In mathematics, given a locally

Lebesgue integrable In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...

function

f

\mathbb^k

, a point

x

in the domain of

f

is a Lebesgue point if :

\lim_\frac\int_ \!, f(y)-f(x), \,\mathrmy=0.

Here,

B(x,r)

is a ball centered at

x

with radius

r > 0

, and

\lambda (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. The

Lebesgue differentiation theorem In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for ...

states that, given any

f\in L^1(\mathbb^k)

almost every In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion t ...

x

is a Lebesgue point of

f

References

{{DEFAULTSORT:Lebesgue Point Mathematical analysis