Metrics Defined via Discrepancy Functions
We introduce the notion of a discrepancy function, as an extended real-valued function that assigns to a pair (A,U) of sets a nonnegative extended real number ω(A,U), satisfying specific properties. The pairs (A,U) are certain pairs of sets such that Asubset of or equal toU, and for fixed A, the function ω takes on arbitrarily small nonnegative values as U varies. We present natural examples of discrepancy functions and show how they can be used to define traditional pseudo-metrics, quasimetrics and metrics on hyperspaces of topological spaces and measure spaces.
W. J. Charatonik and M. Insall, "Metrics Defined via Discrepancy Functions," Topology and its Applications, Elsevier, Jan 2007.
The definitive version is available at http://dx.doi.org/10.1016/j.topol.2006.12.008
Mathematics and Statistics
Keywords and Phrases
Discrepancy function; Hyperspace; Metric; Pseudo-metric; Quasimetric; Symmetric; Whitney map
Article - Journal
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