Abstract
Let (E, E, μ) be a measure space and let E+, Eb denote the set of all measurable
numerical functions on E which are positive, bounded respectively. Moreover, let
G: E ×E → [0,∞] be measurable. We show that the set of all q ∈ E+ for which
{G(x, ·)q : x ∈ E} is uniformly integrable coincides with the set of all q ∈ E+ for
which the mapping f 7→ G(fq) :=
R
G(·, y)f(y)q(y) dμ(y) is a compact operator
on the space Eb (equipped with the sup-norm) provided each of these two sets
contains strictly positive functions.