Lebesgue point explained

In mathematics, given a locally Lebesgue integrable function

f on R ^{k} , a point x in the domain of f is a Lebesgue point if^[1] \lim _{r → 0 ^{+}} (1)/(λ (B(x,r))) \int _{B(x,r)} |f(y) - f(x)| d y =0 .

Here,

B(x,r) is a ball centered at x with radius r > 0 , and λ (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(R} ^{k)} , almost every x is a Lebesgue point of f .^[3]

Notes and References

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