It is well known that certain results such as the Radon-Nikodym Theorem, which are valid in totally σ -finite measure spaces, do not extend to measure spaces in which μ is not totally σ -finite. (See §2 for notation.) Given an arbitrary measure space (X, S, μ) and a signed measure ν on (X, S), then if ν ≪ μ for X, ν ≪ μ when restricted to any e ∊ Sf and the classical finite Radon-Nikodym theorem produces a measurable function ge(x), vanishing outside e, with

The problem of developing an abstract integration theory has been approached from many angles (6). The most general of several definitions based on the norm topology is that of Birkhoff (1), which includes the well-known and widely used Bochner integral (3).

The original Birkhoff formulation was based on the notion of unconditional convergence of an infinite series of elements in a Banach space and the closed convex extensions of certain approximating sums.

Among the variety of integrals which have been devised for integrating vector-valued functions the most widely used is that of Bochner (2), perhaps because of the simplicity of its formulation. Other approaches, including one by Birkhoff (1), have yielded more general integrals yet none of them seems to have supplanted the Bochner integral to a significant extent.