src/HOL/Probability/Caratheodory.thy

 header {*Caratheodory Extension Theorem*}
 
 theory Caratheodory
-  imports Sigma_Algebra SeriesPlus
+  imports Sigma_Algebra Positive_Infinite_Real
 begin
 
 text{*From the Hurd/Coble measure theory development, translated by Lawrence Paulson.*}
 
 subsection {* Measure Spaces *}
 
-text {*A measure assigns a nonnegative real to every measurable set. 
-       It is countably additive for disjoint sets.*}
-
-record 'a measure_space = "'a algebra" +
-  measure:: "'a set \<Rightarrow> real"
-
-definition
-  disjoint_family_on  where
-  "disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
-
-abbreviation
-  "disjoint_family A \<equiv> disjoint_family_on A UNIV"
-
-definition
-  positive  where
-  "positive M f \<longleftrightarrow> f {} = (0::real) & (\<forall>x \<in> sets M. 0 \<le> f x)"
+definition "positive f \<longleftrightarrow> f {} = (0::pinfreal)" -- "Positive is enforced by the type"
 
 definition
   additive  where
-  "additive M f \<longleftrightarrow> 
-    (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} 
+  "additive M f \<longleftrightarrow>
+    (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {}
     \<longrightarrow> f (x \<union> y) = f x + f y)"
 
 definition
   countably_additive  where
-  "countably_additive M f \<longleftrightarrow> 
-    (\<forall>A. range A \<subseteq> sets M \<longrightarrow> 
+  "countably_additive M f \<longleftrightarrow>
+    (\<forall>A. range A \<subseteq> sets M \<longrightarrow>
          disjoint_family A \<longrightarrow>
-         (\<Union>i. A i) \<in> sets M \<longrightarrow> 
-         (\<lambda>n. f (A n))  sums  f (\<Union>i. A i))"
+         (\<Union>i. A i) \<in> sets M \<longrightarrow>
+         (\<Sum>\<^isub>\<infinity> n. f (A n)) = f (\<Union>i. A i))"
 
 definition
   increasing  where
   "increasing M f \<longleftrightarrow> (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<subseteq> y \<longrightarrow> f x \<le> f y)"
 
 definition
   subadditive  where
-  "subadditive M f \<longleftrightarrow> 
-    (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} 
+  "subadditive M f \<longleftrightarrow>
+    (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {}
     \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
 
 definition
   countably_subadditive  where
-  "countably_subadditive M f \<longleftrightarrow> 
-    (\<forall>A. range A \<subseteq> sets M \<longrightarrow> 
+  "countably_subadditive M f \<longleftrightarrow>
+    (\<forall>A. range A \<subseteq> sets M \<longrightarrow>
          disjoint_family A \<longrightarrow>
-         (\<Union>i. A i) \<in> sets M \<longrightarrow> 
-         summable (f o A) \<longrightarrow>
-         f (\<Union>i. A i) \<le> suminf (\<lambda>n. f (A n)))"
+         (\<Union>i. A i) \<in> sets M \<longrightarrow>
+         f (\<Union>i. A i) \<le> psuminf (\<lambda>n. f (A n)))"
 
 definition
   lambda_system where
-  "lambda_system M f = 
+  "lambda_system M f =
     {l. l \<in> sets M & (\<forall>x \<in> sets M. f (l \<inter> x) + f ((space M - l) \<inter> x) = f x)}"
 
 definition
   outer_measure_space where
-  "outer_measure_space M f  \<longleftrightarrow> 
-     positive M f & increasing M f & countably_subadditive M f"
+  "outer_measure_space M f  \<longleftrightarrow>
+     positive f \<and> increasing M f \<and> countably_subadditive M f"
 
 definition
   measure_set where
   "measure_set M f X =
-     {r . \<exists>A. range A \<subseteq> sets M & disjoint_family A & X \<subseteq> (\<Union>i. A i) & (f \<circ> A) sums r}"
-
+     {r . \<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>\<^isub>\<infinity> i. f (A i)) = r}"
 
 locale measure_space = sigma_algebra +
-  assumes positive: "!!a. a \<in> sets M \<Longrightarrow> 0 \<le> measure M a"
-      and empty_measure [simp]: "measure M {} = (0::real)"
-      and ca: "countably_additive M (measure M)"
-
-subsection {* Basic Lemmas *}
-
-lemma positive_imp_0: "positive M f \<Longrightarrow> f {} = 0"
-  by (simp add: positive_def) 
-
-lemma positive_imp_pos: "positive M f \<Longrightarrow> x \<in> sets M \<Longrightarrow> 0 \<le> f x"
-  by (simp add: positive_def) 
+  fixes \<mu> :: "'a set \<Rightarrow> pinfreal"
+  assumes empty_measure [simp]: "\<mu> {} = 0"
+      and ca: "countably_additive M \<mu>"
 
 lemma increasingD:
      "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M \<Longrightarrow> f x \<le> f y"
   by (auto simp add: increasing_def)
 
 lemma subadditiveD:
-     "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M 
+     "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M
       \<Longrightarrow> f (x \<union> y) \<le> f x + f y"
   by (auto simp add: subadditive_def)
 
 lemma additiveD:
-     "additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M 
+     "additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M
       \<Longrightarrow> f (x \<union> y) = f x + f y"
   by (auto simp add: additive_def)
 
 lemma countably_additiveD:
   "countably_additive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A
-   \<Longrightarrow> (\<Union>i. A i) \<in> sets M \<Longrightarrow> (\<lambda>n. f (A n))  sums  f (\<Union>i. A i)"
+   \<Longrightarrow> (\<Union>i. A i) \<in> sets M \<Longrightarrow> (\<Sum>\<^isub>\<infinity> n. f (A n)) = f (\<Union>i. A i)"
   by (simp add: countably_additive_def)
 
-lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"
-  by blast
-
-lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"
-  by blast
-
-lemma disjoint_family_subset:
-     "disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B"
-  by (force simp add: disjoint_family_on_def)
-
-subsection {* A Two-Element Series *}
-
-definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
-  where "binaryset A B = (\<lambda>\<^isup>x. {})(0 := A, Suc 0 := B)"
-
-lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"
-  apply (simp add: binaryset_def)
-  apply (rule set_ext)
-  apply (auto simp add: image_iff)
-  done
-
-lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"
-  by (simp add: UNION_eq_Union_image range_binaryset_eq)
+section "Extend binary sets"
 
 lemma LIMSEQ_binaryset:
   assumes f: "f {} = 0"
   shows  "(\<lambda>n. \<Sum>i = 0..<n. f (binaryset A B i)) ----> f A + f B"
 proof -

 qed
 
 lemma binaryset_sums:
   assumes f: "f {} = 0"
   shows  "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"
-    by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f]) 
+    by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f])
 
 lemma suminf_binaryset_eq:
      "f {} = 0 \<Longrightarrow> suminf (\<lambda>n. f (binaryset A B n)) = f A + f B"
   by (metis binaryset_sums sums_unique)
 
+lemma binaryset_psuminf:
+  assumes "f {} = 0"
+  shows "(\<Sum>\<^isub>\<infinity> n. f (binaryset A B n)) = f A + f B" (is "?suminf = ?sum")
+proof -
+  have *: "{..<2} = {0, 1::nat}" by auto
+  have "\<forall>n\<ge>2. f (binaryset A B n) = 0"
+    unfolding binaryset_def
+    using assms by auto
+  hence "?suminf = (\<Sum>N<2. f (binaryset A B N))"
+    by (rule psuminf_finite)
+  also have "... = ?sum" unfolding * binaryset_def
+    by simp
+  finally show ?thesis .
+qed
 
 subsection {* Lambda Systems *}
 
 lemma (in algebra) lambda_system_eq:
-    "lambda_system M f = 
+    "lambda_system M f =
         {l. l \<in> sets M & (\<forall>x \<in> sets M. f (x \<inter> l) + f (x - l) = f x)}"
 proof -
   have [simp]: "!!l x. l \<in> sets M \<Longrightarrow> x \<in> sets M \<Longrightarrow> (space M - l) \<inter> x = x - l"
     by (metis Int_Diff Int_absorb1 Int_commute sets_into_space)
   show ?thesis
     by (auto simp add: lambda_system_def) (metis Int_commute)+
 qed
 
