Processing math: 57%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

9.2: More on L-Integrals and Absolute Continuity

( \newcommand{\kernel}{\mathrm{null}\,}\)

I. In this section, we presuppose the "starred" §10 in Chapter 7. First, however, we add some new ideas that do not require any starred material. The notation is as in §1.

Definition

Given F:E1E,pE1, and qE, we write

qDF(p)

and call q an F-derivate at p iff

q=limkF(xk)F(p)xkp

for at least one sequence xkp(xkp).

If F has a derivative at p, it is the only F-derivate at p; otherwise, there may be many derivates at p (finite or not).

Such derivates must exist if E=E1(E). Indeed, given any pE1, let

xk=p+1kp;

let

yk=F(xk)F(p)xkp,k=1,2,

By the compactness of E (Chapter 4, §6, example (d)), {yk} must have a subsequence {yki} with a limit qE (e.g., take q=lim_yk), and so qDF(p).

We also obtain the following lemma.

Lemma 9.2.1

If F:E1E has no negative derivates on AQ, where A= [a,b] and mQ=0, and if no derivate of F on A equals , then F on A.

Proof

First, suppose F has no negative derivates on A at all. Fix ε>0 and set

G(x)=F(x)+εx.

Seeking a contradiction, suppose ap<qb, yet G(q)<G(p). Then if

r=12(p+q),

one of the intervals [p,r] and [r,q] (call it [p1,q1]) satisfies G(q1)<G(p1).

Let

r1=12(p1+q1).

Again, one of [p1,r1] and [r1,q1] (call it [p2,q2]) satisfies G(q2)<G(p2). Let

r2=12(p2+q2),

and so on.

Thus obtain contracting intervals [pn,qn], with

G(qn)<G(pn),n=1,2,.

Now, by Theorem 5 of Chapter 4, §6, let

pon=1[pn,qn].

Then set xn=qn if G(qn)<G(po), and xn=pn otherwise. Then

G(xn)G(po)xnpo<0

and xnpo. By the compactness of E, fix a subsequence

G(xnk)G(po)xnkpocE,

say. Then c0 is a G-derivate at poA.

But this is impossible; for by our choice of G and our assumption, all derivates of G are >0. (Why?)

This contradiction shows that ap<qb implies G(p)G(q), i.e.,

F(p)+εpF(q)+εq.

Making ε0, we obtain F(p)F(q) when ap<qb, i.e., F on A.

Now, for the general case, let Q be the set of all pA that have at least one DF(p)<0; so mQ=0.

Let g be as in Problem 8 of §1; so g= on Q. Given ε>0, set

G=F+εg.

As g, we have

(x,pA)G(x)G(p)xpF(x)F(p)xp.

Hence DG(p)0 if pQ.

If, however, pQ, then g(p)= implies DG(p)0. (Why?) Thus all DG(p) are 0; so by what was proved above, G on A. It follows, as before, that F on A, also. The lemma is proved.

We now proceed to prove Theorems 3 and 4 of §1. To do this, we shall need only one "starred" theorem (Theorem 3 of Chapter 7, §10).

Proof of Theorem 3 of §1. (1) First, let f be bounded:

|f|Kon A.

Via components and by Corollary 1 of Chapter 8, §6, all reduces to the real positive case f0 on A. (Explain!)

Then (Theorem 1(f) of Chapter 8, §5) ax<yb implies

LxafLyaf,

i.e., F(x)F(y); so F and F0 on A.

Now, by Theorem 3 of Chapter 7, §10, F is a.e. differentiable on A. Thus exactly as in Theorem 2 in §1, we set

fn(t)=F(t+1n)F(t)1nF(t) a.e.

Since all fn are m-measurable on A (why?), so is F. Moreover, as |f|K, we obtain (as in Lemma 1 of §1)

|fn(x)|=n(Lx+1/nxf)nKn=K.

Thus by Theorem 5 from Chapter 8, §6 (with g=K),

LxaF=limnLxafn=Lxaf

(Lemma 1 of §1). Hence

Lxa(Ff)=0,xA,

and so (Problem 10 in §1) F=f (a.e.) as claimed.

