Almost Basically Scattered Vector Measures
✍ Scribed by L. Drewnowski
- Publisher
- John Wiley and Sons
- Year
- 1985
- Tongue
- English
- Weight
- 723 KB
- Volume
- 120
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
In this paper we introduce the notion of an almost basically scattered vector measure. This class of measures is strictly broader than the class of basically scattered vector measures defined and investigated by ILuroN, TURETT a n d U m [9] (see also [7] for some further results), but nevertheless it has many properties in common with its predecessor. Indeed, all the main results in [9] remain valid, modulo some slight changes, for this larger class of measures.
This new notion arosed from our attempt to find a "converse" to Theorem I in [9] which eventually led us to Theorem 9.a of the present paper, characterizing just almost basically scattered measures. Prior to this, in Example 6., we show that if 2 is any reflexive subspace of Li=L, [O, I] and Q : L,-L,/Z is the quotient map, then the vector measure G defined by G(A)=&(x4) for BOREL subsets of [O, 11 is almost basically scattered. Moreover, and this is perhaps somewhat surprising, we prove that the BANACH space L,(G) of all G-integrable functions coincides (and hence is also isomorphic) with L,. Some of our results indicate that the class of almost basically scattered measures is more natural than its predecessor. For instance, Corollary 9.c shows that the indefinite integrals with respect to an almost basically scattered measure P are almost basically scattered, and Remark 9.e says that the assertion cannot beimproved even if P is assumed to be bwically scattered.
We also investigate a domination relation < between two vector measures, and in Corollary 14.c we prove that if F is almost basically scattered, then G < F holds if and only if E,(#) 3 5?,(P).
Finally, we briefly discuss similar relations between a vector measure and a given sequence in a BANACR space. In Theorem 18. we show that no non-atomic almost basically scattered measure is dominated by the unit vector basis of co.
The reader is referred to [I] and [6] for the theory of vector measures, and to
[lo] and [ll] for bases and basic sequences in BANACH spaces.
- Throughout this paper, 2 is a o-field of subsets of a set S ; X, Y are BA-XACH spaces over the scalar field K = R or C; F : Z-X is a fixed (countably additive) vector measure. Let A CZ. Then IPI(A) is the total variation of P on A , and