INTRODUCTION
The concept of lattice measure was initiated by Gabor (1964).
For the extension of this theme literature took some time. Afterward, the concepts
of lattice sigma algebra and lattice measure on a lattice sigma algebra launched
by Tanaka (2009). Recently the concepts of lattice measurable
set, lattice measure space and lattice σfinite measure were established
by Kumar et al. (2011a, b).
The perception of measurable Borel lattices was introduced and studied by Kumar
et al. (2011a). Further RadonNikodym theorem for signed lattice
measure was expanded by Kumar et al. (2011a).
A class of super lattice measurable sets was introduced by Pramada
et al. (2011). Lebesgue decomposition and its uniqueness of a signed
lattice measure were studied successfully by Kumar et al.
(2012). A class of positive lattice measurable sets and positive lattice
measurable functions was obtained by Pramada et al.
(2012a). A characterization of complex integrable lattice functions and
μfree lattices was made by Pramada et al. (2012c).
Further recently a characterization of boolean valued star and mega lattice
functions was obtained by Pramada et al. (2012b).
This manuscript is aimed to the study of concept of product lattice measurable
functions and their various characterizations. In particular these functions
are observed by defined over topological spaces. Also it has been investigated
the characteristics of Slattice measurable function and Tlattice measurable
functions. The concept of iterated integral of a product lattice measurable
function has been defined in order to identify that the two iterated integrals
of a product lattice measurable function are finite and equal. It is also aimed
to get a condition that product lattice measurable function is lattice measurable
is obtained and the condition that product lattice measurable function is lattice
measurable cannot be dropped.
PRELIMINARIES
This section briefly reviews the wellknown facts of Birkhoff
(1967) lattice theory.
The system (L,
, ∨), where L is a non empty set, ∧ and ∨ are two binary operations
on L, is called a lattice if ∧ 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 x∧(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 x^{c} such that:
x∨x^{c} = 1(L5) 
the law of excluded middle 
x∧x^{c} = 0(L6) 
the law of noncontradiction 
Let L be a lattice and c: L→L be an operator. Then c is called a lattice
complement in L if the following conditions are satisfied.
(L5) and (L6): 
∀x∈L, x∨x^{c} = 1 and x∧x^{c}
= 0 
(L7) the law of contrapositive: 
∀x, y∈L, x<y implies x^{c}>y^{c} 
(L8) the law of double negation: 
∀x∈L(x^{c})^{c} = x 
Definition 1: If a lattice L satisfies the following conditions, then
it is called a lattice salgebra:
(1) 
for all h∈L, h^{c}∈L 
(2) 
if h_{n} ∈ L for n = 1, 2, 3 ....., then: 
We denote σ(L) = ß, as the lattice σalgebra generated by L.
Example 1 (Halmos, 1974): 1: {Φ, X}(that
is the empty set together with entire set) is a lattice σalgebra. 2: P
(X) power set of any nonempty set X is a lattice σalgebra.
Example 2: Let X =
and L = {measurable subsets of }
with usual ordering (≤).
Here, L is a lattice and σ(L) = ß is a lattice σalgebra generated
by L, where,
is a extended real number system.
Example 3: Let X be any nonempty set, L = {All topologies on X}. Here,
L is a complete lattice but not a σalgebra.
Example 4 (Halmos, 1974): Let X =
and L = {E⊂/E
is finite or E^{c} is finite}.
Here L is lattice algebra but not lattice σalgebra.
Definition 2: The entire set X together with a lattice σalgebra
