# 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]}

## References

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

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