  A classical result by J. Diestel establishes that the composition of a
summing operator with a (strongly measurable) Pettis integrable function gives
a Bochner integrable function. In this paper we show that a much more general
result is possible regarding the improvement of the integrability of vector
valued functions by the summability of the operator. After proving a general
result, we center our attention in the particular case given by the
$(p,\sigma)$-absolutely continuous operators, that allows to prove a lot of
special results on integration improvement for selected cases of classical
Banach spaces ---including $C(K)$, $L^p$ and Hilbert spaces--- and operators
---$p$-summing, $(q,p)$-summing and $p$-approximable operators---.
