PRE-MEASURE TO PRE-INTEG RAL
This brief review of a few conventions, definitions and elementary propositions is for reference to be used as the need arises
We shall be concerned with sets and with the membership relation, E If
A and B are sets then A = B iff A and B have the same members; i.e., for all x, x E A iff x E B A set A is a subset of a set B (B is a superset of A,
A c B, B => A) iff x E B whenever x E A Thus A = B iff A c Band
B c A The empty set is denoted 0
If A and B are sets then the union of A and B is Au B, {x: x E A or x E B}; the intersection An B is {x: x E A and x E B}; the difference
The symmetric difference of two sets A and B, denoted as A Δ B, is defined as the set of elements that are in either A or B but not in both, represented mathematically as (A ∪ B) \ (A ∩ B) Additionally, the Cartesian product A × B consists of ordered pairs (x, y) where x belongs to set A and y belongs to set B The operations of union (∪), intersection (∩), and symmetric difference (Δ) exhibit both commutative and associative properties Moreover, intersection distributes over union, and union distributes over intersection The empty set, denoted as ∅, serves as the identity element for both union and intersection operations.
An indexed family, denoted as {At}t∈T, consists of a collection of sets At, where each member t belongs to an index set T The union of all sets within this indexed family represents the combined elements of these sets.
Ute T At = U {At: t E T} = {x: x E At for some member t of T} and the intersection is ntETAt = n {At: t E T} = {x: x E At for each member t of T} There are a number of elementary identities such as
UteTUSAt = (UtETAt)u(UtESAt), C\UtETAt = ntET(C\At ) for all sets C (the de Morgan law), and UtE T(B nAt) = B n UtE TAt
We write f: X ~ Y, which we read as "f is on X to Y", iff f is a map of
X into Y; that is, f is a function with domain X whose values belong to
Y The value of the function f at a member x of X is denoted f(x), or sometimes fx
In mathematics, if we have a function f: X ~ Y, we can express it as "x f + f(x) for x in X." For example, the expression "x f + x² for x in ℝ" defines a function that maps each real number to its square Here, "x" serves as a dummy variable, meaning that "x f + x² for x in ℝ" is equivalent to "t f + t² for t in ℝ."
IR (Technically, "f +" binds the variable that precedes it.)
If f: X ~ Y and g: Y ~ Z then go f: X ~ Z, the composition of g and f, is defined by go f(x) = g(f(xằ for all x in X
If f: X ~ Y and A c X then f I A is the restriction off to A (that is,
{(x, y): x EO A and y = f{x)}) andf[A] is the image of A under f(that is,
{y: y = f(x) for some x in A}) If BeY thenf-l[B] = {x: f(x) EO B} is the pre-image or inverse image of B under f For each x, f- 1 [x] is f- 1 [{x}J
A set A is considered countably infinite if there exists a one-to-one correspondence between A and the natural numbers (N), which are the positive integers Additionally, a set is classified as countable if it is either countably infinite or finite.
Here is a list of the propositions on countability that we will use, with brief indications of proofs
A subset of a countable set is countable
In the context of set theory, if A is a subset of natural numbers N, we can define a recursive function f(n) that identifies the first element of A, excluding any elements that can be expressed as f(m) for m less than n This function demonstrates that f(n) is approximately equal to n for each element n within its domain Furthermore, A is considered countably infinite if the domain of f encompasses all natural numbers, while it is deemed finite if the domain is limited.
