The paper of Olivier Gossner in Security Protocols in 1998 has some definitions that confuse me too much. I will cite here these definitions and my questions and I hope someone is familiar with these notions.
$\textit{Question 1:}$ $I$ 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 mixed set of strategies is defined as $\Sigma^i=\Delta(S^i)$ which is a standard way in game theory, but why do we need the notion of compactness from topology?
$\textit{Question 2:}$ The information structure $\mathfrak{I}=((X^i),\mu)$ is given by a finite set of signals $X^i$ for each $i$ and by a probability measure $\mu$ over $X$. When $x$ is drawn according to$\mu$, player $i$ is informed about the coordinate $x^i$. Why do we need to define the information structure as a measure set which is finite and what does it mean that we now the probability measure?
$\textit{Question 3:}$ A communication mechanism is a triple $\mathfrak{C}=((T_i)_i, (Y_i)_i , l )$, where $T_i$ is $i's$ finite set of messages, $Y_i$ is $i's$ finite set of signals, and $l: T\to \Delta(Y)$ is the signal function. When $t$ is the profile of messages sent by the players, $y\in Y$ is drawn according to $l(t)$ and player $i$ is informed of $y_i$. $\mathfrak{T}_i=\Delta(T_i)$ represents the set of mixed messages for player $i$ and $l$ is extended to $\mathfrak{T}$ by $l(\tau)( y)=\mathbb{E}_{\tau} l(t)( y)$. I am totally lost in this point. What is this $\tau$ probability measure and what is the meaning of $l(t)(y)$, does this mean $l(t,y)$? I have never seen this symbolism $l(t)(y)$ again. Apparently, the way that the communication mechanism is defined comes from the measure theory, but how did he end up with the $l$ function defined under a $\tau$ probability measure?
I also struggle to understand the definitions $2.1$ to $2.5$ but I will stop here in order to find some help with the basic. Thank you in advance!