Gauge-measurable functions
Augusto C. Ponce, Jean Van Schaftingen
TL;DR
This paper explores a gauge-based definition of measurable functions, connecting it to gauge integrability and establishing a dominated convergence property using elementary real analysis tools.
Contribution
It introduces a gauge-based notion of measurability and relates it to gauge integrability, providing new insights with elementary methods.
Findings
Gauge-measurable functions are characterized using elementary real analysis.
A dominated convergence property for gauge-measurable functions is established.
The relation between gauge-measurable and gauge-integrable functions is clarified.
Abstract
In 1973, E.J. McShane proposed an alternative definition of the Lebesgue integral based on Riemann sums, where gauges are used 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.
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See pages 1-21 of gaugmeas.pdf