Suppose $\mu$ is a probability measure and $f$ is its characteristic function. Suppose $f$ has finite first order derivative at $t=0$, can you conclude that $\mu$ has finite expectation?
I know that if $f$ has a finite derivative of even order $k$ at $t=0$, then $\mu$ has a finite moment of order $k$. But I am not sure whether $\mu$ has finite expectation and how to prove it.
Can anyone help me?