Egorov theorem

2020 Mathematics Subject Classification: Primary: 28A [MSN][ZBL]

A theorem on the relation between the concepts of almost-everywhere convergence and uniform convergence of a sequence of functions. In literature it is sometimes cited as Egorov-Severini's theorem since it was proved independently and almost contemporarily by the two authors (see refs. [Ego], [Sev]).

Let $\mu$ be a $\sigma$-additive measure defined on a set $X$ endowed with a $\sigma$-algebra ${\mathcal A}$, i.e. $(X,{\mathcal A})$ is a measurable space. Let $E\in{\mathcal A}$, $\mu(E)<+\infty$, and let $f_k:E\to\mathbb{R}$ be a sequence of $\mu$-measurable functions converging $\mu$-almost-everywhere to a function $f$. Then, for every $\varepsilon>0$ there exists a measurable set $E_\varepsilon\subset E$ such that $\mu(E\setminus E_\varepsilon)<\varepsilon$, and the sequence $f_k$ converges to $f$ uniformly on $E_\varepsilon$.

The result is in general false if the condition $\mu(E)<+\infty$ is dropped. Despite of this, Luzin noted that if $X$, ${\mathcal A}$, $\mu$, $f_k$ and $f$ are as above, and $E\in{\mathcal A}$ is the countable union of sets $E_n$ with finite measure, then there exist a sequence $\{A_n\}\subset\mathcal{A}$ and $H\in{\mathcal A}$, with $\mu(H)=0$, such that $E=(\cup_nA_n)\cup H$, and $f_k$ converges uniformly to $f$ on each $A_n$.

A typical application is when $\mu$ is a positive Radon measure defined on a topological space $X$ (cf. Measure in a topological vector space) and $E$ is a compact set. The case of the Lebesgue measure on the line was first proved by D.F. Egorov ([Ego]).

Egorov's theorem has various generalizations. For instance, it works for sequences of measurable functions defined on a measure space $(X,{\mathcal A},\mu)$ with values into a separable metric space $Y$. The conclusion of Egorov's theorem might be false if $Y$ is not metrizable.

Another generalization is due to G. Mokobodzki (see [DeMe], [Rev]): Let $\mu$, ${\mathcal A}$ and $E$ be as above, and let $U$ be a set of $\mu$-measurable finite functions that is compact in the topology of pointwise convergence. Then there is a sequence $\{A_n\}$ of disjoint sets belonging to ${\mathcal A}$ such that the support of $\mu$ is contained in $\cup_nA_n$ and such that, for every $n$, the restrictions to $A_n$ of the elements of $U$ is compact in the topology of uniform convergence.

References

[Bou]	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
[DeMe]	C. Dellacherie, P.A. Meyer, "Probabilities and potential" , C , North-Holland (1988) (Translated from French) MR0939365 Zbl 0716.60001
[Ego]	D.F. Egorov, "Sur les suites de fonctions mesurables" C.R. Acad. Sci. Paris , 152 (1911) pp. 244–246
[Ha]	P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Sev]	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
[Rev]	D. Revuz, "Markov chains" , North-Holland (1975) MR0415773 Zbl 0332.60045
[Sev]	C. Severini, "Sulle successioni di funzioni ortogonali" (Italian), Atti Acc. Gioenia, (5) 3 10 S, (1910) pp. 1−7 Zbl 41.0475.04