GENERAL THEORY 
81 
iTEGRALS 
¡red only proper multiple 
1 the integrand f(x i ••• x m ) 
improper. When m— 1, 
Lapter 14. An important 
develop is the inversion 
mproper integrals. The 
I may be simplified and 
ties of improper multiple 
■way space At each 
ite value assigned to it. 
which lie in 21 we shall 
,nd this case is by far the 
y. We shall call $ the 
!. At the point x = ~, 
n 
et f—ns. At the other 
of 2i is a point of infinite 
per integrals have been 
e. 
A of norm 8 of into 
Such divisions may be 
> containing points of 21, 
e the cells containing a 
limited in 21«. Hence / 
egral in 21s. The limits, 
for all possible complete divisions A of norm 8, are called the 
lower and upper integrals of f in 21, and are denoted by 
(2 
or more shortly by 
When the limits 1) are finite, the corresponding integrals 2) 
are convergent. We also say f admits a lower or an upper improper 
integral in 21- When the two integrals 2) are equal, we say that 
f is integrable in 21 and denote their common value by 
We call 8) the improper mtegral of f in 21; we also say that 
f admits an improper integral in 21 and that the integral 8) is 
convergent. 
The definition of an improper integral just given is an extension 
of that given in Yol. I, Cdiapter 14. It is the natural develop 
ment of the idea of an improper integral which goes back to the 
beginnings of the calculus. 
It is convenient to speak of the symbols 2) as upper and lower 
integrals, even when the limits 1) do not exist. A similar remark 
applies to the symbol 3). 
Let us replace / by |/| in one of the symbols 2), 3). The 
resulting symbol is called the adjoint of the integral in question. 
We write — — 
When the adjoint of one of the integrals 2), 3) is convergent, 
the first integral is said to be absolutely convergent. Thus if 4) is 
convergent, the second integral in 2) is absolutely convergent, etc. 
29. Type II. Let A, ¿a>0. We introduce a truncated func 
tion / A(li defined as follows : 
An —f(l\ x m) when — A </< g 
iTEGRALS 
¡red only proper multiple 
i the integrand f(x 1 ••• x m ) 
improper. When m— 1, 
Lapter 14. An important 
develop is the inversion 
mproper integrals. The 
I may be simplified and 
ties of improper multiple 
■way space At each 
ite value assigned to it. 
which lie in 21 we shall 
,nd this case is by far the 
y. We shall call $ the 
!. At the point x = —, 
n 
et f—ns. At the other 
of 2i is a point of infinite 
per integrals have been 
e. 
A of norm 8 of into 
Such divisions may be 
> containing points of 21, 
e the cells containing a 
limited in 2ls> Hence / 
egral in 2ts* The limits, 
= ^ 
when/< — A 
when f>g.