Continuity of the Fourier–Stieltjes transform on a Pontryagin dual

The Fourier–Stieltjes transform of a tight finite positive measure on the Pontryagin dual is continuous on the original locally compact group. Together with its positive-definiteness, this supplies the measure-to-function direction of Bochner representation on locally compact abelian groups for regular measures. A compact set carries almost all of the measure; character evaluation is uniformly continuous there near each fixed group element, while the complement contributes at most its small mass.

This is the standard compact truncation argument for Fourier–Stieltjes transforms; see G. B. Folland, A Course in Abstract Harmonic Analysis, Chapter 4.

theorem MeasureTheory.FiniteMeasure.continuous_pontryaginMeasureTransform {G : Type u_1} [AddCommGroup G] [TopologicalSpace G] [LocallyCompactSpace G] [MeasurableSpace (PontryaginDual (Multiplicative G))] [OpensMeasurableSpace (PontryaginDual (Multiplicative G))] (μ : FiniteMeasure (PontryaginDual (Multiplicative G))) (hμ : IsTightMeasureSet {↑μ}) : Continuous μ.pontryaginMeasureTransform

The transform of a tight finite measure on the Pontryagin dual is continuous. Tightness is automatic for finite Radon measures and is the regularity needed when the dual is not metrizable.