A Banach Space Characterization of Purely Atomic Measure Spaces

It is well known [4, p. 265; 3] that the space 7,i[0, l] is not isomorphic with a conjugate space. At the other extreme, it is also well known that h is isometric with the conjugate space of Co. Each of these is an example of a space of all real-valued integrable functions over a measure space (T, p), a major difference between them being that the measure space underlying 7_i[0, l] has no atoms, while that underlying h is purely atomic. It is natural to conjecture that a space Li(P, p) is isomorphic with a conjugate space if and only if (P, p) is purely atomic; we will show that this conjecture is false, although it is true for separable Li spaces. We prove this result, together with one of our characterizations of purely atomic (T, p), by using the notion of differentiability of vector valued functions of bounded variation on [0, l]. (This was the method employed by Gelfand [4] in proving the result cited above.) A related result is given in terms of locally uniformly convex spaces [8]. Let (P, p) be a measure space. (We do not assume that P is measurable.) An atom A ET is a measurable set such that 0, and for each measurable set BEA, either p.(P)=0 or p(B) =n(A). We will consider two atoms to be the "same" if they differ by a set of measure zero. A set 5 of positive finite measure is purely atomic if the set S-~U{.4 ES: A is an atom} has measure zero. (Since atoms are essentially disjoint, p is countably additive, and p(S) < «, S can contain at most countably many atoms, and hence the above set is measurable.) We say that the measure space (T, u) is purely atomic if every subset SET of positive finite measure is purely atomic. We denote by Û the collection of all atoms A ET. There are doubtless other possible definitions of "purely atomic"; that the one given here is reasonable is shown by the following lemma.