58 
IMPROPER MULTIPLE INTEGRALS 
21 C f, aft = 21/, - when c < 0. 
Similarly 
Thus 3), 4) give 
f cf=cf f 
-.f 
c c 
We now need only to pass to the limit a, /3 = ao . 
60. Ze£ owe o/ the integrals 
X / ’ x^ 
c > 0 
f c< 0. 
(1 
converge. If f — g, except at a discrete set T) in 21, both integrals 
converge and are equal. A similar theorem holds for the lower 
integrals. 
For let us suppose the first integral in 1) converges. Let 
21 = A -f- T); 
then 
Now 
JT/=//+//=//. 
= Um 4,./=X / - 
(2 
(3 
Thus the second integral in 1) converges, and 2), 3) show that 
the integrals in 1) are equal. 
61. 1. Let 
$ A f 9 (1 
—21 
converge. Let f>g except possibly at a discrete set. Let 
= Lv(f[^ a£21 !7i a f) 5 f = 21^ a3 ~ T) a(3 i Q a 0 = 21^ a(3 ® a /3* 
If - 
fa0 = O, g a ^=0, as a, /3 = CO,