Is a Measure 0 Null Set Also Dense?

In summary, the conversation discusses the difference between a null set and a set of measure 0, and whether all sets of measure 0 are dense. The empty set and the Cantor set are mentioned as examples of counterexamples. The Cantor set is also mentioned as being uncountable.
  • #1
Bachelier
376
0
Does it also imply that the set is dense?
 
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  • #2
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...
 
  • #3
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?
 
  • #4
Bachelier said:
Now the Cantor set is uncountable. right?

Right!
 
  • #5
thanks
 

1. What is a null set of Measure 0?

A null set of Measure 0 is a set that has a Lebesgue measure of 0, meaning it has no volume or "size" in mathematical terms. It is also known as a negligible set or a set of measure 0.

2. How is a null set of Measure 0 different from an empty set?

A null set of Measure 0 and an empty set are conceptually different. An empty set has no elements, while a null set of Measure 0 can have elements, but they have no measure or size.

3. What is the importance of null sets of Measure 0 in mathematics?

Null sets of Measure 0 are important in mathematics because they help define and understand the concept of measure and integration. They also play a crucial role in measure theory, which is a fundamental branch of mathematics.

4. Can a null set of Measure 0 be uncountable?

Yes, a null set of Measure 0 can be uncountable. The cardinality of a set does not affect its measure, so a null set of Measure 0 can have any number of elements, including an uncountable amount.

5. How are null sets of Measure 0 used in real-world applications?

Null sets of Measure 0 are used in various real-world applications, such as in probability and statistics, where they help define the concept of almost sure convergence. They are also used in physics, specifically in the field of fractals, where they help describe and analyze complex geometric patterns.

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