Independence and atoms (Q2750907)

Let \(P\) be a probability measure on a \(\sigma\)-algebra \({\mathcal A}\). The authors present an algorithm, which is hard to apply, in order to check if \({\mathcal A}\) is generated by a countable family of independent events in case \(P\) is purely atomic. They claim a consequence of this algorithm as follows: If the range \(P({\mathcal A})\) contains an interval \([0,\varepsilon]\), then \({\mathcal A}\) contains infinitely many nontrivial independent events.