7.4: Set Functions. Additivity. Continuity
( \newcommand{\kernel}{\mathrm{null}\,}\)
I. The letter "v" in vA may be treated as a certain function symbol that assigns a numerical value (called "volume") to the set A. So far we have defined such "volumes" for all intervals, then for C-simple sets, and even for Cσ-sets in En.
Mathematically this means that the volume function v has been defined first on C (the intervals), then on C′s (C-simple sets), and finally on Cσ.
Thus we have a function v which assigns values ("volumes") not just to single points, as ordinary "point functions" do, but to whole sets, each set being treated as one thing.
In other words, the domain of the function v is not just a set of points, but a set family (C,C′s, or Cσ).
The "volumes" assigned to such sets are the function values (for C and C′s-sets they are real numbers; for Cσ-sets they may reach +∞).This is symbolized by
v:C→E1
or
v:Cσ→E∗;
more precisely,
v:Cσ→[0,∞],
since volume is nonnegative.
It is natural to call v a set function (as opposed to ordinary point functions). As we shall see, there are many other set functions. The function values need not be real; they may be complex numbers or vectors. This agrees with our general definition of a function as a certain set of ordered pairs (Definition 3 in Chapter 1, §§4-7); e.g.,
v=(ABC⋯vAvBvC⋯).
Here the domain consists of certain sets A,B,C,…. This leads us to the following definition.
A set function is a mapping
s:M→E
whose domain is a set family M.
The range space E is assumed to be E1,E∗,C (the complex field), En, or another normed space. Thus s may be real, extended real, complex, or vector valued.
To each set X∈M, the function s assigns a unique function value denoted s(X) or sX (which is an element of the range space E).
We say that s is finite on a set family N⊆M iff
(∀X∈N)|sX|<∞;
briefly, |s|<∞ on N. (This is automatic if s is complex or vector valued.)
We call s semifinite if at least one of ±∞ is excluded as function value, e.g., if s≥0 on M; i.e.,
s:M→[0,∞].
(The symbol ∞ stands for +∞ throughout).
A set function
s:M→E
is called additive (or finitely additive) on N⊆M iff for any finite disjoint union ⋃kAk, we have
∑ksAk=s(⋃kAk),
provided ⋃kAk and all the Ak are N-sets.
If this also holds for countable disjoint unions, s is called σ-additive (or countably additive or completely additive) on N.
If N=M here, we simply say that s is additive (σ-additive, respectively).
Note 1. As ⋃Ak is independent of the order of the Ak,σ-additivity pre-supposes and implies that the series
∑sAk
is permutable (§2) for any disjoint sequence
{Ak}⊆N.
(The partial sums do exist, by our conventions (2*) in Chapter 4, §4.)
The set functions in the examples below are additive; v is even σ-additive (Corollary 1 in §2).
Examples (b)-(d) show that set functions may arise from ordinary "point functions."
(a) The volume function v:C→E1 on C (= intervals in En), discussed above, is called the Lebesgue premeasure (in En).
(b) Let M={all finite intervals I⊂E1}.
Given f:E1→E, set
(∀I∈M)sI=Vf[¯I],
the total variation of f on the closure of I (Chapter 5, §7).
Then s:M→[0,∞] is additive by Theorem 1 of Chapter 5, §7.
(c) Let M and f be as in Example (b).
Suppose f has an antiderivative (Chapter 5, §5) on E1. For each interval X with endpoints a,b∈E1(a≤b), set
sX=∫baf.
This yields a set function s:M→E (real, complex, or vector valued), additive by Corollary 6 in Chapter 5, §5.
(d) Let C={all finite intervals in E1}.
Suppose
α:E1→E1
has finite one-sided limits
α(p+) and α(p−)
at each p∈E1. The Lebesgue-Stieltjes (LS) function
sα:C→E1
(important for Lebesgue-Stieltjes integration) is defined as follows.
Set sα∅=0. For nonvoid intervals, including [a,a]={a}, set
sα[a,b]=α(b+)−α(a−),sα(a,b]=α(b+)−α(a+),sα[a,b)=α(b−)−α(a−), and sα(a,b)=α(b−)−α(a+).
For the properties of sα see Problem 7ff. , below.
(e) Let mX be the mass concentrated in the part X of the physical space S. Then m is a nonnegative set function defined on
2S={ all subsets X⊆S} (§3).
If instead mX were the electric load of X, then m would be sign changing.
II. The rest of this section is redundant for a "limited approach."
Let s:M→E be additive on N⊆M. Let
A,B∈N,A⊆B.
Then we have the following.
(1) If |sA|<∞ and B−A∈N, then
s(B−A)=sB−sA ("subtractivity").
(2) If ∅∈N, then s∅=0 provided |sX|<∞ for at least one X∈N.
(3) If N is a semiring, then sA=±∞ implies |sB|=∞. Hence
|sB|<∞⇒|sA|<∞.
If further s is semifinite then
sA=±∞⇒sB=±∞
(same sign).
- Proof
-
(1) As B⊇A, we have
B=(B−A)∪A (disjoint);
so by additivity,
sB=s(B−A)+sA.
If |sA|<∞, we may transpose to get
sB−sA=s(B−A),
as claimed.
(2) Hence
s∅=s(X−X)=sX−sX=0
if X,∅∈N, and |sX|<∞.
(3) If N is a semiring, then
B−A=n⋃k=1Ak (disjoint)
for some N-sets Ak; so
B=A∪n⋃k=1Ak (disjoint).
