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An individual ergodic theorem for fuzzy random variables
Fuzzy Sets and Systems 13 (1984) 285-290 North-Holland
285
AN INDIVIDUAL ERGODIC THEOREM FOR FUZZY RANDOM VARIABLF~S* Masaaki M I Y A K O S H I and Masaru SHIMBO Division of Information Engineering, Graduate School of Engineering, Hokkaido University, Sapporo, Japan Received June 1983 Revised October 1983 We generalize Birkhott's celebrated individual ergodic theorem to fuzzy random variables. In particular, it is shown that (i) the limiting time average exists, (ii) it is invariant, and (iii) can be replaced by the phase average.
Keywords: Fuzzy random variables, Individual ergodic theorem.
1. Introduefion The concept of fuzzy random variables (frv's) was firstly introduced by Kwakernaak [3] where some fundamental properties can be found. Additional properties of frv's, e.g., a dominated convergence property, a strong law of large numbers for frv's, etc., have been shown by the present authors [5, 6]. The present paper is concerned with the investigation of a strong limit theorem usually referred to as the ergodic theorem in the presence of fuzziness. For the concepts and notations used in this paper the reader is referred to [5, 6, 9].
2. Preliminmies Let X = {x} be a nonempty set and ,A be a fuzzy subset of X. We denote the a-level sets by ~ - , the unit interval [0, 1] by I, and any dense subset in I by D = {8}, respectively. We have the following lemma. Lemma. If {B~, ot ~ I) is a non-increasing class o[ subsets of X such that
A = fa~
EI
~B~,
(1)
* This work has been partially supported by a Grant in Aid for Scientific Research from the Ministry of Education, Japan. 0165-0114/84/$3.00 t~) 1984, Elsevier Science Publishers B.V. (North-Holland)
M. Miyakoshi, M. Shimbo
286 then
= I8
~D
~B8
(2)
and
where the notations a > ~/ and a > 8 of Eq. (3) denote {~/ I 7 ~ I and a > 7} and {8 [ ~ ~ D and a > ~}, respectively.
ProoL It suffices to remember that Eq. (1) means h~, (x) = sup (c~ ^ 96~.(x))
o~EI
Vx e X.
[]
Next we give brief explanations of a class of fuzzy numbers [7, 8], the superior and inferior limits and the limit of a sequence of fuzzy numbers [5, 6]. Let/Sk(R) be a class of fuzzy numbers satisfying the following three conditions: (i) ~ is strictly normal, i.e., =Ix el~ such that ha(x) = 1, (ii) fi is convex [4], (iii) t~, _% ~ (ti)a for ot ~ (0, 1], where t~ is a fuzzy number, aa =asup(t~)a and _% &inf(5)a. Let {6~,n~>l} be a sequence of fuzzy numbers such that ti,~/Sk(R), and lim,__~ 6., and lirnn_~ ~ be the superior and inferior limits of the sequence, respectively. From Miyakoshi and Shimbo [6] (Corollary 1 and Eq. (9)), we have lim ~ =
rt~
El
a
a- s~"~, 1 ~
rt~
a~ "~
,
(4)
lim ~i, = I,
a (Ilium a(n) -a , -lim - g~"']) ,
rt~'~ El rt~ a(n) =tL in~Zwhere____~,~ r~a,)~ and ~")Asup(6~)~.
If lim,__~ 6~ = li___m,__~6~, let the sequence be called convergent. The limit is denoted by lim,__~ 6~ and defined as lim,__~ 6~ Al-~m,__~ 6~ ( = lirn,__~ 6~)[6]. Denoting the arithmetic mean of the first n terms of the sequence by (l/n) ~ = x ak, we have
l~=xak:Io~io£(lk~=l(ak)~)(:I,5~ D (~(1 k~l (ak)g))
(5)
[6]. The second equality of Eq. (5) follows from Eq. (2). 3. An individual ergodic
theorem
for fuzzy random
variables
3.1. Definitions of fuzzy random variables The definitions of an frv and of the expectation of an frv are given as follows [3]. Let (/2, ~/, P) be a probability space and O = {co}.
Individual ergodic theorem
287
Definition 1. A fuzzy random variable ~ is defined as a function from 12 to/5 k(R) satisfying the following two conditions: (i) Xa(to), X~-(o)~(to)a for all a~(O, 1], (ii) ~-, ~ are ag-measurable 1, where J~a (to) =asup ~(to)a and _Xa-(to) =ainf ~(to)aDelinition 2. The expectation of a fuzzy random variable ~, denoted by g~, is defined as a convex fuzzy number [4, 7, 8] such that "=
I~, c~([,_X~.,X~])
( = Is~D ~([,_X,. ,)2,])).
