Difference between revisions of "Signed measure"
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Revision as of 21:35, 27 July 2012
generalized measure
An extended real-valued 
-additive set function that is defined on the 
-algebra of Borel subsets of a domain 
and that is finite on compact sets 
. The difference between two measures one of which is finite on 
is a charge; conversely, all charges may be obtained in this way: for any charge 
there exists a decomposition of 
into two disjoint Borel sets 
and 
such that 
for 
and 
for 
. The measures 
and 
are independent of the choice of 
and 
and are known respectively as the positive and negative variations of the charge 
; the measure 
is called the total variation of 
. With this notation, the so-called Hahn–Jordan decomposition: 
holds, so that the properties of charges may be phrased in terms of measure theory.
References
| [1] | N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian) |
| [2] | P.R. Halmos, "Measure theory" , v. Nostrand (1950) |
Comments
A charge is also called a signed measure [a1], a real measure or a signed content. It can, more generally, be defined on a ring of subsets of a space 
, or, alternatively, on a Riesz space of functions on 
, see [a2].
Any pair 
as above is called a Hahn decomposition of 
with respect to 
. The pair 
, defined above, is also called the Jordan decomposition of 
.
References
| [a1] | E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) |
| [a2] | K. Jacobs, "Measure and integral" , Acad. Press (1978) |
Signed measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Signed_measure&oldid=19071