ON LEBESGUE PSEUDONORMS ON C0(T)

Let T be a locally compact Hausdorff topological space and let o(SBo) denote the CT-ring of all Baire measurable subsets of T. Denote by G> ( T) the Banach space of all continuous functions on T tending to zero at infinity with the usual supremum norm | | | | r. Let further Y be a Banach space and Y * its dual. (All considered Banach spaces are either real or complex.) Definition. We say that a mapping p: G>(T)—>[0, +°°) is a Lebesgue pseudonorm on C ( T) if it has the following properties: i) p(f)=P(\f\X 2) \f\Z\g\*p(f)*p(g), 3) P(af) = \a\p(f) f°r each scalar a, 4) p(f+g)^p(f) + p(g),and 5) ifg, f„eCo(T), n = 1,2,..., and £ l / » l = \o \ then p(f„)^0. n 1 There is a remarkable result, see [7, 24H], which is valid in the more general context of arbitrary Riesz spaces with a linear space topology such that ever