Externalities, Nonconvexities, and Fixed Points

6.2 Approximating TriplesWebeginwiththedefinition of a Baire 1 function.Definition 6 (Baire 1 Functions)Afunctionf : Z → X is ρ Z -s ∗ Baire 1, denoted byf ∈ ρ Z -s ∗ B 1 := ρ Z -s ∗ B 1 (Z, X),if for every s ∗ -open set G ⊂ X, f −1 (G) is an F σ in Z (a countable union of ρ Z -closedsets in Z).Given that X is w ∗ -compact and metrizable and hence norm separable (see Wilansky,1978, p. 145), f ∈ ρ Z -s ∗ B 1 (Z, X) if and only if f is the pointwise limit of asequence of ρ Z -s ∗ -continuous functions (e.g., see Lee, Tang, and Zhao, 2000).Let N (·) ∈ U ρZ -w ∗ be a Nash USCO and let η(·) =n(K(·)) ∈ [N (·)] ρ Z -w ∗ be aminimal Nash USCO for minimal KFC, n(·), and GCS mapping, K(·). ByTheorem3.3 in Jayne and Rogers (2002), we know that for any USCO, and in particular, forany minimal Nash USCO η(·), there exists a Baire 1 function f ∗ ∈ ρ Z -s ∗ B 1 such thatf ∗ (z) ∈ η(z) for all z ∈ Z (i.e., there exists a Baire 1 selector f ∗ of η). Moreover, weknow that this Baire 1 selector comes equipped with a pointwise converging sequenceof continuous functions, and we also know by Theorem 1.2 in Spruny (2007) (see alsoKalenda, 2007), that this sequence of pointwise approximating, continuous functions,{f n } n ⊂ C ρZ -s ∗(Z, X), can be chosen so it is USCO bounded - that is, so that Grf n ⊂Grβ for all n for some USCO, β ∈ U ρZ -w ∗. The minimal Nash USCO, η ∈ [N ] ρ Z -w ∗,together with its Baire 1 selection, f ∗ ∈ ρ Z -s ∗ B 1 , and the Baire 1’s USCO-bounded,point-wise approximating sequence of continuous functions, {f n } n ⊂ C ρZ -s∗(Z, X),form what we will call an approximating triple,(η,f ∗ , {f n } n ) N ,for Nash USCO N ∈ U ρ∗Z-w ∗. Thus, given approximating triple, (η,f∗ , {f n } n ) N ,forN ∈ U ρZ -w ∗ we have η ∈ [N ] ρ Z -w ∗, f ∗ ∈ ρ Z -s ∗ B 1 , and USCO bounded {f n } n ⊂∗(Z, X) such that for all z ∈ ZC ρZ -sf ∗ (z) ∈ η(z) =n(K(z)) and kf n (z) − f ∗ (z)k ∗ → 0.Because k·k ∗ -convergence (or s ∗ -convergence) implies w ∗ -convergence (i.e., convergencewith respect to the convex metric ρ w ∗), the sequence, {f n } n ⊂ C ρZ -s∗(Z, X),is also a sequence of ρ Z -w ∗ -continuous functions (see Theorem 2.6.7 in Megginson,1998). Thus, given approximating triple, (η,f ∗ , {f n } n ) N , for the Nash USCON ∈ U ρZ -w ∗ we have for all z ∈ Zf ∗ (z) ∈ η(z) =n(K(z)) and ρ w ∗( f n (z),f ∗ (z)) → 0.Note that the existence of an approximating triple for any Nash USCO, N ∈ U ρZ -w ∗,isguaranteed by assumptions [A-1], Proposition 4.3 in Drewnowski and Labuda (1990),Theorem 3.3 in Jayne and Rogers (2002), and Theorem 1.2 in Spruny (2007).31