Let (X,𝒜,m1) and (Y,ℬ,m2)
be separable σ-finite measure spaces. A linear transformation T from an order-ideal L
of measurable functions on Y into the space of measurable functions on X is called a
pseudo-integral operator if it is induced by a measure μ on X × Y via the
equation
for sufficiently many functions g. Our main theorem states that T is a pseudo-integral
operator if and only if Tfn→ 0 a.e. whenever 0 ≦ fn≦ f ∈ L and fn→ 0 a.e. We
also study the order structure of the class of pseudo-integral operators showing
that they form a band (order-closed ideal) in the space of order-bounded
operators.
