I am working on the following problem:
Suppose $C \subset \mathbb{R}^d$ is a compact and non-empty set. Let $C_0 = C$ and let $C_t = \{x \in \mathbb{R}^d : d(x,C) \leq t \}$ for all $t >0$. Also, let $g(t) = m(C_t)$, where $m$ is the Lebesgue measure.
(i) Prove that g is right continuous.
(ii) Prove that for $t > 0$, the set $\{x \in \mathbb{R}^d : d(x,C) = t \}$ has Lebesgue measure $0$.
(iii) Using part (ii), show that $g$ is also left continuous.
For part (i), I am thinking to use the fact that the limit of $m(C_t)$ as $T$ goes to 0 is $m(C)$ (I believe that this is true, for example see a very similar claim in Stein and Shakarchi, Measure Theory, chapter 1 exercise 5). Also, for part (ii) I believe that a sketch of the proof is the following: consider a point $x_0$ such that $d(x_0,C)=t$ and then show that the density of ${x: d(x_0,C)