Trouble Understanding the Integral Underlying Random Utility Models
This has nothing to do with any specific model. For any event $A$, let $I_A$ be the indicator function such that $I_A(\omega)=1$ if $\omega\in A$ and $I_A(\omega)=0$ if $\omega\notin A.$
Then $\mathbb{P}(A)=\mathbb{E}[I_A],$
and here the expectation is given in terms of a density function.