![International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 4 Issue 8 || October. 2016 || PP-15-19
www.ijmsi.org 15 | Page
A Note on Pseudo Operations Decomposable Measure
Dr. N. Sarala1
, S. Jothi2
1
Associate Professor, A.D.M. College for women, Nagapattinam, Tamilnadi, India.
2
Guest Lecturer, Thiru.Vi. Ka. Govt. Arts College, Thiruvarur Tamilnadi, India.
ABSTRACT: In this paper we discussed about the pseudo integral for a measurable function based on a strict
pseudo addition and pseudo multiplication. Further more we got several important properties of the pseudo
integral of a measurable function based on a strict pseudo addition decomposable measure.
KEYWORDS: Pseudo addition, Pseudo multiplication, decomposable measure, Pseudo Inverse
I. INTRODUCTION
Taking into account human subjective measures in engineering science fuzzy measures have been
intensively discussed since Sugeno(1) defined a fuzzy measure as a measure having the monotonicity property
instead of additivity. Weber(2) proposed ⊥-decomposable measures where the additivity of measures is
weakened. t-Conorm ⊥ is an appropriate semigroup operation in [O, l]. ⊥-decomposable measures can be
written as
m(A∪B) = m(A) ⊥ m(B).
for the Archimedean case ⊥ is written as a ⊥ b=𝑔(-1)
( 𝑔(a)+ 𝑔(b)), where 𝑔(-1)
is a pseudo-inverse of 𝑔.
By using t-conorm and multiplication *, Weber (2) defined an integral for the Archimedean cases.
Semigroup operation ⊥ can be written as
a ⊥ b = 𝑔-1
( 𝑔(a)+ 𝑔(b)).
Where 𝑔-1
is an inverse of 𝑔. This implies that 𝑔 is an isomorphism of ([0, 1], ⊥) with ([0, (l)], + )
where Lebesgue measure is defined.
In this paper we discuss a pseudo integral based on pseudo addition and pseudo multiplication. In
section 2 we recall the concept of pseudo addition ⊕ and pseudo multiplication ⊙. and also gives the pseudo
integral based on a strict pseudo addition decomposable measure by generalizing the definition of the pseudo
integral of a bounded measurable function. In section 3, several important properties of the pseudo integral of a
measurable function based on the strict pseudo addition decomposable measure were discussed.
II. PRELIMINARIES
Definition: 2.1
Let [a, b] be a closed real interval and ⊕: [a, b] x [a, b] → [a, b] be a 2-place function satisfying the
following conditions:
(i) ⊕ is commutative.
(ii) ⊕ is nondecreasing in each place.
(iii) ⊕ is associative.
(iv) ⊕ has either a (or) b as zero element,
i.e., either ⊕ (a, x) = x (or)
⊕(b, x) = x.
⊕ will be called a pseudo-addition.
Definition: 2.2
A pseudo-multiplication ⊗ is a 2-place function ⊗: [a, b] x [a, b] ⟶ [a, b], satisfying the following conditions;
(i) ⊗ is commutative.
(ii) ⊗ is non decreasing in each place.
(iii) ⊗ is associative.
(iv) There exists a unit element e ∊ [a, b], i.e., ⊗(x, e) = x for all x ∊ [a, b].
Definition: 2.3
A pseudo-multiplication ⊗ with the generator 𝑔 of strict pseudo-addition ⊕ is defined as
⊗(x,y) = 𝑔-1
(𝑔(x) .𝑔(y)).
𝑔(x).𝑔(y) is always in [0, ∞]. This is called distributive pseudo-multiplication.](https://image.slidesharecdn.com/e040801519-161118102845/85/A-Note-on-Pseudo-Operations-Decomposable-Measure-1-320.jpg)
![A Note on Pseudo Operations Decomposable Measure
www.ijmsi.org 16 | Page
fn
Definition: 2.4
A fuzzy measure m derived from a pseudo addition with a zero element a, is a set function from 𝜎-
algebra 𝔅 of X to [a, c], C ∊ (a, b] such that
(i) m(𝛷) = a, m(X) = c.
