  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.
