Martingales and Theorems of Cantor–Young–Bernstein and de la Vallée Poussin

Uniqueness problems for one-dimensional Haar series and for multiple ones have understood in numerous works. It is well-known that the subsequence of the partial sums $S_{2^k}$ of an arbitrary Haar series can be represented as a discrete-time martingale on some filtered probability space $(\Omega,\,\mathcal{F},\,(\mathcal{F}_k ),\, \mathbf{P})$. In paper the concept of a $\mathcal{U}$-set for martingales is presented and some uniqueness theorems for martingales on arbitrary compact filtered probability spaces are established. In particular, it is proved that every set $U \in \cup_{k=0}^\infty \mathcal{F}_k$ with $\mathbf{P} (U)=0$ is a $\mathcal{U}$-set for martingales on a compact space $(\Omega,\,\mathcal{F},\,(\mathcal{F}_k ),\, \mathbf{P})$ (Cantor–Young–Bernstein type theorem). The result above is supplemented by some de la Vallée Poussin type theorems.