Let $(\Omega,\mathcal{F},P)$ be a probability space. Is there a topology on the set of all sub sigma algebras of $\mathcal{F}$, denoted by $\mathcal{F}^*$, that makes the join operation continuous?
More specifically: Let $A_n \rightarrow A$, for some index sets $A_n,A$ and $X_f$ with $f\in A, A_n$ a collection of real valued random variables on $(\Omega,\mathcal{F},P)$.
Is there a topology on $\mathcal{F}^*$ that gives
$$
\lim_n \bigvee_{f\in A_n} \sigma(X_f)= \sigma(X_f: f\in A)
$$
?