On Soft Lebesgue Measure

In this article, we introduce the concept of soft intervals, soft ordering and sequences of soft real numbers and study their properties and some interesting results. Also the notion of soft Lebesgue measure on the soft real numbers has been introduced. A correspondence relationship between the soft Lebesgue measure and the Lebesgue measure has been established.


Introduction
Classical mathematical methods are not enough to solve the problems of daily life and also are not enough to meet the new requirements. Therefore presence of uncertainty is one issue which arises in many scientific discipline including our day to day problems. In order to reduce and retrieve information from the uncertainty, there are several theories have been developing and few of established theories such as vague sets (Gau and Buehrer 1993), fuzzy sets (Zadeh 1965) and rough sets (Pawlak 1982) are made. These approaches were regarded as the most famous mathematical instruments in decision modelling. However, these approaches have their own limitations due to the inadequacy of parameters. Fuzzy set theory has been generalized into soft set theory which is now one of the important branches of modern mathematics. It provides tool to administer various types of uncertainties arising in diverse problems in economics, environmental sciences, sociology etc. Molodtsov [44](1999) proposed the soft set theory considering ample enough parameters to direct uncertainties. Accordingly, problems with uncertainties becoming easier to tackle using the theory of soft sets. Later, Maji et al. [39] defined operations on soft sets in 2003 and studied the nature of soft sets. Molodtsov et al. (2006) [45], he applied successfully in directions such as smoothness of functions, game theory, operations research, Riemann-Integration, Perron integration, probability and theory of measurement. The first practical application of soft set in decision making problems is presented by Maji et al. [40]. Ali et al. [7] gave some new notions such as restricted intersection, restricted union, restricted difference, and extended intersection of soft sets. Jun [31] applied Molodtsov's notion of soft sets to the theory of BCK/BCI-algebras and introduced the notion of soft BCK/BCIalgebras and soft subalgebras and then investigated their basic properties. Jun and Park [32] dealt with the algebraic structure of BCK/BCIalgebras by applying soft set theory. They introduced the notion of soft ideals and idealistic soft BCK/BCI-algebras and gave several examples. Jun et al. [34] introduced the notion of soft p-ideals and p-idealistic soft BCI-algebras and investigated their basic properties. In addition to the theory by Molodtsov [44] and Maji et al. [39] several other researchers studied the nature of soft sets and its applications in various real life problems. For example, [20], [17], [33], [35], [37], [40], [49].
In 2007, Aktas and Cagman [4] first introduced the notion of a soft group. It is worth mentioning that Aktas and Cagman introduced the definition of soft group over the soft set defined by Molodtsov [44]. The study of Aktas and Cagman [4] includes soft subgroups, normal soft subgroups and soft homomorphisms. Sezgin et al. [53] corrected some of the problematic cases in a previous paper by Aktas and Cagman [4], and introduced the concept of normalistic soft groups and the homomorphism of normalistic soft groups, studied their several related properties, and investigated some structures that are preserved under normalistic soft group homomorphisms. Sezgin and Atagün [54] proved that certain De Morgan's laws hold in soft set theory with respect to different operations and extended the theoretical aspect of operations on soft sets. The algebraic structure of set theories dealing with uncertainties has also been studied by various mathematicians. Wen [65], Yuan Xuehai's graduate student, presented the new definitions of soft subgroups and normal soft subgroups and obtained some primary results.
And consequently several other researchers ( [5], [46], [48]) have extended the idea of soft group following the definition of soft group by Aktas and Cagman. Most of the papers on soft group, devoted to present the definition and properties of the soft groups analogous to that of ordinary groups. In 2008, Feng et al. [19] introduced the notation of soft semirings, soft ideals and idealistic soft semirings and investigated several related properties. Also Soft rings and soft ideals are defined by U.Acar, F.Koyuncu, B.Tanay [2] in (2010) and discussed their basic properties. Since then some researchers Y.B.Jun [31] and Celik ea al. [16] have studied other soft algebraic structures as well as their properties.
In 2011, N.Cagman, S.Karatas, S.Enginogiu introduced soft topology in [15] and M.Shabir, M.Naz defined soft topological spaces in [55]. They defined basic notions of soft topological spaces such as soft open and soft closed sets, soft subspace, soft closure, soft neighborhood of a point, soft T i -spaces, for i = 1; 2; 3; 4, soft regular spaces, soft normal spaces and established their several properties. In 2011, S. Hussain and B. Ahmad [29] continued investigating the properties of soft open(closed), soft neighborhood and soft closure. They also defined and discussed the properties of soft interior, soft exterior and soft boundary. Also in 2012, B. Ahmad and S. Hussain [3] explored the structures of soft topology using soft points. A. Kharral and B. Ahmad [36], defined and discussed the several properties of soft images and soft inverse images of soft sets. They also applied these notions to the problem of medical diagnosis in medical systems. Aygunoglu and Aygun [8] introduced the soft continuity of soft mapping, soft product topology and studied soft compactness and generalized Tychono theorem to the soft topological spaces. Min [43] gave some results on soft topological spaces. Zorlutuna et al. [68] also investigated soft interior point and soft neighbourhood. There are several literature available on the structure of soft topological spaces [22], [23], [41], [48], [64], [6], [9], [10], [14], [22], [23], [27], [28], [30], [41], [55], [56], [58] [60], [63], [64] In this article, we have introduced soft real numbers as real functions of real variable. So these classical functions behave like soft elements and their collection behave like soft set. Now on this soft set a soft topology is defined. This topology is good enough to study the properties of soft Lebesgue measure, which is also introduced and studied. To the best of our knowledge, measure on soft sets is defined here is first of its kind. The introduction of classical measure theory can be referred [69], [70], [71].

