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monotone convergence theorem (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

$\displaystyle \lim_{n\rightarrow\infty} \int_X f_n = \int_X f.$

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.



"monotone convergence theorem" is owned by Koro. [ full author list (2) | owner history (1) ]
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See Also: dominated convergence theorem, Fatou's lemma

Other names:  Lebesgue's monotone convergence theorem, Beppo Levi's theorem

Attachments:
proof of monotone convergence theorem (Proof) by paolini
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Cross-references: increasing, limit, Riemann integrable, rational numbers, characteristic function, even, Riemann integral, Lebesgue integral, measurable, function, measurable functions, sequence, monotone increasing, measure space
There are 9 references to this entry.

This is version 6 of monotone convergence theorem, born on 2002-06-14, modified 2007-04-22.
Object id is 3106, canonical name is MonotoneConvergenceTheorem.
Accessed 19273 times total.

Classification:
AMS MSC: 28A20 (Measure and integration :: Classical measure theory :: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence)
 26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)
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