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Prove the following theorem:

Let $\phi: \Bbb R \to \Bbb C$.

$\phi$ is the characteristic function of a real random variable $X:\Omega \to \Bbb R$ if and only if $\phi(0)=1$ $\phi$ is uniformly continuous at $0$ and it's semi positive definite.

I have no idea how Can I attack this problem, please give me some tips, and tools. Thanks

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The "only if" part follows trivially from Fourier theory. The Fourier transform of a Borel probability measure is positive definite, etc. The most clean argument for the "if" part is probably via Gelfand theory. –  Michael Dec 13 '13 at 9:19

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