A distortion function in mathematics and statistics, for example, g : [ 0 , 1 ] → [ 0 , 1 ] {\displaystyle g:[0,1]\to [0,1]} , is a non-decreasing function such that g ( 0 ) = 0 {\displaystyle g(0)=0} and g ( 1 ) = 1 {\displaystyle g(1)=1} . The dual distortion function is g ~ ( x ) = 1 − g ( 1 − x ) {\displaystyle {\tilde {g}}(x)=1-g(1-x)} . Distortion functions are used to define distortion risk measures.

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

Q ( A ) = g ( P ( X ∈ A ) ) . {\displaystyle \mathbb {Q} (A)=g(\mathbb {P} (X\in A)).} ⁴