Asian Journal of Mathematics & Statistics

Year: 2012 | Volume: 5 | Issue: 1 | Page No.: 1-20
DOI: 10.3923/ajms.2012.1.20

Characterization of Complex Integrable Lattice Functions and μ-Free Lattices

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

Abstract: This paper is a study on Complex integrable lattice functions and μ-free lattices. It initiates the concepts of complex integrable lattice function, positive and negative separations of μ and establishes the separation properties of complex integrable lattice functions. Also it introduce the concepts of free lattice, μ-free lattice and demonstrate that μ is a measure on β, μ is a free lattice in β, β is a μ-free lattice of σ (L). Also it was defined the concept almost free lattice and finally confirm that every almost free lattice is a complex integrable lattice function and a σ-additive.

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

INTRODUCTION

Hus (2000) has made a characterization of outer measures associated with lattice measures. Khare and Singh (2005) introduced the concept of weakly tight functions and their decomposition. Khurana (2008) introduced the concept of lattice-valued borel measures. Tanaka (2009) has established the Hahn decomposition theorem of signed lattice measure and introduced the concept of lattice σ-Algebra. Also Tanaka (2008) further established a Hahn decomposition theorem of signed fuzzy measure. Anil Kumar et al. (2011a) contributed on construction of a gamma lattice. Anil Kumar et al. (2011b) established radon-nikodym theorem and its uniqueness of signed lattice measure. Anil Kumar et al. (2011c) obtained Jordan decomposition and its uniqueness of signed lattice measure. Praroopa and Rao (2011a) established a lattice in pre A*-Algebra. Praroopa and Rao (2011b) obtained pre A*-Algebra as a semilattice. Rao and Satyanarayana (2010) made a semilattice structure on pre A*-Algebra. Rao and Kumar (2010) contributed the structure of weakly distributive and sectionally *semilattice. Rao and Praroopa (2011) obtained logic circuits and gates in pre A*-Algebra. Rao and Rao (2010) derived subdirect representations in A*-Algebras. Satyanarayana et al. (2011) obtained prime and maximal ideals of pre A*-Algebra. Recently Kumar et al. (2011) made a characterization of class of measurable borel lattices. Also, Kumar et al. (2011) introduced the concept of lattice boolean valued measurable functions, function lattice, σ-lattice and lattice measurable space.

This study establishes a general frame work for the study of characterization of complex integrable lattice functions and μ-free lattice. Here some concepts in measure theory can be generalized by means of lattice σ-Algebra.

It has been proved that every almost free lattice is a complex integrable lattice function and a σ-additive. Finally some basic elementary properties of complex integrable lattice functions have been obtained.

PRELIMINARIES

This section, briefly reviews the well-known facts about lattice theory specified by Birkhoff (1967). (L, Λ, ∨) 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 study, lattices will be considered 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;
(1)∀ h ∈ L, h^c ∈ L (2) if h_n ε L for n = 1, 2, 3 ....., then:

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

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

Definition 3: A complex positive lattice measure is a complex-valued countably additive lattice function define on a lattice σ-algebra σ(L).

Definition 4: A function lattice is a collection L^f of extended real valued functions defined on a lattice L^f with respect to usual partial ordering on functions. That is if f, g ε L^f then f ∨ g ε L^f, f ∧ g ε L^f.

Definition 5: If f and g are extended real valued lattice measurable functions defined on L^f, 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 ε L.

Definition 6: If f = u+iv, where, u and v are real lattice measurable functions on X, then f is a complex lattice measurable function on X.

Definition 7: Let f = u+iv is a complex lattice measurable function on X, then u, v and |f| are real lattice measurable functions on X.

Definition 8: Let f be a complex lattice measurable function on X, then |f| is a lattice measurable function from X→[0, ∞). If:

then we say that f is a complex integrable lattice function with respect to μ. The set of all complex integrable lattice functions with respect to μ on X is denoted by L¹.

