In this paper we provide two representation theorems for two relevant classes of operators from any $p$-convex order continuous Banach lattice with weak unit into a Banach space: the class of continuous operators and the class of cone absolutely summing operators. We prove that they can be characterized as spaces of vector measures with finite $p$-semivariation and $p$-variation, respectively, with respect to a fixed vector measure. We give in this way a technique for representing operators as integrals with respect to vector measures.
"$p$-variations of vector measures with respect to vector measures and integral representation of operators." Banach J. Math. Anal. 9 (1) 273 - 285, 2015. https://doi.org/10.15352/bjma/09-1-20