# Finding a Hahn Decomposition involving a Dirac Measure

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.

Since the $$\sigma$$-algebra isn't specified, you cannot give an explicit choice for the Hahn-decomposition. (For example $$F= \{X, \emptyset\}$$ gives only a trivial decomposition. One other example, is $$F=\{X, A,A^c,\emptyset\})$$ with $$X= [0,1]$$, $$A= [0,1/2]$$ and $$\mu = \delta_0$$ and $$x=1$$. Then $$X= A \cup A^c$$ is the Hahn-decomposition.)
Assume that $$\{x\} \in F$$, then the decomposition can be determined. If $$\mu(\{x\}) -a \ge 0$$, then $$\nu := \mu -a \delta_x$$ is a non-negative measure and $$\nu^{-} =0$$ is trivial. (Thus the Hahn-decomposition is $$X=P \cup N$$ with $$N = \emptyset$$.) On the other hand, if $$\mu(\{x\}) -a <0$$. Then the Hahn-decompisition is $$X =P \cup N$$ with $$N= \{x\}$$ and $$P = X \setminus \{x\}$$.