Can a Collection of Measurable Functions Have a Non-Measurable Lim Sup?

[hermanni]

Give an example of a collection of functions f_n (x) for 0 < n< infinity each measurable but such that lim sup n->infinity f_n(x) is not measurable.

I think , this collection shouldn't be a sequence of functions otherwise lim sup would be measurable. So I tried with sup first:
Take V as Vitali set and define the collection as X_{a} where X_{a} is a characteristic function of set {a} and a comes from V. clearly sup is X_{V} which is not measurable. Any idea for the lim sup??

 

[Office_Shredder]

How can you take limsup of something that's not a sequence of functions?