μ -Canonical fuzzy measure on (0,1]

Abstract

In this paper, the definition μ-Canonical Fuzzy Measure on (0, 1] is introduced. The characterization of the μ-Canonical fuzzy measure is given by the μ-invariance properties against translations.

Keywords

Borel measure Canonical fuzzy measure distortion function fuzzy measure

1 Introduction and preliminaries

The term fuzzy measure was first introduced in [9]. However, this term referred to a notion named ‘capacity’ which was first introduced in [1]. Over the years the same notion has been referred to by many different names, such as ‘confidence measure’ [2], ‘non-additive probability’ [9, 10], and ‘weighting function’ [11].

Let μ be a finite Borel measure on (0, 1] with μ ((0, 1]) =1 and let F (x) = μ ((0, x]). (F is sometimes called the distribution function of μ.), then F is increasing by the Theorem 1.8 part (a) and right continuous by the Theorem 1.8 part (d) in [3] because $(0, x] = ⋂_{1}^{\infty} (0, x_{n}]$ whenever x_n ↘ x. And F (0) =0 and F (1) =1. Moreover, if b > a, (0, b] = (0, a] ∪ (a, b], so μ ((a, b]) = F (b) - F (a). Conversely, if F : [0, 1] ⟶ [0, 1] is any increasing and right continuous function which F (0) =0 and F (1) =1, then there exists a Borel measure μ on [0, 1] such that μ ((a, b]) = F (b) - F (a), by the Theorem 1.16 in [3].

Our procedure will be to turn this process around construct a measure μ starting from an increasing, right-continuous function F. The special case F (x) = x will yield the usual “length” measure.

The building blocks for our theory will be the left-open, right-closed intervals in (0, 1]-that is, sets of the form (a, b] or ∅, where 0 < a < b ≤ 1. Clearly the intersection of two h-intervals(half-open intervals) is a h-interval, and the complement of a h-interval is a h-interval or the disjoint union of two h-intervals.

Let F : [0, 1] ⟶ [0, 1] be an increasing and right continuous with F (0) =0 and F (1) =1, then there is a measure μ such that μ|_𝒜 = μ₀ and $\begin{matrix} μ (⋃_{1}^{n} (a_{j}, b_{j}]) & = μ_{0} (⋃_{1}^{n} (a_{j}, b_{j}]) \\ = \sum_{1}^{n} [F (b_{j}) - F (a_{j})] \end{matrix}$

where μ₀ is a premeasure on the algebra 𝒜. For the theory of measure and premeasure, we refer the reader to [3]. In this paper, μ is a dependent measure to the increasing and right-continuous function F : [0, 1] ⟶ [0, 1] which F (0) =0 and F (1) =1 and 𝒜 is an algebra on (0, 1].

The set function g : 𝒜 ⟶[0, 1] is called the fuzzy measure if the following conditions are satisfied:

g (∅) =0 and g ((0, 1]) =1.

For every A, B∈ 𝒜 with A ⊂ B, it follows that g (A) ≤ g (B) (the monotonicity).

Also, the fuzzy measure g : 𝒜 ⟶[0, 1] is called continuous if for each monotone (increasing or decreasing) sequence A_n∈ 𝒜 satisfying $lim_{n} A_{n} \in$ 𝒜, it holds that $lim_{n} g (A_{n}) = g (lim_{n} A_{n})$ , and the function f : [0, 1] ⟶ [0, 1] is called the distortion function if the following conditions are satisfied:

f (0) =0 and f (1) =1,

f is non-decreasing.

Let a ∈ (0, ∞]. An extended real function θ : [0, a] ⟶ [0, ∞) is called a T-function iff it is continuous, strictly increasing, and such that θ (0) =0 and θ^-1 ({∞}) = ∅ or {∞}, according to a being finite or not.

ν is called quasi-additive iff there exists a T-function θ, whose domain of definition contains the range of ν, such that the set function θoν defined on the σ-algebra C by $(θ o ν) (E) = θ (ν (E)),$ for all E ∈ C, is additive; ν is called a quasi-measure iff there exists a T-function θ such that θoν is a classical measure on C. The T-function θ is called the proper T-function of ν. For the theory of quasi measure, we refer the reader to [12].

