On Riemann Integrability of Monotone Multivariate Real-Valued Functions

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On Riemann Integrability of Monotone Multivariate Real-Valued Functions

Hilaire Touyem, Issofa Moyouwou, Bertrand Tchantcho

To cite this version:

Hilaire Touyem, Issofa Moyouwou, Bertrand Tchantcho. On Riemann Integrability of Monotone Mul- tivariate Real-Valued Functions. 2021. �hal-03290360�

On Riemann Integrability of Monotone Multivariate Real-Valued Functions

June 30, 2021

Hilaire Touyem ¹ and Issofa Moyouwou², and Bertrand Tchantcho³ Abstract

A monotonicity condition for multivariate real-valued functions is presented as a simple sufficient condition for Riemman integrability. The one dimension well-known result is extended to show that any monotone and bounded real-valued function on a bounded Jordan measurable subset of the n-cartesian product of the set R of real numbers is Riemann integrable. As an application, this result allows to show that some characteristics of aggregation functions are well-defined without further constraints such as continuity.

1 Introduction

Riemann’s definition [Riemann, 1854] of integral has opened up a fairly wide field of research in integration theory. To this end, numerous results have been developed on this issue. A number of these results deal with a necessary and/or sufficient condition for a real-valued functions defined on a bounded and closed interval of R to be Riemann integrable; see [Levine, 1977], [Protter and Charles Jr, 1977] or [Taylor, 2006]. In particular, it is well known that any mono- tone real-valued function defined on a closed and bounded interval of real numbers is Rie- mann integrable, see for example [Protter and Charles Jr, 1977, Section 5.1, Corrolary 2] or [Taylor, 2006, Theorem 1.21]. To the best of our knowledge, no similar monotonicity condition which guarantees the integrability of multivariate real-valued functions defined on multidimen- sional rectangular boxes (also called cells) of Rⁿ has been proposed in the literature. Some authors are even skeptical by asserting that “ In the case of the real line we have seen that also certain discontinuous functions, e.g. bounded monotone functions, are integrable. The notion of monotonicity cannot be extended to Rⁿ”, see [Jacob and Evans, 2016, pp. 411]. There- fore continuity is sometimes used to justify the Riemann integrability of such functions, see

1Department of Mathematics, Faculty of Sciences, University of Yaounde I, PO Box 812, Yaounde, Cameroon.

E–mail: hilairetouyem2012@yahoo.fr

2 Advanced Teachers Training College, University of Yaounde I, PO Box 47 Yaounde, Cameroon. E-mail:

imoyouwou2@yahoo.fr

3Advanced Teachers Training College, University of Yaounde I, PO Box 47 Yaounde, Cameroon. E–mail:

btchantcho@yahoo.fr

[Guzman, 2012, Section 4.2] for more details. Furthermore, the lack of a quite simple exten- sion of the integrability of “monotone” multivariate real-valued function also constraint some authors such as [Grabisch et al., 2009] and [Kurz, 2018], who work on aggregation functions (a class of multivariate real-valued functions defined on multidimensional rectangular boxes with practical application), to either assume the integrability or continuity of functions considered.

In this paper, we propose a monotonicity condition as a sufficient criterion to guarantee the Riemann integrability (and thus, Lebesgue integrability) of multivariate real-valued func- tions defined on a cell of Rⁿ. The concept of monotone function defined on Rⁿ clearly extends the classical monotone function when n “ 1. This rather simple result therefore generalizes the well-known one dimensional monotonicity condition of Riemann integrability of real-valued functions. As an application, our result allows us to show that some characteristics of aggre- gation functions defined using integrals, see for example [Grabisch et al., 2009, Section 10.3]

and [Kurz, 2018], are well defined without additional constraints such as continuity required in [Kurz, 2018].

The rest of this paper is organized as follows. In Section 2, we introduce the preliminary notations and definitions. The Section 3 is devoted to the main result and Section 4 presents an application of our result to aggregation functions.

2 Basic notations and definitions

Throughout this paper, we consider some distinctions on real intervals with boundsαand β as follows:

• rα, βs “ txPR:αďxďβu;

• sα, βr“ txPR:α ăxăβu;

• rα, βr“ txPR:α ďxăβu;

• sα, βs “ txPR:αăxďβu.

Given a positive integer n ě 1 and two n-tuples a “ paiq1ďiďn and b “ pbiq1ďiďn of real numbers such thata_i ăb_i for 1ďiďn, we set Ipa, bq:“ ra₁, b₁s ˆ ¨ ¨ ¨ ˆ ra_n, b_ns.

