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Generalized Sierpinski Functions and Fragmentable Compact
Spaces
Matsuda, Minoru
Reports of Faculty of Science, Shizuoka University. 35, p. 7-15
2001-02-28
http://dx.doi.org/10.14945/00000872
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This document is downloaded at: 2015-01-31T17:48:27Z
Reports of the Faculty of Science, Shizuoka University, Vol.35, pp.7-15 (2001)
Ori,ginal Paper
Generalized, Sierpinski Functions and
Fragmentable Compact Spaees
Minoru MRrsuolt
Defuartment of Mathemntics, Faculty of Science,
Shi,zuoka Uni,uersi,ty, Ohya, Shizuoka 422-8529
(Receiaed Nouember 29, 2000)
Abstract: We present functions named generalized Sierpinski function, each of which
can be regarded as a generalization of the so-called Sierpinski function. Making use of
these functions, we make a direct study of two types of fragmentable compact spaces in
the measure-theoretic aspect.
2000 Mathematics Subject Classification: 46822, 46807.
Key words and phrases: generalized Sierpinski function, H-fragmentable compact
space, weak H -fragmentable compact space
1. Introduction
Let X be a real Banach space and X* its topological dual space. In our recent paper
[7], for a bounded subset A of. X and a weak*-compact subset K of. X*, we have defined
two notions on sets K called A-Pettis sets and A-Radon-Nikodym sets (slight generalizations of Pettis sets and norm-fragmented sets, respectively), and we have made a
simultaneous and unified study of such sets by constructing K-valued functions associated with non-A-Pettis sets or non-1.-Radon-Nikodym sets and making use of them
effectively.
In this paper, we are going to investigate such type of notions in a more general setting
by following the same ideas as in [7] (refer also to [S] and [6]). Now, our setting is as
follows. Let Y be a compact Hausdorff space, C(Y) the Banach space of all real-valued
continuous functions on Y, and let us give our attention to the following two types of
tE-mail: smmmatu
@
ipc.shizuoka.ac
jp
Minoru Mersuna
fragmentability
of Y
associated
with a bounded subset H
of.
C(Y).The notion of
fragmentability has appeared in [3].
Definition L (H-fragmentability of Y\. Let e
fragmentable if for every nonempty closed subset. M of I there exists an open subset G
of Y suchthat M n G+aand diam, (M n G) (- sup{VU) - fbdli !r, lz e. M n G,
f e H\) ( e, and IZ is said to be H-fragrnentable if IZ is (H, e)-fragmentable for every
e)0.
Note that this is correspondent to the notion of .F/-Radon-Nikodym sets in dual Banach
spaces, and (H, e)-fragmentability has been used in l2l to analyze Radon-Nikodym
compacta.
Definition 2 (Weak H -fragmentability of. Y). Let e ) 0. We say that Y is weak (H,
e)-fragrnentable if for every nonempty closed subset M of. Y and every f e Ho (: the
closure of. H with respect to the pointwise convergence topology on Y) there exists an
open subset G of Y such that M i G + fr and OVIM n G) (-- sup {Vbil - fU,)l : y,,
lz€MnG))
e)-fragmentable for every
e)
0.
Note that this is equivalent to the notion of weak precompactness of. H (That is, .F/
does not contain an /r sequence. For further details of weak precompactness subsets I/
of C(IZ), see Theorem 7.3.1 in [8]) ana this is coffespondent to the notion of .F/-Pettis
sets in dual Banach spaces. Further, H -fragmentability of Y naturally implies weak
H -fragmentability of. Y .
Our study in this paper is a direct and simultaneous consideration of measure-theoretic
characterizations of these notions with the aid of Y-valued functions on .I (- [0,1])
constructed in the case where Iz is a non-H -fragmentable space or a non-weak f/fragmentable space. That is, we wish to give some more characterizations of such sets,
which are completely analogous to the results in [7].