(ii) A ∊ 𝔅, m(A) is decomposable,
if A is divided into Ai’S
A = Ai
m(A)= m(Ai).
Theorem 2.1
If the function ⊗ is continuous and strictly increasing in (a, b), then there exists a monotone function 𝑔
in [a, b] such that (e) = 1
⊗(x, y) = 𝑔-1
(𝑔(x)⦁ 𝑔(y))
Proof
From Aczel’s theorem [3] there exists a continuous and strictly monotone function in [a, b] such that
⊗(x, y) =f -1
(f(x) +f(y)).
Transforming f by 𝑔 = exp(-f )
⊗(x, y) =𝑔-1
(𝑔(x) • 𝑔(y)).
Since ⊕(x,e) = f -1
(f (x) +f (e)) = x
for all x ∊ [a,b], f(e)= O.
Thus (e) = exp(-f (e)) = 1.
Remarks: 2.1
For the sake of simplicity we write ⊕(x, y) as x⊕y and ⊗(x, y) as x ⊗ y, respectively. Distributive
pseudo-multiplication ⊗ has the distributive property with pseudo-addition ⊕ :
x ⊗(y ⊕ z) = (x⊗y) ⊕ (x ⊗ z).
(y ⊕ z) ⊗ x = (y ⊗ x) ⊕ (z ⊗ x).
Theorem 2.2
m is ⊕∨ -decomposable ⇒ m monotone non decreasing. m is ⊕∧ -decomposable
⇒ m monotone non increasing.
Proof
(i) For A⊂ B / m(B) = m(A) ⊕∨ m(B-A) ≥ m(A) and
m(B) = m(A) ⊕∧ m(B-A) ≤ m(A).
(ii) The operation ⊕ is commutative and associative thus
m(A∪ B) ⊕ m(A∩B) = m(A∩B) ⊕ m(A-B) ⊕ m(B-A) ⊕ m(A∩B) =
m(A) ⊕ m(B).
⊕∨ has a and ⊕∨ has b as zero element. Thus m is ⊕ decomposable.
Remark: 2.2
A ⊕∨ -decomposable measure can be regarded as a subjective measure expressing the grade of
importance [4] for example, m({A1}) expresses to what extent an attribute A1 is important to evaluate an object.
It is a reasonable assumption that m has monotonicity,
m({A1}) ≤ m({A1, A2})
III. PROPERTIES OF PSEUDO INTEGRAL BASED ON PSEUDO OPERATION
DECOMPOSABLE MEASURE
Theorem 3.1
Let ⊕ be a strict pseudo addition and let x be a 𝜎 finite set of ⊕ measure and m: A → [a,b] a 𝜎 -
⊕ decomposable measure. If { f n} is a sequence of measurable functional on X then
⊙ dm = fn ⊙ dm
Proof
Let hn = fi , n =1,2…..
Then {hn} is an increasing sequence of measurable functional on X.
n
∪
i=1](https://image.slidesharecdn.com/e040801519-161118102845/85/A-Note-on-Pseudo-Operations-Decomposable-Measure-2-320.jpg)
![A Note on Pseudo Operations Decomposable Measure
www.ijmsi.org 17 | Page
We have hn ⊙ dm = hn ⊙ dm
hn ⊙ dm = fn ⊙ dm
Since,
hn = fn
hn ⊙ dm = fi ⊙ dm = fi ⊙ dm
f n ⊙ dm = fn ⊙ dm
Definition: 3.1
Let ⊕ be a continuous pseudo addition and m: A → [a,b] a 𝜎 ⊕ decomposable measure. If m(x) < ∆
then the pseudo integral of an elementary measurable functional Q : x → [a,b] is defined by then
f ⊙ dm = Qn ⊙ dm (9)
Definition: 3.2
Let ⊕ be a strict pseudo addition and m: A → [a,b] a 𝜎 ⊕ decomposable measure. If x is a 𝜎 finite of
⊕ measure and {En} is a ⊕ measure finite and monotone cover of X then the pseudo integral of a measurable
functional f : x → [a,b] is defined by
f ⊙ dm = [ f ]n ⊙ dm (10)
Theorem 3.2
Let ⊕ be a strict pseudo addition and let X be a 𝜎 finite set of ⊕ measure and m: A → [a,b] a 𝜎
⊕ decomposable measure. If f is a measurable functional on X
f ⊙ dm = f ⊙ dm
for any sequence {En} of pairwise disjoint sets from A with x = En
fn(x) =
Proof
A functional sequence [ f ]n is given by
f n ⊙ dm = f n ⊙ dm ⊕ f n ⊙ dm
= f ⊙ dm](https://image.slidesharecdn.com/e040801519-161118102845/85/A-Note-on-Pseudo-Operations-Decomposable-Measure-3-320.jpg)
![A Note on Pseudo Operations Decomposable Measure
www.ijmsi.org 18 | Page
f ⊙ dm = f ⊙ dm
Theorem: 3.3
Let ⊕ be a strict pseudo addition and m: A → [a,b] a 𝜎 ⊕ decomposable measure
and only if f ≠ 0 a.e on E.
