On Properties of Pseudointegrals Based on Pseudoaddition Decomposable Measures

and Applied Analysis 3 Let {x n } n≥1 be a sequence from [a, b]. The sequence {x n } n≥1 is said to be convergent, if for any 0 ≺ ε, there exists positive integer N(ε), such that d ⊕ (x n , x) ≺ ε for all n ≥ N(ε), denoted by x = lim n→∞ x n , and x is said to be the limit of the sequence {x n } n≥1 ; lim n→∞ x n = ∞ ∨ ⊕ n=1 (∧ ⊙ k≥n x k ) (6) is said to be the lower limit of the sequence {x n } n≥1 ; lim n→∞ x n = ∞ ∧ ⊙ n=1 (∨ ⊕ k≥n x k ) (7) is said to be the upper limit of the sequence {x n } n≥1 . It is obvious that lim n→∞ x n ⪯ lim n→∞ x n . Let {f n } n≥1 be a sequence from F(X). The sequence {f n } n≥1 is said to be convergent, if for any 0 ≺ ε, and for each point x 0 ∈ X, there exists positive integer N(ε, x 0 ), such that d ⊕ (f n (x 0 ), f(x 0 )) ≺ ε for all n ≥ N(ε, x 0 ), denoted by f = lim n→∞ f n , and f is said to be the limit functional of the functionals sequence {f n } n≥1 . LetA be a subset ofF(X).TheposetA is said to be upper complete if lim n→∞ f n ∈ A for each increasing sequence {f n } n≥1 from A; the poset A is said to be lower complete if lim n→∞ f n ∈ A for each decreasing sequence {f n } n≥1 from A; the poset A is said to be complete if lim n→∞ f n ∈ A for each sequence {f n } n≥1 from A, where the limit of the sequence of functionals {f n } n≥1 is given by (lim n→∞ f n )(x) = lim n→∞ f n (x) for all x ∈ X. For any continuous pseudoaddition ⊕ and x, y ∈ [a, b] with x ⪯ y, there exists at least one point z ∈ [a, b] such that y = x ⊕ z. If pseudoaddition ⊕ is strict, then there exists only one point z ∈ [a, b] such that y = x ⊕ z for all x, y ∈ [a, b] with x ≺ Δ. Thus we have the following concepts. Definition 5 (see [34]). For any continuous pseudoaddition ⊕ and x, y ∈ [a, b] with x ⪯ y, the paracomplement set y− ⊕ x is a nonempty set of all points z such that y = x ⊕ z. Example 6. Let the total order ⪯ on [0, 1] be the usual order of the real line and let the pseudoaddition ⊕ be the usual multiplication of the real numbers. It is obvious that zero element is 1. If x = 0, then y = 0 and y− ⊕ x = [0, 1]. If x ̸ = 0, then for any 0 ≤ y < x, we have y− ⊕ x = {y/x} ⊆ [0, 1]. Definition 7 (see [34]). For any continuous pseudoaddition ⊕, iff, h ∈ F(X), then define the paracomplement set |f− ⊕ h| as the set of all those functionals φ such that φ (x) = { f (x) − ⊕ h (x) , if h (x) ⪯ f (x) , h (x) − ⊕ f (x) , if f (x) ≺ h (x) , (8) for all x ∈ X. Definition 8 (see [34]). For any strict pseudoaddition ⊕ and x, y ∈ [a, b] with x ⪯ y, the complement y−󸀠 ⊕ x is defined as


