Understanding Vitali Sets

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In summary, the Vitali set is not measureable because there is a contradiction between its measure and the fact that [0,1] has a measure of 1.
  • #1
Artusartos
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I'm not sure if I understood Vitali Sets correctly, so I just want to write what I understood (because I don't know if it's right):

We have an equivalence relation where [itex]x \sim y \iff x-y \in Q[/itex]. So if we look at the interval [0,1], each irrational number will have its own equivalence class...and we will have one equivalence class for all rational numbers, right? Now, using the axiom of choice, we take one element from each equivalence class as a representative and form the set A. And then we form a new collection of sets [itex]A_q = \{q+a | a \in A\}[/itex]. We know that this collection has a countable number of sets, because each set corresponds to one rational number between 0 and 1...and the rational numbers are countable. We also know that the sets are disjoint. Then, when we take the union of these sets, we just need to add their measure. But we know that each set has the same measure, since measure is translation invariant. But we also know that there are infinite number of rational numbers between 0 and 1, so there are infinite amound of sets...so the measure must be infinity. But that can't be true since [0,1] has measure 1. So that's a contradiction, and the vitali set is not measureable.

Do you think my understanding is correct? If not can you please correct me?
 
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  • #2
I think you got it...
 
  • #3
Artusartos said:
So if we look at the interval [0,1], each irrational number will have its own equivalence class..

It isn't clear what you mean by that.

.and we will have one equivalence class for all rational numbers, right?

Yes, if you mean to say that all rational numbers are in the same equivalence class.
 
  • #4
Artusartos said:
Then, when we take the union of these sets, we just need to add their measure. But we know that each set has the same measure, since measure is translation invariant. But we also know that there are infinite number of rational numbers between 0 and 1, so there are infinite amound of sets...so the measure must be infinity. But that can't be true since [0,1] has measure 1. So that's a contradiction, and the vitali set is not measureable.

Do you think my understanding is correct? If not can you please correct me?
Pretty close, but I would state it as follows. As you said, if ##A## is measurable, then each ##A_q## is measurable and has the same measure, due to translation invariance. Also, the ##A_q## form a countable partition of ##[0,1]##, so we must have
$$\sum_{q\in \mathbb{Q}} m(A_q) = 1$$
But all of the ##m(A_q)## are equal to ##m(A)##, so the sum on the left is either zero or infinity, depending on whether ##m(A) = 0## or ##m(A) > 0##. In either case we have a contradiction.
 
  • #5
Artusartos said:
So if we look at the interval [0,1], each irrational number will have its own equivalence class
No, a single equivalence class is of the form ##\{x + q \textrm{ }|\textrm{ } q \in \mathbb{Q}\}##, so every equivalence class contains a countably infinite number of elements. There is one equivalence class containing all of the rationals (and no irrationals). Every other equivalence class contains a countably infinite number of irrationals (and no rationals).

There are of course uncountably many equivalence classes. ##A## contains one element from each equivalence class, by construction. The same is true of each ##A_q##.
 
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  • #6
jbunniii said:
Pretty close, but I would state it as follows. As you said, if ##A## is measurable, then each ##A_q## is measurable and has the same measure, due to translation invariance. Also, the ##A_q## form a countable partition of ##[0,1]##, so we must have
$$\sum_{q\in \mathbb{Q}} m(A_q) = 1$$
But all of the ##m(A_q)## are equal to ##m(A)##, so the sum on the left is either zero or infinity, depending on whether ##m(A) = 0## or ##m(A) > 0##. In either case we have a contradiction.

Thanks
 

1. What is a Vitali set?

A Vitali set is a non-measurable set in mathematics that was discovered by mathematician Giuseppe Vitali in 1905. It is a subset of the real numbers that cannot be assigned a unique measure, making it a key concept in the study of measure theory.

2. How is a Vitali set constructed?

A Vitali set is constructed using the Axiom of Choice, which allows for the creation of sets that do not have a well-defined measure. The process involves partitioning the real numbers into equivalence classes based on their difference from rational numbers, and then selecting one element from each class to form the Vitali set.

3. Why is understanding Vitali sets important?

Understanding Vitali sets is important in the field of measure theory, as they provide a counterexample to the notion that every set can be measured. They also have applications in other areas of mathematics, such as in the study of fractals and analysis.

4. Can Vitali sets be visualized?

No, Vitali sets cannot be visualized as they are non-measurable and therefore cannot be represented graphically. However, they can be described mathematically and their properties can be studied through mathematical analysis.

5. Are Vitali sets unique?

No, there are infinitely many Vitali sets that can be constructed using the Axiom of Choice. This is because there are infinitely many ways to partition the real numbers into equivalence classes based on their difference from rational numbers, and each partition will result in a different Vitali set.

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