms `finite J` by (auto simp: emeasure_limP)
show "?J \<subseteq> Pow ?\<Omega>" by (auto simp: Pi_iff dest: sets_into_space)
-  show "sets (PiP J M P) = sigma_sets ?\<Omega> ?J" "sets (P J) = sigma_sets ?\<Omega> ?J"
+  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
-  with `finite J` have "X \<in> sets (PiP J M P)" by simp
+  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"
using E sets_into_space
by (auto intro!: prod_emb_PiE_same_index)
-  show "emeasure (PiP J M P) X = emeasure (P J) X"
+  show "emeasure (limP J M P) X = emeasure (P J) X"
unfolding X using E
-    by (intro emeasure_PiP assms) simp
+    by (intro emeasure_limP assms) simp
qed (insert `finite J`, auto intro!: prod_algebraI_finite)

lemma emeasure_fun_emb[simp]:
assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" "L \<subseteq> I" and X: "X \<in> sets (PiM J M)"
-  shows "emeasure (PiP L M P) (prod_emb L M J X) = emeasure (PiP J M P) X"
+  shows "emeasure (limP L M P) (prod_emb L M J X) = emeasure (limP J M P) X"
using assms
-  by (subst PiP_finite) (auto simp: PiP_finite finite_subset projective)
+  by (subst limP_finite) (auto simp: limP_finite finite_subset projective)

lemma prod_emb_injective:
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)"
@@ -235,30 +237,30 @@

definition
"\<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 (PiP J M P) X))"
+    (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))"

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)"
-  shows "\<mu>G A = emeasure (PiP J M P) X"
+  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)"
-  have "emeasure (PiP K M P) Y = emeasure (PiP (K \<union> J) M P) (emb (K \<union> J) K Y)"
+  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 (PiP (K \<union> J) M P) (emb (K \<union> J) J X) = emeasure (PiP J M P) X"
+  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 (PiP J M P) X = emeasure (PiP K M P) Y" ..
+  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 (PiP J M P) X"
+  "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"
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 (PiP J M P) X"
+  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)"
unfolding generator_def by auto
@@ -267,10 +269,10 @@

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 (PiP J M P) X"
+  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"
proof -
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"
-    "\<mu>G A = emeasure (PiP J M P) X" by auto
+    "\<mu>G A = emeasure (limP J M P) X" by auto
then show thesis by (intro that) auto
qed

@@ -334,7 +336,7 @@
using J K by simp_all
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))"
by simp
-    also have "\<dots> = emeasure (PiP (J \<union> K) M P) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)"
+    also have "\<dots> = emeasure (limP (J \<union> K) M P) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)"
using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq Un del: prod_emb_Un)
also have "\<dots> = \<mu>G A + \<mu>G B"
using J K JK_disj by (simp add: plus_emeasure[symmetric])
@@ -356,8 +358,8 @@
show "emeasure (Pi\<^isub>M J M) (space (Pi\<^isub>M J M)) = 1" by (rule f.emeasure_space_1)
qed simp_all

-lemma (in product_prob_space) PiP_PiM_finite[simp]:
-  assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" shows "PiP J M (\<lambda>J. PiM J M) = PiM J M"
-  using assms by (simp add: PiP_finite)
+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)

