Submitted to the Annals of Probability ORTHOGONAL MEASURES AND ERGODICITY

We establish a strengthening of the E0 dichotomy analogous to the known strengthening of the perfect set theorem for families of pairwise orthogonal Borel probability measures. We then use this to characterize the class of countable Borel equivalence relations ad-mitting ergodic Borel probability measures which are not strongly ergodic. A Polish space is a separable topological space admitting a compatible complete metric. A subset of such a space is Kσ if it is a countable union of compact sets, and Borel if it is in the σ-algebra generated by the underlying topology. A function between such spaces is Borel if pre-images of open sets are Borel. A Borel measure on such a space is a function µ: B → [0,∞], where B denotes the family of Borel subsets of X, such that µ(∅) = 0 and µ( n∈NBn) = n∈N µ(Bn) whenever (Bn)n∈N is a sequence of pairwise disjoint Borel subsets of X. A Borel probability measure on X is a Borel measure µ on X for which µ(X) = 1. We use P (X) to denote the space of all Borel probability measures on X, equipped with the Polish topology generated by the functions of the form µ 7 → ∫ f dµ, where f varies over all bounded continuous functions f: X → R (see, for example, [Kec95, Theorem 17.23]). A Borel set B ⊆ X is µ-null if µ(B) = 0, and µ-conull if its complement is µ-null. Two Borel measures µ and ν on X are orthogonal, or µ ⊥ ν, if there is a Borel set B ⊆ X which is µ-conull and ν-null. More generally, we say that a sequence (µi)i∈I of Borel measures on X is orthogonal if there is a sequence (Bi)i∈I of pairwise disjoint Borel subsets of X such that Bi is µi-conull for all i ∈ I. In this case, we say that (Bi)i∈I is a witness to the orthogonality of (µi)i∈I