On Fréchet theorem in the set of measure preserving functions over the unit interval (Q1813950)

The Fréchet theorem says that for every measurable almost everywhere finite function \(f\) defined on the interval \([a,b]\) there exists a sequence of continuous functions converging to \(f\) almost everywhere. Let \(m\) denote the Lebesgue measure on [0,1]. A measurable function \(f: [0,1]\to[0,1]\) is said to be measure preserving if \(m(f^{-1}(A))=m(A)\) for every Borel set \(A\subseteq [0,1]\). In this paper the authors prove that for every measure preserving function \(f\) on [0,1] there exists a sequence of piecewise linear measure preserving continuous functions converging to \(f\) almost everywhere.