How Many Non-Measurable Sets Exist?

  • Thread starter fourier jr
  • Start date
  • Tags
    Sets
If B has measure 0, then its complement (which is B's complement but not B itself) has the same property. So you can get a pair of disjoint subsets of the closed unit interval, neither of which is measurable.In summary, there are various examples of non-measurable sets, including the ones involving addition mod an irrational number and the constructions by Vitali and Edward Van Vleck. Another notable example is due to Bernstein, which involves constructing a set that meets every uncountable closed set and its complement also meeting every uncountable closed set, resulting in a pair of disjoint non-measurable subsets of the closed unit interval.
  • #1
fourier jr
765
13
are there lots of examples of non-measurable sets? the one that seems to be in most textbooks involves a type of addition mod an irrational number with equivalence classes, etc etc, which in some books is done geometrically as rotations of a circle through an irrational angle. that example was by vitali, but there is another one by edward van vleck where he constructs two subsets of (0,1) in a way that either the measure of one is 1 & the other has measure 0, or they both have what he called "upper measure" =1, & therefore they're both non-measurable. how many other ones are out there that I don't know about?
 
Physics news on Phys.org
  • #2
My favorite is due to Bernstein.

You construct (by transfinite induction) a set B so that both B and its complement meet every uncountable closed set.
 

FAQ: How Many Non-Measurable Sets Exist?

What is a non-measurable set?

A non-measurable set is a set that cannot be assigned a measure or size. In mathematical terms, it is a set that does not have a well-defined length, area, or volume.

Can you provide an example of a non-measurable set?

One example of a non-measurable set is the Vitali set, which is a subset of the real numbers. It is constructed by taking one element from each equivalence class of the quotient group of the real numbers modulo the rational numbers.

Why are non-measurable sets important in mathematics?

Non-measurable sets are important in mathematics because they challenge traditional concepts of measure and size. They also play a role in the study of more abstract mathematical concepts such as topology and measure theory.

How do non-measurable sets impact everyday life?

Non-measurable sets may seem abstract and disconnected from everyday life, but they have practical applications in fields such as economics and physics. For example, non-measurable sets can be used to model the behavior of an irrational consumer or to describe the distribution of atoms in a gas.

Are there any real-world examples of non-measurable sets?

Yes, there are real-world examples of non-measurable sets. One example is the Banach-Tarski paradox, which states that a solid ball can be decomposed into a finite number of pieces and then reassembled to form two identical copies of the original ball. This paradox relies on the existence of non-measurable sets and has implications for the study of geometry and measure theory.

Similar threads

Doubts on the Vitali Set - Marco's Story
Replies
3
Views
3K
Is It Possible to Construct Non-Measurable Sets Without the Axiom of Choice?
Replies
4
Views
2K
Various Intuitions and Conceptualizations of Measurable Cardinals.
Replies
0
Views
1K
I Are Real Numbers Essential in Scientific Measurements and Models?
3
Replies
85
Views
6K
I Munkres-Analysis on Manifolds: Extended Integrals
Replies
5
Views
2K
Set of discontinuties with measure 0
Replies
1
Views
1K
I Are slices of measurable functions measurable?
Replies
3
Views
990
I Measure of existence? (Stanford Encyclopedia of Philosophy)
Replies
7
Views
1K
Ruler measurements and knowing how many decimal places to include
Replies
9
Views
740
Integrability, basic measure theory: seeking help with confusing result
Replies
7
Views
3K

Hot Threads

Recent Insights

Back
Top