Lebesgue point

In mathematics, given a locally Lebesgue integrable function on , a point in the domain of is a Lebesgue point if[1]

Here, is a ball centered at with radius , and is its Lebesgue measure. The Lebesgue points of are thus points where does not oscillate too much, in an average sense.[2]

The Lebesgue differentiation theorem states that, given any , almost every is a Lebesgue point of .[3]


  1. Bogachev, Vladimir I. (2007), Measure Theory, Volume 1, Springer, p. 351, ISBN 9783540345145.
  2. Martio, Olli; Ryazanov, Vladimir; Srebro, Uri; Yakubov, Eduard (2008), Moduli in Modern Mapping Theory, Springer Monographs in Mathematics, Springer, p. 105, ISBN 9780387855882.
  3. Giaquinta, Mariano; Modica, Giuseppe (2010), Mathematical Analysis: An Introduction to Functions of Several Variables, Springer, p. 80, ISBN 9780817646127.
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