You're thinking of a "characteristic function" in the sense of "indicator function" of a subset (https://en.wikipedia.org/wiki/Indicator_function https://en.wik

You're thinking of a "characteristic function" in the sense of "indicator function" of a subset (https://en.wikipedia.org/wiki/Indicator_function https://en.wikipedia.org/wiki/Indicator_function), which is different thing to the characteristic function of a probability density function.