A new proof of Poltoratskii's theorem

Abstract

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∈L 1( R,dμ), lim ε→0(F fu(E+iε)/F μ(E+iε))=f(E) a.e. w.r.t. μ sing, where μ sing is the part of μ which is singular with respect to Lebesgue measure and F denotes a Borel transform, namely, F fμ(z)=∫(x−z) −1f(x) dμ(x) and F μ ( z)=∫( x− z) −1 dμ( x).