2. QUANTITATIV E RELATION S 3

D*(N; 1~) = 0 , while for 6(N; a) th e constan t i s chosen s o that /

T

S(N; a) da =

0. Similarl y

D(N;a,0)=6(N;(3)-6(N;a)

whenever a an d (3 ar e chose n s o as no t t o coincid e with an y u n. Henc e

D(N) = sup5(JV; (3) - in f 6(N\ a).

Prom thi s i t i s evident tha t

(12) S(N) D(N) 26(N).

2. Quantitativ e relations . W e now consider the possibility o f constructin g

quantitative relation s betwee n th e assertion s (a)-(d ) o f Weyl's criterion .

Suppose tha t F i s of bounded variatio n o n T, and tha t F i s continuous a t th e

points u

n

. The n

N

£ K ) = f F(a)dZ(N;a).

n = l

^

T

On subtractin g N Jj F(a) da fro m bot h sides , we deduce tha t

V F(u n) -N f F(a) da= f F(a) dD(N; a).

n = l J t Jj

Put

A(N;a) = D(N;a) - \ supD(N;/?) -\hdD(N;l3).

Thus supa A(7V; a) = ±D{N) an d infa A(JV ; a) = -%D(N). Bu t A(AT; a) differ s

from D(N\a) b y a constant , s o the integra l o n the righ t abov e i s

= [ F(a)dA{N-a).

JT

By integratin g b y part s i t follow s tha t thi s i s

= - [ A(N;a)dF{a).

JT

This integra l ha s absolut e valu e no t exceedin g

~D(N) J 1 \dF(a)\ = \D(N)Vzr T(F).

That is ,

N

(13)

J2F(un)~N J F(a)da

n = l

^

T

^D(N)VM

T

(F).

Although th e clas s o f Riemann-integrabl e function s include s function s no t o f

bounded variation , fo r practica l purpose s th e inequalit y abov e meet s ou r need s

in passin g fro m (d ) t o (c ) i n Weyl' s criterion .