Let $A,B\in\mathcal A_{\Bbb R}^*$ where $\overline{\lambda}(A)<\infty$ and $\overline{\lambda}(B)<\infty$ and define $\; \overline{\lambda}_{A,B}:\Bbb R\to\Bbb R$ as follows:
$$\overline{\lambda}_{A,B}(x)=\overline{\lambda}(A\cap(B+x))$$
where $B+x=\{b+x:b\in B\}$.
What I need to prove is that $\overline{\lambda}_{A,B}$ is continuous.
And that fact gives a simple proof of the H.S's theorem, I have already proved this fact for $A=[a,b)$ and $B=[c,d)$ and then for $A,B$ of the form disjoint finite union of intervals like before, now for the general case I need to use the aproximation property who says:
For all $A\in\mathcal A_{\Bbb R}^*$ with $\overline{\lambda}(A)<\infty$ and $\epsilon>0$ exists $A'$ a disjoint finite union of intervals of the form $[a,b)$ s.t. $\overline{\lambda}(A\triangle A')<\epsilon.$
Any hint for the last step?