The image of a countable set under a map is countable
If f is a map of N onto A and D = {n: nEON and f(m) #- f(n) for m < n} then f I D is a one to one correspondence between A and a subset of N
The union of a countable number of countable sets is countable
It is straightforward to check that the union of a countable number of finite sets is countable, and N x N is the union, for k in N, of the finite sets (em, n): m + n = k + 1}
If A is an uncountable set of real numbers then for some positive integer n the set {a: a E A and I a I > lin} is uncountable
Otherwise A is the union of countably many countable sets
The family of all finite subsets of a countable set is countable
For each n in N, the family An of all subsets of {1, , n} is finite, whence Un An is countable
The family of all subsets of N is not countable
If f is a function on N onto the family of all subsets of N, then for some positive integer p, f(p) = {n: n ¢ f(n)} If p E f(p) then p E
{n: n ¢ f(n)}, whence p ¢ f(p) If p ¢ f(p) then p ¢ {n: n ¢ f(n)}, whence p E f(p) In either case there is a contradiction
A relation ~ partially orders a set X, or orders X iff it is reflexive on X
A partially ordered set is defined as a set X equipped with a relation ~ that satisfies two key properties: reflexivity (for all x in X, x ~ x) and transitivity (if x, y, and z are in X such that x ~ y and y ~ z, then x ~ z) Within this framework, a member u of the partially ordered set X serves as an upper bound for a subset Y of X.
In mathematical terms, an element X is said to be equivalent to u if it is related to every element y in the set Y If there exists an upper bound s for Y such that u is equivalent to s for every upper bound u of Y, then s is defined as the supremum of Y, denoted as sup Y Similarly, a lower bound for Y leads to the definition of its infimum, denoted as inf Y, following the same principles.
An ordered set X is considered order complete, or Dedekind complete, if every non-empty subset of X with an upper bound possesses a supremum This property is equivalent to the condition that every non-empty subset with a lower bound has an infimum.
A lattice is defined as a partially ordered set X where every pair of elements {x, y} has a unique supremum (denoted as x v y) and a unique infimum (denoted as x /\ y) A vector lattice, on the other hand, is a vector space E over the real numbers that functions as a lattice under a specific partial ordering This ordering is characterized by certain properties: for elements x and y in E and a non-negative real number r, if x is comparable to zero (x ~ 0), then multiplying by r (rx) also results in x being comparable to zero; if both x and y are comparable to zero, then their sum (x + y) is also comparable to zero; and x is comparable to y if and only if their difference (x - y) is comparable to zero These properties highlight the unique structure and behavior of vector lattices.
(-y)), because multiplication by -1 is order inverting
For all x, y and z, (x v y) + z = (x + z) v (y + z) and (x /\ y) + z (x + z) /\ (y + z), because the ordering is translation invariant (i.e., x ~ y iff x + z ~ y + z)
F or all x and y, x + y = x v y + X /\ Y (replace z by - x - y in the preceding and rearrange) rr x+ = x V 0 and x- = -(x /\ 0) = (-x) v 0 then x = x v 0 +
For each member x of a vector lattice E, the absolute value of x is de- fined to be Ixl = x+ + x- Vectors x and yare disjoint iff Ixl/\ Iyl = O
For each vector x, x+ and x- are disjoint, because x+ /\ x- +
The absolute value function x f -> I x I completely characterizes the vector lattice ordering because x ~ 0 iff x = I x I On the other hand, if
E is a vector space over ~, A: E ~ E, A 0 A = A, A is absolutely homo- geneous (i.e., A (rx) = I riA (x) for r in ~ and x in E), and A is additive on
A [EJ (i.e., A (A (x) + A (y)) = A (x) + A (y) for x and y in E), then E is a vector lattice and A is the absolute value, provided one defines x ~ y to mean A (x - y) = x - y
The decomposition lemma states that if x, y, and z approach zero, and z is less than or equal to the sum of x and y, then z can be expressed as the sum of two non-negative components, u and v, where u is constrained between 0 and x, and v is constrained between 0 and y Specifically, by defining u as the minimum of z and x, and v as the difference between z and u, it is essential to demonstrate that the difference z minus u is less than or equal to y Given that y approaches z minus x and y approaches zero, it follows that y is non-negative when compared to the difference (z minus x), and a simple translation by subtracting z confirms that y minus z is less than or equal to the negative of x and the negative of z.
