Transforming random elements and shifting random fields

Consider a locally compact second countable topological transformation group acting measurably on an arbitrary space. We show that the distributions of two random elements X and X' in this space agree on invariant sets if and only if there is a random transformation Gamma such that Gamma X has the same distribution as X'. Applying this to random fields in d dimensions under site shifts, we show further that these equivalent claims are also equivalent to site-average total variation convergence. This convergence result extends to amenable groups.