Let $(X,F,\mu)$ be a finite measure space, i.e. $\mu(X)<\infty$. And let $x\in X$, and let $\delta_x$ be the Dirac measure with respect to $x$, i.e. $\delta_x(E)=1$ if $x\in E$ and $\delta_x(E)=0$ if $x\notin E$. Finally let $\nu$ be the signed measure $\mu-a\delta_x$, where $a=\mu(X)$. Find the Hahn decomposition of $\nu$.
I'm not really sure how to construct Hahn decompositions. I tried rewriting the proof of the Hahn-Jordan decomposition theorem for the case of this particular signed measure, but it didn't give me a concrete pair of sets.