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Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, Volume 14, Issue 4(2), Pages 569–574 (Mi isu550)  

Mathematics

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

M. G. Plotnikov, Ju. A. Plotnikova

Vologda State Academy of Milk Industry, 2, Shmidt str., P.O. Molochnoe, Vologda, 160555, Russia

Abstract: 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.

Key words: set of uniqueness, martingale, filtered probability space, Cantor–Young–Bernstein theorem, de la Vallée Poussin theorem.

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UDC: 517.518.3+519.216.8

Citation: M. G. Plotnikov, Ju. A. Plotnikova, “Martingales and Theorems of Cantor–Young–Bernstein and de la Vallée Poussin”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 14:4(2) (2014), 569–574

Citation in format AMSBIB
\Bibitem{PloPlo14}
\by M.~G.~Plotnikov, Ju.~A.~Plotnikova
\paper Martingales and Theorems of Cantor--Young--Bernstein and de la Vall\'ee Poussin
\jour Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform.
\yr 2014
\vol 14
\issue 4(2)
\pages 569--574
\mathnet{http://mi.mathnet.ru/isu550}


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