Positivity 13 (2009), 129–143
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010129-15, published online September 1, 2008
Ordered representations of spaces
of integrable functions
Antonio Fern´andez, Fernando Mayoral,
Francisco Naranjo and Enrique A. S´anchez–P´erez
Abstract. Let X be a Banach space, (Ω, Σ) a measurable space and let m :
Σ → X be a (countably additive) vector measure. Consider the correspon-
ding space of integrable functions L
(m). In this paper we analyze the set
of (countably additive) vector measures n satisfying that L
In order to do this we deﬁne a (quasi) order relation on this set to obtain
under adequate requirements the simplest representation of the space L
associated to downward directed subsets of the set of all the representations.
Mathematics Subject Classiﬁcation (2000). Primary 46E30, 46G10; Secondary
Keywords. Banach function space, Vector measure, Representation.
It is well-known that every (real) Banach lattice with order continuous norm and a
weak order unit can be represented as a space L
(m) of (real) integrable functions
with respect to a vector measure (see below for the deﬁnitions and [2,Theorem8]
or [11, Proposition 3.9]). In fact, the vector measure can be chosen to be positive
(that is, a vector measure taking values on the positive cone of a Banach lattice);
in the case that the Banach lattice is a Banach function space X(μ) over a ﬁnite
measure space (Ω, Σ,μ), a vector measure that provides such a representation is
given by m :Σ→ X(μ), where m(A):=χ
∈ X(μ), A ∈ Σ. However, this
representation is not unique, in the sense that the spaces of integrable functions
deﬁned by diﬀerent vector measures can coincide, even if the vector measures
take their values on diﬀerent Banach spaces. A rather clarifying example of this
situation is given by the following measures.
This research has been partially supported by La Junta de Andaluc´ıa. The support of D.G.I.
under project MTM2006–11690–C02 (M.E.C. Spain) and FEDER is gratefully acknowledged.