I've been trying to solve the following question: Let X be a random variable s.t. Pr[|X|<+\infty]=1. Then for every epsilon>0 there exists a bounded random variable Y such that P[X\neq Y]<epsilon. The ideia here would be to find a set of epsilon measure so Y would be different than X in that set. However, it is not clear even that such a measurable set exists... Any help?