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Integral representation without additivity. (English) Zbl 0687.28008
Let $$\Sigma$$ denote a nonempty algebra of subsets of a set S, B denote the set of bounded real-valued $$\Sigma$$-measurable functions on S and v denote the monotone (i.e., $$E\subset F\Rightarrow v(E)\leq v(F))$$ real- valued function on $$\Sigma$$ with $$v(\emptyset)=0$$ and $$v(S)=1$$. G. Choquet [Ann. Inst. Fourier 5, 131-295 (1955; Zbl 0064.351)] defined an integration operation with respect to the not necessarily additive set function v. Given a nonnegative-valued function $$a\in B$$ let $$I(a):=\int_{S}a dv=\int^{\infty}_{0}v(\{s\in S:\quad a(s)\geq \alpha \})d\alpha,$$ where the integral on the right side is the extended Riemann integral. It is known that if every $$E\subset S$$ is identified with its indicator function $$E^*$$ then the functional I extends v from $$\Sigma$$ to the family $$B_+$$ of non-negative elements of B, and this extension is monotone, positively homogeneous and co-monotone additive on $$B_+$$ (i.e., if $$a,b\in B_+$$ then $$I(a+b)=I(a)+I(b))$$.
The main result from the paper is a converse result; namely, if $$I:\quad B\to {\mathbb{R}}$$ is monotone, co-monotone additive and $$I(S^*)=1$$, then $$I(a)=\int^{\infty}_{0}v(a\geq \alpha)d\alpha +\int^{0}_{- \infty}(v(a\geq \alpha)-1)d\alpha$$ for all $$a\in B$$ and v defined by $$v(E)=I(E^*)$$. The author gives some extensions of this result and suggests some applications to Bayesian decision theory and subjective probability.

##### MSC:
 28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 60A10 Probabilistic measure theory 62C10 Bayesian problems; characterization of Bayes procedures
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##### References:
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