STRONG LAW OF LARGE NUMBERS FOR UPPER SET-VALUED AND FUZZY-SET VALUED PROBABILITY

. In this paper, we introduce the concepts of upper-lower set-valued probabilities and related upper-lower expectations for random variables. With a new concept of independence for random variables, we show a strong law of large numbers for upper-lower set-valued probabilities. Furthermore, we extend those concepts and theorem to the case of fuzzy-set.


1.
Introduction. It is well-known that the classic strong laws of large numbers (SLLNs) as fundamental limit theorems in probability theory play an important role in the development of probability theory and its application. The additivity of point-valued probability and expectation is the key in the proofs of these classic SLLNs. However, such additivity assumption is not adaptable in many areas of application, such as mathematical economics, statistics, quantum mechanics and finance, because many uncertain phenomena can not be well modeled and interpreted by additive probability and additive expectation. Therefore, a multitude of scholars have extended SLLNs to nonlinear probabilities mainly by two methods.
On one hand, considering the situation that there are instances in Bayesian estimation when the prior probability is not known precisely, L. DeRobertis and J. A. Hartigan [7] suggest using an interval of measures rather than a single prior, and extend the Bayes theorem in this setting. Consider the set-valued probability space (Ω, F, Π), where Π : F → P(R) is a set-valued function satisfying (i) Π(A) = ∅ for every A ∈ F and (ii) Π( with A i ∈ F. Then the concept which seems to be useful in such situations is the expectation of random variables with respect to a set-valued measure(see [1] and [6]). M. L. Puri and D. A. Ralescu [19] define the expectation E Π of random variable X with respect to set-valued measure Π by E Π [X] = Ω XdΠ = Ω XdQ : Q is a selector of Π}. Then under the set-valued probability space (Ω, F, Π), assuming that be a sequence of independent and identically distributed random variables such that Π P where P is a probability measure, M. L. Puri and D. A. Ralescu [19] prove a SLLN that with the notion of independence for random variables under upper expectation and without identical distribution assumption. Motivated by the two different methods of extending classic SLLNs, we wonder that could we get a new SLLN when consider a given set of set-valued probabilities. Fortunately, if the concepts of upper set-valued probability Γ, the expectation E Γ of random variable on (Ω, F, Γ), independence of random variables can be defined appropriately, then applying the SLLN in [4], we obtain the SLLN under (Ω, F, Γ) That is any cluster point of empirical averages n i=1 Xi n lies in the set E Γ [X + 1 ] − E Γ [X − 1 ] almost surely with respect to the upper set-valued measure Γ. Furthermore, applying [22], we can extend our result to upper fuzzy-set valued probability U. For fuzzy set M and the corresponding membership function m M on R, m M (x) = 1 is denoted by x ∈ M . Therefore, we derive the SLLN in the following form The paper is organized as follow: In Section 2, we recall some basic concepts and related lemmas about set-valued probability and upper-lower probabilities which will be useful in this paper. Then we define the upper-lower set-valued probabilities and expectations in this setting. In Section 3, we prove a strong law of large numbers under the framework of upper set-valued probability. In Section 4, we extend the main result of Section 3 to the case of upper-lower fuzzy-set valued probabilities.

Preliminary.
Let Ω be a set, F be a σ−algebra on Ω, and P(R) denote all the subsets of the Euclidean space R. A set-valued measure is a function Π : F −→ P(R) such that: The following lemma due to [1] is fundamental in the framework of this paper.
Obviously, for any A ∈ F, Π(A) is convex. In what follows we only consider the nonatomic closed set-valued probability. It follows that for such measures, Π(∅) = {0}.
Since the set-valued probabilities {Π n } ∞ n=1 we considered in this paper are all nonatomic closed set-valued probabilities, the values of those mappings are convex sets by Lemma 2.1 (1). Then our request that for any A ∈ F, ∞ n=1 Π n (A) is a convex set and ∞ n=1 Π n (A) is a nonempty convex set is corresponding to the classic set-valued probabilities. We will construct a pair of upper-lower set-valued probabilities to illustrate that our definition is natural and reasonable.
Example 2.4. Suppose that P 1 is a measure and P 2 is a probability such that P 1 (A) ≤ P 2 (A) for any A ∈ F. Then define a sequence of nonatomic closed set values probabilities by Π n (A) = n n + 1 P 1 (A), n n + 1 P 2 (A) + 1 n + 1 , f or n ≥ 1.
