On a New Kind of Collection of Subsets Noted by δ–field and Some Concepts Defined on δ–field

Abstract

The objective of this paper is, first, study a new collection of sets such as field and we discuss the properties of this collection. Second, introduce a new concepts related to the field such as measure on field, outer measure on field and we obtain some important results deals with these concepts. Third, introduce the concept of null-additive on field as a generalization of the concept of measure on field. Furthermore, we establish new concept related to - field noted by weakly null-additive on field as a generalizations of the concepts of measure on and null-additive. Finally, we introduce the restriction of a set function on field and many of its properties and characterizations are given.

Some Properties for the Restriction of P^*-field of Sets

“…ϵℳ wheneverB 1 , B 2 , … ϵℳ . Many authors were interested in studying σ-field and σring; for example, see [6], [7], and [8]. In this work, we denote a universal set by 𝒰.…”

λ–Algebra with Some of Their Properties

“…And a collection 𝒦 is said to be β-σ-field if Φ, 𝔓ϵ 𝒦 and 𝒦 is closed under countable intersection [7]. Ibrahim and Hassan in 2019 also introduced the concept of δ-field as a stronger form of these concepts, where a collection 𝒦 is said to δ-field iff Φϵ 𝒦 and if Φ Aϵ𝒦 andA ⊂ B ⊆ 𝔓, then Bϵ𝒦and 𝒦 is closed under countable intersection [8]. The concept of complete measure on σ-field was studied by Robert in 1972, but not necessarily that every measure defined on σ-field is complete.…”