I'm looking at a theorem in my Analysis textbook that says: If $\mu$ is a complex measure on $X$, then $|\mu|(X) < \infty$.
I can't seem to get my head around this being true. The following seems to me like a counterexample: let $\mu$ be any positive measure with $\mu(X) = \infty$. Then $|\mu|(X) \ge |\mu(X)| = \infty$, so $|\mu|(X) = \infty$.
What am I missing? Can someone give me intuition about why this theorem is true?