Vol. 95, No. 1, 1981

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Vector-valued functions as families of scalar-valued functions

Robert F. Geitz and J. Jerry Uhl, Jr.

Vol. 95 (1981), No. 1, 75–83

One time-honored way of studying the properties of a vector measure F with values in a Banach space X, with dual X is to examine properties of the family of scalar measures {⟨x,F: xX,x1}. The purpose of this paper is to undertake a similar study for vector-valued functions. The first theorem proved in this vein was the classical Pettis measurability theorem which states that if X is a separable Banach space and f is an X-valued function such that x,fis measurable for each x in X, then f is a measurable function. What we propose to do is to take a bounded function f with values in X, form the associated family = {⟨x,f: xX,x1} and study how measurability and integrability properties of f are reflected by topological properties of in the spaces L and B(Σ).

Mathematical Subject Classification 2000
Primary: 28B99
Secondary: 46G10
Received: 25 September 1979
Revised: 30 July 1980
Published: 1 July 1981
Robert F. Geitz
J. Jerry Uhl, Jr.