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The Kurzweil-Henstock integral andits extensions : a historical survey
Jean Mawhin
Universite Catholique de Louvain
The Kurzweil-Henstock integral and its extensions : a historical survey – p.1/29
This lecture is dedicated to the memory of my friendSTEFAN SCHWABIK,
an enthusiastic ambassador of the Kurzweil-Henstock integral,and a great friend of many mathematicians of Sao Carlos
The Kurzweil-Henstock integral and its extensions : a historical survey – p.2/29
I. A short history of integration
The Kurzweil-Henstock integral and its extensions : a historical survey – p.3/29
Cauchy
1823 : Résumé des leçons données à l’École royalepolytechnique sur le calcul infinitésimal
“In the integral calculus, it seemed to me necessary to prove in ageneral way the existence of the integral of primitivable functionsbefore letting their various properties to be known.To reach this aim, it was first necessary to establish the notion ofintegral taken between given limit or definite integrals.As those last ones can be sometimes infinite or undeterminated, itwas essential to search in which case they keep a unique and finitevalue”
The Kurzweil-Henstock integral and its extensions : a historical survey – p.4/29
Integral of a continuous function
f : [a, b] → R continuous
P-partition of [a, b] : Π := (xj , Ij)1≤j≤m, Ij = [aj−1, aj ]
a = a0 < a1 < . . . < am−1 < am = b, xj ∈ Ij
length of Ij : |Ij | = aj − aj−1
mesh of Π : M(Π) = max1≤j≤m |Ij |
The Kurzweil-Henstock integral and its extensions : a historical survey – p.5/29
Integral of a continuous function
f : [a, b] → R continuous
P-partition of [a, b] : Π := (xj , Ij)1≤j≤m, Ij = [aj−1, aj ]
a = a0 < a1 < . . . < am−1 < am = b, xj ∈ Ij
length of Ij : |Ij | = aj − aj−1
mesh of Π : M(Π) = max1≤j≤m |Ij |
22-23th lectures : f continuous on [a, b] ⇒ ∃ ! J ∈ R,
∀ ε > 0,∃ η > 0,∀Π : M(Π) ≤ η : |J −∑m
j=1 f(xj)|Ij || ≤ ε
J =∫ b
af(x) dx : definite integral of f on [a, b]
continuity on [a, b] ⇔ uniform continuity on [a, b]
The Kurzweil-Henstock integral and its extensions : a historical survey – p.5/29
Integral of a continuous function
f : [a, b] → R continuous
P-partition of [a, b] : Π := (xj , Ij)1≤j≤m, Ij = [aj−1, aj ]
a = a0 < a1 < . . . < am−1 < am = b, xj ∈ Ij
length of Ij : |Ij | = aj − aj−1
mesh of Π : M(Π) = max1≤j≤m |Ij |
22-23th lectures : f continuous on [a, b] ⇒ ∃ ! J ∈ R,
∀ ε > 0,∃ η > 0,∀Π : M(Π) ≤ η : |J −∑m
j=1 f(xj)|Ij || ≤ ε
J =∫ b
af(x) dx : definite integral of f on [a, b]
continuity on [a, b] ⇔ uniform continuity on [a, b]
f ∈ C1([a, b]) ⇒∫ b
af ′ = f(b) − f(a)
f ∈ C([a, b]) ⇒∫ ·af ∈ C1([a, b]),
(∫ x
af)′
= f(x)
The Kurzweil-Henstock integral and its extensions : a historical survey – p.5/29
Cauchy and Riemann
AUGUSTIN CAUCHY BERNHARD RIEMANN
1789–1857 1826–1866
The Kurzweil-Henstock integral and its extensions : a historical survey – p.6/29
Riemann
1854 : Habilitation thesis University of Göttingen(published 1867)
“The uncertainty which still prevails on some fundamental points ofthe theory of definite integrals forces us to place here a few remarkson the notion of definite integral, and on its possible generality.
And first, what do we mean by∫ b
a f(x) dx ?”
The Kurzweil-Henstock integral and its extensions : a historical survey – p.7/29
Riemann
1854 : Habilitation thesis University of Göttingen(published 1867)
“The uncertainty which still prevails on some fundamental points ofthe theory of definite integrals forces us to place here a few remarkson the notion of definite integral, and on its possible generality.
