On the Continuity of the Support of Bimodules over Maximal Abelian Self-Adjoint Algebras

We relate the convergence of a net of maximal Abelian selfadjoint algebras (masas) to that of the net of their corresponding supports. This is achieved by using a family of capacities on the collection of subsets of $X\times Y$ (where the masas are realized as collections of operators of multiplication by essentially bounded functions on the measure spaces $X$ and $Y$), which extends a capacity studied previously by Haydon and Shulman.