An eigenvector proof of fatou’s lemma for continuous functions

Abstract

In a first course on analysis, we typically teach students that continuous functions on a compact interval are bounded and attain their bounds. We also teach them about the Riemann integral and that the Riemann integral "works" under uniform convergence. In a second ~ourse on analysis, we typically teach students about measure theory and prove the Bounded Convergence Theorem for the Lebesgue integral. We also prove that a Riemann integrable function is Lebesgue integrable; consequently, the Bounded Convergence Theorem is true for the Riemann integral. So the Bounded Convergence Theorem for Riemann integration can be stated entirely in first-course-in-analysis terms, but we typically give a second-course-in-analysis proof of it. In fact, the Bounded Convergence Theorem for the Riemann integral was known many years before the arrival on the scene of measure theory (it was established by Arzela in 1885), and, even after the arrival of measure theory, a number of authors considered the problem of finding a measure-free proof of it and various generalizations of it. The problem is discussed in great detail by Luxemburg [14]. That article contains an extensive bibliography, going back to Arzelh's original proof. Measure-free proofs of the Bounded Convergence Theorem for continuous functions were obtained by E Riesz [16] (using Dini's Theorem), Eberlein [2] (using innerproduct techniques), and Simons [18, 19] (using vector lattice techniques and using a precursor of the techniques discussed in this article, respectively). Other elementary proofs, some of which apply to Riemann integrable functions with Riemann in tegrable limit, are Cunningham [ 1],