Egorov's theorem

In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. The theorem is named after Dmitri Egorov, a Russian physicist and geometer, who published it in 1911.

Egorov's theorem can be used along with compactly supported continuous functions to prove Lusin's theorem for integrable functions.

tatement of the theorem

Let ("M","d") denote a separable metric space (such as the real numbers with the usual distance "d"("a","b") = |"a" − "b"| as metric). Given a sequence ("f_n") of "M"-valued measurable functions on some measure space ("X",&Sigma;,&mu;), and a measurable subset "A" of finite &mu;-measure such that ("f"_"n") converges &mu;-almost everywhere on "A" to a limit function "f", the following result holds: for every &epsilon; > 0, there exists a measurable subset "B" of "A" such that &mu;("B") < &epsilon;, and ("f_n") converges to "f" uniformly on the relative complement "A" "B".

Here, &mu;("B") denotes the &mu;-measure of "B". In words, the theorem says that pointwise convergence almost everywhere on "A" implies the apparently much stronger uniform convergence everywhere except on some subset "B" of arbitrarily small measure. This type of convergence is also called "almost uniform convergence".

Discussion of assumptions

Note that the assumption &mu;("A") < &infin; is necessary. Under Lebesgue measure, consider the sequence of real-valued indicator functions

:$f\_n(x)\; =\; 1\_\{\; [n,n+1]\; \}(x),qquad\; ninmathbb\{N\},\; xinmathbb\{R\},$

defined on the real line. This sequence converges pointwise to the zero function everywhere but does not converge uniformly on R "B" for any set "B" of finite measure.

The separability of the metric space is needed to make sure that for "M"-valued, measuable functions "f" and "g", the distance "d"("f"("x"),"g"("x")) is again a measurable real-valued function of "x".

Proof

For natural numbers "n" and "k", define the set "E_n,k" by the union

:

These sets get smaller as "n" increases, meaning that "E"_"n"+1,"k" is always a subset of "E_n,k", because the first union involves fewer sets. A point "x", for which the sequence ("f_m"("x")) converges to "f"("x"), cannot be in every "E_n,k" for a fixed "k", because "f_m"("x") has to stay closer to "f"("x") than 1/"k" eventually. Hence by the assumption of &mu;-almost everywhere pointwise convergence on "A",

:

for every "k". Since "A" is of finite measure, we have continuity from above; hence there exists, for each "k", some natural number "n_k" such that

:

For "x" in this set we consider the speed of approach into the 1/"k"-neighbourhood of "f"("x") as too slow. Define

: as the set of all those points "x" in "A", for which the speed of approach into at least one of these 1/"k"-neighbourhoods of "f"("x") is too slow. On the set difference "A" "B" we therefore have uniform convergence. Appealing to the sigma additivity of &mu; and using the geometric series, we get:$mu(B)\; lesum\_\{kinmathbb\{Nmu(E\_\{n\_k,k\})lesum\_\{kinmathbb\{Nfracvarepsilon\{2^k\}=varepsilon.$

References

#Richard Beals (2004). "Analysis: An Introduction". New York: Cambridge University Press. ISBN 0-521-60047-2.
#Dmitri Egoroff (1911). Sur les suites des fonctions measurables. C.R. Acad. Sci. Paris, 152:135–157.
#Eric W. Weisstein et al. (2005). [http://mathworld.wolfram.com/EgorovsTheorem.html Egorov's Theorem] . Retrieved April 19, 2005.