COMPACTNESS OF INTEGRAL OPERATORS AND UNIFORM INTEGRABILITY ON MEASURE SPACES

Let (E, epsilon, mu) be a measure space and let epsilon(+), epsilon(b) denote the set of all measurable numerical functions on E which are positive, bounded respectively. Moreover, let G: E x E -> [0, infinity] be measurable. We show that the set of all q is an element of epsilon(+) for which {G(x, center dot)q: x is an element of E} is uniformly integrable coincides with the set of all q is an element of epsilon(+) for which the mapping f bar right arrow G(fq) := integral G(center dot, y)f(y)q(y) d mu(y) is a compact operator on the space epsilon(b) (equipped with the sup-norm) provided each of these two sets contains strictly positive functions.