Lebesgue's density theorem

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