Asian Journal of Applied Sciences

Year: 2012 | Volume: 5 | Issue: 1 | Page No.: 43-51
DOI: 10.3923/ajaps.2012.43.51

Characterization of Class of Positive Lattice Measurable Sets and Positive Lattice Measurable Functions

J. Pramada, J. Venkateswara Rao and D.V.S.R. Anil Kumar

Abstract: This paper is a study on positive lattice measurable sets and positive lattice measurable functions. It establishes the concepts of positive lattice measure, positive lattice measurable set, positive lattice measurable function, lattice Lebesgue integral, countable union of positive lattice measurable function, countable intersection of positive lattice measurable function and that these functions are lattice measurable. Also we have found that positive lattice measure satisfies first and second valuation theorems. Finally we confirm some basic integral properties of positive lattice measurable functions.

Keywords: measure, σ-algebra, measurable functions, Lattice and positive lattice

INTRODUCTION

In the recent past Royden (1981) has made an effort on the concept of function lattice. Tanaka (2009) has established a Decomposition Theorem of Signed Lattice Measure and the concept of lattice σ-Algebra σ(L). Recently, Anil Kumar et al. (2011) made a Characterization of Class of Measurable Borel Lattices. Also, Anil Kumar et al. (2011) introduced the concept of Lattice Boolean Valued Measurable functions.

In this study, we set up the general frame work for the study of the characterization of positive lattice measurable functions. Here some concepts in measure theory can be generalized by means of lattice σ-Algebra σ(L) defined on X. We establish the concepts of complex lattice measure, simple lattice function, lattice Lebesgue integral, countable union of positive lattice measurable function, countable intersection of positive lattice measurable function and prove that these functions are positive lattice measurable. Also, we establish positive lattice measure satisfies first and second valuation theorems. We prove every C_σ-lattice functions and every C_δ-lattice functions are positive lattice measurable. Finally we confirm some basic integral properties of positive lattice measurable functions.

PRELIMINARIES

In this section, we shall briefly review the well-known facts about lattice theory specified (Birkhoff, 1967).

(L, ∧, ∨) is called a lattice if it is enclosed under operations ∧ and ∨ and satisfies, for any elements x, y, z, in L:

A lattice L is called complete if, for any subset A of L, L contains the supremum ∨ A and the infimum ∧ A. If L is complete, then L itself includes the maximum and minimum elements which are often denoted by 1 and 0 or I and O, respectively.

A distributive lattice is called a Boolean lattice if for any element x in L, there exists a unique complement x^c such that:

Let L be a lattice and €: L → L be an operator. Then € is called a lattice complement in L if the following conditions are satisfied.

Throughout this paper, we consider lattices as complete lattices which obey (L1)-(L8) except for (L6) the law of non-contradiction.

Definition 1: Unless otherwise stated, X is the entire set and L is a lattice of any subsets of X. If a lattice L satisfies the following conditions, then it is called a lattice σ-Algebra:

We denote σ(L), as the lattice σ-Algebra generated by L and ordered pair (X, σ(L)) is said to be lattice measurable space.

Note 1: By Definition1, it is clear that σ(L) is closed under finite unions and finite intersections.

Definition 2: Let σ(L) be a lattice σ-algebra of sub sets of a set X. A function μ: σ(L)→ [0,∞ ] is called a positive lattice measure defined on σ(L) if :

where, {A_n} is a disjoint countable collection of members of σ(L) and μ(A) < ∞ for at least one A σ(L)

Definition 3: A lattice measurable space (X, σ(L)) together with a positive lattice measure defined on σ(L) is called a positive lattice measure space. It is denoted by (X, σ(L), μ).

Definition 4: If μ is a positive lattice measure on σ(L) then the numbers of σ(L) are called positive lattice measurable sets or simply positive lattice measurable.

Definition 5: A function f defined on a lattice σ-algebra σ(L) whose range is in [0, ∞] is called a positive lattice measurable function.

Definition 6: A function lattice is a collection L¹ of extended real valued functions defined on a lattice L with respect to usual partial ordering on functions. That is if f, g∈L¹ then f ∨ g∈L¹, f ∧ g∈L¹.

Definition 7: If f and g are extended real valued lattice measurable functions defined on L¹, then f ∨ g, f ∧ g are defined by (f ∨ g)(x) = sup {f(x), g(x)} and (f ∧ g)(x) = inf{f(x), g(x)} for any x∈L.

Definition 8: A complex positive lattice measure is a complex-valued countably additive positive lattice function define on a lattice σ-algebra σ(L).

Definition 9: A function s on a lattice measurable space X whose range consists of only finitely many points in [0, ∞] is called a simple lattice function.

Note 2: (Anil Kumar et al., 2011): Every simple lattice function is lattice measurable.

