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.
Best Answer
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 $P = X \setminus \{x\}$.