Introduction
The classical measure theory is one of the most important theories in mathematics [1,2].Although the additive measures are widely used, they do not allow modelling many phenomena involving interaction between criteria.For this reason, the fuzzy measure proposed by Sugeno is an extension of classical measure in which the additivity is replaced by a weaker condition, that is, monotonicity [3,4].Therefore, fuzzy measure and the corresponding integrals, for example, Choquet and Sugeno, are introduced [5][6][7][8][9][10].
So far, there have been many different fuzzy measures, such as the decomposable measure, the -additive measure, the belief measure, the possibility measure, and the plausibility measure.Among the fuzzy measures mentioned before, the decomposable measure was independently introduced by Dubois and Prade [11] and Weber [12].Since the close relations with the classical measure theory, further developments of decomposable measures and related integrals have been extensive [13][14][15][16][17][18].Decomposable measures include several well-known fuzzy measures such as the -additive measure and probability and possibility measures, and they provide a natural setting for relaxing probabilistic assumptions regarding the modeling of uncertainty [19,20].Decomposable measures and the corresponding integrals are very useful in decision theory and the theory of nonlinear differential and integral equations [21][22][23][24].
In many problems with uncertainty as in the theory of probabilistic metric spaces [20,25,26], multivalued logics [27,28], and general measures [1,4] often we work with many operations different from the usual addition and multiplication of reals.Some of them are triangular norms, triangular conorms, pseudoadditions, pseudomultiplications, and so forth [21,29].Based on the above-mentioned measures, pseudoanalysis as a generalization of the classical analysis is developed, where instead of the field of real numbers a semiring is taken on a real interval [, ] ⊂ [−∞, +∞] endowed with pseudoaddition ⊕ and with pseudomultiplication ⊙ (see [13,19,[30][31][32][33]).The families of the pseudooperations generated by a function  turn out to be solutions of wellknown nonlinear functional equations [22][23][24].
In this paper, we will discuss pseudointegrals based on pseudoaddition decomposable measures.In Section 2, we recall the concepts of the pseudoaddition ⊕ and the pseudomultiplication ⊙, which form a real semiring on the interval [, ] ⊂ [−∞, +∞] and the notion of the -⊕decomposable measure.Then we will give the definition of the pseudointegral of a measurable function based on a strict pseudoaddition decomposable measure by generalizing the definition of the pseudointegral of a bounded measurable function.In Section 3, we will discuss several important properties of the pseudointegral of a measurable function based on the strict pseudoaddition decomposable measure.
Let  be a nonempty set; we will denote by S, A, and B  algebra, -algebra, and Borel -algebra of subsets of a set , respectively.
Denote by F() the set of all functionals from  to [, ].For each  ∈ [, ] the constant functional in F() with value  will also be denoted by .It will be clear from the context which usage is intended.A functional  ∈ F() is said to be finite if () ≺ Δ for all  ∈ .The functional  ∈ F() is said to be bounded if there exists Ω ≺ Δ, such that () ⪯ Ω for all  ∈ .Denote by B() the set of all bounded functionals.
Let {  } ≥1 be a sequence from [, ].The sequence {  } ≥1 is said to be convergent, if for any 0 ≺ , there exists positive integer (), such that  ⊕ (  , ) ≺  for all  ≥ (), denoted by  = lim  → ∞   , and  is said to be the limit of the sequence {  } ≥1 ; lim is said to be the lower limit of the sequence is said to be the upper limit of the sequence {  } ≥1 .It is obvious that lim  → ∞   ⪯ lim  → ∞   .Let {  } ≥1 be a sequence from F().The sequence {  } ≥1 is said to be convergent, if for any 0 ≺ , and for each point  0 ∈ , there exists positive integer (,  0 ), such that  ⊕ (  ( 0 ), ( 0 )) ≺  for all  ≥ (,  0 ), denoted by  = lim  → ∞   , and  is said to be the limit functional of the functionals sequence {  } ≥1 .
Let A be a subset of F().The poset A is said to be upper complete if lim  → ∞   ∈ A for each increasing sequence {  } ≥1 from A; the poset A is said to be lower complete if lim  → ∞   ∈ A for each decreasing sequence {  } ≥1 from A; the poset A is said to be complete if lim  → ∞   ∈ A for each sequence {  } ≥1 from A, where the limit of the sequence of functionals {  } ≥1 is given by (lim  → ∞   )() = lim  → ∞   () for all  ∈ .
Definition 5 (see [34]).For any continuous pseudoaddition ⊕ and ,  ∈ [, ] with  ⪯ , the paracomplement set − ⊕  is a nonempty set of all points  such that  =  ⊕ .Example 6.Let the total order ⪯ on [0, 1] be the usual order of the real line and let the pseudoaddition ⊕ be the usual multiplication of the real numbers.It is obvious that zero element is 1.
Definition 10 (see [34]).For any pseudoaddition ⊕, a nonempty subset K of F() is said to be a functional space with respect to ⊕, denoted by where ⊙ is a distributive pseudomultiplication with respect to ⊕.
It is clear that (F(), ⊕) is the greatest functional space with respect to any pseudoaddition ⊕.Thus the functional space (K, ⊕) with K ⊆ F() is also called a subspace of (F(), ⊕).If (K, ⊕) is a functional space with respect to ⊕, then we just write K instead of (K, ⊕) whenever ⊕ can be determined from the context.Definition 11 (see [34]).For each subset A of F() the upper closure of A, denoted by Â, is the set of all elements of F() having the form lim  → ∞   for some increasing sequence {  } ≥1 from A.
Definition 13 (see [34]).For any continuous pseudoaddition ⊕, a paracomplemented subspace (K, ⊕) is regular if it contains 1 and is closed under ∨ ⊕ ; for any strict pseudoaddition ⊕, a complemented subspace (K, ⊕) is normal if it contains 1 and is closed under It is obvious that regular and normal are closed under ∧ ⊙ .Definition 14 (see [37]).The pseudocharacteristic function of a set  ⊆  is defined with where 0 is zero element for ⊕ and 1 is unit element for ⊙.
Definition 15 (see [21]).A functional  ∈ F() is said to be elementary if it has the following representation: for each   ∈ [, ] and   ∈ A pairwise disjoint and with  = ⋃  =1   , and the set of such elementary functionals will be denoted by E().It is obvious that   ∈ E(), for all  ⊆ .
Definition 16 (see [21]).A set function  : A → [, ] (or semiclosed interval) is called a -⊕-decomposable measure if it satisfies the following conditions: (1) (0) = 0; (2) () ⪯ () for all ,  ∈ A with  ⊂ ; (3) (∪) = ()⊕() for all ,  ∈ A and ∩ = 0; A pair (, A) consisting of a nonempty set  and a -algebra of subsets of  is called a measurable space.A functional  :  → [,] is said to be a measurable functional if Then E(S) will denote the set of those elements  ∈ E() for which In particular, this means that E(A) = M(A)∩E().Denote by B(A) the set of all bounded measurable functionals.
Let {  } ≥1 be a sequence of measurable functionals of a.e.pseudofinite on .If there exists a measurable functional  of a.e.pseudofinite on , such that lim for arbitrary 0 ≺  ≺ Δ, then the functionals sequence {  } ≥1 is said to be convergent to  with respect to ⊕-measure, denoted by   ⇒ .If the functionals sequence {  } ≥1 does not converge to  with respect to ⊕-measure, denote by     .

