The paper of Bergemann and Morris proves a theorem based on some foundations about the information sets and their expansions. I am trying to understand theorem one intuition, more precisely I cite the appropriate parts that we need to know, that is:
$\textbf{Definition 4:}$ (Combination). The information structure $S^∗ = (T^∗, \pi^∗:\Theta\to\Delta(T_i^*))$ is a combination of information structures $S^1 = (T^1, \pi^1)$ and $S^2 = (T^2, \pi^2)$ if
$$T_i^*=T_i^1\times T_i^2,\quad\text{for each $i$}$$ and $$\sum_{t_i^2\in T_i^2}\pi^*(t_i^1,t_i^2|\theta)=\pi(t_i^1|\theta),\quad\text{for each $t_i^1\in T_i^1$ and $\theta\in\Theta$} $$
$$\sum_{t_i^1\in T_i^1}\pi^*(t_i^1,t_i^2|\theta)=\pi(t_i^2|\theta),\quad\text{for each $t_i^2\in T_i^2$ and $\theta\in\Theta$} $$
Note that the above definition places no restrictions on whether signals $t_i^1\in T_i^1$ and $t_i^2\in T_i^2$ are independent or correlated, conditional on $\theta$, under $\pi^∗$. Thus any pair of information structures $S^1$ and $S^2$ will have many combined information structures.
$\textbf{Definition 5:}$ (Expansion). An information structure $S^∗$ is an expansion of $S^1$ if $S^∗$ is a combination of $S^1$ and some other information structure $S^2$.
Suppose strategy profile $\beta$ was played in $(G, S^∗)$, where $S^∗$ is a combination of two information structures $S^1$ and $S^2$. Now, if the analyst did not observe the signals of the combined information structure $S^∗$, but only the signals of $S^1$, then the behavior under the strategy profile $\beta$ would induce a decision rule for $(G, S^1)$. Formally, the strategy profile $\beta$ for $(G, S^∗)$ induces the decision rule σ for $(G, S^1)$:
\begin{equation}\sigma(a|t_i^1,\theta):=\frac{\sum_{t_i^2\in T_i^2}\pi^*(t_i^1,t_i^2|\theta)\Pi_{j=1}^i\beta_{j}(a_j|t_i^1,t_i^2)}{\pi(t_i^1|\theta)}\end{equation} for each $a\in A$ whenever $\pi^1(t_i^1|\theta) > 0$.
Based on the above the authors give the following therem that makes a connection between the Bayesian Nash equilibrium and the Bayesian Correlated equilibrium, that is
$\textbf{Theorem 1:}$ A decision rule $\sigma$ is a Bayes correlated equilibrium of $(G, S)$ if and only if, for some expansion $S^∗$ of $S$, there is a Bayes Nash equilibrium of $(G, S^∗)$ that induces $\sigma$.
I have the following questions
$\textbf{Question 1:}$ They define the the rule $\sigma(a|t_i^1,\theta)$ in a way to make the connection between the sets of BNE and BCE and I can not understand why they did so?
$\textbf{Question 2:}$ What is the interpretation of $\sigma(a|t_i^1,\theta)$, in terms of porbability someone can say it is the Bayesian rule, however all these parameters as $\pi^*$, $\beta$ and $\pi(t_i^1|\theta)$ have some intuitive interpretation what is it?
$\textbf{Question 3:}$ Why it is so special this obedient rule $\sigma(a|t_i^1,\theta)$ for both the BNE and BCE and does it suffice to make the transition from the one set of solutuions to the other? I am a little confused because it seems like the two sets are somehow the one the subset of the other.
$\textbf{Question 4:}$ Are BNE and BCE directly or indirectly connected? If not, what is the purpose to do this here?