ß is said to be lattice measurable space, it is denoted by the ordered
pair (,
ß).
Example 5: Let X =
and L = {All Lebesgue measurable sub sets of }.
Then it can be verified that (,
ß) is a lattice measurable space.
Definition 3: If the mapping μ: ß→R∪{∞} satisfies
the following properties, then μ is called a lattice measure on the lattice
σalgebra σ(L).
(1) 
μ(φ) = μ(0) = 0 
(2) 
For all h, g∈ß, such that μ(h), μ(g)≥0 and h≤g⇒μ(h)≤μ(g) 
(3) 
For all h, g∈ß, μ(h∨g)+μ(h∧g) = μ(h)+μ(g) 
(4) 
If h_{n}⊂ß, n∈N such that h_{1}≤h_{2}≤
... ≤h_{n}≤ ...., then: 
Note 1: Let μ_{1} and μ_{2} be lattice measures
defined on the same lattice σalgebra ß. If one of them is finite,
then the set function μ(E) = μ_{1 } (E)μ_{2 }(E),
Eεß is well defined and is countably additive on ß.
Example 6 (Royden, 1981): Let X be any set and
ß = P(X) be the class of all sub sets of X. Define for any Aεß,
μ(A) = +∞ if A is infinite = A if A is finite, where A is the
number of elements in A.
Then μ is a countable additive set function defined on ß and hence
μ is a lattice measure on ß.
Definition 4: A set A is said to be lattice measurable set or lattice
measurable if A belongs to ß.
Example 7 (Kumar et al., 2011a): The
interval (a, ∞) is a lattice measurable under usual ordering.
Example 8 (Kumar et al., 2011a, b):
The closed interval [0, 1]<
is lattice measurable under usual ordering.
Let X = ,
L= {Lebesgue measurable subsets of }
with usual ordering (≤) clearly σ(L) is a lattice σalgebra generated
by L. Here, [0, 1] is a member of σ(L). Hence, it is a Lattice measurable
set.
Example 9 (Kumar et al., 2011a, b):
Every Borel lattice is a lattice measurable.
Definition 5: The lattice measurable space (X, ß) together with
a lattice measure μ is called a lattice measure space and it is denoted
by (X, β, μ).
Example 10: Suppose
is a set of real numbers μ is the lattice Lebesgue measure on
and ß is the family of all Lebesgue measurable subsets of real numbers.
Then (,
ß, μ) is a lattice measure space.
Example 11: Let
be the set of real numbers, ß be the class of all Borel lattices and μ
be a lattice Lebesgue measure on .
Then (,
ß, μ) is a lattice measure space.
Definition 6: Let (X,
, μ) be a lattice measure space. If μ(X) is finite, then μ is
called lattice finite measure.
Example 12: The lattice Lebesgue measure on the closed interval [0,
1] is a lattice finite measure.
Example 13: When a coin is tossed, either head or tail comes when the
coin falls. Let us assume that these are the only possibilities. Let X = {H,
T}, H for head and T for tail. Let ß = {φ, {H}, {T}, X}. Define the mapping
P: ß→[0, 1] by P (φ) = 0 P ({H}) = P ({T}) = ½,
P (X) = 1. Then P is a lattice finite measure on the lattice measurable space
(X, ß).
Definition 7: If μ is a lattice finite measure, then (X, ,
μ) is called a lattice finite measure space.
Example 14: Let ß be the class of all Lebesgue measurable sets
of [0, 1] and μ be a lattice Lebesgue measure on [0, 1]. Then ([0, 1],
, μ) is a lattice finite measure space.
Definition 8: Let (X,
, μ) be a lattice measure space. If there exists a sequence of lattices
measurable sets {x_{n}} such that (i) x =
and (ii) μ(x_{n}) is finite, then μ is called a lattice σfinite
measure.
Example 15: The lattice Lebesgue measure on (,
μ) is a lattice σfinite measure since:
and μ(n,n) = 2n is finite for every n.
Definition 9: If μ be a lattice σfinite measure, then (X,,
μ) is called lattice σfinite measure space.
Example 16: Let ß be the class of all Lebesgue measurable sets
on:
and μ be a lattice measure on .
Then (,,
μ) is a lattice σfinite measure space.

