Speedups of ergodic group extensions of -actions

We define what it means to 'speed up' a-measure-preserving dynamical system, and prove that given any ergodic extension T-sigma of a -measure-preserving action by a locally compact, second countable group G, and given any second G-extension S-sigma of an aperiodic -measure-preserving action, there is a relative speedup of T-sigma, which is relatively isomorphic to S-sigma. Furthermore, we show that given any neighbourhood of the identity element of G, the aforementioned speedup can be constructed so that the transfer function associated with the isomorphism between the speedup and S-sigma almost surely takes values only in that neighbourhood.