Lebesgue point

The value of a real variable such that for a given Lebesgue-summable function on one has

According to the Lebesgue theorem, the set of points at which this relation holds (the so-called Lebesgue set) has full (Lebesgue) measure on , that is, at almost-every point , namely at all Lebesgue points, the function differs little in the mean from its values at neighbouring points . The concept of a Lebesgue point has analogues for functions of several variables (see 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

[1]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)

References

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