A Bound for the Lebesgue Measure of a Bounded Slab 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 $\lVert\,\cdot\,\rVert$, distance, dot product and topology on $\mathbb{R}^{n}$, together with the notions of open, closed and bounded subsets, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$, Lebesgue measure $\lambda_{n}$, the absolute value and natural powers of real numbers, and the constant $\sigma_{n}$ with $\sigma_{n}^{2}=n$ are all as fixed there.

Let $z,\nu\in\mathbb{R}^{n}$ with $\lVert\nu\rVert=1$, and let $R,\delta\in\mathbb{R}$ with $0<R$ and $0<\delta$. Put

$$\Sigma=\{y\in\mathbb{R}^{n}:\lVert y-z\rVert\le R\ \text{ and }\ |\nu\cdot(y-z)|\le\delta\}.$$

Then $\Sigma\in\mathcal{B}(\mathbb{R}^{n})$ and (i)

$$\lambda_{n}(\Sigma)\le 2\,\sigma_{n}\,\delta\,(2R)^{n-1},$$

where the factor $(2R)^{n-1}$ is to be read as $1$ when $n=1$.