Is a Measure 0 Null Set Also Dense?

  • Context: Graduate 
  • Thread starter Thread starter Bachelier
  • Start date Start date
  • Tags Tags
    Measure Set
Click For Summary

Discussion Overview

The discussion revolves around the properties of null sets, specifically whether a measure 0 null set can also be considered dense. Participants explore definitions and examples related to measure 0 sets and their density.

Discussion Character

  • Debate/contested, Conceptual clarification

Main Points Raised

  • Some participants question whether a null set is synonymous with a set of measure 0.
  • There is a suggestion that not all sets of measure 0 are dense, with the empty set presented as a counterexample.
  • Participants discuss the Cantor set, noting it is uncountable and questioning its classification in relation to measure 0 and density.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between measure 0 sets and density, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

There are limitations in the definitions and examples provided, particularly regarding the relationship between null sets and density, as well as the implications of specific counterexamples like the empty set and the Cantor set.

Bachelier
Messages
375
Reaction score
0
Does it also imply that the set is dense?
 
Physics news on Phys.org
What is a null set? Isn't it just a synonym for a set of measure 0?

And what is your question? Are you asking whether all sets of measure 0 are dense? The empty set would be a counterexample...
 
micromass said:
What is a null set? Isn't it just a synonym for a set of measure 0?

And what is your question? Are you asking whether all sets of measure 0 are dense? The empty set would be a counterexample...

and so is the Cantor set I assume. I used to think they are both the same. They are indeed 2 different notions.

Now the Cantor set is uncountable. right?
 
Bachelier said:
Now the Cantor set is uncountable. right?

Right!
 
thanks
 

Similar threads

Undergrad Measure with respect to complete measure is also complete
  • · Replies 3 ·
Replies
3
Views
2K
MHB Prove that the algebra generated is dense
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Are slices of measurable functions measurable?
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Dense set equivalent definitions
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Basic measurement theory question
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad On theorem 1.19 in Folland's and completion of measure
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Connection between absolute continuity of function and measure
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Open balls dense in closed balls in Euclidean space
  • · Replies 2 ·
Replies
2
Views
3K
MHB Every Closed Set in R has a Countable Dense Subset
  • · Replies 7 ·
Replies
7
Views
5K
Undergrad Baire Category Theorem: Question About Countable Dense Open Sets
  • · Replies 3 ·
Replies
3
Views
3K