Difference between revisions of "Support of a measure"
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 57</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Federer, "Geometric measure theory" , Springer (1969) pp. 60; 62; 71; 108</TD></TR></table> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 57 {{MR|0210528}} {{ZBL|0142.01701}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Federer, "Geometric measure theory" , Springer (1969) pp. 60; 62; 71; 108 {{MR|0257325}} {{ZBL|0176.00801}} </TD></TR></table> |
Revision as of 12:13, 27 September 2012
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 
,
 |
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 MR0210528 Zbl 0142.01701 |
| [a2] | H. Federer, "Geometric measure theory" , Springer (1969) pp. 60; 62; 71; 108 MR0257325 Zbl 0176.00801 |
Support of a measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Support_of_a_measure&oldid=28274