for all a in R, {x in E: limsup{f_n: n in N}(x)>a},
={x in E: limsup{f_n(x): n in N}>a}
= Union {x in E: f_n(x) > a}
= a measurable set
Thank you. But that's for lim sup (I already knew that proof). For lim inf I can't just take the intersection as it messes up difference between >= and >. I think I have to go about by investigating the complements but I'm not sure.