Beyond the scope of super level measures

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.1 (1-28)

Available online at www.sciencedirect.com

ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss

Beyond the scope of super level measures Lenka Halˇcinová a,∗ , Ondrej Hutník a , Jozef Kisel’ák a,b , Jaroslav Šupina a a Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University in Košice, Jesenná 5, SK 040 01 Košice, Slovakia b Linz Institute of Technology and Department of Applied Statistics, Johannes Kepler University in Linz, Altenberger Straße 69, 4040 Linz, Austria

Received 1 December 2016; received in revised form 10 February 2018; accepted 7 March 2018

Abstract We expand the theoretical background of the recently introduced outer measure spaces theory of Do and Thiele in harmonic and time-frequency analysis context. In the context of non-additive measures and integrals, we propose a certain framework for a natural extension of the basic ingredients of the theory (i.e., the concept of size, outer essential supremum and the corresponding super level measure) which, besides the covering the previously considered cases, permits us to introduce a further substantial extension of a class of non-additive integrals. All these notions are studied in detail and exemplified. © 2018 Elsevier B.V. All rights reserved. Keywords: Size; Outer essential supremum; Super level measure; Borel set; Choquet integral; Shilkret integral; Non-additive measure

1. Introduction This paper deals with a detailed study of the concept of super level measures introduced by D O and T HIELE in [9]. A motivation for introducing an Lp -theory for outer measure spaces by these authors can be seen in the efficiency of encoding the functions already in the classical Lebesgue theory of measure and integral. Classical coding describes functions as assignment of a value to every point of a basic set X = ∅. In the case of Lp -functions, the set of such assignments has a very large cardinality, which is only reduced after consideration of equivalence classes of Lp -functions. This detour over sets of large cardinality can be avoided by coding functions via their averages over dyadic cubes (alternatively, Euclidean balls). There are only countable many such averages, and by Lebesgue differentiation theorem these averages contain the complete information of the equivalence class of the Lp -function. This is the main idea behind the term size = an average over generating sets on which an outer measure is defined. For the convenience of the reader, we briefly sketch its original content here, without the pretence of being rigorous. An outer measure space (X, μ, s) consists of: * Corresponding author.

E-mail addresses: [email protected] (L. Halˇcinová), [email protected] (O. Hutník), [email protected] (J. Kisel’ák), [email protected] (J. Šupina). https://doi.org/10.1016/j.fss.2018.03.007 0165-0114/© 2018 Elsevier B.V. All rights reserved.

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.2 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

2

(i) a non-empty set X; (ii) an outer measure μ generated using countable coverings from a pre-measure on a fixed collection E of non-empty subsets of X, which in particular covers X; (iii) a size s which assigns a non-negative number to each pair (f, E), where f : X → C is Borel measurable and E ∈ E, such that s is monotone and quasi-sublinear with a scaling property. Given an outer measure space (X, μ, s), a Borel measurable function f : X → C and α > 0, it is then possible to replace the outer measure of upper level sets {x ∈ X : |f |(x) > α} with a more subtly defined quantity μ(s(f ) > α), being the infimum of all values μ(F ), where F runs through all Borel subsets of X which satisfy1 sup s(f 1X\F )(E) ≤ α. E∈E

If the predefined averages s are of L∞ -type, the super level measure specializes to the outer measure of the upper level set, but in general the two quantities are quite different. Practically, outer measure spaces give a very convenient language for reduction to Carleson measure properties, where the theory of sizes and integrals based on the corresponding super level measure serves therein as a tool for an explanation of the important role of Carleson measures in connection with time-frequency analysis. Description of our aims Despite the great success of Lp -theory for outer measures in applications, there is a need to investigate the rich concept of sizes, super level measures and integrals based on them in its own sake. There are many reasons for doing so, we mention only few of them, which seem to be interesting and challenging, and the present paper is devoted to: (A) The first principal aim is to describe and transfer very successful techniques and ideas from one area of mathematics (harmonic and time-frequency analysis) to other mathematical areas (non-additive measure and integrals, optimization, fuzzy set theory). From further development and an extension of the original ideas in the new context all the involved areas can profit. (B) The basic concept of a size enables to aggregate functions on basic sets in order to produce one representative value. A typical example, which immediately comes to mind, is an integral (usually in the Lebesgue sense). Such Lp -based sizes are, in fact, the only sizes discussed in the available literature in connection with time-frequency analysis context. However, in connection with aim (A), other interesting sizes based on various aggregation techniques have to be considered and studied, e.g. non-additive integrals, weighted means, etc. For this reason we collect many important examples here. Moreover, further properties, various new operations and inner relations of the new families of sizes will be investigated. (C) The concept of super level measure (as introduced in [9] and sketched above) relates the basic collection of sets (with a pre-measure σ ) with the measure μ generated by σ and the value of size function. The procedure passing abstractly from the pre-measure to the outer measure is natural in many situations, but it seems to be too restrictive in general. For instance, although for L∞ -based sizes the concept of super level measure reduces to the classical measure of upper level sets, the generating procedure (of μ from σ ) cannot produce an arbitrary monotone set function (in fact, it produces only a sub-additive measure μ in the original setting). Thus, the original concept of super level measure does not allow a generalization of many classical (non-additive) integrals. This can be achieved by relaxing the merge between the basic collection and μ, which one enables to take other aspects of the theory into play. Considering such an extension we compute the outer essential supremum and the corresponding super level measure for many of the sizes we collect here and certain operations with them. Outline and organization of the paper In Section 2 we give necessary definitions and notations, and we provide a list of basic collections and their properties we work with. The reader may skip this section and later go back to it if needed. Section 3 deals with the main ingredient of the theory – sizes. In contrast to the original definition of size, we suppose the size s is defined on all Borel subsets of X instead of a basic collection E only. On the one hand, this 1 By 1 we denote the characteristic function of the set A, i.e., 1 (x) = 1 if x ∈ A and 1 (x) = 0 otherwise. A A A

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.3 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

3

can be seen as a technical condition, but on the other hand it enables us to define natural operations with sizes, which may not be defined on the same collection. We collect a list of examples of sizes considering sizes of Lp -type, of non-additive integrals-based type, of mean-based type, sizes on the upper-half plane , and sizes of discrete type. We will use the examples for later considerations. In Section 4 we investigate the notion of the outer essential supremum of a Borel-measurable function f over a Borel set F with respect to a size s and a collection E, we denote it by outsupF s(f ) E . We state some conditions under which the outer essential supremum can be computed effectively, for instance, when the outer essential supremum coincides with the initial size, i.e., outsupF s(f ) E = s(f )(F ). This is the case of all integral sizes (additive as well as non-additive ones) considered in this paper. In the discrete case, i.e., when X is a finite set, we show that computation of the outer essential supremum of f over F is closely related to averaging over singletons from F . In this case we also demonstrate an interesting fact that the outer essential supremum outsupF s(f ) E can dramatically change its behaviour when the values of f depend or do not depend on the sets from a collection E. For instance, we show that all the p -based averages  s˜ p,ν (f )(E) =

1  |f (x)|p ν# (E)

1

p

x∈E

with ν# being the counting measure on Borel sets of a finite set X, behave the same way, i.e., the corresponding outer essential supremum outsupF s˜ p,ν# (f ) coincides with maxx∈F |f (x)| no matter which p ∈ (0, +∞] is chosen. As a consequence, the corresponding super level measure coincides with the standard level measure for each p > 0. On the other hand, for the size  sp,ν (f )(E) =

1  |f (x)|p ν# (E)

1

p

x∈X

the outer essential supremum does depend on the value of p, because it coincides with the p -average of f on F , thus the corresponding super level measure may not be the standard level measure. The super level measure studied in Section 5 is a possible “by-size-generalization” of the standard level measure concept. We again make a small modification of the original concept of Do and Thiele [9] when considering a monotone measure μ on all Borel sets in X instead of an outer measure μ generated by a pre-measure σ on a collection E. This small modification enables us to consider a wider class of non-additive set functions for which the super level measure can be computed explicitly. An advantage of our concept is that we are able to provide a generalization of many non-additive integrals (not only the Choquet and the Shilkret one!). We provide a computation of super level measure for many of the considered sizes. Probably the most interesting result is that the standard weighted Lp -based sizes ⎛ ⎞1 p  1 (L) p s,p (f )(E) = ⎝ |f (x)| d(x)⎠ (E) E

with  being the Lebesgue measure are indistinguishable in the sense, that for each p ≥ 1 the corresponding super level measure coincides with the standard level measure ({x ∈ X; |f (x)| > α}) without any dependence on the parameter p. Thus, all Lp - as well as p -based sizes yield the same result: the super level measure is nothing but the standard level measure no matter which p is chosen. The results are summarized in Table 1 in Appendix. Moreover, we show that there is a large class of sizes for which the super level measure μ(s(f ) E > α) coincides with the level measure of the Borel set where |f | is greater than α. On the other hand, we show that the super level measure concept is a full-value generalization of the standard level measure in the sense, that there is a super level measure of a function f with respect to a size s which cannot be a standard level measure of any function g. Indeed, this verifies that the outer Lp -spaces considered in [9] (and the works of other authors after them) are new and they cannot be obtained from the existing ones. For further purposes of a generalization of classical non-additive integrals, we introduce an equivalence relation between triples of measures, sizes and functions, which provides a generalization of the equality almost everywhere of two functions, and we study three cases of such indistinguishability in detail.

