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Singular measures

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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
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article