Horizontal Egorov property of Riesz spaces
Author:
Mikhail Popov
Journal:
Proc. Amer. Math. Soc. 149 (2021), 323-332
MSC (2010):
Primary 46A40; Secondary 46B42
DOI:
https://doi.org/10.1090/proc/15235
Published electronically:
October 20, 2020
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We say that a Riesz space has the horizontal Egorov property if for every net
in
, order convergent to
with
for all
, there exists a net
of fragments of
laterally convergent to
such that for every
, the net
-uniformly tends to zero. Our main result asserts that every Dedekind complete Riesz space which satisfies the weak distributive law possesses the horizontal Egorov property. A Riesz space
is said to satisfy the weak distributive law if for every
the Boolean algebra
of fragments of
satisfies the weak distributive law; that is, whenever
is a sequence of partitions of
, there is a partition
of
such that every element of
is finitely covered by each of
(e.g., every measurable Boolean algebra is so). Using a new technical tool, we show that for every net
order convergent to
in a Riesz space with the horizontal Egorov property there are a horizontally vanishing net
and a net
, which uniformly tends to zero for every fixed
such that
for all
.
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Additional Information
Mikhail Popov
Affiliation:
Institute of Exact and Technical Sciences, Pomeranian University in Słupsk, ul. Arciszewskiego 22d, PL-76-200 Słupsk, Poland; Vasyl Stefanyk Precarpathian National University, 57 Shevchenko str, Ivano-Frankivsk, 76018 Ukraine.
Email:
misham.popov@gmail.com
DOI:
https://doi.org/10.1090/proc/15235
Keywords:
Riesz space,
order convergence,
uniform order convergence
Received by editor(s):
March 14, 2020
Received by editor(s) in revised form:
April 18, 2020, June 8, 2020, and June 11, 2020
Published electronically:
October 20, 2020
Communicated by:
Stephen Dilworth
Article copyright:
© Copyright 2020
American Mathematical Society