By additivity,
sB=sA+n∑k=1sAk;
so by our conventions,
|sA|=∞⇒|sB|=∞.
If, further, s is semifinite, one of ±∞ is excluded. Thus sA and sB, if infinite, must have the same sign. This completes the proof.◻
In §§1 and 2, we showed how to extend the notion of volume from intervals to a larger set family, preserving additivity. We now generalize this idea.
If
s:C→E
is additive on C, an arbitrary semiring, there is a unique set function
¯s:Cs→E,
additive on Cs, with ¯s=s on C, i.e.,
¯sX=sX for X∈C.
We call ¯s the additive extension of s to Cs=C′s (Corollary 2 in §3).
- Proof
-
If s≥0(s:C→[0,∞]), proceed as in Lemma 1 and Corollary 2, all of §1.
The general proof (which may be omitted or deferred) is as follows.
Each X∈C′s has the form
X=m⋃i=1Xi(disjoint),Xi∈C.
Thus if ¯s is to be additive, the only way to define it is to set
¯sX=m∑i=1sXi.
This already makes ¯s unique, provided we show that
m∑i=1sXi
does not depend on the particular decomposition
X=m⋃i=1Xi
(otherwise, all is ambiguous).
Then take any other decomposition
X=n⋃k=1Yk (disjoint),Yk∈C.
Additivity implies
(∀i,k)sXi=n∑k=1s(Xi∩Yk) and sYk=m∑i=1s(Xi∩Yk).
(Verify!) Hence
m∑i=1sXi=∑i,ks(Xi∩Yk)=n∑k=1sYk.
Thus, indeed, it does not matter which particular decomposition we choose, and our definition of ¯s is unambiguous.
If X∈C, we may choose (say)
X=1⋃i=1Xi,X1=X;
so
¯sX=sX1=sX;
i.e., ¯s=s on C, as required.
Finally, for the additivity of ¯s, let
A=m⋃k=1Bk (disjoint),A,Bk∈C′s.
Here we may set
Bk=nk⋃i=1Cki (disjoint),Cki∈C.
Then
A=⋃k,iCki (disjoint);
so by our definition of ¯s,
¯sA=∑k,isCki=m∑k=1(nk∑i=1sCki)=m∑k=1¯sBk,
as required.◻
Continuity. We write Xn↗X to mean that
X=∞⋃n=1Xn
and {Xn}↑, i.e.,
Xn⊆Xn+1,n=1,2,….
Similarly, Xn↘X iff
X=∞⋂n=1Xn
and {Xn}↓, i.e.,
Xn⊇Xn+1,n=1,2,….
In both cases, we set
X=limn→∞Xn.
This suggests the following definition.
A set function s:M→E is said to be
(i) left continuous (on M) iff
sX=limn→∞sXn
whenever Xn↗X and X,Xn∈M;
(ii) right continuous iff
sX=limn→∞sXn
whenever Xn↘X, with X,Xn∈M and |sXj|<∞.
Thus in case (i),
limn→∞sXn=s∞⋃n=1Xn
if all Xn and ⋃∞n=1Xn are M-sets.
In case (ii),
limn→∞sXn=s∞⋂n=1Xn
if all Xn and ⋂∞n=1Xn are in M, and |sX1|<∞.
Note 2. The last restriction applies to right continuity only. (We choose simply to exclude from consideration sequences {Xn}↓, with |sX1|=∞; see Problem 4.)
If s:C→E is σ-additive and semifinite on C,a semiring, then s is both left and right continuous (briefly, continuous).
- Proof
-
We sketch the proof for rings; for semirings, see Problem 1.
Left continuity. Let Xn↗X with Xn,X∈C and
X=∞⋃n=1Xn.
If sXn=±∞ for some n, then (Lemma 3)
sX=sXm=±∞ for m≥n,
since X⊇Xm⊇Xn; so
limsXm=±∞=sX,
as claimed.
Thus assume all sXn finite; so s∅=0, by Lemma 2.
Set X0=∅. As is easily seen,
X=∞⋃n=1Xn=∞⋃n=1(Xn−Xn−1) (disjoint),
and
(∀n)Xn−Xn−1∈C (a ring).
Also,
(∀m≥n)Xm=m⋃n=1(Xn−Xn−1) (disjoint).
(Verify!) Thus by additivity,
sXm=m∑n=1s(Xn−Xn−1),
and by the assumed σ-additivity,
sX=s∞⋃n=1(Xn−Xn−1)=∞∑n=1s(Xn−Xn−1)=limm→∞m∑n=1s(Xn−Xn−1)=limm→∞sXm,
as claimed.
Right continuity. Let Xn↘X with X,Xn∈C,
X=∞⋂n=1Xn,
and
|sX1|<∞.
As X⊆Xn⊆X1, Lemma 3 yields that
(∀n)|sXn|<∞
and |sX|<∞.
As
X=∞⋂k=1Xk,
we have
(∀n)Xn=X∪∞⋃k=n+1(Xk−1−Xk) (disjoint).
(Verify!) Thus by σ-additivity,
(∀n)sXn=sX+∞∑k=n+1s(Xk−1−Xk),
with |sX|<∞,|sXn|<∞ (see above).
Hence the sum
∞∑k=n+1s(Xk−1−Xk)=sXn−sX
is finite. Therefore, it tends to 0 as n→∞ (being the "remainder term" of a convergent series). Thus n→∞ yields
limn→∞sXn=sX+lim∞∑k=n+1s(Xk−1−Xk)=sX,
as claimed.◻