Lebesgue's density theorem

In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set

A

⊆

R

n

{\displaystyle A\subseteq \mathbb {R} ^{n}}

, the "density" of

A

{\displaystyle A}

is 0 or 1 at almost every point in

R

n

{\displaystyle \mathbb {R} ^{n}}

. Additionally, the "density" of

A

{\displaystyle A}

is 1 at almost every point of

A

{\displaystyle A}

. Intuitively, this means that the boundary of

A

{\displaystyle A}

, the set of points in

A

{\displaystyle A}

for which all neighborhoods are partially in

A

{\displaystyle A}

and partially outside

A

{\displaystyle A}

, is of measure zero.

Statement

Let

μ

{\displaystyle \mu }

be the Lebesgue measure on the Euclidean space and

A

⊆

R

n

{\displaystyle A\subseteq \mathbb {R} ^{n}}

be a Lebesgue measurable set. Let

x

∈

R

n

{\displaystyle x\in \mathbb {R} ^{n}}

and let

B

{\displaystyle B}

ε

(

x

)

{\displaystyle (x)}

denote the open ball of radius

ε

{\displaystyle \varepsilon }

centered at

x

{\displaystyle x}

. Define the density at a point

x

{\displaystyle x}

d

A

(

x

)

=

lim

ε

→

0

μ

(

A

∩

B

ε

(

x

)

)

μ

(

B

ε

(

x

)

)

{\displaystyle \qquad \qquad d_{A}(x)=\lim _{\varepsilon \to 0}{\frac {\mu (A\cap B_{\varepsilon }(x))}{\mu (B_{\varepsilon }(x))}}}

For example, given a square in the plane, the density at every point inside the square is 1, on the edges is 1/2, and at the corners is 1/4. The set of points in the plane at which the density is neither 0 nor 1 is non-empty (the square boundary), but it is of measure zero.

The Lebesgue density theorem is a particular case of the Lebesgue differentiation theorem.

Thus, this theorem is also true for every finite Borel measure on

A

⊆

R

n

{\displaystyle A\subseteq \mathbb {R} ^{n}}

instead of Lebesgue measure, as proven in sections 2.8–2.9 of Federer's Geometric Measure Theory, 1969.

See also

Lebesgue differentiation theorem – Mathematical theorem in real analysis

References

This article incorporates material from Lebesgue density theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.