This is from Measure, Integration, and Real Analysis by Sheldon Axler

4 Give an example of a Borel measurable function f: [0,1] (0, c0) such that L(f, [0,1]) = 0. [Recall that L(f, [0, 1]) denotes the lower Riemann integral, which was defined in Section 1A. If A is Lebesgue measure on [0,1], then the previous exercise states that f f dA > 0 for this function f, which is what we expect of a positive function. Thus even though both L(f,[0,1]) and [ f dA are defined by taking the supremum of approximations from below, Lebesgue measure captures the right behavior for this function f and the lower Riemann integral does not.] 2.40 Definition Borel measurable function Suppose X C R. A function f : X - R is called Borel measurable if f-1(B) is a Borel set for every Borel set B C R.1.7 Definition lower and upper Riemann integrals Suppose f: [a, b] - R is a bounded function. The lower Riemann integral L(f, [a, b] ) and the upper Riemann integral U (f, [a, b]) of f are defined by L(f, [a, b]) = sup L(f, P, [a, b]) P and U(f, a, b]) = infu(f, P, [a, b]), P where the supremum and infimum above are taken over all partitions P of [a, b]