Definition 1 A cell in Rⁿ is a cartesian product C :“ śn

i“1E_i such that for each 1 ďiď n, E_i P trα_i, β_is,sα_i, β_ir,rα_i, β_ir,sα_i, β_isu for some real numbers α_i and β_i.

In this case, the interior of C is the cartesian product intpCq “ śn

i“1sα_i, β_ir and its (n- dimensional) volume is the positive number volpCq “ śn

i“1pβ_i´α_iq. Moreover, if volpCq ą 0 and E_i “ rα_i, β_is for all 1ďiďn, C is called a non-degenerate closed cell.

Definition 2 A cell-partition of C is a collection P “ pC_jq1ďjďq of disjoint cells whose union is C; that is Cj XCk “ H if j ‰k and Y^q_j“1Cj “C.

Definition 3 Astep function defined on a cellC is a real valued functionf such that for some cell-partitionP “ pC_jq1ďjďq of C, f is constant on the interior of eachC_j; that is, there exists a sequence λ“ pλ_jq1ďjďq of real numbers such that fpxq “ λ_j for all xPintpC_jq, j “1,2,¨ ¨ ¨, q .

In this case,f is said to be a step function associated with the partitionP and the collection λ; and the (Riemann) integral of f is the number

ż

C

f :“

q

ÿ

j“1

λ_jvolpC_jq.

It is straightforward that for any other cell-partitionP¹ “ pC_j¹q1ďjďq¹ of C, ż

C

f “

q¹

ÿ

j“1

ż

C

f_|

C1 j

wheref_|

C1 j

pxq “fpxqif xP C_j¹; and f_|

C1 j

pxq “0 if xRC_j¹.

Definition 4 A real-valued function f :Ipa, bq ÝÑR is monotone if one of the two following conditions is satisfied:

• pC1q: for all x, y PIpa, bq:xĺyùñfpxq ďfpyq;

• pC2q: for all x, y PIpa, bq:xĺyùñfpxq ěfpyq;

where the relation xĺy means that x_i ďy_i for all iP t1,2,¨ ¨ ¨, nu.

3 Main result

Step functions onIpa, bq ĂRⁿconstitute the simplest family of integrable real-valued functions on Ipa, bq. Moreover, step functions are used to test whether a real-valued function on Ipa, bq is integrable or not; see [Protter and Charles Jr, 1977] and [Taylor, 2006] for more detailed on theory of integration. To ease the presentation, we use the following characterization of an integrable function, see [Jacob and Evans, 2016, Theorem 18. 13] with slightly modification.

Definition 5 A real-valued functionf onIpa, bqis integrable if there exists two sequencesph_kqk

and pg_kqk of step functions on Ipa, bq such that

• for all non negative integers k and for all xPIpa, bq, h_kpxq ďfpxq ďg_kpxq;

• ş

Ipa,bqpg_k´h_kq tends to 0 as k tends to infinity.

Theorem 1 All monotone functions on Ipa, bq are integrable.

Proof. Suppose thatf :Ipa, bq ÑRis monotone and assume that f satisfiespC1q) (otherwise consider ´f). We show that f is integrable on Ipa, bq by constructing two sequences of step functionsph_pqand pg_pqon Ipa, bq such that:

h_p ďf ďg_p and lim

pÑ`8

ż

Ipa,bq

pg_p´h_pq “ 0.

To do this, let pbe a positive integer. We split each ra_i, b_is into 2^p intervals of equal length andIpa, bqintop2^pqⁿcells of equal volume by considering the sequencea^p_i,j “a_i`₂^jppb_i´a_iqwith

0ďj ď2^p, the collectionpC_i,j^p q1ďjď2^p of intervals, the n-cartesian product R_p “ t1,2,¨ ¨ ¨ ,2^puⁿ and the collectionpC^p,kqkPRp of cells inIpa, bqdefined by:

C_i,j^p “ ra^p_i,j´1, a^p_i,jr for 1ďj ă2^p and C_i,j^p “ ra^p_i,j´1, a^p_i,js forj “2^p C^p,k “ C_1,k^p

1 ˆC_2,k^p

2 ˆ ¨ ¨ ¨ ˆC_n,k^p

n,for all k“ pk₁, k₂,¨ ¨ ¨ , k_nq PR_p. By construction,

tC_i,j^p , j “1,¨ ¨ ¨ ,2^pu “ trai, a^p_i,1r,¨ ¨ ¨,ra^p_i,2p´1, bisu

is a partition of ra_i, b_is; tC^p,k, k P R_pu is a cell-partition of Ipa, bq and for all 1 ď j ď 2^p, C_i,j^p “ C_i,2j´1^{p`1</sup> YC<sub>i,2j</sub><sup>p`1}. Moreover for each k P R_p, we split C^p,k into 2ⁿ disjoint cells of equal volume as follows:

C^p,k “

n

ź

i“1

´ C_i,2k^p`1

i´1

ďC_i,2k^p`1

i

¯

“ ď

lPSp,k

C^p`1,l with S_p,k“ tlPR_p : l_i P t2k_i´1,2k_iu,1ďiďnu.

