A domain-theoretic approach to integration in Hausdorff spaces
J. D. Howroyd
Abstract: In this paper we generalize the construction of a domain-theoretic integral,
introduced by Professor Abbas Edalat, in locally compact separable Hausdorff
spaces, to general Hausdorff spaces embedded in a domain. Our main example
of such spaces comprises general metric spaces embedded in the rounded ideal
completion of the partially ordered set of formal balls. We go on to discuss
analytic subsets of a general Hausdorff space, and give a sufficient condition
for a measure supported on an analytic set to be approximated by a sequence of
simple valuations. In particular, this condition is always satisfied in a
metric space embedded in the rounded ideal completion of its formal ball space.
We finish with a comments section, where we highlight some potential areas for
future research and discuss some questions of computability.
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