characterization of compact subsets of fuzzy sets

8
Fu~'y$¢Is and Sys~n~ 29 ( t ~ ) 341-348 341 No~h-H~and CHARACTE~J~A~ON OF COMPACT SUBSE'I~ OF FUZZY $E~ Phil DIAMOND Departr~nt of Ma~hema~, University of QueenMand, ~.. Luc~ 4067, Queensla.~d, Ag~tr~li~ Peter KLOEDEN $~ol of Matkematlcal and Physical Sciences, Mu~lo¢~ Um'~e~'icy, Murdnch 6150, Wes~er~ Auslra~i~ R~ccivedDecemI~r1986 Rov'~d July 1987 Abstract: A characterizationof eon~pactsabots is p~scnted for the ~etric since of normal fuzzy convexfuzzy ~¢t~ on the b~se space R~, ~he metric fo~ wh:ch~ the supfemmn ov©rt ~ Hau~rff dis:aa~.~sbetween co~¢spondiag level se~s. It is shown that a ct~¢~ subS¢~ is compaa~if and only if R is uniformlys~pport-boen~ed and the g~e.s~ndh~g ~;:~ of support functions is equfl~:flcontinuous in the ~embershipgrade variable. Eeywords: l~on~al fury c~nvcxfuzzysets; ~mpact subsets;suppo~ funcZ[ons; [unctions of botmdedvariation. 1. Int~cdon Many applications of fuzzy ~ets re-strict attention to the convenient metric space (~n, D) of normal, finzy convex sets on the base space R n, with D the supremum over the Hansdorff distances between corresponding level sets. We mention in particular the fuzzy random variables of P:.n'i and Ralescu [8], the f~zzy differential equation~ of KaIev~ [4] the fuzcy .dyp.am.;~1 ~y~t¢ats of Kioeden [5], and the ch~olic itera~ons ot t~zy se~s of Diamot.~d [1] and Kloeden [6]. In these papers specific resul¢~ ~xe often obtained for compact subsets of ~, which raises the question of how to characterize such compact subsets. The purpose o~ this note is to present a convenient characterization of compact subsets of the metric space (~, D). Our m~n result is theft a closed ~ubset of is compact if and only if the support sets are uniformly bounded in R " and the support fun~ti~s of Purl and Ralescu are equfleftcontinuous in the membe~hip grade variable e¢ uniformly on the unit sphere $~-1 of R ~. To this end we note that the ~uppo~.t ~unctions provides a means of embedding all of ~he apace ~ in a Banach ~pace, which we exhibit explicitly, net jug the ~ubspace ~p of 'Lipschitzia'~: fuzzy sets considered by Purl and Ralescu ~]. Our proof in par~ ts based on the proofs of HER}" ~or related sesults ir~,,e~'~ng fun~uns o~ ~'~ed variation [7, Section 36.5] Various definitions an~ preliminary results are se~ out in Section 2o Then in 0165-0114/8~/$3.50 © D89, E~,ier Seiance~abli~ers ~.• (Not~h-HoH~d)

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Fu~'y $¢Is and Sys~n~ 29 ( t ~ ) 341-348 341 No~h-H~and

C H A R A C T E ~ J ~ A ~ O N O F C O M P A C T SUBSE ' I~ O F F U Z Z Y $ E ~

Phil DIAMOND Departr~nt of Ma~hema~, University of QueenMand, ~.. Luc~ 4067, Queensla.~d, Ag~tr~li~

Peter KLOEDEN $ ~ o l of Matkematlcal and Physical Sciences, Mu~lo¢~ Um'~e~'icy, Murdnch 6150, Wes~er~ Auslra~i~

R~ccived DecemI~r 1986 Rov'~d July 1987

Abstract: A characterization of eon~pact sabots is p~scnted for the ~etric since of normal fuzzy convex fuzzy ~¢t~ on the b~se space R ~, ~he metric fo~ wh:ch ~ the supfemmn ov©r t~ Hau~rff dis:aa~.~s between co~¢spondiag level se~s. It is shown that a ct~¢~ subS¢~ is compaa~ if and only if R is uniformly s~pport-boen~ed and the g~e.s~ndh~g ~;:~ of support functions is equfl~:flcontinuous in the ~embership grade variable.

