Positivity 11 (2007), 461–467
2007 Birkh¨auser Verlag Basel/Switzerland
Compactness in Vector-valued Banach Function
Jan van Neerven
Abstract. We give a new proof of a recent characterization by Diaz and May-
oral of compactness in the Lebesgue-Bochner spaces L
,whereX is a Banach
space and 1 ≤ p<∞, and extend the result to vector-valued Banach function
,whereE is a Banach function space with order continuous norm.
Mathematics Subject Classiﬁcation (2000). Primary: 46E40, Secondary: 46E30;
46B50; 47D06; 60B05.
Keywords. Compactness, vector-valued Banach function spaces, order contin-
uous norm, almost order boundedness, uniform integrability.
Let X be a Banach space. The problem of describing the compact sets in the
Lebesgue-Bochner spaces L
,1≤ p<∞, goes back to the work of Riesz, Fr´echet,
Vitali in the scalar-valued case, cf. , and has been considered by many authors,
cf. [2, 4, 5, 11, 12]. In a recent paper, Diaz and Mayoral  proved that if the
underlying measure space is ﬁnite, then a subset K of L
is relatively compact
if and only if K is uniformly p-integrable, scalarly relatively compact, and either
uniformly tight or ﬂatly concentrated. Their proof relies on the Diestel-Ruess-
Schachermayer characterization  of weak compactness in L
and the notion of
Bocce oscillation, which was studied recently by Girardi  and Balder-Girardi-
Jalby  in the context of compactness in L
. The purpose of this note is to
present an extension of the Diaz-Mayoral result to vector-valued Banach function
, with a proof based on Prohorov’s tightness theorem.
We begin with some preliminaries on Banach lattices and Banach function
spaces. Our terminology is standard and follows .
A Banach lattice E is said to have order continuous norm if every net in E
which decreases to 0 converges to 0. Every separable Banach function space E
has this property. Indeed, because such spaces are Dedekind complete [9, Lemma
The author is supported by the ‘VIDI subsidie’ 639.032.201 in the ‘Vernieuwingsimpuls’ pro-
gramme of the Netherlands Organization for Scientiﬁc Research (NWO) and by the Research
Training Network HPRN-CT-2002-00281.