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?