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

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SUMMARY

The discussion focuses on constructing a collection of measurable functions \( f_n(x) \) such that the limit superior \( \limsup_{n \to \infty} f_n(x) \) is not measurable. The Vitali set is utilized as a key example, with the characteristic function \( X_a \) defined for elements \( a \) from the Vitali set. The participants explore the implications of using a decreasing sequence of subsets of \( \mathbb{Q} \) to maintain measurability for each \( A_n \) while ensuring that the intersection \( \bigcap_n A_n \) remains the Vitali set, which is not measurable.

PREREQUISITES
  • 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.
  • Basic concepts of characteristic functions in set theory.
NEXT STEPS
  • Study the properties of the Vitali set in relation to Lebesgue measure.
  • Explore the construction of characteristic functions for various sets.
  • Investigate limit superior and limit inferior in the context of sequences of functions.
  • Learn about measurable sets and functions in the framework of measure theory.
USEFUL FOR

Mathematicians, particularly those specializing in measure theory, analysts studying properties of functions, and students exploring advanced concepts in real analysis.

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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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Is it possible to choose a decreasing sequence (Q_n)_{n&gt;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?
 

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