VERSIONS OF THE
FUNDAMENTAL THEOREM
OF CALCULUS 1666 - 2008
A talk by
Gunther Hormann
in the DIANA Seminar
Fakultat fur Mathematik, Universitat Wien
corrected and updated version, March 16, 2009
§0. FTC before 1800
Cambridge 1664-1669: lecture courses by Isaac Barrow on mathematics, optics,
and geometry; basic idea that
\nding tangents is inverse to quadrature (calculation of areas)"
is mentioned, but without clear notation, notions or algorithms; Isaac Newton attends
as a student, and is explicitly mentioned by Barrow (in 1670) to have made corrections
and additions to the course material; in Newton's manuscript from 1666 we nd the rst
explicit statement of the form
dA
= f(x),
dx
where A = area underneath the graph of f.
Leibniz 1686 (in Hannover): convinced of
the importance of appropriate symbolic
R
notation and notions; introduces integral sign
the term `function'; states FTC in the form
(to remind of the notion of a sum) and
Z
d
f(x) dx = f(x).
dx
1
Task of integration: for Leibniz: nd an (explicit) antiderivative (or primitive);
for Newton: nd (power) series expansion of the integrand, interchange sum and integral.
(For more information and references see [Vol88, HW96, Sti02, Wu08, Wu09].)
Basic notation and conventions
We x a, b ∈ R with a < b. We will work exclusively on the bounded closed interval
[a, b] and consider functions f : [a, b] → R (everywhere dened and with nite real value
at each point).
A partition of [a, b] is a nite collection of contiguous subintervals [xk−1 , xk ] (k =
1, . . . , N) with a = x0 < x1 < · · · < xN = b. The points x0 , . . . , xN are the division points
of the partition.
A tagged partition of [a, b] is a set
P = {(c1 , [x0 , x1 ]), . . . , (cN , [xN−1 , xN ])},
where ck ∈ [xk−1 , xk ] (k = 1, . . . , N) and x0 , . . . , xN are the division points of a partition.
We call ck the tag of the interval [xk−1 , xk ].
If P is a tagged partition, we call
µ(P) := max{xk − xk−1 | k = 1, . . . , N}
the mesh size of P (which clearly depends only on the underlying partition).
§1. FTC according to Cauchy (1823)
1.1. DEF A Cauchy partition of [a, b] is a tagged partition P with tags ck = xk−1
for all k = 1, . . . , N (tags are the left endpoints of the subintervals). The Cauchy sum of
f corresponding to P is the real number
C
X
P
f ∆x :=
N
X
f(xk−1 )(xk − xk−1 ).
k=1
Cauchy used the concept of Cauchy sequences and uniform continuity to prove the following result (cf. [Bur07, Theorem 2.2.1]), which marks the rst appearance of a systematic
integration theory (i.e. abstract denition of an integral and existence proof for a whole
class of functions | in contrast to traditional calculations of many specic areas by a
variety of ingenious ad-hoc tricks).
2
1.2. THM If f : [a, b] → R is continuous then there exists a unique real number A
with the following property: ∀ε > 0 ∃δ > 0 such that, for any Cauchy partition P with
mesh size µ(P) < δ,
X
f ∆x − A < ε.
C
P
Zb
The unique number A is called the Cauchy integral of f and we write C f(x) dx = A.
a
The following two results present the two aspects of the fundamental theorem of calculus
separately, namely that of
\recovering a function by integration of its derivative"
and
\recovering a function by dierentiation of its integral".
The proofs of these statements are nowadays standard in introductory analysis courses
on functions of a real variable (nevertheless, details can also be found in [Bur07, Sections
2.3 and 2.4]).
1.3. THM (FTC - Part I) If F : [a, b] → R is continuously dierentiable, then
Zx
C F 0 (t) dt = F(x) − F(a).
∀x ∈ [a, b] :
a
1.4. THM (FTC - Part II) If f : [a, b] → R is continuous, and we dene F : [a, b] → R
by
Zx
F(x) := C f(t) dt
(x ∈ [a, b]),
a
then F is (continuously) dierentiable on [a, b] and
F 0 (x) = f(x).
