Toward an optimal theory of integration for functions taking values in quasi-Banach spaces

Abstract

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.