I'm trying to understand the first claim in the proof of the Birkhoff ergodic theorem.
The setup is:
$T$ is measure-preserving on a $\sigma$-finite measure space $(\Omega, \mathcal{F}, \mu)$,
$f: \Omega \to \mathbf{R}^*$ is a measurable function, and
$r<s$ are rational.
Why is the set of points $x$ such that $$\liminf \left[ \frac{1}{n} \sum_{i=0}^{n-1} f(T^i(x))\right] < r < s < \limsup \left[ \frac{1}{n} \sum_{i=0}^{n-1} f(T^i(x))\right]$$ measurable?