∀x ∈ [a, b] :
Searching for conditions on functions that would guarantee convergence of the Fourier
series Dirichlet was lead to the question of an extension of Cauchy integration to a wider
class of functions (around 1829). He observed that for many examples of discontinuous
functions the Cauchy sums still converge, while for the, nowadays so-called, Dirichlet
function (the characteristic function of Q ∩ [0, 1]) he failed to decide whether it was
possible to apply an \integration process" to it. Around 1847-1849 he had discussions
about this question with the student Riemann. A few years later Riemann set out to nd
a notion of integral that would handle examples like the Dirichlet function. Although
Riemann failed to achieve this particular goal he developed a signicant extension to
Cauchy's theory of integration.
3
§2. FTC with the Riemann integral (1854)
2.1. DEF Let P = {(c1, [x0, x1]), . . . , (cN, [xN−1, xN])} be a tagged partition of [a, b].
The Riemann sum of f corresponding to P is given by
R
X
f ∆x =
P
N
X
f(ck )(xk − xk−1 ).
k=1
A (bounded ) function f : [a, b] → R is said to be Riemann integrable if there exists a
real number A with the following property: ∀ε > 0 ∃δ > 0 such that, for any tagged
partition P with mesh size µ(P) < δ,
1
X
f ∆x − A < ε.
R
P
Zb
In this case A is unique and is called the Riemann integral of f; we write R f(x) dx = A.
a
2.2. REM The following are standard introductory text book results:
(i) If f is continuous on [a, b], then it is Riemann
integrable.
Rb
R
By an easy exercise we then obtain that C a f = R ab f.
(ii) Monotone functions on [a, b] are R-integrable.
(iii) One could dene Cauchy integrability similarly as in Denition 2.1, but with
Cauchy partitions replacing arbitrary tagged partitions. As reported in [Tal09]
it was shown by Gillespie in 1915 that Cauchy and Riemann integrability are, in
fact, equivalent.
Riemann knew that a countable number of discontinuities does not prevent R-integrability.
But here we will break the strict chronological ow and already state Lebesgue's criterion for Riemann integrability from 1902, which allows for sharper statements of the
FTC for the R-integral. (Elementary proofs of the following three theorems, i.e. on an
introductory course level only with the additional notion of Lebesgue null subsets of
[a, b], can be found in [Bur07, Theorems 3.6.1, 3.7.1, and 3.7.2].)
2
1 It
can be shown that boundedness of f is implied by the stated condition on the Riemann sums
([KS04, Proposition 2.11]).
2 N ⊆ [a, b] is a null set, if ∀ε > 0 we can nd a countable collection of intervals I , I , . . . which cover
1 2
P∞
N, i.e. N ⊆ ∪∞
I
,
and
satisfy
length
(I
)
<
ε
.
A
property
is
said
to
hold
almost
everywhere on
k
k
k=1
k=1
[a, b], if there is a null set N ⊆ [a, b] such that it holds for all x ∈ [a, b] \ N.
4
2.3. THM
Let f : [a, b] → R be bounded. Then f is Riemann integrable i f is
continuous almost everywhere.
2.4. THM (FTC - Part I) If F : [a, b] → R is dierentiable, and F 0 is bounded and
continuous almost everywhere on [a, b] (equivalently, F 0 is R-integrable), then
Zx
R F 0 (t) dt = F(x) − F(a).
∀x ∈ [a, b] :
a
2.5. THM (FTC - Part II) If f : [a, b] → R is R-integrable, and we dene F : [a, b] → R
by
Zx
F(x) := R f(t) dt
(x ∈ [a, b]),
a
then F is Lipschitz continuous on [a, b]. At points x ∈ [a, b] of continuity of f the function
F is dierentiable and F 0 (x) = f(x) holds. Thus, we obtain
F0 = f
almost everywhere on [a, b].
Some, by now folklore, deciencies of the Riemann integral concern non-uniformly converging sequences of functions, in particular the inability to integrate certain limit functions of convergent Fourier series. But it was an example by Volterra (1881) of a(n everywhere) dierentiable function with bounded, but non-R-integrable, derivative which
illustrated a gap in connecting FTC I with II (details of the example are in Appendix A) .
Lebesgue set out to nd a notion of integral that would be able to handle every bounded
derivative. He founded a far-reaching new approach to integration theory and easily
deals with a problem Riemann had to leave unsolved half a century ago: integrating the
Dirichlet function.
3
§3. FTC with the Lebesgue integral (1904)
Based on the concept of Lebesgue measure λ and L-measurable functions (on R) there
exist nowadays a few alternative approaches to dene Lebesgue integrability for functions
f : [a, b] → R (besides the original Lebesgue idea, e.g., via simple functions and monotone
limits or Young's with Lebesgue sums for measurable partitions).