Main Results
Lemma 20 (see [21]).Let ⊕ be a continuous pseudoaddition and  : A → [, ] a -⊕-decomposable measure.If () ≺ Δ, then for all , ℎ ∈ B(A), we have } ( = 1, 2) be two different ⊕-measure finite and monotone covers of  and let { ()   } ( = 1, 2) be two different positive integer sequences with ⊙ } is an increasing sequence, we have for every positive integer .Let  ∈ A with () ≺ Δ and  is an arbitrary positive integer.If  (1)   > , then we have Since { −  (1)   } is a decreasing sequence and by Theorem 3.3 in [38], we have lim which implies that In particular, let  =  (2)   and  =  (2)   .Then we have for every positive integer .Hence, we get that lim On the contrary, using a similar argument, we can obtain lim In Theorem 21, put  (1)    =  and  (2)

𝑙
= .Then we can easily see that the pseudointegral in Definition 19 has a unique value.In particular, we can get some elementary properties of the pseudointegral in the following theorem.
Proof.If {  } is an increasing sequence, then lim If {  } is a decreasing sequence, then lim Thus, we have lim Hence, by Corollary 27, we get that which implies that Consequently, we obtain that lim Since  1 = ( 1 −  ⊕   ) ⊕   and ⊕ is continuous, we have which implies that lim By (3) of Theorem 22, we have Thus, we get that Theorem 31.Let ⊕ be a strict pseudoaddition, and let  be a -finite set of ⊕-measure and  : A → [, ] a -⊕decomposable measure.If {  } is a sequence of measurable functionals on , then Proof.Let ℎ  = ⊕  =1   ,  = 1, 2, . ... Then {ℎ  } is an increasing sequence of measurable functionals on .By Theorem 29, we have lim By (3) of Theorem 22, we have then  = ⊕ ∞ =1   and By Theorem 31, we have Hence, we obtain that    ⊙ . (115)

Conclusions
In this paper, we mainly discussed pseudointegral based on pseudoaddition decomposable measure.Particularly, we have given the definition of the pseudointegral of a measurable function based on a strict pseudoaddition decomposable measure by generalizing the definition of the pseudointegral of a bounded measurable function.Furthermore, we have derived several important properties of the pseudointegral of a measurable function based on strict pseudoaddition decomposable measure.Finally, we have obtained that some theorems on the integral and the limit can be exchanged.
Recently, pseudoanalysis has obtained rapid development in the mechanical, chemical, biological, medical, and computer fields and has solved some uncertainty problems of knowledge.Pseudoanalysis theory has important applications in the field of computer image processing [39,40]; for example, it can analyze and grasp the variation range of the image gray value, solve the relationship between the grey value and image color change, and take appropriate grey value to achieve better image processing effect.With the development of computer technology, pseudoanalysis will also get more and more widely used in computer science.We also hope that our results in this paper may lead to significant, new, and innovative results in other related fields.

Corollary 27 .
If the condition (3) of Theorem 26 is replaced by   →  a.e. on , then the conclusion of Theorem 26 holds.

)
[38]rem 30.Let ⊕ be a strict pseudoaddition, and let  be a -finite set of ⊕-measure and  : A → [, ] a -⊕decomposable measure.If {  } is a decreasing sequence of finite measurable functionals and pseudointegral of  1 is finite on , then Proof.Let {  } be a decreasing sequence of measurable functionals on .By Lemma 28, we get that the sequence of measurable functionals {  } is convergent.Let  = lim  → ∞   .By Theorem 3.5 in[38], we have  ∈ M(A).Since { 1 −  ⊕   } is an increasing sequence of measurable functionals, by Theorem 29, we have lim

)
Theorem 32.Let ⊕ be a strict pseudoaddition, and let  be a -finite set of ⊕-measure and  : A → [, ] a -⊕decomposable measure.If  is a measurable functional on , Then {ℎ  } is an increasing sequence of measurable functionals on .By proof of Theorem 29, we have lim Theorem 33.Let ⊕ be a strict pseudoaddition, and let  be a -finite set of ⊕-measure and  : A → [, ] a -⊕decomposable measure.If {  } is a sequence of measurable functionals on , then Proof.Let ℎ  = ∧ ⊙ ∞ =   ,  = 1, 2, . ...