1. thank you! Xn rightarrow X as n rightarrow + infinity almost surely if and only if the set N =
{w: lim n rightarrow + infinity Xn(w) X(w)} has probability 0.
Solution
Ex n tends to Ex as n tends to positive infinity almost surely if and only if the set N
equals w : limit n tends to positive infinity Ex n of w not equals Ex of w has probability zero.