We recall Lebesgue's original approach from 1902: It starts with a bounded measurable
f such that α 6 f(x) 6 β, partitions the range by α = y0 < · · · < yM = β, which
3
An example of a dierentiable function
on [0, 1] with unbounded derivative is f(x) = x2 sin(π/x2 )
R1 0
(0 < x 6 1), f(0) = 0. However, here 0 f exists as an improper Riemann integral and FTC I holds. In
view of the Henstock-Kurzweil theory (see §§4-5) this is no accident.
5
gives a complicated measurable \partition" of [a, b] by setting Ek = f−1 ([yk−1 , yk [)
(k = 1, . . . , M), and considers the lower and upper sum
M
X
yk−1 λ(Ek ) 6
M
X
yk λ(Ek ).
k=1
k=1
He shows that the supremum, say A, of the lower sums over all possible partitions of the
range [α, β] is nite and equals the corresponding inmum of the upper sums. In this
Zb
sense, f is Lebesgue integrable (on [a, b]) and we dene L f(x) dx := A.
a
For a measurable unbounded, but non-negative, function f we consider a sequence of
truncations fk (x) = f(x) if 0R 6 f(x) 6 k, and fk (x) = k if f(x) > k. If the monotone
sequence of real numbers (L ab fk )k∈N is bounded, then f is L-integrable and
Zb
Zb
L f(x) dx := lim L fk (x) dx
k→∞
a
a
For the general case of a measurable, possibly unbounded, function f : [a, b] → R we
consider the representation as dierence of two measurable non-negative functions in
the form f = f+ − f− , where f+ (x) := max(f(x), 0), f− (x) := − min(f(x), 0). Then f is
L-integrable i both f+ and f− are L-integrable; in this case we set
Zb
Zb
Zb
L f(x) dx := L f+ (x) dx − L f− (x) dx
a
a
a
We collect a few basic properties (proofs are standard material of measure theory courses,
see also [Bur07, Gor94, KS04]; part (i) is an immediate from Lebesgue's denition.).
3.1. Proposition
(i) Let f : [a, b] → R be bounded. Then f is Lebesgue integrable i f is Lebesgue
measurable.
(ii) Let f : [a, b] → R be L-measurable. Then f is L-integrable i |f| is L-integrable.
Zb
Zb
a
a
(iii) Every R-integrable function f is L-integrable and R f(x) dx = L f(x) dx.
We note that the derivative of a dierentiable function F is L-measurable, since it is obtained as the pointwise limit of a sequence of measurable functions by F 0 (x) = limn→∞ n(F(x+
1/n) − F(x)). This observation in combination with the standard convergence theorems
of Lebesgue's integration theory, yields a direct proof of the following FTC variant (cf.
[Bur07, Theorem 6.4.2]).
6
3.2. THM (FTC - Part I) If F : [a, b] → R is dierentiable and F 0 is bounded, then
Zx
∀x ∈ [a, b] :
L F 0 (t) dt = F(x) − F(a).
a
Michael Grosser has pointed out to me the following statement of FTC I from a classic
textbook (cf. [Nat54, page 270]), which is stronger
than that in Theorem 3.2: If F is
Rx 0
0
dierentiable, and F is L-integrable, then L a F (t) dt = F(x) − F(a) holds for all x ∈
[a, b].
For a further improvement of the FTC we recall the following notion due to Vitali (1904).
A function F : [a, b] → R is absolutely continuous if ∀ε > 0 ∃δ > 0 such that the following
holds:
for all nite disjoint collections
of subintervals ]ak , bk [ ⊆ [a, b] (k = 1 . . . , n) with
Pn
Pn
k=1 (bk − ak ) < δ we have
k=1 |F(bk ) − F(ak )| < ε.
3.3. Note: dierentiable with bounded derivative ⇒ Lipschitz continuous ⇒
absolutely continuous ⇒ continuous; absolutely continuous ⇒ bounded variation;
none of these implications holds in the reversed direction!
[Exercise: give proofs and nd counter examples.]
Warning: dierentiable 6⇒ absolutely continuous.
4
However, as a consequence of Lebesgue's Dierentiation Theorem for monotone functions
(1904) together with Jordan's representation of functions of bounded variation as the
dierence of two monotone (increasing) functions (1894), we deduce the following result.
(Again, [Bur07, Gor94, KS04] may serve as references for this and the results following below.)