The paper is orgaruzed as follows. In $2, we show the existence of generalized
Sierpinski functions under some general conditions by the same argument as in [a]. Each
of them can be regarded as a generalization of the well-known function, so"called
Sierpinski function. Making use of this result, in 53, we can present basic generalized
Sierpinski functions which are very useful in analyzing directly H -fragmentable compact
spaces and weak H-fragmentable compact spaes. In view of this consideration, in $4, we
can give some measure-theoretic characterizations of these spaces, from which we can
recognize their similarity and difference in the measure-theoretic aspect. Thus this paper
may be regarded as a complete analogue of [7], although it is written under a more
general setting. In other words, this also indicates that our method introducing general-
Generalized Sierpinski Functions and Fragmentable Compact Spaces
ized Sierpinski functions makes it possible to analyze such general spaces in a direct
mamer lby the complete same argument as in[7]),which is an appealing point of our
paper.
2。
Generalized Sierpinski functions
Let us begin with the following:
Definition 3(Generalized Sierpinski functions).Let T and S be compact Hausdorff
spaces,力 be a continuous surieCtiOn fron1 7￢ to S,and νbe a lRadon probability measure
on S.Then a function λ:S→ r is said to be a`″ %ι ,ヮ ′
″′
グS′ι
脇〔
ψ;ルπσ
πθ%associated
ψ焼
ν)if the fol10wing three conditions are satisfied.
(1)The function力 is a〃 (S)‐ ン (T)measurable ftlnction such that力 (ν )(the image
measure of νby tt is a Radon probability measure on T,where〃 (S)(resp.´ (r))
・algebra of S(resp.T).
denotes the Borel σ
with a pair(力
,
(2)力 (λ (ν ))=ν 。
(3)/(力 (力 (′ )))=/(′ ),力 (ν
R′ %α ′
滋
)‐
aoe.for each/∈
C(r)。
1.If r is a compact metrizable space,the condition(3)in Definition 3 may be
replaced by
a.e.on T,
since C(r)is separable.
(彙
)λ (力 (′ ))=′ ,力 (ν
)・
Here,we wish to show the existence of generalized Sierpinski functions associated
with such a pair(力
,
ν Before stating it, let us see the Sierpinski function in our
)。
framewOrko Let r be the cantor space{0,1}Ⅳ ,S the unit interval f,and
λthe Lebesgue
measure on fo Let%≧ :l and let rpz denote the%th]Rademacher function on r defined by
場(s)=Σ
k=1
for every s∈ f,where r(力
,グ
1ル
(-1)力
0,o(S)
1)/22,グ /2つ if%≧ 1,1≦ ′
≦2π -l and f(π ,2つ
)=[(グ ー
=
[(22-1)/211]if%≧ 0.Define力 :{0,1}Ⅳ → f by力 ({場 }た 1)=Σ 鷹1場 /2π for every′ =
1∈ {0,1}Ⅳ .Thenヵ is a continuous suriectiOno cOnsider the function(so‐ called
{為 }″ ≧
Sierpinski functiOnl λ:r→ {o,1}Ⅳ defined by
力
(s)=(0-場 (s))/2}π ≧
1
for every s ∈ f.Then we get that
(1)The function λis a″ (r)・ ´({o,1}り measurable fШ lction such that λ
(λ )iS a
nOrmalized Haar measure on{0,1}Ⅳ
(2)力 (λ (λ ))):= λ,and
,
Minom MATSUDA
(3)λ (力 (′ ))=′ ,λ (λ a.eo on{0,1}性
)‐
Hence the Sierpirlski function can be regarded as a generalized Sierpinski function
associated with this pair(た ,λ ),Since(0,1}Ⅳ is a cOmpact metrizable space.
