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 A ⊆ U , 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.