Are $L_\infty$ functions measurable/integrable?

Question

Lemma 2.6 of of Einsiedler & Ward's Ergodic Theory with a view towards Number Theory (p.15) involves:

$$ \int f d\mu, $$

where $f \in \mathcal{L}^{\infty}$.

The measure $\mu$ could be the Lebesgue measure and $f$ could be a non-integrable but limited function on $[0,1]$ (like the classic example for not Lebesgue integrable, the indicator function of representatives of equivalence classes of reals over rationals). In this case the given integral is meaningless. Where am I wrong?

Answer 1

In the paragraph above Lemma 2.6, the authors explicitly state "[we write] $\mathcal L^\infty$ for the space of measurable bounded functions"