The dimension of ergodic random sequences

Let mu be a computable ergodic shift-invariant measure over {0, 1}(N). Providing a constructive proof of Shannon-McMillan-Breiman theorem, V'yugin proved that if x is an element of{0, 1}(N) is Martin-Lof random w.r.t. mu then the strong effective dimension Dim(x) of x equals the entropy of mu. Whether its effective dimension dim(x) also equals the entropy was left as an open problem. In this paper we settle this problem, providing a positive answer. A key step in the proof consists in extending recent results on Birkhoff's ergodic theorem for Martin-Lof random sequences. At the same time, we present extensions of some previous results. As pointed out by a referee the main result can also be derived from results by Hochman [8], using rather different considerations.