Lebesgue point

In mathematics, given a locally Lebesgue integrable function ${\displaystyle f}$ on ${\displaystyle \mathbb {R} ^{k}}$, a point ${\displaystyle x}$ in the domain of ${\displaystyle f}$ is a Lebesgue point if[1]

${\displaystyle \lim _{r\rightarrow 0^{+}}{\frac {1}{\lambda (B(x,r))}}\int _{B(x,r)}\!|f(y)-f(x)|\,\mathrm {d} y=0.}$

Here, ${\displaystyle B(x,r)}$ is a ball centered at ${\displaystyle x}$ with radius ${\displaystyle r>0}$, and ${\displaystyle \lambda (B(x,r))}$ is its Lebesgue measure. The Lebesgue points of ${\displaystyle f}$ are thus points where ${\displaystyle f}$ does not oscillate too much, in an average sense.[2]

The Lebesgue differentiation theorem states that, given any ${\displaystyle f\in L^{1}(\mathbb {R} ^{k})}$, almost every ${\displaystyle x}$ is a Lebesgue point of ${\displaystyle f}$.[3]

References

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.