Density point

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In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set A ⊆ R n {\displaystyle A\subseteq \mathbb {R} ^{n}} {\displaystyle A\subseteq \mathbb {R} ^{n}}, the "density" of A {\displaystyle A} {\displaystyle A} is 0 or 1 at almost every point in R n {\displaystyle \mathbb {R} ^{n}} {\displaystyle \mathbb {R} ^{n}}. Additionally, the "density" of A {\displaystyle A} {\displaystyle A} is 1 at almost every point of A {\displaystyle A} {\displaystyle A}. Intuitively, this means that the boundary of A {\displaystyle A} {\displaystyle A}, the set of points in A {\displaystyle A} {\displaystyle A} for which all neighborhoods are partially in A {\displaystyle A} {\displaystyle A} and partially outside A {\displaystyle A} {\displaystyle A}, is of measure zero.

Lebesgue's density theorem, applied to the inside of a square, its corners, edges, inside, and outside
Lebesgue's density theorem, applied to the inside of a square, its corners, edges, inside, and outside

The definition

Let μ {\displaystyle \mu } {\displaystyle \mu } be the Lebesgue measure on the Euclidean space and A ⊆ R n {\displaystyle A\subseteq \mathbb {R} ^{n}} {\displaystyle A\subseteq \mathbb {R} ^{n}} be a Lebesgue measurable set. Let x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} {\displaystyle x\in \mathbb {R} ^{n}} and let B {\displaystyle B} {\displaystyle B}ε ( x ) {\displaystyle (x)} {\displaystyle (x)} denote the open ball of radius ε {\displaystyle \varepsilon } {\displaystyle \varepsilon } centered at x {\displaystyle x} {\displaystyle x}. Define

d ε ( x ) = μ ( A ∩ B ε ( x ) ) μ ( B ε ( x ) ) {\displaystyle \qquad \qquad d_{\varepsilon }(x)={\frac {\mu (A\cap B_{\varepsilon }(x))}{\mu (B_{\varepsilon }(x))}}} {\displaystyle \qquad \qquad d_{\varepsilon }(x)={\frac {\mu (A\cap B_{\varepsilon }(x))}{\mu (B_{\varepsilon }(x))}}}


Lebesgue's density theorem asserts that for almost every point x {\displaystyle x} {\displaystyle x} of A ⊆ R n {\displaystyle A\subseteq \mathbb {R} ^{n}} {\displaystyle A\subseteq \mathbb {R} ^{n}} the density

d ( x ) = lim ε → 0 d ε ( x ) {\displaystyle \qquad \qquad \qquad d(x)=\lim _{\varepsilon \to 0}d_{\varepsilon }(x)} {\displaystyle \qquad \qquad \qquad d(x)=\lim _{\varepsilon \to 0}d_{\varepsilon }(x)}

exists and is equal to 0 or 1.

What the Lebesgues density theorem states

For every measurable set A {\displaystyle A} {\displaystyle A}, the density of A {\displaystyle A} {\displaystyle A} is 0 or 1 almost everywhere[1]. If 0 < μ ( A ) < ∞ {\displaystyle 0<\mu (A)<\infty } {\displaystyle 0<\mu (A)<\infty }, then there are always points of A ⊆ R n {\displaystyle A\subseteq \mathbb {R} ^{n}} {\displaystyle A\subseteq \mathbb {R} ^{n}} where the density either does not exist or exists but is neither 0 nor 1.[2].

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}} {\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

References

  1. Mattila, Pertti (1999). Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. ISBN 978-0-521-65595-8.
  2. Croft, Hallard (1982). "Three lattice-point problems of Steinhaus". Quarterly J. Math. Oxford (2). 33: 71–83.

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