A pointwise selection principle for metric semigroup valued functions

Let ∅ = T ⊂ R, (X, d,+) be an additive commutative semigroup with metric d satisfying d(x + z, y + z) = d(x, y) for all x, y, z ∈ X, and XT the set of all functions from T into X. If n ∈ N and f, g ∈ XT , we set ν(n,f,g,T ) = sup n i=1 d(f (ti )+g(si ), g(ti ) + f (si )), where the supremum is taken over all numbers s1, . . . , sn, t1, . . . , tn from T such that s1 t1 s2 t2 · · · sn tn. We prove the following pointwise selection theorem: If a sequence of functions {fj } j∈N ⊂ XT is such that the closure in X of the set {fj (t)} j∈N is compact for each t ∈ T , and lim n→∞ 1 n lim N→∞ sup j,k N,j =k ν(n,fj ,fk,T ) = 0, then it contains a subsequence which converges pointwise on T . We show by examples that this result is sharp and present two of its variants. © 2007 Elsevier Inc. All rights reserved