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In mathematics, given a locally Lebesgue integrable function f {\displaystyle f} on R k {\displaystyle \mathbb {R} ^{k}} , a point x {\displaystyle x} in the domain of f {\displaystyle f} is a Lebesgue point if
Here, B {\displaystyle B} is a ball centered at x {\displaystyle x} with radius r > 0 {\displaystyle r>0} , and | B | {\displaystyle |B|} is its Lebesgue measure. The Lebesgue points of f {\displaystyle f} are thus points where f {\displaystyle f} does not oscillate too much, in an average sense.
The Lebesgue differentiation theorem states that, given any f ∈ L 1 {\displaystyle f\in L^{1}} , almost every x {\displaystyle x} is a Lebesgue point of f {\displaystyle f}.