We present an example of a set ∧ ∈ double-struck T sign satisfying the following two conditions: (1) there exists a nonzero positive singular measure on the unit circle double-struck T sign with spectrum in ∧; (2) if the spectrum of f ∈ L¹ (double-struck T sign) is contained in ∧ and f vanishes on a set of positive measure, then f = 0.