Let $\mu_1$ and $\mu_2$ be finite signed measures on the measurable space $(X, \mathcal A)$. Define signed measures $\mu_1 \vee \mu_2$ and $\mu_1 \wedge \mu_2$ on $(X, \mathcal A)$ by $\mu_1 \vee \mu_2 = \mu_1 + (\mu_2 - \mu_1)^+$ and $\mu_1 \wedge \mu_2 = \mu_1 - (\mu_1 - \mu_2)^+$.
(a) Show that $\mu_1 \vee \mu_2$ is the smallest of those finite signed measures $\nu$ that satisfy $\nu(A) \geq \mu_1(A)$ and $\nu(A) \geq \mu_2(A)$ for all $A \in \mathcal A$.
(b) Find and prove an analogous characterization of $\mu_1 \wedge \mu_2$.
When it comes to doing (a), I let $(P,N)$ be the Hahn decomposition of $X$ with respect to $\mu_2 - \mu_1$. Then $(\mu_1 \vee \mu_2)(A) = \mu_1 (A \cap N) + \mu_2 (A \cap P)$. I have no idea where to go from this point.