Let $X$ be a measure space, and let $0\leq f_1\leq f_2\leq\cdots$ be a monotone increasing sequence of nonnegative measurable functions. Let $f\colon X \to \mathbb{R}\cup \{\infty\}$ be the function defined by $f(x) = \lim_{n\rightarrow\infty} f_n(x)$ . Then $f$ is measurable, and $$\lim_{n\rightarrow\infty} \int_X f_n = \int_X f.$$

Remark. This theorem is the first of several theorems which allow us to ``exchange integration and limits''. It requires the use of the Lebesgue integral: with the Riemann integral, we cannot even formulate the theorem, lacking, as we do, the concept of ``almost everywhere''. For instance, the characteristic function of the rational numbers in $[0,1]$ is not Riemann integrable, despite being the limit of an increasing sequence of Riemann integrable functions.