$\textit{Definition:}$ $I$ is a finite set of players and and $G=((S^i)_i,g)$ is a compact game, that is given by a compact set of strategies $S^i$ for each player $i$ and by a continuous payoff function $g:S=\times S^i \to \mathbb{R}^{I}$. Also the set of mixed strategies is defined as $\Sigma^i=\Delta(S^i)$ which is a standard way in game theory and $g$ is extended to $\Sigma$ by $g(\sigma)=\mathbb{E}_{\sigma}g(s)$.
$\textit{Question 1:}$ Why do we need the notion of compact game (set) from topology? Can anybody give the intuition and/or an example?
$\textit{Question 2:}$ From the best of my knowledge, the index $\sigma$ in the operator of the expected value, i.e. $\mathbb{E}_{\sigma}$ denotes the probability measure of the environment that we work. In this case $\sigma$ stands for the mixed strategy which is a probability distribution over the set of pure strategies (if I am not mistaken). Does this mean that $\sigma$ coincides with the probability measure?
I updated my question. Thank you in advance!
While it's hard to be certain about this assumption without seeing the full context, compactness is often assumed in order to guarantee existence of a solution to a problem. I'm assuming that that optimality of $g$ over $(S^i)_{i\in I}$ describes a solution of your game? If so, then the game is guaranteed to possess a solution, since continuous functions always achieve their extrema on compact sets.
If the assumption of compactness is dropped, we must use much more technical arguments to verify that a solution exists.