 lemma (in algebra) lambda_system_empty:
-    "positive M f \<Longrightarrow> {} \<in> lambda_system M f"
-  by (auto simp add: positive_def lambda_system_eq) 
+  "positive f \<Longrightarrow> {} \<in> lambda_system M f"
+  by (auto simp add: positive_def lambda_system_eq)
 
 lemma lambda_system_sets:
     "x \<in> lambda_system M f \<Longrightarrow> x \<in> sets M"
   by (simp add:  lambda_system_def)
 
 lemma (in algebra) lambda_system_Compl:
-  fixes f:: "'a set \<Rightarrow> real"
+  fixes f:: "'a set \<Rightarrow> pinfreal"
   assumes x: "x \<in> lambda_system M f"
   shows "space M - x \<in> lambda_system M f"
   proof -
     have "x \<subseteq> space M"
       by (metis sets_into_space lambda_system_sets x)
     hence "space M - (space M - x) = x"
-      by (metis double_diff equalityE) 
+      by (metis double_diff equalityE)
     with x show ?thesis
-      by (force simp add: lambda_system_def)
+      by (force simp add: lambda_system_def ac_simps)
   qed
 
 lemma (in algebra) lambda_system_Int:
-  fixes f:: "'a set \<Rightarrow> real"
+  fixes f:: "'a set \<Rightarrow> pinfreal"
   assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
   shows "x \<inter> y \<in> lambda_system M f"
   proof -
     from xl yl show ?thesis
       proof (auto simp add: positive_def lambda_system_eq Int)

         moreover
         have "u - x \<inter> y - y = u - y" by blast
         ultimately
         have ey: "f (u - x \<inter> y) = f (u \<inter> y - x) + f (u - y)" using fy
           by force
-        have "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) 
+        have "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y)
               = (f (u \<inter> (x \<inter> y)) + f (u \<inter> y - x)) + f (u - y)"
-          by (simp add: ey) 
+          by (simp add: ey ac_simps)
         also have "... =  (f ((u \<inter> y) \<inter> x) + f (u \<inter> y - x)) + f (u - y)"
-          by (simp add: Int_ac) 
+          by (simp add: Int_ac)
         also have "... = f (u \<inter> y) + f (u - y)"
           using fx [THEN bspec, of "u \<inter> y"] Int y u
           by force
         also have "... = f u"
-          by (metis fy u) 
+          by (metis fy u)
         finally show "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) = f u" .
       qed
   qed
 
 
 lemma (in algebra) lambda_system_Un:
-  fixes f:: "'a set \<Rightarrow> real"
+  fixes f:: "'a set \<Rightarrow> pinfreal"
   assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
   shows "x \<union> y \<in> lambda_system M f"
 proof -
   have "(space M - x) \<inter> (space M - y) \<in> sets M"
-    by (metis Diff_Un Un compl_sets lambda_system_sets xl yl) 
+    by (metis Diff_Un Un compl_sets lambda_system_sets xl yl)
   moreover
   have "x \<union> y = space M - ((space M - x) \<inter> (space M - y))"
     by auto  (metis subsetD lambda_system_sets sets_into_space xl yl)+
   ultimately show ?thesis
-    by (metis lambda_system_Compl lambda_system_Int xl yl) 
+    by (metis lambda_system_Compl lambda_system_Int xl yl)
 qed
 
 lemma (in algebra) lambda_system_algebra:
-    "positive M f \<Longrightarrow> algebra (M (|sets := lambda_system M f|))"
-  apply (auto simp add: algebra_def) 
+  "positive f \<Longrightarrow> algebra (M (|sets := lambda_system M f|))"
+  apply (auto simp add: algebra_def)
   apply (metis lambda_system_sets set_mp sets_into_space)
   apply (metis lambda_system_empty)
   apply (metis lambda_system_Compl)
-  apply (metis lambda_system_Un) 
+  apply (metis lambda_system_Un)
   done
 
 lemma (in algebra) lambda_system_strong_additive:
   assumes z: "z \<in> sets M" and disj: "x \<inter> y = {}"
       and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
   shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)"
   proof -
     have "z \<inter> x = (z \<inter> (x \<union> y)) \<inter> x" using disj by blast
     moreover
     have "z \<inter> y = (z \<inter> (x \<union> y)) - x" using disj by blast
-    moreover 
+    moreover
     have "(z \<inter> (x \<union> y)) \<in> sets M"
-      by (metis Int Un lambda_system_sets xl yl z) 
+      by (metis Int Un lambda_system_sets xl yl z)
     ultimately show ?thesis using xl yl
       by (simp add: lambda_system_eq)
   qed
-
-lemma (in algebra) Int_space_eq1 [simp]: "x \<in> sets M \<Longrightarrow> space M \<inter> x = x"
-  by (metis Int_absorb1 sets_into_space)
-
-lemma (in algebra) Int_space_eq2 [simp]: "x \<in> sets M \<Longrightarrow> x \<inter> space M = x"
-  by (metis Int_absorb2 sets_into_space)
 
 lemma (in algebra) lambda_system_additive:
      "additive (M (|sets := lambda_system M f|)) f"
   proof (auto simp add: additive_def)
     fix x and y
     assume disj: "x \<inter> y = {}"
        and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
     hence  "x \<in> sets M" "y \<in> sets M" by (blast intro: lambda_system_sets)+
-    thus "f (x \<union> y) = f x + f y" 
+    thus "f (x \<union> y) = f x + f y"
       using lambda_system_strong_additive [OF top disj xl yl]
       by (simp add: Un)
   qed
 
 
 lemma (in algebra) countably_subadditive_subadditive:
-  assumes f: "positive M f" and cs: "countably_subadditive M f"
+  assumes f: "positive f" and cs: "countably_subadditive M f"
   shows  "subadditive M f"
 proof (auto simp add: subadditive_def)
   fix x y
   assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}"
   hence "disjoint_family (binaryset x y)"
     by (auto simp add: disjoint_family_on_def binaryset_def)
   hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
          (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
-         summable (f o (binaryset x y)) \<longrightarrow>
-         f (\<Union>i. binaryset x y i) \<le> suminf (\<lambda>n. f (binaryset x y n))"
+         f (\<Union>i. binaryset x y i) \<le> (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
     using cs by (simp add: countably_subadditive_def)
   hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
-         summable (f o (binaryset x y)) \<longrightarrow>
-         f (x \<union> y) \<le> suminf (\<lambda>n. f (binaryset x y n))"
+         f (x \<union> y) \<le> (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
     by (simp add: range_binaryset_eq UN_binaryset_eq)
-  thus "f (x \<union> y) \<le>  f x + f y" using f x y binaryset_sums
-    by (auto simp add: Un sums_iff positive_def o_def)
-qed
-
-
-definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set "
-  where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"
-
-lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)"
+  thus "f (x \<union> y) \<le>  f x + f y" using f x y
+    by (auto simp add: Un o_def binaryset_psuminf positive_def)
+qed
+
+lemma (in algebra) additive_sum:
+  fixes A:: "nat \<Rightarrow> 'a set"
+  assumes f: "positive f" and ad: "additive M f"
+      and A: "range A \<subseteq> sets M"
+      and disj: "disjoint_family A"
+  shows  "setsum (f \<circ> A) {0..<n} = f (\<Union>i\<in>{0..<n}. A i)"
 proof (induct n)
-  case 0 show ?case by simp
+  case 0 show ?case using f by (simp add: positive_def)
 next
   case (Suc n)
-  thus ?case by (simp add: atLeastLessThanSuc disjointed_def)
-qed
-
-lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)"
-  apply (rule UN_finite2_eq [where k=0])
-  apply (simp add: finite_UN_disjointed_eq)
-  done
-
-lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}"
-  by (auto simp add: disjointed_def)
-
-lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
-  by (simp add: disjoint_family_on_def)
-     (metis neq_iff Int_commute less_disjoint_disjointed)
-
-lemma disjointed_subset: "disjointed A n \<subseteq> A n"
-  by (auto simp add: disjointed_def)
-
-
-lemma (in algebra) UNION_in_sets:
-  fixes A:: "nat \<Rightarrow> 'a set"
-  assumes A: "range A \<subseteq> sets M "
-  shows  "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
-proof (induct n)
-  case 0 show ?case by simp
-next
-  case (Suc n) 
-  thus ?case
-    by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff)
-qed
-
-lemma (in algebra) range_disjointed_sets:
-  assumes A: "range A \<subseteq> sets M "
-  shows  "range (disjointed A) \<subseteq> sets M"
-proof (auto simp add: disjointed_def) 
-  fix n
-  show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> sets M" using UNION_in_sets
-    by (metis A Diff UNIV_I image_subset_iff)
-qed
-
-lemma sigma_algebra_disjoint_iff: 
-     "sigma_algebra M \<longleftrightarrow> 
-      algebra M &
-      (\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> 
-           (\<Union>i::nat. A i) \<in> sets M)"
-proof (auto simp add: sigma_algebra_iff)
-  fix A :: "nat \<Rightarrow> 'a set"
-  assume M: "algebra M"
-     and A: "range A \<subseteq> sets M"
-     and UnA: "\<forall>A. range A \<subseteq> sets M \<longrightarrow>
-               disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
-  hence "range (disjointed A) \<subseteq> sets M \<longrightarrow>
-         disjoint_family (disjointed A) \<longrightarrow>
-         (\<Union>i. disjointed A i) \<in> sets M" by blast
-  hence "(\<Union>i. disjointed A i) \<in> sets M"
-    by (simp add: algebra.range_disjointed_sets M A disjoint_family_disjointed) 
-  thus "(\<Union>i::nat. A i) \<in> sets M" by (simp add: UN_disjointed_eq)
-qed
-
-
-lemma (in algebra) additive_sum:
-  fixes A:: "nat \<Rightarrow> 'a set"
-  assumes f: "positive M f" and ad: "additive M f"
-      and A: "range A \<subseteq> sets M"
-      and disj: "disjoint_family A"
-  shows  "setsum (f o A) {0..<n} = f (\<Union>i\<in>{0..<n}. A i)"
-proof (induct n)
-  case 0 show ?case using f by (simp add: positive_def) 
-next
-  case (Suc n) 
-  have "A n \<inter> (\<Union>i\<in>{0..<n}. A i) = {}" using disj 
+  have "A n \<inter> (\<Union>i\<in>{0..<n}. A i) = {}" using disj
     by (auto simp add: disjoint_family_on_def neq_iff) blast
-  moreover 
-  have "A n \<in> sets M" using A by blast 
+  moreover
+  have "A n \<in> sets M" using A by blast
   moreover have "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
     by (metis A UNION_in_sets atLeast0LessThan)
-  moreover 
+  moreover
   ultimately have "f (A n \<union> (\<Union>i\<in>{0..<n}. A i)) = f (A n) + f(\<Union>i\<in>{0..<n}. A i)"
-    using ad UNION_in_sets A by (auto simp add: additive_def) 
+    using ad UNION_in_sets A by (auto simp add: additive_def)
   with Suc.hyps show ?case using ad
-    by (auto simp add: atLeastLessThanSuc additive_def) 
+    by (auto simp add: atLeastLessThanSuc additive_def)
 qed
 
