A remark on topologies for characteristic functions

Abstract

Let M1 denote the set of Borel probability measures on the real line and M1 the space of their Fourier transforms, or ‘characteristic functions’. It is well known that the natural correspondence between M1 and M1 is in fact a homeomorphism, provided M1 is endowed with the usual weak topology relative to bounded continuous functions (which topology can be metrized by the construction of Lévy) and that M1 is given the topology (also metric) appropriate to locally uniform convergence, which we call the ‘compact-open’ topology. In particular, ifwhere μn ∈ M1 (and so φn ∈ M1), then φn(λ)→φ0(λ) uniformly on all compact sets if and only if μn ⇒ μ0 (weak convergence). However, the continuity theorem for characteristic functions implies that even if it is only known that φn(λ)→φ0 pointwise (for all λ), the conclusion that μn ⇒ μ0 is still valid. This raises a question as to whether the pointwise (i.e. product) topology for M1 might not, in fact, be equivalent to the compact-open one, since the corresponding notions of sequential convergence do coincide. The purpose of this note is to show that the answer is negative, even in a much more general setting.