OCTOGON MATHEMATICAL MAGAZINE
Vol. 17, No.1, April 2009, pp 90-105
ISSN 1222-5657, ISBN 978-973-88255-5-0,
www.hetfalu.ro/octogon
90
Helly‘s theorem on fuzzy valued functions
J.Earnest Lazarus Piriyakumar and R. Helen6
ABSTRACT. Some recently developed notions of fuzzy valued functions, fuzzy
distribution functions proposed by H.C.Wu [9] are employed to establish Hellys
theorem on fuzzy valued functions. To establish Hellys theorem, and for a
convenient discussion of fuzzy random variables a more strong sense of measurability
for fuzzy valued functions is introduced.
1. INTRODUCTION
The notion of fuzzy random variables, with necessary theoretical framework
was introduced by Kwakernaak [5] and Puri and Ralescu [8]. The notion of
normality of fuzzy random variables was also discussed by M.L.Puri et.al [7].
To make fuzzy random variables amenable to statistical analysis for
imprecise data where dimness of perception is prevalent, H.C.Wu[9-12] has
contributed a variety of research papers on fuzzy random variables which
expose various rudiments of fuzzy random variables such as, weak and strong
law of large numbers, weak and strong convergence with probability, fuzzy
distribution functions, fuzzy probability density functions, fuzzy expectation,
fuzzy variance and fuzzy valued functions governed by strong measurability
conditions. In [9] H.C.Wu has introduced the notion of fuzzy distribution
functions for fuzzy random variables.
Since the α-level set of a closed fuzzy number is a compact interval, in order
to make the end points of the α-level set of a fuzzy random variables to be
the usual random variables H.C.Wu [9-12] has introduced the concept of
strong measurability for fuzzy random variables. In this paper, Hellys
theorem, and Helly Bray theorem for fuzzy valued functions and fuzzy
probability distribution function are introduced. These results are based on
the concept of strong measurability for fuzzy random variables.
In section 2, we introduce some preliminaries related to fuzzy numbers, such
as strong and weak convergence of sequence of fuzzy numbers to a fuzzy
6
Received: 07.02.2009
2000 Mathematics Subject Classification. 03E05
Key words and phrases. Canonical fuzzy number, Fuzzy real number, Fuzzy random variables, Fuzzy probability distribution functions and Fuzzy valued functions.
Helly‘s theorem on fuzzy valued functions
91
numbers. Section 3 is devoted to fuzzy random variables and its fuzzy
distribution functions. The theoretical settings of fuzzy random variables are
derived from H.C. Wu [9-12].
In section 3 we introduce the definition of a fuzzy valued function and other
notions related to the measurability of a fuzzy valued function. The notion
of fuzzy distribution function is also introduced in this section. This section
conclude with the definitions of strong and weak convergence in distribution
of fuzzy random variables. In section 4 we present the Hellys theorem and
Helly- Bray theorem for fuzzy valued functions and fuzzy distribution
functions. Throughout this paper we denote the indicator function of the set
A by 1A .
2. FUZZY NUMBERS
In this section we provide some limit properties of fuzzy numbers by
applying the Hausdorff metric. We also introduce the notions of fuzzy real
numbers and fuzzy random variables.
Definition 2.1.
(i) Let f be a real valued function on a topological space. If {x; f (x) ≥ α} is
closed for each α, then f is said to be upper semi continuous.
(ii) A real valued function f is said to be upper semi continuous at y if and
only if ∀ε > 0, ∃δ > 0 such that |x − y| < δ implies f (x) < f (y) + ε
(iii) f (x) is said to be lower semicontinuous if −f (x) is upper semicontinuous.
Definition 2.2. Let F : X → Rn be a set valued mapping. F is said to be
continuous at x0 ∈ X if F is both upper semi continuous and lower semi
continuous at x0 .
Theorem 2.1. (Bazarra and Shetty [2]) Let S be a compact set in Rn . If f
is upper semicontinuous on S then f assumes maximum over S and if f is
lower semi-continuous on S then f assumes minimum over S.