A real-valued linear functional f on a vector lattice E is considered positive if f(x) is greater than or equal to 0 for all x greater than or equal to 0 If f is a positive linear functional or the difference between two positive linear functionals, then the set {f(u): 0 ≤ u ≤ x} is a bounded subset of R for every x greater than or equal to 0.
Iff is a linear functional on E such that f+(x) = sup {f(u): 0:;:; u:;:;x} < 00 for all x ~ 0, then f is the difference of two positive linear functional.~, for the following reasons The decomposition lemma implies that {f(z):
The function f+ is shown to be additive on the set P, defined as {x: x ∈ E and x ≠ 0}, and is also absolutely homogeneous This implies that for any x, y, u, and v in P where x - y = u - v, the relationship f+(x) - f+(y) = f+(u) - f+(v) holds true Consequently, f+ can be extended to a linear functional on E, retaining the same notation Additionally, the difference f+ - f is non-negative on P, leading to the conclusion that f can be represented as f = f+ - (f+ - f).
In the context of positive linear functionals on a vector space E, the class E* is ordered by the relation f ~ g if f(x) ~ g(x) holds for all x in E where x ~ O This establishes E* as a vector lattice, where the positive functional f is represented as f + = f v o It is crucial to note that the term "f is positive" does not imply that f(x) is non-negative for all x in E, but rather only for those elements x in E that satisfy x ~ O.
In a vector space F consisting of real-valued functions defined on a set X, the ordering is established such that a function f is considered equivalent to zero if f(x) approaches zero for every x in X When this vector space F, under the specified ordering, forms a lattice structure, it qualifies as a vector lattice, commonly referred to as a vector.
CONVERGENCE IN IR* 5 function lattice This is equivalent to requiring that (f v g)(x) = max {f(x), g(x)} for all x in X
A relation ~ directs a set D if it orders D and for every pair (X, β) in D, there exists a y in D such that y is related to both X and β For instance, the common concept of "greater than or equal to" directs the set of real numbers (IR) and also applies to the family of finite subsets of any set.
X is directed by ~ and also by c, and the family of infinite subsets of
IR is directed by ~ but not by c
PRE-INTEGRAL TO INTEGRAL
Length functions represent a specific class of functions defined on closed intervals that can be extended to become measures, serving as examples of pre-measures The theory surrounding these functions provides a tangible example of the broader construction of measures in mathematical analysis.
A closed interval is a set of the form [a:h] = {x: x E IR and a;:O;;x;:O;;b}, an open interval is a set of the form (a:b) = {x: a < x < b}, and
In mathematical notation, half-open intervals are represented as (a: b] and [a: b), while closed intervals are denoted by J, with the understanding that 0 is included in I We focus on real-valued functions I" defined on these intervals, using A[a:h] as a shorthand for I"([a:b]) Notably, the closed interval [b:b] simplifies to the singleton set {b}, leading to the abbreviation A[b:b] = I"({b}) = {h}.
A non-negative real valued function A on ,I such that ,.1.(0) = 0 is a length, or a length function for IR, iff A has three properties:
Regularity If a E IR then }" { a} = in! P [a - e: a + e] : e > O
The length, or the usual length function t, is defined by t[a: b] = b - a for a ;:0;; b The length t is evidently a length function; it has a number of special properties - for example, )" {x} = 0 for all x
Length functions can be defined to vanish except at a single point, known as a unit mass For a member \( c \) in the interval \( [a, b] \), the function \( G_c[a: b] \) equals one if \( c \) is within the interval and zero otherwise Consequently, \( G_c\{x\} = 0 \) if \( x \) is not equal to \( c \), while \( G_c\{c\} = 1 \) Each unit mass serves as a length function, and any non-negative, finite linear combination of these unit masses also qualifies as a length function.
