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Research Article
 

Characterization of Class of Super Lattice Measurable Sets



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

This study is an investigation on super lattice measurable sets. It characterizes super lattice, super lattice measurable set, elementary lattice, monotone class and establishes that the union, intersection, difference of two super lattice measurable sets is a super lattice measurable set. Also it ascertains that the class of elementary lattice is closed under union, intersection and difference. Finally, it confirms that the product of lattice σ- algebra is the smallest monotone class contains all elementary lattices.

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J. Pramada, J. Venkateswara Rao and D.V.S.R. Anil Kumar, 2011. Characterization of Class of Super Lattice Measurable Sets. Journal of Applied Sciences, 11: 3525-3529.

DOI: 10.3923/jas.2011.3525.3529

URL: https://scialert.net/abstract/?doi=jas.2011.3525.3529
 
Received: July 18, 2011; Accepted: October 01, 2011; Published: December 12, 2011

INTRODUCTION

Gabor (1964) has introduced the concept of product lattice. In the recent past (Royden, 1981) has made an effort on the concept of function lattice. Tanaka (2009) has established a Hahn Decomposition Theorem of Signed Lattice Measure and introduced the concept of lattice σ-algebra σ(L). Recently, Anil Kumar et al. (2011) made a categorization of Class of Measurable Borel Lattices. Also, Anil Kumar et al. (2011) made an investigation on Lattice Boolean Valued Measurable functions and defined the concepts of lattice measurable space, lattice measurable set, σ-lattice and δ-lattice.

In this study we establish the general frame work for the study of the characterization of super lattice measurable sets. Here some concepts in measure theory can be generalized by means of a lattice σ-algebra σ(L). We establish super lattice, super lattice measurable set, elementary lattice, monotone class. We studied the characterization of super lattice measurable sets.

In this study, we establish union, intersection, difference of two super lattice measurable sets. Also we establish that class of elementary lattice is closed under union, intersection and difference. Finally we confirm that the product lattice σ-algebra is the smallest monotone class contains all elementary lattices.

PRELIMINARIES

Here, we shall briefly review the well-known facts about lattice theory specified by Birkhoff (1967).

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

(L1) commutative law: x∧y = y∧x and x∨y = y∨x
(L2) associative law: x∧ (y∧z) = (x∧y) ∧z and x∨ (y∨z) = (x∨y) ∨z
(L3) absorption law: x∨ (y∧x) = x and ∧ (y∨x) = x. Hereafter, the lattice (L, ∧, ∨) will often be written as L for simplicity. A lattice (L, ∧, ∨) is called distributive if, for any x, y, z, in L
(L4) distributive law holds: x∨ (y∧z) = (x∨y) ∧ (x∨z) and x∧ (y∨z) = (x∧y) ∨ (x∧z)

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 xc such that:

(L5) the law of excluded middle: x∨xc = 1
(L6) the law of non-contradiction: x∧xc = 0

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.

(L5) and (L6): ∀x∈L, x∨xc = 1 and x∧xc = 0
(L7) the law of contrapositive: ∀x, y∈L, x<y implies xc >yc
(L8) the law of double negation: ∀x∈L, (xc)c = x

Throughout this paper, we consider lattices as complete lattices which obey (L1), (L8) except for (L6) the law of non-contradiction. Unless otherwise stated, X is the entire set and L is a lattice of any subsets of X.

Definition 1: If a lattice L satisfies the following conditions, then it is called a lattice σ-Algebra:

∀ h∈, hc L
If hn∈L for n = 1, 2, 3....., then hn∈L

We denote σ(L), as the lattice σ-Algebra generated by L.

Note 1: By definition 1, 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, {An} is a disjoint countable collection of members of σ(L) and μ(A)<∞ for at least one A∈ σ(L).

Definition 3: The ordered pair (X, σ(L)) is said to be lattice measurable space.

Definition 4: A lattice A is said to be lattice measurable set if A belongs to σ(L).

Definition 5: A function lattice is a collection L1 of extended real valued functions defined on a lattice L1 with respect to usual partial ordering on functions. That is if f, g∈L1 then f∨g∈L1, f∨g∈L1.

Definition 6: If f and g are extended real valued lattice measurable functions defined on Lf, 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 7: Let E be a lattice then the complement of E is defined as Ec = {x∈Ec/x∉E}.