And first, what do we mean by∫ b
a f(x) dx ?”
f : [a, b] → R is R-integrable on [a, b] if ∃ J ∈ R,
∀ ε > 0, ∃ η > 0,∀Π,M(Π) ≤ η : |J −∑m
j=1 f(xj)|Ij || ≤ ε
S(f,Π) :=∑m
j=1 f(xj)|Ij | : Riemann sum for f and Π
J =∫ b
af is the R-integral of f on [a, b]
R-integrable functions are the ones for which CAUCHY’s limit
process made for continuous functions works
The Kurzweil-Henstock integral and its extensions : a historical survey – p.7/29
Range of R-integration
RIEMANN : “Let us search now the range and the limit of thepreceding definition and let us ask the question :in which case is a function integrable ?And in which case not integrable ?”
The Kurzweil-Henstock integral and its extensions : a historical survey – p.8/29
Range of R-integration
RIEMANN : “Let us search now the range and the limit of thepreceding definition and let us ask the question :in which case is a function integrable ?And in which case not integrable ?”
although modeled on CAUCHY’s process for (uniformly) continuous
functions, R-integrable functions may have a dense set of
discontinuities
however, 1Q is not R-integrable on any interval
R-integrable functions are characterized in terms of some ‘measure’
of their set of discontinuities
The Kurzweil-Henstock integral and its extensions : a historical survey – p.8/29
Range of R-integration
RIEMANN : “Let us search now the range and the limit of thepreceding definition and let us ask the question :in which case is a function integrable ?And in which case not integrable ?”
although modeled on CAUCHY’s process for (uniformly) continuous
functions, R-integrable functions may have a dense set of
discontinuities
however, 1Q is not R-integrable on any interval
R-integrable functions are characterized in terms of some ‘measure’
of their set of discontinuities
indefinite R-integral of f not differentiable at points of discontinuity
of f
∃ bounded derivatives not R-integrable (VOLTERRA)
The Kurzweil-Henstock integral and its extensions : a historical survey – p.8/29
Lebesgue
1902 : PhD thesis, Annali di Mat. Pura Appl.
“In the case of continuous functions, the notions of [indefinite]integral and of primitive are identical.Riemann has defined the integral of some discontinuous functions,but all derivatives are not integrable in Riemann sense.The problem of the primitive functions is therefore not solved by[R-]integration, and one can wish to have a definition of the integralcontaining as special case that of Riemann and solving the problemof the primitives”
The Kurzweil-Henstock integral and its extensions : a historical survey – p.9/29
L-integral
based upon a concept of measure of a bounded set A ⊂ R
introduced by BOREL and developed by LEBESGUE
outer measure µe(A) of A ⊂ [c, d] : inf∑∞
j=1(dj − cj) for
all sequences {[cj , dj ]}j∈N : A ⊂ ∪∞j=1[cj , dj ]
inner measure µi(A) = (d − c) − µe([c, d] \ A)
A measurable : µe(A) = µi(A) (measure µ(A) of A )
The Kurzweil-Henstock integral and its extensions : a historical survey – p.10/29
L-integral
based upon a concept of measure of a bounded set A ⊂ R
introduced by BOREL and developed by LEBESGUE
outer measure µe(A) of A ⊂ [c, d] : inf∑∞
j=1(dj − cj) for
all sequences {[cj , dj ]}j∈N : A ⊂ ∪∞j=1[cj , dj ]
inner measure µi(A) = (d − c) − µe([c, d] \ A)
A measurable : µe(A) = µi(A) (measure µ(A) of A )
f : [a, b] → R bounded is L- integrable on [a, b] if