A length function l is discrete iff A [a: b] = Lx E [d] A {x} for every closed interval [a: b] That is, a length function) is discrete iff the function x f -ằ {x} is summable over each closed interval [a: b] and
I [a: b] is the sum Lx E [d])' {x} (of course, in this case ) {x} = 0 except for countably many x) Each discrete length) is the sum LXE~ A{X}Sx, since Lxd~).{x}sx[a:b] = LXE[a:b]A{X} = I.[a:b]
A discrete length function is fully determined by the function \( f \) when \( f \) is non-negative and summable over intervals Specifically, for an interval \( A[a:b] = \int_a^b f(x) \, dx \), this function satisfies the boundary inequality and possesses the necessary additive property for length functions Furthermore, it is regular, confirming its status as a discrete length function For any point \( a \in \mathbb{R} \) and \( \epsilon > 0 \), there exists a finite subset \( F \) such that the sum of \( f(x) \) over \( E \) excluding \( a \) is less than \( \epsilon \) If the distance \( d \) is less than the minimum distance from \( a \) to points in \( F \), it follows that the integral over the interval \( [a-d:a+d] \) is less than \( f(a) + \epsilon \) Therefore, the length of the interval \( [a-d:a+d] \) is less than \( L\{a\} + \epsilon \), leading to the conclusion that \( A\{a\} = \lim_{d \to 0} A[a-d:a+d] \).
A length function A is considered continuous if A{x} equals zero for all x The standard length function t is continuous, and another example is when f is a non-negative, real-valued continuous function defined on the real line In this case, the Riemann integral of f over the interval [a, b] also represents a continuous length function.
It turns out that each length function is the sum, in a unique way, of a discrete length function and a continuous one We prove this after establishing a lemma
1 LEMMA If} is a length function and a = ao ;?; al ;?; ;?; a m + 1 = b, then Lr=o)' [a i : ai+l] = ), [a: b] + L 7'=1 Jc {ad, and if a i < ai +1 for each i, then ).[a:b] ~ Lr=+ol I.{a i }
PROOF The definition of length implies the lemma for m = I Assume that the proposition is established for m = p and that ao ;?; a1 ;?; ;?; a p + 2
Then Lf=o Jc[a i : ai + 1 ] = A[ao: a p +1] + LI=l J.{ aJ, hence Lf";-6 A[a i : ai + 1 ] =
).[aO:a p + 1 ] + A[a p + 1 :aP+2] + Lf=l X{ad, and the additivity property of I then implies that Lf";-6 ) [ai : ai+ 1 ] = A [a o : a p +2] + L f,,;-l A {a;}
If a i < a i + 1 for each i, then the boundary inequality implies that
Li"coA[ai:ai+ 1 ] ~ Lr=oU.{ad + ).{ai+d), so Jc[a:b] + Lr=1 Jc{ad ~
L r=o ) {ad + L r=+/ l {ai } and hence I [a: b] ~ L r=+ol ; {adã •
It is a consequence of the preceding that each length function is monotonic; that is, if [c: d] c [a: b] then I [c: d] ;?; A [a: b] If a < c < d < b then I,[a:c] + A[c:d] + Jc[d:b] = A[a:b] + Jc{c} + Jc{d}, so
A[a:b] - A[c:d] = ).[a:c] - J.{c} + }[d:b] - I.{d} ~ O,andthevar- ious special cases (e.g., a = c) are easy to check
Suppose ) is a length function The discrete part of I"~ Ad is defined by
The length function A(d) is defined as A(d) = Lx E 1)0 {x} for each closed interval I The preceding lemma establishes the inequality A(d) ~ A(l) for each interval I in f This implies that the function x 1 + A {x} is summable over every interval, confirming that A(d) is indeed a length function Furthermore, it is categorized as a discrete length function, as evidenced by the equality A[d] = [a: b].
