Density Point of an Arbitrary Subset of $\mathbb{R}^n$

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 and closed balls $\bar{B}(x,r)$ of $\mathbb{R}^{n}$, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$, and Lebesgue measure $\lambda_{n}$ are as fixed there. Write $\lambda_{n}^{\ast}$ for Lebesgue outer measure, which is defined on every subset of $\mathbb{R}^{n}$. For every $x\in\mathbb{R}^{n}$ and every $r\in\mathbb{R}$ with $0<r$ the closed ball $\bar{B}(x,r)$ belongs to $\mathcal{B}(\mathbb{R}^{n})$ by The Lebesgue Measure of a Closed Ball in $\mathbb{R}^n$ (i), so the real number $\lambda_{n}(\bar{B}(x,r))$ appearing below is defined.

Let $E\subseteq\mathbb{R}^{n}$ be an arbitrary subset and let $x\in\mathbb{R}^{n}$.

Definition. (i) The point $x$ is a density point of $E$ if for every $\alpha\in\mathbb{R}$ with $\alpha<1$ there is $\rho\in\mathbb{R}$ with $0<\rho$ such that

$$\alpha\,\lambda_{n}\bigl(\bar{B}(x,r)\bigr)\le\lambda_{n}^{\ast}\bigl(E\cap\bar{B}(x,r)\bigr)\qquad\text{for every }r\in\mathbb{R}\text{ with }0<r<\rho .$$