Definitions In our course we defined a discrete probability space as a tuple $\left(\Omega,P\right)$, where $P:\mathcal{P}(\Omega)\rightarrow\left[0,1\right]$ and $\Omega$ is at most countable, such that $P\left(\Omega\right)=1$ and for a finite our countable infinite set $I$ we have $P\left(\bigcup_{i\in I}A_{i}\right)=\sum_{i\in I}P\left(A_{i}\right)$ where all $A_{i}$ are pairwise disjoint.
Motivation Now we did several exercise that implicitly involved product spaces. For all these exercises $\Omega$ was finite.
Problem To avoid having to construct for each example separately a product space, I wanted to do it once and for all in the abstract, by proving the following theorem: Let $\left(\Omega_{1},P_{1}\right),\ldots,\left(\Omega_{n},P_{n}\right)$ be discrete probability spaces. Then $\left(\Omega_{1}\times\ldots\times\Omega_{n},P\right)$ where $P:\Omega_{1}\times\ldots\times\Omega_{n}\rightarrow\left[0,1\right],\ P\left(\omega_{1}\ldots\omega_{n}\right)=P_{1}\left(\omega_{1}\right)\cdot\ldots\cdot P_{n}\left(\omega_{n}\right)$ is a discrete probability space as well.
But in trying to prove the second property ($\sigma$-addivity) of discrete probability spaces, I ran into problems: How do I prove, that $ P\left(\bigcup_{i\in I}A_{i}\right)=P_{1}\left(\text{pr}_{1}\left[\bigcup_{i\in I}A_{i}\right]\right)\cdot\ldots\cdot P_{n}\left(\text{pr}_{n}\left[\bigcup_{i\in I}A_{i}\right]\right)=$ $=\sum_{i\in I}P_{1}\left(\text{pr}_{1}\left[A_{i}\right]\right)\cdot\ldots\cdot\sum_{i\in I}P_{n}\left(\text{pr}_{n}\left[A_{i}\right]\right)\quad\left(\star\right) $
equals $\sum_{i\in I}\left(P_{1}\left(\text{pr}_{1}\left[A_{i}\right]\right)\cdot\ldots\cdot P_{n}\left(\text{pr}_{n}\left[A_{i}\right]\right)\right)=\sum_{i\in I}P\left(A_{i}\right)$ ? ($\text{pr}_{j}\left[\cdot\right]$ denotes the $j$-th projection mapping)
If the $\Omega_i$'s are finite I can prove my theorem, since together with the disjointness of the $A_{j}$'s, this implies that $I$ has to be finite, so everything is fine. But for countable infinite $\Omega_i$'s, the sum from $\left(\star\right)$ can only be evaluated using the Cauchy product for series, and this number wouldn't be the same as $\sum_{i\in I}\left(P_{1}\left(\text{pr}_{1}\left[A_{i}\right]\right)\cdot\ldots\cdot P_{n}\left(\text{pr}_{n}\left[A_{i}\right]\right)\right)$, I think.
Can product spaces maybe be defined in a meaningful way only under the conditions from above ?