end```
```--- a/src/HOL/Probability/Projective_Limit.thy	Fri Nov 16 11:22:22 2012 +0100
+++ b/src/HOL/Probability/Projective_Limit.thy	Fri Nov 16 11:34:34 2012 +0100
@@ -189,13 +189,13 @@
for I::"'i set" and P
begin

-abbreviation "PiB \<equiv> (\<lambda>J P. PiP J (\<lambda>_. borel) P)"
+abbreviation "lim\<^isub>B \<equiv> (\<lambda>J P. limP J (\<lambda>_. borel) P)"

lemma
-  emeasure_PiB_emb_not_empty:
+  emeasure_limB_emb_not_empty:
assumes "I \<noteq> {}"
assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
-  shows "emeasure (PiB I P) (emb I J (Pi\<^isub>E J B)) = emeasure (PiB J P) (Pi\<^isub>E J B)"
+  shows "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (lim\<^isub>B J P) (Pi\<^isub>E J B)"
proof -
let ?\<Omega> = "\<Pi>\<^isub>E i\<in>I. space borel"
let ?G = generator
@@ -208,7 +208,7 @@
fix A assume "A \<in> ?G"
with generatorE guess J X . note JX = this
interpret prob_space "P J" using prob_space[OF `finite J`] .
-    show "\<mu>G A \<noteq> \<infinity>" using JX by (simp add: PiP_finite)
+    show "\<mu>G A \<noteq> \<infinity>" using JX by (simp add: limP_finite)
next
fix Z assume Z: "range Z \<subseteq> ?G" "decseq Z" "(\<Inter>i. Z i) = {}"
then have "decseq (\<lambda>i. \<mu>G (Z i))"
@@ -222,7 +222,7 @@
ultimately have "0 < ?a" by auto
hence "?a \<noteq> -\<infinity>" by auto
have "\<forall>n. \<exists>J B. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> B \<in> sets (Pi\<^isub>M J (\<lambda>_. borel)) \<and>
-        Z n = emb I J B \<and> \<mu>G (Z n) = emeasure (PiB J P) B"
+        Z n = emb I J B \<and> \<mu>G (Z n) = emeasure (lim\<^isub>B J P) B"
using Z by (intro allI generator_Ex) auto
then obtain J' B' where J': "\<And>n. J' n \<noteq> {}" "\<And>n. finite (J' n)" "\<And>n. J' n \<subseteq> I"
"\<And>n. B' n \<in> sets (\<Pi>\<^isub>M i\<in>J' n. borel)"
@@ -243,10 +243,10 @@
unfolding J_def B_def by (subst prod_emb_trans) (insert Z, auto)
interpret prob_space "P (J i)" for i using prob_space by simp
have "?a \<le> \<mu>G (Z 0)" by (auto intro: INF_lower)
-      also have "\<dots> < \<infinity>" using J by (auto simp: Z_eq \<mu>G_eq PiP_finite proj_sets)
+      also have "\<dots> < \<infinity>" using J by (auto simp: Z_eq \<mu>G_eq limP_finite proj_sets)
finally have "?a \<noteq> \<infinity>" by simp
have "\<And>n. \<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>" unfolding Z_eq using J J_mono
-        by (subst \<mu>G_eq) (auto simp: PiP_finite proj_sets \<mu>G_eq)
+        by (subst \<mu>G_eq) (auto simp: limP_finite proj_sets \<mu>G_eq)

interpret finite_set_sequence J by unfold_locales simp
def Utn \<equiv> Un_to_nat
@@ -380,20 +380,20 @@
(\<Inter> i\<in>{1..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" .
hence "Y n \<in> ?G" using J J_mono K_sets `n \<ge> 1` by (intro generatorI[OF _ _ _ _ Y_emb]) auto
hence "\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>" unfolding Y_emb using J J_mono K_sets `n \<ge> 1`
-          by (subst \<mu>G_eq) (auto simp: PiP_finite proj_sets \<mu>G_eq)
-        interpret finite_measure "(PiP (J n) (\<lambda>_. borel) P)"
+          by (subst \<mu>G_eq) (auto simp: limP_finite proj_sets \<mu>G_eq)
+        interpret finite_measure "(limP (J n) (\<lambda>_. borel) P)"
proof
-          have "emeasure (PiP (J n) (\<lambda>_. borel) P) (J n \<rightarrow>\<^isub>E space borel) \<noteq> \<infinity>"
-            using J by (subst emeasure_PiP) auto
-          thus  "emeasure (PiP (J n) (\<lambda>_. borel) P) (space (PiP (J n) (\<lambda>_. borel) P)) \<noteq> \<infinity>"
