Defining the Choquet Integral in Mathematica

Question

I would like to define and use the Choquet integral in Mathematica.

I think the first step should be to define a representation of a non-additive (normalized) measure $\nu:\Omega \rightarrow [0,1]$. Thus, I tried to define some sort of distribution function $F_\nu(x)$ that represents the measure of the sets $(- \infty, x]$. A so-called distorted measure $\nu = h \circ P$, where $h:[0,1] \rightarrow [0,1]$ is an increasing function and $P$ a probability measure would do the job.

For instance, letting $h:[0,1] \rightarrow [0,1]$ be $h(x)=x^2$ and using the continuous uniform distribution for the sake of simplicity:

h[x_] = x^2
F[x_] = CDF[\[ProbabilityDistribution[{"CDF", \[CDF[UniformDistribution[], t]]}, {t, -∞, ∞}]], x]

But here I got stuck since I do not know how I need to tell Matheatica to evaluate something like $\int{\nu(\{\omega \in \Omega| f(\omega) \geq x\}) dx}$.

Any ideas to get this running (maybe there is a much more elegant solution)?

Answer 1

Possibly:

ChoquetIntegral[h_, dist_] := NIntegrate[InverseCDF[dist, x] h'[x], {x, 0, 1}]

For your example:

ChoquetIntegral[#^2&, UniformDistribution[]]

0.666667