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Hahn decomposition for the Riesz charge of \(\delta\) subharmonic functions. (English) Zbl 1023.31005
From the introduction: Let \(w\) be a \(\delta\)-subharmonic function and \[ t\mapsto N_w(t,x)={1\over 2\pi}\int^{2\pi}_0 w(z+te^{i\theta}) d \theta,\;t>0. \] Consider the set \({\mathcal E}_+\) of points \(z\) such that the function \(t\mapsto N_w(t,z)\) has intervals \([r_n(z), R_n(z)]\), \(0<r_n(z)< R_n (z)\), \(R_n(z)\searrow 0\) as \(n\to\infty\), on which \(N_w(r_n,z)\leq N_w (R_n,z)\). By \(\mu[w]\) we denote the Riesz charge of \(w(z)\). Then the set \({\mathcal E}_+\) is Borelian, and the restriction \(\mu[w]|_{{\mathcal E}_+}\) is a non-negative measure.

MSC:
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
30D30 Meromorphic functions of one complex variable (general theory)
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