Quote:
Originally Posted by lastcardcharlie
Can an event with probability zero occur?
I wrote about this back in 2008. See
this post and its follow-ups.
Here are some additional comments. The phrase "an event with probability zero" is well-defined, mathematically. Given a probability space (Ω,
F, P), an "event with probability zero" is any element A ∈
F such that P(A) = 0. However, the phrase "A occurs" is not well-defined, mathematically. The events are not occurring. They are just sitting there, in the collection
F.
Let us, therefore, try to rigorously define the concept of "occurring". Let us say that an "occurrence assignment" is a map O:
F → {0,1} -- if O(A) = 0, then A "did not occur"; and if O(A) = 1, then A "did occur" -- which is logically consistent, i.e.
- O(Ø) = 0,
- if A ⊂ B and O(A) = 1, then O(B) = 1.
- if O(A) = 1 and O(B) = 1, then O(A ∩ B) = 1,
- for all A ∈ F, either O(A) = 1 or O(Ac) = 1.
Note that these imply that for all A ∈
F, O(A) = 1 iff O(A
c) = 0.
Occurrence assignments always exist. Given any fixed ω ∈ Ω, we can define O
ω(A) = 1
A(ω). In other words, A occurs if and only if ω ∈ A. But are all occurrence assignments of this form? I think not. I believe it can be proven (using Zorn's lemma) that if
G ⊂
F satisfies (i) Ω ∈
G, (ii) A ∈
G and A ⊂ B implies B ∈
G, and (iii) A ∈
G and B ∈
G implies A ∩ B ∈
G, then there exists an occurrence assignment O such that O(A) = 1 for all A ∈
G.
Now that the phrase "A occurs" is rigorously defined, let us reconsider the original question. Suppose we want to believe that events of probability zero cannot occur, and events of probability one must occur. To show that this position is sensible, we must show that given any arbitrary probability space (Ω,
F, P), it is always possible to find an occurrence assignment O such that P(A) = 0 implies O(A) = 0, and P(A) = 1 implies O(A) = 1. Can we prove this? Yes, if I am right about the Zorn's lemma idea above. We simply take
G = {A ∈
F: P(A) = 1}.
For example, consider Ω = [0,1],
F the Borel subsets of Ω, and P the uniform measure. Can we find an occurrence assignment O such that P(A) = 0 implies O(A) = 0, and P(A) = 1 implies O(A) = 1? Well, not if we restrict our attention to occurrence assignments of the form O
ω(A) = 1
A(ω). For each of these, we will have P({ω}) = 0, but O({ω}) = 1. However, Zorn's lemma should guarantee the existence of an occurrence assignment such that O({w}) = 0 for all ω ∈ [0,1], even if we cannot give an explicit description of it.
For some related info, see
this Wiki link.