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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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