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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.

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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.

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    what does positively collinear mean?2012-12-13
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    It means that there is a positive real number $\lambda>0$ such that $z'=\lambda z$ (or $z=\lambda z'$.)2012-12-13
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    @john Has this helped you at all? Do you want me to finish the argument?2012-12-14