The optimal form of selection principles for functions of a real variable

Abstract

Let T be a nonempty set of real numbers, X a metric space with metric d and X T the set of all functions from T into X. If f ∈ X T and n is a positive integer, we set ν ( n , f ) = sup ∑ i = 1 n d ( f ( b i ) , f ( a i ) ) , where the supremum is taken over all numbers a 1 , … , a n , b 1 , … , b n from T such that a 1 ⩽ b 1 ⩽ a 2 ⩽ b 2 ⩽ ⋯ ⩽ a n ⩽ b n . The sequence { ν ( n , f ) } n = 1 ∞ is called the modulus of variation of f in the sense of Chanturiya. We prove the following pointwise selection principle: If a sequence of functions { f j } j = 1 ∞ ⊂ X T is such that the closure in X of the set { f j ( t ) } j = 1 ∞ is compact for each t ∈ T and (∗) lim n → ∞ ( 1 n lim sup j → ∞ ν ( n , f j ) ) = 0 , then there exists a subsequence of { f j } j = 1 ∞ , which converges in X pointwise on T to a function f ∈ X T satisfying lim n → ∞ ν ( n , f ) / n = 0 . We show that condition (∗) is optimal (the best possible) and that all known pointwise selection theorems follow from this result (including Helly's theorem). Also, we establish several variants of the above theorem for the almost everywhere convergence and weak pointwise convergence when X is a reflexive separable Banach space.