The problem on the measure of the union of line segments in the plane with restrictions on the set of their ends

Some time ago M.A. Patrakeev asked the following question. Let $A$ and $B$ be zero-measure subsets of the unit segment. Let $\varphi$ be bijection between $A$ and $B$. Denote by $S(A,B,\varphi)$ the union of all segments in the plane with the endpoints $(a,0)$ and $(\varphi(a),1)$ for some $a\in A$. The question is what the measure of the set $S(A,B,\varphi)$. We answer this question.