Positivity 7: 149–159, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Convergence for Sequences of Functions and an
Egorov Type Theorem
and P. KIRIAKOULI
Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece
Sogdianis 6, Ano Ilissia, 15771 Athens, Greece
Received 29 December 2000; accepted 4 March 2001
Abstract. For every ordinal ξ<ω
we deﬁne a new type of convergence for sequences of functions
(ξ-uniform pointwise) which is intermediate between uniform and pointwise convergence. Using this
type of convergence we obtain an Egorov type theorem for sequences of measurable functions.
The classical theorem of Egorov  plays an important role in measure theory.
It shows that for a sequence of measurable functions, in a ﬁnite measure space,
pointwise almost everywhere convergence implies almost uniform convergence.
Bartle in  gave necessary and sufﬁcient conditions for a sequence of measur-
able functions, on an arbitrary measure space, to be almost uniformly convergent.
We are interested in investigating a hierarchy of weaker conditions determined by
Mercourakis in  introduced a new type of pointwise convergence (uniformly
pointwise convergence) which is weaker than uniform convergence and stronger
than pointwise convergence.
Later Mercourakis in  extended this convergence with the deﬁnition of m-
uniformly pointwise convergence for every 1 m ω.Form = 1 this con-
vergence coincides with uniformly pointwise convergence and (m + 1)-uniformly
pointwise is weaker than m-uniformly pointwise and stronger than pointwise con-
In the ﬁrst section of this paper we deﬁne a new type of pointwise convergence
which extends the above convergence for all ordinal ξ<ω
is the ﬁrst
uncountable ordinal numbre).
In the second section we deﬁne the almost ξ -uniformly pointwise convergence
for all ξ<ω
. We give necessary and sufﬁcient conditions for a sequence of meas-
urable functions to be ξ -almost uniformly pointwise convergent, which extends the
result of Bartle for the almost uniform convergence (cf. Propositions 2.6, 2.8 and
2.11), Corollaries 2.7, 2.10 and 2.12 and Theorem 2.13).