We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply the nonexistence of Borel complete sections with certain

We survey some research aiming at a theory of effective structures of size the continuum. The main notion is the one of a Borel presentation, where the domain, equality and further relations and functions are Borel. We include the case of uncountable languages where the signature is Borel. We discuss the main open questions in the area.

Given a Polish space X , a countable Borel equivalence relation E on X , and a Borel cocycle ρ : E → (0,∞), we characterize the circumstances under which there is a probability measure μ on X such that ρ(φ−1(x), x) = [d(φ∗μ)/dμ](x) μ-almost everywhere, for every Borel injection φ whose graph is contained in E.