W. Rudin has the following exercise, "to convince the reader of the power of Lebesgue integration".

Let $0 \leq f_n \leq 1$ be continuous functions from $[0,1]$ to $\mathbb R$, such that they converge pointwise to $0$. Prove that their integrals converge to $0$, without using any Lebesgue theory.

How to do this?

  • $\begingroup$ You can exchange limit and Riemann integral for uniformly convergent sequences of Riemann integrable functions. $\endgroup$ – Michael Greinecker Sep 25 '12 at 11:59
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    $\begingroup$ @MichaelGreinecker But only pointwise convergence is assumed. $\endgroup$ – Julián Aguirre Sep 25 '12 at 12:11
  • $\begingroup$ Rudin cites two beautiful papers dating back to 1967 and 1970. I have them, and the bounded convergence theorem for Riemann integrals is proved in details. But it is a really technical proof, too long to be copied here. $\endgroup$ – Siminore Sep 25 '12 at 12:22

In my opinion, the best solution is contained in a paper by Luxemburg, Arzelà's dominated convergence theorem for the Riemann integral, American Math. Monthly 78 (1971), available here but not for free.

It is very nice to read this paper, and the proof is essentially elementary. Please do not ask me to copy it in my answer, since it takes a few pages :-)

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    $\begingroup$ A more recent paper is A Concise, Elementary Proof of Arzela's Bounded Convergence Theorem by Nadish de Silva, American Math. Monthly 117 (2010). See also this $\endgroup$ – Julián Aguirre Sep 25 '12 at 12:47
  • $\begingroup$ relevant $\endgroup$ – leo Dec 23 '12 at 21:13

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