Here a question for you guys. Let $S_1,S_2,\dots, S_l\subset \mathbb{R}^{n}$ be some affine spaces of dimension $n-1$, and let $P_1,P_2,\dots,P_j\in\mathbb{R}^n$ be some points. Let $\mathcal{S}=\bigcup_{i=1}^l S_i$ and $\mathcal{P}=\bigcup_{i=1}^j P_i$. I'd like to prove that the Lebesgue measure of $\mathcal{S}\cup\mathcal{P}=0$. My idea was to show that the Hausdorff dimension of $\mathcal{S}\cup\mathcal{P}$ is $n-1$, in that case the result would follow immediately. However, I am not sure whether for affine spaces the linear dimension coincides with the Hausdorff dimension.
Thanks!
Much easier is to directly prove that a point and a hyperplane each have Lebesgue measure zero.
A single point can be covered by boxes of decreasing radius approaching zero, whose volumes approach zero. Therefore, a single point has Lebesgue measure zero.
Next, an $n-1$ dimensional cube $Q$ contained in a hyperplane can be covered by a box of the form $Q \times J$, where $J$ is an interval of any length in a perpendicular line to the hyperplane. By letting $\text{Length}(J)$ approach zero, the volumes of these boxes approach zero, so the Lebesgue measure of an $n-1$ dimensional cube contained in a hyperplane equals zero.
Next, an $n-1$ dimensional hyperplane is a countable unit of $n-1$ dimensional unit cubes, so the Lebesgue measure of a hyperplane equals zero.
Finally, any finite (or countable) union of sets of Lebesgue measure zero has Lebesgue measure zero.