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Difference between revisions of "Lebesgue point"

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{{MSC|49Q15}}
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{{MSC|26B05|28A20,49Q15}}
  
 
[[Category:Classical measure theory]]
 
[[Category:Classical measure theory]]
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Let $f: \mathbb R^n \to \mathbb R^k$ be an absolutely locally integrable function (with respect to the [[Lebesgue measure]] $\lambda$). A Lebesgue point $x$ for $f$ is a point where
 
Let $f: \mathbb R^n \to \mathbb R^k$ be an absolutely locally integrable function (with respect to the [[Lebesgue measure]] $\lambda$). A Lebesgue point $x$ for $f$ is a point where
 
\[
 
\[
\lim_{r\downarrow 0} \frac{1}{\lambda (B_r (x))} \int_{B_r (x)} |f(y)-f(x)|\, dy = 0
+
\lim_{r\downarrow 0} \frac{1}{\lambda (B_r (x))} \int_{B_r (x)} |f(y)-f(x)|\, dy = 0\, .
 
\]
 
\]
 
Note that a Lebesgue point is, therefore, a point where $f$ is [[Approximate continuity|approximately continuous]]. Viceversa, if $f$ is essentially bounded, then any point of approximate continuity is a Lebesgue point.  
 
Note that a Lebesgue point is, therefore, a point where $f$ is [[Approximate continuity|approximately continuous]]. Viceversa, if $f$ is essentially bounded, then any point of approximate continuity is a Lebesgue point.  
  
The following theorem of Lebesgue holds.
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The following theorem of Lebesgue holds (see Section 1.7.2 of {{Cite|EG}}).
  
 
'''Theorem 1'''
 
'''Theorem 1'''
 
Let $f$ be as above. Then $\lambda$-a.e. $x$ is a Lebesgue point for $f$.  
 
Let $f$ be as above. Then $\lambda$-a.e. $x$ is a Lebesgue point for $f$.  
  
The set of Lebesgue points of $f$ is called [[Lebesgue set]].  
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The set of Lebesgue points of $f$ is called ''Lebesgue set''.  
  
 
====Comments====
 
====Comments====
This concept and assertions of the type of the Lebesgue theorem lie at the foundation of various investigations of problems on convergence almost-everywhere and, in particular, of the investigations concerning singular integrals. A generalizazion is possible for Radon measures in the Euclidean space (see [[Differentiation of measures]]).
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This concept (and more in general assertions of the type of the Lebesgue theorem) lie at the foundation of various investigations of problems on convergence almost-everywhere and, in particular, of the investigations concerning singular integrals. A generalizazion is possible for Radon measures in the Euclidean space (see [[Differentiation of measures]]).
  
 
====References====
 
====References====
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|valign="top"|{{Ref|KF}}|| A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)
 
|valign="top"|{{Ref|KF}}|| A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)
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|-
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|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy,  "Measure theory  and fine properties of functions" Studies in Advanced  Mathematics. CRC  Press, Boca Raton, FL,  1992. {{MR|1158660}}  {{ZBL|0804.2800}}
 
|-
 
|-
 
|valign="top"|{{Ref|St}}|| E.M. Stein,  "Singular integrals and differentiability properties of functions" , Princeton Univ. Press  (1970)
 
|valign="top"|{{Ref|St}}|| E.M. Stein,  "Singular integrals and differentiability properties of functions" , Princeton Univ. Press  (1970)
 
|-
 
|-
 
|}
 
|}

Latest revision as of 12:03, 14 December 2012

2020 Mathematics Subject Classification: Primary: 26B05 Secondary: 28A2049Q15 [MSN][ZBL]

Let $f: \mathbb R^n \to \mathbb R^k$ be an absolutely locally integrable function (with respect to the Lebesgue measure $\lambda$). A Lebesgue point $x$ for $f$ is a point where \[ \lim_{r\downarrow 0} \frac{1}{\lambda (B_r (x))} \int_{B_r (x)} |f(y)-f(x)|\, dy = 0\, . \] Note that a Lebesgue point is, therefore, a point where $f$ is approximately continuous. Viceversa, if $f$ is essentially bounded, then any point of approximate continuity is a Lebesgue point.

The following theorem of Lebesgue holds (see Section 1.7.2 of [EG]).

Theorem 1 Let $f$ be as above. Then $\lambda$-a.e. $x$ is a Lebesgue point for $f$.

The set of Lebesgue points of $f$ is called Lebesgue set.

Comments

This concept (and more in general assertions of the type of the Lebesgue theorem) lie at the foundation of various investigations of problems on convergence almost-everywhere and, in particular, of the investigations concerning singular integrals. A generalizazion is possible for Radon measures in the Euclidean space (see Differentiation of measures).

References

[KF] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961)
[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[St] E.M. Stein, "Singular integrals and differentiability properties of functions" , Princeton Univ. Press (1970)
How to Cite This Entry:
Lebesgue point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_point&oldid=27419
This article was adapted from an original article by K.I. Oskolkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article