- Email: [email protected]
Upper and lower fuzzy measures
F~,y. Sets and Sys~©ms3~ (1989) |91-20~ No~h-HoH~
UPPER
AND
LO~R
191
FUZZY
MEASURES
M. D E L G A D O , S. M O R A L
Received Febnmry 1987 Revised January 1988
Abs~'ac~:Puff m~dR~e~cu [5] sho~ed that a pc~b~ty me~u~©is not, in the gener~ c~u:, fuzzy me,~ur~. This p~per deals with dc~cgn~n~nga general fnuncwo~kwhere both p ~ y and fuzzy measures are ~nc|uded.~ order to do that, the upper (knver) fuzzy measures, which need not be ~ower(u~cr) continuous, are con~dered. The pc~b~ty ~ncce~y) ~ w~ be cnmpies of s~h upper (|ower) fury mc~u~c~. Finely the extenL~on~of ~ e concep~ to fuzzy sets are also considered.
Keywords:Pu~bi~gy me,urea; fuzzy me~m'~; dusty; continuity.
In 1982 Puri and Ra|escu [5] showed h'~at a possibility m e ~ u r e (see Zadeh [8]) is a ~f"f"f"f"f"f"f"f"f"~ measure (see Sugeno [7]) only when the referential is finite or in particular trivial cL~s. This can be extended to the p|ansibility and belief measures associated with an evidence (see Shafer [6]). There seems therefore that, in the field of fuzzy representation of uncertainty, there is no unity of concept, no single structure for all of them. In this paper, our aim is to provide e framework where the plausibility and belief measures, as well as Sugeno's fuzzy measures, are included. If we have a monotonous functional g from the set ~ ( X ) over [0,1], then we always can consider its dual functional g, defined by R(A) -- 1 - g(A). In this way, when we have a subset A from X, we shall have two uncertainty values: g(A) and g(A). A coherence c c n ~ t i o n for a couple of dual fuzzy measures is that Min{g(A), R(A)} ~ obtained always for the same measure, g , , and therefore, Max{g(A), g(A)} for another, g*. In this case, we can associate with a subset ,4 c X an uncertainty interwd [g.(A), g*(A)]: g . ( A ) will be the lower uncertainty v~due and g*(A) the upper uncertainty value. Intuitively, we can say g. is a pessimistic representation of the information and g* an optimistic one. That is the case for the necessity and possibifiW measures, belief and plausibility measures and so on. Now, we pose the following question: Must we demand that an optimistic (pessimistic) measure, be lower-continuous (upper
192
M. DeYgedo, £ Moral
2. " l ~ d , ~ i n l ~ o~ lower ~
u l l ~ r l'~my m~mm'~
The following concept of f u z ~ measure is due to Sugeno [7]. D e ~ L [~t X be a set and ~ ~ P(X) a o-algebra. Then an application g: ~--* [0, 1] is said to be a fuzzy measure if and only if it satisfies: O) g(~) = o, g ( x ) = I. (2) A = B :~ g(A) <~g(B) (3) {An}~A :> g(A)=Lim~®g(An) (4) {A,,}~A ::~ g(A)=Limn_.®g(A#) An h~aportant class of measures are the possibility measures, introduced by Zadeh [8]. D e ~ i f l o n 2. A normalized possibility measure in X is an application/7: ~ ( X ) ~ [0, 1] ~ven by J~r(A) = Supf(a) a~A
where f is a function f :Y--* [0, 1] with Supa~xf(a) = 1. The possibility measures are widely used and are always fuzzy in the sense of Definition 1 when X is a finite set. However, Puri and Ralescu [5] pointed out that, in general, a possibility measure defined in an infinite set is not a fuzzy measure. It is easy to prove that a possibility measure always verifies (1), (2) and (3), bm only in trivial c~es (4) is satisfied. We will e×tend the concept of fuzzy measure to a context containing the possibility measures. First of all, we shall determine the algebraic structure where these ~ew measures are to be defined. De.on satisfies:
3o Lot X be a set. A a*-algebra in X is a class ~ * , ' - ~ ( X ) that
Oi) A , B ~ , ~ * ~ A U B ¢ , , ~ * ; (iii) A n ¢ ~ * , V n c N ^ {A#}~A -'~ A ¢ , ~ * . A o,-algebra in X is a class ,d, ~ ~ ( X ) with the properties: Oi) A , B ¢ ~ . ~ A N B ¢ ~ . ; (iii) An e ,~,, Vn ~ N ^ {An}~A ::~ A ¢ ~ , De.on 4. If ~ . and ~ * are a o.-algebra a::d a o*-algebra respectively, then we say that they are dual if and only if A¢~.
¢:~ / ~ ¢ . ~ *
NoSe 1. To every a.-algebra ,~, is associated a dual o*-algebra ,~*, and reciprocally.
Upperandlower)~azzy mwaures
193
No~e 2, A ~-algebra is a ~-algebra (o*-algebra) that is auto-duaL Note 3. We shall call the couple (X, d * ) an upper measurable space and (X, d . ) a lower measurable space. If .@,, .~* are dual then (X, .@,, .@*) is s~d to be an extreme measurable space.
E n m p l e 1. Let us consider in R the following couple of dual o,. or*-algebras:
~
= { A ~ R IA =(~, +®), ~ E R } U {~} O {R}, ~ , . = {A = R [A = (-®, ~], ~ R } U {~} O {R}.