∀ c < d in range of f, f−1([c, d)) is measurable
∃ J ∈ R,∀ ε > 0,∃ η > 0,∀ P-partitionΠ = (yj , [bj−1, bj ])1≤j≤m of [inf [a,b] f, sup[a,b] f ], M(Π) ≤ η :
|J −∑m
j=1 yjµ[
f−1([bj−1, bj))]
| ≤ ε
J =∫ b
a f(x) dx is the L-integral of f on [a, b]
The Kurzweil-Henstock integral and its extensions : a historical survey – p.10/29
Borel and Lebesgue
EMILE BOREL HENRI LEBESGUE
1871–1956 1875–1941
The Kurzweil-Henstock integral and its extensions : a historical survey – p.11/29
Comparing the R- and L- integrals
approximating sums depend upon measure theory
f R-integrable ⇔ µ(set of discontinuities of f) = 0
f differentiable on [a, b] , f ′ bounded ⇒∫ b
af ′ = f(b)− f(a)
f L-integrable on [a, b] ⇒∫ ·a f differentiable with derivative f
outside of a subset of [a, b] of measure zero
The Kurzweil-Henstock integral and its extensions : a historical survey – p.12/29
Comparing the R- and L- integrals
approximating sums depend upon measure theory
f R-integrable ⇔ µ(set of discontinuities of f) = 0
f differentiable on [a, b] , f ′ bounded ⇒∫ b
af ′ = f(b)− f(a)
f L-integrable on [a, b] ⇒∫ ·a f differentiable with derivative f
outside of a subset of [a, b] of measure zero
extension to unbounded functions
f R- or L-integrable ⇒ |f | R- or L-integrable
f primitivable on [a, b] is L-integrable on [a, b] ⇔ F hasbounded variation on [a, b]
f(x) = 2x sin 1x2 −
2x cos 1
x2 if x 6= 0, f(0) = 0
f = F ′ with F (x)x2 sin 1x2 if x 6= 0, F (0) = 0
f is not L-integrable near 0
The Kurzweil-Henstock integral and its extensions : a historical survey – p.12/29
Denjoy-Perron integral
1912 : DENJOY (transfinite induction argument from L-integral) :
D-integral integrating all derivatives
1914 : PERRON (inspired by DE LA VALLEE-POUSSIN’s
characterization of L-integrability) :
P-integral integrating all derivatives
F+[F−] over-function [under-function] of f on [a, b] if
F±(a) = 0, F ′+(x) ≥ f(x) [F ′
−(x) ≤ f(x)] ∀x ∈ [a, b]
f P-integrable on [a, b] : supF−F−(b) = infF+
F+(b)
common value = P-integral of f on [a, b]
The Kurzweil-Henstock integral and its extensions : a historical survey – p.13/29
Denjoy-Perron integral
1912 : DENJOY (transfinite induction argument from L-integral) :
D-integral integrating all derivatives
1914 : PERRON (inspired by DE LA VALLEE-POUSSIN’s
characterization of L-integrability) :
P-integral integrating all derivatives
F+[F−] over-function [under-function] of f on [a, b] if
F±(a) = 0, F ′+(x) ≥ f(x) [F ′
−(x) ≤ f(x)] ∀x ∈ [a, b]
f P-integrable on [a, b] : supF−F−(b) = infF+
F+(b)
common value = P-integral of f on [a, b]
f D-integrable on [a, b] ⇔ f P-integrable on [a, b]
f L-integrable on [a, b] ⇔ f and |f | DP-integrable on [a, b]
first half of XXth century : many equivalent definitions of L- and
DP-integral
The Kurzweil-Henstock integral and its extensions : a historical survey – p.13/29
Denjoy and Perron
ARNAUD DENJOY OSKAR PERRON
1884–1974 1880–1975
The Kurzweil-Henstock integral and its extensions : a historical survey – p.14/29
KH-integral
1957 : KURZWEIL, new definition of P-integral of f : [a, b] → R
f K-integrable on [a, b] : ∃ J ∈ R, ∀ ε > 0, ∃ δ : [a, b] → R+,∀Π, xj − δ(xj) ≤ aj−1 < aj ≤ xj + δ(xj) (1 ≤ j ≤ m),
|J − S(f,Π)| ≤ ε
Π called δ-fine, δ called gauge on [a, b]
K-integral ⇔ P-integral
The Kurzweil-Henstock integral and its extensions : a historical survey – p.15/29
KH-integral
