In many texts charecteristic function is defined as a Fourier transform of probability density (if random variable admits a density function). Also we can define a charecteristic function as Fourier transform of probability measuer instead (like in Jacod J., Protter P. Probability essentials, second edition, page 104). Are these definitions equal? I'm a bit confused as not all random variables have probability density but probability measuer should always be defined.
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Yes, they are the same. For a random variable which admits a density function $f$, its distribution measure $\mu$ is given by $\mu(A) = \int_A f dm$, or more compactly $d\mu = f dm$, where $m$ is Lebesgue measure. It is clear that in this case, the Fourier transform of $\mu$ is the same as the Fourier transform of $f$.