  We show that a positive operator between $L^p$-spaces is given by integration
against a kernel function if and only if the image of each positive function
has a lower semi-continuous representative with respect to a suitable topology.
This is a consequence of a new characterization of kernel operators on general
Banach lattices as those operators whose range can be represented over a fixed
countable set of positive vectors. Similar results are shown to hold for
operators that merely dominate a non-trivial kernel operator.
