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.