International Journal of Approximate Reasoning

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1 JID:IJ ID:808 /FL [mg; v.; Prn:4/08/20; :2] P. (-2) International Journal of pproximate Reasoning ( ) Contents lists available at ScienceDirect International Journal of pproximate Reasoning On linearity of -integral and -integrable functions space 4 4 Yao Ouyang a, Jun Li b,, Radko Mesiar c,d a Faculty of Science, Huzhou University, Huzhou, Zheiang 000, China 8 b School of Sciences, Communication University of China, Beiing 00024, China 8 c Slovak University of Technology in Bratislava, Faculty of Civil Engineering, Radlinského, Bratislava, Slovakia d UTI CS, Pod Vodárenskou věží 4, Prague, Czech Republic a r t i c l e i n f o a b s t r a c t 24 rticle history: This paper investigates the linearity and integrability of the (+, )-based -integrals on Received 4 June 20 subadditive monotone measure spaces. It is shown that all nonnegative -integrable 25 Received in revised form 8 ugust 20 functions form a convex cone and the restriction of the -integral to the convex cone ccepted 8 ugust 20 2 is a positive homogeneous linear functional. We extend the -integral to the general 2 vailable online xxxx 28 real-valued measurable functions. The generalized -integrals are shown to be symmetric and fully homogeneous, and to remain additive for all -integrable functions. Thus for Keywords: 0 Monotone measure a subadditive monotone measure the generalized -integral is linear functional defined 0 Subadditivity on the linear space which consists of all -integrable functions. We define a p-norm Pan-integral on the linear space consisting of all p-th order -integrable functions, and when the Linearity monotone measure μ is continuous we obtain a complete normed linear space L p (X, μ) Pan-integrable space equipped with the p-norm, i.e., an analogue of classical Lebesgue space L p. 4 L p space 4 20 Published by Elsevier Inc Introduction 4 In nonlinear integral theory there are three types of important integrals, the Choquet integral [2], the -integral (based 4 on the usual addition + and multiplication on reals) [] and the concave integral [2,]. These integrals coincide with 4 the Lebesgue integral in all cases for σ -additive measures, i.e., these three nonlinear integrals (with respect to monotone 4 44 measures) are particular generalizations of the Lebesgue integral. In general, they are significantly different from each other, the Choquet integral deals with finite chains of sets, the -integral is based on finite partitions (disoint set systems, similar to the Lebesgue integral) while the concave integral is related to arbitrary finite set systems, see [2]. These integrals 46 4 have been widely studied and many important results have been obtained, for more details see [,4,8,5,2,28,,5], etc., 4 48 and the structure theory of integral has been enriched and developed in depth, see [,0,,,8,2] It is well-known that L p space theory is a crucial aspect of classical measure theory [,5,6]. Since the above mentioned 4 50 three integrals are based on monotone measures and lack additivity, in general case the L p space theory is not true for 50 5 these integrals. Denneberg [] showed a subadditivity theorem for the Choquet integral under the assumption that the 5 52 monotone measure is submodular. The corresponding L p space theory for the Choquet integral was developed. The similar 52 5 results were presented by Shirali [0], and the related researches were presented in recent paper by Pap [2]. Note that the 5 54 Choquet integral is a level set-based integral, which ensures its comonotonic additivity a property weaker than additivity. 54 * 5 Corresponding author. 5 addresses: oyy@zhu.edu.cn (Y. Ouyang), liun@cuc.edu.cn (J. Li), radko.mesiar@stuba.sk (R. Mesiar) X/ 20 Published by Elsevier Inc.

