Let $(\Omega,\mathcal A, P)$ be a probability space and let $\Sigma(\mathcal A) \subset 2^{\mathcal A}$ be the collection of all sub-$\sigma$-fields of $\mathcal A$. Then, $\Sigma(\mathcal A)$ is closed under arbitrary intersections (including the empty intersection which by convention is the ground set $\mathcal A$).
That is, $\Sigma(\mathcal A)$ is a so-called closure system, hence by a well-known result is a complete lattice, when ordered by inclusion. See e.g. J.B. Nation's notes on Lattice Theory, Theorem 2.5. The closure operator associated with this closure system is $\sigma(\cdot)$ mapping any collection of subsets of $\mathcal A$ to the smallest $\sigma$-field containing it. The meet and join of $\Sigma(\mathcal A)$ can be expressed as $$ \bigwedge_{t \in T} \mathcal F_t = \bigcap_{t \in T} \mathcal F_t, \quad \bigvee_{t \in T} \mathcal F_t = \sigma \Big(\bigcup_{t \in T} \mathcal F_t \Big), $$ for any collection $\{\mathcal F_t\}_{t \in T}$ of elements of $\Sigma(\mathcal A)$.
Now, consider a collection of random variables, $X_t, t \in T$, on the above probability space and for any $S \subset T$ define $$ \mathcal F_S := \sigma ( X_t, t \in S) := \sigma\Big( \bigcup_{t \in S} X^{-1}(\mathcal B)\Big) $$ where $\mathcal B$ is the Borel $\sigma$-field of the real line. In other words, $\mathcal F_S$ is the smallest $\sigma$-field with respect to which all the random variables $X_t, t \in S$ are measurable. We now consider the collection $$ \mathfrak F = \{ \mathcal F_S :\; S \subset T\}. $$ It is not hard to see that $\mathfrak F$ is a join semi-lattice. In fact, $\mathcal F_S \vee \mathcal F_{S'} = \mathcal F_{S \cup S'}$ for any $S, S' \subset T$. The dual relation, however, does not seem to hold in general: $$ \mathcal F_S \wedge \mathcal F_{S'} \stackrel{?}{=} \mathcal F_{S \cap S'}. $$
Questions:
Are there (simpler) conditions on the collection of random variables that make the above meet identity hold? A consequence would that the map $S \mapsto \mathcal F_S$ will be a lattice embedding of $2^T$ inside $\Sigma(\mathcal A)$.
More generally, are there (simpler) conditions on the collection of random variables under consideration that make $\mathfrak F$ a closure system? A consequence would be that $\mathfrak F$ is a complete sublattice of $\Sigma(\mathcal A)$.
