On the role of the point at infinity in Deny's principle of positivity of mass for Riesz potentials

First introduced by J. Deny, the classical principle of positivity of mass states that if $\kappa_\alpha\mu\leqslant\kappa_\alpha\nu$ everywhere on $\mathbb{R}^n$, then $\mu(\mathbb{R}^n)\leqslant\nu(\mathbb{R}^n)$. Here $\mu,\nu$ are positive Radon measures on $\mathbb{R}^n$, $n\geqslant2$, and $\kappa_\alpha\mu$ is the potential of $\mu$ with respect to the Riesz kernel $|x-y|^{\alpha-n}$ of order $\alpha\in(0,2]$, $\alpha<n$. We strengthen Deny's principle by showing that $\mu(\mathbb{R}^n)\leqslant\nu(\mathbb{R}^n)$ still holds even if $\kappa_\alpha\mu\leqslant\kappa_\alpha\nu$ is fulfilled only on a proper subset $A$ of $\mathbb{R}^n$ that is not inner $\alpha$-thin at infinity; and moreover, this condition on $A$ cannot in general be improved. Hence, if $\xi$ is a signed measure on $\mathbb{R}^n$ with $\int1\,d\xi>0$, then $\kappa_\alpha\xi>0$ everywhere on $\mathbb{R}^n$, except for a subset which is inner $\alpha$-thin at infinity. The analysis performed is based on the author's recent theories of inner Riesz balayage and inner Riesz equilibrium measures (Potential Anal., 2022), the inner equilibrium measure being understood in an extended sense where both the energy and the total mass may be infinite.