A generalisation of Henstock-Kurzweil integral to compact metric spaces

Abstract

We introduce the notion of a gauge and of a tagged partition (subordinate to a given gauge) by intersections of open and closed sets of a compact metric space extending the corresponding notions in Henstock-Kurzweil integration of real-valued functions with respect to the Lebesgue measure on the unit interval. We show that, for the integration of bounded functions with respect to a normalised Borel measure $μ$ on a compact metric space, the notion of a gauge and an associated tagged partition, arise naturally from a normalised simple valuation way-below the Borel measure. Then we consider the integration of unbounded functions with respect to a normalised Borel measure on a compact metric space, for which the Lebesgue integral may fail to exist. A pair of a tagged partition and a gauge defines a simple valuation and we introduce a partial order on these pairs, emulating the partial order of simple valuations in the probabilistic power domain. We define the $D_μ$-integral of a real-valued function with respect to a Borel measure using the limit of the net of the integrals of the simple valuations induced by pairs of tagged partitions and gauges for the function. The $D_μ$-integral of functions on a compact metric space with respect to a normalised Borel measure satisfies the basic properties of an integral and generalises the Henstock-Kurzweil integral. We show that when the Lebesgue integral of the function exists then the $D_μ$-integral also exists and they have the same value. We provide a family of real-valued functions on the Cantor space that are $D_μ$-integrable but not Lebesgue integrable.