Co-countable sets of uniqueness for series of independent random variables

Given a sequence of independent random variables (fk) 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 ckfk which are constant on S are constant almost everywhere on F. As a consequence, if the functions fk are not constant almost everywhere, then there is a countable set S ⊂ Ω such that the only series k ckfk 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