3.4. Lemma Absolutely continuous functions are dierentiable almost everywhere.
3.5. Convention: If F is dierentiable almost everywhere, how do we dene a deriva-
tive of F as a function on all of [a, b]? At points x ∈ [a, b] of dierentiability we clearly
take F 0 (x). If N ⊆ [a, b] denotes the Lebesgue null set of points, where F is not dierentiable, we prescribe for each y ∈ N the value F 0 (y) := 0. In the sequel we will always refer
to the derivative function in this sense. Since L-integrability and the values of L-integrals
are not aected by changes on subsets of measure zero, the subsequent statements do
not depend on the precise choice of function values on the exception set N.
3.6. THM (FTC - Part I'; 1904) If F : [a, b] → R is absolutely continuous, then F 0 is
L-integrable and
Zx
∀x ∈ [a, b] :
L F 0 (t) dt = F(x) − F(a).
a
4 E.g.
consider the function in footnote 3.
7
3.7. REM If F is nondecreasing on [a, b] then F 0 is L-integrable and
Zx
L F 0 (t) dt 6 F(x) − F(a).
∀x ∈ [a, b] :
a
3.8. THM (FTC - Part II; 1904) If
F : [a, b] → R by
f : [a, b] → R is L-integrable, and we dene
Zx
F(x) := L f(t) dt
(x ∈ [a, b]),
a
then F is absolutely continuous on [a, b] and
F0 = f
almost everywhere on [a, b].
3.9. REM
(i) Observe that FTC I' and II do not leave a gap between them in the sense that one
may now conclude:
f is L-integrable ⇐⇒ ∃ F absolutely continuous: F 0 = f almost everywhere.
(ii) Improperly R-integrable non-negative functions are L-integrable and the values
of the integrals agree. (Can be deduced from [KS04, Remarks on bottom of page 167
combined with Theorem 4.79 (on page 189)]; a direct proof for the case of an unbounded
domain can be found in [Els05, Kapitel IV, Satz 6.3])
This is no longer true, if f changes sign: f 0 , where f is as in footnote 3, yields
an example of an improperly R-integrable function which is not L-integrable (on
[0, 1]). (This is an easy addition to [Bur07, Exercises 6.2.4. c and d].)
(iii) One of the key technologies now available to analysis and probability theory through
Lebesgue-sytle integration theory is the group of famous convergence theorems
(Levi 1906: monotone convergence; Fatou's lemma 1906; Lebesgue's dominated
convergence 1910). A prominent consequence of these is the completeness of the
spaces Lp ([a, b]).
The fact that (everywhere) dierentiable functions with non-L-integrable derivative exist
raised a new question concerning the FTC: Can we extend Lebesgue's theory to an
integration
process which guarantees for any dierentiable function F that F 0 is integrable
Rx 0
and that a F (t) dt = F(x) − F(a) holds?
8
§4. Integration à la
Denjoy (1912),
Perron (1914),
and Kurzweil (1957) - Henstock (1961)
The good news are:
For functions f : [a, b] → R the three integration theories mentioned in the title of
this Section are all equivalent. (Detailed proofs are discussed in [Gor94].)
The issue of an FTC valid for all dierentiable functions F is addressed successfully.
(So is also the compatibility with Lebesgue integration.)
4.1. The three dierent approaches and their equivalence:
Denjoy 1912: f is D-integrable if ∃F ∈ ACG∗ ([a, b]) [= space of functions that are
generalized absolutely
continuous in the restricted sense ] such that F 0 = f almost
Rb
everywhere; then D a f(x) dx := F(b) − F(a).
The technically demanding task is the denition of ACG∗ ([a, b]) (which is already
a simplication due to Lusin 1912-1913; [Gor94, Chapters 4 and 7])
Perron 1914: based on the notion of upper and lower derivatives (dened by taking
only lim sup and lim inf of the dierence quotients) and the concept of major and
minor functions for f; e.g. U : [a, b] → R continuous is called major function for f,
if U(a) = 0 and the lower derivatives satisfy DU(x) > f(x) for all x.
If inf {U(b) | U major function of f}R = sup {u(b) | u minor function of f}, then
the common value is dened to be P ab f(x) dx.
(Cf. [Gor94, Chapter 8] or [KS04, Section 4.2].)
