Lebesgue point (original) (raw)

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

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

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

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

^ 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.