Positivity 13 (2009), 61–87
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010061-27, published online April 5, 2008
Vector measures: where are their integrals?
Guillermo P. Curbera
, Olvido Delgado
and Werner J. Ricker
Abstract. Let ν be a vector measure with values in a Banach space Z.The
integration map I
(ν) → Z,givenbyf →
fdν for f ∈ L
has a formal extension to its bidual operator I
. So, we may
consider the “integral” of any element f
). Our aim
is to identify when these integrals lie in more tractable subspaces Y of Z
For Z a Banach function space X, we consider this question when Y is any
one of the subspaces of X
given by the corresponding identiﬁcations of X,
(the K¨othe bidual of X)andX
(the topological dual of the K¨othe dual
of X). Also, we consider certain kernel operators T and study the extended
for the particular vector measure ν deﬁned by ν(A):=T (χ
Mathematics Subject Classiﬁcation (2000). Primary 46G10, 47B38; Secondary
Keywords. Banach lattices and function spaces, vector measure, integration
The general theory of vector measures and integration with respect to them is well
established; see [1,19,20,29], for example. In recent years it has become evident
that many classical operators from various branches of analysis can be viewed as
integration operators with respect to suitable vector measures; see [7–10, 14,15,24,
27] and the references therein, for example. Accordingly, such integration operators
are becoming objects of ever ﬁner investigations.
Recall, for a vector measure ν deﬁned on a measurable space (Ω, Σ) and with
values in a Banach space Z, that a measurable function f :Ω→ R is scalarly
Support of the D.G.I. #MTM2006-13000-C03-01 (Spain) is gratefully acknowledged.
Support of the Generalitat Valenciana (TSGD-07) and the Ministerio de Educaci´on y Ciencia
(TSGD-08) (Spain), is gratefully acknowledged.
Support of the German Research Council (DFG) is gratefully acknowledged.