A marriage theorem with Lebesgue measure (Q1803871)

Let \(S\subseteq[0,1]\times[0,1]\) be such that every point has density one and \(| S(J)|\geq| J|\) for every measurable \(J\subseteq[0,1]\). Then for every \(\varepsilon>0\) there is a one-to-one \(f\subseteq S\) defined on some set of measure \(1-\varepsilon\) such that \(| f(J)|=| J|\) for every measurable subset of the domain. Also, if \(\lambda<1\) then there is an \(f\subseteq S\) such that it is a one-to-one function defined a. e. in \([0,1]\) and \(\lambda| J|\leq| f(J)|\leq| J|\) for every measurable \(J\subseteq[0,1]\).