Fuzzy Sets and Systems 100 Supplement (1999) 63-71 North-Holland

63

METRIC SPACES OF FUZZY SETS

Phil DIAMOND Mathematics Department, Uniuersity of Queensland, St. Lucia, QLD 4067 Australia

Peter KLOEDEN Sclrool of Mathematical and Physical Sciences, Murdoch University, Murdoch, WA 6150 Australia

Received 31 March 1988 Revised 18 July 1938

Abstract: Two classes of metrics are introduced for spaces of fuzzy sets. Their equivalence is discussed and basic properties established. A characterisation of compact and locally compact subsets is given in terms of boundedness and p-mean equileft-continuity, and the spaces shown to be locally compact, complete and separable metric spaces.

Keywords: Fuzzy sets; L,, metrics; compact; locally compact.

AMS Subject Classifications: 52A30, 94DO5.

1. Introduction

Applications of fuzzy set theory very often involve the metric space (‘V, d,), of normal fuzzy convex fuzzy sets over R”, where d, denotes the supremum of the Hausdorff distances between corresponding level sets. This metric has been found very convenient in studying, for example, fuzzy random variables (Puri and Ralescu [ll]), fuzzy differential equations (Kaleva [S]), dynamical systems (Kloeden [7]) and chaotic iterations of fuzzy sets (Diamond [l], Kloeden [8]). Indeed, many properties and applications of (‘Z”‘, d,) can be regarded as generalisations of results involving the space Yr&(R”) of nonempty convex compacts endowed with the Hausdorff metric 6,. Both are complete metric spaces, and very recently compact sets have been completely characterised in V? (Diamond and Kloeden [3]), thus extending the Blaschke selection theorem (see Lay [9]) to fuzzy sets.

However, the d, metric fails to extend the Hausdorff metric topology in at least one important respect: the metric space (%‘I, d,) fails to be separable (Klement, Puri and Ralescu [6]). A different metric dl was introduced by Klement, Puri and Ralescu [6], such that the metric topology was separable, to prove a strong law of large numbers (SLLN) for fuzzy random variables. Both the d,, d, metrics share some of the less desirable properties of 6,. For example, variances of random variables taking values in (Y&JR”), 6,) are not additive. Lyashenko [lo] observed that an L2 metric on Yi&(R”), defined by support functions, induces an appropriately additive variance, and this idea has been extended to the very special case of triangular fuzzy numbers by Diamond [2].

Reprinted _fiom Fuzzy Sets and Systems 35 (1990) 241-249 0165-0114/99/$-see front matter @ 1999 Elsevier Science B.V. All rights reserved

64 P. Diamond, P. KloedenlFuzzy Sets and Systems 100 Supplement (1999) 63-71

The purpose of this note is to introduce and investigate two classes of metrics in 8”. The first class d,, extends the Hausdorff metric and includes dl, d,. The second class pP is based upon L,, metrics for the support functions of compact convex sets (Vitale [12], and references therein) and includes the important Lz case which induces additive variance. Our principal result is that, for each lsp<m, the metric spaces (Gs”, d,,) are equivalent to the corresponding p,, topology, and are complete, separable, locally compact metric spaces. Conse- quently, many important theorems (like the SLLN of Klement, Puri and Ralescu [6]) hold in all these equivalent spaces. A characterisation of the compact subsets in these spaces is also given.

Various definitions and preliminaries are set out in Section 2. Section 3 con- tains equivalence proofs, while the last section addresses compactness properties.

2. Preliminaries

As in [3], we restrict attention to the class of fuzzy sets I!!?, consisting of functions u:R”+ I = [0, l] for which

(1) u is normal, i.e. there exists an x0 E R” such that u(xo) = 1; (2) u is fuzzy convex, i.e. for any x, y E R” and 0s 1s 1,

u(l2x -I- (1 - A)y) 2 mW(x), U(Y )I ;

(3) u is uppersemicontinuous; (4) the closure of {x E R”: u(x) > 0}, denoted by [u]‘, is compact. These properties imply that for each 0 <x G 1, the a-level set [u]~ = {x E

R”: u(x) 3 a} is a nonempty compact convex subset of R”, as is the support set [u]“. The linear structure of .%&,(R”) induces addition u + u and scalar multiplication cu, c E R+, on 55” in terms of the a-level sets, by

[u + v]” = [U]& + [v]g [culn = c[u]”

for each 0 G cy < 1. To each u E 5%” there corresponds a support function u* E C(Z x S”-l, R),

where Y-r is the unit sphere in R” (see [3] for details),

u*(cu, x) = OW$W (a, x), (YE I, x E S”_l.