Now,let us prove the follo宙 ng:
ι
′rα%グ sみ ι ttCσ ′
`η
比豚あグ 軍脇θ
6,力 ι
′
」
′αωπ″%%ο 雰 協″ b%/ra%r"Sα %″ νι
′αRα あ%夕 知滋み
妙
`ガ
%ι %″ 0%Sa乙 夕
″
%ヵ物π″ぉおα♂7%ι πルι
グS′ι
%ε J笏 %λ 6sο 磁 ″
グ ′
λαρα
ψれ壺グ
ルι
"′
“
(力 ,ソ
Theore】 ml(Existence of generalized Sierpinski fmctions).ι
)。
Praぼ The ar_ent developed here is almost the same as in[4](or[5]).So we give
an outline of the proof for the sake of ilnportance.Let(S,Σ ν
,ν )(resp.(T,Σ μ
,μ ))be the
completion of(S,多
(S),ν )(resp.(T,多 (r),μ )).Then,by virtue of the well・
known result
(1,2,5)in[8]or PropOSition B。 l in[1]),we have a Radon probability measure μ on
T such that力 し)=ν and Zl(r,Σ μ
l(S,Σ ν
,ν )}.Define a linear
,μ )={g。 力:θ ∈ ι
operator y:Ll(s,Σ ν
,ν )→ Zl(r,Σ μ
,μ )by 7(g)=g。 力for every g∈ ιl(S,Σ ν
,ν ).Then
aoe.on r for every
the operator 7 is a suriectiVe isometry such that y中 び)(力 (′ ))=/(′ )μ ‐
(cf。
0/2)=7*“ )07*α )(in
,μ )and 7*“
/∈ L(T,Σ μ
L(S,Σ ν
,ν ))forバ ,/2∈ JL(T,Σ μ
,
μ Here 7・ denotes the dual operator of 7.
)。
Let′ be a lifting on L (S,Σ
ν
,ν ).For each s(≡ S,the bounded linear functional Ls on
c(r)defined byム び)=′ (7*び ))(S)fOr every/∈ C(r)is multiplicat市 e and so,we have
a function λ:S→ r such that/(λ (S))=′ (7*び ))(S)fOr every/∈ C(r)and every s∈
S.Hence we have that′ び。
劾 =/。 λfOr every/∈ C(r).Thus,by宙 rtue of the lifting
λis a〃 (S)Ⅲ 〃(r)measurable function such that λ
(ν )iS Radono So the
condition(1)is satisfied.Moreover,/(λ (S))=7*び )(s)ν ‐
aoe.on s for evelγ /∈ C(r).
theory,we get that
a.e.On r for every
し)=ν and 7中 び)(力 ))=/(′ )μ ‐
a.e.on r for every/∈ C(r).Hence we have
)))=/(′ )μ ‐
Combining this with the result that力
/∈ c(r),we have that/(λ
(力 (′
(′
that
!)rutoo1''1 -_ l)rwao))dp(t)
f,ru'llankxs)
Ifl,lant xtl
- [,rus))du(s)
for every/∈ C(r).Thus we get that μ =π ν
),and so the conditions(2)and(3)are
satisfied.Hence the proof is completed.
Rι %α ,脅 2.Since it holds that/(λ
that for every E∈ 〃 (S)
(力 (′
a.e.On r for every/∈ Crr),we have
)))=ノ (′ )μ ¨
Generalized Sierpinski Functions and Fragmentable Compact Spaces
11に /(′ )み
(′
)=11に /(λ
=ル
(力 (′ )))″ (′ )
(λ (S))激
し)(S)=ル (λ (s))ん (S)
for every/∈ C(r).
Now,for the convenience of readers,let us indicate here the typical case where the
existence of generalized Sierpinski functions can be guaranteed.
