Can a nonempty set has probability zero?

  • #1
Reworded version:

I think I need to re-word as follow:

Can a nonempty set X with P(X)=0?

Suppose F to be a non-empty set with [tex]P(F)\neq 0[/tex]. Call its closure be F'.

Now let set theoretic different F'\F be A. Clearly, A could be non-empty.
Is this the case where P(A)=0?

Is not, how do you relate P(F) and P(F')?


=================================


Original version:
Can a nonempty set X has P(X)=0?

My thought is suppose a nonempty set F then its closure F'\F=A where P(A)=0.

Is this true??
 
Last edited:

Answers and Replies

  • #2
lavinia
Science Advisor
Gold Member
3,236
623
Can a nonempty set X has P(X)=0?

My thought is suppose a nonempty set F then its closure F'\F=A where P(A)=0.

Is this true??
No. The closure of a set of probability 0 could have probability 1
 
  • #3
No. The closure of a set of probability 0 could have probability 1
I think I need to re-word as follow:

Suppose F to be a non-empty set with [tex]P(F)\neq 0[/tex]. Call its closure be F'.

Now let set theoretic different F'\F be A. Clearly, A could be non-empty.
Is this the case where P(A)=0?

Is not, how do you relate P(F) and P(F')?
 
Last edited:
  • #4
statdad
Homework Helper
1,495
35
Think of an absolutely continuous distribution (Gaussian as one example). The interval (0,1) could have non-zero probability. Its closure is [0,1], also with non-zero probability, but the difference you reference is [tex] \{0,1\}[/tex] which has zero probability.
 
  • #5
662
1
No. The closure of a set of probability 0 could have probability 1
Of course: throw a dart at the real line. Then P(hitting a rational)=0, but

P(hitting some element in Cl(Q))=1

Then this is also an example of a set being non-empty, yet having probability zero.
 
Last edited:
  • #6
Thanks for those commented above.
Now I understand that the probability of a set and probability its closure equals depends on the probability measure (or probability mass function) and of course the set itself.
 

Related Threads on Can a nonempty set has probability zero?

Replies
4
Views
342
Replies
4
Views
973
Replies
10
Views
1K
  • Last Post
3
Replies
62
Views
7K
  • Last Post
Replies
6
Views
3K
  • Last Post
Replies
1
Views
605
Replies
1
Views
907
  • Last Post
Replies
4
Views
649
Top