Definiton of information sets in rational expecations models
Two notes.
A. &quot;Conditioning on information&quot; has always been applied in economics without much attention to probability theory-rigor, because it has (indeed) such a strong intuitive sense: &quot;based on the information I have (where &quot;information&quot; here means data, processing algorithms, psychological makeup, almost anything) I somehow form through a black-box process an expectation for the value of some variable&quot;.
B. Conditioning is always done with respect to a sigma-algebra. But in economics it is customary to write just the generator of the sigma-algebra, in our case $I_{t−1,i}$ and expect to be understood as $\sigma\left(I_{t−1,i}\right)$. So all you have to do is imagine $I_{t−1,i}$ as a set that can generate a sigma algebra. In that case, each element of the set can be anything really.
Note that the $\Omega$ in the probability space and the sigma-algebras that can come from it can literally be anything. We restrict $v$ to be a random variable, namely a function whose range is some numeric set, like the Reals or the Naturals, but the domain of $v$ can be, again, literally anything, and as multi-dimensional and as non-numeric as we like.