Let y be a compact Hausdorff space and suppose that there exists a system(7(η ′
):
%=0,1,¨ 。;′ =0,…
,2″
-1}of nOnempty dosed subsets of y such that y(%+1,2グ )
7(aグ )α %グ 7(%+1,2′ )∩ 7(%+1,2′ +1)=O for%=0,1,… and
′=0,… ,2″ -1.Then,letting 4π =∪ {7(%,2グ +1):0≦ ′≦ 22 1-1}and鳥 =
∪{7(%,2′ ):0≦ グ≦22 1-1},T=∩ 鷹1(4π ∪a)is a nonempty compact subset of
y.Define ψ:T→ {0,1}Ⅳ by ψ )={為 }た l where均 =l if′ ∈ 4″ and物 =O if′ ∈
鳥.Then ψis a∞ ntinuous suriectiOn.Further,consider a functionノ :{0,1}″ →r deined
byノ 0)=Σ 鷹 1場ノ22,where″ ={為 }″ ≧l∈ (0,1}Ⅳ .Thenノ iS a cOntinuous suriectiOn.Let
∪ 7(%+1,2グ +1)⊂
(′
力=ノ 。
ψ :r→ f.Then we have a generalized Sierpinski function
with this pair(力 ,λ
(a)ノ け
0)=λ
)。
λ(:r→ r)associated
So,letting μ =λ (λ ),we have that
,
0)1“ ′
.に D力 )み し)=1/(as))み ●)
for every E∈ 〃(r)and every/∈ C(y)。
These are derived from Remark 2 and the fact that
μ is regarded as a lRadon probability
measure(on y)supported by T。
3.Basic generalized Sierpinski functions associated with
non‐ 』
7‐ fragmentable spaces or non― weak J‐ fragmentable spaces
ln what fonows,an notations and terminology,unless otherwise stated,are as in[7].
Let y be a compact Hausdorff space.If/:J→ y is a〃 (I)¨ 〃(y)measШ rable
ftmctiOn,we obtain a bounded linear Operator 2レ :C(y)→ Zl given by lみ (の =σ O/fOr
every σ∈ C(7).For such a function/,we get a system{22Tナ (ル レ,JD:%=0,1,… ;グ =
0,・ ",2″ -1},which is a tree in C(y)ホ 。
In the following,this is called a tree associated
With/。
Now, let us present a fundamental result which makes it possible to analyze
″‐
frattentable spaces and weak ″ ‐
frattentable spaces in a direct and simultaneous
manner fron■ the measure‐ theoretic view‐ point.This is a complete analogue of Proposi‐
tion 5 in[7].
Theorem 2。 Lι ′yみ ′αωttcσ ′麟
α
Oグ 勁
“
α
%グ ″ み
′αι
ο
%%滋 グsπおι
′o/C(y).
L/1inoru MATSUDA
ι
勿′yた %ο ′二
ι
πttι /aJJa"グ ばs滋 ″zι πた(Pl α
%″ fQl λ
ι
ο
a
均%召 zι πttι ル
・ηわ
“
q
お′α夕asグ″υ
″ αsys″ π {/2,J:%=0,1,… ;′ =0,… ,2″ -1}勿
′π%%ι ′
lPl ηttπ ″
〃α
%″ αsys″ %{7(%,グ ):%=0,1,… ;グ =0,… ,22-1}グ %ο %ι %″ 夕
彰グS%ゐ ο
お
`わ
yS%磁
′
滋′
グ
(a)7(%+1,2グ )∪ 7(%+1,2グ +1)⊂ 7(%,′ ),
(b)%∈ 7(%+1,2′ )α %″ υ∈ 7(%+1,2グ +1)シ ψ夕 ん (%)一 ん,J(υ )≧ c/a″ %=0,1,…
α%グ グ=0,・ “ ,2″ -1.
厖
グS′ι
lQl tt υ
鶴カ
グル%`励 %力 :f→ yw勧 グ
fPl,ω ιttυ ια多%ι πル′
ψグ
η ttθ
(1)S〃ψasa
,ゴ
"グ
商″
S.
力JJa"グ 響 夕%″ 夕
θassOσ 勿″″ ′
あ力なαπJ手 (c/2)け π夕
(a)動 ι″′
.
