Let $E$ be a rearrangement invariant (r.i.) function space on $[0,1]$, and let $Z_E$ consist of all measurable functions $f$ on $(0,\infty )$ such that $f^*\chi _{[0,1]}\in E$ and $f^*\chi _{[1,\infty )}\in L^2$. We reveal close connections between proper