Continous measure-preserving maps onto Peano spaces

In this paper we obtain measure-theoretic versions of some topological existence theorems relating to continuous maps, in particular the Hahn-Mazurkiewitz Theorem. Let X be a Peano Space and a Borel measure on X with (X) = 1. There is a continuous measure-preserving surjection from the unit interval (with Lebesque measure) to X if and only if the support of is X.