+          have "emeasure (limP (J n) (\<lambda>_. borel) P) (J n \<rightarrow>\<^isub>E space borel) \<noteq> \<infinity>"
+            using J by (subst emeasure_limP) auto
+          thus  "emeasure (limP (J n) (\<lambda>_. borel) P) (space (limP (J n) (\<lambda>_. borel) P)) \<noteq> \<infinity>"
qed
-        have "\<mu>G (Z n) = PiP (J n) (\<lambda>_. borel) P (B n)"
+        have "\<mu>G (Z n) = limP (J n) (\<lambda>_. borel) P (B n)"
unfolding Z_eq using J by (auto simp: \<mu>G_eq)
moreover have "\<mu>G (Y n) =
-          PiP (J n) (\<lambda>_. borel) P (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))"
+          limP (J n) (\<lambda>_. borel) P (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))"
unfolding Y_emb using J J_mono K_sets `n \<ge> 1` by (subst \<mu>G_eq) auto
-        moreover have "\<mu>G (Z n - Y n) = PiP (J n) (\<lambda>_. borel) P
+        moreover have "\<mu>G (Z n - Y n) = limP (J n) (\<lambda>_. borel) P
(B n - (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))"
unfolding Z_eq Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets `n \<ge> 1`
by (subst \<mu>G_eq) (auto intro!: Diff)
@@ -420,7 +420,7 @@
unfolding Z'_def Z_eq by simp
also have "\<dots> = P (J i) (B i - K i)"
apply (subst \<mu>G_eq) using J K_sets apply auto
-            apply (subst PiP_finite) apply auto
+            apply (subst limP_finite) apply auto
done
also have "\<dots> = P (J i) (B i) - P (J i) (K i)"
apply (subst emeasure_Diff) using K_sets J `K _ \<subseteq> B _` apply (auto simp: proj_sets)
@@ -593,10 +593,10 @@
qed
then guess \<mu> .. note \<mu> = this
def f \<equiv> "finmap_of J B"
-  show "emeasure (PiB I P) (emb I J (Pi\<^isub>E J B)) = emeasure (PiB J P) (Pi\<^isub>E J B)"
-  proof (subst emeasure_extend_measure_Pair[OF PiP_def, of I "\<lambda>_. borel" \<mu>])
-    show "positive (sets (PiB I P)) \<mu>" "countably_additive (sets (PiB I P)) \<mu>"
-      using \<mu> unfolding sets_PiP sets_PiM_generator by (auto simp: measure_space_def)
+  show "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (lim\<^isub>B J P) (Pi\<^isub>E J B)"
+  proof (subst emeasure_extend_measure_Pair[OF limP_def, of I "\<lambda>_. borel" \<mu>])
+    show "positive (sets (lim\<^isub>B I P)) \<mu>" "countably_additive (sets (lim\<^isub>B I P)) \<mu>"
+      using \<mu> unfolding sets_limP sets_PiM_generator by (auto simp: measure_space_def)
next
show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> B \<in> J \<rightarrow> sets borel"
using assms by (auto simp: f_def)
@@ -610,11 +610,11 @@
hence "\<mu> (emb I J (Pi\<^isub>E J X)) = \<mu>G (emb I J (Pi\<^isub>E J X))" using \<mu> by simp
also have "\<dots> = emeasure (P J) (Pi\<^isub>E J X)"
using JX assms proj_sets
-      by (subst \<mu>G_eq) (auto simp: \<mu>G_eq PiP_finite intro: sets_PiM_I_finite)
+      by (subst \<mu>G_eq) (auto simp: \<mu>G_eq limP_finite intro: sets_PiM_I_finite)
finally show "\<mu> (emb I J (Pi\<^isub>E J X)) = emeasure (P J) (Pi\<^isub>E J X)" .
next
-    show "emeasure (P J) (Pi\<^isub>E J B) = emeasure (PiP J (\<lambda>_. borel) P) (Pi\<^isub>E J B)"
-      using assms by (simp add: f_def PiP_finite Pi_def)
+    show "emeasure (P J) (Pi\<^isub>E J B) = emeasure (limP J (\<lambda>_. borel) P) (Pi\<^isub>E J B)"
+      using assms by (simp add: f_def limP_finite Pi_def)
qed
qed

@@ -631,56 +631,56 @@
hide_const (open) domain
hide_const (open) enum_basis_finmap

-sublocale polish_projective \<subseteq> P!: prob_space "(PiB I P)"
