  Let \mu be a computable ergodic shift-invariant measure over the Cantor
space. Providing a constructive proof of Shannon-McMillan-Breiman theorem,
V'yugin proved that if a sequence x is Martin-L\"of 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 problem
question. 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-L\"of random sequences.
