GENERAL THEORY
53
2. The following example is instructive as showing that when
the conditions imposed in 1 are not fulfilled, the relation 1) may
not hold.
Example. Since ^ ^
I — =+0O,
x
there exists, for any 6 n >0, a 0<6 n+1 <5„, such that if we set
C bn dx n
) — * x n+1>
then
as b n = 0. Let now
b n+1 37
0^1 < 0^2 < "• == °°,
/ = 1 for the rational points in 2Ï = (0,1),
= - for the irrational.
x
ß
Then
f /=1
Let
tb
II
■31«
*-<
Let A n denote the points of 5i in (h n , 1) and the irrational points
in (ô B+1 , 6 n ).
Then C n
I > ^n+l = + 30.
^L A n
But obviously the set A n is conjugate to On the other hand,
J> =1 ’
lim f f =
n = 0o
while
+ QO.
56. If the integral
converges, then
If
J 21
e > 0, o- > 0, i* f
l-K®
for any unmixed part $ of 21 such that
<€
(1
(2