Definition 2.3. Let X be a universal set. Then a fuzzy subset A of X is
defined by its membership function µA : X → [0, 1]. We denote
Aα = {x; µ(x) ≥ α} as the α-level set of A where A0 is the closure of the set
{x; µA (x) 6= 0}.
Definition 2.4.
(i) A is called a normal fuzzy set if their exist x such that µA (x) = 1.
92
J.Earnest Lazarus Piriyakumar and R. Helen
(ii) A is called a convex fuzzy set if
µA (λx + (1 − λ)y) ≥ min{µA (x), µA (y)} for λ ∈ [0, 1].
Theorem 2.2. [13] A is a convex fuzzy set if and only if {x; µA (x) ≥ α} is a
convex set for all α.
Definition 2.5. Let X = R
(i) m
e is called a fuzzy number if m
e is a normal convex fuzzy set and the
α-level set m
e α is bounded ∀α 6= 0.
(ii) m
e is called a closed fuzzy number if m
e is a fuzzy number and its
membership function µm
is
upper
semicontinuous.
e
(iii) m
e is called a bounded fuzzy number if m
e is a fuzzy number and its
membership function µm
has
compact
support.
e
Theorem 2.3. [10] If m
e is a closed fuzzy number then the α-level set of m
e
is a closed interval, which is denoted by
L U
m
eα = m
e α, m
eα
Definition 2.6. m
e is called a canonical fuzzy number, if it is a closed and
bounded fuzzy
number
on
increasing
L L and its membership function is strictly
U
U
e1 ,m
e0 .
the interval m
e0,m
e 1 and strictly decreasing on the interval m
Theorem 2.4. [10] Suppose that e
a is a canonical fuzzy number. Let
U . Then g(α) and h(α) are continuous functions of α.
g(α) = e
aL
and
h(α)
=
e
a
α
α
Theorem 2.5. (Zadeh [14] resolution identity)
(i) Let A be a fuzzy set with membership function µA and
Aα = {x; µA (x) ≥ α}. Then
µA (x) = sup α1Aα (x)
α∈[0,1]
(ii) (Negoita and Relescu [6]) Let A be a set and {Aα : 0 ≤ α ≤ 1} be a
family of subsets of A such that the following conditions are satisfied.
a. A0 = A
b. Aα ⊆ Aβ for α > β
T
c. Aα = ∞
n=1 Aαn for αn ↑ α
Then the function µ : A → [0, 1] defined by µ(x) = sup α1Aα (x) has the
α∈[0,1]
property that
Helly‘s theorem on fuzzy valued functions
93
Aα = {x; µ (x) ≥ α} ∀α ∈ [0, 1]
With the help of α-level sets of a fuzzy set A we can construct closed fuzzy
number. Let g and h be two functions from [0, 1] into R.
Let {Aα = [g(α), h(α)] : 0 ≤ α ≤ 1} be a family of closed intervals. Then we
can induce a fuzzy set A with the membership function
µA (r) = sup α1Aα (r)
0≤α≤1
via the form of resolution identity.
Theorem 2.6. [11] Let {Aα : 0 ≤ α ≤ 1} be a set of decreasing closed
intervals. i.e., Aα ⊆ Aβ for α > β. Then f (α) = α1Aα is upper
semicontinuous for any fixed r.
Theorem 2.7. [11] Let e
a and eb be two canonical fuzzy numbers. Then
e
e
e
e
a ⊕ b, e
a ⊖ b and e
a ⊗ b are also canonical fuzzy numbers. Further more we
have for
i
α ∈h[0, 1]
eL aU + ebU
e
aL
e
a⊕b = e
α + bα , e
α
α
i
α h
U
U −e
ebL , e
b
a
−
aL
e
a ⊖ eb = e
α
α α
α
oi
o
n
n
α h
U
U eL aU e
U
U eL aU e
eL aLebU , e
eL aLebU , e
aL
aL
e
a ⊗ eb = min e
α bα
α α aα bα , e
α bα , e
α bα ; max e
α α aα bα , e
α bα , e
α
Let A ⊆ Rn and B ⊆ Rn . The Hausdorff metric is defined by
dH (A, B) = max sup inf ka − bk , sup inf ka − bk
a∈A b∈B
b∈B a∈A
Let ℑ be the set of all fuzzy numbers and ℑb be the set of all bounded fuzzy
numbers. Puri and Ralescu [8] have defined the metric dℑ in ℑ as
dℑ e
a, eb = sup dH e
aα , ebα
0<α≤1
e
and dℑb in ℑb e
a, eb = sup dH af
α , bα .
