A system of events in some finite probability space $(\mathcal S, P)$ is:
$\mathcal E$ = $[ E_i ; i \in I ]$
$\mathcal F$ = $[ F_j ; j \in J ]$
The join of $\mathcal E$ and $\mathcal F$ is $\mathcal E$$\land$$\mathcal F$ = $[ E_i \cap F_j ; (i, j) \in I x J]$.
a.) Prove that both $\mathcal E$ and $\mathcal F$ are amalgamations of $\mathcal E$$\land$$\mathcal F$.
b.) Suppose $\mathcal G $ = $[G_i ; k \in K]$ is another system in $(\mathcal S, P)$.
Show that each of $\mathcal E$ and $\mathcal F$ is an amalgamation of $\mathcal G$.
Show that $\mathcal E$$\land$$\mathcal F$ is an amalgamation of $\mathcal G$.
Attempts: I know that $\mathcal E$ is an amalgamation of $\mathcal F$ if an only if for each $j \in J$ there is an $i \in I$ such that $P(E_i \cap F_j)$ = $P(F_j)$
For part b.) Supposing that E, F, and G are events in $\mathcal E, \mathcal F, and \mathcal G$ and $P(G \cap E \cap F)$ > 0. Then $P(G \cap E)$, $P(F \cap F)$ > 0. Assuming $\mathcal E$ and $\mathcal F$ are amalgamations of $\mathcal G$, G is essentially contained in E and in F.
This is question 2 here.