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Integral representation of continuous comonotonically additive functionals. (English) Zbl 0905.28006
It is shown that for any quasi integral \(I\) on a Stone lattice \(L\) with \(I(1)= 1\) there exists a unique upper-continuous capacity \(\mu\) on the collection \(\Sigma\) of all upper contours of all functions belonging to \(L\) satisfying \(I(a)= \int_X ad\mu\), \(a\in L\). Moreover, \(I\) introduced by \(I(a)= \int_X ad\mu\), \(a\in L\), for some upper-continuous capacity \(\mu\) on \(\Sigma\) is a quasi integral on \(L\). Here \(\int_X ad\mu\) is defined as the Choquet integral \[ \int^\infty_0 \mu(a\geq t)dt+ \int^0_{-\infty} (\mu(a\geq t)- 1)dt,\quad a\in L. \] As an application it is shown that the set consisting of all upper continuous capacities on a compact space equipped with the weak topology is a compact Hausdorff space, which is in addition metrizable, if the underlying compact space is metrizable.

28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
28A12 Contents, measures, outer measures, capacities
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
Full Text: DOI
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