if f ∊ M(A) then for any E ∊ A
f ⊙ dm = 0
Proof
(1) suppose f ⊙ dm = 0
for arbitrary 0 < 𝛿
let E 𝛿 = {x ∊ E / 𝛿 ≤ f (x)}∊ A
we get
𝛿 ⊙ m(E 𝛿) ≤ f ⊙ dm ≤ f ⊙ dm ⊕ f ⊙ dm
= f ⊙ dm = 0
Thus we have m(E 𝛿) = 0
Since 0 < 𝛿 is arbitrary,
m(𝛿[0 < f ] ∩ E ) = 0
suppose f =0 a.e on E that is
m(E ∩ 𝛿[0 < f ]) = 0
we have
f ⊙ dm = f ⊙ dm ⊕ f ⊙ dm = 0
2) If there exists 𝛺 < ∆ suh that
f (x) ≤ 𝛺 for all x ∊ E
then
f ⊙ dm ≤ 𝛺 ⊙ m(E)
i.e.
f ⊙ dm = 0 [ f ]n ⊙ dm for any f ∊ 𝜇 (A)
f ⊙ dm = [ f ]n ⊙ dm
f ⊙ dm = [ f ]n ⊙ dm
= [ f ]n ⊙ dm = 0
IV. CONCLUSION
In this paper we mainly discussed the integral based on pseudo addition and pseudo multiplication can
be used obtain generalized formulation for decision making further more we have derived several important
properties of the pseudo integral of a measurable function based of strict pseudo addition decomposable
measure. Finally we have obtained that some theorems on the integral and the limit can be changed.](https://image.slidesharecdn.com/e040801519-161118102845/85/A-Note-on-Pseudo-Operations-Decomposable-Measure-4-320.jpg)
![A Note on Pseudo Operations Decomposable Measure
www.ijmsi.org 19 | Page
REFERENCES
[1]. M. SUGENO, "Theory of Fuzzy Integrals and Its Applications",Tokyo Institute of Technology, 1974.
[2]. S. WEBER, ⊥- "Decomposable measures and integrals for Archimedian t- conorms ⊥", J. Math. Anal. Appl. 101 (1984).
[3]. C. H. LING, "Representation of associative functions, Publ. Math. Debrecen 12 (1965)", 189-2 12.
[4]. M. SUGENO, "Fuzzy measures and fuzzy integrals: A survey, in Fuzzy Automata and Decision Process" (M. M. Gupta), North-
Holland, Amsterdam, 1977.
