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8.4: Integration of Elementary Functions

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In Chapter 5, integration was treated as antidifferentiation. Now we adopt another, measure-theoretical approach.

Lebesgue's original theory was based on Lebesgue measure (Chapter 7, §8). The more general modern treatment develops the integral for functions f:SE in an arbitrary measure space. Henceforth, (S,M,m) is fixed, and the range space E is E1,E,C,En, or another complete normed space. Recall that in such a space, i|ai|< implies that ai converges and is permutable (Chapter 7, §2).

We start with elementary maps, including simple maps as a special case.

Definition

Let f:SE be elementary on AM; so f=ai on Ai for some M-partition

A=iAi (disjoint).

(Note that there may be many such partitions.)

We say that f is integrable (with respect to m), or m-integrable, on A iff

|ai|mAi<.

(The notation "|ai|mAi" always makes sense by our conventions (2*) in Chapter 4, §4.) If m is Lebesgue measure, then we say that f is Lebesgue integrable, or L-integrable.

We then define Af, the m-integral of f on A, by

Af=Afdm=iaimAi.

(The notation "dm" is used to specify the measure m.)

The "classical" notation for Afdm is Af(x)dm(x).

Note 1. The assumption

|ai|mAi<

implies

(i)|ai|mAi<;

so ai=0 if mAi=, and mAi=0 if |ai|=. Thus by our conventions, all "bad" terms aimAi vanish. Hence the sum in (1) makes sense and is finite.

Note 2. This sum is also independent of the particular choice of {Ai}. For if {Bk} is another M-partition of A, with f=bk on Bk, say, then f=ai=bk on AiBk whenever AiBk. Also,

(i)Ai=k(AiBk) (disjoint);

so

(i)aimAi=kaim(AiBk),

and hence (see Theorem 2 of Chapter 7, §2, and Problem 11 there)

iaimAi=ikaim(AiBk)=kibkm(AiBk)=kbkmBk.

(Explain!)

This makes our definition (1) unambiguous and allows us to choose any M-partition {Ai}, with f constant on each Ai, when forming integrals (1).

Corollary 8.4.1

Let f:SE be elementary and integrable on AM. Then the following statements are true.

(i) |f|< a.e. on A.

(ii) f and |f| are elementary and integrable on any M-set BA, and

|Bf|B|f|A|f|.

(iii) The set B=A(f0) is σ-finite (Definition 4 in Chapter 7, §5), and

Af=Bf.

(iv) If f=a (constant) on A,

Af=amA.

(v) A|f|=0 iff f=0 a.e. on A.

(vi) If mQ=0, then

Af=AQf

(so we may neglect sets of measure 0 in integrals).

(vii) For any k in the scalar field of E,kf is elementary and integrable, and

Akf=kAf.

Note that if f is scalar valued, k may be a vector. If E=E, we assume kE1.

Proof

(i) By Note 1, |f|=|ai|= only on those Ai with mAi=0. Let Q be the union of all such Ai. Then mQ=0 and |f|< on AQ, proving (i).

(ii) If {Ai} is an M-partition of A,{BAi} is one for B. (Verify!) We have f=ai and |f|=|ai| on BAiAi.

Also,

|ai|m(BAi)|ai|mAi<.

(Why?) Thus f and |f| are elementary and integrable on B, and (ii) easily follows by formula (1).

(iii) By Note 1, f=0 on Ai if mAi=. Thus f0 on Ai only if mAi<. Let {Aik} be the subsequence of those Ai on which f0; so

(k)mAik<.

Also,

B=A(f0)=kAikM (σ-finite!).

By (ii), f is elementary and integrable on B. Also,

Bf=kaikmAik,

while

Af=iaimAi.

These sums differ only by terms with ai=0. Thus (iii) follows.

The proof of (iv)-(vii) is left to the reader.

Note 3. If f:SE is elementary and sign-constant on A, we also allow that

Af=iaimAi=±.

Thus here Af exists even if f is not integrable. Apart from claims of integrability and σ-finiteness, Corollary 1(ii)-(vii) hold for such f, with the same proofs.

Example

Let m be Lebesgue measure in E1. Define f=1 on R (rationals) and f=0 on E1R; see Chapter 4, §1, Example (c). Let A=[0,1].

By Corollary 1 in Chapter 7, §8, ARM and m(AR)=0. Also, ARM.

Thus {AR,AR} is an M-partition of A, with f=1 on AR and f=0 on AR.

