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

Focusing first on the inner $α$-harmonic measure $\varepsilon_y^A$ ($\varepsilon_y$ being the unit Dirac measure, and $μ^A$ the inner $α$-Riesz balayage of a Radon measure $μ$ 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 $y\mapsto\varepsilon_y^A$ outside the inner $α$-irregular points for $A$, and obtain necessary and sufficient conditions for $\varepsilon_y^A$ to be of finite energy (more generally, for $\varepsilon_y^A$ to be absolutely continuous with respect to inner capacity) as well as for $\varepsilon_y^A(\mathbb R^n)\equiv1$ to hold. Those criteria are given in terms of the newly defined concepts of $α$-thinness and $α$-ultrathinness at infinity that generalize the concepts of thinness at infinity by Doob and Brelot, respectively. Further, we extend some of these results to $μ^A$ general by verifying the formula $μ^A=\int\varepsilon_y^A\,dμ(y)$. We also show that there is a $K_σ$-set $A_0\subset A$ such that $μ^A=μ^{A_0}$ for all $μ$, 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 $α=2$, the results obtained also present a further development of the theory of inner Newtonian capacities and of inner Newtonian balayage, originated by Cartan.