Kurzweil 1957, Henstock 1961: Kurzweil studies ODEs (with non-Lipschitz rightR
hand side;cf. [Kur57]) in integrated form x(t) = 0t f(x(τ), τ) dτ and investigates
convergence upon approximating f by a sequence fk , which in turn leads to an ap-
proximation of the integral; he introduces Riemann sums with an additional gauge
condition, mixes this with Perron's constructions and calls this the generalized
Perron integral, but mentions that it could also be called the generalized Riemann
integral,
and furthermore claims that his new integral is, in fact, equivalent to Perron integration
...
A systematic investigation and clarication comes with Henstock from 1961 onwards. We will describe the denition in detail below and call it the HK-integral
(cf. [Gor94, Chapter 9], [KS04, Chapter 4], or [Bur07, Chapter 8]).
9
History of equivalence proofs: D =⇒ P [Hake 1921], P =⇒ D [Aleksandrov 1924,
Looman 1925], P ⇐⇒ HK [Kurzweil and Henstock 1957-1960's].
(Direct proofs of D ⇐⇒ HK seem very hard and have not been achieved until a series of
articles during the late 1980's.)
4.2. DEF A gauge on [a, b] is a positive function δ : [a, b] → R, i.e. δ(t) > 0 for all
t ∈ [a, b]. A tagged partition P = {(c1 , [x0 , x1 ]), . . . , (cN , [xN−1 , xN ])} of [a, b] is said to
be δ-ne if we have for k = 1, . . . , N:
ck − δ(ck ) < xk−1 6 ck 6 xk < ck + δ(ck ).
4.3. REM
(i) Observe that for any δ-ne tagged partition P we have, in particular, that
0 6 xk − xk−1 < 2δ(ck )
(k = 1, . . . , N).
The (varying!) size of δ controls the \locally allowed" maximal length of subintervals in the partition.
(ii) If δ1 and δ2 are gauges on [a, b] such that δ1 6 δ2 , then any δ1 -ne tagged partition
is also δ2 -ne.
(iii) If δ > 0 is constant, then any tagged partition with mesh size µ(P) < δ is δ-ne.
More generally, this is still true in the case 0 < µ(P) 6 inf a6t6b δ(t).
(iv) Due to a lemma by Cousin (1895) δ-ne tagged partitions always exist. (Exercise:
proof by contradiction constructing a sequence of nested intervals.)
4.4. DEF A function f : [a, b] → R is said to be HK-integrable if there exists a real
number A with the following property: ∀ε > 0 ∃ a gauge δ on [a, b] such that, for any
δ-ne tagged partition P, we have
X
f ∆x − A < ε.
R
P
Zb
In this case A is unique and is called the HK-integral of f; we write HK f(x) dx = A.
a
4.5. REM (Basic properties in comparison with other notions of integral)
(i) If f is improperly R-integrable on [a, b] then f is HK-integrable and the values of
the integrals are equal.
(Follows from [KS04, Theorem 4.46].)
(ii) HK-integrable ⇒ L-measurable ([KS04, Corollary 4.86] or [Bur07, Theorem 8.8.1,3]).
10
(iii) If f is L-integrable then f is also HK-integrable and the values of the integrals
agree.
([KS04, Theorem 4.46] or [Bur07, Theorem 8.7.1].)
(iv) Let f > 0 and L-measurable: f is HK-integrable i f is L-integrable.
In this case the values of the integrals agree.
([KS04, Theorem 4.79].)
(v) Warning: f HK-integrable 6⇒ |f| HK-integrable
(use examples from improper R-integral).
(vi) Let HK([a, b]) denote the set of classes of HK-integrable functions modulo equality (Lebesgue) almost everywhere. We obtain a normed vector space with the
Alexiewicz norm
Zx
kfk := sup HK f(y) dy
a6x6b =kFkL∞ ,
where
Rx
F(x)=HK f(y) dy .
a
a
Note that
L1 ([a, b]) ,→ HK([a, b]) continuously, since for f ∈ L1 ([a, b]) we have
Rx
Rx
kfk = HK a f(y) dy 6 L a |f(y)| dy = kfkL1 .
However, (HK([a, b]), k.k) is not complete ([KS04, Example 4.106]). (Although
HK-analogues of the Lebesgue-type convergence theorems doRexist, these do not
overcome deciencies related to (v) above: we cannot use HK |f| as a norm.)
§5. FTC with the HK integral
We want to present at least the one proof which shows why the additional gauge condition for Riemann sums in the denition of the HK-integral ensures integrability of
any derivative. As a preparation we state a simple consequence of dierentiability, also
known as straddle lemma.
