We introduce the notion of uniform distribution on a metric compactum. The desired distribution is defined as the limit of a sequence of the classical uniform distributions on finite sets which are uniformly distributed on the compactum in the geometric sense. We show that a uniform distribution exists on the metrically homogeneous compacta and the canonically closed subsets of a Euclidean space whose boundary has Lebesgue measure zero. If a compactum (satisfying some metric constraints) admits a uniform distribution then so does its every canonically closed subset that has zero uniform measure of the boundary. We prove that compacta, admitting a uniform distribution, are dimensionally homogeneous in the sense of box-dimension.

我们介绍了度量公制上的均匀分布的概念。期望的分布被定义为在几何意义上均匀分布在紧致线上的有限集上经典均匀分布的序列的极限。我们显示出均匀分布存在于度量均质的紧致结构和欧氏空间的规范封闭子集上，该空间的边界具有勒贝格测度为零。如果一个紧密集（满足某些度量标准约束）接受一个均匀分布，那么它的每个规范封闭的子集也将具有零个统一的边界度量。我们证明，在盒装尺寸的意义上，允许均匀分布的粉饼在尺寸上是均匀的。