Sets Having Finite Fuzzy Measure in Real Hilbert Spaces

Abstract

A new type of translation invariant and lower semi continuous fuzzy measure on the class of subsets of a real Hilbert space is introduced. It measures a subset of the Hilbert space as a projection of the set along a fixed vector in the Hilbert space. It is proved that corresponding to each subset of the Hilbert space, the fuzzy measure is determined by one vector of the Hilbert space. Then it is proved that the fuzzy measure of a closed convex subset of the Hilbert space can be obtained in terms of two elements of the subset itself. It is also proved that this fuzzy measure satisfies a condition similar to the null additivity.