 
 lemma countably_subadditiveD:
   "countably_subadditive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow>
-   (\<Union>i. A i) \<in> sets M \<Longrightarrow> summable (f o A) \<Longrightarrow> f (\<Union>i. A i) \<le> suminf (f o A)" 
+   (\<Union>i. A i) \<in> sets M \<Longrightarrow> f (\<Union>i. A i) \<le> psuminf (f o A)"
   by (auto simp add: countably_subadditive_def o_def)
 
-lemma (in algebra) increasing_additive_summable:
-  fixes A:: "nat \<Rightarrow> 'a set"
-  assumes f: "positive M f" and ad: "additive M f"
+lemma (in algebra) increasing_additive_bound:
+  fixes A:: "nat \<Rightarrow> 'a set" and  f :: "'a set \<Rightarrow> pinfreal"
+  assumes f: "positive f" and ad: "additive M f"
       and inc: "increasing M f"
       and A: "range A \<subseteq> sets M"
       and disj: "disjoint_family A"
-  shows  "summable (f o A)"
-proof (rule pos_summable) 
-  fix n
-  show "0 \<le> (f \<circ> A) n" using f A
-    by (force simp add: positive_def)
-  next
-  fix n
-  have "setsum (f \<circ> A) {0..<n} = f (\<Union>i\<in>{0..<n}. A i)"
-    by (rule additive_sum [OF f ad A disj]) 
+  shows  "psuminf (f \<circ> A) \<le> f (space M)"
+proof (safe intro!: psuminf_bound)
+  fix N
+  have "setsum (f \<circ> A) {0..<N} = f (\<Union>i\<in>{0..<N}. A i)"
+    by (rule additive_sum [OF f ad A disj])
   also have "... \<le> f (space M)" using space_closed A
-    by (blast intro: increasingD [OF inc] UNION_in_sets top) 
-  finally show "setsum (f \<circ> A) {0..<n} \<le> f (space M)" .
-qed
-
-lemma lambda_system_positive:
-     "positive M f \<Longrightarrow> positive (M (|sets := lambda_system M f|)) f"
-  by (simp add: positive_def lambda_system_def) 
+    by (blast intro: increasingD [OF inc] UNION_in_sets top)
+  finally show "setsum (f \<circ> A) {..<N} \<le> f (space M)" by (simp add: atLeast0LessThan)
+qed
 
 lemma lambda_system_increasing:
    "increasing M f \<Longrightarrow> increasing (M (|sets := lambda_system M f|)) f"
-  by (simp add: increasing_def lambda_system_def) 
+  by (simp add: increasing_def lambda_system_def)
 
 lemma (in algebra) lambda_system_strong_sum:
-  fixes A:: "nat \<Rightarrow> 'a set"
-  assumes f: "positive M f" and a: "a \<in> sets M"
+  fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> pinfreal"
+  assumes f: "positive f" and a: "a \<in> sets M"
       and A: "range A \<subseteq> lambda_system M f"
       and disj: "disjoint_family A"
   shows  "(\<Sum>i = 0..<n. f (a \<inter>A i)) = f (a \<inter> (\<Union>i\<in>{0..<n}. A i))"
 proof (induct n)
-  case 0 show ?case using f by (simp add: positive_def) 
+  case 0 show ?case using f by (simp add: positive_def)
 next
-  case (Suc n) 
+  case (Suc n)
   have 2: "A n \<inter> UNION {0..<n} A = {}" using disj
-    by (force simp add: disjoint_family_on_def neq_iff) 
+    by (force simp add: disjoint_family_on_def neq_iff)
   have 3: "A n \<in> lambda_system M f" using A
     by blast
   have 4: "UNION {0..<n} A \<in> lambda_system M f"
-    using A algebra.UNION_in_sets [OF local.lambda_system_algebra [OF f]] 
+    using A algebra.UNION_in_sets [OF local.lambda_system_algebra, of f, OF f]
     by simp
   from Suc.hyps show ?case
     by (simp add: atLeastLessThanSuc lambda_system_strong_additive [OF a 2 3 4])
 qed
 
