  For any adic transformation $T$ defined on the path space $X$ of an ordered
Bratteli diagram, endowed with a Markov measure $\mu$, we construct an explicit
dimension space (which corresponds to a matrix values random walk on
$\mathbb{Z}$) whose Poisson boundary can be identified as a $\mathbb{Z}$-space
with the dynamical system $(X,\mu,T)$. We give a couple of examples to show how
dimension spaces can be used in the study of nonsingular transformations.
