What allows us to use Fubini in this passage? A question concerning Liggett's 1985 book.

Question

In the book Interacting Particle system, chapter 1 pg 43 one reads

It is often the case that $\Omega$ is an unbounded operator. Considering that this is the case, what allows us to use Fubini's theorem in the passage

$$\Bbb{E}^{\eta_r}\bigg[\int_0^{t-r} \Omega f (\eta_s)\, ds\bigg] = \int_0^{t-r}\Bbb{E}^{\eta_r}\big[ \Omega f (\eta_s)\big]\, ds ?$$

Answer 1

It suffices to note that the operator $\Omega: C(X) \to C(X)$ acts on the space of continuous functions on $X$ a compact metric space. Since every continuous function on a compact set is bounded we obtain that $\sup_\eta \Omega f(\eta) = M_f < \infty$. This allows you to use Fubini