This result, the Herglotz-Riesz representation theorem, was proved independently by Gustav Herglotz and Frigyes Riesz in 1911.
It can be used to give a related formula and characterization for any holomorphic function on the unit disc with positive real part.
Such functions had already been characterized in 1907 by Constantin Carathéodory in terms of the positive definiteness of their Taylor coefficients.
Conversely if f is positive and harmonic and rn increases to 1, define Then where is a probability measure.
By a compactness argument (or equivalently in this case Helly's selection theorem for Stieltjes integrals), a subsequence of these probability measures has a weak limit which is also a probability measure μ.