57TheScienceReportsofthc KanazawaUniversity，Vol. V，No.2，pp 57‑59，May，1957A Remark01'1"theRiemann~SumBy
Shigeru TAKAHASllI均(R邑ciev
edFebrnary6，1959)
1. Introduction
め，ー∞く t<‑ト∞， be Lebesgue integrable on theintervallnthepresentnotelet f(1) and periodic with period1. Then weput，for every positiv
eintegerム(0，F，，(t，f)=す2f(:fサ〕
kwhich isknown as the Riemann‑sumof the functionf(t).
ln [2] B.Jessenhas proved t
hati
f 7lk1
1/"十[， then wehave，foralmosta
11t，(1. 1)

limF
71，，(t， f)= If(t)dt.
k→∞dSince then v
ariou
ssufficientcon
ditionsof(1.1)for an increasing sequence {nk}without assuming any arithmetica
l property havebeengiven，but inthecasewhere/ぐ t)巴 Ll' (0，1)theessentia
lpointisto deduce from them the converg
開 ceo
f theseries

手J¥Pn，川 ) ‑f fC~めどf川condit
ions wehave，for almosta
11t，lim.Fn!
，(t+Iz，， f)=If(めどlt，k→.
J。

where {l1k} isanysequenceo
frea
l numbers. (c園f.[1]).
On theoth百rhand iti
snotknownwhetherornotthereexist
safunctionof:theclass
L2 (0，1) such t
hatits Riemann‑sum does not converge to t
he integr
al almostverywhere.

告告 t
h邑 following
The purposeo
fthis notei
sto prov

1Z/c} beany increasing sequenceo
f primenumbers. ThenthereexistTheorem. Let {asequence ofreal numbers {llk} and a functionf(t) of the class Ll'CO，1) forany

p乙1such thatthetranslated Riemann‑sum F"，(t+lz，c/f) diverges almosteverywhere
1nt呂sk→+∞*7:‑)

2. Proofofthetheorem
hetheorem. Letus put，for止=1，2，……，
1n this paragraph weprov日 t持)骨持)

Mathe
mati
calI
nstitnte，Facultyo
fScience，KanazawaUniversity1nthesameway we canprov
ethistheoremforthecasewhere{1lk}isanyincreasingsequenceo
fpositiveintegers.(c.f.[4J)58S. TAKAHASHI
log‑2(忌ak= (2k‑1andk(2.1')

N，， =fl n"，.

Thenby the abovedefinitions，the sefIes宮 内 cos2nN
onverges to司 function
mtcm=lのforalmosta
11 twhich belongst
othec
lass L託0，1)for anyT乙1(c.L [6J)
. Ont
hesaprimenumber，wehaveotherhand since1Z" i(2.2)f)= 2
jam cos2nN"，/.
m=kFurther let us considerthe sequence {F叫(tofor anyfixe
dto，O~to三五1 ， andxpanda
11real凹 mb
告白川。三三x三三1， 呂 sfollows;邑

x=さコ仇 (x') N/c‑l('fk(X)=0，1，…… ，nk‑1)，

and put
ヂ間十l(X) 刃 -~m+l ，(2. 3)

H，刈 か(2.3')

skdyuEmwω+ん to}‑[1ωs2n{OThen by(2.1)，(2， 3)andthe r色lationIcos+ゆ ‑cos2π {Oぷゆ + Nふん} <2mz;;;r!
l <2π+1)‑1，1wehave，￨豆 A
k‑1/2‑F(2. 4)

and，for k>l，¥J1H
仇 ル ‑10れ￨山一 1/2，4')

whereA i
8a constant ind邑pendento
fto・ Sincecos2n{Dαfunctions，by (2.3)[同岬

1Sa sequence of independent+N叫ん}] isa
1so a sequenc
号 。
f ind色pend岳民t

ω. N.Kolmogoroff [む to H，，(t。 ，weobtain
functions. Applying thelemmaofA(O:
Cx孟1，Hk(to，(2.5)

I>1)1= 十 四 .

By (2.4) and (2.5)，wehave for anyε(0く εく 1)

手1(O:Cx<l，F(2.

Next letつ+x，f)>1一ε)1=十 四 @どの} beasequenceo
findependentrandomvariablesonaprobabilityspace

2，B，J and for缶百 ch忌(5(2. 7 ) ω〉
くx}= ζr，By (2.6) and (2.

wehave2jP{F叫(10十九 (ω))>1‑e}=十四，kand this implies，bythεBorel‑Cantelli's lemma，P{
limFn，(t。
十九〔ω))>l‑e}=1ok +∞forO<x<1.

59

A RemarkO?l tlze Rieman71‑sum
0 in (2.8)is arbitrary，there exists a point 白)0 such thatSince 1i(O<t<l，lim Ftz，(t+h(ω。
))>1ー ε)1=1.
1''''∞(2. 9)I
fwe put Ilk=tk(ω。
)， then by (2.2) (2.9) andthefact that lim IF;，，(t十 Ilk， f)dt=Ok!
‑+c。、/

フ

we can prove the theorem.

f(め anda sequence of realW e can not seen that there exist a bounded function.
numbers川

suchthat Fk(t+h"，.
f) does not converge tofi(似 almosteverywherein tas k→+∞.

Referencesnalysis (XL)，]om. Math.，1(1953)pp.80‑
86[1] S. Izumi，Notes onFomier a[2] B.Jessen，Ontheapproximation oftheLebesglleintegral
sbyRiemann.sums，Ann. Math.，
35(1934)pp.2
48‑
251.
[3] A. N. Kolmogoro，
狂 UberdasGesetz desiterierten Logarithmus，Math. Ann.，101(1929)pp126‑
135.especiallyHilfssatz11.
[4] S. Takahashi，Onthe complete convergence ofthe Riemann.sllm，llnderthepress.
[5] S. Yano，A remark onthe Riemanns
llms，TohokllMath.Jom.，2(1950)pp.1‑
3日[6] A.Zygmund，Trigonometricals
er児 島 Warsaw(1935)especialJy p.
216.