You may use the points below.

If {i : i = 1,2, ... } is a family of integrable random variables such that i + 0 almost surely as oo and limi* Ei 0, is the family {&i : = 1, 2, ... } uniformly integrable? If yes, give a proof; if no, give a counterexample. : Uniform integrability Def A family of r.v.'s {i : i el} is said to be uniformly integrable if sup E [[Fil1{16}]>c}] +0 as c+00. ier Lemma A necessary and sufficient condition for a family of rvs {&i : El} to be UI is that supier El&ic}] +0 as c+00. ier Lemma A necessary and sufficient condition for a family of rvs {&i : El} to be UI is that supier El&i