On the absolutely continuous component of a weak limit of measures on ℝ supported on discrete sets

Let μ 1 , μ 2 , … \mu _1,\mu _2,\ldots be a sequence of positive Borel measures on R \mathbb {R} each of which is supported on a set having no finite limit points. Suppose the sequence μ n \mu _n weakly converges to a Borel measure ν \nu . Let ν a c \nu _{\mathrm {ac}} be the absolutely continuous component of ν \nu , and X ⊂ R X\subset \mathbb {R} the essential support of ν a c \nu _{\mathrm {ac}} . We characterize the set X X in terms of the limiting behavior of the Hilbert transforms of the measures  μ n \mu _n . Potential applications include those in spectral theory.