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.4 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

4

2. Preliminaries There is a rich literature on various concepts of generalizations of the classical (additive) measure theory. Omitting additivity provides a starting point for non-additive theories, such as capacity theory [8], subadditive measure theory [10], null-additive set functions theory [22], etc. The concept of outer measure spaces developed in [9] needs to make some a priori choices. In particular, we need to choose a basic collection of subsets, a pre-measure defined on them and a size function. In order to make the paper self-contained as much as possible, we provide in this section all the necessary definitions, properties and numerous examples. Basic collections A starting point for our study is the basic collection E of subsets of a non-empty set X. To avoid too abstract setting, we shall always assume that X is a topological space and EB is the σ -algebra of Borel sets of X. The pair (X, EB ) will be called a Borel space associated to X. For a non-empty subcollection E ⊆ EB , the pair (X, E) will be called a sub-Borel space.2 Note that we do not suppose any further requirements on sets of E, such as ∅ ∈ E, X ∈ E, or closedness with respect to union, or intersection. Remark 2.1. In the paper [9] authors apply a concrete-to-abstract principle: they specify a concrete generating function σ : E → [0, +∞) called a pre-measure and then define outer measure μ abstractly by using countable coverings by these concrete sets  X ⊇ E → μ(E) := inf σ (E  ). (1) E

E  ∈E

The infimum is taken over all countable subcollections E of E which cover the set E. Such an outer measure μ is monotone and countably subadditive on the collection of all subsets of X. In general, our approach does not follow the concrete-to-abstract principle of Do and Thiele [9], i.e., a concrete pre-measure on the small collection need not be given and the measure we use to define the super level measure need not be generated by a pre-measure. However, the original approach is naturally included in our theory when considering measures μ generated by a given pre-measure σ on E. Examples of collections There are many examples of various basic collections in the real-analytical as well as complex-analytical literature. For the convenience of the reader, in what follows we collect some of them and, moreover, we indicate a concrete pre-measure σ and the generated outer measure μ to demonstrate the original ideas of Do and Thiele [9] (generalizing the Lebesgue outer measure case). a) Lebesgue measure via dyadic cubes. Let X be the Euclidean space Rm for some m ≥ 1, and Ecube be the set of all dyadic cubes Q = [2k n1 , 2k (n1 + 1)) × · · · × [2k nm , 2k (nm + 1)) with integers k, n1 , . . . , nm . The pre-measure σ (Q) = 2mk generates the classical Lebesgue outer measure ∗ on Rm and ∗ (Q) = σ (Q) for every dyadic cube Q. Note that the collection Ak of all the unions of dyadic cubes of side 2−k , k ∈ Z, is a σ -algebra for each k ∈ Z. b) Lebesgue measure via balls. Let X be the Euclidean space Rm for some m ≥ 1, and Eball be the set of all open balls Br (x) with radius r and centre x ∈ Rm . Then the pre-measure σ (Br (x)) = r m generates the same outer measure3 ∗ on Rm . c) Tents on the upper half-plane in the complex plane. In harmonic analysis, let X = R × (0, +∞) be the open upper half-plane  in the complex plane C, and Etent be the set of all tents (or, Carleson boxes)

 T (a, b) = (x, y) ∈  : y < b, |a − x| < b − y , (a, b) ∈ . 2 Our terminology differs from the notion of sub-Borel space used in [2, Definition 1.3.4], where, given a Borel space (X, E ) and a subset Y of B X, a pair (Y, EB ∩ Y ) is called a sub-Borel space of (X, EB ). 3 More precisely, a multiple of Lebesgue outer measure ∗ .

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.5 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

5

For any such tent T define the pre-measure σ (T (a, b)) = b. Note that tents form a much more restricted collection than dyadic cubes. A generalization of tents on the upper 3-half space R × R × (0, +∞) is used frequently, see [9, Section 5]. d) Closed unit disc. Let X = D be the closed unit disc in the complex plane and Edisc be a collection of sets defined by

(θ, h) := z ∈ D : 1 − h < |z| < 1, θ < arg z < θ + h , θ ∈ [0, 2π ], h ∈ (0, 1), with the pre-measure σ ((θ, h)) = h. e) Pseudohyperbolic balls on the upper half-space. For a positive integer n ≥ 2, let X = Rn−1 × R+ be the upper half-space in Rn . Consider the collection Ehyper of pseudohyperbolic discs Dδ (z) with centre z ∈ X and radius δ ∈ (0, 1), where   |z − w| Dδ (z) := w ∈ X : <δ . |z − w| 2δ For any Dδ (z) ∈ Ehyper we define σ (Dδ (z)) = 1−δ 2 zn with zn ∈ R+ being the n-th coordinate of z. f) Tri-tile collection. In time-frequency analysis, a (standard) tile T = IT × ωT is a rectangle of area 1, where the spatial interval IT is a standard dyadic interval and the frequency interval ωT is a standard dyadic interval. A tri-tile P = (P1 , P2 , P3 ) is an ordered triple of tiles Pj , j = 1, 2, 3 with the property IP1 = IP2 = IP3 . Denote by ωP := co{3 ωP1 , 3 ωP2 , 3 ωP3 } the convex hull of the intervals 3 ωPj , j = 1, 2, 3, and by P the collection of rank one tri-tiles. The generating collection Etile is the set of trees, where the set E ⊂ P is a tree with top data (IE , ξE ), if

IP ⊂ I E ,

ξE ∈ ωP for each P ∈ P.

The pre-measure σ : Etile → [0, +∞) is given by σ (E) = (IE ). g) Finite collection. In many computer-supported applications finite spaces are considered, such as the number of attributes in a database, various sets of criteria, sets of rules, sets of players, etc. For this reason, for n ∈ N let X = {1, 2, . . . , n} be the universe. A natural collection here is the collection of all subsets of X, i.e., Epower = 2X . Also, we use the collection Esing of all singletons, i.e., Esing = {{x} : x ∈ X}, with a pre-measure σ given by σ ({x}) = 1. Clearly, the outer measure μ generated by σ is the counting measure. Covering property Although in general we do not suppose any further requirements on collection E, sometimes we need to specify a suitable collection for considered purposes. Very often we shall ask collection E to satisfy the covering property    (∃{Ei ; i ∈ N} ⊆ E) X = Ei ∧ (∀i ∈ N) Ei ⊆ Ei+1 . (COV) i∈N

One can see that collections Epower , EB , Ecube , Eball and Etent satisfy (COV). However, collections Edisc , Ehyper and Esing are not the case. Measures Throughout the text, the term measure4 will be used in its most general sense, i.e., a set function m : EB → [0, +∞] on a Borel space (X, EB ), which enables “to measure” appropriate sets with the only (natural) condition m(∅) = 0. Other properties of measure m, such as monotonicity, additivity, sub-additivity, continuity, maxitivity, etc. will be added as the corresponding adjectives, i.e., a monotone measure, an additive measure, a sub-additive measure, a continuous measure, a maxitive measure, etc. In this text, the Lebesgue (σ -additive) measure will always be denoted by . 4 Alternatively, a set function m defined on a Borel space (X, E ) for X being a finite non-empty set and E = 2X is usually called a game, B B see [13]. Frequently a term capacity is also used. However, it can be misleading due to its different usage in various literature, see e.g. Choquet capacity [8], Sobolev capacity [18], Besov capacity [21], etc. Thus, we will not use this terminology in the paper.

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.6 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

6

3. Sizes: averaging the functions on collections In this section, we describe a concrete procedure to assign averages over basic sets to a class of Borel-measurable functions following the concept introduced in [9]. Indeed, given a sub-Borel space (X, E), the average (size) s of a Borel-measurable function f on the collection E should reflect some natural properties, namely monotonicity, scaling and quasi-sublinearity. Although the main idea is to have the basic collection E as small as possible, in many examples the domain of functions as values of a size are naturally wider than E. For example, supremum can be computed on any set, integral average can be computed on any Borel set, etc. Therefore, our definition formally differs from the original one, see [9, Definition 2.3], in that the notion of size is defined here on the algebra EB (of all Borel subsets of X) instead of a basic collection E. In some sense, this serves as a technical condition and it does not negate the basic idea of averaging over generating sets. In fact, restriction of a size s to a subcollection E ⊆ EB will produce a size again, thus the original definition is included in our one. In such a case we will say “s is a size on E”. Now the formal definition follows. Definition 3.1. Let (X, EB ) be a Borel space and B(X) be the set of all complex-valued Borel-measurable functions on X. A size is a map s : B(X) → [0, +∞]EB

such that for any f, g ∈ B(X) and E ∈ EB it holds (i) if |f | ≤ |g|, then s(f )(E) ≤ s(g)(E); (ii) s(λf )(E) = |λ| s(f )(E) for each λ ∈ C; (iii) s(f + g)(E) ≤ Cs s(f )(E) + s(g)(E) for some fixed Cs ≥ 1 depending only on s. Monotonicity property (i) presents a non-negative response to any increase of the (function) arguments. This seems to be a quite acceptable property. From monotonicity we immediately get s(f )(E) = s(|f |)(E) for all f ∈ B(X) and E ∈ EB .

Hence, the theory is essentially one of nonnegative functions: the size needs initially be only defined for non-negative Borel functions and can then be extended via the above identity to all functions. Property (ii) is usually called the scaling property of a size, and as demonstrated later, it is the most restrictive requirement for a mapping to be a size. Here, and in what follows, we use the standard convention 0 · (+∞) = (+∞) · 0 = 0 when necessary. Finally, from property (iii) it can be seen that we do not require averages to be linear, but merely sublinear or even quasi-sublinear. In practise, many sizes s will be sublinear, which means that the constant Cs can be chosen to be 1. However, the general constant Cs ≥ 1 allows for certain more general examples, such as Lp -based averages with p ∈ (0, 1), see Remark 3.2, and other. 3.1. Operations with sizes For a given list of sizes one can generate a multiple of different ones using various procedures. Several of them are described in the following lines. Given a non-empty set X and a size s, for each α ∈ C we may define a scalar multiple of s as (α  s)(f )(E) := |α| · s(f )(E) for f ∈ B(X) and E ∈ EB under the stated convention 0 · (+∞) = 0. Moreover, if ν : EB → [0, +∞] is a measure, we may define other useful multiple of a size s, denoted ν  s, by setting5 5 Since both objects (measures and sizes) allow zero as well as infinite values, the multiple ν  s should be understood as follows

⎧ ⎪ ⎨0, (ν  s)(f )(E) = ν(E) · s(f )(E), ⎪ ⎩ +∞,

min{ν(E), s(f )(E)} = 0, max{ν(E), s(f )(E)} < +∞, max{ν(E), s(f )(E)} = +∞ and min{ν(E), s(f )(E)} > 0.

