A note on Riemann integrability

Abstract

In this note we define Riemann integrabillty for real valued functions defined on a compact metric space accompanied by a finite Borel measure. If the measure of each open ball equals the measure of its corresponding closed ball, then a bounded function is Riemann integrable if and only if its set of points of discontinuity has measure zero.

Keywords

• Riemann integrable functions on a compact metric space
• compact metric space with Borel measure.