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Abstract. Let X be a locally compact Hausdorff space. We define a quasimeasure in X, a quasi-integral on C 0 (X), and a quasi-integral on Cc(X). We show that all quasi-integrals on C 0 (X) are bounded, continuity properties of the quasi-integral on Cc(X), representation of quasi-integrals on Cc(X) in terms of quasi-measures, and unique extension of quasi-integrals on Cc(X) to C 0 (X).

Abstract. Quasi-linear functionals are shown to be uniformly continuous and decomposable into a difference of two quasi-integrals. A predual space for the quasi-linear functionals inducing the weak*-topology is given. General constructions of quasi-linear functionals by solid set-functions and q-functions are given.