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.7 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

7

(ν  s)(f )(E) := ν(E) · s(f )(E) for f ∈ B(X) and E ∈ EB . In many examples a multiple by measure takes the form standard way ⎧ ⎪ ν(E) = +∞, ⎨0, 1 (E) := +∞, ν(E) = 0, ⎪ ν ⎩ 1 , otherwise. ν(E)

1 ν

 s, where

1 ν

is defined in a

We shall use also other sizes associated to a size s. Namely, relativization and regularization are defined by s˜ (f )(E) := s(f 1E )(E), sF (f )(E) := s(f )(F )

for f ∈ B(X) and E, F ∈ EB . It can be seen from the definition that the regularization does not depend on the set E from the basic collection E ⊆ EB , just on the set F ∈ EB , usually taken to be the set X. A natural example of a regularized size is any (Banach) function quasi-norm. From now on, we may use various combinations of the defined sizes. For instance, we shall often consider sizes of the form ν  sX . Furthermore, we may define new sizes pointwisely. Indeed, a pointwise addition and pointwise maximum of sizes s1 , s2 are given by (s1 ⊕ s2 )(f )(E) := s1 (f )(E) + s2 (f )(E), max{s1 , s2 }(f )(E) := max{s1 (f )(E), s2 (f )(E)} for f ∈ B(X) and E ∈ EB . Note that given two sub-Borel spaces (X, E1 ) and (X, E2 ), and sizes si on Ei with i ∈ {1, 2}, the addition and maximum of s1 , s2 are defined for sets E ∈ E1 ∩ E2 . Also, for two sizes si on Ei , where E1 ∩ E2 = ∅ we may define a concatenation of s1 and s2 on E1 ∪ E2 as follows  s1 (f )(E), E ∈ E1 , (s1  s2 )(f )(E) := s2 (f )(E), E ∈ E2 . One can easily verify that all the new mappings are really sizes. 3.2. Examples Several examples related to various basic collections are provided in what follows. Note that (X, EB) is always assumed to be a Borel space, unless stated otherwise. Elementary sizes a) Zero size. The zero size szero : B(X) → [0, +∞]EB is defined by the formula szero (f )(E) = 0. b) Evaluation size. Let e ∈ X be an arbitrary (fixed) element. The evaluation size se : B(X) → [0, +∞]EB is defined by the formula se (f )(E) := |f |(e).

Note that if card(X) > 1 and f ∈ B(X) is such that f (e) = 0, then se (f 1X\{e} )(X \ {e}) = 0, but se (f )(X \ {e}) = |f |(e). For a relative counterpart of evaluation size we have s˜ e(f )(E) = 1E (e) · |f |(e). EB defined by c) Lebesgue integral. Let m be a σ -additive measure on EB . The mapping s(L) int,m : B(X) → [0, +∞] the formula  (L) sint,m (f )(E) := (L) |f (x)| dm(x) E

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.8 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

8

is a size, where the integral is taken in the Lebesgue sense (therefore L before the integral sign as well as the (L) superscript in sint,m ). Note that its regularization is    (L) sint,m (f )(E) = (L) |f (x)| dm(x). X

X

d) Sum. Let X be a countable set with discrete topology, and m be a counting measure (we denote it by ν# in this (L) paper). Then the previously defined size sint,ν# reduces to 

ssum (f )(E) :=

|f (x)|,

f ∈ CX , E ⊆ X,

x∈E

and the corresponding regularized size takes the form (ssum )X (f )(E) =



|f (x)|.

x∈X

e) Encoding the functions. The mapping senc : B(X) → [0, +∞]Esing defined by the formula senc (f )({x}) := |f (x)|,

x ∈ X,

is a size on Esing . Mean-based sizes a) Weighting the Lebesgue integral. In Lebesgue theory, for every Borel function f ∈ B(X) and E ∈ EB the mapping given by  (L) s,p (f )(E) :=

1 (L)  sint, 



⎛ 1 p =⎝ (f p )(E)

1 (E)



⎞1

p

|f (x)| d(x)⎠ p

(2)

E (L)

(L)

frequently appears, where p > 0. We shall consider mainly size s,1 on Ecube and size s,1 on Eball . Both sizes are L1 -based averages determining f by the Lebesgue differentiation theorem, i.e., the size (2) contains enough information to reproduce f , thus being a more efficient code than pointwise a.e. information. b) Supremum. On the other hand, the most commonly (implicitly) used size is s∞ (f )(E) = sup |f (x)| = sup |f |[E], x∈E

which is an L∞ -based average. This average takes only the function f and the set E into account and does not depend on any further external input (e.g. on a measure of basic sets). Note that certain combination of a) and b) can be found in many cases of function spaces theory. For example, when defining Banach quasi-norm spaces, BMO spaces, rearrangement invariant spaces, Lorentz spaces, Orlicz spaces, Marcinkiewicz spaces, etc., see [7]. c) Weighting the Lebesgue integral on . Similarly, in the upper half-plane  we make use of Lp -based averages with respect to the weighted Lebesgue measure on . Indeed, we assign a value to each T ∈ Etent by averaging a Borel-measurable function f on the tent via the mapping ⎛ ⎜1 b

⎞1/p



s(f )(T (a, b)) = ⎝

|f (x, y)|p

dx dy ⎟ ⎠ y

.

T (a,b)

d) Weighting the sum. For a non-empty finite set X with discrete topology we define the mapping sν,p : B(X) → [0, +∞]EB by  sν,p (f )(E) :=

1  p 1 p  (ssum )X (f )(E) ν

(3)

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.9 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

9

with p ∈ R, p > 0. Here ν is an arbitrary measure defined on EB . Also, we make use of the standard (discrete) p-mean of f on E of the form  1  p 1 s˜ ν,p (f )(E) = (4)  ssum (f p )(E) . ν Observe that all the mentioned sizes in this paragraph may be written in a unified form    1 (L) sϕ,ψ (f )(E) := ψ  sint,ν2 (ϕ ◦ |f |)(E) ν1 for suitably chosen bijections ϕ, ψ : [0, +∞) → [0, +∞) and measures ν1 and ν2 defined on EB . Note that for a special case of quasi-arithmetic integral means, i.e., when ψ = ϕ −1 , the only mean which is homogeneous of order 1 is the power (Hölder) mean corresponding to ϕ(t) = t p for p = 0 and ϕ(t) = log t for p = 0. Moreover, the power mean is subadditive if and only if p ≥ 1. Thus, Lp -based sizes with p ≥ 1 are exactly the sizes corresponding to power mean. However, when considering general case ψ = ϕ −1 it is possible to find some broad classes of pairs (ϕ, ψ) of non-power functions for which sϕ,ψ is sublinear, see [20], but the scaling condition (homogeneity of ϕ and ψ ) seems to be a very restrictive one. Remark 3.2. Notice, that even if the Lp -based average sϕ,ϕ −1 with ϕ(t) = t p for p ∈ (0, 1) is not subadditive, it does generate an Lp -based size. This follows from the Minkowski-type integral inequality6 in the case 0 < p < 1, which implies property (iii) in the definition of size. Non-additive integrals-based sizes Previous examples have dealt with the Lebesgue integral being an additive integral based on a (σ -)additive measure. Recall that this concept has significant limitations because the (σ -)additivity excludes any interaction between subdomains of X. In the following we introduce sizes which are based on nonadditive measures and integrals originally defined for non-negative real-valued measurable functions (with respect to a non-additive set function m defined on a σ -algebra A of subsets of a non-empty set X). In what follows, X is a topological space, A = EB and m : EB → [0, +∞] is a monotone measure. a) Choquet integral. The Choquet integral is given by ∞

 f dm :=

(Ch) A

m ({x ∈ A ∩ X : f (x) ≥ α}) dα, 0

where the integral on the right-hand side is the improper Riemann integral. If a monotone measure m is submodular, i.e., m(E ∪ F ) + m(E ∩ F ) ≤ m(E) + m(F ) for all E, F ∈ EB , (Ch)

then the Choquet integral is sublinear, cf. [22, Theorem 7.7], and thus the mapping sint,m : B(X) → [0, +∞]EB of a Borel-measurable function f : X → C over a Borel set E ⊆ X given by  (Ch) sint,m (f )(E) := (Ch) |f | dm E

is a size – monotonicity and scaling property follows from [27, Theorem 11.2]. However, the sublinearity of the Choquet integral will not play much of a role for our purposes, because submodularity of m is not necessary for (Ch) sint,m being a size (just for being a sublinear size). 6 I.e., the inequality

  f p + gp ≤ f + gp ≤ Cp f p + gp with the best possible constant Cp = 21/p−1 > 1, see [6, Theorem 2.3.2] for the Lebesgue measure, i.e., ν1 = ν2 = , and [16, p. 199] for any general measure, i.e., ν = ν1 = ν2 .

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.10 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

10

b) Shilkret integral. The Shilkret integral (originally considered for maxitive measures) is an aggregation function defined by  (Sh) f dm := sup {α · m ({x ∈ A ∩ X : f (x) ≥ α})} . α>0

A

Then the mapping (Sh)



sint,m (f )(E) := (Sh)

|f | dm E

defined for any f ∈ B(X) and E ∈ EB is a sublinear size if and only if the monotone measure m is finite and maxitive, i.e., m(E ∪ F ) = max{m(E), m(F )} for all disjoint sets E, F ∈ EB , (Sh)

cf. [24, p. 112–113], see also [3, Theorem 2.18]. Let us mention that maxitivity of m is not necessary for sint,m being a quasi-sublinear size. c) Lower integral. Given a measurable function f : X → [0, +∞) and a set B ∈ EB , the lower integral of f with respect to m on B is defined as  (low) f dm := lim Lε , ε→0+

B

where



Lε = inf

∞ 

αi · m(Bi ) : f ≤

i=1

∞ 

 αi · 1Bi ≤ f + ε, Bi ∈ EB ∩ B, αi ≥ 0, i = 1, 2, . . .

i=1

for ε > 0, see [27, Chapter 12]. Here, EB ∩ B = {A ∩ B; A ∈ EB }. Again, the mapping  (low) sint,m (f )(E) := (low) |f | dm E

defined for any f ∈ B(X) and E ∈ EB is a size. Indeed, monotonicity and scaling property follows from [27, Theorem 12.3], sublinearity is the result of [27, Theorem 12.1]. Note that the Choquet and Shilkret integrals belong to the class of decomposition integrals proposed in E VEN and L EHRER [11]. Moreover, a (hierarchical) family of the so-called copula-based integrals [19] contains both the Choquet and the Shilkret integral. Thus, we might consider averages based on other families of integrals as well. For instance, an extension of Choquet integral with respect to a pre-measure provided by Š IPOŠ [26], or the class of semicopula-based integrals investigated in detail in [5] which contains the Shilkret integral. Tri-tile collection sizes Considering the rank one tri-tile collection P, the generating collection Etile of trees, we define for each j = 1, 2, 3 the corresponding size sj : CP → [0, +∞]Etile on functions f : P → C by ⎛ ⎞1/2  1 sj (f )(E) := ⎝ |f (P )|2 (IP )⎠ + sup |f (P )|, (IE ) P ∈E P ∈E\Ej

where each Ej is a tree with the same top data as E ∈ E such that  E= E \ (Ej ∪ Ek ). 1≤j
Observe that the addition of sizes (discrete 2 -based as well as ∞ -based size) to define the sizes sj has been used.

