mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, given a locally

Lebesgue integrable In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...

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 In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...

. 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 limiting average taken around the point. The theorem is named for Henri Lebesgu ...

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 to ...

x

is a Lebesgue point of

f

References

{{DEFAULTSORT:Lebesgue Point Mathematical analysis