I'm trying to solve exercise 2.2.3 b) from Einsiedler and Ward's "Ergodic theory with a view towards number theory". The exercise considers an arbitrary metric space $X$ with a probability measure $\mu$ defined on its $\sigma$-algebra of Borel sets, and a Borel transformation $T$ of $X$ preserving $\mu$. In this setting we want to show, that the set of points $x\in X$ which are in the closure of their orbit $\{T^n(x): n\geqslant 1\}$ has measure 1.
So far I've reduced the problem to showing, that there are no metric probability preserving systems $(X, \mu, T)$ for which $\forall_{x\in X}d(x, T(x))\geqslant 1$. This can be done by considering a possible counterexample $(X, T)$, noticing that for some $\varepsilon>0$ a set of positive measure $A$ must satisfy $$\forall_{a\in A}d(a, T(a))\geqslant \varepsilon,$$ and then considering the metric space $A$ with the scaled metric $d'=\frac{d}{\varepsilon}$ and scaled measure $\mu'=\frac{\mu}{\mu(A)}$, and the transformation $T'$ defined as $$T'(a)=T^n(a)\quad\text{where $n$ is the smallest positive integer such that} \quad T^n(a)\in A.$$ This transformation (defined almost everywhere by Poincaré recurrence theorem) can be checked to preserve $\mu$.
We can also assume that $T$ is invertible by considering the natural extension of the supposed counterexample $(X, T)$ before applying this reasoning.
Using Poincaré recurrence theorem, we can also see that in this case we must have $\mu'(B(a, 1))=0$ for all points $a\in A$. This clearly gives a contradiction if the space $X$ (or $A$) is separable, but in the general case I don't see a way to proceed.
Since the question is posed as an exercise, I believe it should have some relatively easy solution. Any ideas or directions to literature (or at least assuring me that the statement of the exercise is actually correct) would be very appreciated.