Then u* is well-defined for all u E zp” and satisfies the following properties: (1) u* is uniformly bounded on I x P-l,

]~*(a; x)] d O;u& ]a] for all (Y E Z and all x E S,‘-r;

(2) u*(., x) is nonincreasing and left-continuous in (Y E Z for each x E S”-‘; (3) u*((Y, .) is Lipschitz continuous in x uniformly in (Y E I,

lu*(w x) - u*(@i YN 6 (JJJ$bl) Ix -YL

for all (Y E Z and all x, y E S,r-r.

P. Diamond, P. KIoedenlFuzzy Sets and Systems 100 Supplement (1999) 63-71 65

In particular, if a is a nonempty compact, convex set in R” and x,, its characteristic function, then x/*l is the usual support function of A with domain s”-‘.

Let 6, denote the Hausforff metric in Xc&‘*),

&(A, B)=min($~;~gIla -611, j~!;~~ll~-bll].

Wedenoteby S,, lGp< 03, the L, metric on X,,(R”),

where p(e) is unit Lebesgue measure on Y-i. Then for each cy E I and U, u E V,

&([U]n, [u]“) = s;np-, Iu*(K x> - u*(G x)1.

Definition 1. For each 1 <p < 00 define

dp(u, v) = (j-’ &,([u]S [v]=)p da)“‘. 0

and d&u, ~)=su~~~~~~~~([uJ~ [VI@).

Clearly dP is defined for all U, v E %” by properties (l)-(3) immediately above, and d,(u, v) = lim,,, dP(u, v), with d, s d4 if p c q.

The other class of metrics is defined directly from L, metrics on support functions.

Definition 2. For 1 up < ~0 put

P,(K u) = (I’ $,([4”, [vla)p d@)r”. 0

Again, properties (l)-(3) b a ove imply that pp is well-defined on 8”. Observe that p,, <prl for all lCpSq< co, and pt, G dP s d, for 1 G p < 03. We shall see later as a consequence of of Theorem 2 stated below, that lim,,, p,, = pm = d,.

Theorem 1. (%‘I, d,,)! (ST, p,), 1 Gp < ~0, are metric spaces.

.

Proof. The following argument is a modification of that of Proposition 3.1 in [6], which is for the dI .metric. Symmetry of both dP, pP is clear, while the triangle inequality follows easily from Minkowski’s inequality. It remains to show that dJu, v) = 0 implies u = u, and likewise for pp. The result for d,, is a trivial extension of [6]. If p,(u, v) = 0, then a,([~]~, [u]“) = 0 a.e. in 1. But S, is a metric on X&(R”), so [u]“= [ulm a.e. The argument of [6] again applies to give equality for all (Y, and hence the result follows.

The following estimate will be central to much of our considerations:

66 P. Diamond, P. Kloedenl Fuzzy Sets and Systems 100 Supplement (1999) 63-71

Theorem 2 (Vitale [12]). Let K, L E X&R”) with H = diam(K U L).

c/G, L)(WK L)) @+“-lYp G 6,(K, L) 6 6,(K, L)

where lGp<@,

c,(K, L) = (B(p + 1, n - l)/(H’9?(f, i(n - l))))‘/“,

and B( *, *) is the beta function.

It follows that lim,,, p,,(u, v) = d,(u, v). For, if u, 21 are identical

Then

singletons, ail metric distances are zero and equality holds trivially, while if this is not the case

But [u]“U [vlaz [u]” U [u]‘, so c,([u]S [u]~) a c,,([u]“, [u]“) = rP(u, u), say. It thus suffices to show that lim,,,, rP(u, u) = 1 and this follows since

lim B(p + 1, n - 1) 1’P = lim (r(p)/r(p + n))‘@ = 1. ,‘-W ,,--‘m

Finally, the following notions will help characterise compactness in ‘Z”, and may be found in further detail in [3]. Say that U c %‘” is uniformly support bounded if there is a constant K > 0 such that the support sets lie within a ball of radius K in R” for all u IS U. A family of support functions U* = {u*: u E CI} is called equi-left-continuous in CY E I uniformly in x E Y-l if, for each E > 0, there exists S = S(E) >O such that u*(/3, X) < ~*(a; x) < u*(p, x) + E for all p - 6 < cusp, XES,‘-’ and u’ E U*. In addition, a set U of %” is said to have the Blaschke property iff it is uniformly support bounded and U* is equi-left- continuous. Diamond and Kloeden [3] showed that a closed set in (C, d,) is compact iff it has the Blaschke property.