 
 lemma (in sigma_algebra) lambda_system_caratheodory:
   assumes oms: "outer_measure_space M f"
       and A: "range A \<subseteq> lambda_system M f"
       and disj: "disjoint_family A"
-  shows  "(\<Union>i. A i) \<in> lambda_system M f & (f \<circ> A)  sums  f (\<Union>i. A i)"
+  shows  "(\<Union>i. A i) \<in> lambda_system M f \<and> psuminf (f \<circ> A) = f (\<Union>i. A i)"
 proof -
-  have pos: "positive M f" and inc: "increasing M f" 
-   and csa: "countably_subadditive M f" 
+  have pos: "positive f" and inc: "increasing M f"
+   and csa: "countably_subadditive M f"
     by (metis oms outer_measure_space_def)+
   have sa: "subadditive M f"
-    by (metis countably_subadditive_subadditive csa pos) 
-  have A': "range A \<subseteq> sets (M(|sets := lambda_system M f|))" using A 
+    by (metis countably_subadditive_subadditive csa pos)
+  have A': "range A \<subseteq> sets (M(|sets := lambda_system M f|))" using A
     by simp
   have alg_ls: "algebra (M(|sets := lambda_system M f|))"
-    by (rule lambda_system_algebra [OF pos]) 
+    by (rule lambda_system_algebra) (rule pos)
   have A'': "range A \<subseteq> sets M"
      by (metis A image_subset_iff lambda_system_sets)
-  have sumfA: "summable (f \<circ> A)" 
-    by (metis algebra.increasing_additive_summable [OF alg_ls]
-          lambda_system_positive lambda_system_additive lambda_system_increasing
-          A' oms outer_measure_space_def disj)
+
   have U_in: "(\<Union>i. A i) \<in> sets M"
     by (metis A'' countable_UN)
-  have U_eq: "f (\<Union>i. A i) = suminf (f o A)" 
+  have U_eq: "f (\<Union>i. A i) = psuminf (f o A)"
     proof (rule antisym)
-      show "f (\<Union>i. A i) \<le> suminf (f \<circ> A)"
-        by (rule countably_subadditiveD [OF csa A'' disj U_in sumfA]) 
-      show "suminf (f \<circ> A) \<le> f (\<Union>i. A i)"
-        by (rule suminf_le [OF sumfA]) 
+      show "f (\<Union>i. A i) \<le> psuminf (f \<circ> A)"
+        by (rule countably_subadditiveD [OF csa A'' disj U_in])
+      show "psuminf (f \<circ> A) \<le> f (\<Union>i. A i)"
+        by (rule psuminf_bound, unfold atLeast0LessThan[symmetric])
            (metis algebra.additive_sum [OF alg_ls] pos disj UN_Un Un_UNIV_right
-                  lambda_system_positive lambda_system_additive 
-                  subset_Un_eq increasingD [OF inc] A' A'' UNION_in_sets U_in) 
+                  lambda_system_additive subset_Un_eq increasingD [OF inc]
+                  A' A'' UNION_in_sets U_in)
     qed
   {
-    fix a 
-    assume a [iff]: "a \<in> sets M" 
+    fix a
+    assume a [iff]: "a \<in> sets M"
     have "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) = f a"
     proof -
-      have summ: "summable (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)" using pos A'' 
-        apply -
-        apply (rule summable_comparison_test [OF _ sumfA]) 
-        apply (rule_tac x="0" in exI) 
-        apply (simp add: positive_def) 
-        apply (auto simp add: )
-        apply (subst abs_of_nonneg)
-        apply (metis A'' Int UNIV_I a image_subset_iff)
-        apply (blast intro:  increasingD [OF inc])
-        done
       show ?thesis
       proof (rule antisym)
         have "range (\<lambda>i. a \<inter> A i) \<subseteq> sets M" using A''
           by blast
-        moreover 
+        moreover
         have "disjoint_family (\<lambda>i. a \<inter> A i)" using disj
-          by (auto simp add: disjoint_family_on_def) 
-        moreover 
+          by (auto simp add: disjoint_family_on_def)
+        moreover
         have "a \<inter> (\<Union>i. A i) \<in> sets M"
           by (metis Int U_in a)
-        ultimately 
-        have "f (a \<inter> (\<Union>i. A i)) \<le> suminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)"
-          using countably_subadditiveD [OF csa, of "(\<lambda>i. a \<inter> A i)"] summ
-          by (simp add: o_def) 
-        moreover 
-        have "suminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)  \<le> f a - f (a - (\<Union>i. A i))"
-          proof (rule suminf_le [OF summ])
+        ultimately
+        have "f (a \<inter> (\<Union>i. A i)) \<le> psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)"
+          using countably_subadditiveD [OF csa, of "(\<lambda>i. a \<inter> A i)"]
+          by (simp add: o_def)
+        hence "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le>
+            psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) + f (a - (\<Union>i. A i))"
+          by (rule add_right_mono)
+        moreover
+        have "psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) + f (a - (\<Union>i. A i)) \<le> f a"
+          proof (safe intro!: psuminf_bound_add)
             fix n
             have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
-              by (metis A'' UNION_in_sets) 
+              by (metis A'' UNION_in_sets)
             have le_fa: "f (UNION {0..<n} A \<inter> a) \<le> f a" using A''
               by (blast intro: increasingD [OF inc] A'' UNION_in_sets)
             have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system M f"
-              using algebra.UNION_in_sets [OF lambda_system_algebra [OF pos]]
-              by (simp add: A) 
-            hence eq_fa: "f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i)) = f a"
+              using algebra.UNION_in_sets [OF lambda_system_algebra [of f, OF pos]]
+              by (simp add: A)
+            hence eq_fa: "f a = f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i))"
               by (simp add: lambda_system_eq UNION_in)
             have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))"
-              by (blast intro: increasingD [OF inc] UNION_eq_Union_image 
+              by (blast intro: increasingD [OF inc] UNION_eq_Union_image
                                UNION_in U_in)
-            thus "setsum (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) {0..<n} \<le> f a - f (a - (\<Union>i. A i))"
-              using eq_fa
-              by (simp add: suminf_le [OF summ] lambda_system_strong_sum pos 
-                            a A disj)
+            thus "setsum (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) {..<n} + f (a - (\<Union>i. A i)) \<le> f a"
+              by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric])
           qed
-        ultimately show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a" 
-          by arith
+        ultimately show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a"
+          by (rule order_trans)
       next
-        have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))" 
+        have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))"
           by (blast intro:  increasingD [OF inc] U_in)
         also have "... \<le>  f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))"
           by (blast intro: subadditiveD [OF sa] U_in)
         finally show "f a \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" .
         qed
      qed
   }
   thus  ?thesis
-    by (simp add: lambda_system_eq sums_iff U_eq U_in sumfA)
+    by (simp add: lambda_system_eq sums_iff U_eq U_in)
 qed
 
 lemma (in sigma_algebra) caratheodory_lemma:
   assumes oms: "outer_measure_space M f"
-  shows "measure_space (|space = space M, sets = lambda_system M f, measure = f|)"
+  shows "measure_space (|space = space M, sets = lambda_system M f|) f"
 proof -
-  have pos: "positive M f" 
+  have pos: "positive f"
     by (metis oms outer_measure_space_def)
-  have alg: "algebra (|space = space M, sets = lambda_system M f, measure = f|)"
-    using lambda_system_algebra [OF pos]
-    by (simp add: algebra_def) 
-  then moreover 
-  have "sigma_algebra (|space = space M, sets = lambda_system M f, measure = f|)"
+  have alg: "algebra (|space = space M, sets = lambda_system M f|)"
+    using lambda_system_algebra [of f, OF pos]
+    by (simp add: algebra_def)
+  then moreover
+  have "sigma_algebra (|space = space M, sets = lambda_system M f|)"
     using lambda_system_caratheodory [OF oms]
-    by (simp add: sigma_algebra_disjoint_iff) 
-  moreover 
-  have "measure_space_axioms (|space = space M, sets = lambda_system M f, measure = f|)" 
+    by (simp add: sigma_algebra_disjoint_iff)
+  moreover
+  have "measure_space_axioms (|space = space M, sets = lambda_system M f|) f"
     using pos lambda_system_caratheodory [OF oms]
-    by (simp add: measure_space_axioms_def positive_def lambda_system_sets 
-                  countably_additive_def o_def) 
-  ultimately 
+    by (simp add: measure_space_axioms_def positive_def lambda_system_sets
+                  countably_additive_def o_def)
+  ultimately
   show ?thesis
-    by intro_locales (auto simp add: sigma_algebra_def) 
+    by intro_locales (auto simp add: sigma_algebra_def)
 qed
 
 
 lemma (in algebra) inf_measure_nonempty:
-  assumes f: "positive M f" and b: "b \<in> sets M" and a: "a \<subseteq> b"
+  assumes f: "positive f" and b: "b \<in> sets M" and a: "a \<subseteq> b"
   shows "f b \<in> measure_set M f a"
 proof -
-  have "(f \<circ> (\<lambda>i. {})(0 := b)) sums setsum (f \<circ> (\<lambda>i. {})(0 := b)) {0..<1::nat}"
-    by (rule series_zero)  (simp add: positive_imp_0 [OF f]) 
-  also have "... = f b" 
+  have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = setsum (f \<circ> (\<lambda>i. {})(0 := b)) {..<1::nat}"
+    by (rule psuminf_finite) (simp add: f[unfolded positive_def])
+  also have "... = f b"
     by simp
-  finally have "(f \<circ> (\<lambda>i. {})(0 := b)) sums f b" .
-  thus ?thesis using a
-    by (auto intro!: exI [of _ "(\<lambda>i. {})(0 := b)"] 
-             simp add: measure_set_def disjoint_family_on_def b split_if_mem2) 
-qed  
-
-lemma (in algebra) inf_measure_pos0:
-     "positive M f \<Longrightarrow> x \<in> measure_set M f a \<Longrightarrow> 0 \<le> x"
-apply (auto simp add: positive_def measure_set_def sums_iff intro!: suminf_ge_zero)
-apply blast
-done
-
-lemma (in algebra) inf_measure_pos:
-  shows "positive M f \<Longrightarrow> x \<subseteq> space M \<Longrightarrow> 0 \<le> Inf (measure_set M f x)"
-apply (rule Inf_greatest)
-apply (metis emptyE inf_measure_nonempty top)
-apply (metis inf_measure_pos0) 
-done
+  finally have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = f b" .
+  thus ?thesis using a b
+    by (auto intro!: exI [of _ "(\<lambda>i. {})(0 := b)"]
+             simp: measure_set_def disjoint_family_on_def split_if_mem2 comp_def)
+qed
 
