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.
