TITLE: Monotone Convergence Theorem
QUESTION [6 upvotes]: One simple case of the monotone convergence theorem for integration is:
Let $E \subset \mathbb{R}^n$ and suppose that $f_k : E \rightarrow \mathbb{R}$ is a sequence of non-negative measurable functions which increases monotonically to a limit $f$. Then $f : E \rightarrow \mathbb{R}$ is measurable and
\begin{equation*} \lim_k \int_E f_k = \int_E f \end{equation*}
where here we mean the Lesbegue integral on $\mathbb{R}^n$.
I know one proof of this theorem which is very measure-theoretic, and I was wondering is there a non-measure theoretic proof for the theorem if we only assume that the $f_k$ are Riemann integrable?

REPLY [2 votes]: If you restrict your attention to regulated functions (= uniform limits of step functions, see e.g. Bourbaki), then you can get a nice 'structural proof' of monotone convergence by using Dini's theorem, which states that if a sequence of continuous functions converges pointwisely to 0 on a compact metric space, then it must also converge uniformly. If you use Dini's result by approximating step functions in a smart way with continuous functions, then you arrive at monotone convergence, at least for regulated functions on a compact interval.