Example 1: (Anil Kumar et al., 2011): Every lattice step function is a simple lattice.

Definition 10: Let σ(L) be a lattice σ-algebra defined on a set X. Let μ be a positive lattice measure defined on σ(L). Let s be a simple lattice function on X of the form:

where, α₁, α₂,……α_n are the distinct values of s and A_i = {x ∈ X/s (x) = α_i} 1 ≤ i ≤ n. Let E σ(L) then we define, .

Definition 11: Let f: X → [0, ∞] be a positive lattice measurable function. Let E∈σ(L). Then is define as Sup , the supremum being taken over all simple lattice functions s such that 0≤s≤f. We now call is the lattice Lebesgue integral of f over E with respect to the positive lattice measure μ.

Definition 12: A countable union of positive lattice measurable sets is called a C_σ-lattice.

Definition 13: A countable intersection of positive lattice measurable sets is called a C_δ-lattice.

Definition 14: A countable union of positive lattice measurable functions is called a C_σ-lattice function.

Definition 15: A countable intersection of positive lattice measurable functions is called a C_δ-lattice function.

Note 3: Here, we define a positive lattice measure that is simply called as a lattice measure. The value ∞ is admissible for a positive lattice measure.

Note 4: If {a_n} and {b_n} are monotonic increasing sequences in [0, ∞) and if a_n → a, b_n → b, where 0≤a, b≤∞ then a_nb_n → ab.

Remark 1: Rudin (1987): Let {f_n} be a sequence of lattice measurable functions defined on a domain X and let lim f_n= f, then f is lattice measurable function.

Theorem 1: Rudin (1987): Let f: X → [0, ∞) be a lattice measurable function. Then there exists simple lattice measurable functions s_n on X such that (1) 0 ≤ s₁ ≤ s₂ ≤ ……. ≤ f (2) s_n (x) → f(x) as n → ∞ for every x∈X.

POSITIVE LATTICE MEASURABLE LATTICES AND POSITIVE LATTICE MEASURABLE FUNCTIONS

Definition 16: Let X be a non empty set. Let σ(L) = P(X) (where P(X) is the power set of X). Define μ: σ(L) > [0, ∞] by |E| = number of lattice measurable sets in E, if E is finite, ∞ if E is infinite. Then μ is a lattice measure on P(X) called the lattice counting measure on X.

Result 1: (Kumar et al., 2011): If E₁, E₂ ....... are pair wise disjoint lattice measurable sets and E = E_k , then E is lattice measurable (or) Every σ-lattice is measurable and also .

Result 2: (Kumar et al., 2011): First Valuation Theorem: Suppose that {E_k} is monotonic increasing sequence of lattice measurable sets and E = E_k then m (E) = .

Result 3: (Kumar et al., 2011): If E₁, E₂ ....... are lattice measurable sets then E_k is measurable lattice (or) every δ-lattice is measurable.

Result 4: (Kumar et al., 2011) Second Valuation Theorem: Suppose that {E_k} is a monotonic decreasing sequence of lattice measurable sets and E = E_k then (E) = .

Result 5: If E₁, E₂ ....... are pair wise disjoint lattice measurable sets and E = E_k then m(E) = lim m(E_k).

Proof: Evidently this result is proved by using result 1 and 2.

Note 5: m(E) = means m(E_n) → m(E)

Theorem 2: Let μ be a positive lattice measure defined on a lattice σ-algebra σ(L). Then the following hold (1) (μ) = 0 (2) μ(A₁∨A₂ …….. ∨A_n) = μ(A₁)+μ(A₂)+………μ(A_n) where A₁, A₂, …….. A_n are pairwise disjoint lattice measurable sets (This property is called finite additivity).(3) If A, B are lattice measurable sets such that A < B then μ(A) ≤ μ(B) (This property is called monotonicity).

Proof: Part(1): Since μ is a positive lattice measure, there exists an A ∈ σ(L) such μ(A) < ∞. Let A₁ = A_1, A₂ = A₃= ………= φ then that is Since, μ(A) < ∞, we get . Since, μ(φ) [0, ∞], is possible if and only if μ(φ) = 0.

Part (2): Let A₁,A₂,……..A_n be a pair wise disjoint lattice measurable sets. Take A_n+1 = A_n+2 =………= φ. Then = gives μ(A₁A₂……..A_n) = μ(A₁) +μ(A₂) +………μ(A_n).

Part (3): Let A < B, then B = A ∨ (B-A) and A ∧ (B-A) = ø also B-A = B ∧ A^c ∈ σ(L). Hence μ(B) = μ(A)+μ(B-A) ≥ μ(A) (since μ(B-A) ≥ 0).

