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 S2k of an arbitrary Haar series can be represented as a discrete-time martingale on some filtered probability space (Ω, F, (Fk), P). In paper the concept of a U -set for martingales is presented and some uniqueness theorems for martingales on arbitrary compact filtered probability spaces are established.
