I need some help with a proof of a theorem leading to Jacobi's transformation formula for Lebesgue integral (in the book Measures, Integrals and Martingales from L. Schilling). What I need to prove is that for every $F_{\sigma}$-set $E \subseteq \mathbb{R}^n$ and $\varepsilon > 0$ there exists a $\left\{ \prod_{i = 1}^n [x^j_i - \varepsilon^j, x^j_i + \varepsilon^j [\right\}_{j \in \mathbb{N}_0}$ (a countable family of n-dimensional cubes in $\mathbb{R}^n$) such that $E \subseteq \bigcup_{j \in \mathbb{N}_0} [x^j_i - \varepsilon^j, x^j_i + \varepsilon^j [$ and $\sum_{j \in \mathbb{N}_0} \lambda^n \left( \prod_{i = 1}^n [x^j_i - \varepsilon^j, x^j_i + \varepsilon^j [\right) \leqslant \lambda^d (E) + \varepsilon$ (where $\lambda^d$ is the Lebesgue measure on $\mathbb{R}^d$)
Thanks a lot in advance
The author gives an explanation for that fact. The collection of half-open rectangles in $\mathbb{R}^n$, $\mathcal{S}$, is a semi-ring. So, setting $\mu=\lambda^n$, $$ \mu(E)=\mu^*(E)=\inf\!\left\{\sum_{j\ge 1} \mu(E_j): E\subset\bigcup_{j\ge 1} E_j,E_j\in \mathcal{S}\right\}. $$ (See Eq. (6.1).) Now, by the properties of infima, for every $\epsilon>0$, there exists a sequence $\{E_j^{\epsilon}\}$ in $\mathcal{S}$ such that $$ \sum_{j\ge 1}\mu(E_j^{\epsilon})\le \mu(E)+\epsilon. $$