0<α≤1
Lemma 2.1. [9] Let e
a and eb be two closed fuzzy numbers.
Then we have
94
J.Earnest Lazarus Piriyakumar and R. Helen
o
n
L eL U eU aα − bα , e
aα − bα .
aα , ebα = max e
dH e
Definition 2.7. [9]
(i) Let {e
an } be a sequence of closed (canonical) fuzzy numbers. {e
an } is said
to converge strongly if there is a closed (canonical) fuzzy number e
a with the
following property. ∀ε > 0, ∃N > 0 such that for n > N , we have
dℑ (e
an , e
a) < a (dℑb (e
an , e
a) < ε). We say that the sequence {e
an } converges to
lim e
an =
e
an strongly and it is denoted as n→∞
.
se
a
(ii) Let {e
an } be a sequence of closed (canonical) fuzzy numbers. {e
an } is said
to converge weakly if there is a closed (canonical) fuzzy number with e
an the
following property:
lim (e
an )L
aL
an )U
aU
α and lim (e
α for all α ∈ [0, 1]
α =e
α =e
n→∞
n→∞
we say that the sequence {e
an } converges to e
an weakly and it is denoted as
lim e
an =
n→∞
we
a
. We note that
lim e
an =
n→∞
se
a
is equivalent to lim dℑb (e
an , e
a) = 0.
n→∞
3. FUZZY RANDOM VARIABLE AND ITS DISTRIBUTION FUNCTION
In this section we provide the theoretical framework of fuzzy random
variables and its distribution functions proposed by H.C. Wu [9]. Given a
real number x one can induce a fuzzy number x
e with membership function
µxe (r) such that µxe (x) = 1 and µxe (r) < 1 for r 6= x. We call x
e as a fuzzy
real number induced by the real number x. Let ℑR be a set of all fuzzy real
numbers induced by the real number system R. We define the relation ˜ on
ℑR as x
e1 ∼ x
e2 if and only if x
e1 and x
e2 are induced by the same real number
x. Then ˜ is an equivalence
relation.
This equivalence relation induces the
e
a|e
a∼
equivalence classes [e
x] =
. The quotient set ℑR/N is the set of all
x
equivalence classes. Then the cardinality of ℑR/N is equal to the cardinality
of the real number system R, since the map R → ℑR/N specified by x → [e
x]
is a bijection. We call ℑ
R/N as the fuzzy real number system. For practical purposes we take only
one element x
e from each equivalence class [e
x] to form the fuzzy real number
system ℑR/N R .
i.e. ℑR/N R = {e
x|e
x ∈ [e
x] , x
e is the only element from [e
x]} .
If the fuzzy real number system ℑR/N R consists of canonical fuzzy real
numbers, then we call ℑR/N R as the canonical fuzzy real number system.
Helly‘s theorem on fuzzy valued functions
95
Let (X, M) be a measurable space and (R, B) be a Borel measurable space.
Let f : X → P(R) (power set of R) be a set valued function. According to
Aumann [1] the set valued function f is called measurable if and only if
{(x, y) ; y ∈ f (x)} is calM × B measurable. The function fe(x) is called a
fuzzy valued function if fe : x → ℑ (the set of all fuzzy numbers).
Definition 3.1. [9] Let ℑR/N R be a canonical fuzzy real number system
e : Ω → ℑR/N
e is called a fuzzy
and X
be a closed-fuzzy valued function. X
R
random variable if X is measurable (or equivalently strongly measurable).
Theorem 3.1. [9] Let ℑR/N R be a canonical fuzzy real number system
e : Ω → ℑR/N
e is a fuzzy
and X
be a closed-fuzzy valued function. X
R
eαL and X
eαU are random variables for all
random variable if and only if X
α ∈ [0, 1].
g
eU
If x
e is a canonical fuzzy real number, then x
eL
1 . Let XZ be a fuzzy
1 =x
L
U
e
e
random variable. By theorem 3.1 Xα and Xα are random variables for all α,
eL = X
e U . Let F (x) be a continuous distribution function of a random
and X
1
1
eαU have the same distribution function F (x) for all
eαL and X
variable X. Let X
e
α
e is any fuzzy observation of a fuzzy random variable X
∈ [0, 1]. Ifx
L U
e (w) = x
eα .
