23 | 10 | 2019
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# Riesz type theorems for general integrals

## Abstract

The author gives a general descriptive definition for integration, denoted by$\small{\mathcal P}$, which has as special cases

• the Lebesgue integral for bounded measurable functions,
• the Lebesgue integral,
• the Denjoy-Perron integral$\small{\mathcal D}^*$,
• the wide Denjoy integral $\small{\mathcal D}$,
• the Foran  integral,
• the Iseki integral and
• the$\small S{\mathcal F}$-integral

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

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$\small{\mathcal P}$- integrable functions is of the first category in itself (see Section 5).

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

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