  We show that there is a sequence of subsets of each discrete Heisenberg group
for which the non-singular ergodic theorem holds. The sequence depends only on
the group; it works for any of its non-singular actions. To do this we use a
metric which was recently shown by Le Donne and Rigot to have the Besicovitch
covering property and then apply an adaptation of Hochman's proof of the
multiparameter non-singular ergodic theorem. An exposition of how one proves
non-singular ergodic theorems of this type is also included, along with a new
proof for one of the key steps.