Hence f is elementary and integrable on A, and

Af=1m(AR)+0m(AR)=0.

Thus f is L-integrable (even though it is nowhere continuous).

Theorem 8.4.1 (additivity)

(i) If f:SE is elementary and integrable or elementary and nonnegative on AM, then

Af=kBkf

for any M-partition {Bk} of A.

(ii) If f is elementary and integrable on each set Bk of a finite M-partition

A=kBk,

it is elementary and integrable on all of A, and (2) holds again.

Proof

(i) If f is elementary and integrable or elementary and nonnegative on A=kBk, it is surely so on each Bk by Corollary 2 of §1 and Corollary 1(ii) above.

Thus for each k, we can fix an M-partition Bk=iAki, with f constant (f=aki) on Aki,i=1,2,. Then

A=kBk=kiAki

is an M-partition of A into the disjoint sets AkiM.

Now, by definition,

Bkf=iakimAki

and

Af=k,iakimAki=k(iakimAki)=kBkf

by rules for double series. This proves formula (2).

(ii) If f is elementary and integrable on Bk(k=1,,n), then with the same notation, we have

i|aki|mAki<

(by integrability); hence

nk=1i|aki|mAki<.

This means, however, that f is elementary and integrable on A, and so clause (ii) follows.

Caution. Clause (ii) fails if the partition {Bk} is infinite.

Theorem 8.4.2

(i) If f,g:SE are elementary and nonnegative on A, then

A(f+g)=Af+Ag.

(ii) If f,g:SE are elementary and integrable on A, so is f±g, and

A(f±g)=Af±Ag.

Proof

Arguing as in the proof of Theorem 1 of §1, we can make f and g constant on sets of one and the same M-partition of A, say, f=ai and g=bi on AiM; so

f±g=ai±bi on Ai,i=1,2,.

In case (i), f,g0; so integrability is irrelevant by Note 3, and formula (1) yields

A(f+g)=i(ai+bi)mAi=iaimAi+bimAi=Af+Ag.

In (ii), we similarly obtain

i|ai±bi|mAi|ai|mAi+i|bi|mAi<.

(Why?) Thus f±g is elementary and integrable on A. As before, we also get

A(f±g)=Af±Ag,

simply by rules for addition of convergent series. (Verify!)

Note 4. As we know, the characteristic function CB of a set BS is defined

CB(x)={1,xB,0,xSB.

If g:SE is elementary on A, so that

g=ai on Ai,1,2,,

for some M-partition

A=Ai,

then

g=iaiCAi on A.

(This sum always exists for disjoint sets Ai. Why?) We shall often use this notation.

Screen Shot 2019-07-12 at 5.06.00 PM.png

If m is Lebesgue measure in E1, the integral

Ag=iaimAi

has a simple geometric interpretation; see Figure 33. Let A=[a,b]E1; let g be bounded and nonnegative on E1. Each product aimAi is the area of a rectangle with base Ai and altitude ai. (We assume the Ai to be intervals here.) The total area,

Ag=iaimAi,

can be treated as an approximation to the area under some curve y=f(x), where f is approximated by g (Theorem 3 in §1). Integration historically arose from such approximations.

Integration of elementary extended-real functions. Note 3 can be extended to sign-changing functions as follows.

Definition

If

f=iaiCAi(aiE)

on

A=iAi(AiM),

we set

Af=Af+Af,

with

f+=f00 and f=(f)00;

see §2.

By Theorem 2 in §2, f+ and f are elementary and nonnegative on A; so

Af+ and Af

are defined by Note 3, and so is

Af=Af+Af

by our conventions (2*) in Chapter 4, §4.

We shall have use for formula (3), even if

Af+=Af=;

then we say that Af is unorthodox and equate it to +, by convention; cf. Chapter 4, §4. (Other integrals are called orthodox.) Thus for elementary and (extended) real functions, Af is always defined. (We further develop this idea in §5.)

Note 5. With f as above, we clearly have

f+=a+i and f=ai on Ai,

where

a+i=max(ai,0) and ai=max(ai,0).

Thus

Af+=a+imAi and Af=aimAi,

so that

Af=Af+Af=ia+imAiiaimAi.

If Af+< or Af<, we can subtract the two series termwise (Problem 14 of Chapter 4, §13) to obtain

Af=i(a+iai)mAi=iaimAi

for a+iai=ai. Thus formulas (3) and (4) agree with our previous definitions.


This page titled 8.4: Integration of Elementary Functions 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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