Eeywords: l~on~al fury c~nvcx fuzzy sets; ~mpact subsets; suppo~ funcZ[ons; [unctions of botmded variation.

1. I n t ~ c d o n

Many applications of fuzzy ~ets re-strict attention to the convenient metric space (~n, D) of normal, finzy convex sets on the base space R n, with D the supremum over the Hansdorff distances between corresponding level sets. We mention in particular the fuzzy random variables of P:.n'i and Ralescu [8], the f~zzy differential equation~ of KaIev~ [4] the fuzcy .dyp.am.;~1 ~y~t¢ats of Kioeden [5], and the ch~olic itera~ons ot t ~ z y se~s of Diamot.~d [1] and Kloeden [6]. In these papers specific resul¢~ ~xe often obtained for compact subsets of ~ , which raises the question of how to characterize such compact subsets.

The purpose o~ this note is to present a convenient characterization of compact subsets of the metric space ( ~ , D). Our m ~ n result is theft a closed ~ubset of is compact if and only if the support sets are uniformly bounded in R " and the support fun~t i~s of Purl and Ralescu are equfleftcontinuous in the membe~hip grade variable e¢ uniformly on the unit sphere $~-1 of R ~. To this end we note that the ~uppo~.t ~unctions provides a means of embedding all of ~he apace ~ in a Banach ~pace, which we exhibit explicitly, net j ug the ~ubspace ~ p of 'Lipschitzia'~: fuzzy sets considered by Purl and Ralescu ~]. Our proof in par~ ts based on the proofs of HER}" ~or related sesults ir~,,e~'~ng f u n ~ u n s o~ ~ ' ~ e d variation [7, Section 36.5]

Various definitions an~ preliminary results are se~ out in Section 2o Then in

0165-0114/8~/$3.50 © D89, E~,ier Seiance ~abli~ers ~.• (Not~h-HoH~d)

342 P. D ~ d , P. K/oede~

Section 3 we state and prove our ~aain result. F~nally we prescott a shnple example of a n~neompact snbse~ in Section 4.

2. P ~ a r ~ e s

We restrict our attentio~ to the metric space (~", D) of fuzzy sets on the base space R n, adapting in what follows definitions and results from Kaleva [3, 4] and Purl and ga[escu [8]. Let 1 = [0, 1]. A fuzzy set u ~ ~n is a function u : R ~-71 for which

(i) u is normal, i.e. there exists an xo¢ R n such that U(Xo) "~: 1; (ii) u is fuaey convex, i.e. for any x,y ¢ R" and 0 ~< ~. ~< 1,

u ( ~ + (1 - ^ ) y ) ~> ~ r , {u(x), u g l y ,

(iii) u ~ uppersemicominuous; ~id (iv) the closo~'e of {x ¢ R ' ; u(x) > 0}, denoted b~' [~]c, is compact. Let u ~ ~n. Then fo~" each 0 < oc~ < 1 the ~-le,'el set [u]'~ of u, defined by

[~i ~'~ = {x ~ R"; u(x) ~ ~). is = non~mpty ,~o.~..~act ~ ~x s~bset of R", as is the support set [t~] ° of u. Let d be_ the Ha¢~dor~ ~.~aetrie for nonempty compact subsets of R ~. Then

D(u, v) ffi 0 ~ p a ( M =, [v]~),

where u, v ~ ~", defines a metric on ~ . Moreover, (~n, D) is a complete metric space.

