Hahn Decomposition $(P,P^c)$

Question

Let $\mu$ be a signed measure with Hahn decomposition $(P,P^c)$. Show that $(-\mu)^+=\mu^-$ and $(-\mu)^-=\mu^+$ and that $(P^c,P)$ is a Hahn decomposition of $-\mu$

I would like to know the following would help me solve this exercise, and if so, how could I apply it:

$\mu^+(E):=\mu(E\cap P)\ge0$, and $\mu^-(E):=-\mu(E\cap P^c)\ge0$

Answer 1

Suppose $E \subseteq P$. By assumption $\mu(E) \geq 0$, so $ -\mu(E) \leq 0$.

Similarly, if $E \subseteq P^c$, then $-\mu(E) \geq 0$. So, $(P^c,P)$ decomposes $-\mu$.

Now, by the above, $(-\mu)^+(E) = -\mu(E \cap P^c) = \mu^-(E)$. And similarly $(-\mu)^- = \mu^+$.