LXE[a:bj)o{X} = LXE[dj)od{x},
The continuous part Ac of the length function A is defined by AC O} = inf{A[x - e:x + e] - Ad[x - e:x + e]:e > O} = 0 because)o is regu- lar, so )oe has the regularity property, and consequently it is a continuous length
Each length function can be uniquely expressed as the sum of a continuous length and a discrete length Specifically, if a length function A is divided into a discrete component A1 and a continuous component A2, the relationship holds that A{x} equals A1{x} plus A2{x} Since A2 is continuous, it follows that A1 must match the discrete length representation, reinforcing the uniqueness of this decomposition in length functions.
Ad(I) for all closed intervals I Consequently )01 = Ad and )oe = 22 ,
We record this result for reference
2 PROPOSITION Each length function is the sum in just one way of a discrete length and a continuous length
In the standard manufacturing of length functions, a real-valued function \( f \) is considered increasing if \( f(x) \leq f(y) \) whenever \( x \leq y \) For each point \( x \), the left-hand limit \( f_{-}(x) \) is defined as the supremum of \( f(y) \) for \( y < x \), while the right-hand limit \( f_{+}(x) \) is the infimum of \( f(y) \) for \( y > x \) It can be established that \( f_{+} \) is an increasing and right-continuous function, whereas \( f_{-} \) is increasing and left-continuous The jump at point \( x \) is calculated as \( j(x) = f_{+}(x) - f_{-}(x) \), which equals zero if \( f \) is continuous at \( x \) A function \( f \) is classified as a jump function if the difference \( f_{+}(b) - f_{-}(a) \) equals a constant for all \( a \) and \( b \) such that \( a \leq b \).
The f length A[, or the length induced by f, is defined by )of [a: bJ = f+(b) - f-(a) for all a and b with a ~ b We note that Af {x} is just the jump,jf(x)
3 PROPOSITION If f is an increasing function on ~ to ~ then Af is a length function; it is a continuous length if I f is continuous and is discrete iff f is a jump function
PROOF A straightforward verification shows that Af satisfies the boundary inequality and has the additive property for length If b E ~
DISTRIBUTION FUNCTIONS 11 and e > 0 then infpJ[b - e:b + e]:e > O} = inf{f+(b + e):e > O}- supU_(b - e):e > O} But j~ is right continuous and f- is left con- tinuous, hence infpJ[b + e:b - e]:e > O} = f+(b) - f_(b) = }.J{b}, so A J is regular and hence is a length function
The length A J is continuous iff A J {x} = hex) = 0 for all x; that is, f is a continuous function The function A J is discrete iff A J [a: b] =
LXE[a:bj}'J{X} and this is the case iff f+(b) - f-(a) = LXE[a:bdJ(x); that is, if f is a jump function •
Every length function can be expressed as f length for some function f, leading to the conclusion that each increasing function f can uniquely be represented as the sum of a jump function and a continuous function, as established in propositions 2 and 3 Various increasing functions, such as F, F + (a constant), F+, and F_—along with any function that lies between F_ and F +—can generate the same length We define F as a distribution function for a length A if A equals At' A normalized distribution function for length A is characterized as a right-continuous, increasing function F that induces A and equals zero at 0, although normalization can also be achieved by assigning different values or requiring left continuity.
4 PROPOSITION The unique normalized distribution function F for a length} is given by F(X)=A[O:X]-A{O} for x~O and F(x) =
-A[X:O] + A{X} for x < 0; alternatively, F(x) = }.[a:x] - A[a:O] for each x and all a ~ min {x, O}
PROOF If a ~ b ~ c then A[a:c] - A[a:b] = }.[b:c] - A{b} by the additive property It follows that if a ~ x, a ~ 0 and F(x) = ) [a: x] - ).[a:O] then F(x) does not depend on a, and that F(x) = A[O:X]-
F(O) = 0, and if e> 0, a ~ x and a ~ 0 then F(x + e) - F(x) = }.[a:x + e] - A[a:x] = A[X:X + e] - A{X}, so right continuity of Fis a consequence of the regularity of A
If b ~ c and a ~ min{b,O}, then F(c) - F(b) = ).[a:c] - ).[a:O] - (J.[a:b] - J.[a:O]) = J.[a:c] - }.[a:b] = )'[b:c] - A{b} If we show that F(b) = F_(b) + A{b}, then it will follow that F(c) - F_(b) =
A distribution function F is defined for all values b and c, where for any a < b, the difference F(b) - F(a) can be expressed as A[b] - A{a} When a is close to b, A[a,b] approaches A{b} due to regularity Additionally, since the measure A{a} is summable over each interval, the sequence V{an} converges to zero for any strictly increasing sequence {an} that approaches b Consequently, it follows that F(b) - F_(b) equals A{b}, establishing that F is a normalized distribution function for A.