Fig. 1(ac): 
(a) 2 point lattice L, (b) 4 point lattice M and (c) LxM;
the Cartesian product of lattices L and M 
Definition 10 (Gabor, 1964): Let X and Y be two
lattices. Then their Cartesian product denoted by XxY is defined as XxY = {(x,
y)/xεX, yεY}. It is called product lattice.
Example 17: Let L and M be two lattices shown in the Fig.
1.
Consider LxM in Fig. 1, w h ere, l = (x_{2}, y_{4}),
d = (x_{2}, y_{2}), e = (x_{1}, y_{4}), f =
(x_{2}, y_{3}), a = (x_{1}, y_{2}), b = (x_{2},
y_{1}), c = (x_{1}, y_{3}) and O = (x_{1}, y_{1}).
Definition 11: The lattice measure m defined on SxT is called the product
of the lattice measures μ and λ and is denoted by μxλ.
Example 18: If μ is a lattice measure on R, then m = μxμ
is a product lattice measure on RxR.
Definition 12: If A<X and B<Y, then AxB<XxY. Any lattice of
the form AxB is called super lattice in XxY.
Example 19: If A⊂B and C⊂D, then (AxC)⊂(BxD). 

Let (x, y) be any element of AxC. Then by definition of product
lattice we have xεA, yεC. 

But it is given that A⊂B and C⊂D. 

Therefore xεB and yεD. 

That is (x, y) is an element of BxD. Hence, (AxC)⊂(BxD) is a super
lattice in BxD. 
Remark 1: Counting measure: Let X be a nonempty set. Let σ (L)
= P (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.
Definition 13 (Pramada et al., 2011):
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 measurable functions with
respect to μ on X is denoted by L^{1}.
Definition 14 (Pramada et al., 2011):
Let f = u+iv where u and v are real lattice measurable functions on X. Let fεL^{1}.
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}.
Definition 15 (Kumar et al., 2011a):
If E is a lattice measurable set and then the characteristic function χ_{E}(x)
is defined as if χ_{E}(x) = 1, if xεE = 0, if x∉E.
Remark 2: Let (X, S) (Y, T) be lattice measurable spaces.
Then S is a lattice σalgebra in X and T is a lattice σalgebra in
Y.
Definition 16: If AεS and BεT, then the lattice of the form
AxB is called super lattice measurable set where S, T are lattice σalgebras
on X and Y, respectively.
Example 19: Every member of SxT is a super lattice measurable set.
Definition 17: Let E<XxY where xεX, yεY. We define xsection
lattice of E by E_{x} = {y/ (x, y)εE} and ysection lattice of
E_{y} = {x/(x, y)εE}.
Note 2: E_{x}<Y and E_{y}<X.
Definition 18 (Kumar et al., 2011a):
Let f be an extended real valued measurable function on the lattice of real
numbers such that {xεL/f(x)>α} is lattice for each αεL.
Then f is lattice measurable function.
Definition 19 (Kumar et al., 2011a):
A function s on a lattice measurable space X whose range consists of only finitely
many points in [0, ∞] is called a simple lattice measurable function.
Theorem 1 (Pramada et al., 2011): If
EεSxT, then E_{x}εT and E_{y}εS for every xεX
and yεY.
Theorem 2 (Rudin, 1987) and (Pramada
et al., 2011): Let f: X→[0,∞] be a lattice measurable
function. Then there exists simple lattice measurable functions s_{n}
on X such that:
i: 
