# A question in information strucutres and probability measures - How are they connected?

Suppose that $$\mathcal{I}=(X,\sigma^{\mathcal{X}},\mu)$$ is an information strucutre, which is a probability space, where

1. $$X=X^1\times X^2$$ is the cartesian product of the individual finite sets of signals $$X^1$$ and $$X^2$$
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 $$1$$
3. $$\sigma^\mathcal{X}$$ denoes the sigma algebra of the space $$X$$ and
4. $$\mu$$ 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 $$x^2\in X^2$$?

• In a card game (bridge, poker) the dealer selects a card and gives it to you, you are then "informed" of the content of the card (i.e. you can see its value) while your opponent does not see it and is not "informed". Sep 27, 2021 at 17:29