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We present a new approach to define a suitable integral for functions withvalues in quasi-Banach spaces. The integrals of Bochner and Riemann havedeficiencies in the non-locally convex setting. The study of an integral for$p$-Banach spaces initiated by Vogt is neither totally satisfactory, sincethere are quasi-Banach spaces which are $p$-convex for all $0<p<1$, so it isnot always possible to choose an optimal $p$ to develop the integration. Ourmethod puts the emphasis on the galb of the space, which permits a precisedefinition of its convexity. The integration works for all spaces of galbsknown in the literature. We finish with a fundamental theorem of calculus forour integral.

Toward an optimal theory of integration for quasi-Banach-space-valued functions