Positivity 10 (2006), 261–284
© 2006 Birkh
auser Verlag Basel/Switzerland
1385-1292/020261-24, published online April 26, 2006
Forcing Divergence When the Supremum Is Not
and JOSEPH M. ROSENBLATT
12, Plymouth House, Devonshire Drive, Greenwich, London SE10 8LE, UK.
Department of Mathematics, University of Illinois at Urbana, Urbana, IL 61801, USA.
Received 1 April 2003; accepted 30 May 2003
Abstract. If a sequence of functions (f
) has an integrable supremum and converges
almost everywhere, then an operator sequence (T
) will yield a sequence (T
) that con-
verges almost everywhere too, under some very general assumptions about the sequence
). However, if the supremum of (f
) is not integrable then this can fail to be the case.
It is shown that if a sequence of functions (f
) has a supremum sup
| that is not inte-
grable, then one can always construct a variety of sequences of positive contractions (T
such that lim sup |T
|=∞a.e. These operators can be conditional expectations with
respect to an increasing sequence of ﬁnite σ -algebras, the conditional expectation with
respect to one ﬁxed σ -algebra, or averages with respect to a measure-preserving transfor-
mation. General discussion of these constructions, history of previous results of this type,
and some open questions are also given.
Mathematics Subject Classiﬁcation 2000: 60G42, 28D05
Key words: Conditional expectations, erogdic averages, integrable, supremum
In this article, let (X,β,p) be a non-atomic separable probability space.
The assumptions that p is non-atomic and separable will not always be
needed, but since it is essential to some of the constructions, we take that
consistently as a hypothesis.
This article is about what happens when one applies a sequence of oper-
ators to a sequence of integrable functions term by term. Under suitable
general conditions, the result may be an almost everywhere convergent
sequence. But even when the functions are converging almost everywhere,
and the operator sequence itself is well-behaved when applied to one inte-
grable function, convergence can break down badly.
For example, the Dunford–Zygmund–Fava theorem states the positive
result that for any L
(Dunford , Fava 
and Zygmund ), and any function f ∈ L
(X), with 1 <p<∞, the