Hausdorff dimension in probability theory

Let (ft, (B, u) be a probability measure space on which is defined a sto- chastic process {x,} with finite state space. In 1-3 we define a notion of fractional dimension, in terms of and {x}, for any set M c t. If t is the unit interval, if is Lebesgue measure, and if = x(o)s is, for each , the base s expansion of , the definition reduces to the classical one due to Hausdorff. In 4 and 5 we obtain, under the assumption that {x} is a Markov chain, the dimensions of certain sets defined in terms of the asymp- totic relative frequencies of the various transitions i--. j. In 7 these theorems are specialized to the case in which {x.} is independent. In the classical case these results become extensions of theorems due to Eggleston [4, 5] and Volkmann [16, 17]. In 6 we use the preceding theorems to ob- rain a result on "generalized Lipschitz conditions" on certain measures, a result which reduces in the classical case to one of Kinney In 8 the dimensions obtained in the first part of the paper are shown to be related to entropy and certain allied concepts of information theory.