Definition 9: Let f = u+iv, where, u and v are real lattice measurable functions on X. Let f ε L¹ then we define:

for every lattice measurable set E, where, u⁺ = max {u, 0}, u¯ = -min {u, 0} and v⁺ = max {v, 0}, v¯ = -min {v, 0}.

Note 2: u⁺, u¯ are called positive and negative separations of u. These are measurable, real and non negative.

Note 3: u⁺≤|u|≤|f| and similarly u¯, v⁺ and v¯ are all bounded by |f|.

Note 4: If f ε L¹, then:

and hence each of the four integrals on the right hand side of the definition 9 are finite.

Remark 1: Suppose f: X→(-∞, ∞) is a lattice measurable function then f⁺ and f¯ are lattice measurable functions from X→[0, ∞). Hence:

are defined. Define as provided at least one of the integrals and is finite.

Remark 2: If f ε L¹ and f = u+iv, where, u, v are real lattice measurable functions then:

Definition 10: A property is said to hold almost everywhere if the lattice of points where it fails to hold is a lattice of measure zero.

Definition 11: An extended real valued function f defined on a lattice measurable set E is said to be lattice measurable function if the set f^-1 [α, ∞) = {x ε E/f (x)>α} is a lattice measurable for every real number α.

Example 1: Let f and g be a lattice measurable functions, then {x ε E/f (x)≠g (x)} is lattice measurable set. Let μ ({x ε E/f (x)≠g (x)}) = 0 then we say that f = g almost every where with respect to μ on X. Here onwards write this as f ~ g.

Note 5: f ~ g is an equivalence relation.

Definition 12: Let E be a lattice. Then the complement of E is defined as E^c = {x ε E^c/x ∉ E}.

Definition 13: σ-lattice: countable union of lattice measurable sets.

Theorem 1: (Rudin, 1987). Let f_n: X→[0, ∞) is a lattice measurable function for n = 1, 2, 3 ….. and:

for x ε X then:

Result 1: (Rudin, 1987): If f≥0, c is a constant, 0≤c≤∞ then:

Result 2: (Rudin, 1987): The limit of every point wise converges sequence of complex measurable functions is measurable.

Result 3: (Rudin, 1987): If f and g are measurable within range (-∞, ∞) then so are max {f, g} and min {f, g}. In particular f⁺ = max {f, 0} and f¯ = -min {f, 0} are measurable.

Result 4: (Royden, 1981): If E is measurable set if and only if E^c is also measurable.

Remark 3: Let f, g be lattice measurable functions defined on X then {x ε E/f (x)≠g (x)} and {x ε E/f (x) = g (x)} are lattice measurable sets.

Proof: By (1) we have {x ε E/f (x)>g (x)} and {x ε E/f (x)<g (x)} are lattice measurable sets. Now {x ε E/f (x)≠g (x)} = {x ε E/f (x)>g (x)} ∨ {x ε E/f (x)<g (x)} is lattice measurable set (By definition 13). Also, {x ε E/f (x) = g (x)} = X-{x ε E/f (x)≠g (x)} = {x ε E/f (x)≠g (x)}^c is lattice measurable set (By result 4).

Note 5: 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. Also positive lattice measure is simply called lattice measure.

CHARACTERIZATION OF COMPLEX INTEGRABLE LATTICE FUNCTIONS

Theorem 1: Suppose f and g ε L¹ and α, β are complex numbers. Then αf+βg ε L¹ and:

Proof: Suppose f and g ε L¹. Then f and g are complex integrable lattice measurable functions on X and:

are finite. Hence, αf+βg is a complex lattice measurable function on X and:

(By theorem 1 and result 1). Thus αf+βg ε L¹.

Claim 1: Let f and g are real.

Then:

Let h = f+g then h⁺-h¯ = f⁺-f¯+g⁺-g¯. That is h⁺+f¯+g¯ = f⁺+g⁺-h¯. By theorem 1:

As f, g, f+g ε L¹ and each of these integral is finite we get:

implies:

implies:

implies:

Hence the claim.

