Ok I got this problem, that I cannot say weather this is true or not.
Let's define the figures or however you call them: $$ \mathcal{E}^n = \{[a_1,b_1)\times\cdots\times [a_n,b_n)\ | \ a_1,b_1,\cdots,a_n,b_n\in\mathbb{R}\}\subset P(\mathbb{R}^n)$$
Now we create the closure of this: $$ \overline{\mathcal{F}}^n = \left\{\bigcup_{i=1}^\infty A_i \ \middle | \ (A_i) \subset \mathcal{E}^n\right\} $$
Given the Borel-Algebra throught the Open Sets $\mathcal{O}$ of $\mathbb{R}^n$ like this:
$$ \mathcal{B} :=\mathcal{B}(\mathbb{R}^n) := \bigcap_{{\substack{\mathcal{O}\subset \mathcal{A}\\ \mathcal{A} \ : \ \text{$\sigma$-Algebra}}}} \mathcal{A} $$
My Prof said that this was true: $$ \mathcal{B} = \left\{\bigcup_{i=1}^\infty A_i\setminus B_i \ \middle | \ A_i,B_i\in\overline{\mathcal{F}}^n\right\}$$
I've not seen this anywhere else, so I'm a bit confused because I always thought you cannot simply construct the Borelalgebra. So my question is just is this correct or should I rather try to disprove this formula?