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\title{THE BOUNDED CONVERGENCE THEOREM FOR THE RIEMANN INTEGRAL}
\author{Russell A. Gordon, Department of Mathematics, Whitman College,
Walla Wala, WA 99362, e-mail: {\tt gordon@whitman.edu} }
\markboth{Santa Barbara Symposium -- R.~A.~Gordon}
{Santa Barbara Symposium -- R.~A.~Gordon}
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\begin{document}\label{c-conf-rg}
\maketitle
\def\liml{\lim\limits}
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\let\dt=\delta
\let\cn=\colon
The Bounded Convergence Theorem for the Lebesgue integral is the
simplest of the many convergence theorems for the Lebesgue integral.
It states the following:
If $\{f_n\}$ is a uniformly bounded sequence of Lebesgue integrable
functions that converges pointwise on $[a,b]$ to a function $f$, then
$f$ is Lebesgue integrable on $[a,b]$ and
$\int_a^bf=\liml_{n\to\infty}\int_a^bf_n$.
The proof of this theorem is not difficult and involves only the
definition of the Lebesgue integral and the basic properties of
Lebesgue measure. The Bounded Convergence Theorem for the Riemann
integral then follows as a corollary.
If $\{f_n\}$ is a uniformly bounded sequence of Riemann integrable
functions that converges pointwise on $[a,b]$ to a Riemann integrable
function $f$, then $\int_a^bf=\liml_{n\to\infty}\int_a^bf_n$.
Note that in this case it is necessary to include the Riemann
integrabillity of the limit function as part of the hypothesis; the
failure of the Riemann integral to be closed under limit operations
was one of the motivating forces for the development of the Lebesgue
integral. However, there are several good reasons to have a proof of
this theorem that does not involve the Lebesgue integral. First of
all, for aesthetic purposes, a result that only involves the Riemann
integral should not require the Lebesgue integral in its proof.
Secondly, from a historical perspective, the Bounded Convergence
Theorem for the Riemann integral was proved before the Lebesgue
integral was defined. Finally, it would be much easier to present
this result to undergraduates if no measure theory were required.
The simplest convergence theorem for the Riemann integral involves
uniform convergence and it is easy to prove this well-known result.
It is just as easy to show that uniform convergence is not necessary.
An enlightening example is the following: let $\{c_n\}$ be a sequence
of positive numbers and for each positive integer $n$, let
$$ f_n(x) = \begin{cases} c_n\sin(n\pi x), & \text{if $0\le x\le 1/n$}; \\
0, &\text{if $x>1/n$.}\end{cases} $$
This sequence converges pointwise to the zero function
$f(x)=0$ for all $x\ge0$, and the
convergence is uniform if and only if $\{c_n\}$ converges to $0$.
Since
$$ \int_0^1 f_n(x)\,dx = \int_0^{1/n} c_n\sin(n\pi x)\,dx
= \frac{c_n}{ n\pi}\int_0^\pi \sin\theta\,d\theta
= \frac{2c_n}{ n\pi}, $$
this sequence of integrals can converge to $0$ even if $\{c_n\}$ does
not converge to $0$. In particular, the sequence of integrals
converges to $0$ if $\{c_n\}$ is bounded and this provides evidence
for the Bounded Convergence Theorem for the Riemann integral. The
convergence in this example is only problematic at $0$; the
convergence is uniform on $[a,1]$ for each $0\dt>0$ for all $n$, then there exists a point $z$ in
$[a,b]$ that belongs to infinitely many of the figures $V_n$.
\end{itemize}
Lewin [2] gave an elementary proof of (2). Although his proof is free
of the jargon of measure theory, there are a number of facts about
figures that seem obvious yet are difficult and/or tedious to prove.
However, the result is important enough and involves a number of ideas
that will be seen in more advanced analysis courses that it is worth
finding a way to introduce this result to undergraduates.
\begin{thebibliography}{9}
\bibitem{[1]}{1} P. S. Bullen and R. Vyborny, {\em Arzela's dominated convergence
theorem for the Riemann integral}, Bollettino U. M. I. (7A) 10-A
(1996), 347-353.
\bibitem{[2]}{1} J. W. Lewin, {\em A truly elementary approach to the bounded
convergence theorem}, Amer. Math. Monthly {\bf 93} (1986),
395-397.
\bibitem{[3]}{3} W. A. J. Luxemburg, {\em Arzela's dominated convergence theorem
for the Riemann integral}, Amer. Math. Monthly {\bf 78} (1971),
970-979.
\bibitem{[4]}{4} W. F. Osgood, {\em Non-uniform convergence and the integration
of series term by term}, Amer. J. Math. {\bf 19} (1897), 155-190.
\end{thebibliography}
\end{document}