We shall call ( ~ , , ~ ) the upper Betel o,, o*-algebras. It is immediate that the smallest or-algebra containing ~ ' U ~ l , is the Betel a-algebra in R, ~.
Example 2. Another couple of o,, or*-algebras in ~ is
@~ = i:~ = ~ I A = ( - % ~), ~ }
u {~} u { a } ,
~ , = {A ~
u {~} u (~}.
I ~ = [~, +®), ~ )
These will be called the lower Betel or,, or*-aigebras. Defmit]on $. L~:t (X, ~ * ) be an upper measurable space. An upper measure (UFM) is an application g*: ~*--* |0, 1] that satisfies:
O) g*(~) = o, g * ( x ) = 1; (ii) A ~ B O g*(A)<~g*(B);
(iii) {A,,}T,4 :=> g*(A)=Lim,~..,®g*(A,,). DeRm~oa 6. Let (~Y, d . ) be a lower measurable space. A lower fuzzy measure (LFM) in (X, .#.) is an application g. :~.--~ [0, 1] that satisfies: (i) g.(~) =0, g.(x) = 1; (ii) A c B :~> g,(A)<~g,(B); (iii) {A,}~A ~ g.(A)=Lim,...®g.(A,,). Note 4. If ,~., ,~* are dual or., or*-algebras and g* an UFM in (X, ~ * ) then Me dual measure of g*(g,(A)= 1 - g * ( A ) ) is a LFM in (X, ,~.) and reciprocally. The couple (g., g*) will be called a couple of extreme fuzzy measures. Detdfien 7. The couple of extreme fuzzy measures (g,, g*) is said to be ordered if and only if it verifies:
VAe~.,B~*,
A~B => g.(A)~
Note $. Let us observe that a possibility measure,/7, is an UFM in (X, ~P(X)) and its dual measure, IV, is called a necessity measure, being a LFM.
194
M. Delgado, $. Moral
3. ]a the Theory of Evidence (see Dempster [4], Sharer [6]) a basic probability assignment m, in a finite set U, is a probability distribution in .@(U) with m(~) = 0. Associated with a basic probability assignment in U, there are two fuzzy measures in ~(U): - The p|ausibiiity measure ~ven by P|(A)--
re(B).
~ Br'Ul~
- The belief measure given by
Bel(A)=
X
re(B).
BcA
In general, when X is a not necessarily finite set, we can consider that a basic probability assignment M is a probability measure in (~(X), ~ ) , where ~ c ~ ( ~ X ) ) is a e-algebra. In this situation it is not very difficult to prove the following results: • (a) The class d * = { A = X ] { B ] B N A q: ~} ~ ~} is a o*-aigebrao (b) The class ~ . -- {A = X [ {B ] B c A } E ~ } is a o.-algebra. (¢) ~ , and ~f* are dual (d) The plausibility measure associated with M, Pl:d*---~[0, 1], given by PI(A)=M(BIBNA *~}, is an UFM. (e) The belief measure associated with M; ~ , t : a . - - * [0, I], given ~y Be](A) = M{B [B =A), is a LFM. (f) (Be|, Pl) is a couple of ordered ,f,2zzy measures.
3. Mea~r~J~e fnncfions h t w e e n extreme mea~rable spaces
In this section we will consider the problem of how to transform an upper (lower) fuzzy measure by an application between two extreme measurable spaces. This can be applied to the extension of extreme fuzzy measures to fuzzy sets. 8. Let (X1, a~t., af~) and (X2, af2., ~ ) be two extreme measurable spaces. An application h : X I ~ X 2 is said to be measurable if and only if it sa~es
or equivalently,
h-~(A) e .~,,
'CA e .~2,.
9. Let (X~, ~f~,, .~*) (i -- 1, 2) be two extren~e measurable spaces and a measmabie function. If (gt.,g~) is a couple of ex~eme fuzzy
h:X:'~X2
Upperand lowerfuzzy mcasurca
195
measures in (X1, ~ t . , ~f~), then we shah ca]| extreme fuzzy measures induced by h in (X2, .~2., ,.~) th,~ fo|lowing ones: - x~'M: g~(A) = g,~;,-'(A)),
- LFM: 8~(A)=Z~(h-'(A)). In the scque! we shah only consider ~-v~ued measurable functions. ~ , J ~ o n 10. Let (X, ~., d*) ~ an extreme measurable space. We say that the application h :X---, R is lower measurable if and only if it is measurable considering the lower tT,, o*-algebras, ~2. and ~ , in R (see Example 2). ~ n U . Let (X, ~ . , ~*) be, an extreme measurable space. We say the application h : X - ~ R is upper meas|lrable if and only if it is measurable considering the upper o,, o*-algebras, ~ t , and ~ , in R (see Example 1).
Proposition 1. h is upper measurable if and only if - h is lower measurable. ~fo
It is immediate from the equality ( - h ) - l ( -oo, ~) -- h - l ( - ~ , +co).
~ e 2. If hn is upper measurable for every n E N and {hn} ~h, fhen h is upper measurable. ]~roof. Fh'st of all, let us observe that since {hn}Th, h(a) > ~ ¢~ :~n E ~ with h.(a) > ~,
(1)
h~(a)>~ ¢> hm(a)>ot, Ym~n.