 lemma (in algebra) additive_increasing:
-  assumes posf: "positive M f" and addf: "additive M f" 
+  assumes posf: "positive f" and addf: "additive M f"
   shows "increasing M f"
-proof (auto simp add: increasing_def) 
+proof (auto simp add: increasing_def)
   fix x y
   assume xy: "x \<in> sets M" "y \<in> sets M" "x \<subseteq> y"
-  have "f x \<le> f x + f (y-x)" using posf
-    by (simp add: positive_def) (metis Diff xy(1,2))
+  have "f x \<le> f x + f (y-x)" ..
   also have "... = f (x \<union> (y-x))" using addf
     by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
   also have "... = f y"
     by (metis Un_Diff_cancel Un_absorb1 xy(3))
   finally show "f x \<le> f y" .
 qed
 
 lemma (in algebra) countably_additive_additive:
-  assumes posf: "positive M f" and ca: "countably_additive M f" 
+  assumes posf: "positive f" and ca: "countably_additive M f"
   shows "additive M f"
-proof (auto simp add: additive_def) 
+proof (auto simp add: additive_def)
   fix x y
   assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}"
   hence "disjoint_family (binaryset x y)"
-    by (auto simp add: disjoint_family_on_def binaryset_def) 
-  hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow> 
-         (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow> 
-         f (\<Union>i. binaryset x y i) = suminf (\<lambda>n. f (binaryset x y n))"
+    by (auto simp add: disjoint_family_on_def binaryset_def)
+  hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
+         (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
+         f (\<Union>i. binaryset x y i) = (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
     using ca
-    by (simp add: countably_additive_def) (metis UN_binaryset_eq sums_unique) 
-  hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow> 
-         f (x \<union> y) = suminf (\<lambda>n. f (binaryset x y n))"
+    by (simp add: countably_additive_def)
+  hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
+         f (x \<union> y) = (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
     by (simp add: range_binaryset_eq UN_binaryset_eq)
   thus "f (x \<union> y) = f x + f y" using posf x y
-    by (simp add: Un suminf_binaryset_eq positive_def)
-qed 
- 
+    by (auto simp add: Un binaryset_psuminf positive_def)
+qed
+
 lemma (in algebra) inf_measure_agrees:
-  assumes posf: "positive M f" and ca: "countably_additive M f" 
-      and s: "s \<in> sets M"  
+  assumes posf: "positive f" and ca: "countably_additive M f"
+      and s: "s \<in> sets M"
   shows "Inf (measure_set M f s) = f s"
-proof (rule Inf_eq) 
+  unfolding Inf_pinfreal_def
+proof (safe intro!: Greatest_equality)
   fix z
   assume z: "z \<in> measure_set M f s"
-  from this obtain A where 
+  from this obtain A where
     A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
-    and "s \<subseteq> (\<Union>x. A x)" and sm: "summable (f \<circ> A)"
-    and si: "suminf (f \<circ> A) = z"
-    by (auto simp add: measure_set_def sums_iff) 
+    and "s \<subseteq> (\<Union>x. A x)" and si: "psuminf (f \<circ> A) = z"
+    by (auto simp add: measure_set_def comp_def)
   hence seq: "s = (\<Union>i. A i \<inter> s)" by blast
   have inc: "increasing M f"
     by (metis additive_increasing ca countably_additive_additive posf)
-  have sums: "(\<lambda>i. f (A i \<inter> s)) sums f (\<Union>i. A i \<inter> s)"
-    proof (rule countably_additiveD [OF ca]) 
+  have sums: "psuminf (\<lambda>i. f (A i \<inter> s)) = f (\<Union>i. A i \<inter> s)"
+    proof (rule countably_additiveD [OF ca])
       show "range (\<lambda>n. A n \<inter> s) \<subseteq> sets M" using A s
         by blast
       show "disjoint_family (\<lambda>n. A n \<inter> s)" using disj
         by (auto simp add: disjoint_family_on_def)
       show "(\<Union>i. A i \<inter> s) \<in> sets M" using A s
         by (metis UN_extend_simps(4) s seq)
     qed
-  hence "f s = suminf (\<lambda>i. f (A i \<inter> s))"
+  hence "f s = psuminf (\<lambda>i. f (A i \<inter> s))"
     using seq [symmetric] by (simp add: sums_iff)
-  also have "... \<le> suminf (f \<circ> A)" 
-    proof (rule summable_le [OF _ _ sm]) 
-      show "\<forall>n. f (A n \<inter> s) \<le> (f \<circ> A) n" using A s
-        by (force intro: increasingD [OF inc]) 
-      show "summable (\<lambda>i. f (A i \<inter> s))" using sums
-        by (simp add: sums_iff) 
+  also have "... \<le> psuminf (f \<circ> A)"
+    proof (rule psuminf_le)
+      fix n show "f (A n \<inter> s) \<le> (f \<circ> A) n" using A s
+        by (force intro: increasingD [OF inc])
     qed
-  also have "... = z" by (rule si) 
+  also have "... = z" by (rule si)
   finally show "f s \<le> z" .
 next
   fix y
-  assume y: "!!u. u \<in> measure_set M f s \<Longrightarrow> y \<le> u"
+  assume y: "\<forall>u \<in> measure_set M f s. y \<le> u"
   thus "y \<le> f s"
-    by (blast intro: inf_measure_nonempty [OF posf s subset_refl])
+    by (blast intro: inf_measure_nonempty [of f, OF posf s subset_refl])
 qed
 
 lemma (in algebra) inf_measure_empty:
-  assumes posf: "positive M f"
+  assumes posf: "positive f"
   shows "Inf (measure_set M f {}) = 0"
 proof (rule antisym)
-  show "0 \<le> Inf (measure_set M f {})"
-    by (metis empty_subsetI inf_measure_pos posf) 
   show "Inf (measure_set M f {}) \<le> 0"
-    by (metis Inf_lower empty_sets inf_measure_pos0 inf_measure_nonempty posf
-              positive_imp_0 subset_refl) 
-qed
+    by (metis complete_lattice_class.Inf_lower empty_sets inf_measure_nonempty[OF posf] subset_refl posf[unfolded positive_def])
+qed simp
 
 lemma (in algebra) inf_measure_positive:
-  "positive M f \<Longrightarrow> 
-   positive (| space = space M, sets = Pow (space M) |)
-                  (\<lambda>x. Inf (measure_set M f x))"
-  by (simp add: positive_def inf_measure_empty inf_measure_pos) 
+  "positive f \<Longrightarrow>
+   positive (\<lambda>x. Inf (measure_set M f x))"
+  by (simp add: positive_def inf_measure_empty) 
 
 lemma (in algebra) inf_measure_increasing:
-  assumes posf: "positive M f"
+  assumes posf: "positive f"
   shows "increasing (| space = space M, sets = Pow (space M) |)
                     (\<lambda>x. Inf (measure_set M f x))"
-apply (auto simp add: increasing_def) 
-apply (rule Inf_greatest, metis emptyE inf_measure_nonempty top posf)
-apply (rule Inf_lower) 
+apply (auto simp add: increasing_def)
+apply (rule complete_lattice_class.Inf_greatest)
+apply (rule complete_lattice_class.Inf_lower)
 apply (clarsimp simp add: measure_set_def, rule_tac x=A in exI, blast)
-apply (blast intro: inf_measure_pos0 posf)
 done
 
