  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.
