  We study substitutions on countably infinite alphabet (without
compactification) as Borel dynamical systems. We construct stationary and
non-stationary generalized Bratteli-Vershik models for a class of such
substitutions, known as left determined. In this setting of Borel dynamics,
using a stationary generalized Bratteli-Vershik model, we provide a new and
canonical construction of shift-invariant measures (both finite and infinite)
for the associated class of subshifts.
