The authors investigate the class of Dirichlet spaces \(\mathcal{D}_\mu\) with superharmonic weights induced by a positive Borel measure \(\mu\) on \(\mathbb D\) and the Möbius invariant function space \(M(\mathcal{D}_\mu)\) generated by the Hilbert function space \(\mathcal{D}_\mu\). The study contains four sections, starting with a presentation of the topic, remembering facts about \(Q_k\) and BMOA spaces. In the second section it is shown that \(\text{BMOA}=M(\mathcal{D}_\mu)\) if and only if \(\mu\) is a finite measure (Theorem 2.1). The case of infinite measure is analyzed in the last section. Applying this result, under the assumption that the weight function \(K\) is concave, the function \(K\) is characterized such that \(Q_k=\text{BMOA}\) (Theorem 2.2). In the third section a precise link between two distinct measures \(\mu\) and \(\nu\) is given such that \(\mathcal{D}_\mu=\mathcal D_\nu\). Also the relations between \(\mathcal{D}_\mu\), \(M(\mathcal{D}_\mu)\) and the Dirichlet space are investigated (Theorem 3.3). In the last section, the inner functions in \(M(\mathcal{D}_\mu)\) with infinite positive Borel measure \(\mu\) are investigated. It is proved that any inner function in \(M(\mathcal{D}_\mu)\) must be a Blaschke product. Also a criterion for Carleson-Newman Blaschke products to belong to \(M(\mathcal{D}_\mu)\) is given. To do this, the definitions of an interpolating sequence and an interpolating Blaschke product are given. The main result of the section is stated in Theorem 4.4.