Now define h_p and g_p as follows: for all x P Ipa, bq there exists k P R_p such that x P C^p,k; pose

h_ppxq “ fpC^p,kq and g_ppxq “ fpC^p,kq for all xPC^p,k where C^p,k “ pa^p_1,k

1´1, a^p_2,k

2´1,¨ ¨ ¨ , a^p_n,kn´1q and C^p,k “ pa^p_1,k

1, a^p_2,k

2,¨ ¨ ¨ , a^p_n,knq. Note that hp

and g_p are both step functions onIpa, bq. Moreover, for allk PR_p and for allxP C^p,k, we have C^p,kĺxĺC^p,k. Since f is monotone, thenfpC^p,kq ďfpxq ďfpC^p,kq. Hence for allxPIpa, bq, h_ppxq ďfpxq ďg_ppxq.

To complete the proof, we show that lim

pÑ`8

ş

Ipa,bqpg_p ´h_pq “ 0. For this purpose, let δ_p “ ş

Ipa,bqpg_p´h_pq. By the definition ofh_p and g_p we compute:

δ_p “ ÿ

kPRp

rfpC^p,kq ´fpC^p,kqs ˆvolpC^p,kq “ v₀ 2^np

ÿ

kPRp

rfpC^p,kq ´fpC^p,kqs (1) wherev₀ “volpIpa, bqq. Since C^p,k:k PR_p(

is a cell-partition of Ipa, bq, it follows that δp`1 “

ÿ

kPRp

ż

Ipa,bq

pgp`1´hp`1q_|

Cp,k. (2)

Furthermore, for eachk PR_p,pg_{p`1</sub> ´h<sub>p`1}q_|

Cp,k

is null out ofC^p,k andC^p,k is the disjoint union of cells C^p`1,l for l PSp,k. Thus, for eachk “ pk1, k2,¨ ¨ ¨ , knq PRp we have:

ż

Ipa,bq

pg_{p`1</sub>´h<sub>p`1}q|_Cp,k

“ ÿ

lPSp,k

´

g<sub>p`1</sub>pC<sup>p`1,l</sup>q ´h<sub>p`1</sub>pC<sup>p`1,l</sup>q

¯

ˆvolpC^p`1,lq

“ v₀ 2^npp`1q

» –

ÿ

lPtl¹,l²u

´

fpC^{p`1,l</sup>q ´fpC<sup>p`1,l}q

¯

` ÿ

lPS_p,k^˚

´

fpC^{p`1,l</sup>q ´fpC<sup>p`1,l}q

¯ fi fl

where,

l¹ “ p2k_i´1q_1ďiďn, l² “ p2k_iq1ďiďn and S_p,k^˚ “S_p,kztl¹, l²u. Note that,

C^{p`1,l1</sup> “C<sup>p,k</sup>, C<sup>p`1,l2} “C^{p`1,l1</sup>, C<sup>p`1,l2} “C^p,k (3) and for alll PS_p,k^˚ we have:

C^p,k ĺC^{p`1,l</sup> ĺC<sup>p`1,l} ĺC^p,k. (4)

So, thanks to the monotonicity of f and Equation (4) we get:

fpC^p,kq ď fpC^{p`1,l</sup>q ďfpC<sup>p`1,l}q ď fpC^p,kq, This implies

fpC^{p`1,l</sup>q ´fpC<sup>p`1,l}q ďfpC^p,kq ´fpC^p,kq. (5)

By combining equations (3) and (5), we obtain:

