Sometimes it seems to me, that when we are considering independent variables on a sample space of the form $\Omega^n$, we use the assumed independence of variables to show that $P'(\omega_1\ldots,\omega_n)=P(\omega_1)\cdot \ldots \dots P(\omega_n)$, where $P'$ is a yet unknown measure on $\Omega^n$.
Whereas I would find it natural (supposing we know of a probability distribution $P$ on $\Omega$) to instead define a distribution on $\Omega^n$, derived from $P$ (as the multiplication of individual probabilities), and then show that our $X_i$'s are indeed independent.
So why is the first way of doing it more commonly used and hot do I show using the independence that $P'(\omega_1\ldots,\omega_n)=P(\omega_1)\cdot \ldots \dots P(\omega_n)$ ? (I managed to do it the other way around and show independence using the above product measure)
To end it with Russell: "My mathematical lecturers never showed me any reason to suppose probability theory was anything but a tissue of fallacies".