Einsiedler & Ward's book Ergodic Theory with a view towards Number Theory (GTM259) and am stuck at one of the exercises:
Exercise 6.1.1 (p.156): Let $T: X \to X $ be a homeomorphism on a compact metric space $X$, and let $\mu$ be a $T$-invariant Borel probability measure on $X$. Show that for $\mu$-almost every $x\in X$, the forward and backward orbits of $x$ have the same closures and both contain $x$. In other words:
$$x\in \overline {\{T^{n}x : n \ge 1 \}} = \overline {\{T^{-n}x : n \ge 1 \}}, \text{ a.e. } x\in X$$
I can show the case if $T$ is ergodic with respect to $\mu$ (in this case for a.e $x$, the above closures are the whole space $X$), and I am trying to use this fact together with ergodic decomposition to show the general case (but have little progress).
Any hint would be highly appreciated. Please give me just a hint.