(6)
where the symbol [ , ] denotes a closed interval. 3.2. Individual ergodic theorem Theorem (Birkhoff's individual ergodic theorem [1, 2]). If T is a measurepreseroing transformation on the probability space (1-2,sg, P) and if X ~ LI(O), then: ,-1 X ( Tkto ) converges almost surely (a.s.), (B1) (1/n)~k=0 (B2) the limit function X* is contained in LI(O) and is invariant, i.e., X*(T"to) = X*(to)
a.s.
(n = O, +1, ±2 . . . . ),
(B3) ~X* = ~gX. Moreover, if T is an ergodic measure-preseroing transformation, (B4) X*(to) = ~gX a.s. Note that the following auxiliary results may also be found in [2]: (B5) The properties corresponding to (B1)-(B4) are valid for another unilateral form of the sum (1/n)Y.~,=lX(T-k~o) and a bilateral form of the sum (1/2n) Y.~-~_,X(Tkto). (B6) X*(co) = X.(to) a.s., where X . is the limit function of (l/n) Y.~=I X(T-kco) • 3.3. A n individual ergodic theorem for fuzzy random variables
Theorem. If
T is a measure-preseroing transformation on a probability space (O, ag, P), then: (1) for any fro ~ on (0, sg, P), n--1
1 ~, ~(Tkto ) converges
a.s., 2
nk=0
(2) the limit, denoted by ~*, is an fro and invariant, i.e., I~*(T%J) = ~*(oJ)
a.s.
(n = O, +1, +2 . . . . ),
a W e assume that _Xa,Xa e L l ( / ] ) for all a e (0, 1], that is, g I_Xal< oo and g symbol g X denotes the expectation of X with respect to P-measure.
IR~l
the
2The value ~*(~o) of oJ for which the limit does not converge is defined as ~*(~o)----.1/0, for convenience' sake.
288
M. Miyakoshi, M. Shimbo
(3) ~ * -- g~, (4) if T is an ergodic measure-weserving transformation, ~ * ( ~ ) = ~gli
a.s.
Proof. Let us consider a sequence of fuzzy numbers, ~(rk~o), n / > 1 . k=O
Proof of (1). We have to show that
-lira --~ '
~
_ - 1 .-1 y. ~:(T%a) = lim
n---.~ / l k = O
a.s.
n k=O
n~
According to [6] and [8], since 1 .-1
1 .-1
~(wko,)
_
1
==- Y~ ¢(Tk,,,)~ = 6
/I k=O
=
Y~ ¢(rk,,,)
n" k = 0
*a
'
we have from Eqs. (4) and (5) lim 1
~(Tk~o)=
a(F~(~o)),
r t - - , ~ 11 k = O
lim-
el
n~
a(&(,~)),
Y. {:(Tk~o)=
t"/ k = 0
el
where
j=0
and / A'* (~°' a=sup E / i n f
E/~
/ 1 k-I i=]ffo~ ( T ~ ) a ) ) •
Noting that 11
1 "-*
n--1
sup t ~ 7 o ~(Tk°J)a) = ~ k_~0~ - (Tk°J), n--1
inf
1
~(T%J)a =-k=O
_Xa(Tkoj),
?1 k = O
and n--1
_
1 ,~V ~ - ( T % J ) ,
_ ~ _Xa-(Tka~)e - Y. ~(TkaJ)a,
1 "-'
1 .-1
t/ k=O
/'/ k = O
I"/ k = O
we obtain from Eq. (4)
1 Ln~
!1 k = O
Xa _ (T%J), 1 ~ n~
1
-
y.
~-(T~o,)
n~
,
11 k = O
A~(oj) = [lim 1 " -Y~ ' X ~ ( T % ) , lira -l " - f ~ ( r % ) L n " ~ ' a I'1 k = O
]
n k=O
] .
