  We present a new approach to define a suitable integral for functions with
values in quasi-Banach spaces. The integrals of Bochner and Riemann have
deficiencies in the non-locally convex setting. The study of an integral for
$p$-Banach spaces initiated by Vogt is neither totally satisfactory, since
there are quasi-Banach spaces which are $p$-convex for all $0<p<1$, so it is
not always possible to choose an optimal $p$ to develop the integration. Our
method puts the emphasis on the galb of the space, which permits a precise
definition of its convexity. The integration works for all spaces of galbs
known in the literature. We finish with a fundamental theorem of calculus for
our integral.
