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?



Nonsingular transformation commuting with approximately measure preserving transformation
No. I don't think there can be any result of that type. Let $X=\{0,1\}^{\mathbb N}$ and let $S$ be the dyadic odometer (I like to write elements of $X$ so they are infinite on the left; in this case, the map $S$ is just add 1 with carry to the left -- the way it should be!). Equip $X$ with the measure that is the product measure of $(\frac 12,\frac 12)$ on each coordinate except the $N$th; and $(\epsilon,1-\epsilon)$ on the $N$th coordinate. Let $T$ be $S^{2^N}$. Then $S$ is measure preserving everywhere except on a set of measure $2^{-N}$: the cylinder set $[111111111]$, while $T$ is very far from measure-preserving everywhere.


