The author examines several fractional dimensions for subsets of a separable metric space. He takes under consideration the following kinds of dimensions defined by means of a Hausdorff function \(h:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}:\) the Hausdorff dimension (\(h\)-dim), diameter-based packing dimension (\(h\)-Dim), capacity dimension (\(h\text{-dim}_{C}\)) and lower and upper box dimensions. It is well known (Tricot's theorem) that for \(F\) being a Borel subset of a Euclidean space and \(h\) being the identity function, \(h\)-dim\(F\) and \(h\)-Dim\(F\) can be defined in terms of the lower and upper Lipschitz exponents of a finite Borel measure \(\mu\) positive on \(F\). A generalization of this result to separable metric spaces is still an open problem. The author makes a step to solve the problem and gives estimations of \(h\)-dim\(F\), \(h\)-Dim\(F\), and \(h\text{-dim}_{C}\) in terms of the lower and upper Lipschitz exponents. He uses these results to prove the equivalence of Frostman's condition \(h\)-dim\(F\)=\(h\text{-dim}_{C}\) and Tricot's condition and to examine dimensions of Borel measures. Moreover, under the additional assumption that \(F\) is totally bounded or that a measure \(\mu\) has the totally bounded support the author establishes inequalities between various dimensions of \(F\) or of \(\mu\), respectively.