Abstract

In the context of Kolmogorov's algorithmic approach to the foundations of probability, Martin‐Löf defined the concept of an individual random sequence using the concept of a constructive measure 1 set. Alternate characterizations use constructive martingales and measures of impossibility. We prove a direct conversion of a constructive martingale into a measure of impossibility and vice versa such that their success sets, for a suitably defined class of computable probability measures, are equal. The direct conversion is then generalized to give a new characterization of constructive dimensions, in particular, the constructive Hausdorff dimension, the constructive packing dimension, and their generalizations, the constructive scaled dimension and the constructive scaled strong dimension (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)