Let u,v ¢ g" and c ¢ R. Then addition u + v and multiplication by a scalar c can be defi~eO, in ~n in terms of the ~-l~el se~ by

"; [u+vl"=[/~l~+[vp and [c:d"--cL'd" for each 0 ~ < e ~ < 1, where

A ~-B={a+ti 'aeA, beB} and cAffi{ca;a~A}

for none£npty s~bse~.s A and B of tl~ ~. This defines a linear s~ructure ow ~n, but does notmake ~" a vector space. Nor is there a norm on ~" which is equivalent to D. Nevertheless Ka|eva has us¢~ a theoycm of R[idstrOm to show that ~ can be embedded isomorphicaUy isometrica;ly a~ a convex cone in soa~e Banach space. We shall g~ve arz example of such a Banach space below and use its pn-~perfies in the proof of our main result.

P~fi and Ra~escn showed that the subspace ~ of ~ , consisting of tho~e u ¢ ~ wRh Lipschitzian oMev¢l sets [u]% !_e. wi~h

~([u] ~, [ul ~ ) -<. K l~ - ~1

for all e ,~ ~ I and some K ~ R +, could be embedded isomorphically isometrically in the Banaeh ~pace C(I x S'~-~), where S '~-~ is the unit sphere in R ~. They used the embedding] : ~[i~--~ C(l x S ~-~) defined by u* =/(u) where u* is the suppo~

Comet subs¢~ of ~zzy se~ 343

function

u*(~, x) = sup ta, x) U , D a~iuV '

for ( ~ , x ) ¢ 1 × $ '~-1. (We shall ~ u* instead of their s,°) In fact u* is we|! defined by (2.1) for all u e ~f" and satisfiv~ lhe foIloMng properties:

(a) ~¢ is ~niform|y bounded on I × S ' - I :

lu*(~, x)[ ~<s~p~ lal;

(b) u*(., x) is nonincreasing and leftcontinuous i~ ~ for each x ~ S ' - t ; (c I u*(cc, "1 is Lipschitz continuous in x unitormly in at ~ 1:

(d) for each ar ~. I and u,v ~ ~g',

d([uV, [v] ' ) - sup., lu*(~, x)-- v*(~, x)l. x c S -

Let V(y ) be the total variation [2, 7] on I of a function y : 1---> R. Since for any e~ ¢ ~" and x ~ $ n - t u*(., x) is nonincreasing in at ¢ I it has total variation on 1

V(u*(., x) ) ffi u*(O, x) - u*(1, x). (2.2)

Hence

sup V(u*t.,x))~<2 sup [u*(vr, x)l<~2 sup ]a],

which implies that e: ~ e L , (3 "-~, BV(I)) where ~V(/) is the Bariach space of functions of bon~ded variation on I with norm V(y) + [y(0)[. Moreover from (c) it follows that u*eL®(l, C(S'~-a)). Hence for any u ¢ ~ , u* defined by (2.1) belongs to the space

X --- L . ( S ' % BV(O) n L. i f , C(S'-~))

v~hich is a B~naeh space with the norm

I lzll = sup v ( z ~ . . x ) ~ + s u p sup Iz (~ ,~ ) l - (~,,,~)

(Here we use L®(X, Y) to denote the space of bounded mappings f from a compact space X into a nnrmed space Y, endowed with the norm [if]i~.-- sup.,,~ [If(x)ll~.. )

Thus we can extend the Puff and Ralese, u embedding ] to the whole space $~. Using the sa~ue ter~[nology we hwe u~=](u), where j : $~-- ,~ . Besi¢les preserving the ~ e a r ~tructure of ~ , it also z,-~t~fies

D(u, ~,) ~ l[/(u) -](u)H ~< 3D(u, v) (z4)

which follows from property (d) of u*, (2.2) and the definition of the norm 42.3).

344 P. Diamond, P. Kl~den

C.~a~e~aently j is not an isometzy, but by (2.4) it ~ a homcomorphism, which ~s su~ficien~ for our requirements in t~is paper.

