  For a cardinal δ, a set A ⊆ ℝ is δ-universally Baire if for all partial orders ℙ of cardinality δ, there exist trees S and T on ω × λ (for some λ) such that A = p[T] and, if G ⊆ ℙ is V-generic, then p[T]V[G] = ℝV[G] − p[S]V[G]. A set A ⊆ ℝ is universally Baire if it is δ-universally Baire for all δ. The key point is that universally Baire sets have canonical interpretations in generic extensions V[G]. Let Γ∞ be the collection of sets of reals that are universally Baire. This collection has strong closure properties. For example, Woodin showed that if there is a proper class of Woodin cardinals and A ∈ Γ∞ then (1) L(A,ℝ) ⊧ AD+ and (2) P(ℝ) ∩ L(A,ℝ) ⊆ Γ∞. Here AD+ is a (potential) strengthening of AD designed for models of the form L(P(ℝ)) (see Woodin (1999)). 