# Lebesgue set

From Encyclopedia of Mathematics

*of a locally summable function defined on an open set *

The set of points at which

where is a closed cube containing the point and is the Lebesgue measure. Here the function can be real- or vector-valued.

#### Comments

When is real-valued and locally integrable, the complement of its Lebesgue set has (Lebesgue) measure zero. This is used in the study of differentiability via maximal functions, cf. [a1].

#### References

[a1] | E.M. Stein, "Singular integrals and differentiability properties of functions" , Princeton Univ. Press (1970) |

[a2] | W. Rudin, "Real and complex analysis" , McGraw-Hill (1978) pp. 24 |

**How to Cite This Entry:**

Lebesgue set.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_set&oldid=15208

This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article