Note 2: (Ec)c = E.

Definition 8: A countable union of lattice measurable sets is called a σ-lattice.

Definition 9: A countable intersection of lattice measurable sets is called a δ-lattice.

Definition 10: Let X and Y be two lattices then their Cartesian product denoted by XxY is defined as XxY = {(x, y)/x∈X, y∈Y}.

Definition 11: If A<X, B<Y then AxB<XxY. Any lattice of the form AxB is called super lattice in XxY.

Remark 1 (Rudin, 1987): Let (X, S), (Y, T) be lattice measurable spaces. Then S is a lattice σ-algebra in X and T is a lattice σ-algebra in Y.

Definition 12: If A∈S and B∈T then the lattice of the form AxB is called super lattice measurable set.

Definition 13: If Q =R1∨R2∨...∨Rn where each Ri is a super lattice measurable set and Ri∧Rj = φ for i≠j, then Q is called elementary lattice. The class of all elementary lattices is denoted by LE.

Definition 14: SxT is defined to be smallest lattice σ- algebra in XxY which contains every super lattice measurable set.

Definition 15: If Ai, Bi∈ σ(L) such that Ai<Ai+1, Bi>Bi+1 for i = 1, 2, 3, …. and

then A∈ σ(L) and B∈ σ(L), this lattice σ-algebra σ(L) is a monotone class.

Example 1: XxY is a monotone class.

Definition 16: Let E<XxY where x∈X, y∈Y, we define x- section lattice of E by Ex = {y/ (x, y)∈E} and y-section lattice of Ey = {x/(x, y)∈E}.

Note 3: Ex<Y and Ey<X.

CHARACTERIZATION OF CLASS OF SUPER LATTICE MEASURABLE SETS

Result 1: The union of two super lattice measurable sets is a super lattice measurable set.

Proof: Let (A1xB1), (A2xB2) be two super lattice measurable sets, clearly A1, A2∈S implies A1-A2∈S, A1∧ A2∈S and A1∨A2∈S (Since S is a lattice σ-algebra). Also B1, B2∈T implies B1-B2∈T, B1∧B2∈T and B1∨B2∈T (Since T is a lattice σ-algebra). Now (A1xB1)∨(A2xB2) = (A1∨A2)x(B1∨B2). By definition of σ-lattice (A1∨A2), (B1∨B2) are lattice measurable sets implies (A1∨A2)x(B1∨B2) is a super lattice measurable set. Therefore (A1xB1)∨(A2xB2) is a super lattice measurable set.

Result 2: The intersection of two super lattice measurable sets is a super lattice measurable set.

Proof: Let (A1xB1), (A2xB2) be two super lattice measurable sets, clearly A1, A2∈S implies A1-A2∈S, A1∧ A2∈S and A1∧A2∈S (Since S is a lattice σ- algebra). Also B1, B2∈T implies B1-B2∈T, B1∧B2∈T and B1∨B2∈T (Since T is a lattice σ- algebra). Now (A1xB1)∧(A2xB2) = (A1∧A2) x(B1∧B2). By definition of δ- lattice (A1∧A2), (B1∧B2) are lattice measurable sets implies (A1∧A2)x(B1∧B2) is a super lattice measurable set. Therefore (A1xB1)∧(A2xB2) is a super lattice measurable set.

Result 3: The difference of two super lattice measurable sets is a super lattice measurable set.

Proof: Let (A1xB1), (A2xB2) be two super lattice measurable sets, clearly A1, A2∈S implies A1-A2∈S, A1∧ A2∈S and A1∨A2∈S (Since S is a lattice σ- algebra). Also B1, B2∈T implies B1-B2∈T, B1∧B2∈T and B1∨B2∈T (Since T is a lattice σ-algebra) implies (A1-A2)xB1 is a super lattice measurable set. (A1∧A2)x( B1-B2) is a super lattice measurable set implies ((A1-A2)xB1)∨((A1∧A2)x( B1-B2)) is a super lattice measurable set (By definition of σ-lattice) implies (A1xB1)-(A2xB2) is a super lattice measurable set.

Theorem 1: If E∈S xT, then Ex∈T and Ey∈S for every x∈ X and y∈Y.

