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Article 27

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\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

 (see Real Functions - Current Topics).

 

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.