Characteristic function of random variable is always integrable

Question

Let $X$ be a random variable and let $phi$ be the characteristic function of $X$ . Is the integration of $|phi|$ over $mathbb R$ always finite?
I know how the characteristic function is defined. But how do I integrate it?

Kavi Rama Murthy · Answer

The characteristic function of $X=0$ is $1$ at very point and this is not integrable.

Characteristic function of random variable is always integrable

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