Constructing a Non-Measurable Lim Sup Using a Collection of Functions

[fzgorums]

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??

 

[ystael]

Is it possible to choose a decreasing sequence $$(Q_n)_{n>0}$$ of subsets of $$\mathbb{Q}$$ so that $$A_n = Q_n + V$$ is measurable for each $$n$$ but $$\textstyle\bigcap_n A_n = V$$?