Choquet integral

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A Choquet integral is a way of measuring the expected utility of an event where the event is uncertain. It is applied specifically to capacities, and is used in the field of decision theory. A Choquet integral of a function $f : S \rightarrow R^+$ with respect to a capacity $ν$ is defined by: $\int f d\nu := \int^\infty_0 \nu (\{s | f (s) \geq x\}) dx$ .

It is important to note that the Choquet integral does not satisfy additivity: $\int f d\nu + \int g d\nu \neq \int (f + g) d\nu$ .

Using the Choquet integral to denote the expected utility of belief functions measured with capacities is a way to reconcile the Ellsberg paradox and the Allais paradox.