# Riesz type theorems for general integrals

### Real Analysis Exchange 22 (1997), no. 2, 714-733

#### Mathematical reviews subject classification: 26A39, 26A46, 26A45, 46A50, 54E52.

## Abstract

The author gives a general descriptive definition for integration, denoted by, which has as special cases

- the Lebesgue integral for bounded measurable functions,
- the Lebesgue integral,
- the Denjoy-Perron integral,
- the wide Denjoy integral ,
- the Foran integral,
- the Iseki integral and
- the-integral

(see Real Functions - Current Topics).

This-integral will admit Riesz type representation theorems (introducing an Alexiewicz norm, and identifying f with g whenever f = g a.e. on [a,b]).

As a consequence of Theorem 2, it follows the classical Riesz representation theorem for the linear and continuous functionals on (C([a,b]),||· ||_{∞}).

In addition it is shown that the space of- integrable functions is of the first category in itself (see Section 5).

Also a characterization of the weak convergence on this space is given.