(2) If f is not bounded, we still can reduce all to the case f0,f:E1E so that F and F0 on A.

If so, we use "truncation": For n=1,2,, set

gn={f on A(fn), and 0 elsewhere.

Then (see Problem 12 in §1) the gn are L-measurable and bounded, hence L-integrable on A, with gnf and

0gnf

on A. By the first part of the proof, then,

ddxLxagn=gn a.e. on A,n=1,2,.

Also, set (n)

Fn(x)=Lxa(fgn)0;

so Fn is monotone () on A. (Why?)

Thus by Theorem 3 in Chapter 7, §10, each Fn has a derivative at almost every xA,

Fn(x)=ddx(LxafLxagn)=F(x)gn(x)0a.e. on A.

Making n and recalling that gnf on A, we obtain

F(x)f(x)0a.e. on A.

Thus

Lxa(Ff)0.

But as F (see above), Problem 11 of §1 yields

LxaFF(x)F(a)=Lxaf;

so

Lxa(Ff)=LxaFLxaf0.

Combining, we get

(xA)Lxa(Ff)=0;

so by Problem 10 of §1, F=f a.e. on A, as required.

Proof of Theorem 4 of §1. Via components, all again reduces to a real f.

Let (n)

gn={f on A(fn),0 on A(f>n);

so gnf (pointwise), gnf,gnn, and |gn||f|.

This makes each gn L-integrable on A. Thus as before, by Theorem 5 of Chapter 8, §6,

limnLxagn=Lxaf,xA.

Now, set

Fn(x)=F(x)Lxagn.

Then by Theorem 3 of §1 (already proved),

Fn(x)=F(x)ddxLxagn=f(x)gn(x)0a.e. on A

(since gnf).

Thus Fn has solely nonnegative derivates on AQ(mQ=0). Also, as gnn, we get

1xpLxagnn,

even if x<p. (Why?) Hence

ΔFnΔxΔFΔxn,

as

Fn(x)=F(x)Lxagn.

Thus none of the Fn-derivates on A can be .

By Lemma 1, then, Fn is monotone () on A; so Fn(x)Fn(a), i.e.,

F(x)LxagnF(a)Laagn=F(a),

or

F(x)F(a)Lxagn,xA,n=1,2,.

Hence by (1),

F(x)F(a)Lxaf,xA.

For the reverse inequality, apply the same formula to f. Thus we obtain the desired result:

F(x)=F(a)+Lxaf for xA.

Note 1. Formula (2) is equivalent to F=Lf on A (see the last part of §1). For if (2) holds, then

F(x)=c+Lxaf,

with c=F(a); so F=Lf by definition.

Conversely, if

F(x)=c+Lxaf,

set x=a to find c=F(a).

II. A bsolute continuity redefined.

Definition

A map f:E1E is absolutely continuous on an interval IE1 iff for every ε>0, there is δ>0 such that

\sum_{i=1}^{r}\left(b_{i}-a_{i}\right)<\delta \text { implies } \sum_{i=1}^{r}\left|f\left(b_{i}\right)-f\left(a_{i}\right)\right|<\varepsilon

for any disjoint intervals \left(a_{i}, b_{i}\right), with a_{i}, b_{i} \in I.

From now on, this replaces the "weaker" definition given in Chapter 5, §8. The reader will easily verify the next three "routine" propositions.

Theorem \PageIndex{1}

If f, g, h : E^{1} \rightarrow E^{*}(C) are absolutely continuous on A=[a, b] so are

f \pm g, h f, \text { and }|f|.

So also is f / h if

(\exists \varepsilon>0) \quad|h| \geq \varepsilon \text { on } A.

All this also holds if f, g : E^{1} \rightarrow E are vector valued and h is scalar valued. Finally, if E \subseteq E^{*}, then

f \vee g, f \wedge g, f^{+}, \text {and } f^{-}

are absolutely continuous along with f and g.

Corollary \PageIndex{1}

A function F : E^{1} \rightarrow E^{n}\left(C^{n}\right) is absolutely continuous on A= [a, b] iff all its components F_{1}, \ldots, F_{n} are.