Claim 2: Suppose f and g are complex. Let f = u₁+iv₁, g = u₂+iv₂ then u₁, v₁, u₂ and v₂ are real and are in L¹. Therefore:

(by claim 1):

Case 1: Let α≥0:

(Since α≥0).

Case 2: Let α = -1.

and:

Case 3: Let α = i. Let f = u+iv:

Case 4: Let α = a+ib and f ε L¹ then:

Therefore:

and:

are true for any f, g ε L¹ and α is any complex number. Hence:

Theorem 2: If f ε L¹, then:

Proof: Put:

then z is a complex number. If z = 0 then:

therefore:

but:

Hence:

So the result is true if z = 0.

Suppose z≠0, take:

then |α| = 1 and |z| = αz. Let u be a real part of αf, then u≤|αf| = |α||f| = |f| (Since |α| = 1). Therefore:

As and are real. Hence:

gives:

(where u is a imaginary part of αf. Therefore:

From Eq. 3 and 4 we get:

(Since u≤|f|).

Theorem 3: Suppose {f_n} is a sequence of complex lattice measurable functions on X such that f (x) = lim f_n (x) exists for every x ε X, if there is a function g ε L¹ such that |f_n (x)|≤g (x) where, n = 1, 2, 3, …… x ε X. Then:

Proof:
Part 1: by hypothesis, g: X→(0, ∞) and:

(Since |g| = g and g ε L¹ implies:

By Result 2. Since the limit of every pointwise convergent sequence of a complex lattice measurable function is lattice measurable function, we get f is lattice measurable function. Also since |f_n (x)|≤g (x) for all n, for all x ε X we get |f (x)|≤g (x) for all x ε X implies |f|≤g on X. Therefore:

Therefore f ε L¹.

Part 2: As |f_n|≤g, |f|≤g we get |f_n-f|≤2g. Apply Fatou’s lemma to the functions 2g-|f_n-f| since lim 2g-|f_n-f| = 2g, we get:

Since:

is finite, we may subtract it and get:

implies:

If a sequence of non negative real numbers fails to converges to 0 then its upper limit is positive therefore:

gives:

Part 3: As |f_n-f|≤2g, f_n-f ε L₁. Theorem 2 implies that:

Therefore:

Therefore:

implies:

implies:

Remark 4: Suppose f ~ g, then:

for every L ε σ (L).

Proof: Let N = {x ε L/f (x)≠g (x)}, then E = (E-N) ∨ (E ∧ N) (disjoint union) and μ (E ∧ N) = 0, f = g on E-N. Therefore:

Thus generally speaking lattices of measure zero are negligible in integration.

CHARACTERIZATION OF μ-FREE LATTICES

Definition 14: A lattice measure μ on a lattice σ-algebra σ (L) is called a free lattice if all the lattice measurable sets of measure zero are lattice measurable.

Definition 15: Let μ be a lattice measure on a lattice σ-algebra σ (L). A lattice σ-algebra β containing σ (L) is called μ-free lattice of σ (L) if μ is a lattice measure on β, μ is a free lattice on β and β is the smallest with respect to this property that β = {E<X/ there exists A, B ε σ (L) such that A<E<B and μ (B-A) = 0}.

Definition 16: Almost free lattice. Let {f_n} be a sequence of complex lattice measurable functions on X which converges almost everywhere only on a lattice measurable set E<X, then {f_n} converges and μ (E^c) = 0 where, μ is a free lattice on X.

Definition 17: Let E be a lattice and {f_n} be a sequence of lattice measurable functions defined on E. Say that {f_n} converges pointwise to f on E if f_n (x)→f (x) for all x ε E.

Definition 18: Almost everywhere converges. If there is a sub lattice measurable set B of E such that m(B) = 0 and f_n (x)→f(x) pointwise on E-B then we say that f_n(x)→f(x) almost everywhere on E.

Definition 19: The lattice measurable space (X, σ (L)) together with a lattice measure μ is called a lattice measure space and it is denoted by (X, σ (L) μ).

