diff -r ecffea78d381 -r 635d73673b5e src/HOL/Probability/Measure_Space.thy
--- a/src/HOL/Probability/Measure_Space.thy Thu Nov 15 14:04:23 2012 +0100
+++ b/src/HOL/Probability/Measure_Space.thy Thu Nov 15 10:49:58 2012 +0100
@@ -169,6 +169,27 @@
finally show "f x \ f y" by simp
qed
+lemma (in ring_of_sets) subadditive:
+ assumes f: "positive M f" "additive M f" and A: "range A \ M" and S: "finite S"
+ shows "f (\i\S. A i) \ (\i\S. f (A i))"
+using S
+proof (induct S)
+ case empty thus ?case using f by (auto simp: positive_def)
+next
+ case (insert x F)
+ hence in_M: "A x \ M" "(\ i\F. A i) \ M" "(\ i\F. A i) - A x \ M" using A by force+
+ have subs: "(\ i\F. A i) - A x \ (\ i\F. A i)" by auto
+ have "(\ i\(insert x F). A i) = A x \ ((\ i\F. A i) - A x)" by auto
+ hence "f (\ i\(insert x F). A i) = f (A x \ ((\ i\F. A i) - A x))"
+ by simp
+ also have "\ = f (A x) + f ((\ i\F. A i) - A x)"
+ using f(2) by (rule additiveD) (insert in_M, auto)
+ also have "\ \ f (A x) + f (\ i\F. A i)"
+ using additive_increasing[OF f] in_M subs by (auto simp: increasing_def intro: add_left_mono)
+ also have "\ \ f (A x) + (\i\F. f (A i))" using insert by (auto intro: add_left_mono)
+ finally show "f (\ i\(insert x F). A i) \ (\i\(insert x F). f (A i))" using insert by simp
+qed
+
lemma (in ring_of_sets) countably_additive_additive:
assumes posf: "positive M f" and ca: "countably_additive M f"
shows "additive M f"