X
e then the α-level set x
eα is x
eα = x
eα , x
By a fuzzy observation we mean an imprecise data. We can see that x
eL
α and
U
L
U
e
e
x
eα are the observations of Xα and Xα respectively. From theorem 2.4
L
eU
eαL (w) = x
eU
X
α are continuous with respect to α for
fixed w.
L eαU and Xα (w) = x
L
U
shrinking
eα , x
eα is the
Thus x
eα , x
eα is continuously
with respect to α. x
L, x
L and x
U, x
U . Therefore for any real number
disjoint
union
of
x
e
e
e
e
α 1
α
1
L U
eL
eU
x∈ x
eα , x
eα we have x = x
β or x = x
β for some β ≥ α. This confirm the
existence of a suitable α-level set for which x coincides with the lower end of
that α-level set or with the upper end
set. Hence for any
ofthat α-level
L U
L
with x.
eU
eβ or F x
x∈ x
eα , x
eα , we can associate an F x
β
If fe is a fuzzy valued function then feα is a set valued function for all
α ∈ [0, 1]. fe is called (fuzzy-valued) measurable if and only if feα is
(set-valued) measurable for all α ∈ [0, 1].
To make fuzzy random variables more governable mathematically a more
strong sense of measurability for fuzzy valued function is required. fe(x) is
called a closed-fuzzy-valued function if fe : X → ℑcl (the set of all closed
fuzzy numbers). Let fe(x) be a closed fuzzyvalued function defined on X.
From H.C. Wu [11] the following two statements are equivalent.
96
J.Earnest Lazarus Piriyakumar and R. Helen
(i) feαL (x) and feαU (x) are (real- valued) measurable for all α ∈ [0, 1] .
(ii) fe(x) is (fuzzy-valued) measurable and one of feαL (x) and feαU (x) is (real
valued) measurable for all α ∈ [0, 1] .
Then fe(x) is called strongly measurable if one of the above two conditions is
satisfied. It is easy to see that the strong measurability implies measurability.
Let (X,M, µ ) be a measure space and (R, B) be a Borel measurable space.
Let f : X → P (R) be a set valued function. For K ⊆ R, the inverse image of
f is defined by
f −1 (k) = {x ∈ X, f (x) ∩ k 6= ∅}
Let (X,m, µ) be a complete σ-finite measure space. From Hiai and Umegaki
[3] the following two statements are equivalent.
a) For each Borel set k ⊆ R, f −1 (k) is measurable (ie f −1 (k) ∈ M)
b) {(x, y); y ∈ f (x)} is M × B measurable.
If we construct an interval
"
Aα = min
(
)#
inf F x
eL
eU
, max
sup F x
eL
eU
β , inf F x
β
β , sup F x
β
α≤β≤1
α≤β≤1
α≤β≤1
α≤β≤1
Then this interval
will contain all of the distributions associated with each of
L
L
x) the fuzzy distribution function of the fuzzy
e1 . We denote by Fe (e
x∈ x
eα , x
random variable x
e. Then we define the membership function of Fe (e
x) for any
fixed x
e by
(r)
F (e
x)
µe
= sup α1Aα (r)
α≤β≤1
we also say that the fuzzy distribution function Fe (e
x) is induced by the
distribution function F (x). Since F (x) is continuous from theorem 2.4 and
2.1, we can write Aα as
L
U
U
Aα = min min F x
eL
,
max
max
F
x
e
,
min
F
x
e
,
max
F
x
e
β
β
β
β
α≤β≤1
α≤β≤1
α≤β≤1
For typographical reasons we employ the following notations.
