Support of a measure
The set , where
is a locally compact Hausdorff space on which the regular Borel measure
is given and
is the largest open set for which
. In other words,
is the smallest closed set on which
is concentrated. (Here,
is concentrated on
if
.) If
is compact, then
is called of compact support.
Comments
The support of a measure on a topological space
can be defined whenever the union of all
-zero open subsets is still of measure zero. This is the case if
has a countable base, or if
is tight or Radon (see Regular measure), but it is not always the case if
is only locally compact and
is not tight.
Of course, one can always define, for a measure on a topological space
with topology
,
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But then it is not necessarily true that , contradicting the intuitive idea of a support.
References
[a1] | W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 57 |
[a2] | H. Federer, "Geometric measure theory" , Springer (1969) pp. 60; 62; 71; 108 |
Support of a measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Support_of_a_measure&oldid=28274