 
 lemma (in algebra) inf_measure_le:
-  assumes posf: "positive M f" and inc: "increasing M f" 
-      and x: "x \<in> {r . \<exists>A. range A \<subseteq> sets M & s \<subseteq> (\<Union>i. A i) & (f \<circ> A) sums r}"
+  assumes posf: "positive f" and inc: "increasing M f"
+      and x: "x \<in> {r . \<exists>A. range A \<subseteq> sets M \<and> s \<subseteq> (\<Union>i. A i) \<and> psuminf (f \<circ> A) = r}"
   shows "Inf (measure_set M f s) \<le> x"
 proof -
   from x
-  obtain A where A: "range A \<subseteq> sets M" and ss: "s \<subseteq> (\<Union>i. A i)" 
-             and sm: "summable (f \<circ> A)" and xeq: "suminf (f \<circ> A) = x"
-    by (auto simp add: sums_iff)
+  obtain A where A: "range A \<subseteq> sets M" and ss: "s \<subseteq> (\<Union>i. A i)"
+             and xeq: "psuminf (f \<circ> A) = x"
+    by auto
   have dA: "range (disjointed A) \<subseteq> sets M"
     by (metis A range_disjointed_sets)
-  have "\<forall>n. \<bar>(f o disjointed A) n\<bar> \<le> (f \<circ> A) n"
-    proof (auto)
-      fix n
-      have "\<bar>f (disjointed A n)\<bar> = f (disjointed A n)" using posf dA
-        by (auto simp add: positive_def image_subset_iff)
-      also have "... \<le> f (A n)" 
-        by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A)
-      finally show "\<bar>f (disjointed A n)\<bar> \<le> f (A n)" .
-    qed
-  from Series.summable_le2 [OF this sm]
-  have sda:  "summable (f o disjointed A)"  
-             "suminf (f o disjointed A) \<le> suminf (f \<circ> A)"
-    by blast+
-  hence ley: "suminf (f o disjointed A) \<le> x"
-    by (metis xeq) 
-  from sda have "(f \<circ> disjointed A) sums suminf (f \<circ> disjointed A)"
-    by (simp add: sums_iff) 
-  hence y: "suminf (f o disjointed A) \<in> measure_set M f s"
+  have "\<forall>n.(f o disjointed A) n \<le> (f \<circ> A) n" unfolding comp_def
+    by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A comp_def)
+  hence sda: "psuminf (f o disjointed A) \<le> psuminf (f \<circ> A)"
+    by (blast intro: psuminf_le)
+  hence ley: "psuminf (f o disjointed A) \<le> x"
+    by (metis xeq)
+  hence y: "psuminf (f o disjointed A) \<in> measure_set M f s"
     apply (auto simp add: measure_set_def)
-    apply (rule_tac x="disjointed A" in exI) 
-    apply (simp add: disjoint_family_disjointed UN_disjointed_eq ss dA)
+    apply (rule_tac x="disjointed A" in exI)
+    apply (simp add: disjoint_family_disjointed UN_disjointed_eq ss dA comp_def)
     done
   show ?thesis
-    by (blast intro: y order_trans [OF _ ley] inf_measure_pos0 posf)
+    by (blast intro: y order_trans [OF _ ley] posf complete_lattice_class.Inf_lower)
 qed
 
 lemma (in algebra) inf_measure_close:
-  assumes posf: "positive M f" and e: "0 < e" and ss: "s \<subseteq> (space M)"
-  shows "\<exists>A l. range A \<subseteq> sets M & disjoint_family A & s \<subseteq> (\<Union>i. A i) & 
-               (f \<circ> A) sums l & l \<le> Inf (measure_set M f s) + e"
-proof -
-  have " measure_set M f s \<noteq> {}" 
-    by (metis emptyE ss inf_measure_nonempty [OF posf top])
-  hence "\<exists>l \<in> measure_set M f s. l < Inf (measure_set M f s) + e" 
-    by (rule Inf_close [OF _ e])
-  thus ?thesis 
-    by (auto simp add: measure_set_def, rule_tac x=" A" in exI, auto)
+  assumes posf: "positive f" and e: "0 < e" and ss: "s \<subseteq> (space M)"
+  shows "\<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> s \<subseteq> (\<Union>i. A i) \<and>
+               psuminf (f \<circ> A) \<le> Inf (measure_set M f s) + e"
+proof (cases "Inf (measure_set M f s) = \<omega>")
+  case False
+  obtain l where "l \<in> measure_set M f s" "l \<le> Inf (measure_set M f s) + e"
+    using Inf_close[OF False e] by auto
+  thus ?thesis
+    by (auto intro!: exI[of _ l] simp: measure_set_def comp_def)
+next
+  case True
+  have "measure_set M f s \<noteq> {}"
+    by (metis emptyE ss inf_measure_nonempty [of f, OF posf top])
+  then obtain l where "l \<in> measure_set M f s" by auto
+  moreover from True have "l \<le> Inf (measure_set M f s) + e" by simp
+  ultimately show ?thesis
+    by (auto intro!: exI[of _ l] simp: measure_set_def comp_def)
 qed
 
