Weak containment of measure preserving group actions

This paper is a contribution to the study of the global structure of measure preserving actions of countable (discrete) groups on non-atomic standard probability spaces. For such a group Γ and space (X,μ), we let A(Γ, X, μ) be the space of measure preserving actions of Γ on (X,μ). In the book [K] a hierarchical notion of complexity of such actions, called weak containment, was introduced, motivated by analogous notions of weak containment of unitary representations. Roughly speaking an action a ∈ A(Γ, X, μ) is weakly contained in an action b ∈ A(Γ, X, μ), in symbols a b, if the action of a on finitely many group elements and finitely many Borel sets in X can be simulated with arbitrarily small error by the action of b. We also denote by a ' b ⇐⇒ a b & b a the corresponding notion of weak equivalence. This notion of equivalence is much coarser than the notion of isomorphism (conjugacy) a ∼= b. It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space A(Γ, X, μ)/∼= of actions modulo isomorphism. On the other hand, weak equivalence is a smooth equivalence relation and the space of weak equivalence classes A(Γ, X, μ)/' is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, combinatorial parameters, etc., turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to