Monotone Convergence Theorem for the Riemann Integral

The monotone convergence theorem holds for the Riemann integral, provided (of course) it is assumed that the limit function is Riemann integrable. It might be thought, though, that this would be difficult to prove and inappropriate for an undergraduate course. In fact the identity is elementary: in the Lebesgue theory it is only the integrability of the limit function that is deep. This article shows how to prove the monotone convergence theorem for Riemann integrals using a simple compactness argument (i.e., invoking Cousin’s lemma). This material could reasonably and appropriately be used in classroom presentations where the students are indoctrinated on this antiquated, but still popular, integration theory. The monotone convergence theorem is usually stated and proved for the Lebesgue integral, but there is little difficulty in formulating and proving a version for the Riemann integral. doi:10.4169/000298910X492835 June–July 2010] NOTES 547 This content downloaded on Tue, 18 Dec 2012 15:44:16 PM All use subject to JSTOR Terms and Conditions Monotone Convergence Theorem. Let { fn} be a nondecreasing sequence of Riemann integrable functions on the interval [a, b]. Suppose that f (x) = lim n→∞ fn(x) for every x in [a, b]. Then, provided f is also Riemann integrable on [a, b], ∫ b a f (x) dx = lim n→∞ ∫ b