For G a locally compact
abelian group with dual Γ, let μ be a (finite regular Borel) measure on G with
Fourier-Stieltjes transform μ. Doss has recently shown that when Γ is (algebraically)
a totally ordered abelian group and μ is square integrable on the negative half Γ− of
Γ then its singular component σ has σ= 0 on Γ−; in particular μE = 0 for each
common null set E of the analytic measures (those with transforms 0 on Γ−), such E
being Haar-null.
In the similar (but usually distinct) case in which Γ is partially ordered by a
nonzero homomorphism ψ : Γ → R with Γ−= ψ−1 the common null sets E
are known, and our purpose is to note in this setting how function algebra results
apply to show μE = 0 when μ∈ L2(Γ−), and when μ salisfies sometimes weaker (but
more obscure) hypotheses.
![(− ∞, 0]](a120x.png)
the common null sets 