2 JID:IJ ID:808 /FL [mg; v.; Prn:4/08/20; :2] P.2 (-2) 2 Y. Ouyang et al. / International Journal of pproximate Reasoning ( ) In [] the linearity and generalized linearity of fuzzy integral (including the Choquet integral and the Sugeno integral) 2 were discussed (see also []). Unlike the Choquet integral, the -integral is a partition-based integral. The relationships 2 between these two types of integrals were discussed in [6,25,] (see also [5]). s a result, the -integral even lacks 4 comonotonic additivity. From this point of view, it requires to examine whether or not the L p space theory holds for the 4 5 -integral. In this paper we will investigate this problem. 5 6 The concept of -integral introduced by Yang ([5,]) involves two binary operations, the -addition and multiplication 6 of real numbers, i.e., it considers the commutative isotonic semiring (R +,, ) (see [4,2,2,5,6]). 8 related concept of generalized Lebesgue integral based on a generalized ring was proposed and discussed (see []). In 8 this paper we only consider the -integrals based on the standard addition + and multiplication. 0 This paper is structured as follows. fter this introduction, in the next section we recall some basic facts about the monotone 0 measures and the -integrals. In Sections and 4 we consider the -integral for nonnegative measurable functions. We show that on a subadditive monotone measure space (X,, μ) the -integral with respect to μ is additive. Noting that the -integral based on the standard addition + and multiplication is positively homogeneous, then all nonnegative 4 -integrable functions form a convex cone in the usual sense and thus the restriction of the -integral to the convex 4 5 cone is a positive homogeneous linear functional. In Section 5, in the same way as the symmetric Choquet integral ([,2]) 5 6 we extend the -integral for nonnegative measurable functions to the class of general real-valued measurable functions 6 (not necessarily nonnegative). We get a type of symmetric and fully homogeneous nonlinear integral. For a subadditive 8 monotone measure μ the generalized -integral (with respect to μ) remains additive for all -integrable functions, and 8 hence it is a linear functional defined on the linear space consisting of all -integrable functions. In Section 6, we show 20 that all p-th order -integrable functions form a linear space under the conditions that the monotone measure is subadditive 20 2 and continuous from below. On such the linear space, by using the Minkowski type inequality for -integral [8], 2 we can define a p-norm in the usual manner as Lebesgue space L p and then we obtain a complete normed linear space 2 L p (X, μ) equipped with the p-norm, i.e., it is analogous to Lebesgue space L p. Thus, in the framework of the generalized integral the classical L p space is generalized Preliminaries Let X be a nonempty set and a σ -algebra of subsets of X. set function μ : [0, + ] is called a monotone measure [5] on (X, ), if it satisfies the following conditions: () μ( ) = 0 and μ(x) > 0; (2) μ() μ(b) whenever B and, B. When μ is a monotone measure, the triple (X,, μ) is called a monotone measure space ([2,5]). Note: In the literature, the monotone measure is also known as a monotone set function, a capacity, a fuzzy measure, or a nonadditive probability (constrained by μ(x) =, sometimes also assuming the continuity of μ), etc. (see [2,,5,,2,,4,6]). 6 6 In this paper we always assume that μ is a monotone measure on (X, ). Recall that μ is said to be (i) subadditive, if μ( B) μ() + μ(b) for any, B ; (ii) submodular, if μ( B) + μ( B) μ() + μ(b) for any, B ; (iii) supermodular, if μ( B) + μ( B) μ() + μ(b) for any, B ; (iv) null-additive [5], if for any, B, μ(b) = 0implies μ( B) = μ(); (v) continuous from below, if for any { n } n=, 2... implies μ( n= n) = lim n μ( n ); 42 4 (vi) continuous from above, if for any { n } n=, 2... and μ( ) < imply μ( n= n) = lim n μ( n ); 4 44 (vii) continuous if it is continuous both from below and from above Obviously, the submodularity of μ implies the subadditivity and both imply the null-additivity, but not vice versa In [] (see also [2,5,]) the concept of -integral was introduced, which involves two binary operations, the - 46 addition and -multiplication of real numbers (see [20,2,2,5]). In this paper we only consider the -integrals based on the standard addition + and multiplication. 