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 for every k and every α, then the set S can be taken inside any measurable set F with μ(F)>b.