The Lebesgue Density Theorem in $\mathbb{R}^n$

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the Euclidean norm, distance, topology, notion of openness and closed balls $\bar{B}(x,r)$ of $\mathbb{R}^{n}$, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$, Lebesgue measure $\lambda_{n}$, and the notion of a null subset of $\mathbb{R}^{n}$ are as fixed there. Write $\lambda_{n}^{\ast}$ for Lebesgue outer measure, and let density point have the meaning fixed in that definition.

Let $E\subseteq\mathbb{R}^{n}$ be an arbitrary subset. Then the following holds.

1. (Almost every point of a set is a density point of it) (i) The set of those $x\in E$ which are not density points of $E$ is null.