Preliminaries
Let X be an universal set, A be parametric set and P (X) be the power set of X.

Definition 2.4. [39] Let F and H be two soft sets over an universal set
Definition 2.6. [39] Two soft sets F and H are said to be soft equal if F (t) = H(t) for all t ∈ A and it is denoted by F = s H. Also a, b ∈ s F are said to be soft equal if a(t) = b(t) for all t ∈ A.

Soft Real Sets, Soft Real Numbers and Some Definitions
From this section, we take [0, 1] as a parametric set and R as a universal set. Also we have introduced the notation of collection of all soft real numbers, collection of all soft real set, soft interval and investigated its structural properties. Now, we define the soft real number, soft real set and soft intervals on the universal set R as follows.
So δ A ∈ S(R) and we write A for δ A . If A ∈ S(R) then α ∈ s A implies α(t) ∈ A(t) for all t ∈ [0, 1]. Hence α : [0, 1] → R and so α ∈ N (R). Therefore A ⊂ N (R).   Proof. Let 1 and 2 be any two distinct elements in N (R).

Sequences of Soft Real Numbers
In this section, we have studied sequence of soft real numbers and investigated some interesting results on it.  {x n } is said to be convergent to a soft real number l if for all > 0 there exists N ∈ N such that x n ∈ (l c , l ⊕ c ) s for all n ≥ N . We say that l is a soft limit of the sequence of soft real numbers {x n } and defined by x n → s l.  Proof. Let {x n } converges to a soft real number l so for the c (t) = for all t ∈ [0, 1], ( > 0) there exists N ∈ N such that x n ∈ (l c , l ⊕ c ) s for all n ≥ N . So x n (t) ∈ (l c , l ⊕ c ) s (t) for all t ∈ [0, 1] and for all n ≥ N which implies that x n (t) ∈ (l(t) − , l(t) + ) for all t ∈ [0, 1] and for all n ≥ N . Hence x n (t) → l(t) as n → ∞. For the converse part, the proof is easy and follows from Definition 4.2. Theorem 4.6. Let {x n } and {y n } be two sequences in N (R) converges to the soft real numbers x and y respectively, then (1) x n ⊕ y n → s x ⊕ y (2) x n y n → s x y (3) x n y n → s x y (4) x n y n → s x y Proof. The proof of the theorem follows from Theorem 4.4.
In this section, we have introduced the notation of soft Lebesgue measure and studied its analogous properties. Lemma 5.2. Let A ⊂ N (R) and {(a n , b n ) s : n ∈ I} be a soft open cover of A then {(a n , b n ) s (t)} = {(a n ∧ b n )(t), (a n ∨ b n )(t)} be a cover of A(t) for all t ∈ [0, 1].
Proof. Let α ∈ A(t), so there exists y ∈ s A such that y(t) = α. Since {(a n , b n ) s : n ∈ I} is a soft cover of A then y ∈ s (a n , b n ) s for some n. Therefore α = y(t) ∈ (a n , b n ) s (t) = ((a n ∧ b n )(t), (a n ∨ b n )(t)). Now since α ∈ A(t) is arbitrary then {(a n , b n ) s (t)} = {(a n ∧ b n )(t), (a n ∨ b n )(t)} is a cover of A(t) for all t ∈ [0, 1].
