A new proof of Poltoratskiiʹs theorem

We provide a new simple proof to the celebrated theorem of Poltoratskii concerning ratios of Borel transforms of measures. That is, we show that for any complex Borel measure μ on R and any f∈L1(R,dμ),limε→0(Ffu(E+iε)/Fμ(E+iε))=f(E) a.e. w.r.t. μsing, where μsing is the part of μ which is singular with respect to Lebesgue measure and F denotes a Borel transform, namely, Ffμ(z)=∫(x−z)−1f(x) dμ(x) and Fμ(z)=∫(x−z)−1dμ(x).