Definiton of information sets in rational expecations models

I am struggling with the notion of ‘information sets’ in the context of rational expectation models in economics. I found interesting notes on the web (http://www2.econ.iastate.edu/tesfatsi/reintro.pdf) but I am not sure whether I am understanding the concept well. Let me explain my concerns along with the first example in the notes given in the link.

Consider a small model given in the following three equations:
$$ y_t =y_t^*+ap_{t-1}+bmathbb{E}_{t-1}p_t
p_t =m_t+varepsilon_t
mathbb{E}_{t}p_{t+1}=mathbb{E}(p_{t+1}vert I_t)$$

where

$y_t^*$ notes the log of potential real GDP in period $t$

$y_t$ denotes the log of actual real GDP in period $t$

$mathbb{E}_{t}p_{t+1}$ denotes the subjective forward-looking expectation of a representative agent in period $t$ regarding the price level in period $t+1$

$m_t$ denotes the log of the nominal money supply in period $t$

$varepsilon_t$ is a stochastic shock at time $t$

$I_t$ denotes a period-t information set that is available to the representative agent at the end of the period $t$.

So my possibly stupid question is: What is $I_t$ or how is it defined?

To be more precise, let me outline what I think $I_t$ is.

First of all, because most economists apply the law of iterated expectations and other propositions that can be applied to conditional expectations, I suggest $I_t$ has to be a $sigma$-Field because otherwise, one wouldn’t be able to apply these propositions.

But how is this $sigma$-Field defined?

Following the notes, Leigh Tesfatsion writes that the equations plus classification of variables and admissibility conditions together with the true variable values a,b and the deterministic exogenous process $(m_t)_{t in mathbb{N}}$ have to be part of the information set, as well as the properties of the probability distribution and properties of the stochastic shocks $(varepsilon_t)_{tin mathbb{N}}$ and the values of past realizations of all variables.

Typically it is assumed that $varepsilon=(varepsilon_t)_{tin mathbb{N}}$ is a stochastic process defined on the probability space $(Omega,mathcal{F},mathbb{P})$.
Thus I would say $I_{t}$ has to be a $sigma$-Field over $Omega$ and thus it has to be a system of subsets of $Omega$, thus it cannot include specific equations, specific variable values nor variable classifications, or am I wrong?

Let $mathbb{F}=(mathcal{F}_t)_{tin mathbb{N}}$ be a filtration on $(Omega,mathcal{F},mathbb{P})$, given as $mathcal{F}_t=sigma({varepsilon_s:sleq t})$.

I thought $I_{t}$ to be the history of the stochastic process, i.e. $I_{t}=mathcal{F}_{t}$, is this correct?

If not, could you provide me a (mathematical rigorous) definition of the information set $I_{t}$ or could you provide me some literature related to this issue?

Thanks in advance

Frank