Lebesgue point

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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}{|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 |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.[3]

References[edit]

  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 .