(7)
Individual ergodic theorem
289
Since X~-, ~ e Ll(~), from Birkhoff's individual ergodic theorem, we have X~,X~e-* L I ( ~ ) satisfying (B1), (B2) and (B3). Hence there exist two measurable sets /q= and _N~ with null P-measure such that n--1
lim 1 ~ ~ _ ( T k t o ) = ~ ( t o ) n ~
Vta~l~r
/l k=0
and n--1
lim 1 ~ _Xa(Tkco)=X.=(cu) n ~
VoJ~_N=.
y/, k = 0
Putting N(°)-a U,E(o.ll (_N, U/~,), we have N ( ° ) e ~ , P(N(m) = 0 and for all ¢a¢N (°) and r e (0, 1]
1 .~1
lim -n~
X'~(Tk¢o)= X'*r(¢O) and
17, k=O
_.-1
lim 1 ~. ~X_(Tk~o) = X'~(¢o), n~
where r denotes a rational number in I. Thus, from Eqs. (7) and (8), we have for all r e ( 0 , 1] and ~ N F,(to)
= [X'~(to), ~ ( t o ) ]
and
(8)
11, k = O
(°)
a,(to) [_X~(oJ), X"~(tu)]. =
Consequently, we have for all toe N ~°)
I~ -1"~ I ~j(Tk~)= lira _1"~' ~(Tk(a ) = I n ~
n
k=0
n ~
/I k=0
r([_X'~(~), X'*v(~)])._
(9)
E1
This is exactly what we want to prove. Proof of (2). By Eq. (9), we can write
~*(,~)
=
[ r(E_XT,(,o),~*(o~)]).
(10)
ar
Hence, from the lemma in Section 2, having tj*(co)a = n [x~(¢o), x--~(¢o)] = [sup _x~F(¢o), inf X * ( ~ ) ] , ~>r
LO¢ ; > r
o¢>r
we obtain in turn the following: for all a e (0, 1] (a) sup ~*(tu)a, inf ~*(tu)a e ~*(ta)a, (b) sup ~*(to)a, inf ~*(to)a are ~ - m e a s u r a b l e and contained in LI(~), (c) ~*(tu) is convex and strictly normal. From the invariance of ~ and ~ , we have X'~(T"to) = . ~ ( t o ) _X*(T",o) = _X*(~)
a.s. a.s.
(n = 0, +1, ±2 . . . . ).
Hence, we can construct a measurable set N ~1) with null P-measure such that for all toCN ~1~ and all r e ( 0 , 1] [X~('/~to), X"~F(T~to)] = [_X~F(to),,~(to)]
(n = 0, ±1, +2 . . . . ).
M. Miyakoshi, M. Shimbo
290
Consequently, we have for all toe N TM
Proof of (3): Follows immediately from (B3) and Eq. (6). Proof of (4): By (B4), we have X*F(to)=~_X~
a.s.
and
X~(to)=~X¢
a.s.
Hence, (4) follows immediately from Eqs. (10) and (6), and this completes the proof of the theorem. [] Finally we note that it is easy to obtain the following properties from (B5) and (B6), though the proofs are omitted here: (5) Similar properties corresponding to (1)-(4) of the theorem are valid for another unilateral form of the sum (l/n) ~ = 1 f(T-%J) and a bilateral form of the sum (1/2n) Y.~,21_,~5(Tk~o). (6) Denoting the limit of ( l / n ) ~ , = 1 ~(T-koJ) by ~5., we have ~*(oJ) = ~.(~o)
a.s.
References [1] P.R. Halmos, Lecture on ergodic theory (The Mathematical Society of Japan, 1956). [2] K. Itoh, Kakuritsuron (Probability Theory) (Iwanami, Tokyo, 1958); in Japanese. [3] H. Kwakernaak, Fuzzy random variables-I, II, Information Sciences 15 (1978) 1-29; 17 (1979) 253-278. 14] R. Lowen, Convex fuzzy sets, Fuzzy Sets and Systems 3 (1980) 291-310. [5] M. Miyakoshi and M. Shimbo, Some properties of finite and countable fuzzy random variables, Preprints of IFAC Symposium on Fuzzy Information, Knowledge Representation and Decision Analysis, Marseille, France (1983) 415-419. [6] M. Miyakoshi and M. Shimbo, A strong law of large numbers for fuzzy random variables, Fuzzy Sets and Systems 12 (1984) 133-142. 1-7] M. Mizumoto and K. Tanaka, Some properties of fuzzy numbers, in: M.M. Gupta et al., Eds., Advances in Fuzzy Set Theory and Applications (North-Holland, Amsterdam, 1979) 153-164. [8] H.T. Nguyen, A note on the extension principle for fuzzy sets, Journal of Mathematical Analysis and Applications 64 (1978) 369-380. [9] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Information Sciences 8 (1975) 199-249.