(•)
U
F min x
eβ = min min F x
,
min
F
x
e
eL
β
β
α≤β≤1
α≤β≤1
α≤β≤1
97
Helly‘s theorem on fuzzy valued functions
F
max
(•)
x
eβ
= max
max F
α≤β≤1
x
eL
β
, max F
α≤β≤1
x
eU
β
e and Ye be two fuzzy random variables. We say that X
e and Ye are
Let X
independent if and only if each random variable in the set
o
n
eαL , X
eαU ; 0 ≤ α ≤ 1 is independent of each random variable in the set
X
o
n
e and Ye are identically distributed if
YeαL , YeαU ; 0 ≤ α ≤ 1 . We say that X
eαL and YeαL are identically distributed and X
eαU and YeαU are
and only if X
identically distributed for all α ∈ [0, 1] .
e and {X
en } be fuzzy random variables defined on the
Definition 3.2. Let X
same probability space (Ω, A, P).
L
en
en } converges in distribution to X
e level-wise if X
and
(i) We say that {X
α
U
eαU respectively for all α ∈ (0, 1] .
eαL and X
en
converge in distribution to X
X
α
en and
Let Fen (e
x) and Fe (e
x) be the respective fuzzy distribution functions of X
e
X.
en } converge in distribution to X
e strongly if
(ii) We say that {X
lim
limn→∞ Fen (e
x) =
e
sF (e
x) i.e.
U
L
U
L
e
e
e
e
=0
x) − Fα (e
x)
x) − Fα (e
x) , Fn (e
sup max Fn (e
n→∞ α≤β≤1
α
α
en } converges in distribution to X
e weakly if
(iii) We say that {X
limn→∞ Fen (e
x) =
e
wF (e
x) i.e.
L
x) = FeαL (e
x) and
lim Fen (e
n→∞
lim
n→∞
Fen
U
α
α
(e
x) = FeαU (e
x) for all α ∈ [0, 1] .
98
J.Earnest Lazarus Piriyakumar and R. Helen
4. HELLYS THEOREMS
In this section based on the theoretical framework of section 3 and section 4
we have established the Hellys theorem and Helly Bray theorem for fuzzy
valued functions and fuzzy distribution functions.
Theorem 4. (Hellys Theorem) If (i) non-decreasing sequence of fuzzy
probability distribution function {Fn (x)} converges to the fuzzy probability
distribution function F (x) (ii) the fuzzy valued function g(x) is everywhere
continuous and (iii) a, b are continuity points of F (x) then
lim
n→∞
=
Zb
a
(•)
(•)
gαL (x) dFnmin xβ ∧ gαU (x) dFnmin xβ =
Zb
(•)
(•)
gαL (x) dF min xβ ∧ gαU (x) dF max xβ
a
(3.1)
Proof. By stipulation Fn xL
and
F
is non-decreasing.
xU
n
β
β
U
is also decreasing.
∴ F xL
β and F xβ
L
For all n ≥ 1, Fn xL
β + h − Fn xβ ≥ 0; if h > 0.
U
Fn xU
β + h − Fn xβ ≥ 0; if h > 0.
Letting n → ∞ we have
L
F xL
β + h − F xβ ≥ 0; if h > 0
U
F xU
β + h − F xβ ≥ 0; if h > 0
we take a = x0 < x1 < x2 < ..... < xk = b where x0 , x1 , ..... are the continuity
points of the fuzzy probability distribution function F . then for α ≤ β ≤ 1
Zb
a
(•)
(•)
gαL (x) dF min xβ ∧ gαU (x) dF max xβ =
x
k−1 Zi+1
X
(•)
(•)
L
min
U
max
=
gα (x) dF
xβ ∧ gα (x) dF
xβ =
i=0 x
i
Helly‘s theorem on fuzzy valued functions
=
k−1
X
i=0
∧
k−1
X
i=0
99
 xi+1

x
Z
Zi+1
(•)
(•)

gαL (x) − gαL (xi ) dF min xβ +
gαL (xi ) dF min xβ  ∧
xi
xi

 xi+1
x
Z
Zi+1
(•)
(•)

gαU (x) − gαU (xi ) dF max xβ +
gαU (xi ) dF max xβ  =
xi
xi
=
x
k−1 Zi+1
X
(•)
gαL (x) − gαL (xi ) dF min xβ +
i=0 x
i
k−1
X
+
gαL (xi )
i=0
x
k−1 Zi+1
X
∧
i=0 x
i
+
=
x
k−1 Zi+1
X
i=0 x
i
+
k−1
X
i=0
∧
xi
gαU
(xi )
x
Zi+1
xi
(•)
dF max xβ =
(•)
gαL (x) − gαL (xi ) dF min xβ +
gαL (xi ) F min (xi+1 ) − F min (xi ) ∧
x
k−1 Zi+1
X
i=0 x
i
+
(•)
dF min xβ ∧
(•)
gαU (x) − gαU (xi ) dF max xβ +
k−1
X
i=0
x
Zi+1
k−1
X
i=0
(•)
gαU (x) − gαU (xi ) dF max xβ +
gαU (xi ) (F max (xi+1 ) − F max (xi ))
By stipulation the fuzzy valued function g is continuous every where. i.e for
each α ∈ [0, 1] .
