Criteria for (sub-)harmonicity and continuation of (sub-)harmonic functions

We obtain criteria for harmonicity and subharmonicity of a function in a domain in $\mathbb{R}^d$, $d\geqslant2$, in terms of special Arens–Singer and Jensen measures. We also establish a criterion for (sub-)harmonicity of a $\delta$-subharmonic function in terms of the associated Riesz charge and special Arens–Singer and Jensen functions. To this end, we use the theorem of this article on continuation of (sub-)harmonic functions to polar sets.