  Consider a Banach space valued measurable function $f$ and an operator $u$
from the space where {$f$} takes values. If $f $ is Pettis integrable, a
classical result due to J. Diestel shows that composing it with $u$ gives a
Bochner integrable function $u \circ f$ whenever $u$ is absolutely summing. In
a previous work we have shown that a well-known interpolation technique for
operator ideals allows to prove under some requirements that a composition of a
$p$-Pettis integrable function with a $q$-summing operator provides an
$r$-Bochner integrable function. In this paper a new abstract inclusion theorem
for classes of {abstract} summing operators is shown and applied to the class
of interpolated operator ideals. Together with the results of the
{aforementioned} paper, it provides more results on the relation about the
integrability of the function $u \circ f$ and the summability properties of
$u$.