48 real-valued function f : X (, + ) is said to be -measurable on (X, ) (or simply measurable when there is no confusion) if f (B) for every Borel set B of real numbers. Let F + be the collection of all nonnegative real-valued 50 measurable functions on (X, ). We recall the following definition. 5 Definition 2.. Let (X,, μ) be a monotone measure space, and f F +. The -integral of f on with respect 5 54 to μ, is defined by 54 { [ fdμ = sup ( inf f (x)) μ( E) ]}, (2.) x E E ˆP E E where ˆP is the set of all finite measurable partitions of X. When = X, 60 X fdμ is written as fdμ. If fdμ <, 6 then we say that f is -integrable on. When f is -integrable on X, we simply say that f is -integrable. 6

3 JID:IJ ID:808 /FL [mg; v.; Prn:4/08/20; :2] P. (-2) Y. Ouyang et al. / International Journal of pproximate Reasoning ( ) Note: By a finite measurable partition of X we mean that it is a finite disoint system of sets { i } n such that i= 2 i = for i and n 2 i= i = X. The above -integral of f can be expressed in the following form: } 5 { fdμ = sup λ i μ( i ) : λ i χ i f, { i } n i= is a partition of X,λ i 0,n N, i= i= where χ i is the characteristic function of i. Let f, g be real-valued measurable function on (X,, μ). We say that f and g are equal almost everywhere with respect to μ on, and denoted by f = g μ-a.e. on (simply, f = g a.e. on ), if μ({x : f (x) g(x)}) = We present some basic properties of -integrals, some of them can be found in [5] (see also []). 2 4 Proposition 2.2. Let (X,, μ) be a monotone measure space and f, g F +. Then we have the following 4 5 (i) if μ() = 0 then fdμ = 0; 5 6 (ii) if f = 0 a.e. on, then fdμ = 0. Further, if μ is continuous from below, then fdμ = 0 if and only if f = 0 a.e. on ; 6 (iii) if f gon, then fdμ gdμ; (iv) for, fdμ = f χ dμ; (v) fdμ = 20 { f >0} fdμ, where { f > 0} ={x X : f (x) > 0}; 20 (vi) if μ is null-additive and f = ga.e. on, then fdμ = Proof. We only prove (vi). Let ={x, f (x) g(x)}, then μ( ) = 0 and for any subset B of we have μ(b) = 28 μ(b \ ). Thus, for any n i= λ iχ Bi f (x) χ, n i= λ iχ Bi \ g(x) χ. Moreover, 28 gdμ λ i μ(b i \ ) = λ i μ(b i ), 2 i= i= 2 4 which implies that gdμ fdμ. In a similar way fdμ gdμ, and thus (vi) holds. 4 6 Note 2.. () When μ is continuous from below, from the above proposition (ii), we know that for any measurable functions 6 f, g F + and any real number p > 0, f g p dμ = 0if and only if f = g a.e. on. 8 (2) Due to the above proposition (v), we will not distinguish the partition of X and the partition of { f > 0} when we 8 discuss the -integral of the function f. Sometimes we may ust say that { i } n is a partition. i= () By (vi), if μ is null-additive then, from the -integral point of view, we do not distinguish f and g whenever f = g 4 a.e. on X The additivity of the -integral for nonnegative measurable functions The main task of this section is to prove the following result Theorem.. Let (X,, μ) be a monotone measure space. If μ is subadditive, then the -integral is additive w.r.t. the integrands, i.e., for any f, g F +, 4 4 ( f + g)dμ = gdμ Proof. Here we only consider the case that f + g is -integrable, that is, ( f + g)dμ is finite. The case of ( f + g)dμ = will be considered in the next section. For an arbitrary but fixed ε > 0, there exist a partition { i } n 5 i= and a sequence of nonnegative numbers {λ i} n such that i= 5 both n i= λ iχ i f + g and 5 5 ( f + g) dμ < λ i μ( i ) + ε 2 i= (.)

