REMARKS ON INTERSECTION-EQUIVALENCE AND CAPACITY-EQUIVALENCE

Two random sets are intersection-equivalent if the probabilities that they hit any given set are comparable; two fixed sets are capacity-equ ivalent if they have positive capacity in the same (distance-dependent ) kernels. We survey recent results on these equivalence relations, em phasizing their connections with Hausdorff dimension. We also describe an example which illustrates the role of probabilistic uniformity in capacity estimates for random sets.