Editing Lebesgue point

In [[mathematics]], given a locally [[Lebesgue integrable]] function <math>f</math> on <math>\mathbb{R}^k</math>, a point <math>x</math> in the domain of <math>f</math> is a '''Lebesgue point''' if<ref>{{citation|title=Measure Theory, Volume 1|first=Vladimir I.|last=Bogachev|publisher=Springer|year=2007|isbn=9783540345145|page=351|url=https://books.google.com/books?id=CoSIe7h5mTsC&pg=PA351}}.</ref>
:<math>\lim_{r\rightarrow 0^+}\frac{1}{\lambda (B(x,r))}\int_{B(x,r)} \!|f(y)-f(x)|\,\mathrm{d}y=0.</math>

Here, <math>B(x,r)</math> is a ball centered at <math>x</math> with radius <math>r > 0</math>, and <math>\lambda (B(x,r))</math> is its [[Lebesgue measure]].  The Lebesgue points of <math>f</math> are thus points where <math>f</math> does not oscillate too much, in an average sense.<ref>{{citation|title=Moduli in Modern Mapping Theory|series=Springer Monographs in Mathematics|first1=Olli|last1=Martio|first2=Vladimir|last2=Ryazanov|first3=Uri|last3=Srebro|first4=Eduard|last4=Yakubov|publisher=Springer|year=2008|isbn=9780387855882|page=105|url=https://books.google.com/books?id=y3oGDTHi-6oC&pg=PA105}}.</ref>

The [[Lebesgue differentiation theorem]] states that, given any <math>f\in L^1(\mathbb{R}^k)</math>, [[almost everywhere|almost every]] <math>x</math> is a Lebesgue point of <math>f</math>.<ref>{{citation|title=Mathematical Analysis: An Introduction to Functions of Several Variables|first1=Mariano|last1=Giaquinta|first2=Giuseppe|last2=Modica|publisher=Springer|year=2010|isbn=9780817646127|page=80|url=https://books.google.com/books?id=0YE_AAAAQBAJ&pg=PA80}}.</ref>

==References==
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{{DEFAULTSORT:Lebesgue Point}}
[[Category:Mathematical analysis]]