In [5], Honda introduced the concept of canonical fuzzy measure by using the Lebesgue measure on (0, 1]. Now, we extend this concept for every Borel measure on (0, 1], which is dependent to increasing and right-continuous function F.

2 Main results

Now, we need to some extra definitions for expression of our main results.

Definition 2.1. The fuzzy measure g on an algebra 𝒜 on [0, 1] is called μ-canonical if there exists a distortion function f : [0, 1] ⟶ [0, 1] satisfying that g = foμ.

Let 𝒞 be the family of subsets in (0, 1] of the form (a₁, b₁] ∪ (a₂, b₂] ∪ … ∪ (a_n, b_n], where 0 ≤ a₁ ≤ b₁ ≤ a₂ ≤ b₂ ≤ … ≤ a_n ≤ b_n, n = 1, 2, . . .. That is the class 𝒞 is the set of all finite disjoint unions of left-open and right-closed subintervals in (0, 1]. This family 𝒞 forms an algebra ([4], Chap. 1, Sec. 4).

Definition 2.2. Let g be a fuzzy measure on 𝒞 and μ be a measure, then g is called μ-weakly invariant against translation if for every (a, b]∈ 𝒞, it follows that $g ((a, b]) = g ((0, μ ((a, b])]) .$

Theorem 2.3. Let g be a μ-weakly invariant against translation on 𝒞. Then there is a distortion function f : [0, 1] ⟶ [0, 1] such that g ((a, b]) = f (μ ((a, b])). Furthermore, if μ is the Lebesgue measure, then f is unique.

Proof. Let f (x) = g ( (0, μ ((0, x]) ] ), for each x ∈ (0, 1]. For x ≤ y, we have (0, x] ⊆ (0, y] which implies that μ ((0, x]) ≤ μ ((0, y]) and hence f (x) ≤ f (y). Also, we have $\begin{matrix} f (0) & = & g ((0, \underset{0}{\underset{︸}{μ (\underset{\emptyset}{\underset{︸}{(0, 0]}})}}]) = g (\emptyset) = 0 \\ f (1) & = & g ((0, \underset{1}{\underset{︸}{μ (\underset{(0, 1]}{\underset{︸}{(0, 1]}})}}]) = g ((0, 1]) = 1 . \end{matrix}$ Also, for all (a, b]∈ 𝒜 we have $\begin{matrix} fo μ ((a, b]) & = & f (F (b) - F (a)) \\ = & g ((0, μ ((0, F (b) - F (a)])]) \\ = & g ((0, F (b) - F (a)]) \\ = & g ((0, μ ((a, b])]) \\ = & g ((a, b]) . □ \end{matrix}$

Remark 2.4. In the previous theorem, let ℬ be the generated σ-algebra by left-open and right-closed subintervals in [0, 1] and suppose that g and μ are fuzzy measure and increasing measure on ℬ, respectively where $g ((a, b]) = {\begin{matrix} 1; & μ ((a, b]) = 1 \\ 0; & μ ((a, b]) \neq 1 \end{matrix} .$ Also, g is the μ-weakly invariant against translation measure.

Define $f (x) : = {\begin{matrix} 0; & x \neq 1 \\ 1; & x = 1 \end{matrix}$ and $h (x) : = go θ (x); {\begin{matrix} θ : [0, 1] & ⟶ & ℬ \\ x & ⟼ & (0, x] \end{matrix} .$ We can see that foμ = g = hoμ but f ¬ = h.

Definition 2.5. Let g and μ be arbitrary fuzzy measure and Borel measure on 𝒞, respectively. Then we say that g is μ-strongly invariant against translation if for every A = (a, b] and $B = \cup_{i = 1}^{n} (a_{i}, b_{i}]$ in 𝒞, satisfying 0 ≤ a ≤ b ≤ a₁ ≤ b₁ ≤ a₂ ≤ b₂ ≤ … ≤ a_n ≤ b_n, it holds that $\begin{matrix} g (A \cup B) & = & g ((F (a), F (b)] \cup (⋃_{i = 1}^{n} (F (a_{i}) - F (a_{1}) \\ + F (b), F (b_{i}) - F (a_{1}) + F (b)])) . \end{matrix}$