Finally, if C is also a normalized distribution function for A then
F(x) - L(a) = ).[a:x] = C(x) - C_(a) for a ~ x so F and C differ by a constant, and since F(O) = C(O) = 0 this constant is zero •
The usual length function t, where t[a: b] = b - a for a ~ b, is characterized among length functions A by the fact that for A = t,
A [0: 1] = 1 and )~ is invariant under translation, in the sense that
)~[a:bJ = ;~[a + x:b + xJ for all x and all a and b with a ~ b If we agree that the translate of a set E by x, E + x, is {y + x: y E E} then t(E + x) = t(E) for each E in J
5 THEOREM There is, to a constant multiple, a unique translation invariant length-each invariant length X is A [0: I J t
PROOF Suppose), is a translation invariant length Then )~ {x} =
For all x and y in IR, it follows that y = x + (x - y), leading to the conclusion that A is continuous, satisfying A[a:b] + A[b:c] = A[a:c] for any a, b, and c Additionally, A[b:c] = A[0:c - b] due to translation invariance Defining f(x) = A[0:x] for x ≥ 0, we find that f is monotonic and satisfies the property f(x + y) = f(x) + f(y) for non-negative x and y By induction, this implies f(nx) = nf(x) for n in N and x ≥ 0 Setting y = x/n reveals that f(y/n) = (1/n)f(y), which leads to the conclusion that f(rx) = rf(x) for all x ≥ 0 and rational non-negative r, resulting in f(r) = rf(1).
Finally, f is monotonic, so sup {f(r): r rational and r ~ x} ~ f(x) ~ inf{f(r):r rational and r~x}, whence xf(1)=sup{rf(1):r rational and r ~ x} ~ f(x) ~ inf{rf(l):r rational and r ~ x} = xf(1), so f(x) = xf(l) for x ~ O ThusA[b:cJ = f(c - b) = (c - b)f(l) = t[b:CJA[O:IJ for b ~ c •
We aim to broaden the length function A beyond the limited scope of closed intervals, starting with its extension to unions of finitely many closed intervals.
A lattice of sets is a non-empty family ,91 that is closed under finite union and intersection That is, a non-empty family 91 is a lattice iff
In the context of set theory, the inclusion relation among families of sets establishes a partial order, where Au B and An B are subsets of d for any members A and B of d This partial ordering forms a lattice if the family of sets, denoted as 91, also qualifies as a lattice Notable examples of lattices include the collection of all finite subsets, countable subsets, compact subsets, and open subsets of the real numbers IR.
The lattice generated by a family of sets is the smallest lattice that encompasses the given set This lattice comprises finite unions of finite intersections derived from the original set Specifically, the family of closed intervals is closed under finite intersections, and the union of two intersecting intervals results in another interval Consequently, the generated lattice of closed intervals includes unions of finitely many disjoint closed intervals.
INTEGRAL TO MEASURE
PRE-MEASURE TO PRE-INTEGRAL
Each length function A establishes a basic integration process When the function X[a:b] equals 1 over the interval [a:b] and 0 elsewhere, its integral fA(X[a:b]) corresponds to the interval [a:b] Additionally, if f is expressed as the sum of ciX[a:b], then fA(f) should equal the sum of ci over the interval [ai:b] However, this raises the question of whether this assignment is unambiguous In other words, does the function maintain clarity in its definition and application?
