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 coutable 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

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