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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 Cs (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,Cs, or Cσ).

The "volumes" assigned to such sets are the function values (for C and Cs-sets they are real numbers; for Cσ-sets they may reach +).This is symbolized by

v:CE1

or

v:CσE;

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=(ABCvAvBvC).

Here the domain consists of certain sets A,B,C,. This leads us to the following definition.

Definition 1

A set function is a mapping

s:ME

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 XM, 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 NM iff

(XN)|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 s0 on M; i.e.,

s:M[0,].

(The symbol stands for + throughout).

Definition 2

A set function

s:ME

is called additive (or finitely additive) on NM 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."

Examples

(a) The volume function v:CE1 on C (= intervals in En), discussed above, is called the Lebesgue premeasure (in En).

(b) Let M={all finite intervals IE1}.

Given f:E1E, set

(IM)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,bE1(ab), set

sX=baf.

This yields a set function s:ME (real, complex, or vector valued), additive by Corollary 6 in Chapter 5, §5.

(d) Let C={all finite intervals in E1}.

Suppose

α:E1E1

has finite one-sided limits

α(p+) and α(p)

at each pE1. The Lebesgue-Stieltjes (LS) function

sα:CE1

(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 XS} (§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."

lemmas

Let s:ME be additive on NM. Let

A,BN,AB.

Then we have the following.

(1) If |sA|< and BAN, then

s(BA)=sBsA ("subtractivity").

(2) If N, then s=0 provided |sX|< for at least one XN.

(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 BA, we have

B=(BA)A (disjoint);

so by additivity,

sB=s(BA)+sA.

If |sA|<, we may transpose to get

sBsA=s(BA),

as claimed.

(2) Hence

s=s(XX)=sXsX=0

if X,N, and |sX|<.

(3) If N is a semiring, then

BA=nk=1Ak (disjoint)

for some N-sets Ak; so

B=Ank=1Ak (disjoint).

By additivity,

sB=sA+nk=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.

Theorem 7.4.1

If

s:CE

is additive on C, an arbitrary semiring, there is a unique set function

¯s:CsE,

additive on Cs, with ¯s=s on C, i.e.,

¯sX=sX for XC.

We call ¯s the additive extension of s to Cs=Cs (Corollary 2 in §3).

Proof

If s0(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 XCs has the form

X=mi=1Xi(disjoint),XiC.

Thus if ¯s is to be additive, the only way to define it is to set

¯sX=mi=1sXi.

This already makes ¯s unique, provided we show that

mi=1sXi

does not depend on the particular decomposition

X=mi=1Xi

(otherwise, all is ambiguous).

Then take any other decomposition

X=nk=1Yk (disjoint),YkC.

Additivity implies

(i,k)sXi=nk=1s(XiYk) and sYk=mi=1s(XiYk).

(Verify!) Hence

mi=1sXi=i,ks(XiYk)=nk=1sYk.

Thus, indeed, it does not matter which particular decomposition we choose, and our definition of ¯s is unambiguous.

If XC, we may choose (say)

X=1i=1Xi,X1=X;

so

¯sX=sX1=sX;

i.e., ¯s=s on C, as required.

Finally, for the additivity of ¯s, let

A=mk=1Bk (disjoint),A,BkCs.

Here we may set

Bk=nki=1Cki (disjoint),CkiC.

Then

A=k,iCki (disjoint);

so by our definition of ¯s,

¯sA=k,isCki=mk=1(nki=1sCki)=mk=1¯sBk,

as required.

Continuity. We write XnX to mean that

X=n=1Xn

and {Xn}, i.e.,

XnXn+1,n=1,2,.

Similarly, XnX iff

X=n=1Xn

and {Xn}, i.e.,

XnXn+1,n=1,2,.

In both cases, we set

X=limnXn.

This suggests the following definition.

Definition 3

A set function s:ME is said to be

(i) left continuous (on M) iff

sX=limnsXn

whenever XnX and X,XnM;

(ii) right continuous iff

sX=limnsXn

whenever XnX, with X,XnM and |sXj|<.

Thus in case (i),

limnsXn=sn=1Xn

if all Xn and n=1Xn are M-sets.

In case (ii),

limnsXn=sn=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.)

Theorem 7.4.2

If s:CE 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 XnX with Xn,XC and

X=n=1Xn.

If sXn=± for some n, then (Lemma 3)

sX=sXm=± for mn,

since XXmXn; 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(XnXn1) (disjoint),

and

(n)XnXn1C (a ring).

Also,

(mn)Xm=mn=1(XnXn1) (disjoint).

(Verify!) Thus by additivity,

sXm=mn=1s(XnXn1),

and by the assumed σ-additivity,

sX=sn=1(XnXn1)=n=1s(XnXn1)=limmmn=1s(XnXn1)=limmsXm,

as claimed.

Right continuity. Let XnX with X,XnC,

X=n=1Xn,

and

|sX1|<.

As XXnX1, Lemma 3 yields that

(n)|sXn|<

and |sX|<.

As

X=k=1Xk,

we have

(n)Xn=Xk=n+1(Xk1Xk) (disjoint).

(Verify!) Thus by σ-additivity,

(n)sXn=sX+k=n+1s(Xk1Xk),

with |sX|<,|sXn|< (see above).

Hence the sum

k=n+1s(Xk1Xk)=sXnsX

is finite. Therefore, it tends to 0 as n (being the "remainder term" of a convergent series). Thus n yields

limnsXn=sX+limk=n+1s(Xk1Xk)=sX,

as claimed.


This page titled 7.4: Set Functions. Additivity. Continuity is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Elias Zakon (The Trilla Group (support by Saylor Foundation)) via source content that was edited to the style and standards of the LibreTexts platform.

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