It is well known that if $T$ is a nonsingular transformation of a standard probability space $(X,\mu)$ and there exists an ergodic measure preserving transformation $S$ of $(X,\mu)$ such that $T$ commutes with $S$, then $T$ is measure preserving.

I would like to know of there is an approximate version of this fact. More explicitly, let $T$ be a nonsingular transformation of $(X,\mu)$ and suppose that for some small $\epsilon$ there exists an ergodic nonsingular tranformation $S$ which commutes with $T$ and such that there is a set $A \subseteq X$ with $\mu(A) > 1-\epsilon$ and $\frac{\mathrm{d}S \mu}{\mathrm{d}\mu}(x) = 1$ for all $x \in A$. Does this imply that $T$ is close to being measure preserving in some sense, for example do we have that the quantity $||\frac{\mathrm{d}T \mu}{\mathrm{d}\mu} - 1||_1$ or the quantity $- \int_X \log \frac{\mathrm{d}T \mu}{\mathrm{d}\mu}(x) \hspace{2 pt} \mathrm{d}\mu(x)$ is small?

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