Riesz type theorems for general integrals

Abstract

The author gives a general descriptive definition for integration, denoted by \({\mathcal P}\), which has as special cases the Lebesgue integral for bounded measurable functions, the Lebesgue integral, the Denjoy-Perron integral \({\mathcal D}^*\), the wide Denjoy integral \({\mathcal D}\), the Foran integral, the Iseki integral and the \(S{\mathcal F}\)-integral (\cite{Ene1}). This \({\mathcal P}\)-integral will admit Riesz type representation theorems (introducing an Alexiewicz norm, and identifying \(f\) with \(g\) whenever \(f = g\) \(a.e.\) on \([a,b]\)). The classical Riesz representation theorem for the linear and continuous functionals on \((C([a,b]),\|\cdot\|_\infty)\) is a consequence of Theorem 2. In addition it is shown that the space of \({\mathcal P}\)-integrable functions is of the first category in itself (see Section 5). Also a characterization of weak convergence on this space is given.