I'm considering the following situation. Suppose $\mathcal{A}$ is a semiring of sets in $\mathbb{R}^n$ of the form $(a_1,b_1]\times\cdots\times(a_n,b_n]$. Then there is the unique Lebesgue measure $\lambda$ such that $\lambda((a_1,b_n]\times(a_n,b_n])=\prod_{i=1}^n (b_i-a_i)$ for all the sets in the semiring. I'll denote the completion of $\lambda$ as $\lambda$ as well.
If I denote by $\mathcal{C}$ the subfamily of boxes with equal side lengths, i.e., elements of form $(c_1,c_1+L]\times\cdots\times(c_n,c_n+L]$. Then for any $C\subseteq \mathbb{R}^n$, why is it that $\lambda^*(C)=\inf\left\{\sum_i \lambda(A_i)\mid C\subseteq\bigcup_i A_i, \ A_i\in\mathcal{C}\right\}?$