Theorem 7.4.7: Riemann implies Lebesgue Integrable
If f is a bounded function defined on [a, b] such that f is Riemann integrable, then f is Lebesgue integrable andThe proof is simple. Recall that for a given function f we defined I*(f) to be the infimum over all upper sums and I*(f) to be the supremum over all lower sums.
f(x) dx = [a,b] f(x) dx
Since every step function is also a simple function, every upper sum is a simple function that is bigger than f, and every lower function is a simple function less than f. Therefore:
I*(f) I*(f)L I*(f)L I*(f)But if f is Riemann integrable, the first and last quantities agree, so that f must be Lebesgue integrable as well with the same value for the integral.