Let Dμ,p be a Dirichlet type space induced by a positive parameter p and a positive Borel measure μ on the open unit disk. Denote by M(Dμ,p) the Möbius invariant function space generated by Dμ,p. It is known that if the measure μ is finite, then M(Dμ,p) is equal to the well-known Möbius invariant space Qp. In this paper, we investigate Dμ,p and M(Dμ,p) spaces when the measures μ are not necessarily finite. We give the relation between M(Dμ,p) and the Bloch space. We characterize inner functions in M(Dμ,p) spaces. We also consider a Carleson measure problem for Dμ,p spaces.