(2)
From (1) it is easy to check that
h-~((a, + ® ) ) = { a E X l h ( a ) > a } = { a e- X [ : ~ n e N with h.(a) > ~}
= nU {a~xlh~(a)>~} =Uh='((~, +®)). ~N Taking into account (2) it follows that
(~='((~, ÷®))}T~-~((~, +®)); and as h~((~, +®)) ¢ ~*, Vn ~ N, then ~-1((~, +~)) ¢ ~*, which ends the proof.
I~o~n 3. If h~ is lower measurable for every n ~ N and {h~}~h, then h is lower measurable. ~ .
It follows from Propositions i and 2 and the following property:
{~}~ ~ {-~}l-~.
~
M. Oelgado, $. Moral
4. If hi and hz are upper measurable functions then Sup(hi, h2) = hi v he ~ upper measurable too. Proof. This property follows from the equality
(h~ v h~)-~((~, ÷~)) = hi~((~, +oo))u h~'~((~,+®)). I~ro~a $. If hi and hz are lower measurable then Inf(hl, h2)= hl ^ h2 /s lower measurable too. Proof. It is hnmediate from
(hi ^ h2)-1((-% ~)) = hil((-®, ~)) n hi'((-®, ~)). Note 6. Let (X, ~ , , ~ * ) be an extreme measurable space. Denote by ~ X ) the set of fuzzy subsets of X and consider in it the operator: - Union: ~AvB= ~,4 V ~a; - Intersection: ,UAna: ~,~ ^/~a; Complementary: tzA = 1 - ~A; and the relation of inclusion: A c B ¢~ t~A(x)~<~8(x), Vx ¢ X. Then we can fuzzffy the o,, Ù*-algebras ~ * and ..~.considering - Fuzzy a*-algebra: -
~ * = {A ¢ ~(X) ! ~,4 is upper measurable}; -
F~
o,-a]gebra~ •~ , = {A ~ ~(X) I~A is lower measurable}.
R is immediate to prove the following results: (a) If ~A is constant in X, then A ¢ ~ , and A e ~*. (b) A , B ¢ ~ , => A f 3 B ¢ ~ , ; A , B ¢ ~ * :.':> A t 3 B ¢ ~ * . (c) A ~ ¢ . ~ * , V n e N , and {A,,}'[A :-~ A ¢ ~ * ; A,,¢f~,,VneN,
( ~ } ~ =>~ . . (d) ~ *
and
~ A~..
Insh0rt, we can consider ,~,, ~ * as the dual fuzzy o,, ~*-algebras associated w;~ ~ , , ~*. In the sequel, we shall show how every upper (lower) measurable application can be app~oxhnated by some special and more shnple upper (lower) measurable
12. An application h :X--* R is said to be simple if and only if h = ~IA + ~, where ~, ~ ¢ ~ and/,4 is the ~emVersi~ip function of A ~ X. For these func~ons we have the following property. The proof is trivial and will be onfiUed.
Upperand levy fuzzy measures
197
6. Let ~I~ + [3 be a simple funclion ~ d in an ex~eme measurab~ space (X, M., ~*). Then: (a) ¢IA + ~ is upper measurable if and only if (~ > 0 and A ¢ ~ * ) or (~ < 0 and A ~
~ , ) or (~ =
0).
(b) ~I~ + ~ is lower measurable if and only if (~ > 0 and A ¢ ~ , ) or (~ < O and A ¢
~*) or (~ =
0).
D e l e t i o n 13o (a) A function h : X - ~ R is said to be upper elemental if and o ~ y if h = Mexl~iG.h~, where h~ is simple and upper measurable for every i ¢ {1 . . . . . n}. (b) A function h : X - - ~ is said to be lower elemental if and only if h = M i n ~ . h ~ , where h~ is simple and lnwer measurable for every i ¢ {1 . . . . . n}. Finally we can state and show the approximatinn property. Propesf~ion 7. Let (X, ~ . , ~ * ) be an extreme measurable space and h a function h : X - * R. Then: (a) h is upper measurable and lower bounded ~ there exists {h.}~'h, where h~ is an upper elemental function for every n ¢ N. (b) h is lower measurable and upper bounded =~ there exists {h.} ~h, where h~ is a lower measurable funcffon for every n ¢ N. ][~eof. We shall show only (a) because the proof of (b) is very similar. Sufficiency: This is ~mmediate from Proposition 2 and the fact that an elemental function is always lower bounded. Necessity: If h is an upper measurable function, we shall define h~, for every n ¢ N, in the following way. Write h,~=~.~l~-M,
i = 0 , 1. . . . . n2 "+~,
where a~.~= (i/2~), A,~ = {a ¢ X I h(a) > ~m} and M is a lower bound of h. The applications h~ are simple and upper measurable becau~ a~,~> 0 and A,~ ¢ ..~* (~om the measurability of h). Therefore
h~=
M~h~
O~i~n2,~+t
is upper elemental for every n ¢ N. We have only to prove that {h.} is non-decreasing and {h,}--,h. Since h~ = h ¢ . + ~ it follows that h.+~ ~ h~, that is h~ is non-decreasing. Furthermore ifa c X a n d h(a)<~n, then h(a) - h . ( a ) ~< (1/2"),
that is {h,}~h, wh~h ends the proof.
