Positivity 5: 65–74, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
The Lyapunov Theorem for Measures Valued in
Orlicz Sequence Spaces
Raimundstrasse 10, 04177 Leipzig, Germany
(Received 15 September 1999; accepted 8 November 1999)
Abstract. It is proved that the closure of the range of a nonatomic countably additive measure with
values in an Orlicz space l
], is convex.
Mathematics Subject Classiﬁcation (2000): 46B20, 46G10
Key words: Banach space, vector measure, Orlicz sequence spaces, Lyapunov convexity theorem
The famous Lyapunov theorem ([5, 6], Theorem 5.5, or , p. 264) states that the
range of a nonatomic vector measure valued in a ﬁnite dimensional space is convex
and compact. In 1992 V.M. Kadets and G. Schechtman  discovered the following
inﬁnite-dimensional generalization: the closure of the range of every nonatomic
(p = 2, 1 p<∞)orc
-valued vector measure proved to be convex. The
aim of the present paper is to characterize Orlicz sequence spaces for which the
generalization of the Lyapunov theorem mentioned above is valid too.
Throughout the note by ‘X-valued measure’ we mean a σ-additive X-values
measure µ deﬁned on a σ-ﬁeld
of subsets of a set . A Banach space X is said
to be a Lyapunov space (X ∈ LPr) if the closure of every nonatomic X-valued
measure range is convex.
We shall show that all Orlicz spaces containing no isomorphic copies of l
Lyapunov spaces. We shall consider also vector-valued Orlicz sequence spaces.
First let us outline some results on Lyapunov spaces.
LEMMA 1 .LetX be a Banach space. The following statements are equivalent:
(*) X ∈ LPr.
(**) There exists a triple (,
is a nonatomic
measure, and a linear bounded map T : L
,λ)→ X such that
(a) T is σ(L
) − σ(X,X
(b) there exists ε>0 such that Tf ελ(suppf) for any ‘sign’ f ∈ L
i.e., any function taking only the values 0 and ±1.