No abstract

It is shown that, for every sequence (fn) of stochastically independent functions defined on [0, 1]-of mean zero and variance one, uniformly bounded by M -if the series ∞ n=1 anfn converges to some constant on a set of positive measure, then there are only finitely many non-null coefficients an, extending similar results by Stechkin and Ul'yanov on the Rademacher system. The best constant C M is computed such that for every such sequence (fn) any set of measure strictly less than C M is a set of uniqueness for (fn).

Given a sequence of independent random variables (f k ) on a standard Borel space Ω with probability measure µ and a measurable set F , the existence of a countable set S ⊂ F is shown, with the property that series k c k f k which are constant on S are constant almost everywhere on F . As a consequence, if the functions f k are not constant almost everywhere, then there is a countable set S ⊂ Ω such that the only series k c k f k which is null on S is the null series; moreover, if there exists b < 1 such that µ(f −1 k ({α})) b for every k and every α, then the set S can be taken inside any measurable set F with µ(F ) > b.  2004 Elsevier Inc. All rights reserved.