285
AN INDIVIDUAL ERGODIC THEOREM FOR FUZZY RANDOM VARIABLF~S* Masaaki M I Y A K O S H I and Masaru SHIMBO Division of Information Engineering, Graduate School of Engineering, Hokkaido University, Sapporo, Japan Received June 1983 Revised October 1983 We generalize Birkhott's celebrated individual ergodic theorem to fuzzy random variables. In particular, it is shown that (i) the limiting time average exists, (ii) it is invariant, and (iii) can be replaced by the phase average.
Keywords: Fuzzy random variables, Individual ergodic theorem.
1. Introduefion The concept of fuzzy random variables (frv's) was firstly introduced by Kwakernaak [3] where some fundamental properties can be found. Additional properties of frv's, e.g., a dominated convergence property, a strong law of large numbers for frv's, etc., have been shown by the present authors [5, 6]. The present paper is concerned with the investigation of a strong limit theorem usually referred to as the ergodic theorem in the presence of fuzziness. For the concepts and notations used in this paper the reader is referred to [5, 6, 9].
2. Preliminmies Let X = {x} be a nonempty set and ,A be a fuzzy subset of X. We denote the a-level sets by ~ - , the unit interval [0, 1] by I, and any dense subset in I by D = {8}, respectively. We have the following lemma. Lemma. If {B~, ot ~ I) is a non-increasing class o[ subsets of X such that
A = fa~
EI
~B~,
(1)
* This work has been partially supported by a Grant in Aid for Scientific Research from the Ministry of Education, Japan. 0165-0114/84/$3.00 t~) 1984, Elsevier Science Publishers B.V. (North-Holland)
M. Miyakoshi, M. Shimbo
286 then
= I8
~D
~B8
(2)
and
where the notations a > ~/ and a > 8 of Eq. (3) denote {~/ I 7 ~ I and a > 7} and {8 [ ~ ~ D and a > ~}, respectively.
ProoL It suffices to remember that Eq. (1) means h~, (x) = sup (c~ ^ 96~.(x))
o~EI
Vx e X.
[]
Next we give brief explanations of a class of fuzzy numbers [7, 8], the superior and inferior limits and the limit of a sequence of fuzzy numbers [5, 6]. Let/Sk(R) be a class of fuzzy numbers satisfying the following three conditions: (i) ~ is strictly normal, i.e., =Ix el~ such that ha(x) = 1, (ii) fi is convex [4], (iii) t~, _% ~ (ti)a for ot ~ (0, 1], where t~ is a fuzzy number, aa =asup(t~)a and _% &inf(5)a. Let {6~,n~>l} be a sequence of fuzzy numbers such that ti,~/Sk(R), and lim,__~ 6., and lirnn_~ ~ be the superior and inferior limits of the sequence, respectively. From Miyakoshi and Shimbo [6] (Corollary 1 and Eq. (9)), we have lim ~ =
rt~
El
a
a- s~"~, 1 ~
rt~
a~ "~
,
(4)
lim ~i, = I,
a (Ilium a(n) -a , -lim - g~"']) ,
rt~'~ El rt~ a(n) =tL in~Zwhere____~,~ r~a,)~ and ~")Asup(6~)~.
If lim,__~ 6~ = li___m,__~6~, let the sequence be called convergent. The limit is denoted by lim,__~ 6~ and defined as lim,__~ 6~ Al-~m,__~ 6~ ( = lirn,__~ 6~)[6]. Denoting the arithmetic mean of the first n terms of the sequence by (l/n) ~ = x ak, we have
l~=xak:Io~io£(lk~=l(ak)~)(:I,5~ D (~(1 k~l (ak)g))
(5)
[6]. The second equality of Eq. (5) follows from Eq. (2). 3. An individual ergodic
theorem
for fuzzy random
variables
3.1. Definitions of fuzzy random variables The definitions of an frv and of the expectation of an frv are given as follows [3]. Let (/2, ~/, P) be a probability space and O = {co}.
Individual ergodic theorem
287
Definition 1. A fuzzy random variable ~ is defined as a function from 12 to/5 k(R) satisfying the following two conditions: (i) Xa(to), X~-(o)~(to)a for all a~(O, 1], (ii) ~-, ~ are ag-measurable 1, where J~a (to) =asup ~(to)a and _Xa-(to) =ainf ~(to)aDelinition 2. The expectation of a fuzzy random variable ~, denoted by g~, is defined as a convex fuzzy number [4, 7, 8] such that "=
I~, c~([,_X~.,X~])
( = Is~D ~([,_X,. ,)2,])).
