On a New Kind of Collection of Subsets Noted by Î´â€“field and Some Concepts Defined on Î´â€“field

Hassan H Ebrahim,Ibrahim S Ahmed

doi:10.30526/32.2.2140

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.

Highlights

The theory of measure is an important subject in mathematics
The notion of –field was studied by Robret and Dietmar, where be a nonempty set
The concept of monotone measure was studied by Peipe, Minhao and Jun in 2018, where be a –field, a set function

Summary

Introduction

The theory of measure is an important subject in mathematics. In 1972, Robret [1], discusses many details about measure and proves some important results in measure theory. In 2016, Juha used the concept of –field to define measure, where be a –field, a measure on is a set function. Used power set to define outer measure, where be a non-empty set, a set function ( ) , - is called outer measure, if ( ). The concept of monotone measure was studied by Peipe, Minhao and Jun in 2018, where be a –field, a set function. 2. The Main Results Let be a nonempty set. A collection ( ) is said to be – of a set if the following conditions are satisfied: 1-.

Then there exist

Let and

Proof Let

Conclusions