Suppose that $\mathcal{I}=(X,\sigma^{\mathcal{X}},\mu)$ is an information strucutre, which is a probability space, where
- $X=X^1\times X^2$ is the cartesian product of the individual finite sets of signals $X^1$ and $X^2$
- $X^1=(s^1=v_1+e_1,\sum_{i=1}^{2}k_i^1\times s^i)$, $X^2=(s^2=v_2+e_2, \sum_{i=1}^{2}k_i^2\times s^i)$ where $(v_i,e_i)$ are uncorrelated, $e_1$, $e_2$ iid and $v_i\sim N(v^i,\sigma^2_{v^i})$ and $\rho=cov(v_1,v_2)/(\sigma_{v^1}\times\sigma_{v^2})\neq 0$. Alo $k^i$ denote the weights s.t they sum to $1$
- $\sigma^\mathcal{X}$ denoes the sigma algebra of the space $X$ and
- $\mu$ the probability measure. If $x\in X$ is drawn according to $\mu$, then player $i$ is informed about $x^i\in X^i$ and in general we know that $\mu$ is defined as it follows: $$\mu: \sigma^\mathcal{X}\to [0,1]$$
$\mu$ is a system of beliefs that provides rationality to the players according to the probability measure that is the same for both of them.
My question is the following:
what is the meaninig of the proposition that if $x\in X$ is drawn according to $\mu$, then player $i$ is informed about $x^i\in X^i$? For example the first cordinate of the sets $X^i$ are the standard signals in the literature and the second coordinates like an estimation between the signal of player $i$ and $-i$ (namely player $1$ and $2$), what is the meaning of $\mu(x)$, where $x=(x^1,x^2)$ where $x^1\in X^1$ and $x^2\in X^2$?