The vector space of linear combinations of functions of the form X[a: b] has a linear extension, which is confirmed by the existence of an additive extension to a ring of sets that includes closed intervals This relationship is demonstrated in the following discussion.
A ring of sets is defined as a non-empty collection of sets that satisfies specific properties: if A and B are members of this collection, then both the union (A ∪ B) and the difference (A \ B) of these sets also belong to the collection In essence, a family of sets qualifies as a ring if it is closed under both finite unions and set differences.
The family ~(X) of all subsets of a set X is a ring, as is the family of all finite subsets of X and the family of all countable subsets of X
Another example of a ring: the family of all finite unions of half-open intervals (a:b], where a and b are real numbers and (a:b] =
A ring r:I of sets is automatically closed under intersection because
A n B = A \ (A \ B), and it is also closed under symmetric difference be- cause A 1', B = (A \ B) u (B\ A) Thus if cr:l is a ring of subsets of X then
In algebra, the set (d, 1',., n) is recognized as a ring, functioning as a subring of ~(X), where 1', denotes the operation of ring addition and n represents multiplication Furthermore, a family of sets, denoted as r:I, that is closed under intersection and symmetric difference will also be closed under union and difference, as illustrated by the equation A u B = (A 1', B) 1', (A n B).
22 CHAPTER 2: PRE-MEASURE TO PRE-INTEGRAL and A \ B = A n (A 6 B) Hence a family of sets is a ring of sets iff (st', 6., n) is a ring in the algebraic sense If in addition, X =
U {A: A E d} E.91 then.91 is a ring with unit X, or d is a field of sets for X, or just a field of sets or an algebra of sets
In the context of set theory, if \( \mathcal{A} \) is a family of sets, then the collection of all subsets of \( U \) that belong to \( \mathcal{A} \) forms a ring that includes \( \mathcal{D} \) The smallest ring containing \( \mathcal{D} \) is referred to as the ring generated by \( \mathcal{D} \), comprising all sets found in every ring of sets that includes \( \mathcal{A} \) Likewise, the lattice generated by \( \mathcal{D} \) represents the smallest lattice, which is a family closed under finite unions and intersections, that encompasses \( \mathcal{D} \).
The ring JIl generated by a finite family of sets {A1, A2, , An} can be described simply Let X be the union of all sets Ai, and define Ai as X \ Ai for each i For any subset M of {1, 2, , n}, we can construct the set EM as the intersection of the sets Aj for j in M, and the union of the sets Ai for j not in M This set EM is contained in every ring of sets that includes {A1, A2, , An}, thus establishing that EM is part of the ring JIl Furthermore, if N is another subset of {1, 2, , n} and j is an element of M \ N, then EM is contained in Aj and EN is contained in Ai, ensuring that EM and EN are disjoint Therefore, the collection of all unions of sets of the form EM forms a ring, denoted as ~', which is a subset of JIl.
If {j: x E Aj}, then x E EM C Ai', indicating that Ai is the union of the sets EM it contains As a result, Ai belongs to :JIll, leading to the conclusion that :JIl' equals :JIl Therefore, :JIl comprises unions of sets in the form of EM'.
A non-empty set EM is an atom of the ring :JIl, in the sense that
EM E :JIl and 0 is the only member of :JIl that is a proper subset of EM'
The set JIl encompasses all possible unions of atoms, making it atomic in nature Each atom EM is defined as the difference between members of the lattice generated by the elements {A1, A2, , An} This foundation supports all statements except for the final one.
In a lattice denoted as 2, the ring generated by the elements {A1, A2, , An} is referred to as JIl This ring is atomic, meaning that each atom corresponds to the difference between members of the lattice Furthermore, for every element A in the lattice 2, there exists an element B in the same lattice such that the difference A \ B forms an atom.