Hence a complex function F : E^{1} \rightarrow C is absolutely continuous iff its real and imaginary parts, F_{re} and F_{im}, are.

Corollary \PageIndex{2}

If f : E^{1} \rightarrow E is absolutely continuous on A=[a, b], it is bounded, is uniformly continuous, and has bounded variation, V_{f}[a, b]<\infty all on A.

Lemma \PageIndex{2}

If F : E^{1} \rightarrow E^{n}\left(C^{n}\right) is of bounded variation on A=[a, b], then

(i) F is a.e. differentiable on A, and

(ii) F^{\prime} is L-integrable on A.

Proof

Via components (Theorem 4 of Chapter 5, §7), all reduces to the real case, F : E^{1} \rightarrow E^{1}.

Then since V_{F}[A]<\infty, we have

F=g-h

for some nondecreasing g and h (Theorem 3 in Chapter 5, §7).

Now, by Theorem 3 from Chapter 7, §10, g and h are a.e. differentiable on A. Hence so is

g-h=F.

Moreover, g^{\prime} \geq 0 and h^{\prime} \geq 0 since g \uparrow and h \uparrow.

Thus for the L-integrability of F^{\prime}, proceed as in Problem 11 in §1, i.e., show that F^{\prime} is measurable on A and that

L \int_{a}^{b} F^{\prime}=L \int_{a}^{b} g^{\prime}-L \int_{a}^{b} h^{\prime}

is finite. This yields the result.\quad \square

Theorem \PageIndex{2} (Lebesgue)

If F : E^{1} \rightarrow E^{n}\left(C^{n}\right) is absolutely continuous on A=[a, b], then the following are true:

(i*) F is a.e. differentiable, and F^{\prime} is L-integrable, on A.

(ii*) If, in addition, F^{\prime}=0 a.e. on A, then F is constant on A.

Proof

Assertion (i*) is immediate from Lemma 2, since any absolutely continuous function is of bounded variation by Corollary 2.

(ii*) Now let F^{\prime}=0 a.e. on A. Fix any

B=[a, c] \subseteq A

and let Z consist of all p \in B at which the derivative F^{\prime}=0.

Given \varepsilon>0, let \mathcal{K} be the set of all closed intervals [p, x], p<x, such that

\left|\frac{\Delta F}{\Delta x}\right|=\left|\frac{F(x)-F(p)}{x-p}\right|<\varepsilon.

By assumption,

\lim _{x \rightarrow p} \frac{\Delta F}{\Delta x}=0 \quad(p \in Z),

and m(B-Z)=0; B=[a, c] \in \mathcal{M}^{*}. If p \in Z, and x-p is small enough, then

\left|\frac{\Delta F}{\Delta x}\right|<\varepsilon,

i.e., [p, x] \in \mathcal{K}.

It easily follows that \mathcal{K} covers Z in the Vitali sense (verify!); so for any
\delta>0, Theorem 2 of Chapter 7, §10 yields disjoint intervals

I_{k}=\left[p_{k}, x_{k}\right] \in \mathcal{K}, I_{k} \subseteq B,

with

m^{*}\left(Z-\bigcup_{k=1}^{q} I_{k}\right)<\delta,

hence also

m\left(B-\bigcup_{k=1}^{q} I_{k}\right)<\delta

(for m(B-Z)=0). But

\begin{aligned} B-\bigcup_{k=1}^{q} I_{k} &=[a, c]-\bigcup_{k=1}^{q-1}\left[p_{k}, x_{k}\right] \\ &=\left[a, p_{1}\right) \cup \bigcup_{k=1}^{q-1}\left[x_{k}, p_{k+1}\right) \cup\left[x_{q}, c\right] \quad\left(\text { if } x_{k}<p_{k}<x_{k+1}\right); \end{aligned}

so

m\left(B-\bigcup_{k=1}^{q} I_{k}\right)=\left(p_{1}-a\right)+\sum_{k=1}^{q-1}\left(p_{k+1}-x_{k}\right)+\left(c-x_{q}\right)<\delta.

