Almost compactness and decomposability of integral operators

Let $(X,\mu )$, $(Y,v)$ be finite measure spaces and $1 < q \leqslant \infty$, $1 \leqslant p \leqslant q$. An integral operator $\operatorname{Int}(k):{L^q}(v) \to {L^p}(\mu )$ becomes compact, if we cut away a suitably chosen subset of $X$ of arbitrarily small measure. As a consequence we prove that $\operatorname{Int}(k)$ may be written as the sum of a Carleman operator and an orderbounded integral operator, where the orderbounded part may be chosen to be compact and of arbitrarily small norm.