Now let U be a nonempty ~b~,~t of ~ and let U* = j (U) be its embedded hn~ge in ~. We say that U ~s unifor,~y support-bounded if there exists a K a R ÷ such that

for all ~ e U, Le. ff the support se~s [u]0 l~e within a sphere of radius K ~lo~/. ~he orJ~h~ ~ ~ for ~1 u ~ U. Note that ~hL~ ~mpl[es t h ~ U* is uniforra b ~o.,~ded in the ,~-norm and ~hat U* is eq~iconti~'uous in x ~ $~"~ uniformly in c~ ¢ I:

for all x, y ~ S ~-~, ~ ~ U and cr ~ L Finally, we sa~ that U* Js equileftcontinez~,~ in ¢ ¢ 1 uniformly in x ~ S ~-~ if for

each e > 0 there exists a 6 -- 6(e) > 0 such C'~at

u*(#, x) <<. u*(o~, x; < u*(t3, ~:~ + e

for a~ p - - ~ < o ~ < ~ , ~e 'S ~-~ and u*~U*.

The L~ain p~p~se of this paper is to establish the following characterization of compact subsets of the metric space tl~', D) of f u r y sets on the !~ase space R% Lz places our proof resembles that of similar resutts of Helly [7, :~¢ction 36.5]' for more general classes of functions of bou~,0ed variation. We present o~r ~ o o f jr, sorae cle~a~l for compie~ehess a~d to Ladicatc clearly the role of the characteristics under consideration.

" l [ ~ . m . A ctosed set U of ( ~ , D) is compact if and only if U is anij~rmly zupporz~boanded and U* is equi~'eftcot~tinuous in ~ ~. ! u~i]brmly in x ~ S n-~-.

Proof. Part 1 (Necesslty). Let U be a no,empty compact subset of ~ . $~nce j is homeomorphJsm, U* = j ( U ) is a compact subset of the Banach space ~. Hence U* is uniformly bounded iR ~he ~ norm, that is there exists a K ¢ R + such that

tl~*l[ -'~= K for all u* E U*.

xeSa-t ~a~fl so

I a \ sup laj = sup ~ o , , - - - ~ sup u*(O,x)<~K

for aH u e U, that is U ~z un~fo~ly support-boun~ed.

Suppose that U* is not ¢quilefieontinuous in at ~ l uniformly in x ¢ S "~'~. Th~n there egists an e~ > 0, and sequ~e~s u~ ¢ U*, a~.--* a t - in ~ x~-ffi* x in S ' -z such that

u.(at . , x . ) ~ u,~fat, ~.j + eo ( ~ ! )

for all n. Now U is uniformly su~port-bounded so

,u.*¢,~., ~o~ -- u~*(at., x)l ~ g ix - x,,l.

t, nd hen6~

u. (at., x . ) ~ . . ,~ . . . . x) + K Ix - x ~ l

for ~dl n. We combine this whh (3.1) to obtain

ro + ~*(at. z ) ~ u *.( at., x ) + g, Ix - x . i (3.2)

for all n. Since U* is comp~ g and go is a [3anach space, U* is ~qnenfiaHy ¢ompacL

Hence there exists a ~ubsequence u.*o) which co~-~¢,rges to a limit u* ~ U*, that is

Uu .o ) - u*ll/l~ ~ for all n(y)>~N(,)

for any e > 0. This implies that

lu.*~j)(#, v) - u*(#, y)l < 8

for all n(] ) ~ ~:~e) and all (#, y ) e I × S "-a. Thus with j$ -- at.o) and y - x we have

for n(]) ~ N ( ~ e o ) , and w i t h / $ = at a ~ d y ~ x ,

for n(j)-~ N(~ro). Combining these with (3.2} we get

u*(at, x) + :~eo < u *.~;~(at, x) + eo

u.o)(ot.to, x) + g Ix - a=~(,I

< u.~: (at. x) + g Ix - x.o~l + ¼co

arad henc~

u*(at, r ) + i~o < u*~¢~(at, x) + g Ix - x.o~l

for n(i) ~ Ndeo), In addition, u*o~(at, x) < u*(~', x) ~- ¼e0 for v(/) ~ N(~eo), so

~eo < t¢ Ix - x,,, ,:

for n(D >~ N(~eo). This is however impossi01e as x.--* x. Consequently U* must be equileRcontinuous in at ~ 1 uniformly in x ¢ S ~-x.