+sublocale polish_projective \<subseteq> P!: prob_space "(lim\<^isub>B I P)"
proof
-  show "emeasure (PiB I P) (space (PiB I P)) = 1"
+  show "emeasure (lim\<^isub>B I P) (space (lim\<^isub>B I P)) = 1"
proof cases
assume "I = {}"
interpret prob_space "P {}" using prob_space by simp
show ?thesis
-      by (simp add: space_PiM_empty PiP_finite emeasure_space_1 `I = {}`)
+      by (simp add: space_PiM_empty limP_finite emeasure_space_1 `I = {}`)
next
assume "I \<noteq> {}"
then obtain i where "i \<in> I" by auto
interpret prob_space "P {i}" using prob_space by simp
-    have R: "(space (PiB I P)) = (emb I {i} (Pi\<^isub>E {i} (\<lambda>_. space borel)))"
+    have R: "(space (lim\<^isub>B I P)) = (emb I {i} (Pi\<^isub>E {i} (\<lambda>_. space borel)))"
by (auto simp: prod_emb_def space_PiM)
moreover have "extensional {i} = space (P {i})" by (simp add: proj_space space_PiM)
ultimately show ?thesis using `i \<in> I`
apply (subst R)
-      apply (subst emeasure_PiB_emb_not_empty)
-      apply (auto simp: PiP_finite emeasure_space_1)
+      apply (subst emeasure_limB_emb_not_empty)
+      apply (auto simp: limP_finite emeasure_space_1)
done
qed
qed

context polish_projective begin

-lemma emeasure_PiB_emb:
+lemma emeasure_limB_emb:
assumes X: "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
-  shows "emeasure (PiB I P) (emb I J (Pi\<^isub>E J B)) = emeasure (P J) (Pi\<^isub>E J B)"
+  shows "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (P J) (Pi\<^isub>E J B)"
proof cases
interpret prob_space "P {}" using prob_space by simp
assume "J = {}"
-  moreover have "emb I {} {\<lambda>x. undefined} = space (PiB I P)"
+  moreover have "emb I {} {\<lambda>x. undefined} = space (lim\<^isub>B I P)"
by (auto simp: space_PiM prod_emb_def)
-  moreover have "{\<lambda>x. undefined} = space (PiB {} P)"
+  moreover have "{\<lambda>x. undefined} = space (lim\<^isub>B {} P)"
by (auto simp: space_PiM prod_emb_def)
ultimately show ?thesis
-    by (simp add: P.emeasure_space_1 PiP_finite emeasure_space_1 del: space_PiP)
+    by (simp add: P.emeasure_space_1 limP_finite emeasure_space_1 del: space_limP)
next
assume "J \<noteq> {}" with X show ?thesis
-    by (subst emeasure_PiB_emb_not_empty) (auto simp: PiP_finite)
+    by (subst emeasure_limB_emb_not_empty) (auto simp: limP_finite)
qed

-lemma measure_PiB_emb:
+lemma measure_limB_emb:
assumes "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
-  shows "measure (PiB I P) (emb I J (Pi\<^isub>E J B)) = measure (P J) (Pi\<^isub>E J B)"
+  shows "measure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = measure (P J) (Pi\<^isub>E J B)"
proof -
interpret prob_space "P J" using prob_space assms by simp
show ?thesis
-    using emeasure_PiB_emb[OF assms]
-    unfolding emeasure_eq_measure PiP_finite[OF `finite J` `J \<subseteq> I`] P.emeasure_eq_measure
+    using emeasure_limB_emb[OF assms]
+    unfolding emeasure_eq_measure limP_finite[OF `finite J` `J \<subseteq> I`] P.emeasure_eq_measure
by simp
qed

@@ -693,9 +693,9 @@
proof qed

lemma (in polish_product_prob_space)
-  PiP_eq_PiM:
-  "I \<noteq> {} \<Longrightarrow> PiP I (\<lambda>_. borel) (\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)) =
+  limP_eq_PiM:
+  "I \<noteq> {} \<Longrightarrow> lim\<^isub>P I (\<lambda>_. borel) (\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)) =
PiM I (\<lambda>_. borel)"
-  by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_PiB_emb)
+  by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_limB_emb)

end```