3, Equivalence of metrics

Our principal result is:

Theorem 3. For each given p, 1 up < 00, d,, and pP induce equivalent topologies on ZY and yield complete, separable and locally compact metric spaces, in which closed sets with the Blaschke property are compact.

The proof will be accomplished by showing the equivalence of the pP topology to that induced by d,,, and then demonstrating that (W, d,,) has the requisite properties, through the following three lemmas. The local compactness is established as a corollary in the next section.

Lemma 1. d,,, p,, induce the same topology on V’.

Proof. From Theorem 2, for u, u E %“, pP(u, v) s d,,(u, u), and

c,([u]“, [v]“)dP+,,-,(u, +‘+n-‘)‘p s P&G u)

P. Diamond, P. Kloedenl Fuzzy Sets and Systems 100 Supplement (1999) 63-71 67

provided [u]” and [v]” are not the same singleton set. Suppose v is fixed and {z+} a sequence in P. If [v]” is not a singleton in R”, then dP(uk, v)-, 0 iff pP(uk, v)+O as k-,03, since ~$+,~-r(u~, V) ~d,,(u~, v). If v is a singleton, this argument breaks down if uk = V, when the constant c,, is infinite. But then d&k, v) = &d”k, v) = 0.

Lemma 2. For 1 <p C 03, (V’, d,,) is separable.

Proof. This is adapted from [6], but uses a somewhat different construction to ensure that the approximating fuzzy sets are fuzzy convex. Take any u E 5~7” and E > 0. Since [u]’ is compact, there exists a minimal cover {St} of cubes si=&‘=r[aij, b,), i=l,. . . , r, with aij, b, E Q, 0 < b, - aij < ~/(4n’“) and [u]’ c

tJisl q. From fuzzy convexity, for each 0~ CY < 1, [u]” has a minimal sub- cover {Sic(y)} E {Si}. Write vk = [u]” U (U; q). Note that &([u]‘, LJ; St) S de,

s-([“]S Ui(a) si(a)) < t6 and &(vk, UL &) 4 in. Write ai = SUP,,~, U(X) and relabel S1 , . . . , S,sothatO=a,Gcu,<... C cu, = 1. Define the fuzzy set +. by

#o(x) = ( Xi if X E Sj, 1 G i S r , 0 otherwise.

For any 0 < (Y 4 1, there exists 1 G k G r so that Lyk-1 G (Y C &kuk. If k is the largest integer such that u is constant on [u]“U (U:,!$), then &([u]S [$o]“) = 6,(&, l_l; Si) < aE. If ai(-1 = (Y < @k, then

6m([uInl [$o]“) 6 Sm([u]“-‘. $J $) = a,( vk-1, $J $) < :E,

and similarly if &&l< (Y = @k. For (Yk_l < (Y < ffk,

Define the fuzzy convex set @ E 8” by [#I u = CO[C$~]S 0 < CY < 1, where Co denotes the closed convex hull. Then

Thus d,($, 90) = (IA L([#lS [&]“)p da)“@ < 4~ and so d,(u, 4) s dP(u, #o) + d&h $0) < 3~.

Now let M > 4(r - l)diam([u]‘). For i = 1, . . . , r, relabel ~yi, . . . , a;, and exclude duplicates if necessary, so that 0 c a0 < LYE. . - < as = 1, with s <r. If CY~ $ Q, choose /3, E Q such that

max{&i_,, pi - @IMP} <pi < CU;

and if Cui E Q, set pi = Cui. Define q E 8” by

pi if 6(X) = &i, ‘@) = (0 otherwise.

68 P. Diamond, P. KloedenIFuzzy Sets and Systems 100 Supplement (1999) 63-71

Observe that the class of all such q is countable, and that

Finally, d,,(u, q) s dr,(u, #) + d,($, q) < E and the result follows.

Lemma 3. Every closed set in (I?, d,) which has the Blaschke property is compact.

Proof. Each d,, topology, 16~ < 00, embeds continuously into (‘P, d,), since d,, 6 d,. Consequently, the compact sets in the latter space, which are precisely the closed Blaschke sets [3], are compact in (%“I, d,)).