(6)
where the symbol [ , ] denotes a closed interval. 3.2. Individual ergodic theorem Theorem (Birkhoff's individual ergodic theorem [1, 2]). If T is a measurepreseroing transformation on the probability space (1-2,sg, P) and if X ~ LI(O), then: ,-1 X ( Tkto ) converges almost surely (a.s.), (B1) (1/n)~k=0 (B2) the limit function X* is contained in LI(O) and is invariant, i.e., X*(T"to) = X*(to)
a.s.
(n = O, +1, ±2 . . . . ),
(B3) ~X* = ~gX. Moreover, if T is an ergodic measure-preseroing transformation, (B4) X*(to) = ~gX a.s. Note that the following auxiliary results may also be found in [2]: (B5) The properties corresponding to (B1)-(B4) are valid for another unilateral form of the sum (1/n)Y.~,=lX(T-k~o) and a bilateral form of the sum (1/2n) Y.~-~_,X(Tkto). (B6) X*(co) = X.(to) a.s., where X . is the limit function of (l/n) Y.~=I X(T-kco) • 3.3. A n individual ergodic theorem for fuzzy random variables
Theorem. If
T is a measure-preseroing transformation on a probability space (O, ag, P), then: (1) for any fro ~ on (0, sg, P), n--1
1 ~, ~(Tkto ) converges
a.s., 2
nk=0
(2) the limit, denoted by ~*, is an fro and invariant, i.e., I~*(T%J) = ~*(oJ)
a.s.
(n = O, +1, +2 . . . . ),
a W e assume that _Xa,Xa e L l ( / ] ) for all a e (0, 1], that is, g I_Xal< oo and g symbol g X denotes the expectation of X with respect to P-measure.
IR~l
the
2The value ~*(~o) of oJ for which the limit does not converge is defined as ~*(~o)----.1/0, for convenience' sake.
288
M. Miyakoshi, M. Shimbo
(3) ~ * -- g~, (4) if T is an ergodic measure-weserving transformation, ~ * ( ~ ) = ~gli
a.s.
Proof. Let us consider a sequence of fuzzy numbers, ~(rk~o), n / > 1 . k=O
Proof of (1). We have to show that
-lira --~ '
~
_ - 1 .-1 y. ~:(T%a) = lim
n---.~ / l k = O
a.s.
n k=O
n~
According to [6] and [8], since 1 .-1
1 .-1
~(wko,)
_
1
==- Y~ ¢(Tk,,,)~ = 6
/I k=O
=
Y~ ¢(rk,,,)
n" k = 0
*a
'
we have from Eqs. (4) and (5) lim 1
~(Tk~o)=
a(F~(~o)),
r t - - , ~ 11 k = O
lim-
el
n~
a(&(,~)),
Y. {:(Tk~o)=
t"/ k = 0
el
where
j=0
and / A'* (~°' a=sup E / i n f
E/~
/ 1 k-I i=]ffo~ ( T ~ ) a ) ) •
Noting that 11
1 "-*
n--1
sup t ~ 7 o ~(Tk°J)a) = ~ k_~0~ - (Tk°J), n--1
inf
1
~(T%J)a =-k=O
_Xa(Tkoj),
?1 k = O
and n--1
_
1 ,~V ~ - ( T % J ) ,
_ ~ _Xa-(Tka~)e - Y. ~(TkaJ)a,
1 "-'
1 .-1
t/ k=O
/'/ k = O
I"/ k = O
we obtain from Eq. (4)
1 Ln~
!1 k = O
Xa _ (T%J), 1 ~ n~
1
-
y.
~-(T~o,)
n~
,
11 k = O
A~(oj) = [lim 1 " -Y~ ' X ~ ( T % ) , lira -l " - f ~ ( r % ) L n " ~ ' a I'1 k = O
]
n k=O
] .
