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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](http://en.wikipedia.org/wiki/Radon%E2%80%93Nikodym_theorem).2011-04-01
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    In brief, if $\mu$ is the probability measure, $F$ the corresponding distribution function, and $f$ the probability density function, then $E[e^{itX} ] = \int {e^{itx} d\mu (x)} = \int {e^{itx} dF(x)} = \int {e^{itx} f(x)dx}$.2011-04-03

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