198
M. Delgado, $. Moral
4, E ~
of e x ~ e u e m e ~ e s
to ~
se~
Let (X, ~f,, .~*) be an extreme measurable space and (g,, g*) a couple of extreme fuzzy measures defined over it. Then we have (see Note 6) two extreme o,, o*-algebras in X:
~ * = {A e ~(X) ] ~.4 is upper measurable}, .~, = {A e .~(X) [ ~A is lower measerable}. We want to define an UFM in (X, ~ * ) and a LFM in (X, •.). We have two ways in order to do that (see [2]). The first is based on the Sugeno integral [7]: ~ ~ 14o The Sugeno extension of a couple of extreme measures (g., g*) i~ defined as ~*(A) = Inf ( , v g*(A')), ~e[0.1]
vA ¢ 3 "
where A~, = {x e X [ ~A(x) > oQ, and g,(A) = Sup ( e ^ g,(Ao)), ~e[o,1]
w G~ ,
where A~ = {x G x [ ~A(x) >~~}. Proposition 8° Under the above conditions the couple (g,, g*) verifies: (1) I f .~, = M* and g , = g*, then .~, = .~* and g , = g*. (2) g/~A is constantly equal to c ¢ [0, 1] then g,(A ) = g*(A ) - c.
(3) A ~ e (4) (5) (6) (7)
=> (g,(A)~
I f {An}~,A, where An ~ ~ , , then g,(A) = Limn-~=g,(An). I f {An}~A, where An ¢ fg*, then g*(A) = Limn...®g*(A.). I f A G ~ , , then g,(A) + g*(.~)= l. I f (g,, g*) is ordered, then VA ¢ z~,, B ¢ .~*, A = B => g,(A) ~
l~,~f.. It is immediate from the properties of Sugeno fuzzy integral. The other extension we propose is based on Choquet expectation for capacities [3]. D e ~ 15. The Chequer extension of a couple of extreme measures (g,, g*) defined as
~*(~)=
g*(A;)de,
~,(~)=
g,(~.)de.
9. The couple (G,, G*) verifies the followingproperties: (1) I f ~ , = .~* and g, = g*, then M, = M* and G, = G*. (2) ~ f ~ is constantly equal to c¢[0, 1], then (~,(A) = ¢*(A) = c.
(3) A ~ ~ => (O,(A) ~<~.(~)) ^ (~3~=(A)~
Upperand lowerfuzzy measm~ (4) (5) (6) (7)
199
If {A,}~A, where An ¢ ,~., then IAmn.-,, ~,.(A,) = ~ . ( A ) . If {An}~A, where An ¢ ,M*, then Linln-.® ~*(An) = (~*(A). I r A ¢ M., then ~ . ( A ) + ~*(A) = 1. If (g., g*) is ordered, then VA ¢ ,~,, B ¢ ,~*, A c B ~ ~ , ( A ) ~ ~*(B).
Proof. This proposition can be easily shown from Me properties of I ~Desgue inte~al. 5. Conch~ona
In this paper, a general stn~cture for opfimisfic-pesshnis~c measures, where the Sugeno fuzzy measures a~e also included, is developed. The following points have been considered. The determination of the algebraic structures where these measures are defined: the upper and lower a,, a*-algebras. - The definition of upper fuzzy measures (optimistic point of view, where plausibility, possibility and probability measures are included) and lower fuzzy measures (pessimistic point of view, ~,here belief, necessity and probabiliW measures are included). - The o,, a*-algebras and extreme fuzzy measures are extended to fuzzy sets. Finally, Figure 1 shows the relationships among the upper and lower fuzzy measures and the different types of fuzzy measures as classified by Bannon [1]. -
A1 = {lower fuzzy m~asures} A1 n A2 = {fu~Y measures} 51 : {belief measures} B1 n B2 = C1 = D1 : El :
{probability measures} {necessity measures} {simple support functions} {vacuous belief fu,~c~;o,~s}
A2 = {upper fu~/measures} 8 2 : {plausibility measures} C2 = {possibility measures} D2 = {0-1 po•ibility measures} E2 = {maximum possibility measure}
Fig. 1. Upperand 2owerfuzzymeasuresexceptDirac measures.
M. Delgado, $. Moral
200
A~~gemen~ Whe authors are gratefu| to the referees for their usefu| cormnents and suggestions.
References [1] G. Bannon, Distinction betw©el, several su'osets of ~
m~nres,
~.~zzy Sets and Systems $
(1981) 291-3o5. [2] MJ. Bolafi~, M.T. L~unata, S. Moral, Altcmativas para |a extcnsi6n de medidas differ, Ac~s X V Congreso SEIOE, Oviedo (1985). I3] C~. C ~ u © t , Theory of capacities, Ann. Inst. Fourier 5 (1953) 131-292. [~] A.P. Dempster, Upper and lower probabilities induced by a mu]tivalued mapping, Ann. ?dath. Staff. 38 (1967) 325-339. [5] M.L. Puri and D. Ra!e~u, A p~ibmty measure is not a fuzzy,measure, Fuzzy Set~ and Systems 7 (1982) 311-313. [6] G. Sharer, A Mathema~cal Theory of Evidence (Princeton University Press, Princeton, NJ, |976). [7] M. Sugeno, 'Bl¢ory of fu~v integrals and its applications, Thesis, Tokio Institute of Technology (1974). [8] L.A. Zadeh, Ftl~y sets as a basis for a theory of possibility, Fuzzy Se~s and Systems I (1978) 3-28.
