We consider the notion of multiple gap as a finite set of ideals that cannot
be separated. We study the different types of such objects that can be found in
the Boolean algebra of subsets of the natural numbers modulo finite sets.
Di Piazza and Preiss asked whether every Pettis integrable function defined
on [0,1] and taking values in a weakly compactly generated Banach space is
McShane integrable. In this paper we answer this question in the negative.