  It is well known due to Hahn and Mazurkiewicz that every Peano continuum is a
continuous image of the unit interval. We prove that an assignment, which takes
as an input a Peano continuum and produces as an output a continuous mapping
whose range is the Peano continuum, can be realized in a Borel measurable way.
Similarly, we find a Borel measurable assignment which takes any nonempty
compact metric space and assigns a continuous mapping from the Cantor set onto
that space. To this end we use the Burgess selection theorem. Finally, a Borel
measurable way of assigning an arc joining two selected points in a Peano
continuum is found.
