At last we come to Lebesgue’s condition for Riemann-integrability in terms of Lebesgue measure. It asserts, simply enough, that a bounded function defined on an -dimensional interval is Riemann integrable on that interval if and only if the set of discontinuities of has measure zero. Our proof will go proceed by way of our condition in terms of Jordan content.
As in our proof of this latter condition, we define
and by our earlier condition we know that for all . In particular, it holds for for all natural numbers .
If is a point where is discontinuous, then the oscillation must be nonzero, and so for some . That is
Conversely, let’s assume that . Given an , we know that is a closed set contained in . From this, we conclude that . Since this is true for all , the Jordan content condition holds, and is Riemann integrable.