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

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