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 ,

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