Proving the Vitali Set: A Real Line Challenge

In summary, the conversation discusses constructing a Vitali set, which is a subset of the real line containing exactly one point from each coset of the group of rational numbers. The first two equations state that every Lebesgue measurable subset of V is a nullset, and V itself is not Lebesgue measurable. The third equation presents a challenge in constructing a Vitali set, as it needs to be shown that every set of positive Lebesgue outer measure contains a set that is not Lebesgue measurable. The suggested solution involves choosing points from each coset that intersects with a given set S, and then making sure that the set is not too small or too big.
  • #1
jem05
56
0

Homework Statement


Let V a subset of the real line be called a vitali set if V contains precisely one point from each coset of the group of rational numbers. Prove:

Homework Equations


1) every lebesgue measurable subset of V is a nullset.
2) V is not lebesgue measurable
3) every set of positive Lebesgue outer measure contains a set that is not lebesgue measurable.

The Attempt at a Solution



i already did 1 and 2.
i am stuck at 3,
i think i need to consruct a vitali set in the given one which i could do if the set contains an interval,..
but it doesn't need to , all i could know about it is that it is not countable since if it were, then its outermeasure will be zero.

thanks a lot
any help is appreciated
 
Physics news on Phys.org
  • #2
If S is a set of positive Lebesgue outer measure, not every coset of [itex]\mathbb{Q}[/itex] will necessarily intersect S, but uncountably many of them will. Try defining V as follows: from each coset that does intersect S, choose one point from the intersection. Then proceed as you did in 2.

[Edit]: My construction will assure that V is not too small, but you also need to make sure that it isn't too big. But S has outer measure > 0, so there must exist some interval I (say, of length 1) such that [itex]S \cap I[/itex] has outer measure > 0. So replace S by [itex]S \cap I[/itex] and before constructing V.
 
Last edited:

1. What is the Vitali Set and why is it important to prove?

The Vitali Set is a set of real numbers that was first discovered by Giuseppe Vitali in 1905. It is an example of a non-measurable set, meaning it cannot be assigned a definite length or area. Proving the existence of the Vitali Set is important because it demonstrates the limitations of the traditional measure theory and helps mathematicians to develop alternative approaches to measure and integration.

2. What is the Real Line Challenge and how does it relate to the Vitali Set?

The Real Line Challenge is a problem posed by mathematician David Hilbert in 1900, which asks for a proof of the existence of non-measurable sets in the real line. The Vitali Set is one of the possible solutions to this challenge, and proving its existence would provide a solution to the challenge.

3. What have been the main obstacles in proving the existence of the Vitali Set?

One of the main obstacles in proving the Vitali Set is that it requires a rigorous understanding of measure theory and set theory. The concept of non-measurable sets also goes against our intuitive understanding of measurement. Additionally, the Vitali Set is a highly abstract mathematical concept, making it challenging to visualize and work with.

4. How have mathematicians attempted to prove the existence of the Vitali Set?

Several methods have been proposed to prove the existence of the Vitali Set, including the use of non-standard analysis, ultrafilters, and non-measurable functions. However, all attempts have faced significant challenges and have not been universally accepted by the mathematical community.

5. Why is the existence of non-measurable sets important in mathematics?

The existence of non-measurable sets challenges our understanding of measurement and highlights the limitations of traditional measure theory. It also has practical applications in fields such as probability theory, where non-measurable sets are used to construct non-measurable events. The study of non-measurable sets has also led to the development of alternative approaches to measure and integration, which have proven useful in solving other complex mathematical problems.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
3K
  • Calculus and Beyond Homework Help
Replies
1
Views
455
  • Calculus and Beyond Homework Help
Replies
5
Views
849
  • Calculus and Beyond Homework Help
Replies
20
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • General Math
2
Replies
66
Views
4K
  • Calculus and Beyond Homework Help
Replies
8
Views
2K
Back
Top