Abstract:
The generalized Riemann integral on locally compact spaces
Abbas Edalat, Sara Negri
We extend the basic results on the theory of the
generalized Riemann integral to the setting of bounded or locally
finite measures on locally
compact second countable Hausdorff spaces.
The correspondence between Borel measures on X and
continuous valuations on the upper space UX gives rise to a
topological embedding between the space of locally finite measures and
locally finite continuous
valuations, both endowed with the Scott topology.
We construct an approximating chain of simple valuations on the upper
space of a locally compact space, whose least upper bound is the given
locally finite measure. The generalized Riemann integral is defined for
bounded
functions with respect to both bounded and locally finite measures. Also in
this setting,
generalized R-integrability
for a bounded function is proved to be equivalent to the condition that
the
set of its discontinuities has measure zero.
Furthermore, if a bounded function is R-integrable then it is
also Lebesgue integrable and the two integrals coincide.
Finally, we extend R-integration to an open set and provide a sufficient
condition for the
computability of the integral of a bounded almost everywhere
continuous function.
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