[5]. 5. M. kaulzka, A. Okolewski and M. Boczek" On chebyshew type inequalities for generalized sugeno Integral. Fuzzy sets and
systems Vol. 160
[6]. P.R. Halmos and CC. Moore, Measure theory, Springer, Newyork, USA. 1970
[7]. H.L. Royden Real Analysis, Macmillan Newyork, USA 1988
[8]. ZY Wang and G.J. Klir Generalized measure Theory springer boston mass USA, 2009
[9]. R. Mesiar and E. pap "Idempotent" integral as limit of g integral Fuzzy sets and systems vol. 102 no. 3 pp 385 – 392, 1999
[10]. D. Qiu and w. zhang "On decomposable measures" measures induced by metrics journal of applied mathematics vol 2012,
Article ID 701206 8 pages 2012.](https://image.slidesharecdn.com/e040801519-161118102845/85/A-Note-on-Pseudo-Operations-Decomposable-Measure-5-320.jpg)
A Note on Pseudo Operations Decomposable Measure
- 1. International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 4 Issue 8 || October. 2016 || PP-15-19 www.ijmsi.org 15 | Page A Note on Pseudo Operations Decomposable Measure Dr. N. Sarala1 , S. Jothi2 1 Associate Professor, A.D.M. College for women, Nagapattinam, Tamilnadi, India. 2 Guest Lecturer, Thiru.Vi. Ka. Govt. Arts College, Thiruvarur Tamilnadi, India. ABSTRACT: In this paper we discussed about the pseudo integral for a measurable function based on a strict pseudo addition and pseudo multiplication. Further more we got several important properties of the pseudo integral of a measurable function based on a strict pseudo addition decomposable measure. KEYWORDS: Pseudo addition, Pseudo multiplication, decomposable measure, Pseudo Inverse I. INTRODUCTION Taking into account human subjective measures in engineering science fuzzy measures have been intensively discussed since Sugeno(1) defined a fuzzy measure as a measure having the monotonicity property instead of additivity. Weber(2) proposed ⊥-decomposable measures where the additivity of measures is weakened. t-Conorm ⊥ is an appropriate semigroup operation in [O, l]. ⊥-decomposable measures can be written as m(A∪B) = m(A) ⊥ m(B). for the Archimedean case ⊥ is written as a ⊥ b=𝑔(-1) ( 𝑔(a)+ 𝑔(b)), where 𝑔(-1) is a pseudo-inverse of 𝑔. By using t-conorm and multiplication *, Weber (2) defined an integral for the Archimedean cases. Semigroup operation ⊥ can be written as a ⊥ b = 𝑔-1 ( 𝑔(a)+ 𝑔(b)). Where 𝑔-1 is an inverse of 𝑔. This implies that 𝑔 is an isomorphism of ([0, 1], ⊥) with ([0, (l)], + ) where Lebesgue measure is defined. In this paper we discuss a pseudo integral based on pseudo addition and pseudo multiplication. In section 2 we recall the concept of pseudo addition ⊕ and pseudo multiplication ⊙. and also gives the pseudo integral based on a strict pseudo addition decomposable measure by generalizing the definition of the pseudo integral of a bounded measurable function. In section 3, several important properties of the pseudo integral of a measurable function based on the strict pseudo addition decomposable measure were discussed. II. PRELIMINARIES Definition: 2.1 Let [a, b] be a closed real interval and ⊕: [a, b] x [a, b] → [a, b] be a 2-place function satisfying the following conditions: (i) ⊕ is commutative. (ii) ⊕ is nondecreasing in each place. (iii) ⊕ is associative. (iv) ⊕ has either a (or) b as zero element, i.e., either ⊕ (a, x) = x (or) ⊕(b, x) = x. ⊕ will be called a pseudo-addition. Definition: 2.2 A pseudo-multiplication ⊗ is a 2-place function ⊗: [a, b] x [a, b] ⟶ [a, b], satisfying the following conditions; (i) ⊗ is commutative. (ii) ⊗ is non decreasing in each place. (iii) ⊗ is associative. (iv) There exists a unit element e ∊ [a, b], i.e., ⊗(x, e) = x for all x ∊ [a, b]. Definition: 2.3 A pseudo-multiplication ⊗ with the generator 𝑔 of strict pseudo-addition ⊕ is defined as ⊗(x,y) = 𝑔-1 (𝑔(x) .𝑔(y)). 𝑔(x).𝑔(y) is always in [0, ∞]. This is called distributive pseudo-multiplication.
