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, and you be able tocan construct avarious probabilistic spacespaces for it as follows:. Here is one particularly simple example, and some thoughts on how can one modify it.
Simple example
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 $$X(\omega) = \frac{1}{N} \sum_{\text{person} \in \omega} \text{person.age}$$
Things to modify
You can modify $\Omega$ and probability measure to better fit reality. Sigma algebra is not important.
Taking $\Omega$ to be set of subsets of the population models the fact that you are sampling $N$ different people. If you change your mind and decide to drop the difference requirement and sample with replacement, you can then take $\Omega$ to be set of multisets (the $N$-fold Cartesian product).
Assigning equal mass to all the points in $\Omega$ models the fact that when you sample, you are equally likely to draw any $N$ individuals. This may be accurate, for example, if you have a full record of Californians and genuinely choose $N$ of them uniformly at random.
This may be not accurate for a different sampling process, and the probability measure is the place to encode that information. For example, if you will only survey adults, then any $\omega$ that contains minors gets $0$ weight, and all other $\omega$ get a slightly higher weight.