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 the function 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.

Title	monotone convergence theorem
Canonical name	MonotoneConvergenceTheorem
Date of creation	2013-03-22 12:47:27
Last modified on	2013-03-22 12:47:27
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	9
Author	Koro (127)
Entry type	Theorem
Classification	msc 26A42
Classification	msc 28A20
Synonym	Lebesgue’s monotone convergence theorem
Synonym	Beppo Levi’s theorem
Related topic	DominatedConvergenceTheorem
Related topic	FatousLemma