A couple of remarks on the convergence of σ-fields on probability spaces

The following modes of convergence of sub-σ-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, and monotone convergence. It is noted that all preserve independence in the limit, and all are invariant under passage to an equivalent probability measure. Partial results for the case of operator-norm convergence obtain.