Egorov theorem

A theorem on the relation between the concepts of almost-everywhere convergence and uniform convergence of a sequence of functions. Let be a -additive measure defined on a -algebra , let , , and let a sequence of -measurable almost-everywhere finite functions , , converge almost-everywhere to a function . Then for any there exists a measurable set such that , and the sequence converges to uniformly on . For the case where is the Lebesgue measure on the line this was proved by D.F. Egorov [1].

Egorov's theorem has various generalizations extending its potentialities. For example, let be a sequence of measurable mappings of a locally compact space into a metrizable space for which the limit

exists locally almost-everywhere on with respect to a Radon measure . Then is measurable with respect to , and for any compact set and there is a compact set such that , and the restriction of to is continuous and converges to uniformly on . The conclusion of Egorov's theorem may be false if is not metrizable.

References

[1]	D.F. Egorov, "Sur les suites de fonctions mesurables" C.R. Acad. Sci. Paris , 152 (1911) pp. 244–246
[2]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[3]	N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)

Comments

In 1970, G. Mokobodzki obtained a nice generalization of Egorov's theorem (see [a2], [a3]): Let , and be as above. Let be a set of -measurable finite functions that is compact in the topology of pointwise convergence. Then there is a sequence of disjoint sets belonging to such that the support of is contained in and such that, for every , the set of restrictions to of the elements of is compact in the topology of uniform convergence.

Egorov's theorem is related to the Luzin -property.

References

[a1]	P.R. Halmos, "Measure theory" , v. Nostrand (1950)
[a2]	C. Dellacherie, P.A. Meyer, "Probabilities and potential" , C , North-Holland (1988) (Translated from French)
[a3]	D. Revuz, "Markov chains" , North-Holland (1975)