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(u n )~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 .
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