Proof: Let K be the class of all E∈SxT such that Ex∈T for every x∈X that is K = {E∈SxT/ Ex∈T for every x∈X}. Let F = AxB be a super lattice measurable set that is A∈S, B∈T. Also Fx = B if x∈A and Fx = φ if x∉A implies Fx∈T for every x∈X. Therefore F∈K. That is every super lattice measurable set belongs to K. In particular XxY∈K.

Let E∈K. Then y∈(Ec)x if and only if (x, y)∈Ec if and only if (x, y)∉E if and only if y∉Ex if and only if y∈(Ex)c. Therefore (Ec)x = (Ex)c. Since Ex∈T and since T is a lattice σ- algebra we have Ecx∈T. Therefore Ec∈K.

Let Ei∈K (i = 1, 2, 3, ….) and let:

then Y∈Ex if and only if (x, y)∈E if and only if (x, y)∈Ec for some i if and only if y∈(Ei)x. Therefore:

Since T is a lattice σ- algebra, (Ei)x∈T implies Ex∈T. Therefore E∈T.

From

K is a lattice σ-algebra. Since K<SxT we get K = SxT. Hence for any E∈SxT, Ex∈T for every x∈X. In a similar way we can prove Ey∈S for every y∈Y.

Lemma 1: To prove SxT is a monotone class.

Proof: Let Ai, Bi∈SxT, Ai<Ai+1, Bi>Bi+1 for i = 1, 2, 3, …. and:

Since SxT is a lattice σ- algebra implies A∈SxT and:

implies B∈SxT. Since SxT is a lattice σ- algebra. Therefore SxT is a monotone class.

Lemma 2: To prove LE is closed under intersection, difference and unions.

Proof: Let A1xB1, A2xB2 be two super lattice measurable sets. Clearly (A1xB1)∧(A2xB2) = (A1∧A2)x(B1∧B2), we get (A1∧A2)x(B1∧B2) is super lattice measurable set (Since by definition of δ-lattice and every δ-lattice is lattice measurable). Also (A1xB1)-(A2xB2) = ((A1-A2)x B1)∨((A1 ∧A2)x(B1-B2)), we get the difference of two super lattice measurable sets is a union of two disjoint super lattice measurable sets. (Since by σ-lattice and every σ-lattice is lattice measurable). [Note that A1, A2∈S implies A1-A2∈S and A1∧A2∈S since S is a lattice σ-algebra. Also B1-B2∈T, B1∈T implies (A1-A2)xB1, (A1∧A2)x(B1-B2) are super lattice measurable sets and they are disjoint since (A1∧A2)∧(A1- A2) = φ]. Hence (A1xB1)-(A2xB2)∈LE.

Part 1: Closed under intersection. Let P, Q∈LE implies P = R1∨R2∨.....∨Rn, Q = R11∨R12∨.....∨R1m where Ri∧Rj = φ for i≠j, R1i∧R1j = φ for i≠j and Ri’s∧Rj’s are super lattice measurable sets (By result3..). Now P∧Q = (R1∧R11)∨(R1∧R12)∨….∨(R1∧R1m)∨.....∨(Rn∧R11)∨......∨(Rn ∧R1m). Here each Ri∧Rj is super lattice measurable set (By result 3.2) and clearly these are disjoint. Therefore P∧Q∈ LE.

Part 2: Closed under difference. Now P-Q = P∧Qc = (R1∨R2∨......∨Rn)∧(R11∨R12∨......∨R1m)c = (R1∨R2∨......∨Rn)∧(R1c1∧R1c2∧.....∧R1cm) = (R1∧(R1c1∧R1c2∧.....∧R1cm))∨R2∧ (R1c1∧R1c2∧......∧R1cm)∨......∨Rn∧(R1c1∧R1c2∧......∧R1cm)) = {([(R1-R11)-R12]…..)-R1m}∨.....∨{([(Rn-R11)-R12]......)-R1m}.

Since the difference of two super lattice measurable sets is the disjoint union of two super lattice measurable set (By result 3), we get right hand side is the disjoint union of super lattice measurable sets. Hence P-Q∈LE.

Part 3: Closed under union. Now P∨Q = (P-Q)∨Q and (P -Q)∧Q = φ, we get P∨Q∈LE. Therefore LE is closed under intersection, difference and unions.

Theorem 2: SxT is the smallest monotone class which contains all elementary lattices.