UPPER
AND
LO~R
191
FUZZY
MEASURES
M. D E L G A D O , S. M O R A L
Received Febnmry 1987 Revised January 1988
Abs~'ac~:Puff m~dR~e~cu [5] sho~ed that a pc~b~ty me~u~©is not, in the gener~ c~u:, fuzzy me,~ur~. This p~per deals with dc~cgn~n~nga general fnuncwo~kwhere both p ~ y and fuzzy measures are ~nc|uded.~ order to do that, the upper (knver) fuzzy measures, which need not be ~ower(u~cr) continuous, are con~dered. The pc~b~ty ~ncce~y) ~ w~ be cnmpies of s~h upper (|ower) fury mc~u~c~. Finely the extenL~on~of ~ e concep~ to fuzzy sets are also considered.
Keywords:Pu~bi~gy me,urea; fuzzy me~m'~; dusty; continuity.
In 1982 Puri and Ra|escu [5] showed h'~at a possibility m e ~ u r e (see Zadeh [8]) is a ~f"f"f"f"f"f"f"f"f"~ measure (see Sugeno [7]) only when the referential is finite or in particular trivial cL~s. This can be extended to the p|ansibility and belief measures associated with an evidence (see Shafer [6]). There seems therefore that, in the field of fuzzy representation of uncertainty, there is no unity of concept, no single structure for all of them. In this paper, our aim is to provide e framework where the plausibility and belief measures, as well as Sugeno's fuzzy measures, are included. If we have a monotonous functional g from the set ~ ( X ) over [0,1], then we always can consider its dual functional g, defined by R(A) -- 1 - g(A). In this way, when we have a subset A from X, we shall have two uncertainty values: g(A) and g(A). A coherence c c n ~ t i o n for a couple of dual fuzzy measures is that Min{g(A), R(A)} ~ obtained always for the same measure, g , , and therefore, Max{g(A), g(A)} for another, g*. In this case, we can associate with a subset ,4 c X an uncertainty interwd [g.(A), g*(A)]: g . ( A ) will be the lower uncertainty v~due and g*(A) the upper uncertainty value. Intuitively, we can say g. is a pessimistic representation of the information and g* an optimistic one. That is the case for the necessity and possibifiW measures, belief and plausibility measures and so on. Now, we pose the following question: Must we demand that an optimistic (pessimistic) measure, be lower-continuous (upper
192
M. DeYgedo, £ Moral
2. " l ~ d , ~ i n l ~ o~ lower ~
u l l ~ r l'~my m~mm'~
The following concept of f u z ~ measure is due to Sugeno [7]. D e ~ L [~t X be a set and ~ ~ P(X) a o-algebra. Then an application g: ~--* [0, 1] is said to be a fuzzy measure if and only if it satisfies: O) g(~) = o, g ( x ) = I. (2) A = B :~ g(A) <~g(B) (3) {An}~A :> g(A)=Lim~®g(An) (4) {A,,}~A ::~ g(A)=Limn_.®g(A#) An h~aportant class of measures are the possibility measures, introduced by Zadeh [8]. D e ~ i f l o n 2. A normalized possibility measure in X is an application/7: ~ ( X ) ~ [0, 1] ~ven by J~r(A) = Supf(a) a~A
where f is a function f :Y--* [0, 1] with Supa~xf(a) = 1. The possibility measures are widely used and are always fuzzy in the sense of Definition 1 when X is a finite set. However, Puri and Ralescu [5] pointed out that, in general, a possibility measure defined in an infinite set is not a fuzzy measure. It is easy to prove that a possibility measure always verifies (1), (2) and (3), bm only in trivial c~es (4) is satisfied. We will e×tend the concept of fuzzy measure to a context containing the possibility measures. First of all, we shall determine the algebraic structure where these ~ew measures are to be defined. De.on satisfies:
3o Lot X be a set. A a*-algebra in X is a class ~ * , ' - ~ ( X ) that
Oi) A , B ~ , ~ * ~ A U B ¢ , , ~ * ; (iii) A n ¢ ~ * , V n c N ^ {A#}~A -'~ A ¢ , ~ * . A o,-algebra in X is a class ,d, ~ ~ ( X ) with the properties: Oi) A , B ¢ ~ . ~ A N B ¢ ~ . ; (iii) An e ,~,, Vn ~ N ^ {An}~A ::~ A ¢ ~ , De.on 4. If ~ . and ~ * are a o.-algebra a::d a o*-algebra respectively, then we say that they are dual if and only if A¢~.
¢:~ / ~ ¢ . ~ *
NoSe 1. To every a.-algebra ,~, is associated a dual o*-algebra ,~*, and reciprocally.
Upperandlower)~azzy mwaures
193
No~e 2, A ~-algebra is a ~-algebra (o*-algebra) that is auto-duaL Note 3. We shall call the couple (X, d * ) an upper measurable space and (X, d . ) a lower measurable space. If .@,, .~* are dual then (X, .@,, .@*) is s~d to be an extreme measurable space.
E n m p l e 1. Let us consider in R the following couple of dual o,. or*-algebras:
~
= { A ~ R IA =(~, +®), ~ E R } U {~} O {R}, ~ , . = {A = R [A = (-®, ~], ~ R } U {~} O {R}.
