In mathematics, **Lebesgue’s density theorem** states that for any Lebesgue measurable set *A*, the “density” of *A* is 1 at almost every point in *A*. Intuitively, this means that the “edge” of *A*, the set of points in *A* whose “neighborhood” is partially in *A* and partially outside of *A*, is negligible.

Let μ be the Lebesgue measure on the Euclidean space **R**^{n} and *A* be a Lebesgue measurable subset of **R**^{n}. Define the **approximate density** of *A* in a ε-neighborhood of a point *x* in **R**^{n} as

where *B*_{ε} denotes the closed ball of radius ε centered at *x*.

**Lebesgue’s density theorem** asserts that for almost every point of *A* the **density**

exists and is equal to 1.

In other words, for every measurable set *A* the density of *A* is 0 or 1 almost everywhere in **R**^{n}. However, it is a curious fact that if μ(*A*) > 0 and μ(**R**^{n}\*A*) > 0, then there are always points of **R**^{n} where the density is neither 0 nor 1.

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 negligible.