- 2. A Note on Pseudo Operations Decomposable Measure www.ijmsi.org 16 | Page fn Definition: 2.4 A fuzzy measure m derived from a pseudo addition with a zero element a, is a set function from 𝜎- algebra 𝔅 of X to [a, c], C ∊ (a, b] such that (i) m(𝛷) = a, m(X) = c. (ii) A ∊ 𝔅, m(A) is decomposable, if A is divided into Ai’S A = Ai m(A)= m(Ai). Theorem 2.1 If the function ⊗ is continuous and strictly increasing in (a, b), then there exists a monotone function 𝑔 in [a, b] such that (e) = 1 ⊗(x, y) = 𝑔-1 (𝑔(x)⦁ 𝑔(y)) Proof From Aczel’s theorem [3] there exists a continuous and strictly monotone function in [a, b] such that ⊗(x, y) =f -1 (f(x) +f(y)). Transforming f by 𝑔 = exp(-f ) ⊗(x, y) =𝑔-1 (𝑔(x) • 𝑔(y)). Since ⊕(x,e) = f -1 (f (x) +f (e)) = x for all x ∊ [a,b], f(e)= O. Thus (e) = exp(-f (e)) = 1. Remarks: 2.1 For the sake of simplicity we write ⊕(x, y) as x⊕y and ⊗(x, y) as x ⊗ y, respectively. Distributive pseudo-multiplication ⊗ has the distributive property with pseudo-addition ⊕ : x ⊗(y ⊕ z) = (x⊗y) ⊕ (x ⊗ z). (y ⊕ z) ⊗ x = (y ⊗ x) ⊕ (z ⊗ x). Theorem 2.2 m is ⊕∨ -decomposable ⇒ m monotone non decreasing. m is ⊕∧ -decomposable ⇒ m monotone non increasing. Proof (i) For A⊂ B / m(B) = m(A) ⊕∨ m(B-A) ≥ m(A) and m(B) = m(A) ⊕∧ m(B-A) ≤ m(A). (ii) The operation ⊕ is commutative and associative thus m(A∪ B) ⊕ m(A∩B) = m(A∩B) ⊕ m(A-B) ⊕ m(B-A) ⊕ m(A∩B) = m(A) ⊕ m(B). ⊕∨ has a and ⊕∨ has b as zero element. Thus m is ⊕ decomposable. Remark: 2.2 A ⊕∨ -decomposable measure can be regarded as a subjective measure expressing the grade of importance [4] for example, m({A1}) expresses to what extent an attribute A1 is important to evaluate an object. It is a reasonable assumption that m has monotonicity, m({A1}) ≤ m({A1, A2}) III. PROPERTIES OF PSEUDO INTEGRAL BASED ON PSEUDO OPERATION DECOMPOSABLE MEASURE Theorem 3.1 Let ⊕ be a strict pseudo addition and let x be a 𝜎 finite set of ⊕ measure and m: A → [a,b] a 𝜎 - ⊕ decomposable measure. If { f n} is a sequence of measurable functional on X then ⊙ dm = fn ⊙ dm Proof Let hn = fi , n =1,2….. Then {hn} is an increasing sequence of measurable functional on X. n ∪ i=1
- 3. A Note on Pseudo Operations Decomposable Measure www.ijmsi.org 17 | Page We have hn ⊙ dm = hn ⊙ dm hn ⊙ dm = fn ⊙ dm Since, hn = fn hn ⊙ dm = fi ⊙ dm = fi ⊙ dm f n ⊙ dm = fn ⊙ dm Definition: 3.1 Let ⊕ be a continuous pseudo addition and m: A → [a,b] a 𝜎 ⊕ decomposable measure. If m(x) < ∆ then the pseudo integral of an elementary measurable functional Q : x → [a,b] is defined by then f ⊙ dm = Qn ⊙ dm (9) Definition: 3.2 Let ⊕ be a strict pseudo addition and m: A → [a,b] a 𝜎 ⊕ decomposable measure. If x is a 𝜎 finite of ⊕ measure and {En} is a ⊕ measure finite and monotone cover of X then the pseudo integral of a measurable functional f : x → [a,b] is defined by f ⊙ dm = [ f ]n ⊙ dm (10) Theorem 3.2 Let ⊕ be a strict pseudo addition and let X be a 𝜎 finite set of ⊕ measure and m: A → [a,b] a 𝜎 ⊕ decomposable measure. If f is a measurable functional on X f ⊙ dm = f ⊙ dm for any sequence {En} of pairwise disjoint sets from A with x = En fn(x) = Proof A functional sequence [ f ]n is given by f n ⊙ dm = f n ⊙ dm ⊕ f n ⊙ dm = f ⊙ dm
