Distortion function

A distortion function $g:[0,1]\to [0,1]$ is a non-decreasing function such that $g(0)=0$ and $g(1)=1$. The dual distortion function is ${\tilde {g}}(x)=1-g(1-x)$. Distortion functions are used to define distortion risk measures.

Given a probability space $(\Omega ,{\mathcal {F}},\mathbb {P} )$, then for any random variable $X$ and any distortion function $g$ we can define a new probability measure $\mathbb {Q}$ such that for any $A\in {\mathcal {F}}$ it follows that

$\mathbb {Q} (A)=g(\mathbb {P} (X\in A)).$