The following problem arose while trying to justify some "known results" in abstract harmonic analysis on noncommutative groups, for which I couldn't find explicit statements in the literature. It is much more general than what I require, but I would like to know the "correct level of generality" for what I have been able to prove.

To establish terminology: if $\Omega$ is a set and $\Sigma$ a $\sigma$-algebra on it, we say that $(\Omega,\Sigma)$ is *standard* if there is a Polish topology on $\Omega$ that generates $\Sigma$.

Let $(\Omega_1,\tau_1)$ and $(\Omega_2,\tau_2)$ be topological spaces, and let $\Sigma_1$ and $\Sigma_2$ be the $\sigma$-algebras generated by $\tau_1$ and $\tau_2$. Equip $\Omega_1\times\Omega_2$ with the product topology, which I denote by $\tau_{12}$, and let $\Sigma_{12}$ be the $\sigma$-algebra generated by $\tau_{12}$.

It is easy to check that the product $\sigma$-algebra $\Sigma_1\boxtimes\Sigma_2$ is coarser than (i.e. contained in) $\Sigma_{12}$, and in general this can be strict, i.e. the two $\sigma$-algebras might be different.

**Question.** Suppose $(\Omega_1,\Sigma_1)$ and $(\Omega_2,\Sigma_2)$ are both standard. (This implies, since the product of Polish spaces is Polish, that $(\Omega,\Sigma_1\boxtimes\Sigma_2)$ is standard.) Does it follow that $\Sigma_1\boxtimes\Sigma_2=\Sigma_{12}$?

**Remarks.**

The place where I am having difficulties is that the topologies $\tau_1$ and $\tau_2$ need not be Polish (indeed, need not be Hausdorff), even though the $\sigma$-algebras they generate are standard. (For the intended applications one can relate $\tau_1$ and $\tau_2$ to certain Polish topologies, but this requires "opening up a black box" and I am hoping to avoid this.)

I can prove the desired result if I add the extra assumption that $(\Omega_1\times\Omega_2,\Sigma_{12})$ is standard (and this assumption holds in the intended applications). For then the identity map $(\Omega_1\times\Omega_2,\Sigma_{12})\to (\Omega_1\times\Omega_2,\Sigma_1\boxtimes\Sigma_2)$ is a measurable bijection between standard spaces, hence has measurable inverse by (a consequence of) Souslin's theorem. But this feels like a sledgehammer, and also I suspect that my extra assumption is not necessary.