The average age of the population is not a random variable, or at least not an interesting one, even if you make it to be one.
The average age of the fixed size sample, on the other hand, is. And should you be able to construct a probabilistic space for it as follows:
Take $\Omega$ to be the set of subsets of the population of size $N$, with sigma algebra $2^\Omega$. Assign equal mass to all the points in $\Omega$.
The sample average is then a random variable $X: \Omega \to \mathbb{R}$, with $$X(\omega) = \frac{1}{N} \sum_{\text{person} \in \omega} \text{person.age}$$