User:Matteo.focardi/sandbox

A theorem on the relation between the concepts of almost-everywhere convergence and uniform convergence of a sequence of functions. Let $\mu$ be a $\sigma$-additive measure defined on a $\sigma$-algebra , let , $\mu(E)<+\infty$, and let a sequence of $\mu$-measurable almost-everywhere finite functions $f_k(x)$, $x\in E$, $k=1,2,\ldots$, converge almost-everywhere to a function $f(x)$. Then for any $\varepsilon>0$ there exists a measurable set $E_\varepsilon\subset E$ such that $\mu(E\setminus E_\varepsilon)<\varepsilon$, 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) MR1025126 MR0708717 MR0630899 MR0435771 MR0377444 MR0234241 MR0215962 MR0118796 MR1530727 MR0118795 MR0085462 MR0070045 Zbl 0932.46001 Zbl 0672.46001 Zbl 0501.46001 Zbl 0501.46002 Zbl 0235.46001 Zbl 0103.08801
[3]	N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001

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) MR0033869 Zbl 0040.16802
[a2]	C. Dellacherie, P.A. Meyer, "Probabilities and potential" , C , North-Holland (1988) (Translated from French) MR0939365 Zbl 0716.60001
[a3]	D. Revuz, "Markov chains" , North-Holland (1975) MR0415773 Zbl 0332.60045