1957 : KURZWEIL, new definition of P-integral of f : [a, b] → R
f K-integrable on [a, b] : ∃ J ∈ R, ∀ ε > 0, ∃ δ : [a, b] → R+,∀Π, xj − δ(xj) ≤ aj−1 < aj ≤ xj + δ(xj) (1 ≤ j ≤ m),
|J − S(f,Π)| ≤ ε
Π called δ-fine, δ called gauge on [a, b]
K-integral ⇔ P-integral
1961 : independent rediscovery by HENSTOCK
HENSTOCK gives many generalizations and applications
J =∫ b
a f Kurzweil-Henstock or KH-integral or gauge integral of
f on [a, b]
constant gauge in KH-definition ⇔ R-integral
The Kurzweil-Henstock integral and its extensions : a historical survey – p.15/29
Henstock and Kurzweil
RALPH HENSTOCK JAROSLAV KURZWEIL
1923-2007 born in 1928
The Kurzweil-Henstock integral and its extensions : a historical survey – p.16/29
δ-fine P-partitions
constant gauge δ : δ-fine P-partition easily constructed
arbitrary gauge δ : existence of a δ-fine P-partition has to be proved
1895 : done by COUSIN in a different context (Cousin’s lemma)
equivalent to the Borel-Lebesgue property (1894, 1902) for a
compact interval
proof depends upon the non-empty intersection property of a nested
sequence of closed intervals
The Kurzweil-Henstock integral and its extensions : a historical survey – p.17/29
II. A ‘history-fiction’ of integration
The Kurzweil-Henstock integral and its extensions : a historical survey – p.18/29
Another road for Cauchy
CAUCHY’s aim : construct integral calculus for derivatives(fundamental objects in NEWTON-LEIBNIZ’s calculus)
mimick CAUCHY’s approach for continuous functions
f : [a, b] → R differentiable, with derivative f ′ : [a, b] → R
∀ ε > 0,∀x ∈ [a, b],∃ δ(x) > 0,∀y ∈ [a, b], |y − x| ≤ δ(x) :|f(y) − f(x) − f ′(x)(y − x)| ≤ ε|y − x|/(b − a)
|f(z) − f(y) − f ′(x)(z − y)| ≤ ε(z − y)/(b − a)if x − δ(x) ≤ y ≤ x ≤ z ≤ x + δ(x)
Π δ − fine ⇒ |f(aj) − f(aj−1) − f ′(xj)(aj − aj−1)|≤ ε(aj − aj−1)/(b − a) (1 ≤ j ≤ m)
Π δ − fine ⇒ |f(b) − f(a) − S(f ′,Π)| ≤ ε
δ non constant because differentiability on [a, b] 6⇒ uniformdifferentiability on [a, b]
The Kurzweil-Henstock integral and its extensions : a historical survey – p.19/29
Cauchy, Riemann, Weierstrass ?
“CAUCHY” : f : [a, b] → R differentiable ⇒ ∀ ε > 0,∃ gauge δon [a, b],∀ δ-fine Π : |f(b) − f(a) − S(f ′,Π)| ≤ ε
“RIEMANN” : f : [a, b] → R is integrable on [a, b] if ∃ J ∈ R,∀ ε > 0,∃ gauge δ on [a, b],∀ δ-fine Π : |J − S(f,Π)| ≤ ε
J =∫ b
af KH-integral of f on [a, b]
The Kurzweil-Henstock integral and its extensions : a historical survey – p.20/29
Cauchy, Riemann, Weierstrass ?
“CAUCHY” : f : [a, b] → R differentiable ⇒ ∀ ε > 0,∃ gauge δon [a, b],∀ δ-fine Π : |f(b) − f(a) − S(f ′,Π)| ≤ ε
“RIEMANN” : f : [a, b] → R is integrable on [a, b] if ∃ J ∈ R,∀ ε > 0,∃ gauge δ on [a, b],∀ δ-fine Π : |J − S(f,Π)| ≤ ε
J =∫ b
af KH-integral of f on [a, b]
existence of δ-fine P-partition : CAUCHY ? RIEMANN ?
WEIERSTRASS ?
if yes, BOREL-LEBESGUE-COUSIN lemma, Denjoy-Perron’s integral
and Borel-Lebesgue’s measure of a bounded set of R could have
arrived half a century before
The Kurzweil-Henstock integral and its extensions : a historical survey – p.20/29
Cauchy, Riemann, Weierstrass ?
“CAUCHY” : f : [a, b] → R differentiable ⇒ ∀ ε > 0,∃ gauge δon [a, b],∀ δ-fine Π : |f(b) − f(a) − S(f ′,Π)| ≤ ε
“RIEMANN” : f : [a, b] → R is integrable on [a, b] if ∃ J ∈ R,∀ ε > 0,∃ gauge δ on [a, b],∀ δ-fine Π : |J − S(f,Π)| ≤ ε
J =∫ b
af KH-integral of f on [a, b]
existence of δ-fine P-partition : CAUCHY ? RIEMANN ?
