# monotone convergence theorem

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.

Title monotone convergence theorem MonotoneConvergenceTheorem 2013-03-22 12:47:27 2013-03-22 12:47:27 Koro (127) Koro (127) 9 Koro (127) Theorem msc 26A42 msc 28A20 Lebesgue’s monotone convergence theorem Beppo Levi’s theorem DominatedConvergenceTheorem FatousLemma