4 JID:IJ ID:808 /FL [mg; v.; Prn:4/08/20; :2] P.4 (-2) 4 Y. Ouyang et al. / International Journal of pproximate Reasoning ( ) hold. Without loss of generality, we can assume that λ i > 0for all i (otherwise, the summand λ i μ( i ) can be omitted). 2 Since μ( ) < and λ > 0, there exists a positive number l <λ such that (λ l )μ( ) < ε 2 2. Obviously, l χ 2 < λ χ ( f + g) χ. Let δ = λ l and divide the interval [0, l ] as = r (0) < r () <...<r (m ) = l such that max i m (r (i) r(i ) ) <δ. Denote { } (k) = x r (k ) f (x)<r (k), k =,...,m, 0 0 and 2 (m +) = {x f (x) l }. 2 4 Then (i) ) ( = (i ) and m + (i) i= =. Moreover, we conclude that for each x (k), k =, 2,..., m, g(x) 4 5 l 5 r (k ). In fact, if g(x) < l r (k ), then f (x) + g(x)<r (k) + (l r (k ) ) = l + (r (k) r (k ) )<l + δ = λ, which contradicts with the fact that λ χ ( f + g) χ. For x (m +), g(x) 0 = l r (m ) also holds. Denote 2 t (k) 2 = l r (k ), k =,...,m +. Then m r (k ) 25 χ (k) f χ 25 k= 2 and 2 m + t (k) χ 0 (k) g χ. 0 k= 2 Observe that the subadditivity of μ implies 2 m + m μ( 5 ) = μ (k) ( ) μ (k). 5 6 k= k= 6 Thus m + ( ) m + 40 r (k ) μ (k) ( ) + t (k) μ (k) 40 4 k= k= 4 ( ) ( ) 42 m = r (k ) + t (k) μ (k) 4 44 k= m + ( ) = l μ (k) 4 l μ( ) 4 48 k= 48 >λ μ( ) ε For each i n, let l i (0, λ i ) be such that (λ i l i )μ( i ) < ε 5 2i+. By using the technique used above, we can prove that for each n there exist a partition { (k) 5 } m + of k= and a sequence {r (k ) } m + of nonnegative numbers satisfying k= 5 54 m + 54 r (k ) χ (k) f χ k= and 5 m + 5 t (k) χ 6 (k) g χ, 6 k=

5 JID:IJ ID:808 /FL [mg; v.; Prn:4/08/20; :2] P.5 (-2) Y. Ouyang et al. / International Journal of pproximate Reasoning ( ) 5 where t (k) = l r (k ), such that m + ( ) m + 4 r (k ) μ (k) ( ) + t (k) μ (k) >λ 4 μ( ) ε k= k= 5 Thus, we have proven that 8 m 8 + r (k ) χ 0 (k) f χ f, 0 = k= = 2 m + 2 t (k) χ 4 (k) g χ g, 4 = k= = 6 and gdμ 2 m + ( ) m + 2 r (k ) μ (k) ( ) + t (k) μ (k) 2 = k= = k= 2 > ( λ μ( ) ε ) 2 + = > λ μ( ) ε 2 0 = 0 > ( f + g)dμ ε. 5 Letting ε 0, we get fdμ + gdμ ( f + g)dμ as desired. 5 6 On the other hand, for any n α i= iχ i f and m = β χ B g, where α i, β 0, { i } and {B } are two partitions of X, 6 put C i = i B (C i = may hold for some i, ) and γ i = α i + β, i n, m. Then m γ i χ Ci = m m α i χ Ci + β χ Ci i= = i= = = i= 4 4 = m α i χ i + β χ B f + g i= = and ( f + g)dμ m γ i μ(c i ) 4 4 i= = = m m α i μ(c i ) + β μ(c i ) 5 i= = = i= 5 ( m ) m ( n ) α i μ C i + β μ C i i= = = i= = m α i μ( i ) + β μ(b ), 5 5 i= = 6 from which we can get that 6

6 JID:IJ ID:808 /FL [mg; v.; Prn:4/08/20; :2] P.6 (-2) 6 Y. Ouyang et al. / International Journal of pproximate Reasoning ( ) ( f + g)dμ 4 Hence it holds that ( f + g)dμ = fdμ + 4 Note.2. Theorem. tells us that if μ is subadditive then all nonnegative -integrable functions form a convex cone. Corollary.. Let (X,, μ) be a monotone measure space. If μ is subadditive, then for any f F + and any, B with B =, 0 we have 0 fdμ = fdμ. 4 B B 4 6 Proof. Denote g = f χ and h = f χ B. Then, by (iv) of Proposition 2.2 and Theorem., we have 6 fdμ = 8 8 f χ B dμ 2 B 2 = (g + h)dμ = gdμ + hdμ = fdμ. B When X is finite, Theorem. has obvious meaning. Let (X,, μ) be a monotone measure space with X < and,..., k be all minimal atoms. Recall that a minimal atom is a measurable set such that μ() > 0 and for each measurable proper subset B of we have μ(b) = 0. When X is a finite set, each set of positive measure contains at least 6 one minimal atom [24]. If we further assume that μ is subadditive then