For the complex plane C, we consider the family of those signed measures on C with compact support, of finite logarithmic energy, and with zero total mass. We show that the logarithmic potential of such a measure sits in the Beppo Levi space, namely, the extended Dirichlet space of the Sobolev space of order 1 over C, and that the half of its Dirichlet integral equals its logarithmic energy. We then derive the (local) Markov property of the Gaussian field indexed by this family of measures. Exactly analogous considerations will be made for the Beppo Levi space over the upper half plane H and for the Cameron Martin space over the real line R, that yields an alternative simple way to produce the Brownian motion.