On a space of Besicovitch functions

Abstract

Let $C([0,1])$ be the set of all continuous functions mapping the unit interval $[0,1]$ into itself, equipped with the metric $\rho$ of uniform convergence (and the induced topology $\tau$). A function $f\in C([0,1])$ is called Besicovitch if it is nowhere one-sided differentiable (finite or infinite). For the Lebesgue measure $\lambda$ we define the set $B(\lambda)\subset C([0,1])$ by $$B(\lambda)=\{f\vert~\forall~{\rm Borel}~A\subset [0,1]\colon~\lambda(A)=\lambda(f^{-1}(A))\text{ and }f \text{ is Besicovitch}\}.$$ We construct a set $X\subset B(\lambda)$ such that the space $(X,\tau\vert X)$ is homeomorphic to the product topological space $(\prod_{i=0}^{\infty}[0,1),\mu)$.