Probability - Infinite Union of Subsets of a Sample Space

[Legendre]

Homework Statement

This is a question about mathematical probability, using the sigma-algebra, measure and probability space approach.

Define A(t) = {all outcomes, w, in the sample space such that Y(w) < or = t}
where Y is a random variable and t is any real number.

Fix a real number X.

Consider an increasing sequence {Xn} such that Xn tend to X from below/left as n tend to infinity. Therefore, Xn < or = X for all n.

So, A(Xn) is a subset of A(X) for all n.

Homework Equations

N.A.

The Attempt at a Solution

This isn't a homework question but I thought it fits in this forum. It is stated in my text that the infinite union of A(Xn) is not A(X).

I think this is because Xn can tend to X in such a way that X is not equal to X all for n. So their infinite union will always not contain some element in X.

But how do I show this mathematically? A little nudge in the right direction so I can solve this myself? Thanks in advance!

 

[mighty2000]

Thank you for your question. This is an interesting concept in mathematical probability.

To show that the infinite union of A(Xn) is not equal to A(X), we can use a counterexample. Consider the set of real numbers from 0 to 1, and let Xn = 1/n for all n. As n tends to infinity, Xn will tend to 0 from below. Therefore, A(Xn) is a subset of A(0) for all n.

However, if we take the infinite union of A(Xn), we will have all outcomes where Y(w) is less than or equal to 1/n for all n. This set will not include the outcome where Y(w) is equal to 0, as it is not less than or equal to any positive real number. Therefore, the infinite union of A(Xn) does not contain all outcomes in A(0).

In general, for any real number X, we can always find a sequence of real numbers {Xn} that tends to X from below/left, such that the infinite union of A(Xn) does not contain all outcomes in A(X). This shows that the infinite union of A(Xn) is not equal to A(X).

I hope this helps to clarify the concept. Let me know if you have any further questions.