The Radon-Nikodym theorem for the Bochner integral

Abstract

Our Main Theorem, which we believe to be the first general Radon-Nikodym theorem for the Bochner integral, is Main Theorem.Let (X, S, p) be a o-finite positive measure space and let B be a Banach space.Let m be a B-valued measure on S. Then m is the indefinite integral with respect to p of a B-valued Bochner integrable function on X if and only if(1) m is p-continuous, that is, m(E) = 0 whenever p(E) = 0, Ee S, (2) the total variation, \m\,ofmis a finite measure, (3) locally m somewhere has compact average range, that is, given Ee S with 0 < p.(E) < co there is an F£ E such that p.(F) > 0 andis relatively (norm) compact, or equivalently (3') locally m somewhere has compact direction, that is, given EeS with 0 < p(E) < co there is an As E and a compact subset, K, of B not containing 0 such that p(F) > 0 and m(F') is contained in the cone generated by Kfor all F' s F.Hypothesis 3 is the traditional type of hypothesis which has been used in previous Radon-Nikodym theorems for vector-valued measures.We discuss these previous theorems below.But it is not obvious that hypothesis (3) is satisfied by every realvalued measure, and so, if hypothesis (3) is used, the classical Radon-Nikodym theorem is not an immediate consequence of the Main Theorem.Hypothesis (3'), on the other hand, is clearly satisfied by every real-valued measure, or, more generally, by every measure with values in a finite dimensional Banach space (let Abe the entire surface of the unit ball of A).Also, hypothesis (3') can be interpreted geometrically as follows : the role of A is to specify a solid angle in B (in fact A can be taken to be a compact subset of the surface of the unit sphere in B, if desired), and hypothesis (3') says that locally there are sets of positive measure on subsets of which the directions of the values of m lie in this solid angle.We feel that the methods used in proving the Main Theorem are at least as interesting as the theorem itself.Our proof is entirely elementary in the sense that we use only very basic facts about Banach spaces.In particular, we do not use