Co-countable sets of uniqueness for series of independent random variables (Q1763416)

Let be given a sequence of independent random variables \((f_k)\) on standard Borel space \(\Omega\) with probability \(\mu\) and a measurable set \(F\). Then a countable set \(S\subset F\) is shown with the property that series \(\sum_K e_k f_k\) which are constant on \(S\) are constant almost everywhere on \(F\).