Lebesgue point

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)