Gauge-measurable functions

Abstract

In 1973, E. J. McShane introduced an alternative definition of the Lebesgue integral based on Riemann sums, where gauges are used to decide what tagged partitions are allowed. Such an approach does not require any preliminary knowledge of Measure Theory. We investigate in this paper a definition of measurable functions also based on gauges. Its relation to the gauge-integrable functions that satisfy McShane’s definition is obtained using elementary tools from Real Analysis. We show in particular a dominated integration property of gauge-measurable functions.