- 4. A Note on Pseudo Operations Decomposable Measure www.ijmsi.org 18 | Page f ⊙ dm = f ⊙ dm Theorem: 3.3 Let ⊕ be a strict pseudo addition and m: A → [a,b] a 𝜎 ⊕ decomposable measure and only if f ≠ 0 a.e on E. if f ∊ M(A) then for any E ∊ A f ⊙ dm = 0 Proof (1) suppose f ⊙ dm = 0 for arbitrary 0 < 𝛿 let E 𝛿 = {x ∊ E / 𝛿 ≤ f (x)}∊ A we get 𝛿 ⊙ m(E 𝛿) ≤ f ⊙ dm ≤ f ⊙ dm ⊕ f ⊙ dm = f ⊙ dm = 0 Thus we have m(E 𝛿) = 0 Since 0 < 𝛿 is arbitrary, m(𝛿[0 < f ] ∩ E ) = 0 suppose f =0 a.e on E that is m(E ∩ 𝛿[0 < f ]) = 0 we have f ⊙ dm = f ⊙ dm ⊕ f ⊙ dm = 0 2) If there exists 𝛺 < ∆ suh that f (x) ≤ 𝛺 for all x ∊ E then f ⊙ dm ≤ 𝛺 ⊙ m(E) i.e. f ⊙ dm = 0 [ f ]n ⊙ dm for any f ∊ 𝜇 (A) f ⊙ dm = [ f ]n ⊙ dm f ⊙ dm = [ f ]n ⊙ dm = [ f ]n ⊙ dm = 0 IV. CONCLUSION In this paper we mainly discussed the integral based on pseudo addition and pseudo multiplication can be used obtain generalized formulation for decision making further more we have derived several important properties of the pseudo integral of a measurable function based of strict pseudo addition decomposable measure. Finally we have obtained that some theorems on the integral and the limit can be changed.
- 5. A Note on Pseudo Operations Decomposable Measure www.ijmsi.org 19 | Page REFERENCES [1]. M. SUGENO, "Theory of Fuzzy Integrals and Its Applications",Tokyo Institute of Technology, 1974. [2]. S. WEBER, ⊥- "Decomposable measures and integrals for Archimedian t- conorms ⊥", J. Math. Anal. Appl. 101 (1984). [3]. C. H. LING, "Representation of associative functions, Publ. Math. Debrecen 12 (1965)", 189-2 12. [4]. M. SUGENO, "Fuzzy measures and fuzzy integrals: A survey, in Fuzzy Automata and Decision Process" (M. M. Gupta), North- Holland, Amsterdam, 1977. [5]. 5. M. kaulzka, A. Okolewski and M. Boczek" On chebyshew type inequalities for generalized sugeno Integral. Fuzzy sets and systems Vol. 160 [6]. P.R. Halmos and CC. Moore, Measure theory, Springer, Newyork, USA. 1970 [7]. H.L. Royden Real Analysis, Macmillan Newyork, USA 1988 [8]. ZY Wang and G.J. Klir Generalized measure Theory springer boston mass USA, 2009 [9]. R. Mesiar and E. pap "Idempotent" integral as limit of g integral Fuzzy sets and systems vol. 102 no. 3 pp 385 – 392, 1999 [10]. D. Qiu and w. zhang "On decomposable measures" measures induced by metrics journal of applied mathematics vol 2012, Article ID 701206 8 pages 2012.