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⟩ : x∗∈ X∗,∥x∗∥≦ 1}. 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∗,f⟩ is 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
ℱ = {⟨x∗,f⟩ : x∗∈ X∗,∥x∗∥≦ 1} and study how measurability and integrability
properties of f are reflected by topological properties of ℱ in the spaces L∞ and
B(Σ).
