Comments on E 2.5.5 (b). …

Comments on E 2.5.5 (b).

The result of this exercise holds only for "sufficiently regular" measures, such as Lebesgue measure on Euclidean space R^n. Otherwise counter examples can easily be constructed. Let for instance c be counting measure on the sigma algebra S generated by two disjoint singletons {x} and {y} in X={x,y}. Thus S={Ø,X,{x},{y}}. Then the characteristic function f of {x} (and of {y}) is an extreme point on the closed unit ball B of L^1. Show this!

The same argument applies to the counting measure on the (infinite) sigma algebra of all subsets of a set of any countable collection of distinct points, X={x(1),...,x(n),...}. The characteristic function of each one-point set {x(k)} ( 1≤k ) is an extreme point in the closed unit ball of L^1. Other examples are obtained whenever there is a point x in X such that the set {x} has positive measure.