(a n ∨ b n a n ∧ b n ) s :{(a n , b n ) s } is a collection of soft open intervals which cover A}.
Note 5.4. As (a n ∨b n a n ∧b n ) is a function from [0, 1] to R, the summation is well defined and is a soft real number. So the infimum is taken over a set of soft real numbers. Since this set is uniformly bounded by zero, the infimum exists. Hence µ * s : S(R) → [0 s , ∞ s ) s .
Proof. Let t 0 ∈ [0, 1] be fixed. Let {(α n , β n )} be a cover of A(t 0 ) and {(a n , b n ) s } be a soft open cover of A. For each n ∈ N define soft real numbers c n and d n by Then clearly {(c n , d n ) s } be a collection of soft intervals and is a soft cover of Taking infimum of all such collection {(α n , β n )} that cover A(t 0 ), we get µ * s (A)(t 0 ) ≤ µ * (A(t 0 )). Since t 0 ∈ [0, 1] is arbitrary, we have µ * s (A)(t) ≤ µ * (A(t)) for all t ∈ [0, 1]. Now suppose {(a n , b n ) s } be a soft open cover of A then {(a n ∧ b n )(t), (a n ∨ b n )(t)} be a cover of A(t) for all t ∈ [0, 1]. So µ * (A(t)) ≤ ( ∞ n=1 (a n ∨ b n a n ∧ b n ))(t) for all t ∈ [0, 1]. Consider a soft real number s such that (a n ∨ b n a n ∧ b n ). Taking infimum of all collection of soft open intervals {(a n , b n ) s } that cover the soft real set A, we conclude that s ≤ s µ * s (A) i.e., µ * (A(t)) ≤ µ * s (A)(t) for all t ∈ [0, 1]. This complete the proofs.
Proof. The proof follows from Theorem 5.5.
Definition 5.8. Let A ⊂ N (R). Then A is said to be soft Lebegue measurable if for each A 1 ⊆ N (R) such that µ * s (A 1 ) = s µ * s (A ∩ A 1 ) + µ * s (A c ∩ A 1 ). This definition shows that if a soft set A is soft Lebesgue measurable then A(t) is lebesgue measurable for all t ∈ [0, 1] and the converse is also true. So the next theorems are trivial.
Theorem 5.9. (i) Φ and N (R) are both soft measurable and A is soft measurable implies A c is also soft measurable. (ii) Let A ⊂ N (R). If µ * s (A) = s 0 s then A is soft measurable. (iii) Let x 0 ∈ N (R) and A ⊂ N (R). If A is soft measurable then A ⊕ x 0 is also soft measurable. (iv) Every soft interval is soft measurable.
(v) If {A i } is a sequence of soft measurable sets in N (R) then

Ethical approval:
This article does not contain any studies with human participants or animals performed by any of the authors.

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The authors do not receive any funding towards the present study.

Conflict of Interest:
We declare that there is no conflict of interest.