This questions seems to be fairly easy but I got stuck.
Let $\mu$ be a signed measure with domain $\Sigma$. We define a measurable set $E$ as negative with respect to $\mu$ if $\mu(E\,\cap\, A)\leq0$ for any measurable set $A$. In the book I'm reading it is said that the union of two negative sets is a negative set.
To show this I tried to rewrite the union as:
$\mu(E\, \cup\, F) = \mu(E) + \mu(F) - \mu(E\, \cap\, F)$,
where $E$ and $F$ are negative sets, which I know holds when $\mu$ is a probability measure. But I'm not sure if the previous equality is justified for signed measures in general (is it?). Even if it is justified we still need to show that $\mu(E) + \mu(F) \leq \mu(E\, \cap\, F)$, considering $\mu(E\, \cap\, F)\leq0$ and thus $-\mu(E\, \cap\, F)\geq0$.
If $E,F$ are negative with respect to $\mu$, and $A$ is a measurable set, then $$ (E\cup F)\cap A = \Bigl(E\cap (F'\cap A)\Bigr) \; \cup \; \Bigl(F\cap (E'\cap A)\Bigr) \; \cup \; \Bigl(E\cap (F\cap A)\Bigr) \; $$ which is a disjoint union of $3$ sets, each with nonpositive measure.