Difference between revisions of "Singular measures"
From Encyclopedia of Mathematics
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m (moved Mutually-singular measures to Singular measures: More common name. I will redirect the old page.) |
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Revision as of 13:32, 12 August 2012
Two (positive) measures 
and 
, defined on a locally compact space 
, such that 
.
Two measures 
and 
are mutually singular if and only if there exist in 
two disjoint sets 
and 
such that 
is concentrated on 
and 
on 
.
References
| [1] | N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) |
Comments
The second characterization in the main article above holds if 
and 
are 
-additive 
-finite measures on an abstract measurable space, and 
and 
belong to the 
-field.
Mutually-singular measures are also called singular measures or orthogonal measures.
Instead of "concentrated on" one also uses "supported in" (cf. also Support of a measure).
References
| [a1] | P.R. Halmos, "Measure theory" , v. Nostrand (1950) |
| [a2] | E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) |
How to Cite This Entry:
Singular measures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Singular_measures&oldid=14182
Singular measures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Singular_measures&oldid=14182
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article