Proof: Let σ(L) be the smallest monotone class which contains LE. This can be exists, let M be the family of all monotone class containing LE. Since XxY∈M implies M is non-empty. Let F be the intersection of all members of M that is F = ∧M. Then LE<F. We prove F is monotone class. Let Ai. Bi∈F, Ai<Ai+1, Bi>Bi+1 for i = 1, 2, 3, …. and

Then Ai , Bi belongs to every member of M and since every member of M is a monotone class, A, B belongs to every member of M therefore A, B∈F. Therefore F is a monotone class. By lemma1, SxT is a monotone class also it is obvious that LE<SxT and since σ(L) be the smallest monotone class which contains LE we get L:

(1)

Now for any lattice P<XxY define K(P) = {Q<XxY/P-Q∈ σ(L), Q-P∈ σ(L) and P∨Q∈ σ(L)}. Clearly Q∈K(P) iff P-Q, Q -P, P∨Q∈ σ(L) iff Q-P, P-Q, Q∨P∈ σ(L) iff P∈K(Q) that I s:

(2)

Let Ai, Bi∈K(P), Ai<Ai+1, Bi>Bi+1 for i = 1, 2, 3, …. and:

then Ai-P, P-Ai, P∨Ai, Bi-P, P-Bi, P∨Bi all belongs to σ(L) for every i, also:

(3)

(Since σ(L) is a monotone class).

(4)

Since P-Ai∈ σ(L) and P-Ai>P-Ai+1 we get:

(5)

(Since σ(L) is a monotone class) implies P-A∈ σ(L). Again:

(6)

(Since each P∨Ai∈ σ(L) and σ(L) is a monotone class). Hence A∈K(P). Similarly B∈K(P). Therefore K(P) is a monotone class. Fix P∈LE. Let Q be any lattice of LE. Then P-Q, Q-P, P∨Q∈ LE (By lemma2). Hence P-Q, Q-P, P∨Q∈ σ(L) (Since LE< σ(L). Therefore Q∈K(P) that is Q∈K(P) for every Q∈LE that is LE <K(P) and by Eq. 3 K(P) is a monotone class. But σ(L) is the smallest monotone class containing LE. Therefore σ(L)<K(P). Again fix Q∈ σ(L) for any P∈LE. By Eq. 5 we have Q∈K(P). Therefore by Eq. 2 P∈K(Q). Therefore by Eq. 4 LE∈K(Q) and hence by Eq. 5 σ(L)<K(Q). Thus σ(L)<K(Q) for every Q∈ σ(L). Let P, Q∈ σ(L) then Q∈K(P). Therefore Q-P, P-Q, P∨Q∈ σ(L). Therefore σ(L) is closed under difference and since:

Let Q∈ σ(L) then Qc = (XxY)-Q∈ σ(L). By Eq. 6. Let Pi∈ σ(L) for i = 1, 2, 3, ….. Let P = ∨Pi and Qn = P1∨P2∨....∨Pn. Since by Eq. 6 σ(L) is closed under finite union Qn∈ σ(L). Since Qn<Qn+1 and P = ∨Qn, the monotonicity of σ(L) shows that P∈ σ(L). Thus σ(L) is a lattice σ-algebra. Also LE< σ(L) ∈S xT. But SxT is the smallest lattice σ-algebra in XxY contains every super lattice measurable set. Here σ(L) is a lattice σ-algebra containing LE and hence every super lattice measurable set. Therefore σ(L) = SxT.

REFERENCES
Birkhoff, G.D., 1967. Lattice Theory. 3rd Edn., American Mathematical Society, Colloguim Publications, Rhode Island, New Delhi..

Gabor, S., 1964. Introduction to Lattice Theory. Academic Press, New York and London.

Kumar, D.V.S.R.A., J. Venkateswara Rao and E.S.R. Ravi Kumar, 2011. Characterization of class of measurable borel lattices. Int. J. Contemp. Math. Sci., 6: 439-446.
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Royden. H.L., 1981. Real Analysis. 3rd Edn., Macmillan Publishing, New York..

Rudin, W., 1987. Real and Complex Analysis. 3rd Edn., McGraw-Hill, UK.

Tanaka, J., 2009. Hahn decomposition theorem of signed lattice measure. arXiv:0906.0147v1, Cornell University Library. http://arxiv.org/abs/0906.0147.

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