The class of harmonic and subharmonic functions have been studied by many authors, defined in different ways, by the Laplace differential operator, averaging, generalised Laplace operators, etc. The well-known theorem of Blaschke-Privalov gives an excellent criterion for subharmonicity in terms of the generalised Laplace operators: an upper semi-continuous in the domain $$D \subset \mathbb {R}^{n}$$ function u(x), $$u(x)\not \equiv -\infty ,$$ is subharmonic if and only if $$ {\overline{\bigtriangleup }} u(x)\ge 0 \quad \forall x^0\in D{\setminus } u_{-\infty }.$$ One of the notable results is Privalov’s theorem, where he got more deeper result with an exceptional set E: if the function u(x), $$u(x)\not \equiv -\infty $$ , is upper semi-continuous in the domain $$D \subset {\mathbb {R}}^{n}$$ and the following two conditions hold: Then the function u(x) is subharmonic in D. The purpose of this paper is to characterise completely this type of exceptional sets. For this, we introduce the so-called $$ {\underline{S}} $$ and $$ {\overline{S}} $$ singular-sets, which are directly related to the exceptional set of I. Privalov. We prove: $$E \in {\underline{S}}$$ if and only if $$mes E =0;$$ $$E \in {\overline{S}}$$ if and only if $$E^\circ =\emptyset .$$