TITLE: total variation of a measure
QUESTION [5 upvotes]: I ran across this question in my analysis textbook.  I just cannot prove this.  
Suppose $\mu$ is a complex measure on $X$ such that $\mu(X)=1$ and $|\mu|(X)=1$.  Show $\mu$ is a positive real measure.

REPLY [2 votes]: Take some measurable set $A$. Then 
$$\mu(A)+\mu(X\setminus A)=\mu(X)=1$$ while 
$$1\leq|\mu(A)|+|\mu(X\setminus A)|\leq|\mu|(X)=1$$
Now use the fact that for any two complex numbers $z,z'$, $|z+z'|\leq|z|+|z'|$ with equality iff $z$ and $z'$ are positively collinear.