Mutual dimension and random sequences

We also identify conditions under which Billingsley generalizations of the mutual dimensions mdim(S:T) and Mdim(S:T) can be meaningfully defined; we show that under these conditions these generalized mutual dimensions have the “correct” relationships with the Billingsley generalizations of dim(S), Dim(S), dim(T), and Dim(T) that were developed and applied by Lutz and Mayordomo; and we prove a divergence formula for the values of these generalized mutual dimensions.