Let $P$ be a Peano space. Recall that $P$ is a Hausdorff space that is a continuous surjective image of $[0,1]$.
The standard Peano curve $f:[0,1]\to [0,1]^2$ is self-intersecting and the set $\{x\in[0,1]^2||f^{-1}(x)|>1\}$ has measure zero. Recall that $f^{-1}(x)=\{a\in[0,1]|f(a)=x\}$ is the preimage.
Specifically, my question is that given a Peano space $P$, does there always exist a surjective continuous function $f_P:[0,1]\to P$ such that the set $\{x\in P||f_P^{-1}(x)|>1\}$ has measure zero?