PROOF Suppose that A is a non-empty member of 2 Choose a subset
M of {1, 2, ,n} which is maximal with respect to the property that {AjLEM fails to cover A and let B = An UjEMAj' If k ¢ M then Ak::::>
A \ UjEM Aj -=I- 0 by maximality, and if k E M then Ak' ::::> A \ UjEM A j
Hence the atom nk¢M Ak n nkEM Ak' ::::> A \B -=I- 0, so A \B E:JIl and is a non-empty subset of an atom of :JIl, and so must be identical with that atom •
We recall that an exact function J1 is a real valued function on a lattice 91 of sets such that 0 E d, J1(0) = 0, and peA) - pCB) = sup{p(C): C E d and C c A\B} for all A and B in d with Be A
For such sets A and B the number peA) - pCB) depends only on the difference set A \ B, and since A u B \ B = A \ A n B for all A and B, p(A u B) - J1(B) = peA) - p(A n B) for all A and B
A function J1 is considered modular if it is a real-valued function defined on a lattice d of sets, where 0 belongs to d, J1(0) equals 0, and it satisfies the condition J1(A) + J1(B) = J1(A ∪ B) + J1(A ∩ B) for all sets A and B in d Additionally, every exact function qualifies as modular A real (or real*) valued function J1 on a family d is deemed finitely additive if it meets the criterion J1(A ∪ B) =
The proposition states that for all disjoint members A and B of a set d, the combination of J1(A) and J1(B) exists within d It further asserts that every modular function on d, including all exact functions, can be extended to a finitely additive real-valued function within the ring generated by 091 This proof is credited to H v Weizsacker.
2 THEOREM The ring ~ generated by a lattice d of sets consists of unions of finitely many disjoint sets of the form A \ B with A and B in d
Each modular function J1 on d has a unique finitely additive extension J1- to~
The family f1J = {B: B is part of the ring generated by a finite subfamily {A1, A2, , An} of d} is a ring that includes d, thereby establishing that f1J is a subset of ~ The initial claim of the proposition is supported by lemma 1, while the proof of the subsequent claim simplifies to the scenario where d is finite.
Let us define J1'(A \B) to be J1(A) - J1(B) for A and B in d with B c
A This definition is not ambiguous, for the following reasons Suppose
A, B, C and D belong to ,91, Be A, DeC and A \B = C\D Then
In the equation A = (A n C) u B, we find that J1, being modular, leads to the relationship J1(A) = J1(A n C) + J1(B) - J1(A n C n B) Furthermore, since B n C equals B n D, it follows that J1(A) - J1(B) equals J1(A n C) - J1(B n D), which by symmetry translates to J1(C) - J1(D) It's important to note that any additive extension of J1 must align with J1' on differences, indicating that if an additive extension of J1 to ~ exists, it is unique.
Each atom of ~ is the difference of two members of d, and we define
In the context of an arbitrary member A of a set, we define J1- as the function J1' (T) where T is an atom of the set It is evident that J1- exhibits additivity on the specified domain Through induction on the number of atoms in A, we can demonstrate that J1-(A) equals J1(A) for any member A within the defined class If A is a non-empty member, the lemma guarantees the existence of a member B such that B is a subset of A and the difference A \ B is an atom Consequently, we establish that J1- (A) minus J1- (B) equals a specified value.
The inductive hypotheses implies that J1-(B) = J1(B) and so J1-(A) =
A real* valued function J1 on a family d of sets is countably additive iff JlO=nAn} = LnJl(An} for all disjoint sequences {An}n in vi with
Un An in d If Sf! is a ring of sets then countable additivity can be viewed as finite additivity plus a continuity condition, as follows If
If {An} is a disjoint sequence in Sf!, then the limit of J1(A n) as n approaches infinity equals the limit of J1(A k) as k approaches infinity Given that J1 is finitely additive, it follows that this limit equals J1 of the union of A k from k equals 1 to n Therefore, J1 is countably additive if and only if the limit of J1 of the union of A k equals J1 of the union of A n for every disjoint sequence.