GENERAL THEORY 
57 
integral on the 
, the condition 1° 
are satisfied, 
tisfied,f+g is in- 
definitely infinite. 
Example. Let 21 consist of the rational points in (0, 1). 
Let 
/=l + w , g = 1 — n 
at the point x = —. Then 
n 
Now 
/+<7=2 
21, 
in 2f. 
*/, a/3 1 a/3 
embrace only a finite number of points for a given a, /3. On the 
other hand, 
3i/ +ff ,a0 = 2l for /3 > 2. 
Thus the upper content of the last set in 1) does not = 0 as 
a, /3 = go and condition 1° is not fulfilled. Also relation 2) does 
not hold in this case. For 
/(/ + /> = 2 , ff= 0 , Cg = 0. 
59. 
Tfc>0, thenj % cf=cj^ 
if c<0, then cf = c £ f 
provided the integral on either side is convergent. 
For _ __ 
X cf=c S* f iic>0 
Cf a/3 
cf a/3 
= c r f if c < 0. 
—®e/, a/3 
Let c > 0. Since 
therefore 
« < cf < /3 in 2f c/ , a|3 , 
in this set. 
/ "^ — 
c — c 
(1 
(2 
(3 
(4 
Hence any point of 2f c/ , „,3, is a point of 21/, - and conversely. 
Thus *.« 
dations 2) or 3) 
when c > 0.