100
J.Earnest Lazarus Piriyakumar and R. Helen
L
gα (x) − gαL (xi ) <
ε
3
F (b) − F (a)
for xi ≤ x ≤ xi+1 .
We take |θ1 | ≤ 1. Then
Zb
a
θ ε
(•)
(•)
1
+
gαL (x) dF min xβ ∧ gαU (x) dF max xβ ≤
3
+
k−1
X
gαL (xi ) F min (xi+1 ) − F min (xi ) ∧
i=0
∧
k−1
X
i=0
gαU (xi ) (F max (xi+1 ) − F max (xi ))
(3.2)
Similarly
Zb
gαL (x) dFnmin
a
(•)
xβ
∧
gαU
(x) dFnmax
(•)
xβ
≤
k−1
θ2 ε X L
≤
gα (xi ) Fnmin (xi+1 ) − Fnmin (xi ) ∧
+
3
i=0
∧
k−1
X
i=0
gαU (xi ) (Fnmax (xi+1 ) − Fnmax (xi ))
(•)
(•)
Since Fnmin xβ → F min xβ
at continuity points of F and
(•)
(•)
Fnmax xβ → F max xβ at continuity points of F
Fnmin (xi+1 ) − F min (xi+1 ) <
6
and large values of n. Similarly
Fnmax (xi+1 ) − F max (xi+1 ) <
6
P
P
ε
for all i
gαL (xi )
ε
for all i
gαU (xi )
and large values of n.
Hence letting n → ∞ the absolute difference of (3.2) and (3.3) is
(3.3)
101
Helly‘s theorem on fuzzy valued functions
Zb
a
(•)
(•) gαL (x) dFnmin xβ − dF min xβ ∧
(•)
(•) ∧gαU (x) dF max xβ − dF max xβ ≤
≤
k−1
X
ε
gαL (xi ) Fnmin (xi+1 ) − Fnmin (xi ) − F min (xi+1 ) − F min (xi ) ∧
|θ2 − θ1 |+
3
i=0
∧
Then
k−1
X
i=0
gαU (xi ) |Fnmax (xi+1 ) − Fnmax (xi ) − F max (xi+1 ) − F max (xi )|
Zb
(•)
(•) gαL (x) dFnmin xβ − dF min xβ ∧
Zb
a
(•)
(•)
gαL (x) dFnmin xβ ∧ gαU (x) dFnmin xβ =
Zb
(•)
(•)
gαL (x) dF min xβ ∧ gαU (x) dF max xβ
a
(•)
(•) ∧gαU (x) dF max xβ − dF max xβ <∈
This show that
lim
n→∞
=
a
which completes the proof.
Theorem 4.2. (Helly Bray Theorem) If (i) the fuzzy valued function g(x) is
continuous
(ii) the fuzzy probability distribution function Fn (x) → F (x) in each
continuity point of F (x) and
(iii) for any ε > 0 we can find A such that
ZA
−∞
L
gα (x) dFnmin x(•) ∧ gαU (x) dFnmax x(•) +
β
β
102
J.Earnest Lazarus Piriyakumar and R. Helen
+
Z∞
A
L
gα (x) dFnmin x(•) ∧ gαU (x) dFnmax x(•) <∈
β
β
(3.4)
for all n = 1, 2, 3, ......, then
lim
Z∞
(•)
(•)
gαL (x) dFnmin xβ ∧ gαU (x) dFnmax xβ =
=
Z∞
(•)
(•)
gαL (x) dF min xβ ∧ gαU (x) dF max xβ
n→∞
−∞
−∞
Proof. The theorem is proved for Fn (x) → F (x) for all x.
