Extension of Multi Linear Utility Function and Its Fuzzy Integral Representation

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  • 多重線型効用関数の拡張とそのファジ積分標示

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Abstract

In this paper, we formulate a new fuzzy integral of vector valued functinos. First, the map Φ : R^n→R characterizing any multi linear utility function is extended to the map Φ : V^n→R preserving Φ's properties. The extended map Φ is expressed as the canonical inner product between the direct sum of alternating tensor spaces, [○!+]^n_<γ=1>A^γ(V), and the direct sum of their dual spaces, [○!+]^n_<γ=1>A^γ(V^*). Naturally, the map Φ is identified with the Lebesgue integral representation because any measurable function and any measure can be defined by an element of [○!+]^n_<γ=1>A^γ(V) and an element of [○!+]^n_<γ=1>A^γ(V^*), respectively. Next, any such measure is expressed by a fuzzy measure derived from the map Φ because it is a monotone increasing function for each real variable. Consequently, this Lebesgue integral can be considered as an integral with respect to a fuzzy measure.

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