We see that A∩ B = ∅ and since F is increasing, so $\begin{matrix} (F (a), F (b)] \cap (⋃_{i = 1}^{n} (F (a_{i}) - F (a_{1}) \\ + F (b), F (b_{i}) - F (a_{1}) + F (b)]) = \emptyset . \end{matrix}$

Remark 2.6. Every μ-strongly invariant against translation measure is μ-weakly invariant against translation. To show this, it is enough you put B = (a, b] and A = ∅ = (a, a] in 𝒞. Then we have $\begin{matrix} g (A \cup B) & = & g (\underset{\emptyset}{\underset{︸}{(a, a]}} \cup (a, b]) \\ = & g (\underset{\emptyset}{\underset{︸}{(F (a), F (a)]}} \cup (F (a) - F (a), \\ F (b) - F (a)]) \\ = & g ((0, μ ((a, b])]) . \end{matrix}$ If g is μ-strongly invariant against translation, then for each (a₁, b₁] , (a₂, b₂]∈ 𝒞, we have $\begin{matrix} g ((a_{1}, b_{1}] \cup (a_{2}, b_{2}]) \\ = g ((F (a_{1}), F (b_{1})] \cup \\ (F (a_{2}) - F (a_{2}) + F (b_{1}), F (b_{2}) \\ - F (a_{2}) + F (b_{1})]) \\ = g ((F (a_{1}), F (b_{1})] \\ \cup (F (b_{1}), F (b_{2}) - F (a_{2}) \\ + F (b_{1})]) \\ = g ((F (a_{1}), F (b_{2}) - F (a_{2}) \\ + F (b_{1})]) \\ = g ((0, F (b_{2}) - F (a_{2}) \\ + F (b_{1}) - F (a_{1})]) \\ = g ((0, μ ((a_{1}, b_{1}] + μ ((a_{2}, b_{2}]))) \end{matrix}$ and by the induction, it follows that $g (⋃_{i = 1}^{n} (a_{i}, b_{i}]) = g ((0, \sum_{i = 1}^{n} μ ((a_{i}, b_{i}])))$

Theorem 2.7. Let g and μ be arbitrary fuzzy measure and measure on ℬ, respectively. If g is μ-strongly invariant against translation, then there exists the distortion function f : [0, 1] ⟶ [0, 1] such that $g (A) = f (μ (A)) . \forall A \in 𝒞$ Furthermore, if μ is a Lebesgue measure, then f is unique.

Proof. Define f (x) = g ((0, x]). Then for each $A = ⋃_{i = 1}^{n} (a_{i}, b_{i}] \in 𝒞$ , therefore $\begin{matrix} g (A) & = & g (⋃_{i = 1}^{n} (a_{i}, b_{i}]]) \\ = & g ((0, \sum_{i = 1}^{n} μ ((a_{i}, b_{i}]))) \\ = & g ((0, μ (⋃_{i = 1}^{n} (a_{i}, b_{i}])]) \\ = & f (μ (⋃_{i = 1}^{n} (a_{i}, b_{i}])) = f (μ (A)) . \end{matrix}$ If μ is a Lebesgue measure and h is another distortion function on (0, 1] such that g = hoμ, then $f (x) = g ((0, x]) = ho μ ((0, x]) = h (x); \forall x \in (0, 1]$ hence, f is unique. □

Denote by ℬ the Borel σ-algebra on (0, 1]. ℬ is the smallest monotone class containing 𝒞 (See [4], Chap. 1, Sec. 6, Theorem B).

Theorem 2.8. Let g be a fuzzy measure on the Borel σ-algebra ℬ on (0, 1] and μ be a measure on the Borel σ-algebra on (0, 1] such that:

g ((0, x]) = g ((0, x)), for all x ∈ (0, 1]

g is continuous

g is μ-strongly invariant against translation on 𝒞.

