monotone convergence theorem

Let $X$ be a measure space, and let $0\le f_1\le f_2\le \mathrm{\cdots }$ be a monotone increasing sequence of nonnegative measurable functions. Let $f:X\to ℝ\cup \{\mathrm{\infty }\}$ be thefunction defined by $f(x)={lim}_{n\to \mathrm{\infty }}{f}_n(x)$.Then $f$ is measurable, and

$$\underset{n\to \mathrm{\infty }}{lim}{\int }_X{f}_n={\int }_{X}f.$$

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.