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.11 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

11

Negative examples Here we provide examples of often considered mappings which cannot be sizes. a) Diameter of the range of a function. For a Borel function f : X → R the mapping defined by

 s(f )(E) = diam |f |(E) := sup |f (x)| − |f (y)| : x, y ∈ E is not a size on EB , because it is not monotone. b) Support of a function. Let X = {x1 , . . . , xn } be a finite set. For a function f : X → C we denote by supp f := {x ∈ A : f (x) = 0} A

the support of f on A ⊆ X. Then the mapping defined by   s(f )(E) = card supp |f | E

is not a size on Epower , because it does not meet the scaling property. c) Geometric mean. The mappings ⎛ ⎞  1 s(f )(E) = exp ⎝ ln |f (x)| d(x)⎠ (E) E

for f ∈ B(X), E ∈ EB and  s(f )(E) =

!

 |f (x)|

1 ν# (E)

x∈E

for f ∈ CX with X being finite and E ∈ Epower , are not sizes, because they do not satisfy the quasi-sublinearity property. This follows from the inequality stated in Remark 3.2 as both mappings are limit cases of p-mean-based sizes for p → 0+ . d) Sugeno integral. The third prominent non-additive integral (a member of the class of semicopula-based integrals [5]) is that of Sugeno (originally considered for continuous monotone measures m) given by 

 (Su) f dm = sup min α, m ({x ∈ A ∩ X : f (x) ≥ α}) . A

α>0

Since the Sugeno integral is not homogeneous (with respect to non-negative constants), see [27, Chapter 9], the mapping  (Su) sint,m (f )(E) := (Su) |f | dm E (Su)

does not satisfy the scaling property, and the average sint,m based on the Sugeno integral is not a size in general. 4. Outer essential supremum Recall that in our setting we consider a topological space X, a subcollection E ⊆ EB of Borel subsets of X, and a size s : B(X) → [0, +∞]EB . For the sake of brevity, we collect this data into a triple (X, E, s) called a sub-Borel size space keeping the collection and the size explicit in the notation. For a fixed function f ∈ B(X), the values s(f )(E) for all E ∈ E do not provide enough information to determine the essential supremum of f on a Borel set F (excluding the trivial case ∅). It is important to refer back to the function f and to truncate it according to the set F . This set enters as modifying the function f , but not impacting the sets E of the collection E. This is the essence of the definition of the outer essential supremum as defined in [9]. Since the resulting value depends on all the inputs, we indicate them all in the following notation.

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.12 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

12

Definition 4.1. Let X be a topological space. The outer essential supremum of a function f ∈ B(X) over a set F ∈ EB with respect to a size s and a collection E ⊆ EB is defined by outsup s(f ) E := sup s(f 1F )(E). F

E∈E

Usually we omit the indication of collection E from the above notation when there is no possible confusion, or the value of the outer essential supremum is the same for any collection E. The original definition in [9] does not reflect this viewpoint, because the collection E is always the domain of the size s. In our approach, for a fixed function f , size s and set F we may change the collections E which yields different values of outer essential supremum, see for instance Corollary 4.8. Monotonicity, scaling and quasi-sublinearity of the outer essential supremum follow from the corresponding properties for the size. We summarize it in the following proposition, see also [9]. Proposition 4.2. Let (X, E, s) be a sub-Borel size space. Then for every f, g ∈ B(X) and every F ∈ EB we have: (i) if |f | ≤ |g|, then outsupF s(f ) ≤ outsupF s(g); (ii) outsupF s(λf ) = |λ| outsupF s(f ) for each λ ∈ C; (iii) outsupF s(f + g) ≤ Cs (outsupF s(f ) + outsupF s(g)) for some finite constant Cs . Immediately from definition it follows that for the whole set X the outer essential supremum of f coincides with the supremum of s(f )(E) over the whole collection E. However, in general, the outer essential supremum does not coincide with the essential supremum of f on F as it is demonstrated in the following examples and further investigated in the forthcoming text. Observe that the outer essential supremum may coincide with the initial size. Relations between outer essential supremum of several operations with sizes and original sizes is straightforward. Proposition 4.3. Let (X, E, s) be a sub-Borel size space. Then for every f, g ∈ B(X) and every F ∈ EB we have: (i) (ii) (iii) (iv)

if α ∈ C, then outsupF (α  s)(f ) = |α| · outsupF s(f ); outsupF (s1 ⊕ s2 )(f ) = outsupF s1 (f ) + outsupF s2 (f ); outsupF max{s1 , s2 }(f ) = max{outsup F s1 (f ), outsupF s2 (f )};

 outsupF (s1  s2 )(f ) = max outsupF s1 (f ) E1 , outsupF s2 (f ) E2 for E = E1 ∪ E2 with E1 ∩ E2 = ∅.

4.1. Examples In what follows we provide formulas for outer essential supremum of basic sizes. Let (X, EB ) be a Borel space and f ∈ B(X). a) For the zero size szero we get trivially outsupF szero (f ) = 0 for any F ∈ EB . b) For the evaluation size se we obtain outsupF se (f ) = s˜ e (f )(F ) = 1F (e) · |f |(e) for any F ∈ EB . c) For the supremum size s∞ we get outsup s∞ (f ) E = s∞ (f )(F ),

F ∈ EB ,

F

i.e., the outer essential supremum here coincides with the usual supremum under the condition that the set F is covered by sets from E ⊆ EB . Thus, the most existing outer measure theories (e.g. Choquet capacity theory) seem to be subsumed into the new theory. (L) d) For the Lebesgue integral-based size sint, , where X = Rm , we have (L)

(L)

(L)

outsup sint, (f ) Eball = outsup sint, (f ) Ecube = sint, (f )(F ), F

F

F ∈ EB .

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.13 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

13

Moreover, if X is a topological space and a collection E satisfies property (COV), then for any σ -additive measure m we obtain (L)

(L)

outsup sint,m (f ) E = sint,m (f )(F ). F

For further explanation of this equality see Theorem 4.6. e) Similarly, for the size ssum we get outsupF ssum (f ) E = ssum (f )(F ), F ∈ EB , for any collection E satisfying property (COV). f) Let X = Rm with Lebesgue measure  and Enice be a collection of sets in EB that shrinks nicely to7 x ∈ X with respect to . Obviously, collections Eball and Ecube possess this property. If f ∈ B(X), then8 outsup s(L) ,1 (f ) Enice = ess sup |f 1F |,

F ∈ EB .

(5)

F

For f ∈ Lloc 1 (X) the previous equality follows from the generalized version of Lebesgue(–Besicovitch) differentiation theorem, see [28, Theorem 25.17]. If f ∈ / Lloc 1 (X) it follows directly from the definition. There are several directions in which the generalization could be made, e.g. for Radon measures on Rm instead of  and Eball see [12, Section 1.7.1], or considering an appropriate metric space X instead of Rm , etc. g) Equality in (5) may be easily extended to the case p > 1, i.e., for each f ∈ B(X) and F ∈ EB it holds outsup s(L) ,p (f ) Enice = ess sup |f 1F |.

(6)

F

Indeed, for any E ∈ Enice (see e.g. [15, Sections 6.10, 6.11] for inequality between any two power means and arbitrary measurable set E) we have (L)

(L)

s,1 (f 1F )(E) ≤ s,p (f 1F )(E) ≤ ess sup |f 1F |,

and therefore (L)

(L)

outsup s,1 (f ) Enice ≤ outsup s,p (f ) Enice ≤ ess sup |f 1F |. F

F

(L) s,p

This means that the sizes are indistinguishable with respect to outer essential supremum, because for each p ≥ 1 we get the equality (6). 4.2. General observations and further examples The next observation follows immediately from the definition of the outer essential supremum. It enables a relatively fast computation of the outer essential supremum for a class of sizes satisfying a relativization inequality. The resulting outer essential supremum of f on F is the value of size s of function f restricted on F . A key essence is the localization property (∀E ∈ E) s(f 1F )(E) ≤ s˜ (f )(F )

(LOC)

whenever F ∈ EB , which implies the following lemma. 7 A collection E = {E (x) : r ∈ (0, +∞)} in E is said to shrink nicely to x ∈ X with respect to a measure μ, if it satisfies the following r B conditions: 1) Er (x) ⊆ Br (x) for every r > 0; 2) there exists a constant α > 0 such that μ(Er (x)) ≥ α μ(Br (x)) for r > 0, see [28, Definition 25.16]. Sometimes an equivalent concept of measure-metrizable convergence of sets to a point is used, see [1, Chapter 8]. 8 We make use of the notation

 ess supμ f := inf α ≥ 0; {x ∈ X : f (x) > α} is null w.r.t. μ to keep the measure μ in the notation of essential supremum, see [13, Definition 4.2].

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.14 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

14

Lemma 4.4. Let (X, E, s) be a sub-Borel size space and f ∈ B(X). If F ∈ EB satisfies the property (LOC), then outsup s(f ) ≤ s˜ (f )(F ). F (low)

Example 4.5. For a monotone measure m consider the lower integral-based size sint,m . Then for a non-negative measurable function f , a fixed set F ∈ EB and each E ∈ EB it holds   (low) (low) sint,m (f 1F )(E) = (low) |f |1F dm = (low) |f | dm = sint,m (f )(E ∩ F ), E

E∩F

cf. [27, Theorem 12.3]. Since E ∩ F ⊆ F , and   (low) |f | dm ≤ (low) |f | dm, E∩F

F (low)

(low)

(low)

(low)

cf. [27, Theorem 12.4], then sint,m (f 1F )(E) = sint,m (f )(E ∩ F ) ≤ sint,m (f 1F )(F ), so the size sint,m satisfies the property (LOC), and therefore, (low)

(low)

(low)

outsup sint,m (f ) = s˜ int,m (f )(F ) = sint,m (f )(F ) for any F ∈ EB . F

The reversed inequality to the inequality in Lemma 4.4 holds trivially for F ∈ E. However, for sets F from outside the collection E, i.e., for F ∈ EB \ E, some additional assumptions are necessary in order to get the equality. Namely, monotonicity with respect to collection (∀E ∈ E) (F ⊆ E → s˜ (f )(F ) ≤ s(f 1F )(E)).

(MON)

Note that all the considered integral sizes possess this property. Moreover, if {Ei ; i ∈ N} witnesses that E satisfies (COV), then we may require sup s(f 1F )(En ) = s(f 1F )(X).

