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| − | {{MSC|28A}}
| + | [[Category:Linear and multilinear algebra; matrix theory]] |
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| − | [[Category:Classical measure theory]] | |
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| − | 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
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| − | theorem since it was proved independently and almost contemporarily by the two authors (see
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| − | refs. {{Cite|Ego}}, {{Cite|Sev}}).
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| − | Let $\mu$ be a [[Set function|$\sigma$-additive measure]] defined on a set $X$ endowed with a [[Algebra of sets|$\sigma$-algebra]] ${\mathcal A}$, i.e. $(X,{\mathcal A})$ is a [[Measurable space|measurable space]].
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| − | 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 $E_\varepsilon$.
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| − | 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
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| − | $\{A_n\}\subset\mathcal{A}$ 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 $A_n$.
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| − | A typical application is when $\mu$ is a positive [[Radon measure|Radon measure]] defined on a topological space $X$
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| − | (cf. [[Measure in a topological vector space|Measure in a topological vector space]]) and $E$ is a compact set.
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| − | The case of the Lebesgue measure on the line was first proved by D.F. Egorov ({{Cite|Ego}}).
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| − | Egorov's theorem has various generalizations. For instance, it works for sequences of measurable functions defined on a
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| − | measure space [[Measure space|$(X,{\mathcal A},\mu)$]] with values into a separable metric space $Y$. The conclusion of
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| − | Egorov's theorem might be false if $Y$ is not metrizable.
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| − | Another generalization is due to G. Mokobodzki (see {{Cite|DeMe}}, {{Cite|Rev}}):
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| − | 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|pointwise convergence]].
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| − | Then there is a sequence $\{A_n\}$ of disjoint sets belonging to ${\mathcal A}$ such that the
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| − | support of $\mu$ is contained in $\cup_nA_n$ and such that, for every $n$, the restrictions
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| − | to $A_n$ of the elements of $U$ is compact in the topology of
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| − | [[Uniform convergence|uniform convergence]].
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| − | ====References====
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| − | {|
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| − | |valign="top"|{{Ref|Bou}}|| N. Bourbaki, "Elements of mathematics. Integration", Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) {{MR|0583191}} {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}} {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}} {{ZBL|1095.28001}} {{ZBL|0156.06001}}
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| − | |valign="top"|{{Ref|DeMe}}||
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| − | C. Dellacherie, P.A. Meyer, "Probabilities and potential" , '''C''' , North-Holland (1988) (Translated from French) {{MR|0939365}} {{ZBL|0716.60001}}
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| − | |valign="top"|{{Ref|Ego}}|| D.F. Egorov, "Sur les suites de fonctions mesurables" ''C.R. Acad. Sci. Paris'' , '''152''' (1911) pp. 244–246 {{MR|}} {{ZBL|}}
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| − | |valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory", v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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| − | |valign="top"|{{Ref|Sev}}|| A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock (1957–1961) (Translated from Russian) {{MR|1025126}} {{MR|0708717}} {{MR|0630899}} {{MR|0435771}} {{MR|0377444}} {{MR|0234241}} {{MR|0215962}} {{MR|0118796}} {{MR|1530727}} {{MR|0118795}} {{MR|0085462}} {{MR|0070045}} {{ZBL|0932.46001}} {{ZBL|0672.46001}} {{ZBL|0501.46001}} {{ZBL|0501.46002}} {{ZBL|0235.46001}} {{ZBL|0103.08801}}
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| − | |valign="top"|{{Ref|Rev}}|| D. Revuz, "Markov chains" , North-Holland (1975) {{MR|0415773}} {{ZBL|0332.60045}}
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| − | |valign="top"|{{Ref|Sev}}||
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| − | C. Severini, "Sulle successioni di funzioni ortogonali" (Italian), Atti Acc. Gioenia, (5) 3 10 S, (1910) pp. 1−7 {{ZBL|41.0475.04}}
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| − | |}
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