Theorem 3: Let μ be a positive lattice measure defined on a lattice σ-algebra σ(L). Then μ satisfies first valuation theorem (Result 2) and second valuation theorem (Result 3.4.) that is:

(1) Let A = , A_n ∈ σ(L). Let A₁<A₂…….. Then μ(A_n) → μ(A) as n → ∞.

(2) If A= A_n, A_n ∈ σ(L) and A₁>A₂….. with μ (A₁) finite. Then μ (A_n)→ μ (A) as n →.

Part (1): Let A₁ < A₂………B₁ = A_1, B₂ = A₂ - A₁,………B_n = A_n-A_n-1 then B_n = A_n ∧ A_n-1 ε σ(L). B_i ∧ B_j = ø if i ≠ j for x ε B_i ∧ B_j implies x ε A_{i ,} x ∉ A_i-i and x ε A_j , x A_j-i since i ≠ j, assume i < j(similar prove holds if j < i). Then x ∉ A_j-i implies x ∉ A_j (since i<j A_j<A_j-i). Hence x ε A_j , x ∉ A_j a contradiction. Therefore B_i ∧ B_j =φ . Also A_n = B₁ ∨ B₂……….∨ B_n for B₁ ∨ B₂ = A₁ ∨(A2-A₁) = A₂ (since A₁<A₂). For (B₁ B₂) ∨ B₃ = A₂ ∨(A₃-A₂) = A₃ ∨(since A₂<A₃). Hence by induction, B₁ ∨ B₂………∨ B_n = A_n. Therefore, A = A_n= B_n. Hence μ(A_n) = (since B_i^s are pairwise disjoint lattice measurable sets). Therefore μ(A) = μ B_n = = (μ(B₁) + μ(B₂) +………+μ(B_n)) = μ(A_n) that is μ(A_n) → μ(A) as n →∞.

Part (2): Let A₁>A₂…….. Let C_n = A₁ - A_n. Then C₁ < C₂ <……… for x ∈ C_n-1 implies x A₁-A_n-1 implies x ∈ A₁ , x ∉ A_n-1 implies x ∈ A₁, x ∉ A_n (since A_n-1 > A_n ) implies x ∈ C_n. μ(C_n) = μ(A₁ - A_n) = μ(A₁) -μ(A_n) (For: A₁ = A_n ∨ (A₁-A_n) and A_n ∧ (A₁ - A_n) = ø therefore μ(A₁) = μ(A_n)+μ(A₁-A_n) as μ(A₁) is finite and μ(A_n) ≤ μ(A₁) we get that μ(A_n) is finite. Hence, μ(A₁-A_n) = μ(A₁)-μ(A_n)).
Also C_n= A₁- A_n for C_n = A₁-A_n < A₁ for all n x ∈ implies x ∈ C_n for some n implies x ∈ A₁, x ∉ A_n implies x ∈ A₁ , x ∉ " A_n implies x ∈ A₁ - A_n . Similarly, x ∈ A₁ - A_n . Implies x∈A₁, x ∉ A_n implies x ∈ A₁, x ∉ A_n for some n implies x ∈ A₁-A_n implies x ∈ C_n. Hence C_n= A₁ - A_n = A₁-A. Now A < A₁, A₁ = A (A₁ - A) implies μ(A₁) = μ(A) +μ (A₁-A). Implies μ (A₁-A) = μ(A₁) -μ(A) (since μ(A₁) ,μ(A) are finite ). Hence, μ(A₁) - μ(A) = μ (A₁ - A) = = C_n =(μ(A₁) - μ(A_n)) = μ(A₁) - (A_n) since μ(A₁) < ∞, we get μ(A_n) = μ(A) that is μ(A_n) → μ(A) as n → ∞.

Result 6: The condition μ(A₁) < ∞ in the above theorem can not be dropped.

Proof: Consider the set of natural numbers N. Let μ be the lattice counting measure on N (by definition 1). Let A_n = {n, n+1, n+2,…….}. Then A₁>A₂…….. A = {1, 2, 3,……..} = N and so μ(A₁) = ∞. Also μ(A_n) = ∞ for all n. Hence,- μ(A_n) = ∞. But A_n = ø and so μ( A_n) = μ(ø) = 0. That is μ (A_n) does not converge to μ(A) where A = A_n.

Theorem 4: Every Cσ- lattice function is lattice measurable also every Cδ-lattice function is lattice measurable.

