Distortion function

A distortion function in mathematics and statistics, for example, $g:[0,1]\to [0,1]$, 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 $(\mathrm{\Omega },{ℱ},\mathbb{\mathbb{P}})$, then for any random variable $X$ and any distortion function $g$ we can define a new probability measure $\mathbb{\mathbb{Q}}$ such that for any $A\in {ℱ}$ it follows that

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