Tiffs completes the necessity part of the proof. Part 2 (Suff~cieac3~. Let {~.} be a sequence i~ U and {uS} the corresponding

sequ,~nce, in U*. A~so ,let ~ = {ati e 1; i : L2,3 . . . . } and ~z = (x~ e S '-~; / = L, 2. ~ . . . . } be coentably dense subsets of I and S "-~, respectively. Then by the ~ua~ diagonal subseqaence construction. ~ese exist~ a subsequen:e (u*~(,~:~ and a

346 P. Di~mmnd, P. Kl~den

fimction g : ~1 × ~2 "~R such that

uniformly in (e~i, xj) ~ ~ x ~2- For notational simplicity in what follows, write W n - - Un(n) a n d w n = Un(n).

By the uniform support-boundedness of U, there exists a K ~ R* such that

[w*~(eci, x) - w,~(c~. Y)I -< g [x -Yl

for all a~t ¢ ~ and any w~*, that is the w*(o:i, .) are equicominuous in x E S n-t uniformly in act E ~1. Hence the subsequence {w~*(a~t, x)} converges for each et; ¢ ~,. and x ¢ S ~-l. We denote the Jimits by g(e~, x). Then by a theorem of Grave~ [2; Chapter VII, Theorem 25] this convergence is uniform in x ¢ S ~-t. Moreover it is uniform in ec~ E @~ ton.

Now from the ptolmr~ies of the w~* it follows that

(i) [g(~'/,X)i~<K for all e~e~ l a n d x e 8 " - ~ ;

(it) [g(et, x ) -g ( ec , y ) l ~ K I x - y l f o r a l l g ~ t ;

and

(iii) g(etl~x)~g(ec~,x) forall ec~<¢~ m ~t a n d x ~ S ~ - l .

Finelly for each (or, x) ¢ 1 x S~-~ we ~efine

Each of these hmits exists because the g(-, x) are noninc~asing in ec~ ¢ ~t and boundea. Thi~ defines g on all of I x S ~-a. Moreover ~? satisfies the three properties (i). (fi) and (iii) above an all of I x S "-~.

We need now to show that

w~*(~, x)~g(~, ~)

uniformly in (e~, z) ~ I × S ~-~, not just uniformly on ~ x S ~-t. Let N(e) be such that

g(~,, x) - ~ < ~*(~ . x) <g(~. ,~)

for all n(n)~N(e) , uniformly in (c,;:, ,~)¢ &~ × S ~-~. Choqse an ml in ~l with e~ ~< e:. Then, as g(., x) ~s uonincre~sing.

g(ar, x) ~g(~,, x) <g(~ , x) + e, provided e~ is sufficiently close t c a . In addition each wn*(-, x) i* nonincreasing, so

w~ ~, x) <- w*( ,r,, x) <g(a~,x)+ e for n(n )~N(e ) < g ( ~ x ) + ~

for ~ dose enough to ~. Tlms w~(~:, x) < g(ac, x) + 2e (3.3)

for all n(n) ~ N(e), uniformly ~u (,~, x) ¢ l x S ~-~.

Compact subsets of fuzzy sets 347

By the eq~i~eftconUauity of the w* in of unffo~'mly in x ¢ S ~-~ ~t follows that

w*(of, x) ~ w*(~,, x) < w~*(of, x) + e

provided ~r - 6 < ~, ~< u for 6 - ~(e) > 0 uniformly in w,*, a and x. Thus, as g(., x) ~s ~,~nincreask~g,

g(of, x) - ~ ~< g(of,, x) - ~ < w~*(~,, x) < w~*(of, x ) + e (3.4)

for n(n) >~ N(e) and for of - oft > 0 safficiently small, t h ~ is

g(of, x) - 2c < w,~(~, x )

for n(n)>~N(e), ~mif~rmiy in (oc, x ) ¢ 1 × $ n-l. Combining (3.3) and 0 .4 ) we have

