A remark on topologies for characteristic functions

Extract

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, if

where μ_n ∈ M₁ (and so φ_n ∈ M₁), then φ_n(λ)→φ₀(λ) uniformly on all compact sets if and only if μ_n ⇒ μ₀ (weak convergence). However, the continuity theorem for characteristic functions implies that even if it is only known that φ_n(λ)→φ₀ pointwise (for all λ), the conclusion that μ_n ⇒ 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.