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In measure theory, a branch of mathematical analysis, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and pointwise convergence for a sequence of functions. This theorem shows the superiority of the Lebesgue integral over the Riemann integral for many theoretical purposes.
## Statement of the theorem

Let f_{1}, f_{2}, f_{3}, ... denote a sequence of real-valued measurable functions on a measure space $(S,Sigma,mu)$. Assume that the sequence converges pointwise and is dominated by some integrable function g.
Then the pointwise limit is an integrable function and
## Proof of the theorem

Lebesgue's dominated convergence theorem is a special case of the Fatou–Lebesgue theorem. Below is a direct proof, using Fatou's lemma as the essential tool.## Discussion of the assumptions

That the assumption that the sequence is dominated by some integrable g can not be dispensed with may be seen as follows: define f_{n}(x) = n for x in the interval (0,1/n] and f_{n}(x) = 0 otherwise. Any g which dominates the sequence must also dominate the pointwise supremum h = sup_{n} f_{n}. Observe that# sum_{n

1}^{m-1}int_{left(frac1{n+1},frac1nright]}n,dx
# sum_{n

1}^{m-1}frac1{n+1}
toinftyquadtext{as }mtoinfty
# 0neq 1

lim_{ntoinfty}int_0^1 f_n(x),dx,
## Extensions

The theorem applies also to measurable functions with values in a Banach space, with the dominating function still being non-negative and integrable as above.
## See also

## References

- $$

- $$

- $int\_S|g|,dmumath>$

The convergence of the sequence and domination by g can be relaxed to hold only $mu$-almost everywhere.

If f denotes the pointwise limit of the sequence, then f is also measurable and dominated by g, hence integrable. Furthermore,

- $$

- $$

- $$

- $$

- $$

by the divergence of the harmonic series. Hence, the monotonicity of the Lebesgue integral tells us that there exists no integrable function which dominates the sequence on [0,1]. A direct calculation shows that integration and pointwise limit do not commute for this sequence:

- $$

because the pointwise limit of the sequence is the zero function.

- Bounded convergence theorem
- Convergence in mean
- Monotone convergence theorem (does not require domination by an integrable function but assumes monotonicity of the sequence instead)
- Scheffé's lemma
- Uniform integrability
- Vitali convergence theorem (a generalization of Lebesgue's dominated convergence theorem)

- R.G. Bartle, "The Elements of Integration and Lebesgue Measure", Wiley Interscience, 1995.
- H.L. Royden, "Real Analysis", Prentice Hall, 1988.
- D. Williams, "Probability with Martingales", Cambridge University Press, 1991, ISBN 0-521-40605-6

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Last updated on Friday January 25, 2008 at 14:09:09 PST (GMT -0800)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday January 25, 2008 at 14:09:09 PST (GMT -0800)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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