Harmonic measure, equilibrium measure, and thinness at infinity in the theory of Riesz potentials

Focusing first on the inner $\alpha$-harmonic measure ${\epsilon }_y^A$ (${\epsilon }_y$ being the unit Dirac measure, and ${\mu }^A$ the inner $\alpha$-Riesz balayage of a Radon measure $\mu$ to $A\subset {\mathbb{R}}^n$ arbitrary), we describe its Euclidean support, provide a formula for evaluation of its total mass, establish the vague continuity of the map $y\mapsto {\epsilon }_y^A$ outside the inner $\alpha$-irregular points for $A$, and obtain necessary and sufficient conditions for ${\epsilon }_y^A$ to be of finite energy (more generally, for ${\epsilon }_y^A$ to be absolutely continuous with respect to inner capacity) as well as for ${\epsilon }_y^A({\mathbb{R}}^n)\equiv 1$ to hold. Those criteria are given in terms of the newly defined concepts of $\alpha$-thinness and $\alpha$-ultrathinness at infinity that generalize the concepts of thinness at infinity by Doob and Brelot, respectively. Further, we extend some of these results to ${\mu }^A$ general by verifying the formula ${\mu }^A=\int {\epsilon }_y^{A}d\mu (y)$. We also show that there is a $K_{\sigma }$-set $A_0\subset A$ such that ${\mu }^A={\mu }^{A_0}$ for all $\mu$, and give various applications of this theorem. In particular, we prove the vague and strong continuity of the inner swept, resp. equilibrium, measure under the approximation of $A$ arbitrary, thereby strengthening Fuglede’s result established for $A$ Borel (Acta Math., 1960). Being new even for $\alpha =2$, the results obtained also present a further development of the theory of inner Newtonian capacities and of inner Newtonian balayage, originated by Cartan.