WEIERSTRASS ?
if yes, BOREL-LEBESGUE-COUSIN lemma, Denjoy-Perron’s integral
and Borel-Lebesgue’s measure of a bounded set of R could have
arrived half a century before
tragical consequence : DENJOY, PERRON, KURZWEIL and
HENSTOCK disappear in our fiction : DPKH-integral is just the
integral defined by “RIEMANN”
The Kurzweil-Henstock integral and its extensions : a historical survey – p.20/29
Qualities and defects of KH-integral
qualities :∫ b
af ′ = f(b) − f(a) for all differentiable f
improper integrals are real integrals (HAKE’s theorem)
monotone and dominated convergence theorems(nice proof by HENSTOCK)
E ⊂ [a, b] measurable : 1E integrable on [a, b]
measure µ(E) :=∫ b
a 1E
change of variable theorem
The Kurzweil-Henstock integral and its extensions : a historical survey – p.21/29
Qualities and defects of KH-integral
qualities :∫ b
af ′ = f(b) − f(a) for all differentiable f
improper integrals are real integrals (HAKE’s theorem)
monotone and dominated convergence theorems(nice proof by HENSTOCK)
E ⊂ [a, b] measurable : 1E integrable on [a, b]
measure µ(E) :=∫ b
a 1E
change of variable theorem
defects :
restriction property holds only for finite families ofnon-overlapping subintervals, may already fail for a countableunion of such intervals
due to the non-absolute character of the integral
The Kurzweil-Henstock integral and its extensions : a historical survey – p.21/29
We must save the soldier Lebesgue
wanted : an integral with better restriction property
f L-integrable on [a, b] if f and |f | are integrable on [a, b]
f L-integrable on [a, b] ⇒ f L-integrable on any measurableE ⊂ [a, b]
integrability of an unbounded derivative may be lost
Hake’s property may be lost (there exists improper L-integrals)
The Kurzweil-Henstock integral and its extensions : a historical survey – p.22/29
We must save the soldier Lebesgue
wanted : an integral with better restriction property
f L-integrable on [a, b] if f and |f | are integrable on [a, b]
f L-integrable on [a, b] ⇒ f L-integrable on any measurableE ⊂ [a, b]
integrability of an unbounded derivative may be lost
Hake’s property may be lost (there exists improper L-integrals)
can attribute to LEBESGUE the introduction and emphasis on this
important subclass of integrable functions
absolute character makes it a better tool for functional analysis
(Lebesgue spaces Lp(a, b) are Banach spaces)
The Kurzweil-Henstock integral and its extensions : a historical survey – p.22/29
III. Higher dimensions
The Kurzweil-Henstock integral and its extensions : a historical survey – p.23/29
n-dimensional KH-integral
(closed) n-interval I = I1 × . . . × In, |I| n-volume of I
P-partition of I : Π := {(xj , Ij)}1≤j≤m, xj ∈ Ij
Ij ⊂ I non-overlapping n-intervals, ∪mj=1I
j = I
gauge on I : δ : I → R+; Π δ-fine : ∀ j : Ij ⊂ B[xj , δ(xj)]
f : I → R , Riemann sum : S(f,Π) :=∑m
j=1 f(xj)|Ij |
f KH-integrable on I : ∃ J ∈ R,∀ ε > 0, ∃ gauge δ on I,∀ δ−fine Π : |J − S(f,Π)| ≤ ε
J =∫
If is the KH-integral of f on I
E ⊂ I measurable if 1E KH-integrable on I , µ(E) :=∫
I1E
Fubini, monotone and dominated convergence thms
no change of variables thm, restriction to finite union of n-intervals
f L-integrable on I : f and |f | KH-integrable on I
The Kurzweil-Henstock integral and its extensions : a historical survey – p.24/29
n-dim. fundamental thm of calculus
v ∈ C1(A, Rn), A ⊂ Rn, ∂A ‘nice’∫
Adiv v =
∫
∂A〈v, nA〉, nA outer normal on ∂A, |nA| = 1
