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.
Characteristic function: fourier transform of probability measure or density?
1
$\begingroup$
probability-theory
measure-theory
fourier-analysis
-
0In 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
-
2In 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