A Note on Riemann Integrability

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. Let denote the algebra of sets generated by the open and closed subintervals of an interval [a,b]. A bounded real valued function f defined on [a,b] is Riemann integrable if for each positive , there exist two functions and that are linear combinations of characteristic functions of sets in { satisfying <. f <_ and fba , dmsba dm < where m denotes ordinary Lebesgue measure. Riemann integrability may be defined in an analagous way for real valued functions defined on a compact metric space K accompanied by a finite Borel measure. If we make a simple