(Uniqueness of elementary measure). Let $d \geq 1$. Let $m^{\prime}: \mathcal{E}\left(\mathbf{R}^d\right) \rightarrow \mathbf{R}^{+}$be a map from the collection $\mathcal{E}\left(\mathbf{R}^d\right)$ of elementary subsets of $\mathbf{R}^d$ to the nonnegative reals that obeys the non-negativity, finite additivity, and translation invariance properties. Show that there exists a constant $c \in \mathbf{R}^{+}$such that $m^{\prime}(E)=c m(E)$ for all elementary sets $E$. In particular, if we impose the additional normalisation $m^{\prime}\left([0,1)^d\right)=1$, then $m^{\prime} \equiv m$. (Hint: Set $c:=m^{\prime}\left([0,1)^d\right)$, and then compute $m^{\prime}\left(\left[0, \frac{1}{n}\right)^d\right)$ for any positive integer $n$.)