Theorem 5: Let (X, σ (L), μ) be a lattice measure space and let β be the collection of all E<X for which there exists A, B ε σ (L) such that A<E<B and μ (B-A) = 0. Define μ(E) = μ(A). Then β is a lattice σ-algebra, μ is a free lattice in β and μ is a lattice measure on β.

Proof:
Part 1: Let X ε σ(L) also X<X<X, μ (X-X) = μ (φ) = 0 hence X ε β ….. (5).
Let E ε β then there exists A, B ε σ(L) such that A<E<B and μ (B-A) = 0 therefore B^c<E^c<A^c and A^c-B^c = B-A. Since A, B ε σ(L) we get A^c, B^c ε σ(L) also μ (A^c-B^c) = μ (B-A) = 0. Hence E^c ε β …… (6).

Let E_i ε β for every i, 1≤i≤∞. Then there exists A_i, B_i ε σ(L) such that A_i<E_i<B_i and μ (B_i-A_i) = 0 for every i. Let:

then A<E<B and:

Since μ (B_i-A_i) = 0 for every i, we get:

and hence μ (B-A) = 0. Therefore E ε β that is if E_i ε β, then:

From (5) (6) and (7) β is a lattice σ-algebra.

Part 2: To prove μ is free lattice in β.
Let E ε β and μ(E) = 0. Let E₁<E, as E ε β there exists A, B ε σ (L) such that A<E<B and μ (B-A) = 0. Also μ (E) = μ (A)+μ (E-A) = μ (A) (Since μ (E-A)≤μ (B-A) = 0. But μ (E) = 0 therefore μ (A) = 0 also B = A ∨ (B-A) implies μ (B) = μ (A)+μ (B-A) = 0. Since φ<E₁<B and φ, B ε σ (L), μ (B-φ) = μ (B) = 0. Therefore, E₁ ε β. Therefore, μ is a free lattice on β.

Part 3: To prove μ is a lattice measure on β.
Define for any E ε β, μ (E) = μ (A) where A<E<B, A, B ε σ (L), μ (B-A) = 0. First, μ is well defined on β. For, suppose E ε β. Suppose further that there exists A₁, A₂, B₁, B₂ ε σ (L) such that A₁<E<B₁, A₂<E<B₂ and μ (B₁-A₁) = 0, μ (B₂-A₂) = 0. Then A₁-A₂<B₂-A₂. Therefore, μ (A₁-A₂) = 0, similarly μ (A₂-A₁) = 0. Hence μ (A₁) = μ(A₂ ∧ A₁) = μ (A₂). (Since A₁ = (A₁-A₂) ∨ (A₁ ∧ A₂), A₂ = (A₂-A₁) ∨ (A₂ ∧ A₁). Thus, we define μ (E) = μ (A₁) or μ (E) = μ (A₂) (Since μ (A₁) = μ (A₂)). Hence μ is well defined on β.

To show μ is countably additive on β. Let {E_i} be a disjoint countable collection of members of β. Then there exist A_i, B_i ε σ (L) such that A_i<E_i<B_i for all i and μ (B_i-A_i) = 0 also μ (E_i) = μ (A_i). Hence and Therefore:

Since E_i’s are disjoint and since A_i<B_i, A_i’s are disjoint. As A_i’s are in σ (L):

Therefore:

Therefore, μ is a lattice measure on β.

Theorem 6: β is the μ-free lattice of σ(L).

Proof: By part 2 and 3 of the theorem 5 we have μ is a free lattice on β and μ is a lattice measure on σ (L). Now we prove β is the smallest lattice σ-algebra containing σ (L) with respect to the property that β = {E<X/ there exists A, B ε σ (L) such that A<E<B and μ (B-A) = 0}. Let β₁>σ (L) and let μ be a free lattice on β₁. Let E ε β. Then there exists A, B ε σ (L) such that A<E<B and μ (B-A) = 0. Since β₁>σ (L), we have A, B ε β₁. Also E-A ε B-A and μ (B-A) = 0. Hence E-A ε β₁. As A ε β₁ and E-A ε β₁ we get E = A ∨ (E-A) ε β₁. Therefore β<β₁. Thus β is the smallest lattice σ-algebra containing σ (L) such that μ is a free lattice on β. That is β is the μ-free lattice of σ (L).

