  Given positive linear functional l on a vector lattice L of real functions,
and a vector subspace M of L, we construct a vector subspace P(M) of M in such
a way that 1) l is nullcontinuous on P(M), and 2) if l is nullcontinuous on M
then P(M) is all of M. We mention here that this result continues to hold for
quite general modes of convergence, including tau-continuity. Our construction
uses a new method involving the "kernel" of a seminorm.