0≤s_{1}≤s_{2}≤…….≤f 
ii: 
s_{n}(x) →f(x) as n →∞for every xεX 
Theorem 3 (Rudin, 1987) and (Pramada
et al., 2011): Let {f_{n}} be a sequence of lattice measurable
functions on X such that 0≤f_{1}(x)≤f_{2}(x)......≤∞
for every xεX and f_{n}(x)→f(x) as n→∞ for every
xεX. Then f is lattice measurable and:
Note 3: Let E = [a, b]. Then:
CHARACTERIZATION OF LATTICE MEASURABLE FUNCTIONS ON PRODUCT LATTICES
Definition 20: A lattice measurable function f: XxY→Z where z is a
topological space. For each xεX, we define f_{x}: Y→Z by f_{x}(y)
= f(x, y). Then f_{x} is called Ylattice measurable function. For each
yεY, we define f_{y} : X→Z by f_{y}(x) = f(x, y).
Then f_{y} is called Xlattice measurable function.
Theorem 4: Let f be an (SxT) lattice measurable function on XxY, The:
(1): 
For each xεX, f_{x} is a Tlattice measurable
function 
(2): 
For each yεY, f_{y} is a Slattice measurable function 
Proof: Let V be an open set in Z. Let Q = {(x, y) εXxY: f(x, y)εV}.
Since f is SxT lattice measurable, QεSxT.
Q_{x} = {y: (x, y)εQ} = {y: f(x,y)εV}
= {y: f_{x} (y)εV} 
By theorem 1, Q_{x}εT. Therefore f_{x} is a Tlattice
measurable function.
A similar argument shows that f_{y} is an Slattice measurable function.
Theorem 5: Let (X, S, μ) and (Y, T, λ) be a lattice σfinite
measure spaces. Let f be an (SxT)lattice measurable function on product lattice
XxY. Then the following conditions are hold good:
then Φ is Slattice measurable. Ψ is Tlattice measurable and:
2: 
If f is complex and if: 
and
then fεL^{1}(μxλ).
3: 
If fεL^{1}(μxλ) then f_{x}εL^{1}(λ)
for almost all xεX, f_{y}εL^{1}(μ) for almost
all yεY; the functions Φ and Ψ defined by: 
almost every where, are in L^{1}(μ) and L^{1}(λ),
respectively and:
Proof: By theorem 4, we get f_{x} is a Tlattice measurable
function for each xεX and f_{y} is an Slattice measurable function
for each yεY. Hence the definitions of Φ and Ψ make sense.
Part (a): Let QεSxT. Let f =χ_{Q}
Then:
Similarly:
Therefore, by theorem 4:
Hence, we get (a) for characteristic functions.
Let f be a nonnegative (S x T) simple lattice measurable function.
Then:
Let:
Similarly:
Now:
Therefore:
That is:
That is:
Hence, (a) holds for all nonnegative (SxT)simple lattice measurable functions
S.
Let f be any (S x T)lattice measurable function. Then by theorem 2, there
exist (S x T)simple lattice measurable functions s_{n} on X x Y such
that 0≤s_{1}≤s_{2}≤…….
≤f and s_{n} (x, y) →f(x, y) as n→∞ for every (x, y) εX
x Y. Let Φ_{n} be associated with s_{n} in the same way
as Φ is associated to f. we have:
Now:
Therefore, if we apply theorem 3, on (Y, T, λ) then this shows that:
That is, Φ_{n}(x) increase to Φ(x) for every xεX as
n→∞.
Again applying theorem 3, to the integrals of (1) we get:
By interchanging the role of x and y we get:
Therefore:
This completes proof of (a).
Part (b): Let f be complex. Then 0≤f≤∞.
Let:
Given:
Then by using (a) for f, we get:
therefore, fεL^{1}(μxλ).
Proof of (c): First we prove for real f∈L^{1}
(μxλ) 