 lemma (in algebra) inf_measure_countably_subadditive:
-  assumes posf: "positive M f" and inc: "increasing M f" 
+  assumes posf: "positive f" and inc: "increasing M f"
   shows "countably_subadditive (| space = space M, sets = Pow (space M) |)
                   (\<lambda>x. Inf (measure_set M f x))"
-proof (auto simp add: countably_subadditive_def o_def, rule field_le_epsilon)
-  fix A :: "nat \<Rightarrow> 'a set" and e :: real
-    assume A: "range A \<subseteq> Pow (space M)"
-       and disj: "disjoint_family A"
-       and sb: "(\<Union>i. A i) \<subseteq> space M"
-       and sum1: "summable (\<lambda>n. Inf (measure_set M f (A n)))"
-       and e: "0 < e"
-    have "!!n. \<exists>B l. range B \<subseteq> sets M \<and> disjoint_family B \<and> A n \<subseteq> (\<Union>i. B i) \<and>
-                    (f o B) sums l \<and>
-                    l \<le> Inf (measure_set M f (A n)) + e * (1/2)^(Suc n)"
-      apply (rule inf_measure_close [OF posf])
-      apply (metis e half mult_pos_pos zero_less_power) 
-      apply (metis UNIV_I UN_subset_iff sb)
-      done
-    hence "\<exists>BB ll. \<forall>n. range (BB n) \<subseteq> sets M \<and> disjoint_family (BB n) \<and>
-                       A n \<subseteq> (\<Union>i. BB n i) \<and> (f o BB n) sums ll n \<and>
-                       ll n \<le> Inf (measure_set M f (A n)) + e * (1/2)^(Suc n)"
-      by (rule choice2)
-    then obtain BB ll
-      where BB: "!!n. (range (BB n) \<subseteq> sets M)"
-        and disjBB: "!!n. disjoint_family (BB n)" 
-        and sbBB: "!!n. A n \<subseteq> (\<Union>i. BB n i)"
-        and BBsums: "!!n. (f o BB n) sums ll n"
-        and ll: "!!n. ll n \<le> Inf (measure_set M f (A n)) + e * (1/2)^(Suc n)"
-      by auto blast
-    have llpos: "!!n. 0 \<le> ll n"
-        by (metis BBsums sums_iff o_apply posf positive_imp_pos suminf_ge_zero 
-              range_subsetD BB) 
-    have sll: "summable ll &
-               suminf ll \<le> suminf (\<lambda>n. Inf (measure_set M f (A n))) + e"
-      proof -
-        have "(\<lambda>n. e * (1/2)^(Suc n)) sums (e*1)"
-          by (rule sums_mult [OF power_half_series]) 
-        hence sum0: "summable (\<lambda>n. e * (1 / 2) ^ Suc n)"
-          and eqe:  "(\<Sum>n. e * (1 / 2) ^ n / 2) = e"
-          by (auto simp add: sums_iff) 
-        have 0: "suminf (\<lambda>n. Inf (measure_set M f (A n))) +
-                 suminf (\<lambda>n. e * (1/2)^(Suc n)) =
-                 suminf (\<lambda>n. Inf (measure_set M f (A n)) + e * (1/2)^(Suc n))"
-          by (rule suminf_add [OF sum1 sum0]) 
-        have 1: "\<forall>n. \<bar>ll n\<bar> \<le> Inf (measure_set M f (A n)) + e * (1/2) ^ Suc n"
-          by (metis ll llpos abs_of_nonneg)
-        have 2: "summable (\<lambda>n. Inf (measure_set M f (A n)) + e*(1/2)^(Suc n))"
-          by (rule summable_add [OF sum1 sum0]) 
-        have "suminf ll \<le> (\<Sum>n. Inf (measure_set M f (A n)) + e*(1/2) ^ Suc n)"
-          using Series.summable_le2 [OF 1 2] by auto
-        also have "... = (\<Sum>n. Inf (measure_set M f (A n))) + 
-                         (\<Sum>n. e * (1 / 2) ^ Suc n)"
-          by (metis 0) 
-        also have "... = (\<Sum>n. Inf (measure_set M f (A n))) + e"
-          by (simp add: eqe) 
-        finally show ?thesis using  Series.summable_le2 [OF 1 2] by auto
-      qed
-    def C \<equiv> "(split BB) o prod_decode"
-    have C: "!!n. C n \<in> sets M"
-      apply (rule_tac p="prod_decode n" in PairE)
-      apply (simp add: C_def)
-      apply (metis BB subsetD rangeI)  
-      done
-    have sbC: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)"
-      proof (auto simp add: C_def)
-        fix x i
-        assume x: "x \<in> A i"
-        with sbBB [of i] obtain j where "x \<in> BB i j"
-          by blast        
-        thus "\<exists>i. x \<in> split BB (prod_decode i)"
-          by (metis prod_encode_inverse prod.cases)
-      qed 
-    have "(f \<circ> C) = (f \<circ> (\<lambda>(x, y). BB x y)) \<circ> prod_decode"
-      by (rule ext)  (auto simp add: C_def) 
-    also have "... sums suminf ll" 
-      proof (rule suminf_2dimen)
-        show "\<And>m n. 0 \<le> (f \<circ> (\<lambda>(x, y). BB x y)) (m, n)" using posf BB 
-          by (force simp add: positive_def)
-        show "\<And>m. (\<lambda>n. (f \<circ> (\<lambda>(x, y). BB x y)) (m, n)) sums ll m"using BBsums BB
-          by (force simp add: o_def)
-        show "summable ll" using sll
-          by auto
-      qed
-    finally have Csums: "(f \<circ> C) sums suminf ll" .
-    have "Inf (measure_set M f (\<Union>i. A i)) \<le> suminf ll"
-      apply (rule inf_measure_le [OF posf inc], auto)
-      apply (rule_tac x="C" in exI)
-      apply (auto simp add: C sbC Csums) 
-      done
-    also have "... \<le> (\<Sum>n. Inf (measure_set M f (A n))) + e" using sll
-      by blast
-    finally show "Inf (measure_set M f (\<Union>i. A i)) \<le> 
-          (\<Sum>n. Inf (measure_set M f (A n))) + e" .
+  unfolding countably_subadditive_def o_def
+proof (safe, simp, rule pinfreal_le_epsilon)
+  fix A :: "nat \<Rightarrow> 'a set" and e :: pinfreal
+
+  let "?outer n" = "Inf (measure_set M f (A n))"
+  assume A: "range A \<subseteq> Pow (space M)"
+     and disj: "disjoint_family A"
+     and sb: "(\<Union>i. A i) \<subseteq> space M"
+     and e: "0 < e"
+  hence "\<exists>BB. \<forall>n. range (BB n) \<subseteq> sets M \<and> disjoint_family (BB n) \<and>
+                   A n \<subseteq> (\<Union>i. BB n i) \<and>
+                   psuminf (f o BB n) \<le> ?outer n + e * (1/2)^(Suc n)"
+    apply (safe intro!: choice inf_measure_close [of f, OF posf _])
+    using e sb by (cases e, auto simp add: not_le mult_pos_pos)
+  then obtain BB
+    where BB: "\<And>n. (range (BB n) \<subseteq> sets M)"
+      and disjBB: "\<And>n. disjoint_family (BB n)"
+      and sbBB: "\<And>n. A n \<subseteq> (\<Union>i. BB n i)"
+      and BBle: "\<And>n. psuminf (f o BB n) \<le> ?outer n + e * (1/2)^(Suc n)"
+    by auto blast
+  have sll: "(\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n)) \<le> psuminf ?outer + e"
+    proof -
+      have "(\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n)) \<le> (\<Sum>\<^isub>\<infinity> n. ?outer n + e*(1/2) ^ Suc n)"
+        by (rule psuminf_le[OF BBle])
+      also have "... = psuminf ?outer + e"
+        using psuminf_half_series by simp
+      finally show ?thesis .
+    qed
+  def C \<equiv> "(split BB) o prod_decode"
+  have C: "!!n. C n \<in> sets M"
+    apply (rule_tac p="prod_decode n" in PairE)
+    apply (simp add: C_def)
+    apply (metis BB subsetD rangeI)
+    done
+  have sbC: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)"
+    proof (auto simp add: C_def)
+      fix x i
+      assume x: "x \<in> A i"
+      with sbBB [of i] obtain j where "x \<in> BB i j"
+        by blast
+      thus "\<exists>i. x \<in> split BB (prod_decode i)"
+        by (metis prod_encode_inverse prod.cases)
+    qed
+  have "(f \<circ> C) = (f \<circ> (\<lambda>(x, y). BB x y)) \<circ> prod_decode"
+    by (rule ext)  (auto simp add: C_def)
+  moreover have "psuminf ... = (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))" using BBle
+    by (force intro!: psuminf_2dimen simp: o_def)
+  ultimately have Csums: "psuminf (f \<circ> C) = (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))" by simp
+  have "Inf (measure_set M f (\<Union>i. A i)) \<le> (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))"
+    apply (rule inf_measure_le [OF posf(1) inc], auto)
+    apply (rule_tac x="C" in exI)
+    apply (auto simp add: C sbC Csums)
+    done
+  also have "... \<le> (\<Sum>\<^isub>\<infinity>n. Inf (measure_set M f (A n))) + e" using sll
+    by blast
+  finally show "Inf (measure_set M f (\<Union>i. A i)) \<le> psuminf ?outer + e" .
 qed
 
 lemma (in algebra) inf_measure_outer:
-  "positive M f \<Longrightarrow> increasing M f 
+  "\<lbrakk> positive f ; increasing M f \<rbrakk>
    \<Longrightarrow> outer_measure_space (| space = space M, sets = Pow (space M) |)
                           (\<lambda>x. Inf (measure_set M f x))"
-  by (simp add: outer_measure_space_def inf_measure_positive
-                inf_measure_increasing inf_measure_countably_subadditive) 
+  by (simp add: outer_measure_space_def inf_measure_empty
+                inf_measure_increasing inf_measure_countably_subadditive positive_def)
 
 (*MOVE UP*)
 
