Vol. 34, No. 1, 1970

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On the measurability of Perron integrable functions

Washek (Vaclav) Frantisek Pfeffer and John Benson Wilbur

Vol. 34 (1970), No. 1, 131–144

By means of majorants and minorants a Perron-like integral can be defined in an arbitrary topological space. Although for its definition only a finitely additive set function is used, it turns out that if the underlying topological space is Hausdorff and locally compact, then the integral itself gives rise to a regular measure. The natural question, whether every integrable function is measurable with respect to this measure, is the subject of our paper.

In §2 some sufficient conditions for measurability of integrable functions are given and the connection of our measure with the original set function is described. The results of this section are then applied to integration with respect to the natural and monotone convergences. The natural convergence, which can be used in any topological space is discussed in §3. In §4 some elementary properties of the monotone convergence are derived. This convergence can be used in any locally pseudo-metrizable space and it seems to be the most important convergence for the definition of an integral over a differentiable manifold. A proof that for the monotone convergence every integrable function is measurable is given in §5. Finally, §6 contains a few illustrative examples.

Mathematical Subject Classification
Primary: 28.20
Received: 1 May 1969
Published: 1 July 1970
Washek (Vaclav) Frantisek Pfeffer
John Benson Wilbur