Lebesgue point

Last updated December 11, 2022

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

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

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

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

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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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[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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