The uniqueness theorem for measures in $C(K)$ and its application in the theory of stochastic processes.

Let $(\Omega, \Sigma, \mathbf P)$ be a probability space, $K$ – a separable topological space, $\xi\colon\Omega\times K\to\mathbb R$ – a stochastic process with continious realizations. Let us define the distance between the stochastic process $\xi$ and the continious function $a\in C(K)$ as a random variable
$$ \alpha_a(\omega)=\max_{R\in K}|\xi(\omega, R)-a(K)|. $$

The main result of this article is the theorem that the stochastic process can be determined by the $p$-th moments of its distances from continious functions, where $p$ is a fixed real number, $p\ne0, 2, 4, 6,\dots$.