Then there exists a continuous distortion functionf : [0, 1] ⟶ [0, 1] such that $g (A) = f (μ (A)) . \forall A \in ℬ$

Proof. The proof of this theorem is similar to the proof the theorem 3 in [5]. If we set f (x) = g ((0, x]), then it follows that $\begin{matrix} \forall A = ⋃_{1}^{n} (a_{i}, b_{i}] \in 𝒞; \\ g (A) = f (μ (⋃_{1}^{n} (a_{i}, b_{i}])) = f (μ (A)) . \end{matrix}$ Now we show that f is continuous. Let t_n ↘ t and s_n ↗ s are decreasing and increasing sequences in (0, 1], respectively. Then $lim_{n} (0, t_{n}] = ⋂_{1}^{\infty} (0, t_{n}] = (0, t]$ and $lim_{n} (0, s_{n}] = ⋃_{1}^{\infty} (0, s_{n}] = (0, s)$ and since g is continuous, so g ((0, t_n]) ↘ g ((0, t]) and this shows that f (t_n) ↘ f (t). Hence f is right-continuous. Also, by assumptions (1) and continuity of g, we have $f (s_{n}) = g ((0, s_{n}]) ↗ g ((0, s)) = g ((0, s]) = f (s) .$ So f (s_n) ↗ f (s) and therefore f is left-continuous. Then f is continuous. Now, let $D = {A \in ℬ | g (A) = f (μ (A))} .$ Since g is continuous and f is continuous on (0, 1], so $D$ is monotone class and since the Borel σ-algebra ℬ is the smallest monotone class containing 𝒞 ([4], Cha. 1, Sec. 6. Theorem B), hence $D = ℬ$ . □

Definition 2.9. Let g be a fuzzy measure on the σ-algebra 𝒜 on (0, 1] and μ be a Borel measure. Then g is called μ-length invariant, if for all A∈ 𝒜, g (A) = g ((0, μ (A)]).

Theorem 2.10. Let g be an additive fuzzy measure on the σ-algebra 𝒜 on (0, 1], such that g is invariant under translation and μ be a measure. Then g is μ-length invariant against translation if and only if g is μ-strongly invariant.

Proof. (←) Since g is μ-strongly invariant against translation, so there is a distortion function f : [0, 1] ⟶ [0, 1] such that g = foμ and f (0) =0 and f (1) =1. (we can suppose that f (x) = g ((0, x])). Then we have $\forall A \in 𝒜; g (A) = fo μ (A) = g ((0, μ (A)])$ that is g is μ-length invariant against translation.

(→) Since g is μ-length invariant, so $\forall A \in 𝒜; g (A) = g ((0, μ (A)])$ Let $A = ⋃_{i = 1}^{n} (a_{i}, b_{i}] \in 𝒜$ and E = (a, b] and $F = ⋃_{i = 1}^{n} (c_{i}, d_{i}]$ , where $0 \leq a < b \leq c_{1} < d_{1} \leq \dots \leq c_{n} < d_{n} .$ Then we have $\begin{matrix} g (E \cup F) & = & g (E) + g (F) \\ = & g (E) + \sum_{i = 1}^{n} g ((c_{i}, d_{i}]) \\ = & g ((0, μ (E)]) + \sum_{i = 1}^{n} g ((0, μ ((c_{i}, d_{i}])]) \\ = & g ((0, F (b) - F (a)]) \\ + \sum_{i = 1}^{n} g ((0, F (d_{i}) - F (c_{i})]) \\ = & g ((F (a), F (b)]) \\ + \sum_{i = 1}^{n} g ((F (c_{i}), F (d_{i})]) \\ = & g ((F (a), F (b)]) \\ + \sum_{i = 1}^{n} g ((F (c_{i}) - F (c_{1}) + F (b), \\ F (d_{i}) - F (c_{1}) + F (b)]) \\ = & g ((F (a), F (b)]) \\ + g (⋃_{i = 1}^{n} (F (c_{i}) - F (c_{1}) + F (b), \\ F (d_{i}) - F (c_{1}) + F (b)]) \\ = & g ((F (a), F (b)] \\ \cup (⋃_{i = 1}^{n} (F (c_{i}) - F (c_{1}) + F (b), \\ F (d_{i}) - F (c_{1}) + F (b)])) . □ \end{matrix}$

Theorem 2.11. Let g be a fuzzy measure on the algebra 𝒜 on (0, 1] and μ be a dependent measure to the increasing and right-continuous function F : [0, 1] ⟶ [0, 1], that F² = F and F (0) =0. Then g is μ-canonical if and only if g is μ-length invariant.