Now, as F is absolutely continuous, we can choose \delta>0 so that (3) implies

\left|F\left(p_{1}\right)-F(a)\right|+\sum_{k=1}^{q-1}\left|F\left(p_{k+1}\right)-F\left(x_{k}\right)\right|+\left|F(c)-F\left(x_{q}\right)\right|<\varepsilon.

But I_{k} \in \mathcal{K} also implies

\left|F\left(x_{k}\right)-F\left(p_{k}\right)\right|<\varepsilon\left(x_{k}-p_{k}\right)=\varepsilon \cdot m I_{k}.

Hence

\left|\sum_{k=1}^{q}\left[F\left(x_{k}\right)-F\left(p_{k}\right)\right]\right|<\varepsilon \sum_{k=1}^{q} m I_{k} \leq \varepsilon \cdot m B=\varepsilon(c-p).

Combining with (4), we get

|F(c)-F(a)| \leq \varepsilon(1+c-a) \rightarrow 0 \text { as } \varepsilon \rightarrow 0;

so F(c)=F(a). As c \in A was arbitrary, F is constant on A, as claimed.\quad \square

Note 2. This shows that Cantor's function (Problem 6 of Chapter 4, §5) is not absolutely continuous, even though it is continuous and monotone, hence of bounded variation on [0,1]. Indeed (see Problem 2 in §1), it has a zero derivative a.e. on [0,1] but is not constant there. Thus absolute continuity, as now defined, differs from its "weak" counterpart (Chapter 5, §8).

Theorem \PageIndex{3}

A map F : E^{1} \rightarrow E^{1}\left(C^{n}\right) is absolutely continuous on A= [a, b] iff

F=L \int f \quad \text { on } A

for some function f; and then

F(x)=F(a)+L \int_{a}^{x} f, \quad x \in A.

Briefly: Absolutely continuous maps are exactly all L-primitives.

Proof

If F=L \int f, then by Theorem 1 of §1, F is absolutely continuous on A, and by Note 1,

F(x)=F(a)+L \int_{a}^{x} f, \quad x \in A.

Conversely, if F is absolutely continuous, then by Theorem 2, it is a.e. differentiable and F^{\prime}=f is L-integrable (all on A). Let

H(x)=L \int_{a}^{x} f, \quad x \in A.

Then H, too, is absolutely continuous and so is F-H. Also, by Theorem 3 of §1,

H^{\prime}=f=F^{\prime},

and so
(F-H)^{\prime}=0 \quad \text {a.e. on } A.

By Theorem 2, F-H=c; i.e.,

F(x)=c+H(x)=c+L \int_{a}^{x} f,

and so F=L \int f on A, as claimed.\quad \square

Corollary \PageIndex{3}

If f, F : E^{1} \rightarrow E^{*}\left(E^{n}, C^{n}\right), we have

F=L \int f

on an interval I \subseteq E^{1} iff F is absolutely continuous on I and F^{\prime}=f a.e. on I.

(Use Problem 3 in §1 and Theorem 3.)

Note 3. This (or Theorem 3) could serve as a definition. Comparing ordinary primitives

F=\int f

with L-primitives

F=L \int f,

we see that the former require F to be just relatively continuous but allow only a countable "exceptional" set Q, while the latter require absolute continuity but allow Q to even be uncountable, provided m Q=0.

The simplest and "strongest" kind of absolutely continuous functions are so-called Lipschitz maps (see Problem 6). See also Problems 7 and 10.

III. We conclude with another important idea, due to Lebesgue.

Definition

We call p \in E^{1} a Lebesgue point ("L-point") of f : E^{1} \rightarrow E iff

(i) f is L-integrable on some G_{p}(\delta);

(ii) q=f(p) is finite; and

(iii) \lim _{x \rightarrow p} \frac{1}{x-p} L \int_{p}^{x}|f-q|=0.

The Lebesgue set of f consists of all such p.

Corollary \PageIndex{4}

Let

F=L \int f \quad \text { on } A=[a, b].

If p \in A is an L-point of f, then f(p) is the derivative of F at p (but the converse fails).

