Does a Set with Empty Interior Have Measure Zero?

In summary, the Cantor set, specifically the standard middle thirds Cantor set, has both an empty interior and a measure of zero. However, not all Cantor sets have measure zero as stated in the book. The set of irrational numbers between 0 and 1 is also an example of a set with empty interior and measure zero.
  • #1
ak416
122
0
does a set with empty interior have measure zero? I think it does...
 
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  • #2
Doesn't the Cantor set have empty interior?
 
  • #3
ya and I read somewhere that it has measure zero. So this wouldn't be a counterexample.
 
  • #4
ok i may be wrong on this...
 
  • #5
The Cantor set does have an empty interior and also has measure zero.

I never worked with measurable spaces yet, but if I had to guess, I would say a set with empty interior does in fact have measure zero.
 
  • #6
no actually it says in my book that the standard middle thirds cantor set has measure zero, but not any cantor set. So I guess its incorrect.
 
  • #7
come on guys, look at the set of irrational numbers betwen 0 and 1. what is the interior? what is the measure?

take your book and put it under your coffee cup while you think about this.
 

1. What does it mean for a set to have empty interior?

Having empty interior means that there are no points within the set that are contained in an open ball of any positive radius. In other words, there are no points that are isolated from the rest of the set.

2. How is the interior of a set related to its measure?

The interior of a set is related to its measure through the concept of measure zero. If a set has empty interior, then it is said to have measure zero because it does not contain any points that contribute to its measure.

3. Can a set with empty interior have a positive measure?

No, a set with empty interior cannot have a positive measure. This is because the interior of a set is a subset of the set itself, and if the interior is empty, then the set must also have measure zero.

4. Is there a difference between a set with empty interior and an empty set?

Yes, there is a difference between a set with empty interior and an empty set. An empty set has no elements, while a set with empty interior may still have elements, but they are not isolated from the rest of the set.

5. How does the concept of empty interior apply to real-world examples?

The concept of empty interior can be applied to real-world examples such as a line segment. A line segment has empty interior because it does not contain any points that are isolated from the rest of the segment. This means that it has measure zero, even though it has a non-zero length.

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