  In this paper, we establish the existence of $p$-energy norms and the
corresponding $p$-energy measures for scale-irregular Vicsek sets, which may
lack self-similarity. We also investigate the characterizations of $p$-energy
norms in terms of Besov-Lipschitz norms, with their weak monotonicity and the
corresponding Bourgain-Brezis-Mironescu convergence.
