0,

Thus, every function x E N vanishes outside supp N, and for any set D ~ supp N of positive measure one can find a subset Do ~ D, also of positive measure, and a function Xo E N for which Do ~ supp xo. Given an ideal space X over 0, a unit in X is, by definition, a nonnegative function u E X such that supp u = supp X. One can show (ZABR.EJ KO [1974]) that units exist in every ideal space. An element x E X is said to have an absolutely continuous norm if

where PD is the multiplication operator (3.19). The set XO of all functions with an absolutely continuous norm in X is a closed (ideal) subspace of X j we call XO the regular part of X. The subspace XO has many remarkable properties. In particular, convergent sequences in XO admit an easy characterization: a sequence (xn)n in XO converges to x E X (actually, x E XC) if and only if (xn)n converges to x in measure and

lim sup IIPDxn llx = 0, p(D)-O n

i.e. the elements X n have uniformly absolutely continuous norms. Further, the space XO is separable if and only if the underlying space (5 with metric (4.1) is separable. Unfortunately, the subspace XO can be much smaller than the whole space X. An ideal space X is called regular if X = XO, and quasiregular if supp X :: supp XCi the latter condition means that XO is dense in X with respect to convergence in measure.