TITLE: Hahn Decomposition Theorem In Folland
QUESTION [9 upvotes]: I was reading the proof of Hahn Decomposition theorem from the textbook of Folland: precisely I was looking at the following text



I have the following question:

As Highlighted in the text above, why $m$ is finite? It may be infinite as there is no restriction on $X$. Why does the author consider it finite? 
Again why $\nu(A)<\infty$ ? I do not understand also this.
I understand that even for $A$ also we get some $B$ with the property that $\nu(B)>\nu(A)+1/n$ but I do not understand how this leads to a contradiction.

I would be really thankful if someone could help me. Any help will be appreciated.

REPLY [4 votes]: I'm fairly confident that the assumption in the very first line is a typo. It should say $\nu < \infty$ rather than $\nu > -\infty$. It appears like $\nu > -\infty$ is not used anywhere in the proof whereas as you've noted $\nu < \infty$ is used multiple times. 
($\dagger$) As for 3. the point is that for any set, $C$, contained in $N$ with $\nu(C) > 0$ we can find another set $C' \subset C$ with $\nu(C') > \nu(C)$. 
Furthermore, by contradiction it is assumed that such a $C$ exists as otherwise $N$ would be negative. Then given this initial $C$ there exists a sequence $N \supset C \supset A_1 \ldots \supset A_k \supset$ of sets $A_i$ constructed using the result ($\dagger$) about positive sets of $N$. Additionally the $A_i$ are constructed so that $A_{i+1}$ approximates $A_i$ poorly so that the difference $\nu(A_{i+1}) - \nu(A_i)$ is large. Now as $N \supset \cap_{i = 1}^{\infty}A_i$ and $\nu(\cap_{i = 1}^{\infty}A_i) > 0$ by ($\dagger$) again there exists a $D \subset \cap_{i = 1}^{\infty}A_i$ with $\nu(\cap_{i = 1}^{\infty}A_i) > \nu(D) + n^{-1}$ for some $n$. It is shown that $n < n_j$ for some $j$ as well so that $\nu(D) > \nu(A_{j-1}) + n^{-1} > \nu(A_{j-1}) + n_j^{-1}$. This is the desired contradiction because $n_j$ was chosen so that for any $H \subset A_{j-1}$ we have that $\nu(A_{j-1}) + (n_j - 1)^{-1} > \nu(H)$ so take $H = D$ and you are done.