Prologue: Leb esgue’s 1901 pap er that c hanged the

in tegral . . . forev er

Sur une g´ en´ eralisation de l’int ´ egrale d ´ eﬁnie

On a generalizatio n of the deﬁnite in tegral

1

Note b y Mr. H. Lebesgue. Presen ted b y M. Picard.

In the case of contin uo us functions, the notions of the integral and antideriv a-

tives are iden tical. Riemann deﬁned the integral of ce r tain dis contin uous functions,

but all der iv atives are not in tegrable in the sense of Riemann. Research in to the

problem of an tideriv atives is thu s not solved by in tegratio n, and one can desire a

deﬁnition of the integral including as a particula r case that o f Riemann and al-

lowing one to solve the problem of antideriv atives.

(1)

T o deﬁne the in tegral of an

increasing contin uo us function

y ( x ) ( a ≤ x ≤ b )

we divide the in terv al ( a, b ) into subinterv als and s ums the q ua n tities o btained by

m ultiplying the length of each subin terv a l b y one of the v alues o f y when x is in

the subinterv al. If x is in the in terv al ( a

i

, a

i +1

), y v aries b etw een certain limits m

i

,

m

i +1

, and co n versely if y is b et ween m

i

and m

i +1

, x is b etw een a

i

and a

i +1

. So

that ins tead of g iving the division of the v ariation of x , that is to sa y , to giv e the

nu mbers a

i

, we co uld hav e g iven to o ur selves the division o f the v ariation of y , that

is to say , t he n umbers m

i

. F rom here there are tw o manner s of generalizing t he

concept o f the integral. W e know that the ﬁrs t (to b e given the num b e rs a

i

) le a ds

to the deﬁnition g iven by Riema nn and the deﬁnitions of the integral by upp e r a nd

low er s ums given by Mr. Darb oux. Let us see the seco nd. Let the function y range

betw een m and M . Consider the situation

m = m

0

< m

1

< m

2

< · · · < m

p − 1

< M = m

p

y = m when x b elongs to the set E

0

; m

i − 1

< y ≤ m

i

when x be lo ngs to the set

E

i

.

2

W e will deﬁne the measure s λ

0

, λ

i

of these sets. Let us consider one or the

other of the tw o s ums

m

0

λ

0

+

X

m

i

λ

i

; m

0

λ

0

+

X

m

i − 1

λ

i

;

1

This is a translation of Lebesgue’s paper where he ﬁrst reveals his integration theory . Thi s

paper appeared in C omptes Rendus de l’Academie des Scien ces (1901) , pp. 1025–1028, and is

translated by Paul Loy a and Emanue le Delucchi.

2

T ranslator’s footnote: That is, Leb esgue deﬁnes E

0

= y

− 1

( m ) = { x ∈ [ a, b ] ; y ( x ) = m } and

E

i

= y

− 1

( m

i − 1

, m

i

] = { x ∈ [ a, b ] ; m

i − 1

< y ( x ) ≤ m

i

} .

1