So if I'm correct, given probability spaces $(\Omega_i,F_i,P_i)_{i\leq n}$ the following is a probability space: $(\Omega_1\times\ldots\Omega_n,F_1\times\ldots F_n,P_1P_2\ldots P_n):=(\Omega, F, P)$ (where union and complement in $F_1\times\ldots F_n$ is done componentwise so it is still a $\sigma$ algebra).
So I've shown that there is a measurable function on each of these probability spaces (in fact, the measurable function is the exact same) and I want to kind of stitch these together to show that I can make a random variable from the larger probability space. In particular, I'm trying to show that if I have a set of numbers at the end of a simulation (all stochastic), the variance is a random variable.
So far, assuming I'm not wildly misinterpreting what I'm reading, I've managed to do everything but justify that I'm allowed to stitch these together to get the variance formula.
My idea is as follows:
Define $X_i:(\Omega_i,F_i,P_i)\rightarrow \mathbb{R}$ such that $a_i\mapsto a_i$ which is a random variable (not included here). From the properties of measurable functions (not included here), we deduce that $\text{Var}:(\Omega, F, P)\rightarrow \mathbb{R}$ such that $(a_1,a_2,\ldots a_n)\mapsto\text{Var}((a_1,a_2,\ldots a_n))$ is a random variable as it is the sum, product and composition of measurable functions and $(\Omega, F, P)$ is a probability space.
Sorry if this isn't very clear, I'm not big into probability or measure theory. This is for a project I'm doing. If you need any clarification just let me know!
You're not allowed to just change the definition of union and complement to be componentwise. $$F_1\otimes F_2:=\{A\times B:A\in F_1,\,B\in F_2\}$$Is almost never a genuine $\sigma$-algebra. When we craft product measure spaces, we instead take the $\sigma$-algebra generated by these sets, $\sigma(F_1\otimes F_2)$.
Your question is a little confusing. It is true that $(X_1,\cdots,X_n):(\Omega_1\times\cdots\times\Omega_n,\sigma(F_1\otimes\cdots\otimes F_n))\to\Bbb R^n$ is a random variable, but also be warned that you need a little measure theoretic trickery to define $P_1P_2\cdots P_n$ on this space since $\sigma(F_1\otimes\cdots\otimes F_n)$ will contain sets not of the form $A_1\times\cdots\times A_n$, so $P_1(A_1)P_2(A_2)\cdots P_n(A_n)$ won't be applicable.
If you really do want the variance to be a random variable we can make do with the sample variance (really not the same thing as the population variance), since that is simply the random variable mapping $(\omega_1,\cdots,\omega_n)$ to: $$\frac{(X_1(\omega_1)-\mu_1)^2+\cdots+(X_n(\omega_n)-\mu_n)^2}{n-1}$$
Being a random variable $\Omega_1\times\cdots\Omega_n\to\Bbb R$.
Side note:
I've never seen "population" variance treated as a random variable. For this, you'd need to at least consider the domain to be a probability space of random variables... the variance of random variables based on $\Omega$ is not sensibly itself a random variable based on $\Omega$, unless $\Omega$ is some kind of really weird space of random variables of random variables (?? not sure if this can ever be possible).