A $\textbf{box}$ $Q\subset \mathbb{R}^n$ is defined as $Q:= (a_1,b_1) \times \ldots \times (a_n,b_n)$ and the same with closed or half opened intervals where $a_i < b_i \in \mathbb{R}$ and in case of open, half-opened intervals we allow $\pm \infty$. The $\textbf{volume}$ of a box $Q$, call it $\lambda(Q)$, is the product of the lengths of the intervals. An $\textbf{elementary set}$ $E$ is the finite disjoint union of boxes: $E:= \dot\bigcup_{i=1}^m Q_i$ with volume $\lambda(E)= \sum_{i=1}^m \lambda(Q_i)$.
The elementary sets form an algebra $\mathcal{A}$. This volume function $\lambda$ is a premeasure, but I can't prove that it is $\sigma$-additive, meaning for any disjoint sequence $\lbrace{E_i\rbrace}_{i\in \mathbb{N}} \subset \mathcal{A}$ with $\dot\bigcup_{i=1}^\infty E_i \in \mathcal{A}$ it follows that $\lambda(\dot\bigcup_{i=1}^\infty E_i)= \sum_{i=1}^\infty \lambda(E_i)$.