Theorem 7: Every almost free lattice is a complex integrable lattice function and a σ-additive.

Proof: Suppose {f_n} is sequence of complex lattice measurable functions defined almost everywhere on X such that:

and let:

converges for all x, we prove:

and:

Part 1: Let {f_n} is a complex lattice measurable functions defined almost everywhere on X. Let S_n<X be the lattice on which f_n is defined then Let:

implies:

implies μ (S^c) = 0 (Since for every n). If S = φ implies μ (S) = 0 and hence μ (X) = 0 which is not true (generally we assume that μ (X)>0, since otherwise the result becomes trivial). Hence S≠φ. Let:

for all x. By theorem 1:

(By hypothesis). Therefore:

Let E = {x ε S/φ (x)<∞}. Since:

we get μ (E^c) = 0. (If μ (E^c)>0 then:

Since:

we get for every x ε E:

Since:

we get:

for all x ε E. Since:

we get:

Therefore:

Hence, f ε L¹ (μ) on E.

Part 2: Let g_n = f₁+f₂+... +f_n then |g_n| is the partial sum of the series:

We get, |g_n|≤φ for all n and g_n (x) converges to f (x) for all x ε E. By theorem 4 we get:

Since μ (E^c) = 0 we get:

Remark 5: Suppose f_n were defined at every point of X and suppose:

then:

converges almost everywhere only.

Proof: As in the theorem 7 if we take:

for all x ε X then:

this implies φ(x)<∞ almost everywhere. Hence:

converges almost everywhere.

Theorem 8:

then f = 0 almost everywhere on E

for every E ε σ (L) then f = 0 almost everywhere on X

then there is a constant α such that αf = |f| almost everywhere on X

Proof:
Part 1: Let A_n = {x ε E/f (x)>1/n}, n = 1, 2, 3, …. then A_n’s are lattice measurable also:

Therefore:

(Since A_n<E). Therefore, μ (A_n) = 0. Now = {x ε E/f (x)>0} and This shows that μ ({x ε E/f (x)>0}) = 0 implies f (x) = 0 almost everywhere on E.

Part 2: Let f ε L¹ (μ) and let f = u+iv …. (8). Let E = {x ε X/u (x) = 0}. Then, E ε σ (L) (Since u is lattice measurable set):

(From (8), definition 9). Therefore real part of:

(Since u (x)≥0 for all x ε E. As:

(by hypothesis). Hence real part of:

implies:

Therefore by part 1 we have u⁺ = 0 almost everywhere on E but u⁺ = 0 on E^c hence u⁺ = 0 almost everywhere on X. In a similar way we can prove u¯, v⁺, v- are 0 almost everywhere on X. Hence f = 0 almost everywhere on X.

Part 3: Let f ε L¹ (μ) and let:

by theorem 3 we have:

where, u is the real part of αf. Since by hypothesis:

We get:

implies:

Since |f| -u≥0 (u≤|αf|= |f|). We get from part 1 |f|-u = 0 almost everywhere implies |f| = u almost everywhere implies |f| = real part of αf almost everywhere implies |αf| = real part of αf almost everywhere (Since |α| = 1). Hence imaginary part of αf = 0 almost everywhere. Therefore αf = |αf| = |f| almost everywhere on X.

CONCLUSION

This work establishes a wide-ranging outline for the study of characterization of complex integrable lattice functions and μ-free lattice. Here several concepts in measure theory can be generalized by means of lattice σ-Algebra. It has been established that every almost free lattice is a complex integrable lattice function and a σ-additive. Finally various basic elementary properties of complex integrable lattice functions have been achieved.

REFERENCES