I frequently work with population studies, where, let's say, age and sex are collected for $N=1,000,000$ individuals in California. I might ask a simple question: what is the average age in this population?
How can I frame this simple question in terms of measure theoretic probability? I would like to define a probability space and then take the expectation of a random variable to get the answer. Then, the simple sample average of age would be an estimate of the expectation of this random variable.
It seems I need to define a "random experiment" which perhaps is drawing a random person from the population I am studying. Since, for my purposes, a random person is simply a tuple of (Age, Sex), then it seems my sample space $\Omega$ should be all tuples of the form $(x,y)$, where $x \in \mathbb R^+$ and $y \in \{\text{M}, \text{F}\}$. As for the $\sigma$-algebra, I can let $E$ be the product sigma algebra of the Borel sigma algebra on $\mathbb R^+$ and $2^{\{M,F\}}$. This approach encounters difficulty though in choosing the probability measure. I would have liked to use the counting measure normalized by $N$. That is, for example, if $A = (65.3243,\text{M}) \in E$,
$$\mathbb P ( A ) = \frac{\text{number of sampled males aged 65.3243 } }{N}.$$
But since I defined age as a real number ( as opposed to a discrete number of ages), this is actually probability 0, which makes no sense.
Another construction might be to model the actual collected data as the sample space. That is, $\Omega$ consists of $N$ elements, one corresponding to each of the data points. Then the $\sigma$-algebra would be $2^\Omega$ and the probability would be that same counting measure defined above.
In my head, the measure theoretic view of probability seems to fall apart whenever I start sampling things. How can I connect that theoretical view with the simple epidemiological questions I ask on a daily basis?
Edit: Having read this highly relevant link, I see another option. I could define a probability space $(\Omega, E, \mathbb P$), where $\Omega$ models the people in California (i.e. for each person in California there is an element in $\Omega$). $E$ is then then $2^\Omega$, and $\mathbb P$ is the counting measure normalized by $N$.
In order to model a random $N$ sized sample of this population, I should define a new probability space $(\Omega^N, E^N, \mathbb P^N)$, where $\Omega^N$ and $E^N$ are the Cartesian products of $\Omega$ and $E$ respectively. But then, I am still not sure how to define the probability measure.
