Weak containment in the space of actions of a free group

(A) We consider measure preserving actions of an infinite, countable (discrete) group Γ on non-atomic standard measure spaces (X,μ), i.e., standard Borel spaces equipped with a non-atomic probability Borel measure. (All such measure spaces are isomorphic to ([0, 1], λ), where λ is Lebesgue measure.) We denote by A(Γ, X, μ) the space of such actions. If a ∈ A(Γ, X, μ) and γ ∈ Γ, we denote by γ(x) = a(γ, x), the corresponding automorphism of the space (X,μ). The group Aut(X,μ) admits a canonical Polish topology, called the weak topology, which is the topology generated by the maps T 7→ T (A) (A a Borel subset of X) from Aut(X,μ) to the measure algebra MALG(X,μ) of (X,μ), equipped with the metric dμ(A,B) = μ(A∆B) and the corresponding topology. Since A(Γ, X, μ) can be viewed as a subspace of the produced space Aut(X,μ), it inherits the product of the weak topology which we also call the weak topology on A(Γ, X, μ). Note that Aut(X,μ) acts continuously via conjugation on A(Γ, X, μ): Given S ∈ Aut(X,μ) and a ∈ A(Γ, X, μ), we let S · a = SaS−1 be the action of Γ for which γSaS = SγaS−1,∀γ ∈ Γ. Then a, b ∈ A(Γ, X, μ) are conjugate iff they are isomorphic, in symbols a ∼= b. Motivated by the concept of weak containment of unitary representations, we can consider an analogous concept of weak containment of actions (see Kechris [Ke09], Section 10, (C)). We say, for a ∈ A(Γ, X, μ), b ∈ A(Γ, Y, ν), that a is weakly contained in b, in symbols