Neither notation is strictly formal from strictly set-theoretical point of view. An infinite cartesian product is a set of functions from the index set into the union of the producted sets, while a finite cartesian product is usually seen as a set of tuples (which can be seen as ordered pairs of ordered pairs... etc, or just ordered pairs in case of a product of two sets), so if you go down to set-theoretical formalism, they're distinct objects (that is, $A\times \prod_1^\infty X$ is a set of pairs of elements of $X$ and sequences of elements of $X$).
The most formal ay to write it would be just the thing you've written using set-builder notation: $\lbrace \omega\in\Omega\vert \omega_0\in A\rbrace$.
That said, unless you're explicitly dealing with very low-level objects, looking too closely at set-theoretical formalism is neither useful nor enlightening; I think the other two are good enough, if you make it clear what you mean. You might just spell it out in words, or just write something to highlight the abuse like $A\times \Omega\subseteq \Omega$.
Also, an useful notation for cartesian (countable) power is $X^{\mathbf N}$ or $X^{\aleph_0}$ (or even $X^\omega$, but that would be a little too confusing in that context).