Abstract
Lattice-valued fuzzy measures are lattice-valued set functions which assign the bottom element of the lattice to the empty set and the top element of the lattice to the entire universe, satisfying the additive properties and the property of monotonicity. In this paper, we use the lattice-valued fuzzy measures and outer measure definitions and generalize the Caratheodory extension theorem for lattice-valued fuzzy measures.
Highlights
Let m1 and m2 be lattice measures defined on the same lattice σ-algebra σ L
With a lattice-valued fuzzy outer measure m∗ having the following properties, we mean an extended lattice-valued set function defined on LX: i m∗ ∅ L0, ii m∗ μ1 ≤ m∗ μ2 for μ1 ≤ μ2, iii m∗
We must show that σ L is a lattice fuzzy σ-algebra containing σ L and m is a lattice-valued fuzzy measure on σ L. We show it step by stepin the following
Summary
Studies including the fuzzy convergence 1 , fuzzy soft multiset theory 2 , lattices of fuzzy objects 3 , on fuzzy soft sets 4 , fuzzy sets, fuzzy S-open and S-closed mappings 5 , the intuitionistic fuzzy normed space of coefficients 6 , set-valued fixed point theorem for generalized contractive mapping on fuzzy metric spaces 7 , the centre of the space of Banach lattice-valued continuous functions on the generalized Alexandroff duplicate 8 , L, M -fuzzy σ-algebras 9 , fuzzy number-valued fuzzy measure and fuzzy number-valued fuzzy measure space 10–12 , construction of a lattice on the completion space of an algebra and an isomorphism to its Caratheodory extension 13 , fuzzy sets 14, , generalized σalgebras and generalized fuzzy measures , generalized fuzzy sets 17–19 , common fixed points theorems for commutating mappings in fuzzy metric spaces , and fuzzy measure theory have been investigated. Abstract and Applied Analysis is more general than that of 24 Using this new definition, we provide new proof of Caratheodory extension theorem for lattice valued-fuzzy measure. Not many studies have been explored including Caratheodory extension theorem on latticevalued fuzzy measure. In 25 , Sahin used the definitions given in 19 and generalized Caratheodory extension theorem for fuzzy sets. No study has been done related to Caratheodory extension theorem for lattice-valued fuzzy measure.
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