In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set is 0 or 1 at almost every. Intuitively, this means that the boundary for which all neighborhoods, is of measure zero.

Statement

be the Lebesgue measure on the Euclidean spaceε denote the open ball

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

See also

References