A geometric characterization of non-atomic measure spaces

Abstract

Professors J. B. Conway and J. Sztics have kindly pointed out to the author that there is a gap in the proof of Theorem I in the paper [2]. The oversight occurs in the first sentence on p. 58 of [2], where it is asserted that "... the closure is mid-point convex". Due to the fact that the ball U(L°~(/~)) (all unexplained notation is that of [2]) need not satisfy any countability axioms with respect to the weak* topology, the above assertion only becomes convincing under some additional hypothesis such as "the space L~(#) is separable". The purpose of the present note is to indicate how the technique introduced in [2] can be further exploited to yield a valid proof of Theorem I as originally formulated. A persual of the argument given in [2] shows that what is needed for completeness is a proof of the following Assertion. Let (S, Z, #) be a finite non-atomic measure space and let 1 < 2 < 1. Then there is a sequence {f,} C ext{U(L~(~u))) such that