# Gauge-measurable functions

## 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.

## Key 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.