A quantity $\int |E\mathscr{B} f| dP$ (or equivalently $\int|u - P(A: \mathscr{B})| dP, 0 < u < 1)$ associated with a $\sigma$-algebra $\mathscr{B}$ is shown to act as a criterion for a type of convergence of $\sigma$-algebras. This quantity also defines an ordering of $\sigma$-algebras, so that upper and lower limits can be defined in terms of this quantity. Another criterion for the convergence of $\sigma$-algebras is described based on the existence of these limits.