for each minimal atom, there are no nonempty 6 proper subsets B of such that B. In fact, if there is some nonempty proper subset B, then by the definition of minimal atom, μ(b) = 0, μ( \ B) = 0 and the subadditivity of μ implies μ() = 0, a contradiction. Observing this fact, to ensure its measurability, a function f must take constant value on each minimal atom. Thus, for any measurable function f : X [0, ) it holds fdμ = c i μ( i ), i= where c i is the constant value f takes on the minimal atom i. From this fact it is immediate that the -integral is 46 4 additive Further discussion of Theorem In this section we will complete the remaining proof of Theorem.. That is, we show that the equality (.) in Theo- 5 rem. remains valid in the case that f + g is not -integrable. The following proposition will demonstrate our assertion. Proposition 4.. Under the assumptions of Theorem., f + gis -integrable if and only if both f and g are -integrable. From the above Proposition 4., we know that if f + g is not -integrable, then either f or g is not -integrable. Therefore, both sides of (.) in Theorem. are equal to infinity, and hence the equality (.) holds. The proof of Theorem 5. is thereby completed Now we prove Proposition 4.. First, we need two lemmas. In the following the set {x X : f (x) > 0} will be denoted by 60 6 { f > 0} for short. 6

7 JID:IJ ID:808 /FL [mg; v.; Prn:4/08/20; :2] P. (-2) Y. Ouyang et al. / International Journal of pproximate Reasoning ( ) Lemma 4.2. Let (X,, μ) be a monotone measure space, and f, g F +. If { f > 0} {g > 0} =, then ( f + g)dμ 4 4 Proof. If one of the two integrals on the right-hand side of Ineq. (4.) is infinite then, by the monotonicity of the integral, ( f + g)dμ is also equal to infinity, which implies 8 the validity of (4.). 8 So, without loss of generality, we can suppose that both fdμ and gdμ are finite. For any arbitrary but fixed ε > 0, there are a partition { i } k i= of { f > 0}, a partition {B } m = of {g > 0}, and two sequences of positive numbers {λ i} k i= 0 0 and {l } m = such that k i= λ iχ i f, m = l χ B g and both the following two inequalities hold fdμ < λ i μ( i ) + ε i= 5 and 8 8 gdμ < m l μ(b ) + ε 2. = Since { f > 0} {g > 0} =, then { i } k i= {B } m = is a partition of { f + g > 0}. Moreover, we have that k i= λ iχ i + 2 m 2 24 = l χ B f + g, and 24 m ( f + g)dμ λ i μ( i ) + l μ(b ) 28 i= = gdμ ε. Letting ε 0, we get the result as desired. 5 Notice that we further require that μ is subadditive in Lemma 4.2, then Ineq. (4.) becomes an equality. That is, the 5 6 following result holds. 6 8 Lemma 4.. Let f, g F + and { f > 0} {g > 0} =. If μ is subadditive, then 8 ( f + g)dμ = Proof. By Lemma 4.2, it suffices to prove that ( f + g)dμ For any given partition { 4 i } k i= of { f + g > 0} and a sequence of positive numbers {λ i} k i= such that k i= λ iχ i f + g. 4 Since { f > 0} {g > 0} =, we have that { i {f > 0}} k i= (resp. { i {g > 0}} k ) is a partition of { f > 0} (resp. {g > 0}), i= and that 5 λ i χ i { f >0} f and λ i χ i {g>0} g. i= i= Moreover, the subadditivity of μ implies that ) μ( i ) = μ ( i ({ f > 0} {g > 0}) 5 5 μ( i {f > 0}) + μ( i {g > 0}). 6 Therefore 6 (4.) (4.2)

