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 $$(\Omega,\mathcal{F},\mathbb{P})$$, 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
 * $$\mathbb{Q}(A) = g(\mathbb{P}(X \in A)).$$