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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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In the context of measure theory, the role of density is played by the Radon-Nikodym derivative. – Raskolnikov Apr 1 '11 at 15:35
In brief, if $\mu$ is the probability measure, $F$ the corresponding distribution function, and $f$ the probability density function, then $E[e^{itX} ] = \int {e^{itx} d\mu (x)} = \int {e^{itx} dF(x)} = \int {e^{itx} f(x)dx}$. – Shai Covo Apr 3 '11 at 5:04

1 Answer 1

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 $f$.

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