Support of a measure

2020 Mathematics Subject Classification: Primary: 28C15 [MSN][ZBL]

Let $X$ be a topological space with a countable basis, $\mathcal{B}$ a $\sigma$-algebra of subsets of $X$ containing the open sets (and hence also the Borel sets) and $\mu: \mathcal{B}\to [0, \infty]$ a ($\sigma$-additive) measure (cp. with Borel measure). The support of $\mu$ (usually denoted by ${\rm supp}\, (\mu)$) is the complement of the union of all open sets which are $\mu$-null sets, i.e. the smallest closed set $C$ such that $\mu (X\setminus C) =0$.

The existence of a countable base guarantees that

(P) the union of all open $\mu$-null sets is itself a nullset.

If (P) does not hold and $C$ is the complement of the union of all open sets which are $\mu$-null sets, then we do not have $\mu (X\setminus C)=0$. When $X$ does not have a countable base, property (P) can still be inferred from other assumptions. A common one is to assume that $X$ is a locally compact Hausdorff space and $\mu$ is a Radon measure (some authors use for such measures the terminologies tight and Borel regular, compare with Borel measure). For such measures the support is then defined as above.

The support can be defined also in case $\mu$ is a signed measure or a vector measure (in both cases we are assuming $\sigma$-additivity): the support of $\mu$ is then defined as the support of its total variation measure (see Signed measure for the definition).

References

[AFP]	L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Bi]	P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[Bo]	N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[DS]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[Ha]	P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Ma]	P. Mattila, "Geometry of sets and measures in euclidean spaces. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005