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.

The objective of this paper is, first, to introduce and study the concept of α–field as a generalization of field, σ–field and δ–field, and we discuss the properties of this concept. Furthermore, we study the relationships between σ–field and σ–field. As a first σ–field is α–field. second, to introduce the concept of β–field as a generalization of σ–field, β–σ–field and ring. So, we prove that every σ–field is β–field and we obtain some important results deals with this concept. Finally, we introduce and study the concept of restriction of β– field and we prove that, if ℘ is a β– field of a set ℵ and K is a non-empty subsets of ℵ. Then ℘|K is a β–field of a set K.

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The objective of this paper is, firstly, we study a new concept noted by algebra and discuss the properties of this concept. Secondly, we introduce a new concept related to the algebra such as smallest algebra. Thirdly, we introduce the notion of the restriction of algebra on a nonempty subset  of and investigate some of its basic properties. Furthermore, we present the relationships between field, monotone class, field and algebra. Finally, we introduce the concept of measure relative to the algebra and prove that every measure relative to the   is complete.