Proof. (→) If g is canonical, then there is a non-decreasing distortion function f : [0, 1] ⟶ [0, 1] such that f (0) =0 and f (1) =1 and g (A) = f (μ (A)), for all A ∈ 𝒜. Since every A ∈ 𝒜 can be written as union of left-open and right-closed subintervals of (0, 1], so we have $\begin{matrix} \forall A \in 𝒜; \\ g (A) = f (μ (A)) = f (μ ((0, μ (A)])) = g ((0, μ (A)]) \end{matrix}$ because μ ((0, μ (A)]) = μ (A).

(←) Set f (x) = g ((0, x]). Then for each A ∈ 𝒜 we have f (μ (A)) = g ((0, μ (A)]) = g (A). Thus g is μ-canonical. □

Proposition 2.12. Let g be a μ-canonical fuzzy and quasi measure. Then μ is a quasi measure.

Proof. Since g is μ-canonical fuzzy measure, there is a distortion function f : [0, 1] ⟶ [0, 1] such that $g \overset{1}{=} fo μ$ .

Also, g is quasi measure, so there exists T-function θ : [0, 1] ⟶ [0, ∞] with a = 1 and classical measure ν such that $θ og \overset{2}{=} ν$ .

Therefore $ν = θ og = (θ of) o μ . (by 1, 2)$ Clearly, θof is continuous and increasing function and $\begin{matrix} (θ of) : [0, 1] ⟶ [0, \infty] (θ of) (0) = θ (0) = 0 & \\ (θ of)^{- 1} ({\infty}) = f^{- 1} (\emptyset) = \emptyset . \end{matrix}$ Hence μ is quasi measure. □

Proposition 2.13. Let μ be a quasi measure and g be a μ-canonical fuzzy measure with the continuous, one-to-one and onto distortion function f. Then g is quasi measure.

Proof. There exists T-function θ : [0, 1] ⟶ [0, ∞] and classical measure μ such that θoμ = ν. And g = foμ. Let ${\begin{matrix} θ^{*} : [0, 1] ⟶ [0, \infty] \\ θ^{*} : = θ {of}^{- 1} \end{matrix}$ . Then θ^* is continuous and $θ^{*} og = θ o (f^{- 1} og) = θ o μ = ν .$ And also, $\begin{matrix} θ^{*} (0) = θ (0) = 0 & \\ θ^{*^{- 1}} ({\infty}) = (fo θ^{- 1}) ({\infty}) = f (\emptyset) = \emptyset \end{matrix}$

Hence θ^* is the T-function and so g is quasi measure. □

Footnotes

Acknowledgments

The authors are grateful to the anonymous referees and the Editors for their fruitful comments, valuable suggestions and careful corrections.

References

Choquet

, Theory of capacities, Annales de l’institut Fourier5 (1953), 131–295.

Dubois

and Prade

, Fuzzy Sets and Systems, Academic Press, New York, 1980.

Folland

G.B.

, Real Analysis, Modern Techniques and their Applications, Second edition, John Wiley & Sons, Inc., 1999.

Halmos

P.R.

, Measure Theory, Van Nostrand Reinhold, New York, 1969.

Honda

, Canonical fuzzy measure on (0, 1], Fuzzy set and Systems123 (2001), 147–150.

Ross

, Fuzzy Logic with Engineering Applications, 2nd Edition, John Wiley & Sons, New York, 2004.

Schmeidler

, Integral representation without additivity, Proceedings of American Mathematical Society97 (1986), 255–261.

Schmeidler

, Subjective probability and expected utility with-out additivity, Econometrica57 (1989), 571–587.

Sugeno

, Theory of Fuzzy Integrals and its Applications, Ph.D. Thesis, Institute of Technology, Tokyo, 1974.

10.

Sugeno

, Gupta

M.M.

, Saridis

G.N.

and Gaines

B.R.

, Fuzzy measures and fuzzy integrals, a survey, 1977.

11.

Tversky

and Kahneman

, Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty5 (1992), 297–323.

12.

Wang

and Klir

G.J.

, Generalized Measure Theory, Spring, New York, 2009.