Let $(X, \mathcal{B}, \mu, T)$ be a measure-preserving system, with $T$ invertible, where the $\sigma$-algebra $\mathcal{B}$ is a Borel algebra arising from a topology which makes $T$ continuous, and such that $X$ is compact Hausdorff. Then there's a well-defined action of the semigroup $\beta \mathbb{Z}$ on $X$ :

$ \forall p \in \beta \mathbb{Z}, \forall x \in X,\quad T^p x := p\lim\limits_{n \in \mathbb{Z}} T^n x$

If $p$ is a principal ultrafilter, it is clear that $T^p$ is measure-preserving. Is it still true
for *any* ultrafilter on $\mathbb{Z}$ ?

The measure preserving property of a $T^p$ would be a consequence of the corresponding dominated convergence theorem along the ultrafilter $p$ ; as far as I can see, this is not a direct consequence of the (usual) dominated convergence theorem, and I didn't succeed in proving or disproving it. Any ideas ?