 lemma (in algebra) algebra_subset_lambda_system:
-  assumes posf: "positive M f" and inc: "increasing M f" 
+  assumes posf: "positive f" and inc: "increasing M f"
       and add: "additive M f"
   shows "sets M \<subseteq> lambda_system (| space = space M, sets = Pow (space M) |)
                                 (\<lambda>x. Inf (measure_set M f x))"
-proof (auto dest: sets_into_space 
-            simp add: algebra.lambda_system_eq [OF algebra_Pow]) 
+proof (auto dest: sets_into_space
+            simp add: algebra.lambda_system_eq [OF algebra_Pow])
   fix x s
   assume x: "x \<in> sets M"
      and s: "s \<subseteq> space M"
-  have [simp]: "!!x. x \<in> sets M \<Longrightarrow> s \<inter> (space M - x) = s-x" using s 
+  have [simp]: "!!x. x \<in> sets M \<Longrightarrow> s \<inter> (space M - x) = s-x" using s
     by blast
   have "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
         \<le> Inf (measure_set M f s)"
-    proof (rule field_le_epsilon) 
-      fix e :: real
+    proof (rule pinfreal_le_epsilon)
+      fix e :: pinfreal
       assume e: "0 < e"
-      from inf_measure_close [OF posf e s]
-      obtain A l where A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
-                   and sUN: "s \<subseteq> (\<Union>i. A i)" and fsums: "(f \<circ> A) sums l"
-                   and l: "l \<le> Inf (measure_set M f s) + e"
+      from inf_measure_close [of f, OF posf e s]
+      obtain A where A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
+                 and sUN: "s \<subseteq> (\<Union>i. A i)"
+                 and l: "psuminf (f \<circ> A) \<le> Inf (measure_set M f s) + e"
         by auto
       have [simp]: "!!x. x \<in> sets M \<Longrightarrow>
                       (f o (\<lambda>z. z \<inter> (space M - x)) o A) = (f o (\<lambda>z. z - x) o A)"
         by (rule ext, simp, metis A Int_Diff Int_space_eq2 range_subsetD)
       have  [simp]: "!!n. f (A n \<inter> x) + f (A n - x) = f (A n)"
         by (subst additiveD [OF add, symmetric])
            (auto simp add: x range_subsetD [OF A] Int_Diff_Un Int_Diff_disjoint)
-      have fsumb: "summable (f \<circ> A)"
-        by (metis fsums sums_iff) 
       { fix u
         assume u: "u \<in> sets M"
-        have [simp]: "\<And>n. \<bar>f (A n \<inter> u)\<bar> \<le> f (A n)"
-          by (simp add: positive_imp_pos [OF posf]  increasingD [OF inc] 
-                        u Int  range_subsetD [OF A]) 
-        have 1: "summable (f o (\<lambda>z. z\<inter>u) o A)" 
-          by (rule summable_comparison_test [OF _ fsumb]) simp
-        have 2: "Inf (measure_set M f (s\<inter>u)) \<le> suminf (f o (\<lambda>z. z\<inter>u) o A)"
-          proof (rule Inf_lower) 
-            show "suminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A) \<in> measure_set M f (s \<inter> u)"
-              apply (simp add: measure_set_def) 
-              apply (rule_tac x="(\<lambda>z. z \<inter> u) o A" in exI) 
-              apply (auto simp add: disjoint_family_subset [OF disj])
-              apply (blast intro: u range_subsetD [OF A]) 
+        have [simp]: "\<And>n. f (A n \<inter> u) \<le> f (A n)"
+          by (simp add: increasingD [OF inc] u Int range_subsetD [OF A])
+        have 2: "Inf (measure_set M f (s \<inter> u)) \<le> psuminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A)"
+          proof (rule complete_lattice_class.Inf_lower)
+            show "psuminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A) \<in> measure_set M f (s \<inter> u)"
+              apply (simp add: measure_set_def)
+              apply (rule_tac x="(\<lambda>z. z \<inter> u) o A" in exI)
+              apply (auto simp add: disjoint_family_subset [OF disj] o_def)
+              apply (blast intro: u range_subsetD [OF A])
               apply (blast dest: subsetD [OF sUN])
-              apply (metis 1 o_assoc sums_iff) 
               done
-          next
-            show "\<And>x. x \<in> measure_set M f (s \<inter> u) \<Longrightarrow> 0 \<le> x"
-              by (blast intro: inf_measure_pos0 [OF posf]) 
-            qed
-          note 1 2
+          qed
       } note lesum = this
-      have sum1: "summable (f o (\<lambda>z. z\<inter>x) o A)"
-        and inf1: "Inf (measure_set M f (s\<inter>x)) \<le> suminf (f o (\<lambda>z. z\<inter>x) o A)"
-        and sum2: "summable (f o (\<lambda>z. z \<inter> (space M - x)) o A)"
-        and inf2: "Inf (measure_set M f (s \<inter> (space M - x))) 
-                   \<le> suminf (f o (\<lambda>z. z \<inter> (space M - x)) o A)"
+      have inf1: "Inf (measure_set M f (s\<inter>x)) \<le> psuminf (f o (\<lambda>z. z\<inter>x) o A)"
+        and inf2: "Inf (measure_set M f (s \<inter> (space M - x)))
+                   \<le> psuminf (f o (\<lambda>z. z \<inter> (space M - x)) o A)"
         by (metis Diff lesum top x)+
       hence "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
-           \<le> suminf (f o (\<lambda>s. s\<inter>x) o A) + suminf (f o (\<lambda>s. s-x) o A)"
-        by (simp add: x)
-      also have "... \<le> suminf (f o A)" using suminf_add [OF sum1 sum2] 
-        by (simp add: x) (simp add: o_def) 
+           \<le> psuminf (f o (\<lambda>s. s\<inter>x) o A) + psuminf (f o (\<lambda>s. s-x) o A)"
+        by (simp add: x add_mono)
+      also have "... \<le> psuminf (f o A)"
+        by (simp add: x psuminf_add[symmetric] o_def)
       also have "... \<le> Inf (measure_set M f s) + e"
-        by (metis fsums l sums_unique) 
+        by (rule l)
       finally show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
         \<le> Inf (measure_set M f s) + e" .
     qed
-  moreover 
+  moreover
   have "Inf (measure_set M f s)
        \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
     proof -
     have "Inf (measure_set M f s) = Inf (measure_set M f ((s\<inter>x) \<union> (s-x)))"
       by (metis Un_Diff_Int Un_commute)
-    also have "... \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))" 
-      apply (rule subadditiveD) 
-      apply (iprover intro: algebra.countably_subadditive_subadditive algebra_Pow 
+    also have "... \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
+      apply (rule subadditiveD)
+      apply (iprover intro: algebra.countably_subadditive_subadditive algebra_Pow
                inf_measure_positive inf_measure_countably_subadditive posf inc)
-      apply (auto simp add: subsetD [OF s])  
+      apply (auto simp add: subsetD [OF s])
       done
     finally show ?thesis .
     qed
-  ultimately 
+  ultimately
   show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
         = Inf (measure_set M f s)"
     by (rule order_antisym)
 qed
 
 lemma measure_down:
-     "measure_space N \<Longrightarrow> sigma_algebra M \<Longrightarrow> sets M \<subseteq> sets N \<Longrightarrow>
-      (measure M = measure N) \<Longrightarrow> measure_space M"
-  by (simp add: measure_space_def measure_space_axioms_def positive_def 
-                countably_additive_def) 
+     "measure_space N \<mu> \<Longrightarrow> sigma_algebra M \<Longrightarrow> sets M \<subseteq> sets N \<Longrightarrow>
+      (\<nu> = \<mu>) \<Longrightarrow> measure_space M \<nu>"
+  by (simp add: measure_space_def measure_space_axioms_def positive_def
+                countably_additive_def)
      blast
 
 theorem (in algebra) caratheodory:
-  assumes posf: "positive M f" and ca: "countably_additive M f" 
-  shows "\<exists>MS :: 'a measure_space. 
-             (\<forall>s \<in> sets M. measure MS s = f s) \<and>
-             ((|space = space MS, sets = sets MS|) = sigma (space M) (sets M)) \<and>
-             measure_space MS" 
+  assumes posf: "positive f" and ca: "countably_additive M f"
+  shows "\<exists>\<mu> :: 'a set \<Rightarrow> pinfreal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and> measure_space (sigma (space M) (sets M)) \<mu>"
   proof -
     have inc: "increasing M f"
-      by (metis additive_increasing ca countably_additive_additive posf) 
+      by (metis additive_increasing ca countably_additive_additive posf)
     let ?infm = "(\<lambda>x. Inf (measure_set M f x))"
     def ls \<equiv> "lambda_system (|space = space M, sets = Pow (space M)|) ?infm"
-    have mls: "measure_space (|space = space M, sets = ls, measure = ?infm|)"
+    have mls: "measure_space \<lparr>space = space M, sets = ls\<rparr> ?infm"
       using sigma_algebra.caratheodory_lemma
               [OF sigma_algebra_Pow  inf_measure_outer [OF posf inc]]
       by (simp add: ls_def)
-    hence sls: "sigma_algebra (|space = space M, sets = ls, measure = ?infm|)"
-      by (simp add: measure_space_def) 
-    have "sets M \<subseteq> ls" 
+    hence sls: "sigma_algebra (|space = space M, sets = ls|)"
+      by (simp add: measure_space_def)
+    have "sets M \<subseteq> ls"
       by (simp add: ls_def)
          (metis ca posf inc countably_additive_additive algebra_subset_lambda_system)
-    hence sgs_sb: "sigma_sets (space M) (sets M) \<subseteq> ls" 
+    hence sgs_sb: "sigma_sets (space M) (sets M) \<subseteq> ls"
       using sigma_algebra.sigma_sets_subset [OF sls, of "sets M"]
       by simp
-    have "measure_space (|space = space M, 
-                          sets = sigma_sets (space M) (sets M),
-                          measure = ?infm|)"
-      by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets) 
+    have "measure_space (sigma (space M) (sets M)) ?infm"
+      unfolding sigma_def
+      by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets)
          (simp_all add: sgs_sb space_closed)
-    thus ?thesis
-      by (force simp add: sigma_def inf_measure_agrees [OF posf ca]) 
-qed
+    thus ?thesis using inf_measure_agrees [OF posf ca] by (auto intro!: exI[of _ ?infm])
+  qed
 
 end