8 JID:IJ ID:808 /FL [mg; v.; Prn:4/08/20; :2] P.8 (-2) 8 Y. Ouyang et al. / International Journal of pproximate Reasoning ( ) λ i μ( i ) λ i μ( i {f > 0}) + λ i μ( i {g > 0}) i= i= i= 4 4 Thus, 0 0 ( f + g)dμ = sup{ λ i μ( i ) : λ i χ i f + g, { i } n i= F is a partition of X,λ i 0,n N} i= i= 4 4 Proof of Proposition 4.. Denote ={x f (x) g(x)} and B ={x f (x) > g(x)}. Then B = and f = f χ + f χ B, g = 8 g χ + g χ B. Moreover, f χ g χ and g χ B f χ B. Thus, 8 f + g 2g χ + 2 f χ B. 2 Combining the monotonicity and positive homogeneity of the -integral, and Lemma 4., we conclude that 2 ( f + g)dμ 2 2 (2g χ + 2 f χ B ) dμ = 2g χ dμ + 2 f χ B dμ 2 2 = 2 g χ dμ + f χ B dμ 2 gdμ + fdμ. 6 6 Thus if f + g is not -integrable then either f or g is not -integrable. On the other hand, if one of f, g is not integrable, then f + g is not -integrable. Hence f + g is -integrable if and only if both f and g are -integrable. Note 4.4. The following example shows that Ineq. (4.) can be violated if { f > 0} {g > 0}. 4 4 Example 4.5. Let X ={, 2,,...}, = 2 X and the monotone measure μ: [0, ] be defined as μ() = 4 X\ +, where 4 stands for the cardinality of. Suppose that f, g: X [0, ] are defined as f (x) = x and g(x) = for all x X Then, we can show that ( f + g)dμ < Observe first that for any subset, μ() > 0if and only if X \ is a finite set. So, for any partition { 50 i } k of X, there is at i= 50 most one set i with positive measure. For simplicity, we suppose that μ( ) > 0 and t = min{x x }. Then μ( ) t. 52 To ensure k i= λ iχ i f, there must be λ t. Thus, it holds that 52 λ i μ( i ) = λ μ( ) t <. t i= For the arbitrariness of the partition { i }, we conclude that fdμ. In a similar way, we can prove that gdμ and ( f + g)dμ 5. On the other hand, (t )χ {t,t+,t+2,...} f and (t )μ({t, t +, t + 2,...}) = t for each t X. t 5 60 Thus we have that fdμ sup{ t t X} =, and hence t fdμ =. Similarly, we have that gdμ = and 60 6 ( f + g)dμ =. 6

9 JID:IJ ID:808 /FL [mg; v.; Prn:4/08/20; :2] P. (-2) Y. Ouyang et al. / International Journal of pproximate Reasoning ( ) 5. The -integral for real-valued functions The definition of the -integral given in [] (see also [5,]) is restricted to nonnegative measurable functions. In the 4 previous two sections, we have seen that if μ is subadditive then all nonnegative -integrable functions form a convex 4 5 cone. Now we extend the definition of the -integral (Definition 2.) to general real-valued measurable functions. Similar 5 6 to the definition of Lebesgue integral for real-valued measurable functions ([,5,6]), we decompose a real-valued measurable 6 function to be the difference of two nonnegative measurable functions, and then consider the difference of integrals of these 8 two nonnegative measurable functions. 8 Let F denote the class of all real-valued measurable functions on (X, ). Let f F, then f = f + f where f + and 0 f are the positive and negative parts of f, that is f + = max( f, 0) and f = max( f, 0). Both f + and f are nonnegative 0 measurable functions, i.e., f +, f F +. Now we give the following the definition of -integral for general real-valued 2 measurable functions. 2 4 Definition 5.. Let (X,, μ) be a monotone measure space, and f F. If at least one of f + dμ and f dμ takes 4 5 finite value, then we define the -integral of f (with respect to μ) via 5 fdμ = f + dμ f dμ. 20 If both f + and f are -integrable in the sense of Definition 2., i.e., f + dμ < and f dμ <, then we say 20 2 that f is -integrable (or simply, f is integrable). 2 When X, define fdμ = f χ dμ. 24 Obviously, if f is integrable, then fdμ is finite. 