4. Compactness in dp topology

The Blaschke property is sufficient for compactness of a closed set (Lemma 3), in the d,, metric topology, but it is too strong to be also necessary. This is because d,, G d, and we seek a condition more appropriate for L, type spaces, related to equi-left-continuity in the stronger topology. Let u E ‘8”. If for each E > 0, there exists 6 = S(E, u) > 0 such that for all 0 s h < 6,

say that u is p-mean left-continuous. If for nonempty U c 8” this holds uniformly in u E U, we say U is p-mean equi-left-continuous. If, in addition, Cl is uniformly support bounded, then U is said to have the p-Blaschke property. Observe that for the corresponding family of support functions, this property translates as

I ,,I (~*(a, - h, x) - u*((Y, x))” da < E”

for all 0~ h < 6, x E S”-’ and u* E U*, and that in the limit p = 03 this concept is just the Blaschke property of the previous section. However, p-Blaschkee Blaschke (although the converse is true), as the following example shows.

Counterexample. Define U c gl’ by U = (6, u,, u2, . . . } where

u,,(x) = 1 x” ifO=SxSl,

0 ifx$[O, 11,

and a(x) = 1 if x = 1, and 0 otherwise. Clearly U is uniformly support bounded. We show u to be l-mean equi-left-continuous, and thus 1-Blaschke, but not

P. Diamond, P. KIoedenIFuzy Sets and Systems 100 Supplement 11999) 63-71 69

equi-left-continuous. For 0 < h < 1,

I ,,I &,((u,,]~-“, [u,,]“) dc = I’ ((Y” - (a - h)“) da

=(~+I)-‘[a”+l-(~-h)“+l]:,

= (IZ + l)-‘(1 - (1 -h)“+’ -h”+‘)

d i(l - (1 - h)‘- h2)

=h-h2<h

for all 0~ h G 1 and all n 2 1. Thus ]:, d,([~,,)~-‘*, [u,,]“) dar G E if 0 < h < C?(E) = E and this is the required condition. On the other hand,

,,s”,pcl d,([u]“-‘I, [u]~) = ,,SUI~ Id’ - (a - h)“( = 1 - (1 - h)”

and for h 2 0 this has supremum 1 as II + 03, while for h = 0 the supremum is 0. Consequently, u is not equi-left-continuous, and thus not Blaschke.

Lemma 4. Any u E (ST”, d,,), 1 up < ~0, is p-mean left-continuous.

Proof. Let (Y E [0, l] and suppose {a;,} is a nondecreasing sequence converging to a. Then [u]“= nI=, [u]?~ and &.,([u]%, [uln)+O, and the result follows from left-continuity on the compact interval [0, 11.

Theorem 4. A closed set U of (V, d,,), 1 up s 03, is compact iff U has the p-Blaschke property.

Proof. For p = w, the result is that of [3], so suppose 1 G p < w. Necessity. Let u be a compact set in (%‘I, d,,). If u were not uniformly support

bounded, then there would exist a sequence of compact convex sets in R”, I$ = [Uj]“, Uj E U, such that a,( I$, (0)) > j. Clearly { I$} has no subsequence with limit in X,o(R”). But since U is compact, there is a subsequence {ujckj} converging to u E U, and lim,_, I$.(,) = [u]” E .X-JR”) which is impossible. Hence U must be uniformly support bounded.

Let e>O and let ul, u2,. . . , uk E 8”’ be a SE-cover of U, that is for any u E U one of the sequence elements ui satisfies d,,(u, ui) < 4~. Such a sequence exists by compactness of U. By Lemma 4, u,, . . . , uk are p-mean left-continuous and so there exists 6(E) = mini,i<k a(~, Ui) > 0 such that J’f, sm([Uild-‘*, [it]“)” da < (4~)~ for i = 1, . . . , k and 0 s h < 6(e). Thus for u E U, the triangle inequality gives

’ 6,([u]“” p [u]“)’ da)is d d,>(u, Ut) + (6 6w([UtIe-“, [U/I”)” da)“f + dp(ui, U)

so U is p-mean equi-left-continuous.

70 P. Diamond, P. Kloeden I Fuzzy Sets and Systems 100 Supplement (1999) 63-71

Suficiency. This adapts an argument of [3] and is only sketched. Let {uk} be a sequence in U and {uz} the corresponding sequence in U*. Let D, = {q}, & = {Xi} be countable dense subsets of I and s”-’ respectively. The usual diagonalisation construction gives a subsequence {r&)} and a function g : D, x &+R such that Uzck,(ai, aj)+g(a,, Xi) uniformly in (aiui, Xi) E D1 X Dz as k-* 00. For notational simplicity write w: = u&), W, = u,+).

