  Focusing first on the inner $\alpha$-harmonic measure $\varepsilon_y^A$
($\varepsilon_y$ 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
$y\mapsto\varepsilon_y^A$ outside the inner $\alpha$-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 $\alpha$-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
$\mu^A=\int\varepsilon_y^A\,d\mu(y)$. 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.

Classical Analysis and ODEs;Complex Variables