24 Example 5.2. Let X ={x, x 2, x, x 4 }, = P(X) and the monotone measure μ be defined as follows: μ({x }) = μ({x 2 }) =,μ({x }) = 2,μ({x 4 }) = μ({x, x 2 }) =.5, μ({x, x }) = μ({x 2, x 4 }) = μ({x, x 4 }) = 4,μ({x, x 4 }) = 2.5, μ({x 2, x }) =.5,μ({x, x 2, x }) = μ({x, x 2, x 4 }) = 5, μ({x, x, x 4 }) = 4.5,μ({x 2, x, x 4 }) = 6,μ(X) = 6.5. Let f : X R be defined as 2 if x = x, 2 if x = x f (x) = 2, if x = x, if x = x Then f + dμ = μ({x, x }) = 4 and f dμ = μ({x 2, x 4 }) = 4. Thus fdμ = f + dμ f dμ = In the rest of this section, we investigate some basic properties of the generalized -integral Proposition 5.. Let (X,, μ) be a monotone measure space, f, g F and c R be a constant. Then we have the following: (i) cfdμ = c fdμ; (homogeneity) (ii) f gimplies fdμ fdμ; (monotonicity) 46 (iii) If f is integrable then f is also integrable. Moreover, if μ is subadditive then f is integrable if and only if f is integrable Proof. 4 The proofs of (i) and (ii) are standard and thus omitted. For (iii), notice first that 0 max( f +, f ) f. If 50 f dμ <, by the monotonicity of -integral, we then have that both f + dμ < and f dμ <, which 50 5 imply the integrability of f By definition, the integrability of f implies that both f + and f are integrable. Suppose now μ is subadditive, Theorem holds. Thus 5 ( f dμ = f + + f ) dμ = f + dμ + f dμ < The following result shows that the generalized -integral is linear whenever the monotone measure μ is subadditive. 6 (5.)

10 JID:IJ ID:808 /FL [mg; v.; Prn:4/08/20; :2] P.0 (-2) 0 Y. Ouyang et al. / International Journal of pproximate Reasoning ( ) Theorem 5.4. Let (X,, μ) be a monotone measure space. If μ is subadditive, then for any -integrable functions f, g F, ( f + g)dμ = 4 4 Proof. Similar to the proof of classical Lebesgue integral. Remark 5.5. For general measurable functions (not necessarily nonnegative) the result of Lemma 4.2 fails. Let us reconsider Example 5.2. Let g = f χ {x,x 4 } and h = f χ {x2,x }. Then {g 0} {h 0} =. It is easy to see that gdμ = 2μ({x }) 0 0 μ({x 4 }) = 0.5, hdμ = μ({x }) 2μ({x 2 }) = 0. That is, 2 2 (g + h)dμ = fdμ = 0 < 0.5 = gdμ + hdμ Remark 5.6. In Definition 5. we got a symmetric and fully homogeneous integral, the generalized -integral. Recall the 6 case of the Choquet integral, there are two kinds of extensions, the asymmetric Choquet integral and the symmetric Choquet 8 integral. It is well-known that the asymmetric Choquet integral is comonotonic additive and positively homogeneous, while 8 the symmetric Choquet integral is homogeneous but lacks comonotonic additivity []. The asymmetric -integral, another 20 extension of the -integral, is for us a topic for the further study Pan-integrable functions space In this section we will establish an analogue of classical L p space. Let (X,, μ) be a monotone measure space and p <. Let L p (X, μ) be the set of all measurable functions f F such that 2 2 f p dμ <. 0 X 0 Then L p (X, μ) is a linear space in natural linear structure. 2 Let f L p (X, μ). We define the symbol f μ,p by p f μ,p = f p dμ. 6 6 X 8 For a subadditive and continuous from below monotone measure, the Hölder and Minkowski inequalities for classical 8 integrals remain valid for -integrals. The following result is due to [8]. 4 4 Proposition 6.. Let (X,, μ) be a monotone measure space and μ be subadditive and continuous from below. Then, (i) (Hölder) If p, q satisfy 4 p + q = and f Lp (X, μ) and g L q (X, μ), then 4 fg μ, f μ,p g μ,q (ii) (Minkowski) For p and f, g L p (X, μ), then f + g μ,p f μ,p + g μ,p Let us observe that, in general, μ,p is not a norm on the linear space L p (X, μ) s discussed in the classical L p space theory, we write f g for f, g L p (X, μ) if f = g μ-a.e. Note that if μ 5 52 is subadditive, then the relation is an equivalence relation. We define the space L p (X, μ) to be the collection of all 52 equivalence classes in L p (X, μ). Similar to the discussion of classical L p space and by using Theorem., we can get the following result. Theorem 6.2. Let (X,, μ) be a monotone measure space. If μ is subadditive, then L p 5 (X, μ) is a linear space in the usual manner. 