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

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SUMMARY

The discussion centers on the existence of a collection of measurable functions, specifically denoted as f_n(x), where each function is measurable, yet the limit superior (lim sup) as n approaches infinity is not measurable. The example provided utilizes the Vitali set, V, and defines a collection of characteristic functions, X_a, corresponding to elements a in V. The supremum of this collection results in a non-measurable function, X_V. The challenge remains in determining the lim sup of a non-sequential collection of functions.

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  • Understanding of measurable functions and their properties
  • Familiarity with the concept of limit superior in mathematical analysis
  • Knowledge of the Vitali set and its implications in measure theory
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  • Research the properties of limit superior in the context of non-sequential collections of functions
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hermanni
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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??
 
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How can you take limsup of something that's not a sequence of functions?
 

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