Since U is uniformly support bounded, there exists K > 0 such that

Iwi(ai, x, - w,*(ai9 Y)l c ( asy$,” I4 ) Ix -Y I = wx - Y I

for all ai E Dl and any wl E U*. That is, the w:(eiui, *) are equicontinuous on Y-r, uniformly in ai E D,. Hence the sequence { W~(~i~ x)} converges for each Cui E D1 and x E S”-’ in the d, and hence d,, norms, and we denote the limits by g(cU,p x). As in [3] (see also [4]) such convergence is uniform in S”-‘, and moreover is uniform in D, as well, for the sup norm, and thus for d,, norm.

From the properties of the w$ E U* it follows that (1) ]g(a!, x)] < K for all Cui E DI and x E S”-‘; (2) ]g(aivi, X) -g(a;, y)] c K ]X -JJ] for all ai E D1; (3) g(a,, X) Gg(pi, X) for all pi s CU, in D1 and x E P-l.

Then for each (a, X) E I x S"-', define

g(a, X) = lim g(Wi, x), ai E D1. W,--ttYy-

Each such exists because g(*, X) is nonincreasing in ai E D1 and bounded. This defines g on all of I x S,r-', and in such a way that the three properties, immediately above, hold for g on all of I x Y-l. These, together with the left-continuity of g(*, x), show that g(-, *) is the support functioin of a well-defined fuzzy set w whose support lies in lJlrEO [u]“. It remains to show that dp(wk, w)--*O as k-tm.

By p-mean equi-left-continuity, for a monotonic nondecreasing sequence q=a--h,eDI,

I, ,l’ S&v,]“-“‘, [w,$T da < (fey

provided 0 c hi < 6 for 6 = a(~) > 0, uniformly in w, E U. But for k >N(~E), g(cv,, X) - 1~ < ~;(a~, x)<g(~~i~ X) uniformly in S”-~’ and since g is non- decreasing,

g(cu,x)-a~~g(a;,x)-1E<W:(~i,X)<g(~,XX).

Thus &,([~~]a’, [wla) = sup,,sn-I Iwl(ai, X) -g(a, x)1 < 4~. Hence

for all k > N($E).

P. Diamond, P. Kloedenl Fuzzy Sets and Systems 100 Supplement (1999) 63-71 71

Corollary. The space (‘Z”, d,), 1 up < w, is locally compact. Moreover a subset U is locally compact iff every uniformly support bounded and closed subset of U is p-Blaschke.

Proof. For sufficiency, let U c g,l be such that any uniformly support bounded and closed set is p-Blaschke, and take u E U. Since u has compact support, there exists K >O such that d,,(u, (0))~ K. Then N,,(u) = {v: dP(u, v)< q} form a neighbourhood basis of u, and for every w E N,(u), d,,(w, (0)) ~d,,(w, u) + dP(u, (0)) G K + q. So N,(u) is uniformly support bounded, and hence p- Blaschke. So cl(N,,(u)) is compact, and U is locally compact. For necessity, note that (V’, d,,), 1 up <w, is actually a locally compact space, since the same argument shows every point of the metric space has a compact neighbourhood. Since, for 1 G p < ~0, the space is also separable, %” = IJkrl U, where U, E . . - 5 u, c r/k,, c * * * and the r/, are p-Blaschke. So any closed subset of U that is uniformly support bounded lies in one of the uk, for some sufficiently large k, and is thus p-mean equi-left-continuous, and so p-Blaschke.

Remark. The space (%“, d-q) is not locally compact, in contrast to the above.

References

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Systems 29 (1989) 341-348. [4] L.M. Graves, ‘Be Theory of Functions of Real Variables (McGraw-Hill, New York, 1946). [S] 0. Kaleva, Fuzzy differential equations, Fuzzy Sers and Systems 24 (1987) 301-317. [6] E.P. Klement, M.L. Puri and D.A. Ralescu, Limit theorems for fuzzy random variables, Proc.

Roy. Sot. London Ser. A 407 (1986) 171-182. [7] P.E. Kloeden, Fuzzy dynamical systems, Fuzzy Sers and Sysfems 7 (1982) 275-296. [8] P.E. Kloeden, Chaotic mappings on fuzzy sets, Proc. Second IFSA Congress (Tokyo, July 1987)

Vol. 1, 368-371. [9] S.R. Lay, Convex Sefs and their Applications (John Wiley, New York, 1982).

[lo] N.N. Lyashenko, Statistics of random compacts in Euclidean space, J. Soviet Math. 21 (1983) 76-92.

[ll] M.L. Puri and D.A. Ralescu, The concept of normality for fuzzy random variables, Ann. Probab. 13 (1985) 1373-1379.

[12] K.A. Vitale, Lp metrics for compact, convex sets, J. Approx. Theory 45 (1985) 280-287.