Proposition 4

The support of the sum of two measures is contained in the union of their supports.

For, if the restrictions of two measures to an open set are zero, then the same is true of the restriction of their sum.

If $\mu$ and $\nu$ are two positive measures, then the support of $\lambda = \mu + \nu$ is equal to the union of the supports of $\mu$ and $\nu$; for, if $x_0$ is a point of the union and $V$ is any neighborhood of $x_0$, then there exists a continuous function $f \geq 0$ with support contained in $V$ and such that one of the numbers $\mu(f)$, $\nu(f)$ is $> 0$; a fortiori, $\lambda(f) = \mu(f) + \nu(f) > 0$.