(LIM)

n∈N

In Section 4.1 we have demonstrated that the outer essential supremum may coincide with the considered size, i.e., outsupF s(f ) = s(f )(F ) without any restriction of f on the set F . This is not an accident here as it is explained in what follows. Theorem 4.6. Let (X, E, s) be a sub-Borel size space and f ∈ B(X). Then outsup s(f ) E = s˜ (f )(F ) F

in each of the following cases: (i) (X, E, s) satisfies condition (LOC) and F ∈ E; (ii) (X, E, s) satisfies conditions (LOC), (MON) and there is EF ∈ E such that F ⊆ EF ; (iii) (X, E, s) satisfies conditions (LOC), (MON), (COV), (LIM) and F is arbitrary. Hence, in all previous cases if s˜ (f )(F ) = s(f )(F ), then outsupF s(f ) = s(f )(F ). Proof. By Lemma 4.4 we have outsupF s(f ) ≤ s˜ (f )(F ) in all three cases. The reversed inequality holds in (i) trivially. (ii) Let us assume that there is a set EF ∈ E such that F ⊆ EF . By (MON) we obtain outsup s(f ) E = sup s(f 1F )(E) ≥ s(f 1F )(EF ) ≥ s˜ (f )(F ). F

E∈E

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.15 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

15

(iii) Let {Ei ; i ∈ N} witness that E satisfies (COV). By (LIM) and (MON) we have outsup s(f ) E = sup s(f 1F )(E) ≥ sup s(f 1F )(En ) = s(f 1F )(X) ≥ s˜ (f )(F ), F

E∈E

n∈N

which completes the proof. 2 Note that the outer essential supremum for the zero and evaluation size may be computed by Theorem 4.6 as well. Example 4.7. Let us consider non-additive integrals-based sizes s(N) int,m with N = {Ch, Sh}. In this case, for any f ∈ B(X) we get   (N) (N) sint,m (f 1F )(E) = (N) |f |1E∩F dm ≤ (N) |f |1F dm = s˜ int,m (f )(F ), E

F

and therefore, by Theorem 4.6(i) we have (N)

(N)

outsup sint,m (f ) E = s˜ int,m (f )(F ) for any E ⊆ EB and F ∈ E. F

In order to get the equality for an arbitrary F ∈ EB , we consider a collection E satisfying (COV). Let {Ei ; i ∈ N} be a family from property (COV). For a Borel set F and a Borel function f on X we put gn := |f 1F ∩En | and g := |f 1F |. One can verify that gn converges pointwisely to g. Indeed, if x ∈ / F , then gn (x) = g(x) = 0. If x ∈ F , then g(x) = |f (x)|, and there is n0 such that x ∈ En for any n ≥ n0 . Thus, gn (x) = |f (x)| for any n ≥ n0 . Moreover, gn ≤ gn+1 for any n ∈ N. Then by monotone convergence theorem for the Choquet and Shilkret integral (under the condition m is continuous from below), see [17, Theorem 6.1], we have   lim (N) gn dm = (N) g dm. n→∞

X

X

  " However, the sequence (N) X gn dm 

 gn dm = sup (N)

lim (N)

n→∞

X

n∈N

n∈N

is non-decreasing, and therefore

gn dm. X

In terms of sizes we have proved that (N)

(N)

(N)

sup sint,m (f 1F )(En ) = sint,m (f 1F )(X) = s˜ int,m (f )(F )

n∈N

verifying thus the condition (iii) of Theorem 4.6. An immediate observation is that the outer essential supremum can be easily computed for a special class of sizes for which the value of size s on set E from collection E is always the same. For instance, all regularized sizes sX have this property. Corollary 4.8. Let (X, E, s) be a sub-Borel size space, let f ∈ B(X) and F ∈ EB . If there is a constant a ∈ [0, +∞] such that s(f 1F )(E) = a for each E ∈ E, then outsup s(f ) = a. F

Moreover, for any measure ν on E we get   1 a outsup  s (f ) E = . ν inf ν(E) F E∈E

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.16 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

16

Observe that due to the constant value of s(f 1F )(E) on each E ∈ E the set F can be chosen arbitrarily (from the collection E as well as outside the collection E).   (N) Example 4.9. The result of Corollary 4.8 can be applied for sizes sint,m as well. Their outer essential supremum X can be stated in the form   (N) (N) outsup sint,m (f ) E = sint,m (f 1F )(E), F

X

where E is an arbitrary set from a collection E ⊆ EB . Especially, for F ∈ E the outer essential supremum takes the form (N)

(N)

sint,m (f 1F )(F ) = s˜ int,m (f )(F ),

i.e., its value depends on the set F only and not on the collection E. The last observation motivates us to state the following result. Its proof is straightforward and therefore omitted. Proposition 4.10. Let (X, E, s) be a sub-Borel size space, f ∈ B(X) and F ∈ E. If s˜ (f 1F )(E) ≤ s˜ (f )(F ) and s(f 1F )(E) ≤ s˜ (f )(F ) hold for each E ∈ E, then outsup s(f ) E = outsup s˜ (f ) E . F

F

4.3. Discrete sizes In the following we study a discrete case, i.e., when X is a non-empty finite set. To shorten many statements in this paragraph, for X being a non-empty finite set the pair (X, E) will be called a discrete sub-Borel space. Note that in this case each subset of X is Borel measurable, and thus Epower = EB . We recall two types of (discrete) sizes based on the power mean (briefly, the p-mean) and we aim to show an important difference with computing the outer essential supremum for them. Let f : X → C be a finite-valued function defined on a non-empty finite set X. Recall that the mapping sν,p : B(X) → [0, +∞]Epower defined in (3), i.e.,  sν,p (f )(E) =

1  p 1 p  (ssum )X (f )(E) ν

with p ∈ R, p > 0, is a size. Here ν is an arbitrary measure defined on Epower . Let us note that the set E from a collection does not affect the values of function f whose size is computed. The set E is taken into account only when the value ν(E) is calculated. It may look as an “artificial” size, because it does not correspond to the idea of averaging the functions on basic sets. Nevertheless, the mapping sν,p has some interesting properties which deserve to be mentioned. Example 4.11. Considering the counting measure ν# and a collection E ⊆ Epower that involves at least one singleton, for p > 0 we get  outsup sν# ,p (f ) E = F



1

p

|f (x)|

p

 1 = ssum (f p )(F ) p ,

F ∈ Epower .

x∈F

This equality follows from the fact that for F = ∅ it holds sν# ,p (f 1F )(E) ≤ sν# ,p (f 1F )({x}),

x ∈ X,

and the value sν# ,p (f 1F )({x}) is the same for each x ∈ X. Trivially, outsup∅ sν# ,p (f ) = 0. This means that the outer essential supremum of f on F depends also on the value p.

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.17 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

17

However, the situation dramatically changes when the sum (values of function f ) depends on the set E ∈ E as well. Thus, consider now the standard (discrete) p-mean s˜ ν,p given by (4), i.e., the size  s˜ ν,p (f )(E) =

1  p 1 p  ssum (f )(E) . ν

A computation of the outer supremum for this type of sizes is the result of the following proposition. Proposition 4.12. Let (X, E, s) be a discrete sub-Borel size space, let f ∈ B(X) and F ∈ Epower . If the collection E contains all singletons of F , and the following two conditions are satisfied (i) s(f 1F )(E) = s(f 1E∩F )(E) for each E ∈ E; (ii) s(f 1F )(E) ≤ max s(f 1F )({x}) for each E ∈ E, x∈F

then the outer essential supremum of f on F takes the form outsup s(f ) E = max s(f 1F )({x}). F

x∈F

Proof. Let us denote by E the set of all singletons from E. Then the following equalities hold   outsup s(f ) E = sup s(f 1F )(E) = max F

E∈E

sup s(f 1F )(E), sup s(f 1F )(E)

E∈E

E∈E\E

= sup s(f 1F )(E), E∈E

where the last equality is true because of the property (ii). Moreover, from the first assumption (i) for the set E = {x} such that x ∈ / F we get s(f 1F )(E) = s(f 1E∩F )(E) = s(f 1∅ )(E) = 0.

Finally, sup s(f 1F )(E) = sup s(f 1F )({x}) = max s(f 1F )({x}),

E∈E

x∈F

x∈F

which completes the proof. 2 Remark 4.13. Analyzing the proof of Proposition 4.12 one can easily see that if (X, E, s) is an arbitrary sub-Borel size space and maximum in condition (ii) is replaced by supremum, then the conclusion can be stated with supremum as well. Moreover, we can alternatively require the condition s(f 1F )({y}) = 0, or s(f 1F )({y}) ≤ sup s(f 1F )({x}) for y ∈ /F x∈F

instead of the condition (i) of the previous proposition. These alternative conditions may even be satisfied for a larger class of sizes. On the other hand, the requirement (i) seems to be more natural. Another way how to get the required statement is to take a collection of sets E containing only singletons of a non-empty set F . However, this requirement is not very suitable from practical point of view, e.g. when computing the super level measure. In the following example we demonstrate that the condition (i), or its alternatives mentioned in Remark 4.13, can not be omitted from Proposition 4.12. Example 4.14. Let X = {x1 , x2 } and consider the set function ν : Epower → [0, +∞] given by ν(∅) = 0, ν({x1 }) = 1, ν({x2 }) = 0.5 and ν({x1 , x2 }) = 1. Then for the function f (x) = 1 for x = x1 , x2 , the singleton F = {x1 } and the size sν,p given by (3) we get

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.18 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

18

sν,p (f 1F )(∅) = 0, sν,p (f 1F )({x1 }) = f (x1 ) = 1,

f (x1 ) = 2, 0.5 sν,p (f 1F )({x1 , x2 }) = f (x1 ) = 1. sν,p (f 1F )({x2 }) =

For the outer essential supremum we obtain outsup sν,p (f ) Epower = F

sup

sν,p (f 1F )(E) = 2

E∈Epower

= 1 = sν,p (f 1F )({x1 }) = sup sp,ν (f 1F )({x}). x∈F

It is easy to check that none of the above mentioned conditions is satisfied. Corollary 4.15. Let (X, E) be a discrete sub-Borel space, f ∈ B(X) and F ∈ Epower . If E contains all singletons of F , then outsup s˜ ν# ,p (f ) E = max |f (x)|. x∈F

F

Proof. It is enough to verify the assumptions of Proposition 4.12. It is easy to see that s˜ ν# ,p (f 1F )(E) = s˜ ν# ,p (f 1E∩F )(E) holds true for each E ∈ E. The second inequality (ii) is also true for the following reasons: for an arbitrary E ∈ E we get  s˜ ν# ,p (f 1F )(E) = s˜ ν# ,p (f 1E∩F )(E) =

 ≤

 1 |f (x)|p ν# (E)

1

p

x∈E∩F

 1 |f (x)|p ν# (E ∩ F )

1

p

x∈E∩F

≤ max |f (x)|, x∈E∩F

where the last inequality follows from the property of quasi-arithmetic means, i.e., they are bounded from above by maximum. Hence, max |f (x)| ≤ max s˜ ν# ,p (f 1F )({x}).

x∈E∩F

x∈F

Using the fact that s˜ ν# ,p (f 1F )({x}) = |f (x)| for x ∈ F , we get the required result.