Proof

By assumption,

F(x)=c+L \int_{p}^{x} f, \quad x \in G_{p}(\delta),

and

\frac{1}{|\Delta x|}\left|L \int_{p}^{x}(f-q)\right| \leq \frac{1}{|\Delta x|} L \int_{p}^{x}|f-q| \rightarrow 0

as x \rightarrow p. (Here q=f(p) and \Delta x=x-p.)

Thus with x \rightarrow p, we get

\begin{aligned}\left|\frac{F(x)-F(p)}{x-p}-q\right| &=\frac{1}{|x-p|}\left|L \int_{p}^{x} f-(x-p) q\right| \\ &=\frac{1}{|x-p|}\left|L \int_{p}^{x} f-L \int_{p}^{x}(q)\right| \rightarrow 0, \end{aligned}

as required.\quad \square

Corollary \PageIndex{5}

Let f : E^{1} \rightarrow E^{n}\left(C^{n}\right). Then p is an L-point of f iff it is an L-point for each of the n components, f_{1}, \ldots, f_{n}, of f.

Proof

(Exercise!)

Theorem \PageIndex{4}

If f : E^{1} \rightarrow E^{*}\left(E^{n}, C^{n}\right) is L-integrable on A=[a, b], then almost all p \in A are Lebesgue points of f.

Note that this strengthens Theorem 3 of §1.

Proof

By Corollary 5, we need only consider the case f : E^{1} \rightarrow E^{*}.

For any r \in E^{1},|f-r| is L-integrable on A; so by Theorem 3 of §1, setting

F_{r}(x)=L \int_{a}^{x}|f-r|,

we get

F_{r}^{\prime}(p)=\lim _{x \rightarrow p} \frac{1}{|x-p|} L \int_{p}^{x}|f-r|=|f(p)-r|

for almost all p \in A.

Now, for each r, let A_{r} be the set of those p \in A for which (5) fails; so m A_{r}=0. Let \left\{r_{k}\right\} be the sequence of all rationals in E^{1}. Let

Q=\bigcup_{k=1}^{\infty} A_{r_{k}} \cup\{a, b\} \cup A_{\infty},

where

A_{\infty}=A(|f|=\infty);

so m Q=0. (Why?)

To finish, we show that all p \in A-Q are L-points of f. Indeed, fix any p \in A-Q and any \varepsilon>0. Let q=f(p). Fix a rational r such that

|q-r|<\frac{\varepsilon}{3}.

Then

| | f-r|-| f-q| | \leq|(f-r)-(f-q)|=|q-r|<\frac{\varepsilon}{3} \text { on } A-A_{\infty}.

Hence as m A_{\infty}=0, we have

\left|L \int_{p}^{x}\right| f-r\left|-L \int_{p}^{x}\right| f-q| | \leq L \int_{p}^{x}\left(\frac{\varepsilon}{3}\right)=\frac{\varepsilon}{3}|x-p|.

Since

p \notin Q \supseteq \bigcup_{k} A_{r_{k}},

formula (5) applies. So there is \delta>0 such that |x-p|<\delta implies

\left|\left(\frac{1}{|x-p|} L \int_{p}^{x}|f-r|\right)-\right| f(p)-r| |<\frac{\varepsilon}{3}.

As

|f(p)-r|=|q-r|<\frac{\varepsilon}{3},

we get

\begin{aligned} \frac{1}{|x-p|} L \int_{p}^{x}|f-r| & \leq\left|\left(\frac{1}{|x-p|} L \int_{p}^{x}|f-r|\right)-\right| q-r| |+|q-r| \\ &<\frac{\varepsilon}{3}+\frac{\varepsilon}{3}=\frac{2 \varepsilon}{3}. \end{aligned}

Hence

L \int_{p}^{x}|f-r|<\frac{2 \varepsilon}{3}|x-p|.

Combining with (6), we have

\frac{1}{|x-p|} L \int_{p}^{x}|f-q|<\frac{\varepsilon}{3}+\frac{2 \varepsilon}{3}=\varepsilon

whenever |x-p|<\delta. Thus

\lim _{x \rightarrow p} \frac{1}{|x-p|} L \int_{p}^{x}|f-q|=0,

as required.\quad \square


This page titled 9.2: More on L-Integrals and Absolute 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.

Support Center

How can we help?