Dimension of random fractals in metric spaces

We study the local and Hausdorff dimensions of measures in function and sequence spaces and the Hausdorff dimension of such spaces with respect to deterministic and random "scale metrics." Following ideas due to Billingsley and Furstenberg we show that the local dimension of a properly chosen probability measure is an efficient tool for the calculation of the Hausdorff dimension. In particular, the calculation of the Hausdorff dimension of a sequence space with respect to a deterministic scale metric with finite memory is reduced to the calculation of the local dimension of the associated Markov chain that can be found easily; both dimensions coincide with the solution of the generalized Moran equation specified by the scale metric. When the scale metric is random we come to a stochastic analogue of the Moran equation. These results are used as a "leading special case" in the study of the Hausdorff dimension of deterministic and random fractals in general metric spaces.