Letting n → ∞ and from condition (3.4) we have
ZA
−∞
+
Z∞
A
L
gα (x) dF min x(•) ∧ gαU (x) dF max x(•) +
β
β
L
gα (x) dF min x(•) ∧ gαU (x) dF max x(•) <∈
β
β
(3.5)
By theorem (3.1)
lim
n→∞
=
Zb
a
(•)
(•)
gαL (x) dFnmin xβ ∧ gαU (x) dFnmax xβ =
Zb
(•)
(•)
gαL (x) dF min xβ ∧ gαU (x) dF max xβ
a
(3.6)
By the expression (3.4) it follows that for continuity points b > a, the fuzzy
valued function g(x) is integrable over (−∞, ∞), with respect to the fuzzy
probability distribution function F (x). If in (3.6) w e take a = −A and b = A
we have
lim
ZA
n→∞
−A
(•)
(•)
gαL (x) dFnmin xβ ∧ gαU (x) dFnmax xβ =
103
Helly‘s theorem on fuzzy valued functions
ZA
=
−A
Z∞
−∞

=
−
Z∞
+
ZA
−∞

−
−A
Z−A
+
−∞
Then
(3.7)
(•)
(•)
gαL (x) dFnmin xβ ∧ gαU (x) dFnmax xβ −
−∞
Z−A
(•)
(•)
gαL (x) dF min xβ ∧ gαU (x) dF max xβ
(•)
(•)
gαL (x) dF min xβ ∧ gαU (x) dF max xβ =

Z∞ (•)
(•)
+  gαL (x) dFnmin x
∧ gαU (x) dFnmax x
−
ZA
−A
β
β
A

Z∞ (•)
(•)
+  gαL (x) dF min x
∧ gαU (x) dF max x
β
β
A
∞
Z (•)
(•)
gαL (x) dFnmin xβ ∧ gαU (x) dFnmax xβ
−
lim n→∞ −∞
Z∞ (•)
(•)
≤
−
gαL (x) dF min xβ ∧ gαU (x) dF max xβ
−∞

≤ lim 
n→∞
Z−A −∞
+
Z∞ A
L
gα (x) dFnmin x(•) ∧ gαU (x) dFnmax x(•) +
β
β

min (•) U
L
max (•) gα (x) dFn
+
x
x
∧ gα (x) dFn
β
β
A
Z (•)
(•)
+ lim gαL (x) dFnmin xβ − gαL (x) dF min xβ
∧
n→∞ −A
(•)
(•)
∧ gαU (x) dFnmax xβ − gαU (x) dF max xβ
< ε + ε + ε = 3ε
104
J.Earnest Lazarus Piriyakumar and R. Helen
if n and A are both large. Thus we have proved the theorem if
(•)
(•)
(•)
(•)
(•)
Fnmin xβ → F min xβ
and Fnmax xβ → F max xβ
at all xβ
∴ lim
Z∞
n→∞
−∞
=
Z∞
−∞
(•)
(•)
gαL (x) dFnmin xβ ∧ gαU (x) dFnmax xβ =
(•)
(•)
gαL (x) dF min xβ ∧ gαU (x) dF max xβ
Now we can choose only continuity points of F (x) viz., x1 , x2 , ... Between
two discontinuity points one can choose a continuity point and for monotonic
Fn , the points of discontinuities are at most countable. Hence the result is
true if Fnmin (x) → F min (x) and Fnmax (x) → F max (x) at all continuity points
of F , and the proof is complete.
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Department of Mathematics,
Tranquebar Bishop Manickam Lutheran College,
Porayar 609 307, S. India.
Department of Mathematics,
Bharathiyar College of Engineering and Technology,
Karaikal 609 609, S. India
E-mail: [email protected]