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.
Given F:E1→E,p∈E1, and q∈E, we write
q∼DF(p)
and call q an F-derivate at p iff
q=limk→∞F(xk)−F(p)xk−p
for at least one sequence xk→p(xk≠p).
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 p∈E1, let
xk=p+1k→p;
let
yk=F(xk)−F(p)xk−p,k=1,2,…
By the compactness of E∗ (Chapter 4, §6, example (d)), {yk} must have a subsequence {yki} with a limit q∈E∗ (e.g., take q=lim_yk), and so q∼DF(p).
We also obtain the following lemma.
If F:E1→E∗ has no negative derivates on A−Q, 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 a≤p<q≤b, 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
po∈∞⋂n=1[pn,qn].
Then set xn=qn if G(qn)<G(po), and xn=pn otherwise. Then
G(xn)−G(po)xn−po<0
and xn→po. By the compactness of E∗, fix a subsequence
G(xnk)−G(po)xnk−po→c∈E∗,
say. Then c≤0 is a G-derivate at po∈A.
But this is impossible; for by our choice of G and our assumption, all derivates of G are >0. (Why?)
This contradiction shows that a≤p<q≤b implies G(p)≤G(q), i.e.,
F(p)+εp≤F(q)+εq.
Making ε→0, we obtain F(p)≤F(q) when a≤p<q≤b, i.e., F↑ on A.
Now, for the general case, let Q be the set of all p∈A 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,p∈A)G(x)−G(p)x−p≥F(x)−F(p)x−p.
Hence DG(p)≥0 if p∉Q.
If, however, p∈Q, 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 f≥0 on A. (Explain!)
Then (Theorem 1(f) of Chapter 8, §5) a≤x<y≤b implies
L∫xaf≤L∫yaf,
i.e., F(x)≤F(y); so F↑ and F′≥0 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)1n→F′(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(L∫x+1/nxf)≤n⋅Kn=K.
Thus by Theorem 5 from Chapter 8, §6 (with g=K),
L∫xaF′=limn→∞L∫xafn=L∫xaf
(Lemma 1 of §1). Hence
L∫xa(F′−f)=0,x∈A,
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 f≥0,f:E1→E∗ so that F↑ and F′≥0 on A.
If so, we use "truncation": For n=1,2,…, set
gn={f on A(f≤n), and 0 elsewhere.
Then (see Problem 12 in §1) the gn are L-measurable and bounded, hence L-integrable on A, with gn→f and
0≤gn≤f
on A. By the first part of the proof, then,
ddxL∫xagn=gn a.e. on A,n=1,2,….
Also, set (∀n)
Fn(x)=L∫xa(f−gn)≥0;
so Fn is monotone (↑) on A. (Why?)
Thus by Theorem 3 in Chapter 7, §10, each Fn has a derivative at almost every x∈A,
F′n(x)=ddx(L∫xaf−L∫xagn)=F′(x)−gn(x)≥0a.e. on A.
Making n→∞ and recalling that gn→f on A, we obtain
F′(x)−f(x)≥0a.e. on A.
Thus
L∫xa(F′−f)≥0.
But as F↑ (see above), Problem 11 of §1 yields
L∫xaF′≤F(x)−F(a)=L∫xaf;
so
L∫xa(F′−f)=L∫xaF′−L∫xaf≤0.
Combining, we get
(∀x∈A)L∫xa(F′−f)=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(f≤n),0 on A(f>n);
so gn→f (pointwise), gn≤f,gn≤n, and |gn|≤|f|.
This makes each gn L-integrable on A. Thus as before, by Theorem 5 of Chapter 8, §6,
limn→∞L∫xagn=L∫xaf,x∈A.
Now, set
Fn(x)=F(x)−L∫xagn.
Then by Theorem 3 of §1 (already proved),
F′n(x)=F′(x)−ddxL∫xagn=f(x)−gn(x)≥0a.e. on A
(since gn≤f).
Thus Fn has solely nonnegative derivates on A−Q(mQ=0). Also, as gn≤n, we get
1x−pL∫xagn≤n,
even if x<p. (Why?) Hence
ΔFnΔx≥ΔFΔx−n,
as
Fn(x)=F(x)−L∫xagn.
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)−L∫xagn≥F(a)−L∫aagn=F(a),
or
F(x)−F(a)≥L∫xagn,x∈A,n=1,2,….
Hence by (1),
F(x)−F(a)≥L∫xaf,x∈A.
For the reverse inequality, apply the same formula to −f. Thus we obtain the desired result:
F(x)=F(a)+L∫xaf for x∈A.◻
Note 1. Formula (2) is equivalent to F=L∫f on A (see the last part of §1). For if (2) holds, then
F(x)=c+L∫xaf,
with c=F(a); so F=L∫f by definition.
Conversely, if
F(x)=c+L∫xaf,
set x=a to find c=F(a).
II. A bsolute continuity redefined.
A map f:E1→E is absolutely continuous on an interval I⊆E1 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.
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.
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.
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.
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
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 intervalsI_{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).
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
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.
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.
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
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!)
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