TITLE: If $\mu$ is a signed finite measure, then $\|\mu\| = \sup \left\{ \int f d\mu : |f| \leq 1 \right\} $
QUESTION [5 upvotes]: If $\mu$ is a signed finite measure, then 
$$
\|\mu\| = \sup \left\{ \int f d\mu : |f| \leq 1 \right\}
$$
I did the inequality "$\geq$". Somebody help with the other inequality "$\leq$"?

REPLY [7 votes]: By Hahn decomposition theorem, there exist two measurable sets $P,N$ such that $P\cup N=X$ and $P\cap N=\emptyset$, and $\mu(A\cap P)\geq 0$ while $\mu(A\cap N)\leq 0$ for all measurable sets $A$.
Now let
$$
f=1_P-1_N.
$$
We have
$$
\int_Xfd\mu=\mu(P)-\mu(N)=|\mu|(X).
$$
So the reverse inequality follows.