A useful "abstract nonsense" construction in ergodic theory takes a measure-preserving transformation
$T$ of a probability space $(X,\mathcal B,\mu)$ and extends it to an *invertible* measure-preserving transformation $\bar T$ of a probability space $(\bar X,\bar{\mathcal B},\bar\mu)$.

One description of this is in Omri Sarig's notes (section 1.6.4). In his construction he needs to make the assumption that $T(X)=X$, or the weaker assumption, $T(X)$ is measurable. My question is whether this is automatic for Lebesgue spaces.

Hence my precise question:

If $T$ is a measure-preserving transformation of $[0,1]$ (equipped with Lebesgue measure and the $\sigma$-algebra of Lebesgue measurable sets), is $T([0,1])$ necessarily measurable?

Note: $[0,1]\setminus T([0,1])$ does not contain any measurable sets of positive measure by the Poincaré recurrence theorem, so $T([0,1])$ is certainly of outer measure 1.