  We consider the natural definition of DLR measure in the setting of
$\sigma$-finite measures on countable Markov shifts. We prove that the set of
DLR measures contains the set of conformal measures associated with Walters
potentials. In the BIP case, or when the potential normalizes the Ruelle's
operator, we prove that the notions of DLR and conformal coincide. On the
standard renewal shift, we study the problem of describing the cases when the
set of the eigenmeasures jumps from finite to infinite measures when we
consider high and low temperatures, respectively. For this particular shift, we
prove that there always exist finite DLR measures, and we have an expression to
the critical temperature for this volume-type phase transition, which occurs
only for potentials with the infinite first variation.