Proof: Let f, g are positive lattice functions from X →[0, ∞]. Then (by theorem 1.) there exists simple lattice measurable functions s_n, t_n on X such that 0 ≤ s₁ ≤ s₂ ≤……..≤ f, 0 ≤ t₁ ≤ t₂ ≤………≤ g, such that s_n(x)→ f(x), t_n(x)→ g(x) for every x∈X. Hence s_n(x) ∨ t_n(x)→ f(x) ∨ g(x) and s_n(x) ∧ t_n(x) → f(x) ∧ g(x) for every x∈X (by note 4). Hence f ∨ g = lim sup (s_n ∨ t_n) and f ∧ g = lim inf (s_n ∧ t_n). Hence, f ∨ g and f ∧ g are lattice measurable (remark 1). By induction we have every Cσ-lattice function, every Cδ-lattice function are lattice measurable. Cσ-lattice function is lattice measurable also every Cδ-lattice function is lattice measurable.

Remark 1: Let f be a simple lattice measurable function. Let then we have apparently two conditions for integral of f namely = and and = where s is a simple lattice measurable function. But these two assign the same value to the integral.

Proof: Let s be a simple lattice measurable function less than or equal to f. Let . Let E_ij = A_i ∧ B_j where 1 ≤ i ≤ n, 1 ≤ j ≤ m. For any x∈E_ij, s(x) = β_j ≤ f(x) = α_i (since x∈A_i) also E_ij are disjoint lattice measurable sets whose union is X. And B_j = B_j ∧ X = B_j ∧ = (B_j ∧ E_ij) = E_ij = (since B_j ∧ E_ij = E_ij and B_k ∧ C_ij = ø if k ≠ j) similarly, A_i = E_ij now μ (B_j ∧ E) = = = = = {β₁ μ (E_i1 E)+ β₂ μ (E_i2 E)……………….+ β_m μ (E_im E)} ≤ {α_i μ (E_i1 ∧ E)+ α_i μ (E_i2 ∧ E)……………….+ α_i μ (E_im ∧ E)} (since β₁ ≤ α_i in E_i1, β₂ ≤ α_i E_i2 etc ). = (since A_i = E_ij) Hence = μ(A_i E). Therefore = μ(A_i E). But f is itself simple lattice measurable function and hence μ(A_i E) =. Hence, ≤ μ(A_i ∧ E). Therefore μ(A_i ∧ E).But f is itself simple lattice measurable function and hence μ(A_i ∧ E)≤ = . Hence, μ(A_i ∧ E) = (according to definition 10.) But L.H.S = (according to definitions 11). Hence, these two definitions assign the value to the integral.

Theorem 5: Let A, B, E be positive lattice measurable sets and f, g are positive lattice measurable functions. Then the following are true:

Proof: Part(1): = = (s is simple lattice measurable function). Since f ≤ g, s ≤ f implies s ≤ g hence ≤ that is

Part (2): Let 0 ≤ s ≤ f, (s is a simple lattice measurable function).

Let s = Thus (since A < B implies A_i ∧ A < A_i B implies μ(A_i ∧ A) = μ(A_i ∧ B)) = .

Hence,

Part (3): Let f ≤0, c a constant, 0 ≤ c ≤ ∞ . Let 0 ≤ s ≤ f (s is a simple lattice measurable function). Let s = . Then cs = . Therefore μ (A_i ∧ E) = . Therefore = = (since c ≥ 0). To each 0 ≤ s ≤ f, there corresponds 0 ≤ ≤ f where is a simple lattice measurable function. Hence . Therefore = .

Part (4): Suppose f(x) = 0 for all x ∈ E. Then for any 0 ≤ s ≤ f, s is a simple lattice measurable function, s(x) = 0 for all x ∈ E μ (A_i ∧ E) since s(x) = α_i in A_i, s(x) = 0 in E, A_i ∧ E = ø for all i. Therefore μ(A_i ∧ E) = μ(ø) = 0 for all i. Therefore = 0. This is true for every s, such that 0 ≤ s ≤ f, s is a simple lattice measurable function. Therefore = 0 that is = 0. Note that the above result does not depend on μ(E). Hence, even if μ(E) = ∞, the above result is true.

Part (5): Let μE = 0. Let s be a simple lattice measurable function given by s = . Then = = = 0 (since μ(E) = 0 implies μ(E A_i) = 0 fo all i). Therefore = 0 that is = 0. Here, the result does not depend on the value f(x) with x E. Hence, even if f(x) = for all x ∈ E, the above result is true.

Part (6): Let f ≥ 0. Let s be any simple lattice measurable function with, 0 ≤ s ≤ f. Let s = . Then μ(E ∧ A_i)XE is a simple lattice measurable function with values α_i at E ∧ A_i, i = 1 to n. Hence . Therefore . That is .

Part (7): Let E ∈ M and let E be the disjoint union of positive lattice measurable sets E₁ and E₂. Then = .

Remark 2: The last result shows that we could have restricted our definition of integration to integrals over all of X without loosing any generality. If we want to integrate over lattice measurable sets we could use part(6) namely fdμ as the definition.

REFERENCES