We consider examples Aλ = A + λ(φ, · )φ of rank one perturbations with φ a cyclic vector for A. We prove that for any bounded measurable set B ⊂ I, an interval, there exists A,φ so that {E ∈ I | some Aλ has E as an eigenvalue} agrees with B up to sets of Lebesgue measure zero. We also show that there exist examples where Aλ has a.c. spectrum [0,1] for all λ, and for sets of λ’s of positive Lebe...

In this state-of-the-art paper, important advances that have been made during the past five years for both general and interval type-2 fuzzy sets and systems are described. Interest in type-2 subjects is worldwide and touches on a broad range of applications and many interesting theoretical topics. The main focus of this paper is on the theoretical topics, with descriptions of what they are, wh...

A <r-complete Boolean algebra is a Boolean algebra in which for every sequence of elements a$-, i = l, • • • , there is an element U?an, the countable union of the a», such that aiQU?an for every i, and such that if diQx for every i then U?anQx. The dual operation, countable intersection, can be introduced through complementation, and the distributive law afMJi*'a» = Uf {aC\an) and its dual can...

Here, instead of taking a σ-algebra as given and then defining a measure on this σ-algebra (namely, on the measurable sets), we take an outer measure as given and then define measurable sets using this outer measure. Carathéodory’s theorem states that the collection M of ν-measurable sets is a σ-algebra and that the restriction of ν to M is a complete measure. Suppose that (X, ρ) is a metric sp...

We prove that if ε ( m stretchy="false">) stretchy="false">→ 0 <...