Definition of Random Variable (from Durrett)

[CantorSet]

Hi everyone,

I'm confused about Durrett's formal definition of a random variable, as well his formal notions of probability spaces in general. I always try to make abstract definitions concrete through simple examples, but I can't wrap my head around this one:

Durrett defines:

X is a random variable if $$X: \Omega \rightarrow \Re$$ and for every Borel set $$B \subset R$$, we have $$X^{-1}(B) \in F$$, where F is the sigma-algebra on $$\Omega$$.

Well, I tried to make this concrete with a simple example. Suppose we have a box with 100 balls labeled from 1 to 100. We draw 3 balls from this box without replacement. Let X be the sum of these three balls. We know that X is a random variable and it takes on values from 3 to 297.

To interpret this example in the context of Durrett's formal definition, I guess $$\Omega$$ is the set of all subsets of size three from 1 to 100. So $$\Omega = \{\{1,2,3\} , \{1,2,4\}, ...\}$$ The sigma-algebra F is all collections of these subsets. Then, our random variable X basically only maps the subsets of size 3 to the integers from 3 to 279. So far so good.

But the definition says for every Borel set $$B \subset R$$, which we can take as $$B = (0,1)$$, a perfectly legit Borel set, and we would have $$X^{-1}(B) \in F$$. But the problem is the random variable X only maps to discrete points, not intervals. So based on this definition, X wouldn't be a random variable. But I know it is.

Any suggestions on where I may be interpreting things wrong here?

 

[disregardthat]

X^(-1)(B) would in this case be the empty set, which is contained in any sigma-algebra.