Proposition 2

The support of a measure $\mu$ is identical to the support of the measure $|\mu|$; if $\mu$ is real, its support is the union of the supports of the measures $\mu^+$ and $\mu^-$.

For, if the restriction of $\mu$ to an open set $U$ is zero, then so is that of $|\mu|$ (resp. of $\mu^+$ and $\mu^-$ when $\mu$ is real), and conversely.

Note that the supports of $\mu^+$ and $\mu^-$ can be nonempty and identical (cf. Ch. V, § 5, Exer. 5).