ż

Ipa,bq

pg_{p`1</sub>´h<sub>p`1}q|C^p,k

“ v0

2^npp`1q

» –

ÿ

lPtl¹,l²u

´

fpC^{p`1,l</sup>q ´fpC<sup>p`1,l}q

¯

` ÿ

lPS_p,k^˚

´

fpC^{p`1,l</sup>q ´fpC<sup>p`1,l}q

¯ fi fl

ď v₀

2<sup>npp`1q</sup>rfpC<sup>p,k</sup>q ´fpC<sup>p,k</sup>qs ` v₀ 2^npp`1q

ÿ

lPS_p,k^˚

rfpC^p,kq ´fpC^p,kqs

ďv₀ 2ⁿ

fpC^p,kq ´fpC^p,kq

2^np p1` |S_p,k^˚ |q “ 2ⁿ´1 2ⁿ

v₀

2^nprfpC^p,kq ´fpC^p,kqs Finally for all k PR_p, we have

ż

Ipa,bq

pg_{p`1</sub> ´h<sub>p`1}q|_C^p,k ď 2ⁿ´1 2ⁿ

v₀

2^nprfpC^p,kq ´fpC^p,kqs (6) By summing over k PR_p all left-hand-side terms and all right-hand-side terms from (6), equa- tions (1) and (2) imply that

δ_p`1 ď 2ⁿ´1 2ⁿ δ_p. for all positive integer p. Therefore 0ďδ_p ď

ˆ2ⁿ´1 2ⁿ

˙p

δ₀, where δ₀ “v₀pfpbq ´fpaqq. Since

pÑ`8lim

ˆ2ⁿ´1 2ⁿ

˙p

“0, then lim

pÑ`8δ_p “0.

A nonempty subset D of Rⁿ is said to be bounded if it is contained in some cell. The characteristic function ofD is the function χ_D defined on Rⁿ by

χ_Dpxq “

"

1 if xPD 0 otherwise

Definition 6 Letf :DÝÑ Rbe bounded andC a non-degenerate closed cell such thatDĎC.

Thenf is Riemann integrable onDiffp:C ÝÑRis Riemann integrable onC, withfpxq “p fpxq if xPD and 0 otherwise.

Definition 7 [Jacob and Evans, 2016, Definition 19. 23] A bounded setDcontained in a non- degenerate closed cell C of Rⁿ is said to be Jordan measurable if the restriction χ_D|

C of χ_D on C is Riemann integrable.

Theorem 2 Let D be a nonempty Jordan measurable subset of a non-degenerate closed cell C of Rⁿ. Then any monotone and bounded real-valued function f :DÝÑR is integrable.

Proof. Let D be a nonempty Jordan measurable subset of a cell C of Rⁿ and f :D ÝÑ R a monotone and bounded function. Without loss of generality, assume that f satisfies condition pC1q. We show that f is integrable on D by constructing a monotone function g : C ÝÑ R such thatfp“g¨χ_D|

C. The result comes from the fact that,fpis the product of two integrable functions.

Let x P C we set D^´pxq “ tt P D , t ĺ xu. Since f is bounded on a nonempty set D, we defineg :C ÝÑR as follows:

gpxq “

"

suptfptq, tPD^´pxqu if D^´pxq ‰ H

inftfptq, tPDu otherwise (7)

Let us show that g is monotone and fp“g¨χ_D|

C. To do this, consider x, y P D such that xĺy. We note that D^´pxq Ď D^´pyq.

• IfD^´pyq “ H then D^´pxq “ H. So, from Equation (7), we get gpxq “inftfptq, tPDu “ gpyq;

• If D^´pyq ‰ H, two cases are possible. First assume that D^´pxq “ H, then gpxq “ inftfptq, tP Du ďfpt₀q for all t₀ P D^´pyq. Hence,

gpxq ď suptfptq, tPD^´pyqu “gpyq.

Second, suppose that D^´pxq ‰ H, then gpxq “ suptfptq, t P D^´pxqu. Since D^´pxq Ď D^´pyq then,

gpxq “ suptfptq, tPD^´pxqu ďsuptfptq, tPD^´pyqu “gpyq.

Finally for all x, y P C such that x ĺ y we have gpxq ď gpyq. We then conclude that g is monotone on C and it follows from Theorem 1 thatg is integrable on C.

To conclude this proof, it is sufficient to show thatg_|D “f. LetxP D, thenxP D^´pxqand by Equation (7) we can write:

fpxq ď gpxq “suptfptq, tPD^´pxqu. (8) Moreover for all t P D^´pxq we have t ĺ x. This implies that fptq ď fpxq, for all t P D^´pxq. Hence

gpxq “suptfptq, tP D^´pxqu ďfpxq. (9) and from relations (8) and (9) we conclude that gpxq “fpxq, for all xPD, i.e., g_|D “f. This implies thatfp“g¨χ_D|

C. SinceD is a bounded Jordan measurable set then χ_D|

C is integrable onC; so is the product fpof g and χ_D|

C.