Let f be in L^{1} (μxλ) and let f be real.
Then 0≤f^{+}≤ ∞ and 0≤f¯≤∞. 

Let Φ_{1} and Φ_{2} correspond to f^{+}
and f^{–} , respectively as Φ corresponds to f. 

Now fεL^{1} (μxλ) and f+≤ f 

Since, (a) holds for f^{+}, we get that: 
Therefore, Φ_{1}εL^{1 }(μ). 
Similarly, Φ_{2}ε L^{1 }(μ). 
Now f_{x} = (f^{+})_{x}(f¯)_{x}.
Also shows
that f^{+}_{x}εL^{1}(λ) for every x for
which both Φ_{1}(x) and Φ_{2}(x) <∞. 

Similarly (f^{–})_{x}ε L^{1}(λ). 

Therefore f_{x}εL^{1 }(λ) for every x for which
both Φ_{1} (x) and Φ_{2}(x) are<∞. 

Since, Φ_{1}, Φ_{2}, εL^{1}(μ)
we get that Φ_{1}(x) and Φ_{1}(x) are <∞,
almost every where. 

Hence, f_{x}ε L^{1} for almost all xεX. 

For such x, we have Φ (x) = Φ_{1 }(x)Φ_{2}(x) 

Hence, Φε L^{1}(μ) using (a) we get: 
Therefore:
That is:
Similarly, we can prove that:
by using f_{y} in place of f_{x} and Ψ in the place of
φ.
Suppose f is complex and fεL^{1}(μxλ).
Let f = u+iv. Then u, vεL^{1} (μxλ) and u, v are real.
Then applying what we proved above to u, v we get:
where, Φ_{u}, Φ_{v} corresponds to u, v as Φ
corresponds to f.
Thus:
That is:
This proves (c).
Hence the theorem.
Note 4:
can be written as:
The integrals at the ends are the so called iterated integrals of f.
The middle integral is often referred to as a double integral.
Result 1: The two iterated integrals are finite and equal.
Proof: From part (b) and part (c) we get the following useful result.
Let f is (SxT)lattice measurable and let:
Then ΦεL^{1}(μ). ΨεL^{1}(λ) that
is:
Therefore, the two iterated integrals are finite and equal.
Note 5: The order of integration may be reversed for (SxT)lattice measurable
functions f whenever f≥0 or when ever one of the iterated integrals of fis
finite.
Result 2: For lattice σfiniteness μ can not be omitted.
Proof: Let X = [0, 1] = Y, μ = Lebesgue measure on [0,1], λ
= lattice counting measure on Y.
Let f(x, y) = 1 if x = y, f(x, y) = 0 if x≠y:
since, for a given y, f (x, y) = 1 when x = y and 0 at all other x. also Lebesgue
lattice measure of a single point is 0.
That is:
= 1 (since λ is the lattice counting measure)
Hence:
So:
Hence:
To show that f in (SxT)lattice measurable.
(where, S is the class of all Lebesgue lattice measurable sets in [0, 1] and
T consists of all subsets of [0, 1]).
Since f (x, y) = 1 if x = y, f (x, y) = 0 if x ≠ y, we see that f = χ_{D}
where, D is the diagonal of the unit square.
Given a positive integer n:
Let Q_{n} = (I_{1 }x I_{1}) V (I_{2 }x I_{2})
V………….. V ( I_{n }x I_{n})
Where n = 1,
I_{1} = [0, 1], Q_{1} = I_{1} x I_{1} is the
unit square.
When:
n 
= 
2 
I_{1 } 
= 
[0, 1/2], I_{2}= [1/2, 1] 
Q_{2 } 
= 
[0, 1/2] x [0, 1/2] V [1/2, 1] x [1/2, 1] 
Q_{3 } 
= 
[0, 1/3] x [0, 1/3] V [1/3, 2/3] x [1/3, 2/3] V [2/3, 1] x [2/3, 1] etc. 

Thus, Q_{n} is finite union of super lattice measurable
sets and D = ΛQ_{n}. 

Hence, DεSxT. 

Therefore, f = χ_{D} is SxTlattice measurable. 

Since λ is the lattice counting measure, if: 
a disjoint union such that λ (Yn) < ∞ for all
n, then every Y_{n} is a finite set. 

Hence, Y is countable, a contradiction since Y = [0, 1] 

Thus λ is not lattice σfinite. 

Thus the lattice σfiniteness of λ, so μ can not be omitted. 
Result 3: The condition that f is lattice measurable with respect to
S x T can not be dropped.
Proof: Consider X = Y = [0, 1], μ = λ = Lebesgue lattice measure
on [0, 1], S = T = class of all Lebesgue lattice measurable sets in [0, 1].
Let us assume the following consequence of continuum hypothesis: there exists
a onetoone map θ from [0, 1] onto a well ordered set W such that θ
(x) has at most countably many predecessors in W for each xε[0, 1].
Let Q = {(x, y)∈XxY: θ(x) precedes θ(y) in W }.
For each xε[0, 1] Q_{x} = {y: (x, y)εQ}.
YεQ_{x} if and only if (x, y)εQ if and only if θ(x) precedes
θ(y) in W.
Since θ(x) has at most countably many precedessors in W, there will be
only countably many y^{’s}
in [0, 1] such that θ(y) processes θ(x).
Hence, all but countably many y’^{s}
in [0, 1] are such that θ(x) precedes θ(y) that is, Q_{x}
contains all but countably many points of [0, 1].
For each yε[0, 1] Q_{y} = {x: (x, y)εQ}.
That is, xεQ_{y} if and only if (x, y)εQ if and only if θ(x)
precedes θ(y).
But θ(y) has at most countably many predecessors in W.
Hence, Q_{y} contains at most countably many points of [0,1].
Let f = χ_{Q}.
Since Q_{x} and Q_{y} are Borel lattice measurable, we get
that f_{x} and f_{y} are Borel lattice measurable and:
since for any fixed x, f (x, y) =
and Q_{x} contains all but countably many points:
All since
contains at most countably many number of points:
Hence:
In this result. f is not lattice measurable w.r.t. lattice σalgebra S
x T.
Hence, the condition that f is lattice measurable with respect to S x T can
not be dropped.
CONCLUSION
This manuscript illustrate the concept of product lattice measurable functions
and their various characterizations. In particular these functions were defined
over topological spaces. Also it has been introduced and advanced the characteristics
of S lattice measurable function and Tlattice measurable function. The concept
of iterated integral of a product lattice measurable function has been defined
and proved that the two iterated integrals of a product lattice measurable function
are finite and equal. The condition that product lattice measurable function
is lattice measurable is obtained and it has been derived that the condition
that product lattice measurable function is lattice measurable cannot be dropped.