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:

1. (1)

   \( {\overline{\bigtriangleup }}_B u(x^0)\ge 0 \quad \forall x^0\in D {\setminus } [E \cup u_{-\infty }]\), where \(E \subset D\) is a closed in D set, \(mes E =0\);

2. (2)

   \({\overline{\bigtriangleup }}_B u(x^0) > - \infty \quad \forall x^0\in E{\setminus } P, \) where \(P \subset E \) is some polar set.

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 \(E^\circ =\emptyset .\)

许多作者已经研究了谐波和亚谐波函数的类别，并以不同的方式对其进行了定义，包括拉普拉斯微分算子，求平均值，广义拉普拉斯算子等。拉普拉斯算子的形式：域\（D \ subset \ mathbb {R} ^ {n} \）函数u（x）中的上半连续，\（u（x）\ not \ equiv-\ infty， \）是次谐波，且仅当\（{\ overline {\ bigtriangleup}} u（x）\ ge 0 \ quad \ forall x ^ 0 \ in D {\ setminus} u _ {-\ infty}。\）之一时值得注意的结果是普里瓦洛夫定理，他在极好的集合E下得到了更深的结果：如果函数u（x）\ \（u（x）\ not \ equiv-\ infty \）在域\ {D \ subset {\ mathbb {R}} ^ {n} \中是上半连续的），并且以下两个条件成立：

1. （1）

   \（{\ overline {\ bigtriangleup}} _ B u（x ^ 0）\ ge 0 \ quad \ forall x ^ 0 \ in D {\ setminus} [E \ cup u _ {-\ infty}] \），其中\ （E \ subset D \）是D集合中的闭合\（mes E = 0 \） ;

2. （2）

   \（{\ overline {\ bigtriangleup}} _ B u（x ^ 0）>-\ infty \ quad \ forall x ^ 0 \ in E {\ setminus} P，\）其中\（P \ subset E \\）是一些极地集。

然后，函数u（x）在D中为次谐波。本文的目的是完全刻画这种类型的例外集。为此，我们介绍了所谓的\（{\\ underline {S}} \）和\（{\ overline {S}} \}）奇异集，它们与I. Privalov的特殊集合直接相关。我们证明：\（E \ in {\ underline {S}} \\）当且仅当\（mes E = 0; \） \（E \ in {\ overline {S}} \\）当且仅当\（mes E = 0; \ E ^ \ circ = \ emptyset。\）