A couple of remarks on the convergence of $\sigma$-fields on probability   spaces
  The following modes of convergence of sub-$\sigma$-fields on a given probability space have been studied in the literature: weak convergence, strong convergence, convergence with respect to the Hausdorff metric, almost-sure convergence, set-theoretic convergence, monotone convergence. It is noted that all preserve independence, and all are invariant under passage to an equivalent probability measure. Partial results for the case of operator-norm convergence obtain. 
Matija Vidmar
math.PR