Sci.Rep.KallazawaUniv.，
Vo1.14，No.2，pp.59~60
D巴cemb
巴r1969A Note on a Definition of (G)convergen.
ce
担ChikaraWATANABE
DepartmentofM athematics，FaC
1utyofScience，Ka拝。zawaUniv
ersi・ty(Reciev
ed2
3October1969)

InaLectur
己 notegiv
enbyG.Stolzenber
gl，thedefinitionof (G)‑convergence isgiv
enasfollowsle
tX beamet'1icspac
記 and{S;Jbeasequenceo
fc
10se
dsu
bse
tso
fX，thenitissaicl'g巴stoac10sec
lsubsetS i
ffo1' any compact s
巴t K c X，thatthesequenc
e{Si} conve1{SinK}isaconve'1gentsequencei
nComp (瓦) anclS=Ulim (S，nK).
Kiーラ∞

Moreoveritisdenoteclt
hati
fX is←compact，thenafamilyo
fc
los吋 subsetsof
Xi
sno
'
1
mali
nthεaboves
ense.1nou
rformerp
aper
s2)，¥
v
eu
sec
lt
h
eabovep
rop
e
r
t
yfo
'
1afamilyofanalyticset
si
nadomainofC'. Howeve，
'
1 r
ecent
1y，M.Kita3) pointedout
記 abovec
lefinition. Therefo1'ewethatnoconv
ergentse
quenceofp
oint
si
snormalinthamendt
hed
efinitionof (G)‑convergencea
sfol1ows.
，ρ) beam巴tr
ics
p
旦c
ea
nc
l{S;}beas
equ己nceo
fc
10sedsu
bsets
De
fi
ni
tion. Let(Xhato
fX. Wesayt

conv
ergesgeometr
icall
yt
oadosedsubsetS ofX ifうES，therei
sa.sequencε{戸;} ofpointssuch thatPiE三Si and(i) foranypoint1

Jうz一""p.
(ii) foranycompacts
etK andpositiv
e number e，therei
sapositiveintεger。(K，ε) suchthatSnKcS )nKfor vミ;Vo，whereS'，)= リ {q'E三X;ρ(q，q')<ε1Vo=ν

ノC
εqES

Notet
hatfromt
hi
sd色finitionthefollowingpropertiesar
e01
コtaineclimmediatel}
人1. I
fSnK=仇 thenS
川 K=ゆforsu
fficientlylargev.
20 I
fasequence住，}ofp
oints1
うi
ES
ic
onv
er
gestoJ
うE X，thenρεS.
30 I
f{Si}converg邑sgeometricallytoS and T，thenS=T.
LEMMA 1
01fX isaィ ompact，thenαlamily @501closed subsetsisnormalo1)Volumes，limitsandetensionsofa
nalyticvar
ieties，LecturenoteinMath.，No.19，Spr
inger
Ver
lag(1966).
2) Onnor
mar
ityo
faf
amilyofa
nalyticsets，Sci.R巴p.K
anazawaUniv.，12(1967)，209‑213.
im邑nsionaianalyticsets，ibid，13(1968)，73‑
82.
Onafamiiyofpur
edAremarkonthetheoremofBishop，Proc.JapanA
cad.4
5(1969)，243‑
246.
tyo
fS
cience，TokyoUniversity.
3)DopartmentofMathematics，Facu1
瓦咽‑59‑‑

ChikaraWATANABE，60Let{Kν}beasequenceo
f compactsubsets()
fX such thatandU瓦v=)乙 Takeanysequencethereisasubsequence

dof思 SinceCcmp(K1) i
sccnr
cac
tmetr
icspace，ofsuch that'1 inccnv
erge
sto1
Also，sinceComp (K2) i
scompact，therei
sasubs問 uencε(2)}ofn2convergestoT2 iil}c{SiConv
erge
stothespaceinComp(K)，wehaγeassumεT手功。 Let。
〉

EuchthatWec
仁ntinじεthisp
n:ceEs.A diagcnalEequ
Ence.LetT=foranyj

From thepropertyofto prove thatT i
sdosed.

Evidεwe may

bea sequenc巴 of;r.ointsinT Euchtha
ttj→jJEX. PutK =othenforsome

K~瓦 00 We sha1
1 show thattjETjo forinteger
ESv'ρince1うjE Tj!forsomej'， ther
ε isa sequenceo
fanylLettj年 生 ・ Ssuchthat
Letd=psuchthatν(→∞)， ands
incet，EEKjo。

convεrges toT""thereisa0).S
inceS
ν(レintegerνocTJJdJZ〉 品

for ν量;;2inceqνCj)
E5lJ(シ10・ S三lT =min

we mayassumet
hat討/2)for JIミ;;
voheother hand，・ On tdr;l(d(ρj ， 一 ρtETj，contradictio
nand1合jET
ioforanyJSinceQ.E.D.

i8closedtETj.cT.

，l‑dimensionalanalyi
icEetsinadomainbeas
eq
釘enceεetS in
D ofCn.
conv
ergestoa
島‑dimensicnalhe
n{S;}C01官v
ergesD，ttoS.
LEMA2. LetProof. Thisi
sadirectconc
1usionofan乳 nalyticconv
ergence.
0 LetX beametr
icstacea針d {8.
Jbeasequena clusedsubsets X
LEMMA3
whichconv
ergestoS. ihereareec
正 問ta11t M s'l/.chthatfo1'anytE
Hd ιr.
B(長:rη二三lVrd wJ:ereHriis a d.
dime
11sIi釘alιte幻 vall<IYIalll，thena町andthat1Vloreov
er Xisトthen約三五Theproofofthis Lemmai
8thesame asthatheTheoremo
fFromLemma 3，tTHEOR琵 M OF BISHOP. Let{S'i}be asequenceofi
nadomainD ofCn α!
hichconv
ergest.
hevolumesofSiareradiu
sr with centerT inEet
Hd(Sr瓦〉話in.
theLeciure note'of G。
holds.
k胆正iimensionalEets. 1fclosedsetSsetin4) Condi
tion
sfortheanalyticityo
fc
巴rtainsets Mich Math.J.，11 (1964)，289‑304
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