Integral in topological spaces

Abstract

Let X, Y be Banach spaces (or either topological vector spaces) and let us consider the function space C (S, X) of all continuous functions f: S → X, from the compact (locally compact) space S into X, equipped with some appropriate topology. Put C (S, X) = C (S) if X = R. In this work we will mainly be concerned with the problem of representing linear bounded operators T: C (S, X) → Y in an integral form: f ∈ C (S, X), Tf =R S f dµ, for some integration process with respect to a measure µ on the Borel σ−field BS of S. The prototype of such representation is the theorem of F. Riesz according to which every continuous functional T: C (S) → R has the Lebesgue integral form Tf =R S f dµ. This paper is intended to present various extensions of this theorem to the Banach spaces setting alluded to above, and to the context of locally convex spaces.