Another question on Borel sets and projections

Question

Let $A$ be a (bounded) Borel set in $R^n$. Then we know that its projection $A_1$ on $R^{n-1}$ does not have to be Borel. But does $A_1$ have the following property?

Let $\mu$ be a given nonnegative, finite Borel measure on $R^{n-1}$. Then:

$\forall\epsilon>0 \ \exists B,C\subset R^{n-1}$ Borel sets, such that $B\subset A_1\subset C$ and $\mu(C\setminus B)<\epsilon$.

Answer 1

Yes. Projections of Borel sets are analytic sets, and these are measurable with respect to completions of Borel measures. (In fact, it will be possible to get $\mu(C\setminus B)=0$.)