Then for any A ∈ F, we have Therefore, (Γ, Γ) is a pair of upper-lower set-valued probabilities that defined in Definition 2.3.
Proposition 2.5. If Γ is an upper set-valued probability, then , the first and second assertions can be easily proved. Therefore we only prove the third assertion. Assume to be a family of disjointed sets of F. Then by the additivity of each Π n , By the definition of addition of sets, there is a sequence {y i } ∞ i=1 such that x = ∞ i=1 y i and y i ∈ Π n0 (A i ) for every i. Obviously, y i ∈ ∞ n=1 Π n (A i ) for every i. Consequently, y i ∈ Γ(A i ) for every i. Again by the definition of addition of sets, An upper set-valued probability space is a triple (Ω, F, Γ), where Γ is an upper set-valued probability. In order to state and prove a law of large numbers, we need to define the concept of expectation. In the case of set-valued measure, this concept first appeared in [19]. Let X : Ω → R be a random variable on Ω and Π : F → P([0, 1]) be a set-valued probability. Without loss of generality, assume that Π is absolutely continuous with respect to some probability measure P, denoted by Π P ( i.e. P (A) = 0 implies Π(A) = {0}). If E P [|X|] = Ω |X|dP < ∞, the expectation of X with respect to Π is defined as Now we will extend the concept of expectation under the upper set-valued probability space (Ω, F, Γ) .
Definition 2.7. Let (Ω, F, Γ) be an upper set-valued probability space and Γ(·) be the lower set-valued probability corresponding to Γ(·). Let X : Ω → R be a real random variable. Then the upper expectation E Γ [X] and lower expectation E Γ [X] are defined respectively by Proposition 2.8. Assume that X and Y are nonnegative random variables, then the upper expectation has following properties:

Remark 2.9. The subtraction of two bounded sets A and B is defined by
Proposition 2.10. Assume that Γ 1 and Γ 2 are two upper set-valued probabilities such that Γ 1 (A) ⊆ Γ 2 (A) for any A ∈ F, then for any nonnegative random variables X and Y , we have The above propositions are important in the proof of our main theorem in Section 3. To prove them, we need the following lemmas.
n=1 is a sequence of nonatomic closed set-valued probability. By Lemma 2.11, there exist a measure P n1 and a probability P n2 , such that for any fixed n, P n1 (A) ≤ P n2 (A) and Π n (A) = [P n1 (A), P n2 (A)] for any A ∈ F. Therefore, by Definition 2.3, we have Lemma 2.13. Let Π : F → P([0, 1]) be a nonatomic closed set-valued probability. Then for any nonnegative random variable X, where P 1 and P 2 denote the same measure and probability deduced in Lemma 2.11.
Proof. By Lemma 2.11, Π(A) = [P 1 (A), P 2 (A)] for any A ∈ F. Since X is nonnegative and Lemma 2.14. Let (Γ, Γ) be a pair of upper-lower set-valued probabilities. Then for any nonnegative random variable X, denote the same sequences of measures and probabilities deduced in Lemma 2.12.
Proof. We will prove that for any nonnegative random variable X, ] is a convex set. In the following, we will prove this assertion in two steps.
Step 1. Assume X to be a simple nonnegative random variable, that is X = n i=1 a i I Ai , where n i=1 A i = Ω, A i A j = ∅ for any i = j, and a i ≥ 0 for i = 1, 2, · · · , n. Now we will prove that for any j, k ∈ N + , Without loss of generality, we assume that where I 1 and I 2 are index sets such that for any i ∈ I 1 we have P j2 (A i ) ≥ P k2 (A i ) and for any i ∈ I 2 , we have P j2 (A i ) < P k2 (A i ). It is obvious that I 1 I 2 = {1, 2, · · · , n} and I 1 I 2 = ∅. For On the other hand, by Lemma 2.12, we get for any A ∈ F, Consequently, we have By the arbitrariness of j and k, we have that ] is a convex set.
Step 2. Assume X to be a nonnegative random variable. Construct a sequence of simple nonnegative random variables: Then we get X m ≤ X, X m ↑ X and E Pni [X] = lim m→∞ E Pni [X m ] for i = 1, 2. By the same method in Step 1, we only need to prove that for any j, k ∈ N + , Without loss of generality, we assume that lim m→∞ E P k2 [X m ] ≤ lim m→∞ E Pj2 [X m ]. Then for any > 0, there exists a positive integer M , such that for any m > M , E P k2 [X m ] < E Pj2 [X] + . By Step 1, for any m, So it is obvious that for any j, k ∈ N + . This proof is complete. Now, we will prove Proposition 2.8 and Proposition 2.10.
