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Difference between revisions of "Lebesgue point"

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The value of a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057890/l0578901.png" /> such that for a given Lebesgue-summable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057890/l0578902.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057890/l0578903.png" /> one has
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{{MSC|49Q15}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057890/l0578904.png" /></td> </tr></table>
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[[Category:Classical measure theory]]
  
According to the [[Lebesgue theorem|Lebesgue theorem]], the set of points at which this relation holds (the so-called [[Lebesgue set|Lebesgue set]]) has full (Lebesgue) measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057890/l0578905.png" />, that is, at almost-every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057890/l0578906.png" />, namely at all Lebesgue points, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057890/l0578907.png" /> differs little in the mean from its values at neighbouring points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057890/l0578908.png" />. The concept of a Lebesgue point has analogues for functions of several variables (see [[Lebesgue set|Lebesgue set]]). This concept and assertions of the type of the Lebesgue theorem lie at the foundation of various investigations of problems on convergence almost-everywhere, in particular, of the investigations concerning singular integrals.
 
  
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR></table>
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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
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\[
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\lim_{r\downarrow 0} \frac{1}{\lambda (B_r (x))} \int_{B_r (x)} |f(y)-f(x)|\, dy = 0
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\]
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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.
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The following theorem of Lebesgue holds.
  
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'''Theorem 1'''
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Let $f$ be as above. Then $\lambda$-a.e. $x$ is a Lebesgue point for $f$.
  
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The set of Lebesgue points of $f$ is called [[Lebesgue set]].
  
 
====Comments====
 
====Comments====
 
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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]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"E.M. Stein,  "Singular integrals and differentiability properties of functions" , Princeton Univ. Press  (1970)</TD></TR></table>
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{|
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|-
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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)
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|-
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|valign="top"|{{Ref|St}}|| E.M. Stein,  "Singular integrals and differentiability properties of functions" , Princeton Univ. Press  (1970)
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|-
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|}

Revision as of 12:48, 7 August 2012

2020 Mathematics Subject Classification: Primary: 49Q15 [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.

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 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)
[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