Indeed, given the event { ζ$_{0}$ = s$_{0}$, β$_{0}$ = b$_{0}$, . . . , ζ$_{n}$$_{−}$$_{1}$ = s$_{n}$$_{−}$$_{1}$, β$_{n}$$_{−}$$_{1}$ = b$_{n}$$_{−}$$_{1}$ } , the conditional distribution of α$_{0}$, . . . , α$_{n}$$_{−}$$_{1}$ is like the conditional distribution of a sequence ¯ α$_{0}$, . . . , ¯ α$_{n}$$_{−}$$_{1}$ that satisfies (9) given that f$_{k}$ (¯ α$_{0}$ ) = s$_{k}$ for k < n . (Here I use the fact that β$_{n}$ is measurable with respect to ζ$_{0}$, . . . , ζ$_{n}$ .) From (19),(13), and (14) it follows that ζ$_{0}$, β$_{0}$, ζ$_{1}$, β$_{1}$, . . . is an ( x $^{∗}$, y $^{∗}$)-random play of Γ$_{ϵ}$( W ). Therefore, the expected payoff for player 1 in Γ ∗ $_{ϵ}$( W ) under ( x $^{∗}$, y $^{∗}$) is where the first equality follows from (7) and (12), the second equality from (2), and the inequality from the fact that y is ϵ -optimal. Summing up, I have provided a pure strategy y ∗ of player 2 in Γ ∗ $_{ϵ}$( W ) (namely, play y$_{1}$, y$_{3}$, . . . ) that gives expected payoff of at most val Γ( W ) + ϵ against any pure strategy x ∗ of player 1 in Γ ∗ $_{ϵ}$( W ). Therefore, val Γ ∗ $_{ϵ}$( W ) ≤ val Γ( W ) + ϵ . □ Proof of Theorem 1.3. Consider the stochastic game Γ ∗ $_{ϵ}$( W ) defined above. Let W$_{0}$ be a compact subset of W such that val Γ ∗ $_{ϵ}$( W$_{0}$ ) > val Γ ∗ $_{ϵ}$( W ) − ϵ , and whose existence follows from Lemma 4.4. Then where the first inequality follows from the fact that W ⊇ W$_{0}$ , the first equality follows from Lemma 3.1, the second inequality follows from Lemma 4.5, and the third inequality follows from the choice of W$_{0}$ . Similarly, for player 2 we get val Γ( W ) < val Γ ∗ $_{ϵ}$( W ) + 2 ϵ . It follows that val Γ( W ) < val Γ( W ) + 4 ϵ . Since ϵ was arbitrary, it follows that val Γ( W ) = val Γ( W ). □ 16 P ( F ( ζ$_{0}$, ζ$_{1}$, . . . ) ∈ W ) = P (( α$_{0}$, α$_{1}$, . . . ) ∈ W ) = µ$_{x,y}$ ( W ) ≤ val Γ( W ) + ϵ, val Γ( W ) ≥ val Γ( W$_{0}$ ) = val Γ( W$_{0}$ ) ≥ val Γ ∗ $_{ϵ}$( W$_{0}$ ) − ϵ > val Γ ∗ $_{ϵ}$( W ) − 2 ϵ,