5.1. Lemma If f : [a, b] is dierentiable at y ∈ [a, b], then the following holds: ∀ε > 0
∃δ(y) > 0 such that we have
|f(v) − f(u) − f 0 (y)(v − u)| 6 ε(v − u)
whenever u, v ∈ [a, b] and y − δ(y) < u 6 y 6 v < y + δ(y).
Proof: Let ε > 0. Dierentiability of f at y provides a δ(y) > 0 so that for x ∈ [a, b]
with |x − y| < δ(y) we have |f(x) − f(y) − f 0 (y)(x − y)| 6 ε|x − y|.
11
Let u and v be as in the statement, then
|f(v) − f(u) − f 0 (y)(v − u)|
= | f(v) − f(y) − f 0 (y)(v − y) + f(y) − f(u) − f 0 (y)(y − u) |
6 |f(v) − f(y) − f 0 (y)(v − y)| + |(f(y) − f(u) − f 0 (y)(y − u)|
6 ε(v − y) + ε(y − u) = ε(v − u).
5.2. THM (FTC - Part I) If F : [a, b] → R is dierentiable, then F 0 is HK-integrable
and
Zx
∀x ∈ [a, b] :
HK F 0 (t) dt = F(x) − F(a).
a
Proof: Let ε > 0. For each t ∈ [a, b] choose δ(t) > 0 as in the straddle lemma. This denes a gauge δ on [a, b]. Let x ∈ [a, b] be arbitrary. If P = {(c1 , [t0 , t1 ]), . . . , (cN , [tN−1 , tN ])}
is an arbitrary tagged δ-ne partition of [a, x], then
N
X
X
F 0 ∆t − F(x) − F(a) = F 0 (ck )(tk − tk−1 ) − F(x) − F(a) .
R
P
Since F(x) − F(a) =
proceed by
k=1
PN
k=1 (F(tk )
− F(tk−1 )) we may bring all terms into one sum and
N
N
X
X
0
|F(tk ) − F(tk−1 ) − F 0 (ck )(tk − tk−1 )|
F (ck )(tk − tk−1 ) − F(tk ) + F(tk−1 ) 6
{z
}
|
k=1
k=1
6ε
N
X
[straddle
]
lemma!
6ε(tk −tk−1 )
(tk − tk−1 ) = ε(x − a) 6 ε(b − a).
k=1
Hence we have shown
HK-integrability of F 0 (on [a, b], in fact on every [a, x] with a 6
Rx 0
x 6 b) and that HK a F (t) dt = F(x) − F(a) holds.
5.3. THM (FTC - Part I') If
F : [a, b] → R is continuous and dierentiable nearly
everywhere (i.e. except on a countable subset), then F 0 is HK-integrable and
Zx
∀x ∈ [a, b] :
HK F 0 (t) dt = F(x) − F(a).
a
(For a proof see [KS04, Theorem 4.24] or [Bur07, Theorem 8.7.3].)
5.4. THM (FTC - Part II) If f : [a, b] → R is HK-integrable, and we dene F : [a, b] →
R by
Zx
F(x) := HK f(t) dt
a
12
(x ∈ [a, b]),
then F is continuous and dierentiable almost everywhere on [a, b] and
F0 = f
almost everywhere on [a, b].
(For a proof see [KS04, Theorem 4.83] or [Bur07, Theorem 8.8.1].)
§6. FTC with the distributional Denjoy
integral (2008), non-standard functions,
Colombeau generalized functions, . . .
This Section title is simply meant to suggest a possible topic for a master thesis and I
will not elaborate on it (for now). Let me just briey report on the basic concept of a
very recent paper by Talvila in (2008; [Tal08]):
He shows that the completion of HK([a, b]) with respect to the Alexiewicz norm yields
the following subspace of the distributions on the line
AC ([a, b]) := {f ∈ D 0 (]a, b[) | ∃F ∈ C([a, b]) : F(a) = 0 and F 0 = f (in D 0 )}.
Furthermore, AC ([a, b]) is separable, isomorphic (as a Banach space) to (C([a, b]), kkL∞ ),
and has L1 ([a, b]) as a dense subspace.
R
If f ∈ AC ([a, b]) then the integral is dened by ax f := F(x) (note that F(a) = 0) and we
clearly get a (cheap) FTC
Zx
F 0 = F(x) − F(a)
∀x ∈ [a, b].
a
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