Positivity 6: 147–190, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
Maharam extensions of positive operators and
W.A.J. LUXEMBURG and B. DE PAGTER
Department of Mathematics, California Institute of Technology, Pasadena, CA, USA
Department of Mathematics, Delft University of Technology, Delft, The Netherlands
(Received 25 July 2000; accepted 4 March 2001)
Abstract. The principal result of this paper is the construction of simultaneous extensions of col-
lections of positive linear operators between vector lattices to interval preserving operators (i.e.,
Maharam operators). This construction is based on some properties of so-called f -modules. The
properties and structure of these extension spaces is discussed in some detail.
2000 Mathematics Subject Classiﬁcation: 47B65, 46A40, 06F25.
One of the fundamental theorems in measure theory is the classical Radon-Nikodym
theorem: if µ and ν are two σ -ﬁnite σ -additive measures on a measurable space
(X, ) satisfying 0 ≤ ν(A) ≤ µ(A) for all A ∈ , then there exists a bounded -
measurable function m on X such that dν = mdµ. This Radon-Nikodym theorem
can also be formulated in terms of positive linear functionals on some lattice ideal
E of measurable functions on some σ -ﬁnite measure space (X,,λ) as follows.
If ϕ and ψ are normal (i.e., order continuous) linear functionals on E satisfying
0 ≤ ψ(u) ≤ ϕ(u) for all 0 ≤ u ∈ E,thenψ(u) = ϕ(mu) for some bounded meas-
urable function m on X. Equivalently, ψ = ϕ ◦ π
denotes the operator
of multiplication by m in E. For positive linear operators such a Radon-Nikodym
theorem is in general not valid. However, it will follow from the results in the
present paper that, given any collection of positive operators, it is always possible
to enlarge the domain space of the operators involved such that a Radon-Nikodym
theorem holds for the extended operators.
The natural setting to discuss these problems is the framework of vector lattices
(or, Riesz spaces). In this setting a general Radon-Nikodym type theorem has been
obtained for so-called interval preserving operators (or, Maharam operators) in .
We brieﬂy recall this result. Let L and M be Dedekind complete vector lattices and
assume that T : L → M is a positive linear operator (so, 0 ≤ u ∈ L implies that
Tu ≥ 0inM). Such an operator is called interval preserving if 0 ≤ w ≤ Tu in
M implies that w = Tv for some 0 ≤ v ≤ u in L, i.e., T [0,u]=[0,Tu] for all
Dedicated to the memory of our friend and colleague Pay Huijsmans.