4 Application to aggregation functions

Aggregation functions constitute a class of multivariate real-valued functions defined on multi- dimensional rectangular boxes with practical application; see [Grabisch et al., 2009, Definition 1.1] for the following formal definition with a given closed and bounded interval I of R:

Definition 8 An aggregation function on Iⁿ is a function A:IⁿÑI that piq is nondecreasing (in each variable)

piiq fulfills the boundary conditions

xPIinfⁿApxq “infI and sup

xPIⁿ

Apxq “supI .

Note that, the condition pC1q in Definition 4 is equivalent to saying that, the function f is nondecreasing in each variable x_i for i “ 1,2, . . . , n. That is why aggregation functions can be considered as monotone multivariate real-valued functions satisfying condition pC1q; see [Grabisch et al., 2009, Definition 2.1] and [Beliakov et al., 2007, Section 1.1].

We now consider some characteristics of aggregation functions and prove that they are all well-defined on the full set of aggregation functions. Hereafter, we denote by A_npa, bq the set of all aggregation functions defined on the cell ra, bsⁿ for some real numbers a and b such that aăb. Any aggregation function in A_np0,1q is calledsimple aggregation function.

Definition 9 [Grabisch et al., 2009, Definition 10. 37] Let a, bP R such that a ă b, let A be an integrable aggregation function onra, bsⁿ, and consider any subset K Ď rns. The importance index of coordinates in K Ď rns onA is defined by

φ_KpAq:“ 1 pb´aqⁿ

ż

ra,bsⁿ

Apb_Kxq ´Apa_Kxq

b´a dx (10)

whererns “ t1,2, . . . , nu, dx“dx₁¨ ¨ ¨dx₂ and for eachαP ta, bu, α_Kxdenotes then-tuple of ra, bsⁿ whose ith coordinate is α, if iPK, and x_i, otherwise.

Beside importance indices, interaction indices are also presented in [Grabisch et al., 2009, Sec- tion 10.4] among which the following:

Definition 10 [Grabisch et al., 2009, Definition 10. 41] Let a, bPR such that aăb, let A be an integrable aggregation function onra, bsⁿ, and consider any subset K Ď rns. The interaction index of coordinates in K Ď rns onA is defined by

I_KpAq “ 1 pb´aqⁿ

ż

ra,bsⁿ

∆_KApxq

b´a dx (11)

where

∆_KApxq “ ÿ

LĎK

p´1q^|L|Apa_Lb_KzLxq

with y“a_Lb_KzLx“ py₁, y₂, . . . , y_nq been the element of ra, bsⁿ such that y_i “a if iP L , y_i “b if iP KzL and y_i “x_i otherwise.

In both Definition 9 and Definition 10, the mention “integrable” can be omitted as shown below.

Proposition 1 Given K Ď rns and a, b P R such that a ă b, the importance index of coordi- nates inK as well as the interaction index of coordinates in K respectively defined by equations (10) and (11) are well-defined on A_npa, bq without any restriction.

Proof. LetAPA_npa, bq, K Ď rns andcP ra, bs. It is obvious that the mappingxÞÝÑApc_Kxq is an monotone function onra, bsⁿthat satisfies conditionpC1q. Taking into account the linearity of the integral, the result follows from Theorem 1.

Another characteristic of aggregation functions is found in [Kurz, 2018] under the continuity assumption.

Definition 11 [Kurz, 2018, Definition 5.1] LetA be a continuous simple aggregation function and iP rns. The importance measure of coordinate i onA is defined by

ϕ_ipAq:“ 1 n!

ÿ

πPS_n

ż

r0,1sⁿ

rAp1_πěixq ´Ap0_πěixqs ´ rAp1_πąixq ´Ap0_πąixqs dx. (12) where S_n is the set of all permutations on rns and for all π P S_n, π_ěi “ tj P rns, πpjq ě πpiqu and πąi “ tj P rns, πpjq ąπpiqu.

Note that, the importance measureφ, is well known in the literature as a tool for measuring decision-making power, see, [Kurz, 2014][Kurz et al., 2019], or [Kurz et al., 2021].

Proposition 2 GiveniP rns, the importance measure of coordinate idefined by Equation (12) is well-defined over the full set A_np0,1q.

Proof. As above, the result follows from Theorem 1 and the fact that x ÞÝÑ Apc_Kxq is monotone on r0,1sⁿ for K P tπ_ěi, π_ąiuand cP t0,1u.

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