We shall call ( ~ , , ~ ) the upper Betel o,, o*-algebras. It is immediate that the smallest or-algebra containing ~ ' U ~ l , is the Betel a-algebra in R, ~.
Example 2. Another couple of o,, or*-algebras in ~ is
@~ = i:~ = ~ I A = ( - % ~), ~ }
u {~} u { a } ,
~ , = {A ~
u {~} u (~}.
I ~ = [~, +®), ~ )
These will be called the lower Betel or,, or*-aigebras. Defmit]on $. L~:t (X, ~ * ) be an upper measurable space. An upper measure (UFM) is an application g*: ~*--* |0, 1] that satisfies:
O) g*(~) = o, g * ( x ) = 1; (ii) A ~ B O g*(A)<~g*(B);
(iii) {A,,}T,4 :=> g*(A)=Lim,~..,®g*(A,,). DeRm~oa 6. Let (~Y, d . ) be a lower measurable space. A lower fuzzy measure (LFM) in (X, .#.) is an application g. :~.--~ [0, 1] that satisfies: (i) g.(~) =0, g.(x) = 1; (ii) A c B :~> g,(A)<~g,(B); (iii) {A,}~A ~ g.(A)=Lim,...®g.(A,,). Note 4. If ,~., ,~* are dual or., or*-algebras and g* an UFM in (X, ~ * ) then Me dual measure of g*(g,(A)= 1 - g * ( A ) ) is a LFM in (X, ,~.) and reciprocally. The couple (g., g*) will be called a couple of extreme fuzzy measures. Detdfien 7. The couple of extreme fuzzy measures (g,, g*) is said to be ordered if and only if it verifies:
VAe~.,B~*,
A~B => g.(A)~
Note $. Let us observe that a possibility measure,/7, is an UFM in (X, ~P(X)) and its dual measure, IV, is called a necessity measure, being a LFM.
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3. ]a the Theory of Evidence (see Dempster [4], Sharer [6]) a basic probability assignment m, in a finite set U, is a probability distribution in .@(U) with m(~) = 0. Associated with a basic probability assignment in U, there are two fuzzy measures in ~(U): - The p|ausibiiity measure ~ven by P|(A)--
re(B).
~ Br'Ul~
- The belief measure given by
Bel(A)=
X
re(B).
BcA
In general, when X is a not necessarily finite set, we can consider that a basic probability assignment M is a probability measure in (~(X), ~ ) , where ~ c ~ ( ~ X ) ) is a e-algebra. In this situation it is not very difficult to prove the following results: • (a) The class d * = { A = X ] { B ] B N A q: ~} ~ ~} is a o*-aigebrao (b) The class ~ . -- {A = X [ {B ] B c A } E ~ } is a o.-algebra. (¢) ~ , and ~f* are dual (d) The plausibility measure associated with M, Pl:d*---~[0, 1], given by PI(A)=M(BIBNA *~}, is an UFM. (e) The belief measure associated with M; ~ , t : a . - - * [0, I], given ~y Be](A) = M{B [B =A), is a LFM. (f) (Be|, Pl) is a couple of ordered ,f,2zzy measures.
3. Mea~r~J~e fnncfions h t w e e n extreme mea~rable spaces
In this section we will consider the problem of how to transform an upper (lower) fuzzy measure by an application between two extreme measurable spaces. This can be applied to the extension of extreme fuzzy measures to fuzzy sets. 8. Let (X1, a~t., af~) and (X2, af2., ~ ) be two extreme measurable spaces. An application h : X I ~ X 2 is said to be measurable if and only if it sa~es
or equivalently,
h-~(A) e .~,,
'CA e .~2,.
9. Let (X~, ~f~,, .~*) (i -- 1, 2) be two extren~e measurable spaces and a measmabie function. If (gt.,g~) is a couple of ex~eme fuzzy
h:X:'~X2
Upperand lowerfuzzy mcasurca
195
measures in (X1, ~ t . , ~f~), then we shah ca]| extreme fuzzy measures induced by h in (X2, .~2., ,.~) th,~ fo|lowing ones: - x~'M: g~(A) = g,~;,-'(A)),
- LFM: 8~(A)=Z~(h-'(A)). In the scque! we shah only consider ~-v~ued measurable functions. ~ , J ~ o n 10. Let (X, ~., d*) ~ an extreme measurable space. We say that the application h :X---, R is lower measurable if and only if it is measurable considering the lower tT,, o*-algebras, ~2. and ~ , in R (see Example 2). ~ n U . Let (X, ~ . , ~*) be, an extreme measurable space. We say the application h : X - ~ R is upper meas|lrable if and only if it is measurable considering the upper o,, o*-algebras, ~ t , and ~ , in R (see Example 1).
Proposition 1. h is upper measurable if and only if - h is lower measurable. ~fo
It is immediate from the equality ( - h ) - l ( -oo, ~) -- h - l ( - ~ , +co).
~ e 2. If hn is upper measurable for every n E N and {hn} ~h, fhen h is upper measurable. ]~roof. Fh'st of all, let us observe that since {hn}Th, h(a) > ~ ¢~ :~n E ~ with h.(a) > ~,
(1)
h~(a)>~ ¢> hm(a)>ot, Ym~n.
