Choquet integral

The following notation is used:

• ${\displaystyle S}$ – a set.
• ${\displaystyle {\mathcal {F}}}$ – a collection of subsets of ${\displaystyle S}$.
• ${\displaystyle f:S\to \mathbb {R} }$ – a function.
• ${\displaystyle \nu$ :{\mathcal {F}}\to \mathbb {R} ^{+}} – a monotone set function.

Assume that ${\displaystyle f}$ is measurable with respect to ${\displaystyle {\mathcal {F}}}$, that is

${\displaystyle \forall x\in \mathbb {R} \colon \{s\in S\mid f(s)\geq x\}\in {\mathcal {F}}}$

Then the Choquet integral of ${\displaystyle f}$ with respect to ${\displaystyle \nu }$ is defined by:

${\displaystyle (C)\int fd\nu$ :=\int _{-\infty }^{0}(\nu (\{s|f(s)\geq x\})-\nu (S))\,dx+\int _{0}^{\infty }\nu (\{s|f(s)\geq x\})\,dx}

where the integrals on the right-hand side are the usual Riemann integral (the integrands are integrable because they are monotone in ${\displaystyle x}$).

In general the Choquet integral does not satisfy additivity. More specifically, if ${\displaystyle \nu }$ is not a probability measure, it may hold that

${\displaystyle \int f\,d\nu +\int g\,d\nu \neq \int (f+g)\,d\nu .}$

for some functions ${\displaystyle f}$ and ${\displaystyle g}$.

The Choquet integral does satisfy the following properties.

Let ${\displaystyle G}$ denote a cumulative distribution function such that ${\displaystyle G^{-1}}$ is ${\displaystyle dH}$ integrable. Then this following formula is often referred to as Choquet Integral:

${\displaystyle \int _{-\infty }^{\infty }G^{-1}(\alpha )dH(\alpha )=-\int _{-\infty }^{a}H(G(x))dx+\int _{a}^{\infty }{\hat {H}}(1-G(x))dx,}$

where ${\displaystyle {\hat {H}}(x)=H(1)-H(1-x)}$.

• choose ${\displaystyle H(x):=x}$ to get ${\displaystyle \int _{0}^{1}G^{-1}(x)dx=E[X]}$,
• choose ${\displaystyle H(x):=1_{[\alpha ,x]}}$ to get ${\displaystyle \int _{0}^{1}G^{-1}(x)dH(x)=G^{-1}(\alpha )}$

The Choquet integral was applied in image processing, video processing and computer vision. In behavioral decision theory, Amos Tversky and Daniel Kahneman use the Choquet integral and related methods in their formulation of cumulative prospect theory.[6]

• Nonlinear expectation
• Superadditivity
• Subadditivity

• Gilboa, I.; Schmeidler, D. (1992). "Additive Representations of Non-Additive Measures and the Choquet Integral". {{cite journal}}: Cite journal requires |journal= (help)
• Even, Y.; Lehrer, E. (2014). "Decomposition-integral: unifying Choquet and the concave integrals". Economic Theory. 56 (1): 33–58. doi:10.1007/s00199-013-0780-0. MR 3190759. S2CID 1639979.