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