(2)
From (1) it is easy to check that
h-~((a, + ® ) ) = { a E X l h ( a ) > a } = { a e- X [ : ~ n e N with h.(a) > ~}
= nU {a~xlh~(a)>~} =Uh='((~, +®)). ~N Taking into account (2) it follows that
(~='((~, ÷®))}T~-~((~, +®)); and as h~((~, +®)) ¢ ~*, Vn ~ N, then ~-1((~, +~)) ¢ ~*, which ends the proof.
I~o~n 3. If h~ is lower measurable for every n ~ N and {h~}~h, then h is lower measurable. ~ .
It follows from Propositions i and 2 and the following property:
{~}~ ~ {-~}l-~.
~
M. Oelgado, $. Moral
4. If hi and hz are upper measurable functions then Sup(hi, h2) = hi v he ~ upper measurable too. Proof. This property follows from the equality
(h~ v h~)-~((~, ÷~)) = hi~((~, +oo))u h~'~((~,+®)). I~ro~a $. If hi and hz are lower measurable then Inf(hl, h2)= hl ^ h2 /s lower measurable too. Proof. It is hnmediate from
(hi ^ h2)-1((-% ~)) = hil((-®, ~)) n hi'((-®, ~)). Note 6. Let (X, ~ , , ~ * ) be an extreme measurable space. Denote by ~ X ) the set of fuzzy subsets of X and consider in it the operator: - Union: ~AvB= ~,4 V ~a; - Intersection: ,UAna: ~,~ ^/~a; Complementary: tzA = 1 - ~A; and the relation of inclusion: A c B ¢~ t~A(x)~<~8(x), Vx ¢ X. Then we can fuzzffy the o,, Ù*-algebras ~ * and ..~.considering - Fuzzy a*-algebra: -
~ * = {A ¢ ~(X) ! ~,4 is upper measurable}; -
F~
o,-a]gebra~ •~ , = {A ~ ~(X) I~A is lower measurable}.
R is immediate to prove the following results: (a) If ~A is constant in X, then A ¢ ~ , and A e ~*. (b) A , B ¢ ~ , => A f 3 B ¢ ~ , ; A , B ¢ ~ * :.':> A t 3 B ¢ ~ * . (c) A ~ ¢ . ~ * , V n e N , and {A,,}'[A :-~ A ¢ ~ * ; A,,¢f~,,VneN,
( ~ } ~ =>~ . . (d) ~ *
and
~ A~..
Insh0rt, we can consider ,~,, ~ * as the dual fuzzy o,, ~*-algebras associated w;~ ~ , , ~*. In the sequel, we shall show how every upper (lower) measurable application can be app~oxhnated by some special and more shnple upper (lower) measurable
12. An application h :X--* R is said to be simple if and only if h = ~IA + ~, where ~, ~ ¢ ~ and/,4 is the ~emVersi~ip function of A ~ X. For these func~ons we have the following property. The proof is trivial and will be onfiUed.
Upperand levy fuzzy measures
197
6. Let ~I~ + [3 be a simple funclion ~ d in an ex~eme measurab~ space (X, M., ~*). Then: (a) ¢IA + ~ is upper measurable if and only if (~ > 0 and A ¢ ~ * ) or (~ < 0 and A ~
~ , ) or (~ =
0).
(b) ~I~ + ~ is lower measurable if and only if (~ > 0 and A ¢ ~ , ) or (~ < O and A ¢
~*) or (~ =
0).
D e l e t i o n 13o (a) A function h : X - ~ R is said to be upper elemental if and o ~ y if h = Mexl~iG.h~, where h~ is simple and upper measurable for every i ¢ {1 . . . . . n}. (b) A function h : X - - ~ is said to be lower elemental if and only if h = M i n ~ . h ~ , where h~ is simple and lnwer measurable for every i ¢ {1 . . . . . n}. Finally we can state and show the approximatinn property. Propesf~ion 7. Let (X, ~ . , ~ * ) be an extreme measurable space and h a function h : X - * R. Then: (a) h is upper measurable and lower bounded ~ there exists {h.}~'h, where h~ is an upper elemental function for every n ¢ N. (b) h is lower measurable and upper bounded =~ there exists {h.} ~h, where h~ is a lower measurable funcffon for every n ¢ N. ][~eof. We shall show only (a) because the proof of (b) is very similar. Sufficiency: This is ~mmediate from Proposition 2 and the fact that an elemental function is always lower bounded. Necessity: If h is an upper measurable function, we shall define h~, for every n ¢ N, in the following way. Write h,~=~.~l~-M,
i = 0 , 1. . . . . n2 "+~,
where a~.~= (i/2~), A,~ = {a ¢ X I h(a) > ~m} and M is a lower bound of h. The applications h~ are simple and upper measurable becau~ a~,~> 0 and A,~ ¢ ..~* (~om the measurability of h). Therefore
h~=
M~h~
O~i~n2,~+t
is upper elemental for every n ¢ N. We have only to prove that {h.} is non-decreasing and {h,}--,h. Since h~ = h ¢ . + ~ it follows that h.+~ ~ h~, that is h~ is non-decreasing. Furthermore ifa c X a n d h(a)<~n, then h(a) - h . ( a ) ~< (1/2"),
that is {h,}~h, wh~h ends the proof.
