It is known that an invertible *mpt* $S$ is weakly mixing if and only if $S \times T$ is ergodic for any ergodic invertible *mpt* $T$. Is it more generally true that the invariant $\sigma$-field of $S \times T$ is the product of the trivial $\sigma$-field times the invariant $\sigma$-field of $T$ whenever $S$ is weakly mixing ? 



Invariant $\sigma$-field of a product with a weakly mixing transformation
The answer is yes (I am assuming here the spaces are standard and the measures are probability measures).


For this kind of questions I find that it is convenient to think about quotient spaces rather then sub-$\sigma$-algebras. In particular, instead of considering the $\sigma$-algebra of invariants you better consider the space of ergodic components.


Recall that given a probability measure preserving transformation $T$ on $(Y,\nu)$ there exists a space $Z$ and a measurable (a.e defined) map $\pi:Y\to Z$ such that the $\sigma$ algebra of $T$-invariants equals the pull back $\sigma$-algebra under $\pi$. Alternatively, $\pi:Y\to Z$ is characterized by:


* $\pi$ is $T$-invariant.
* presenting the disintegration of $\nu$ wrt $\pi$ as $\nu=\int\_Z \nu\_z d\pi\_\*\nu(z)$, for $\pi\_\*\nu$-a.e $z\in Z$, $\nu\_z$ is ergodic.


The space $(Z,\pi\_\*\nu)$ is uniquely defined by these properties (up to a natural isomorphism) and is called *the space of ergodic components of $T$*.


Its existence is a simple application of Choquet theory. One can also define it as the spectrum of the algebra of $T$-invariants in $L^\infty(Y)$.


Let us go back to the asked question now.
Assume $S$ is a transformation of $(X,\mu)$ and $T$ of $(Y,\nu)$, both are probability measure preserving. Assume the $S$ is weakly mixing. Let $Z$ be the space of ergodic components of $T$ and $\pi:Y\to Z$ the corresponding map. 


Then $\mu\times \nu=\int\_Z \mu\times\nu\_z d\pi\_\*\nu(z)$. Clearly, the map $\pi\circ p\_2:X\times Y\to Z$ is $S\times T$-invariant (where $p\_2$ is the projection $X\times Y\to Y$) and by the weak mixing assumption, for $\pi\_\*\nu$-a.e $z\in Z$, $\mu\times\nu\_z$ is $S\times T$-ergodic. Thus $Z$ is the space of ergodic components of $S\times T$.
It follows that the $\sigma$-algebra of $S\times T$ invariant sets in $X\times Y$ is the pull back $\sigma$-algebra under $\pi\circ p\_2$. This clearly provides a positive answer to the question.