~ p Iw*(of, : ) - ~(of, x)i < 2~ xEb'a-I

for all n(n) ~ N(e). . Since g(., x) is nonincreas~ng, its total variation

V(g(-, x)) = g(O, x) - g (L x)

SO

I",'(,,~(., x))-- v(,~(., x))l = I{w~*(0, z ) -- ~'~*(l, x ) } -- (g(O, X) -- g ( i , X)) I

~< Iw.*(o, ~) -g(O, x)t + Iw*(1, x) -g(x , x)l ~ 2 s u~ lw~*(of, x ) -g (o f , x)l

~.4e for n(n)>~N(e).

Thus we have for n(n) ~ N(e),

IIw~*-gll < 6 e ,

that i3 the snbsequenc¢ w,* converges to g in the ~Y-*,orm. Since U* is closed, ~ U * and since U* =I(U), there exists a u ~ U such that g =j(u). Fi~ully,

D(w~, u) <<- IIi(w.) - / ( t s ) l l = IIw* -gll so the subsequence w, = u0,~) co~,verg~s to ~ in ~'~. ~hat is U is sequentially compact :,nd hence a -,ompa~t subset of the metric space: ( ~ ' , D ) .

"l~is completes the sufficiency part of the proof.

4. An example

We consider an example of a closed but ~ancc~mpact subset of ~t due to Kaleva [3] to illustrate the crhesia for compactness of our theorem. This is th~ ~ t U = {u,. ~ ~ ; r ~ 1}, where u, is defined by its ~-Ievel sets ~ follows:

r l if O~<of~r, [u,] = t ( 1 ~ i ~ r < o ~ l , a n d r e ~ .

Kaleva shows that U is not ~otaiIy bounded, so U i~ nat compact.

348 P. Diamond, P. KIoeden

Here u,* ~ U* satE~ies

+ ! i f x = + l , O ~ a ~ l ,

u~*~a', x ) ffi i f x = - 1 , 0 ~ < ~ < r ,

- if x = - 1 , r < ~ < l ,

for ~ach r ~ L since S ~ = {±1}. "[he set U is unifo~,,,ly support-bounded since [u,] '~ = 1 for each r =. L The set

U* is however not eq: i[ef tconfinuous in ~ e I uniformly in x e S °. To see this take any 0 < z < ~ and x = - l . Then u*(L - 1 ) = - 1 and fcr any 6 > 0 there exi.~t 1 - 6 < e ~< 1 for which

- 1 -- u*(1, - 1 ) ~ u,*(~, - I ) < u,*(l, - 1 ) - e -- - 1 + e

is not satisfied for all u* ~ U*. In fact for those u* with 1 - r < 6 there ~xist 1 - ~ < -~ < ," for which u*(a~, - 1) = 0 > - 1 + e.

Refe~n~zs

[1] F. ~ m o n d , Fu~,y chaos, & M ~ . Anal. AppL (subtaiU,~d). [2] L.M. Graves, The Theory of Functions of Real Vo#iables (McGraw-Hill, New York, 1946). [3] O. Kaleva, ~3n the c~nvergence of fa~z~ sets, Fuzzy Sets and Systems 17 (1985) 54--65. [4] O. Ka[eve, The Caaehy problem for fuzay differential equations, Pn:pfint. [5] P.E. Klceden, Fury dynamical systems, Ftczzy Sets and Systems ? (1982) 275-296. [6] P.B. Kloeden, C-'hao~e rnappi~,:~.~ on fuzzy ~ets, Preprint, Second ~nte~natione! C~a'h~¢ess o' the

lnter~ationel Fezzy Systen~s As~ocia~on, Tokyo Quly 1987). [7] A.N. Ko~og*~!'ov end S.V. Formn, lntroduclory Real Analysis (Dover, New Y~:~, 1975). [8] M.L. P ~ ant[ D.A. l~a!escu, The ~;,~ep~ of normallty for fu,~, random w~ ~es, Ann. Prob~,;:.

D (1985) 13,3-1379.