5 5 When μ is continuous from below, it follows from Proposition 2.2 (ii) that for any f L p (X, μ), f μ,p = 0if and 5 60 only if f = 0a.e. on X, i.e., f is the zero element in L p (X, μ). Combining Proposition 5. and 6., then μ,p is a norm 60 6 on L p (X, μ) and hence L p (X, μ) is a normed linear space. 6

11 JID:IJ ID:808 /FL [mg; v.; Prn:4/08/20; :2] P. (-2) Y. Ouyang et al. / International Journal of pproximate Reasoning ( ) Note: It is common to say or to write that some function f is an element of L p (X, μ), although the element of that 2 space are actually equivalence classes of functions. 2 Using Levi s theorem (the monotone convergence theorem) and Fatou s lemma for the -integrals ([5,], see also 4 []) and the usual arguments as in [,6] we can prove the following result (the details are omitted). 4 6 Theorem 6.. Let (X,, μ) be a monotone measure space. If μ is finite, subadditive and continuous, then L p (X, μ) is a Banach 6 space.. Conclusions 0 0 In this paper, we have studied the linearity of (+, )-based -integrals. For a subadditive monotone measure the - 2 integral is a positively homogeneous linear functional on the convex cone which consists of all nonnegative -integrable 2 functions (Theorem., Note.2 and Proposition 4.). We have also extended the -integral to arbitrary real-valued measurable 4 functions (Definition 5.). The generalized -integral has been shown to be symmetric and fully homogeneous 4 5 (Proposition 5.), and to be linear for subadditive monotone measures (Theorem 5.4). Further assuming that the monotone 5 6 measure is finite and continuous, then we have obtained an analogue of classical L p space, i.e., a complete normed linear 6 space L p (X, μ) consisting of all p-th order -integrable functions (Theorem 6.). In the follow-up study we will investigate other properties of the space L p (X, μ), almost along the same lines as in the familiar case of classical Lebesgue space L p, such as the separability and the dual space of L p 20 (X, μ), etc. 20 We point out that the concave integral coincides with the -integral when μ is subadditive (see [24,25]), thus the results obtained in this paper also hold for concave integrals. Note that an outer measure is subadditive. Thus we can define a Lebesgue-like integral (possesses linearity) from an outer measure, and the L p theory holds for this integral. Moreover, the domain of an outer measure is the power set, measurability restrictions of functions are thus unnecessary for this integral. In future work, we will investigate similar results for general (, )-based -integral. We stress that this is not a trivial generalization, since we even do not know whether or not the positive homogeneity holds for general -integrals. On the other hand, the subadditivity is a rather strong restriction for monotone measure. It is of interest to examine the results of this paper under some weaker conditions, such as the autocontinuity from below of monotone measures ([5]). cknowledgements This work was partially supported by NSFC (nos. 2 and 506) and the NSF of Zheiang Province (no. LY5000), and by the grant PVV-4-00 and VEG /0420/5. References 6 6 [] J.J. Benedetto, W. 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