(7)
Individual ergodic theorem
289
Since X~-, ~ e Ll(~), from Birkhoff's individual ergodic theorem, we have X~,X~e-* L I ( ~ ) satisfying (B1), (B2) and (B3). Hence there exist two measurable sets /q= and _N~ with null P-measure such that n--1
lim 1 ~ ~ _ ( T k t o ) = ~ ( t o ) n ~
Vta~l~r
/l k=0
and n--1
lim 1 ~ _Xa(Tkco)=X.=(cu) n ~
VoJ~_N=.
y/, k = 0
Putting N(°)-a U,E(o.ll (_N, U/~,), we have N ( ° ) e ~ , P(N(m) = 0 and for all ¢a¢N (°) and r e (0, 1]
1 .~1
lim -n~
X'~(Tk¢o)= X'*r(¢O) and
17, k=O
_.-1
lim 1 ~. ~X_(Tk~o) = X'~(¢o), n~
where r denotes a rational number in I. Thus, from Eqs. (7) and (8), we have for all r e ( 0 , 1] and ~ N F,(to)
= [X'~(to), ~ ( t o ) ]
and
(8)
11, k = O
(°)
a,(to) [_X~(oJ), X"~(tu)]. =
Consequently, we have for all toe N ~°)
I~ -1"~ I ~j(Tk~)= lira _1"~' ~(Tk(a ) = I n ~
n
k=0
n ~
/I k=0
r([_X'~(~), X'*v(~)])._
(9)
E1
This is exactly what we want to prove. Proof of (2). By Eq. (9), we can write
~*(,~)
=
[ r(E_XT,(,o),~*(o~)]).
(10)
ar
Hence, from the lemma in Section 2, having tj*(co)a = n [x~(¢o), x--~(¢o)] = [sup _x~F(¢o), inf X * ( ~ ) ] , ~>r
LO¢ ; > r
o¢>r
we obtain in turn the following: for all a e (0, 1] (a) sup ~*(tu)a, inf ~*(tu)a e ~*(ta)a, (b) sup ~*(to)a, inf ~*(to)a are ~ - m e a s u r a b l e and contained in LI(~), (c) ~*(tu) is convex and strictly normal. From the invariance of ~ and ~ , we have X'~(T"to) = . ~ ( t o ) _X*(T",o) = _X*(~)
a.s. a.s.
(n = 0, +1, ±2 . . . . ).
Hence, we can construct a measurable set N ~1) with null P-measure such that for all toCN ~1~ and all r e ( 0 , 1] [X~('/~to), X"~F(T~to)] = [_X~F(to),,~(to)]
(n = 0, ±1, +2 . . . . ).
M. Miyakoshi, M. Shimbo
290
Consequently, we have for all toe N TM
Proof of (3): Follows immediately from (B3) and Eq. (6). Proof of (4): By (B4), we have X*F(to)=~_X~
a.s.
and
X~(to)=~X¢
a.s.
Hence, (4) follows immediately from Eqs. (10) and (6), and this completes the proof of the theorem. [] Finally we note that it is easy to obtain the following properties from (B5) and (B6), though the proofs are omitted here: (5) Similar properties corresponding to (1)-(4) of the theorem are valid for another unilateral form of the sum (l/n) ~ = 1 f(T-%J) and a bilateral form of the sum (1/2n) Y.~,21_,~5(Tk~o). (6) Denoting the limit of ( l / n ) ~ , = 1 ~(T-koJ) by ~5., we have ~*(oJ) = ~.(~o)
a.s.
References [1] P.R. Halmos, Lecture on ergodic theory (The Mathematical Society of Japan, 1956). [2] K. Itoh, Kakuritsuron (Probability Theory) (Iwanami, Tokyo, 1958); in Japanese. [3] H. Kwakernaak, Fuzzy random variables-I, II, Information Sciences 15 (1978) 1-29; 17 (1979) 253-278. 14] R. Lowen, Convex fuzzy sets, Fuzzy Sets and Systems 3 (1980) 291-310. [5] M. Miyakoshi and M. Shimbo, Some properties of finite and countable fuzzy random variables, Preprints of IFAC Symposium on Fuzzy Information, Knowledge Representation and Decision Analysis, Marseille, France (1983) 415-419. [6] M. Miyakoshi and M. Shimbo, A strong law of large numbers for fuzzy random variables, Fuzzy Sets and Systems 12 (1984) 133-142. 1-7] M. Mizumoto and K. Tanaka, Some properties of fuzzy numbers, in: M.M. Gupta et al., Eds., Advances in Fuzzy Set Theory and Applications (North-Holland, Amsterdam, 1979) 153-164. [8] H.T. Nguyen, A note on the extension principle for fuzzy sets, Journal of Mathematical Analysis and Applications 64 (1978) 369-380. [9] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Information Sciences 8 (1975) 199-249.