2

Remark 4.16. Similarly to the case of Lp -mean-based sizes discussed in Section 4.1(vii), the result of Corollary 4.15 says that whenever choosing a p-mean-based size s˜ ν# ,p the corresponding outer essential supremum of function f on F will always be the same – no matter which p ∈ (0, +∞] is taken.9 It means that on the level of outer essential supremum these sizes are indistinguishable and the corresponding outer essential supremum coincides with the maximum. From this point of view, it is irrelevant (for theoretical purposes) to consider other than 1 -based sizes. Of course, for practical purposes it is sometimes more useful to take 2 -averages, or other ones. This observation has important consequences in practical computation of super level measures and integrals based on them. Corollary 4.17. Let X be any topological space and f ∈ B(X). Then for each F ∈ EB it holds outsup senc (f ) Esing = sup |f (x)| = s∞ (f )(F ). F

x∈F

9 When identifying s˜ ν# ,+∞ with s∞ .

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.19 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

19

5. Super level measure In many situations, such as the definition of the essential supremum, or definition of Lp -norms (see below), or the weak type estimates of the Hardy–Littlewood operator, or the construction of certain non-additive integrals, etc., the upper level set of a (measurable) function f of the form Fα = {x ∈ X : |f (x)| > α}, frequently appears. This set is called the (strict) super level set as well as strict α-cut of f , etc. depending on the literature. Measuring this set10 using a monotone measure m yields the non-increasing real-valued function hm,f (α) := m(Fα ) known also as decumulative distribution of f with respect to m, or survival function, see [13, Section 4.2]. Indeed, for each α > 0 we may write  

 hm,f (α) = m {x ∈ X; |f (x)| > α} = inf m(F ) : F ∈ EB , (∀x ∈ X \ F ) |f (x)| ≤ α   = inf m(F ) : F ∈ EB , sup |f (x)| ≤ α x∈X\F





= inf m(F ) : F ∈ EB , outsup s∞ (f ) EB ≤ α .

(7)

X\F

In order to distinguish between the decumulative distribution function and the new more general object introduced in Definition 5.1, we will use the term “standard level measure” of f with respect to m for the function hm,f from now on. Various aggregations of the values of α and hm,f (α) into one representative value yield various non-additive integrals (including the already mentioned Choquet, Shilkret and Sugeno integrals). Also, recall that for 0 < p < +∞ and a measurable function f on X with an outer measure m it holds p f p

∞

 =

|f | dm = p

X

p α p−1 hm,f (α) dα. 0

This equality is a starting point of D O and T HIELE for introducing their Lp -theory for outer measures. Following a concrete-to-abstract principle they generalize the Lp -norm definition by introducing a new quantity to replace the function hm,f (α). Formally replacing the supremum size s∞ by a general size s on E in (7) we get the formula 



inf μ(F ) : F ∈ EB , outsup s(f ) E ≤ α ,

α > 0.

X\F

This new quantity involves predefined averages over basic sets (i.e., the concept of sizes) on the one hand, and an abstractly generated outer measure μ given by (1) from a concrete pre-measure σ on E on the other hand. Recall that the generated outer measure μ is always sub-additive (providing a generalization of Lebesgue outer measure). For our purposes it is still reasonable to allow sizes to do their work on functions, but in order to consider general measures, we leave the concrete-to-abstract setup, i.e., we do not consider a pre-measure σ on a basic collection E and, thus, μ need not be a generated outer measure. In this context, it is enough to consider only monotone measures11 μ on EB . So, we summarize all of this in the following formal definition. Note that we use the same name super level measure for the new quantity as in [9], but in a wider context. 10 Measuring the upper level set appears frequently in function spaces theory, see e.g. [7,8,18,21,27]. 11 This requirement seems to be natural for practical reasons, because monotone measures are used for modelling problems in non-deterministic

environment, especially in decision making, sociology, biology or economic mathematics. In the recent years, this area has been widely developed and a great variety of topics has been explored.

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.20 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

20

Definition 5.1. Let (X, E, s) be a sub-Borel size space and μ : EB → [0, +∞] be a monotone measure. The quantity   μ(s(f ) E > α) := inf μ(F ) : F ∈ EB , outsup s(f ) E ≤ α ,

α > 0,

X\F

is called a super level measure of f ∈ B(X) with respect to measure μ, size s and collection E. We keep indicating the basic collection E in the notation of super level measure analogously as it was done in the case of the outer essential supremum. The following properties of super level measure follow from the corresponding properties of the outer essential supremum, see also [9]. Proposition 5.2. Let (X, E, s) be a sub-Borel size space and μ be a monotone measure. Then (i) if f, g ∈ B(X) such that |f | ≤ |g|, then μ(s(f ) E > α) ≤ μ(s(g) E > α); (ii) for each f ∈ B(X) and each complex number α  we have μ(s(α  f ) E > |α  |α) = μ(s(f ) E > α); (iii) for some finite constant C independent of f, g ∈ B(X) and s we have μ(s(f + g) E > Cα) ≤ μ(s(f ) E > α) + μ(s(g) E > α). 5.1. Examples Let (X, E) be a sub-Borel space, μ be a monotone measure on EB and f ∈ B(X). a) If s is such that outsupF s(f ) E = a for each non-empty F ∈ EB , then  0, α ≥ a, μ(s(f ) E > α) = μ(X), α < a. Clearly, all regularized sizes possess this property. In particular, for szero we get μ(szero (f ) E > α) = 0 for any E ⊆ EB . b) Let e ∈ X be fixed. If |f |(e) ≤ α, then μ(se (f ) E > α) = 0 as well. If |f |(e) > α, then μ(se (f ) E > α) = μ({e}). c) For a finite set X and a collection E containing all singletons of X, we may easily conclude that if outsup s(f ) E = max s(f )({x}) for any set F ∈ Epower , F

x∈F

(8)

then μ(s(f ) E > α) = μ({x ∈ X; s(f )({x}) > α}). Corollary 4.15 shows that s˜ ν# ,p has the property (8). Further examples of such sizes may be easily obtained using concatenation, e.g. for β > 1 the sizes    (β  senc )(f )(E), E ∈ Esing , sβ (f )(E) = (β  senc )  s˜ ν# ,p = s˜ ν# ,p (f )(E), E ∈ EB \ Esing , possess the property (8), and therefore μ(sβ (f ) E > α) = μ({x ∈ X : β |f (x)| > α}), which clearly does not coincide with the standard level measure of f with respect to μ. (L) d) By Theorem 4.6, if collection E satisfies condition (COV), then for any F ∈ EB we have outsupF sint,m (f ) E = " F |f (x)| dm(x). Thus,

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.21 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

⎧ ⎪ ⎨

  μ s(L) int,m (f ) E > α = inf ⎪μ(F ) : F ∈ EB , ⎩

⎫ ⎪ ⎬

 X\F

21

|f (x)| dm(x) ≤ α . ⎪ ⎭

 (L) |f (x)| dm(x) ≤ α, then μ s (f ) E

> α = 0. However, one can easily find simple examint,m X   (L) ples which show that μ sint,m (f ) E > α is non-zero and differs from the standard level measure. For instance,   if X = [0, 1], then for α < 1 we have (1[0,1] > α) = 1, but  s(L) int, (1[0,1] ) E > α = 1 − α. e) For the size ssum we similarly get ⎧ ⎫ ⎨ ⎬  μ(ssum (f ) E > α) = inf μ(F ) : |f (x)| ≤ α ⎩ ⎭ If α is such that



"

x∈X\F

for any collection E satisfying (COV). (N) f) For the non-additive integral size sint,m with N ∈ {Ch, Sh}, where m is an appropriate12 monotone and continuous from below measure, and any collection E satisfying (COV) we get ⎧ ⎫ ⎪ ⎪  ⎨ ⎬   (N) μ sint,m (f ) E > α = inf μ(F ) : F ∈ EB , (N) |f | dm ≤ α . ⎪ ⎪ ⎩ ⎭ X\F

  The same result holds when considering “artificial” sizes of the form s(N) int,m X . In this case it is enough to consider an arbitrary collection E ⊆ EB . Remark 5.3. In general, a computation of super level measure (even if the outer essential supremum is given) is a challenging problem. However, it seems possible to provide a simple algorithm for super level measure computation in some specific situations, for instance on a discrete sub-Borel size space, see the recent paper [4]. 5.2. Super level versus standard level measure Comparing the definition of standard level measure and the definition of super level measure we may observe that the formula (∀x ∈ X \ F ) |f (x)| ≤ α is formally replaced by the formula outsupX\F s(f ) E ≤ α. However, the super level measure may still coincide with the standard level measure, i.e., the measure of the Borel set where |f | is larger than α, as the following assertion shows. It was remarked by D O and T HIELE in [9] without any proof. Proposition 5.4. Let (X, E, s) be a sub-Borel size space and μ be a monotone measure such that outsup s(f ) = ess supμ |f 1F |,

F ∈ EB .

F

Then μ(s(f ) E > α) = μ({x ∈ X : |f (x)| > α}). In particular, if outsupF s(f ) E = sup |f |[F ] with F ∈ EB , then μ(s(f ) E > α) = μ({x ∈ X : |f (x)| > α}). Proof. Let F ∈ EB . If ess supμ |f 1X\F | ≤ α, then there is B ∈ EB such that μ(B) = 0 and (X \ F ) \ B ⊆ {x ∈ X : |f (x)| ≤ α} \ B, and thus {x ∈ X : |f (x)| > α} \ B ⊆ F \ B. From monotonicity of measure μ we then have μ({x ∈ X : |f (x)| > α}) = μ({x ∈ X : |f (x)| > α} \ B) ≤ μ(F \ B) = μ(F ), 12 I.e., submodular for the Choquet-type size and maxitive for the Shilkret-type size, see Section 3.2 paragraph Non-additive integrals-based sizes.