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M. Delgado, $. Moral
4, E ~
of e x ~ e u e m e ~ e s
to ~
se~
Let (X, ~f,, .~*) be an extreme measurable space and (g,, g*) a couple of extreme fuzzy measures defined over it. Then we have (see Note 6) two extreme o,, o*-algebras in X:
~ * = {A e ~(X) ] ~.4 is upper measurable}, .~, = {A e .~(X) [ ~A is lower measerable}. We want to define an UFM in (X, ~ * ) and a LFM in (X, •.). We have two ways in order to do that (see [2]). The first is based on the Sugeno integral [7]: ~ ~ 14o The Sugeno extension of a couple of extreme measures (g., g*) i~ defined as ~*(A) = Inf ( , v g*(A')), ~e[0.1]
vA ¢ 3 "
where A~, = {x e X [ ~A(x) > oQ, and g,(A) = Sup ( e ^ g,(Ao)), ~e[o,1]
w G~ ,
where A~ = {x G x [ ~A(x) >~~}. Proposition 8° Under the above conditions the couple (g,, g*) verifies: (1) I f .~, = M* and g , = g*, then .~, = .~* and g , = g*. (2) g/~A is constantly equal to c ¢ [0, 1] then g,(A ) = g*(A ) - c.
(3) A ~ e (4) (5) (6) (7)
=> (g,(A)~
I f {An}~,A, where An ~ ~ , , then g,(A) = Limn-~=g,(An). I f {An}~A, where An ¢ fg*, then g*(A) = Limn...®g*(A.). I f A G ~ , , then g,(A) + g*(.~)= l. I f (g,, g*) is ordered, then VA ¢ z~,, B ¢ .~*, A = B => g,(A) ~
l~,~f.. It is immediate from the properties of Sugeno fuzzy integral. The other extension we propose is based on Choquet expectation for capacities [3]. D e ~ 15. The Chequer extension of a couple of extreme measures (g,, g*) defined as
~*(~)=
g*(A;)de,
~,(~)=
g,(~.)de.
9. The couple (G,, G*) verifies the followingproperties: (1) I f ~ , = .~* and g, = g*, then M, = M* and G, = G*. (2) ~ f ~ is constantly equal to c¢[0, 1], then (~,(A) = ¢*(A) = c.
(3) A ~ ~ => (O,(A) ~<~.(~)) ^ (~3~=(A)~
Upperand lowerfuzzy measm~ (4) (5) (6) (7)
199
If {A,}~A, where An ¢ ,~., then IAmn.-,, ~,.(A,) = ~ . ( A ) . If {An}~A, where An ¢ ,M*, then Linln-.® ~*(An) = (~*(A). I r A ¢ M., then ~ . ( A ) + ~*(A) = 1. If (g., g*) is ordered, then VA ¢ ,~,, B ¢ ,~*, A c B ~ ~ , ( A ) ~ ~*(B).
Proof. This proposition can be easily shown from Me properties of I ~Desgue inte~al. 5. Conch~ona
In this paper, a general stn~cture for opfimisfic-pesshnis~c measures, where the Sugeno fuzzy measures a~e also included, is developed. The following points have been considered. The determination of the algebraic structures where these measures are defined: the upper and lower a,, a*-algebras. - The definition of upper fuzzy measures (optimistic point of view, where plausibility, possibility and probability measures are included) and lower fuzzy measures (pessimistic point of view, ~,here belief, necessity and probabiliW measures are included). - The o,, a*-algebras and extreme fuzzy measures are extended to fuzzy sets. Finally, Figure 1 shows the relationships among the upper and lower fuzzy measures and the different types of fuzzy measures as classified by Bannon [1]. -
A1 = {lower fuzzy m~asures} A1 n A2 = {fu~Y measures} 51 : {belief measures} B1 n B2 = C1 = D1 : El :
{probability measures} {necessity measures} {simple support functions} {vacuous belief fu,~c~;o,~s}
A2 = {upper fu~/measures} 8 2 : {plausibility measures} C2 = {possibility measures} D2 = {0-1 po•ibility measures} E2 = {maximum possibility measure}
Fig. 1. Upperand 2owerfuzzymeasuresexceptDirac measures.
M. Delgado, $. Moral
200
A~~gemen~ Whe authors are gratefu| to the referees for their usefu| cormnents and suggestions.
References [1] G. Bannon, Distinction betw©el, several su'osets of ~
m~nres,
~.~zzy Sets and Systems $
(1981) 291-3o5. [2] MJ. Bolafi~, M.T. L~unata, S. Moral, Altcmativas para |a extcnsi6n de medidas differ, Ac~s X V Congreso SEIOE, Oviedo (1985). I3] C~. C ~ u © t , Theory of capacities, Ann. Inst. Fourier 5 (1953) 131-292. [~] A.P. Dempster, Upper and lower probabilities induced by a mu]tivalued mapping, Ann. ?dath. Staff. 38 (1967) 325-339. [5] M.L. Puri and D. Ra!e~u, A p~ibmty measure is not a fuzzy,measure, Fuzzy Set~ and Systems 7 (1982) 311-313. [6] G. Sharer, A Mathema~cal Theory of Evidence (Princeton University Press, Princeton, NJ, |976). [7] M. Sugeno, 'Bl¢ory of fu~v integrals and its applications, Thesis, Tokio Institute of Technology (1974). [8] L.A. Zadeh, Ftl~y sets as a basis for a theory of possibility, Fuzzy Se~s and Systems I (1978) 3-28.