Positivity (2005) 9:369–383 © Springer 2005
Operators Represented by Conditional
Expectations and Random Measures
and D.T. RAMBANE
Unit for Business Mathematics and Informatics, North-West University, Potchefstroom
2520, South Africa. (E-mail: firstname.lastname@example.org);
Department of Mathematics and Applied
Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa.
Received 15 September 2004; accepted 19 March 2005
Abstract. On standard measure spaces every order continuous linear map between two
ideals of almost everywhere ﬁnite measurable functions can be represented by a random
measure. An analogue of this theorem is proved for the case of arbitrary σ-ﬁnite measure
spaces. This fact leads to a proof that every order continuous linear map between ideals
of almost everywhere ﬁnite measurable functions on σ -ﬁnite measure spaces is multipli-
cation conditional expectation representable. This sheds further light on the structure of
order continuous operators.
Mathematics Subject Classiﬁcation (2000): 47B38, 47B65
Key words: random measure, multiplication conditional expectation operator, pseudo-integral
operator, order continuous operator
A.R. Sourour [13,12], building on work of N. Kalton [6,7] and
W. Arveson , proved the remarkable result that every order bounded order
continuous linear operator acting between ideals of measurable functions is
generated by a random measure, provided that the measure spaces involved
are standard measure spaces (i.e., spaces which are isomorphic to Borel sub-
sets of complete separable metric spaces, see ). This representation is useful
in studying these operators as was shown by L. Weis in [14–16]. In , the
lattice properties of operators generated by random measures are studied
in general σ -ﬁnite measure spaces and the question arises whether one can
prove the result mentioned above in the general case. An examination of the
ideas involved seems to indicate otherwise.
Financial assistance of the NRF is gratefully acknowledged.
Financial assistance of the University of Venda research fund is gratefully acknowl-