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Let $U\not\equiv \pm\infty$ be a $\delta$-subharmonic function on a closeddisc of radius $R$ centered at zero. In the previous two parts of our paper, weobtained general and explicit estimates of the integral of the positive part ofthe radial maximum growth characteristic ${\mathsf M}_U(t):=\sup\bigl\{U(z)\bigm| |z|=r\bigr\}$ over the increasing integration function$m$ on the segment $[0, r]\subset [0,R)$ through the difference characteristicof Nevanlinna and the quantities associated with the integration function $m$.The third part of our paper contains estimates of these quantities in terms ofthe Hausdorff $h$-measure and $h$-content of compact subset $S\subset [0, r]$such that the integration function $m$ is constant on each open component ofthe connectivity of the complement $[0, r]\setminus S$. The case of thed-dimensional Hausdorff measure is highlighted separately.

Meromorphic functions and differences of subharmonic functions in  integrals and the difference characteristic of Nevanlinna. III. Estimates of  integrals over fractal sets in terms of the Hausdorff measure and content