The purpose of this paper is to exhibit a theory of integration which can be used on a large class of topological spaces and which generalizes the theory of integration over the locally compact Hausdorff spaces. The first part of the paper is devoted to the construction of such a theory. An integral is viewed as a positive linear functional having a certain extension property on a particular function space. The relationship between an integral T and the corresponding set function μ_T is discussed and one gets a theorem similar to the Riesz Representation Theorem as a result. The set function μ_T has several topological properties among which are that μ_T is semi-regular and that every Borel set is μ_T-measurable. Finally, it is shown that most Lebesgue integrals can be constructed using the above approach regardless of the topology on the underlying space.