∃ v : I → Rn differentiable : div v not KH-integrable on I
mimick proof of fundamental theorem for n = 1
∀ ε > 0, ∀x ∈ I, ∃ δ(x) > 0, ∀ y ∈ B[x, δ(x)] :
‖v(y) − v(x) − v′(x)(y − x)‖ ≤ ε2‖y − x‖
∀x ∈ I, wx := v(x) + v′(x)(· − x) ∈ C∞(Rn, Rn)
Π = {(xj , Ij)}1≤j≤m δ-fine ⇒∫
∂Ij〈wxj , nIj〉 =∫
Ij div wxj = div v(xj)|Ij |∫
∂Ij〈v, nIj〉 − div v(xj)|Ij | =∫
∂Ij〈v − wxj , nIj〉
‖v(y) − wxj(y)‖ ≤ ε2‖y − xj‖ ∀ y ∈ Ij , ∀ j = 1, . . . ,m
The Kurzweil-Henstock integral and its extensions : a historical survey – p.25/29
n-dim. fundamental thm of calculus
‖∫
∂I〈v, nI〉 − S(div v,Π)‖ ≤
∑mj=1 ‖
∫
∂Ij〈wxj − v, nIj〉‖
≤ ε2∑m
j=1 d(Ij) |∂Ij | := ε2σ(Π)
σ(Π) irregularity of Π, d(Ij) diameter of Ij
|∂Ij | (n-1)-dimensional measure of ∂Ij
‖∫
∂I〈v, nI〉 − S(div v,Π)‖ ≤ ε if one adds to Π δ-fine the
irregularity restriction σ(Π) ≤ ε−1
geometrical meaning : Ij = Ij1 × . . . × Ij
n,
d(Ij) = max1≤k≤n |Ijk|, |∂Ij | ≤ 2n|Ij |
min1≤k≤n |Ij
k|
σ(Π) ≤ 2n max1≤j≤mmax1≤k≤n |Ij
k|
min1≤k≤n |Ij
k||I| := 2nσ0(Π) |I|
‖∫
∂I〈v, nI〉 − S(div v,Π)‖ ≤ ε if Π satisfies the stronger
irregularity restriction : σ0(Π) ≤ 12nε|I|
The Kurzweil-Henstock integral and its extensions : a historical survey – p.26/29
Generalized KH-integrals on n-intervals
f : I ⊂ Rn → R, I n-interval
1981, M. : M-integrable on I if ∃ J ∈ R,∀ ε > 0,∃ gauge δ
on I,∀ δ-fine Π, σ0(Π) ≤ 12nε|I| : |S(f,Π) − J | ≤ ε
1983, JARNIK, KURZWEIL, SCHWABIK : M1-integrable on I :
replace σ0(Π) ≤ 12nε|I| by σ(Π) ≤ ε−1
1986, PFEFFER : Pf-integrable on I , using irregularity withrespect to a finite family of planes parallel to the coordinate axes
1992, JARNIK, KURZWEIL : ext-integrable on I if f extendedby 0 on some n-interval L ⊃ int L ⊃ I is M-integrable on L
M1-int ⇒ Pf-int ⇔ ext-int ⇒ M-int
all properties of KH-integral except Fubini’s thm; divergence thm for
differentiable vector field; no change of variable thm
The Kurzweil-Henstock integral and its extensions : a historical survey – p.27/29
Generalized KH-integrals on M ⊂ Rn
f : M ⊂ Rn → R, M compact
1985, 1988, JARNIK, KURZWEIL : PU-integral on M ,PU-partition defined from a suitable partition of unity, irregularitymodelled on σ
1991, PFEFFER : v-integral on BV-set M , v continuous outsideof a set of (n-1)-Hausdorff measure zero and almost differentiableoutside a set of σ-finite (n-1)-Hausdorff measure
1991, KURZWEIL, M., PFEFFER : G-integral on BV-set M , BVpartitions of unity; same divergence thm
2001, PFEFFER : R-integral on BV-set M , based on charges
2004, DE PAUW, PFEFFER : apply R-integral to obtain removablesets of singularities of elliptic equations
other results by JURKAT, NONNENMACHER, BUCZOLICH,
PLOTNIKOV, FLEISCHER, KUNCOVA, MALY, MOONENS. . .The Kurzweil-Henstock integral and its extensions : a historical survey – p.28/29
Thank you for your patience !
More details and references in
B. BONGIORNO, The Henstock-Kurzweil integral, Handbook of
Measure Theory, Elsevier, 2002, 587-615
TH. DE PAUW, Autour du theoreme de la divergence, Panorama et
syntheses 18 (2004), 85-121
J. MAWHIN, Two histories of integration theory : riemannesque vsromanesque, Bull. Cl. Sci. Acad. Roy. Belgique (6) 18 (2007) 47-63
W.F. PFEFFER, The Riemann Approach to Integration : LocalGeometric Theory, Cambridge, 1993
W.F. PFEFFER, Derivation and Integration, Cambridge, 2001
W.F. PFEFFER, The Divergence Theorem and Sets of FinitePerimeter, Chapman and Hall/CRC, 2012
The Kurzweil-Henstock integral and its extensions : a historical survey – p.29/29