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.22 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

22

and finally we obtain 



μ(s(f ) E > α) = inf μ(F ) : F ∈ EB , outsup s(f ) E ≤ α X\F

 = inf μ(F ) : F ∈ EB , ess supμ |f 1X\F | ≤ α = μ({x ∈ X : |f (x)| > α}). To prove the particular case, one can consider a measure μ which assigns 1 to each non-empty set. 2 Remark 5.5. If X is a finite set, then supremum coincides with maximum. Thus, in such a case, if outsup s(f ) E = max |f |[F ] F

for any set F , then for the super level measure we have μ(s(f ) E > α) = μ({x ∈ X : |f (x)| > α}). Example 5.6. From Proposition 5.4 we immediately get that the values

  (L) μ(s∞ (f ) E > α), μ(˜sν# ,p (f ) E > α), μ(senc (f ) E > α), μ s,p (f ) E > α

coincide with the value μ({x ∈ X : |f (x)| > α}) for a suitable sub-Borel space (X, E), respectively. Examples in Section 5.1 demonstrate that if a sub-Borel size space (X, E, s) and a monotone measure μ on EB are given, then in general μ(s(f ) E > α) = μ({x ∈ X : |f (x)| > α}). However, we still do not know whether there is a monotone measure m such that for each f ∈ B(X) and each α > 0 we have the equality μ(s(f ) E > α) = m({x ∈ X : |f (x)| > α}). Remark 5.5 shows that the answer is positive for several sizes and measures. However, in general, this is not the case. We begin with a simple observation which answers the question negatively. For a monotone measure μ on Epower denote by N (μ) the family of all null sets w.r.t. μ, i.e., sets N ∈ Epower with μ(N) = 0. Proposition 5.7. Let (X,  E) be a discrete sub-Borel space and let μ be a monotone measure on Epower . If & card F ∈N (μ) X \ F ≥ 2, then for any α > 0 there is f ∈ B(X) such that {x ∈ X : |f (x)| > α} = ∅ and μ(ssum (f ) E > α) > 0. Proof. Put '

A=

X\F

F ∈N (μ)

and let us assume that card(A) ≥ 2. For an arbitrary α > 0 we may define function f as f (x) = 0 for x ∈ / A and f (x) = α for x ∈ A. Immediately we obtain that {x ∈ X : |f (x)| > α} = ∅. If F ∈ N (μ), then X \ F ⊇ A. Thus,   outsup ssum (f ) E = f (x) ≥ f (x) = α · card(A) ≥ 2α > α, X\F

x∈X\F

which yields μ(ssum (f ) E > α) > 0.

x∈A

2

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.23 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

23

From Example 5.1(iii) we can see that although the super level measure is not the standard level measure of f , it still coincides with the standard level measure of another function (in this case g = β f ). The following example shows that this is not true in general, i.e., the super level measure μ(s(f ) E > α) in the example cannot be a standard level measure of any function defined on X with respect to any measure m. Example 5.8. Let X = {a, b, c} and μ be a monotone measure on Epower . We assume that μ is strictly increasing with respect to the following order ≺ on Epower : ∅ ≺ {a} ≺ {b} ≺ {c} ≺ {a, b} ≺ {a, c} ≺ {b, c} ≺ X. We define a function f on X as f (a) = 2, f (b) = 3, f (c) = 4. One can see that varying α > 0, the super level measure μ(ssum (f ) > α) takes all 8 possible values of measure μ. In contrary, varying α > 0 in the standard level measure m({x ∈ X : |g(x)| > α}) for any function g on X and any measure m on Epower one can obtain at most 5 different values. On the strength of Example 5.8, we may conclude that there is a measure μ and a size s on E such that for each function g and each measure m there is a function f and a number α > 0 such that μ(s(f ) E > α) = m({x ∈ X : |g(x)| > α}). This verifies that the super level measure concept is a “full-value” generalization of the standard level measure concept. 5.3. Indistinguishability Results of the previous section motivate us to introduce an equivalent relation between triples of measures, sizes and functions. In fact, this provides a generalization of the equality almost everywhere of two functions. Definition 5.9. Let (X, E) be a sub-Borel space. The triples (μ1 , s1 , f1 ) and (μ2 , s2 , f2 ), where μi : EB → [0, +∞] are monotone measures, si : B(X) → [0, +∞]EB are sizes and fi ∈ B(X) with i ∈ {1, 2}, are called integral equivalent, if μ1 (s1 (f1 ) E > α) = μ2 (s2 (f2 ) E > α) for all α ∈ (0, +∞]. Remark 5.10. In the classical case, i.e., when the super level measure is the standard level measure, integral equivalence can be viewed as a generalization of the stochastic equivalence of random variables. Thus, two random variables (possibly defined on two different probability spaces) are integral equivalent, if and only if they have coincident distribution functions. Fixing pairs of the three inputs in the definition we get the following triple of useful concepts. Indistinguishable functions If (μ, s, f1 ) and (μ, s, f2 ) are integral equivalent, then we say that the functions f1 and f2 are (μ, s)-indistinguishable. Note that for the size s such that outsup s(f ) E = sup |f |[F ],

F ∈ EB ,

F

the (μ, s)-indistinguishability reduces to the μ-indistinguishability of functions f1 and f2 from [25]. Moreover, in such a case for μ being a probability measure, the classical equality μ-almost everywhere of two functions f1 and f2 implies that f1 and f2 are μ-indistinguishable, but not vice versa, i.e., the μ-indistinguishability relation is a generalization of the equality μ-almost everywhere. For the greatest monotone measure μ : EB → [0, +∞] given by  0, E = ∅,  μ (E) = +∞, otherwise, we characterize (μ , s)-indistinguishability of two functions as follows.

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.24 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

24

Proposition 5.11. Let (X, E, s) be a sub-Borel size space. Functions f1 , f2 ∈ B(X) are (μ , s)-indistinguishable if and only if outsup s(f1 ) E = outsup s(f2 ) E . X

(9)

X

Proof. “⇐” From monotonicity of s for each F ∈ EB we have outsup s(fi ) E ≤ outsup s(fi ) E = a, X\F

i ∈ {1, 2}.

X

If α ≥ a, then 



∅ ∈ F ∈ EB : outsup s(fi ) E ≤ α X\F

and we get i ) E > α) = 0 for each i ∈ {1, 2}. Otherwise, if α < a, then F = ∅ does not satisfy the inequality outsupX\F s(fi ) E ≤ α, therefore μ (s(fi ) E > α) = +∞ for each i ∈ {1, 2}. Thus, f1 and f2 are (μ , s)-indistinguishable. “⇒” Let a = outsupX s(f1 ) E < outsupX s(f2 ) E = b without loss of generality. Then for α ∈ (a, b) we have that outsupX s(f1 ) E ≤ α, i.e., μ (s(f1 ) E > α) = 0, but α < outsupX s(f2 ) E , i.e., μ (s(f2 ) E > α) = +∞. Thus, f1 and f2 are not (μ , s)-indistinguishable. 2 μ (s(f

As a consequence, for size s such that outsupF s(f ) E = sup |f |[F ] for a collection E satisfying (COV) we have that f1 and f2 are μ -indistinguishable if and only if sup |f1 | = sup |f2 |. If X is finite, then f1 and f2 are μ -indistinguishable if and only if max |f1 | = max |f2 |. Similarly, for the weakest monotone measure μ : EB → [0, +∞] given by  +∞, E = X, μ (E) = 0, otherwise, we have the following characterization. The proof is very similar to the proof of Proposition 5.11 and therefore omitted. Proposition 5.12. Let (X, E, s) be a sub-Borel size space. Functions f1 , f2 ∈ B(X) are (μ , s)-indistinguishable if and only if outsup s(f1 ) E =

inf

F ∈EB \{X} X\F

inf

outsup s(f2 ) E

F ∈EB \{X} X\F

(10)

and both terms either simultaneously possess the minimal value, or not. The only difference in this characterization is the additional condition that both terms in (10) either simultaneously possess the minimal value, or not. This condition is important. Otherwise, let us suppose without loss of generality that the set   Ai := outsup s(fi ) E : F ∈ EB \ {X} ,

i = 1, 2,

X\F

possess the minimal value only for i = 1. Then for α = min A1 it may happen 0 = μ (s(f1 ) E > α) = μ (s(f2 ) E > α) = +∞. Remark 5.13. Note that the condition (10) does not seem to be “dually” related to the condition (9). However, the condition (9) may be written equivalently in the form sup outsup s(f1 ) E = sup outsup s(f2 ) E ,

F ∈EB

F

F ∈EB

F

which uncovers the required duality (between supremum and infimum).

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.25 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

25

We shall compare the notion of indistinguishability with the equality almost everywhere with respect to monotone measure based on the notion of null set in non-additive setting, see [22] and [13]. Definition 5.14. A set N ∈ EB is called a null set with respect to a measure m, if m(E ∪ N ) = m(E) for all E ∈ EB . We say that functions f1 , f2 ∈ B(X) equal almost everywhere with respect to m, if there exists a null set N ∈ EB with respect to m such that f1 (x) = f2 (x) for all x ∈ X \ N . Theorem 5.15. Let (X, E, s) be a sub-Borel size space and μ be a monotone measure. If f1 , f2 ∈ B(X) equal almost everywhere with respect to μ, then f1 and f2 are (μ, s)-indistinguishable. Proof. Let N ∈ EB be a null set with respect to μ such that f1 (x) = f2 (x) for each x ∈ X \ N . Then for each F ∈ EB we get outsup s(fi ) E ≤ outsup s(fi ) E ,

X\(F ∪N )

i ∈ {1, 2}

X\F

and outsupX\(F ∪N ) s(f1 ) E = outsupX\(F ∪N ) s(f2 ) E . Thus, for each α > 0 we have   μ(s(f1 ) > α) = inf μ(F ∪ N ) : F ∈ EB , outsup s(f1 ) E ≤ α X\(F ∪N )





= inf μ(F ∪ N ) : F ∈ EB , outsup s(f2 ) E ≤ α X\(F ∪N )

= μ(s(f2 ) > α).

2

Theorem 5.15 says that two functions which equal almost everywhere with respect to μ are (μ, s)-indistinguishable no matter which size s is chosen. The converse does not hold in general. For instance, the empty set is the only null set with respect to μ , and therefore functions f1 , f2 equal almost everywhere with respect to μ if and only if f1 (x) = f2 (x) for each x ∈ X. On the other hand, by Proposition 5.11 the functions f1 and f2 are (μ , s)-indistinguishable if and only if the condition (9) holds. For instance, for the size s∞ this is equivalent to sup |f1 | = sup |f2 |. Indistinguishable measures If (μ1 , s, f ) and (μ2 , s, f ) are integral equivalent, then we say that the measures μ1 and μ2 are (s, f )-indistinguishable. For a particular case of constant sizes and functions we may provide the following characterization. It immediately follows from example (a) in Section 5.1. Proposition 5.16. Let (X, E, s) be a sub-Borel size space with f ∈ B(X) being such that outsupF s(f ) E is constant for each non-empty F ∈ EB . Then measures μ1 and μ2 on EB are (s, f )-indistinguishable if and only if μ1 (X) = μ2 (X). As a corollary we have that for the size s∞ and a constant function f = c ∈ [0, +∞] on X two measures μ1 and μ2 are (s∞ , c)-indistinguishable if and only if μ1 (X) = μ2 (X). More general, for the function fA = c · χA with A ∈ EB we get that μ1 and μ2 are (s∞ , fA )-distinguishable if and only if μ1 (A) = μ2 (A). Indistinguishable sizes If (μ, s1 , f ) and (μ, s2 , f ) are integral equivalent, then we say that the sizes s1 and s2 are (μ, f )-indistinguishable. Modifying the assumptions of Proposition 5.11 we easily get the following characterization. Proposition 5.17. Let (X, E) be a sub-Borel space and f ∈ B(X). Sizes s1 , s2 are (μ , f )-indistinguishable if and only if outsupX s1 (f ) E = outsupX s2 (f ) E . Moreover, s1 , s2 are (μ , f )-indistinguishable if and only if inf

outsup s1 (f ) E =

F ∈EB \{X} X\F

inf

outsup s2 (f ) E

F ∈EB \{X} X\F

and both terms either simultaneously possess the minimal value, or not.

