The measurable function $\xi$ is with value of $\mathbb{R}$?

Question

I have a problem I don't understand, in the following lemma:

In the proof of the lemma, the function $\xi$ is with value in $\overline{\mathbb{R}}$. But in the statement of the lemma, the measurable function $\xi$ is with value of $\mathbb{R}$, right?