The Interior of a Convex Set is Convex and Carries Each of Its Borel Subsets up to a Null Set

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 topology of $\mathbb{R}^{n}$, its Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$ and its Lebesgue measure $\lambda_{n}$ are as fixed there.

Let $C\subseteq\mathbb{R}^{n}$ be convex and let $\operatorname{int}C$ be its interior, which is open by The Interior is the Largest Open Subset and therefore belongs to $\mathcal{B}(\mathbb{R}^{n})$. Then the following hold.

1. (The interior is convex)(i) The set $\operatorname{int}C$ is convex.

2. (A Borel subset lies in the interior up to a null set)(ii) For every $E\in\mathcal{B}(\mathbb{R}^{n})$ with $E\subseteq C$ one has

$$\lambda_{n}\bigl(E\setminus\operatorname{int}C\bigr)=0 .$$