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.26 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

26

6. Concluding remarks In this paper we have studied the recent concept of super level measure introduced by D O and T HIELE. However, our approach differs from the original one in at least two points: (I) we enable sizes to be defined on all Borel measurable subsets of X instead of a basic collection E; and more important, (II) measure μ appearing in the definition of super level measure need not be an outer measure generated by a pre-measure on E. In this wider context we have introduced a number of sizes related to various aggregations (e.g. non-additive integralsbased sizes, mean-like-based sizes, etc.) instead of Lp -based averages considered by Do and Thiele [9]. We have further investigated certain properties of outer essential supremum and super level measures for some families of sizes. We have also provided explicit formulas for outer essential supremum and corresponding super level measures for many of the considered sizes. Our results are summarized in Table 1 in Appendix. Integrals based on super level measures Once the super level measure is introduced, the definition of various (in general, non-additive) integrals with respect to a (non-additive) measure is immediate and copies the classical case replacing formally the standard level measure with the super level measure. A novelty is that the new integrals depend on the size to be chosen. On the one hand, the “by-size-generalized” Choquet and Shilkret integrals are already included explicitly in the definition of an Lp -norm for functions on outer measures spaces in [9]: if μ is an outer measure generated by a pre-measure σ on E, the outer Lp -space norm for 1 ≤ p < +∞ is defined by ⎛∞ ⎞1/p  f Lp (X,μ,s) := ⎝ p α p−1 μ(s(f ) E > α) dα ⎠ 0

mimicking the definition of Lp -spaces based on the standard Choquet integral. Also, weak Lp -space (Lorentz space) is defined by

 f Lp,∞ (X,μ,s) := sup α p μ(s(f ) E > α) . α>0

Observe that both these definitions are closely related to non-additive integrals of Choquet and Shilkret (in the super level measure context), and the classical Choquet and Shilkret integrals are only their special cases. On the other hand, a deeper study of the properties of these integrals in a dependence on the chosen size is still missing. Some examples in discrete setting are given in [14]. Let us repeat again that these integrals serve in [9] (as well as in further publications on the topic) as a tool, and they are not a basic object of their research. Naturally, the concept of “by-size-generalized” integrals may be studied from different viewpoints and may provide an interesting background for its own research. An interpretation of various averaging procedures (sizes) when considering super level measure-based integrals as a tool for evaluating data from various sources (such as in behavioural sciences, in classification and multicriteria decision making, etc.) is also interesting itself. Moreover, such a concept has not been studied in the area of non-additive measures and integrals so far, and surely will bring new results, techniques, applications, etc. Acknowledgements We thank the referees for their valuable comments and suggestions improving previous versions of the paper, especially for pointing out the reference [23]. This work was supported by the Slovak Research and Development Agency under the contract No. APVV-16-0337, the Internal University grant No. VVGS-2016-255 and the Internal Faculty grant No. VVGS-PF-2017-255. Jozef Kisel’ák acknowledges LIT-2016-1-SEE-023 project “Modeling complex dependencies: how to make strategic multicriterial decisions? (mODEC)”.

JID:FSS AID:7391 /FLA

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.27 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

27

Table 1 Formulas for outer essential supremum and super level measures for certain sizes. Size

Collection

outsupF s(f ) E

μ(s(f ) E > α)

szero

E

0

se

E

1F (e)|f |(e)

0  0, μ({e}),

senc

Esing

s∞ (f )(F )

μ({x ∈ X : |f (x)| > α})

s∞

E

s∞ (f )(F )

Eball , Ecube

(L) sint, (f )(F )

μ({x ∈ X : inf μ(F ) :

Enice

ess sup |f 1F |

({x ∈ X : |f (x)| > α})

E

sint,m (f )(F )

(L) sint, (L) s,p , p ≥ 1 (Ch)

sint,m

(Ch)

Additional conditions

|f |(e) ≤ α, |f |(e) > α |f (x)| > α}) (L) sint, (f )(F ) ≤ α

F is covered by sets from E

m is a submodular measure



(Ch) inf μ(F ) : sint,m (f )(F ) ≤ α

m is continuous from below E satisfies (COV)

(Sh)

sint,m

(Sh)

m is a maxitive measure



(Sh) inf μ(F ) : sint,m (f )(F ) ≤ α

E

sint,m (f )(F )

(low) sint,m

EB

(low) sint,m (f )(F )

sν# ,p

E



s˜ ν# ,p , p > 0

E

max |f (x)|

μ({x ∈ X : |f (x)| > α})

E ⊆ Epower containing all singletons of F

s

E

max s(f )({x})

μ({x ∈ X : s(f )({x}) > α})

E ⊆ Epower containing all singletons of F

ssum

1/p (f p )(F )

x∈F

x∈F



(low) inf μ(F ) : sint,m (f )(F ) ≤ α



inf μ(F ) :



m is continuous from below E satisfies (COV)

1/p ssum (f p )(F ) ≤α

m is a monotone measure E ⊆ Epower involving at least 1 singleton

Appendix A In Table 1 we collect the explicit formulas for outer essential supremum and super level measures for many sizes considered in the paper. The first column of the table represents specific (or general) size, the second column is the collection on which the computations hold (if not specified, E represents an arbitrary collection from EB ), the third column provides the formula for outer essential supremum of f ∈ B(X) on a set F ∈ EB and the fourth column is the corresponding formula for super level measure. The last column specifies additional conditions (if any) under which the formulas hold. References [1] J.J. Benedetto, W. Czaja, Integration and Modern Analysis, Birkhäuser Advanced Texts, Birkhäuser, Boston, 2009. [2] S.K. Berberian, Borel spaces, in: Functional Analysis and Its Applications, Nice, 1986, in: ICPAM Lecture Notes, World Sci. Publishing, Singapore, 1988, pp. 134–197. [3] M. Boczek, M. Kaluszka, On the Minkowski–Hölder type inequalities for generalized Sugeno integrals with an application, Kybernetika 52 (3) (2016) 329–347. [4] J. Borzová, L. Halˇcinová, J. Šupina, Size-based super level measures on discrete space, submitted for publication. [5] J. Borzová-Molnárová, L. Halˇcinová, O. Hutník, The smallest semicopula-based universal integrals I: properties and characterizations, Fuzzy Sets and Systems 271 (2015) 1–17. [6] V.I. Burenkov, Introduction to the theory of Lp -spaces, available at http://www.math.unipd.it/~burenkov/dispense. [7] R.E. Castillo, H. Rafeiro, An Introductory Course to Lebesgue Spaces, CMS Book in Mathematics, Springer, 2016. [8] G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1953) 131–295. [9] Y. Do, C. Thiele, Lp theory for outer measures and two themes of Lennart Carleson united, Bull. Amer. Math. Soc. 52 (2) (2015) 249–296. [10] L. Drewnowski, Topological rings of sets, continuous set functions, integration I, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 20 (4) (1972) 269–276. [11] Y. Even, E. Lehrer, Decomposition-integral: unifying Choquet and the concave integrals, Econ. Theory 56 (2014) 33–58. [12] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992.

JID:FSS AID:7391 /FLA

28

[m3SC+; v1.282; Prn:19/03/2018; 11:30] P.28 (1-28)

L. Halˇcinová et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

[13] M. Grabisch, Set Functions, Games and Capacities in Decision Making, Theory and Decision Library C, vol. 46, Springer International Publishing, Switzerland, 2016. [14] L. Halˇcinová, Sizes, super level measures and integrals, in: V. Torra, R. Mesiar, B. Baets (Eds.), Aggregation Functions in Theory and in Practice, AGOP 2017, in: Advances in Intelligent Systems and Computing, vol. 581, Springer, Cham, 2018, pp. 181–188. [15] G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, 2nd ed., Cambridge Univ. Press, Cambridge, 1952. [16] E. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York, 1975. [17] J. Kawabe, A unified approach to the monotone convergence theorem for nonlinear integrals, Fuzzy Sets and Systems 304 (2016) 1–19. [18] J. Kinnunen, O. Martio, The Sobolev capacity on metric spaces, Ann. Acad. Sci. Fenn. Math. 21 (1996) 367–382. [19] E.P. Klement, R. Mesiar, F. Spizzichino, A. Stupˇnanová, Universal integrals based on copulas, Fuzzy Optim. Decis. Mak. 13 (3) (2014) 273–286. [20] J. Matkowski, The converse theorem for Minkowski’s inequality, Indag. Math. 15 (1) (2004) 73–84. [21] M. Milman, J. Xiao, The ∞-Besov capacity problem, Math. Nachr. 290 (2017) 2961–2976. [22] E. Pap, Null-Additive Set Functions, Kluwer–Ister Science, Dordrecht–Bratislava, 1995. [23] E. Pap, Sublinear means, in: B. de Baets, R. Mesiar, S. Saminger-Platz, E.P. Klement (Eds.), Functional Equations and Inequalities, 36th Linz Seminar on Fuzzy Set Theory, Johannes Kepler Universität, Linz, 2016, pp. 75–79. [24] N. Shilkret, Maxitive measure and integration, Indag. Math. 33 (1971) 109–116. [25] P. Struk, Extremal fuzzy integrals, Soft Comput. 10 (2006) 502–505. [26] J. Šipoš, Integral with respect to a pre-measure, Math. Slovaca 29 (1979) 141–155. [27] Z. Wang, G. Klir, Generalized Measure Theory, IFSR International Series on Systems Science and Engineering, vol. 25, Springer, 2009. [28] J. Yeh, Real Analysis. Theory of Measure and Integration, 2nd ed., World Scientific, Singapore, 2006.