Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and correspond to (nonlinear in general) functionals that are linear on singly generated subalgebras or singly generated cones of functions. They lack subadditivity, and many standard techniques of measure theory and functional analysis do not apply to them. Nevertheless, we show that many classical results of probability theory hold for topological and deficient topological measures. In particular, we prove a version of Aleksandrov’s theorem for equivalent definitions of weak convergence of deficient topological measures. We also prove a version of Prokhorov’s theorem which relates the existence of a weakly convergent subsequence in any sequence in a family of topological measures to the characteristics of being a uniformly bounded in variation and uniformly tight family. We define Prokhorov and Kantorovich–Rubenstein metrics and show that convergence in either of them implies weak convergence of (deficient) topological measures on metric spaces. We also generalize many known results about various dense and nowhere dense subsets of deficient topological measures. The present paper constitutes a necessary step to further research in probability theory and its applications in the context of (deficient) topological measures and corresponding nonlinear functionals.

在拓扑空间的开放和封闭子集上定义拓扑测度和不足的拓扑测度，归纳规则的Borel测度，并对应于（通常是非线性的）在单个生成的子代数或单个生成的函数锥上呈线性的泛函。它们缺乏亚可加性，许多标准的测量理论和功能分析技术均不适用于它们。但是，我们证明了概率论的许多经典结果适用于拓扑和不足的拓扑测度。特别是，我们证明了Aleksandrov定理的一个版本，该不足定理是不足拓扑度量的弱收敛的等价定义。我们还证明了普罗霍洛夫定理的一个版本，该定理将拓扑度量族中任意序列中弱收敛子序列的存在与变异均匀一致且族群紧密一致的特征联系起来。我们定义了Prokhorov和Kantorovich–Rubenstein度量，并表明它们两者的收敛都意味着度量空间上（不足）拓扑度量的弱收敛。我们还概括了许多有关拓扑措施的各种密集子集和无处密集子集的已知结果。本论文构成了进一步研究概率论及其在（不足）拓扑测度和相应的非线性泛函背景下的应用的必要步骤。我们定义了Prokhorov和Kantorovich–Rubenstein度量，并表明它们两者的收敛都意味着度量空间上（不足）拓扑度量的弱收敛。我们还概括了许多有关拓扑措施的各种密集子集和无处密集子集的已知结果。本论文构成了进一步研究概率论及其在（不足）拓扑测度和相应的非线性泛函背景下的应用的必要步骤。我们定义了Prokhorov和Kantorovich–Rubenstein度量，并表明它们两者的收敛都意味着度量空间上（不足）拓扑度量的弱收敛。我们还概括了许多有关拓扑措施的各种密集子集和无处密集子集的已知结果。本论文构成了进一步研究概率论及其在（不足）拓扑测度和相应的非线性泛函背景下的应用的必要步骤。