Proof. We will only prove the second and third assertions in Proposition 2.8 and the third assertion in Proposition 2.10, since others can be easily proved.
Proposition 2.8 (2). By Lemma 2.14, since X and Y are nonnegative random variables, there exist a sequence of measures {P n1 } ∞ n=1 and a sequence of probabili- Since sup and inf Proposition 2.8 (3). By Lemma 2.14, since X and Y are nonnegative random variables, we have . Then for nonnegative random variables X and Y , we have Next, we will introduce the concepts of upper-lower probability and upper-lower expectation, which will be useful to prove the main theorem in next section. It is easy to check that V(A)+v(A c ) = 1, and V satisfies the following properties: Definition 2.16. Let X 1 , X 2 , · · · , X n+1 be real measurable random variables on (Ω, F). X n+1 is said to be independent of (X 1 , {X n } ∞ n=1 is said to be a sequence of independent random variables under E[·] if X n+1 is independent of (X 1 , X 2 , · · · , X n ) for each n ≥ 1. 3. Main results. Let (Ω, F, Γ) be an upper set-valued probability space. Then we define the concept of independent random variables with respect to E Γ as the extension of Definition 2.16.
Definition 3.1. Let X 1 , X 2 , · · · , X n+1 be real measurable random variables on (Ω, F). X n+1 is said to be independent of (X 1 , , where the product of sets A and B is defined by AB = {ab : a ∈ A, b ∈ B}. {X n } ∞ n=1 is said to be a sequence of independent random variables under E Γ [·] if X n+1 is independent of (X 1 , X 2 , · · · , X n ) for each n ≥ 1.
Remark 3.2. This kind of independence is asymmetric and directional. Namely, X is independent of Y dose not imply automatically that Y is independent of X.
Finally, we need another notation: if x ∈ R, and A ⊆ R, then almost surely with respect to Γ.
is a sequence of nonnegative random variables, then by Lemma 2.14, we obtain that is a sequence of measures and {P n2 } ∞ n=1 is a sequence of probabilities such that for any A ∈ F, are independent random variables under E. Define g(x) = max(x, 0).
If ϕ i is a nonnegative measurable function, then (ϕ i • g)(·) = ϕ i (g(·)) is also a nonnegative measurable function. Then by Definition 3.1, is a sequence of independent random variables under E Γ . Since Therefore, That is Similarly, we get At the beginning, we will introduce some concepts (see [22]) of fuzzy-set valued probability briefly.
Fuzzy sets are sets whose It is known that (F C (R), d) is complete (see [20]). The operations of fuzzy-sets are defined through their corresponding membership functions as follow. (2)Scalar multiplication:    (2). µ( The concept of f −probability is an extension of set-valued probability. Setvalued measures which are related to a f −probability can be defined as follow: if µ is a f −probability, then define For proofs that suppµ and µ α defined above are set-valued probabilities, see [21]. Proof. Since {µ α n } ∞ n=1 is a sequence of nonatomic closed set-valued probabilities and ∞ n=1 µ α n (A) is a convex set for any A ∈ F, by the Definition 2.3, U α (·) is an upper set-valued probability. Similarly, we can prove that U α (A) = ∞ n=1 µ α n (A). Then U α is a lower set-valued probability corresponding to U α . Now, we will give the concept of expectations under upper-lower f −probabilities. ∞ n=1 E µn [X]( denoted by E U [X])can be defined as the upper and lower expectations of random variable X respectively, that is where {E µn [X]} ∞ n=1 are fuzzy sets such that L α (E µn [X]) = E µ α n [X] for any α ∈ (0, 1].
Remark 4.8. We require that Ω |X|dsuppµ n < ∞ for the expectation to exist and the existence and uniqueness of E µn [X] are from [12]. Proof.
The penultimate equality is from the assumption that L α (E U [X]) is a closed set and the last equality is from the definition of expectation of upper set-valued probability.
{X n } ∞ n=1 is said to be a sequence of independent random variables under E U , if X n+1 is independent of (X 1 , X 2 , · · · , X n ) for each n ≥ 1.