Difference between revisions of "Lebesgue point"
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| − | {{MSC|49Q15}} | + | {{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. | + | 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 | + | 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]]). | + | 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) | ||
| + | |- | ||
| + | |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) |
Lebesgue point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_point&oldid=27419