Complete measure
From Encyclopedia of Mathematics
A measure on a
-algebra
for which
and
imply
for every
. Here
is the total variation of
(
for a positive measure).
Comments
Complete measures arise as follows (cf. [a1]). Let be a set,
a
-algebra of subsets of it and
a positive measure on
. It may happen that some set
with
has a subset
not belonging to
. It is natural, then, to define the measure
on such a set
as
.
In general, let be the collection of all sets
for which there exists sets
such that
,
. In this situation, define
. Then
is a
-algebra and
becomes a complete measure on it (this process is called completion).
is then called a complete measure space.
References
[a1] | W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24 |
[a2] | E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) |
How to Cite This Entry:
Complete measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_measure&oldid=46419
Complete measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_measure&oldid=46419
This article was adapted from an original article by A.P. Terekhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article