"グ
(D Jft潔
力)(s)一 二び。
ヵ)(S】 )み (s)≧ c/2
し +1び °
当
日
foreueryn2L.
(c) The set {f"kif e H} X not equimeasurable in I^.
(D Suppose tlat Y is not ueak H-fragmentable. Then the following statements @ and
(s) hold.
(N There exist a positiue number e, a seq?,cence {f"I or,, in H and a system {W(n, i) : n
0, 1, .." ; i : 0, ... , 2n - L| of nonempfii closed subsets of Y such thnt
(a) W(ntL,zi) U W(n*I,zi+L) c W(n, i),
(b) u e, w(n*r,Zi) and u € w (n*L,zi+r) imbly .f"*r(u)
- f"*r(a) 2 e for n = 0, L,
...andi:0,...,2n-L.
(S)
In
uiew of (N, we laae a generalized Si,er?inski function
h:
I -
Y satisfuW
the
fo llowi.ng prop erti,es.
(a) The tree associated with h is an H-separated (e/2)-tree.
O sup‖
ノ∈″
。
島。
劾―島び
1≧
の
‖
習￨ル
+1び
foreueryn2L.
(c) The set { f "h: f e Hl
i^s
(as》 %③
況01≧
c/2
not relatiuely norrn compact in Lr.
Proof. Since the proof can be given by the complete same argument as that of
Proposition 5 in [7], it is omitted.
4. Chara ctefizations of fragmentable compact spaces
Making use of results in preceding sections, we are going to give some measuretheoretic characterizations of H -fragmentable compact spaces and weak H fragmentable compact spaces.
Concerning H -fragmentable compact spaces, we have:
Generalized Sierpinski Functions and Fragmentable Compact
Spaces
L3
αωの α
′肋 箔あ″rs″ α
σ
%グ 〃 bι αめ
%%滋 ″s%ゐο
ο
′グ C(y).
“
yα
動ι
π′
滋 /aJJa"′ 響 S滋 滋 ι
%お α
0%′
%グ
″
″ι
υ
α
″%洗
め
α
μグ
ル
(a)yな 月7姥g%ι π滋み
υ
ι
(b)Faγ ι
ιassOθ 滋″グ紗′
′
み/た
η 〃(r)_〃 (y)協 ι %π bル ル%ι 励%/:J→ Кttθ ″ι
“
%0′ α
%μ δ″θ
ι
。
υ
ι
厖ル″
(C)乃 ″θ
%θ 滅,%/」 J→ Z″ 力
ο
JJs′ 滋′
η 〃(r)‐ 〃(y)2ι ω%π み
Theorem 3.ι ι
′y
bι
.
縫ilプ k:当 1島
(o)」
+1(gO/)(S)一 島
(σ
O/)(S)D裁 (s)}=0。
勁″ι
υ
ι
ルル%σ 励%/:f→ Кttι sι ′{θ O/:J∈ ″}た
η 〃(f)‐ 〃(y)2″ s%π み
ι
σ%′%ι
%ηb厖 ′
%L.
`雰
助 グ The Same ar_ent as in[6]and[7]easily deduces that O)⇒
(C)→
(b).
(b)⇒ (a).This part is crucial and it follows from Theorem 2.
So we have only to show that(a)⇒
To this end,take a〃 (r)‐ 〃(y)measurable
function/:f → И Then,by virtue of the lifting theory,we have a〃 (f)‐ ′(y)
。
measurable function力 :f→ y such that λ
(λ )(=μ )is Radon and{gO/:g∈ 〃}=(θ
λ:g∈ 〃}in Lo By the property of Radon mё asures on y and″ ‐
fraglllentability of
(d)。
y,we easily get that for every positive nurnber c and every element 4 of Z(y)with
μ(4)>0,there exists an element 3 of〃 (Y)with μ(3)>O
and B⊂
z4such that supg∈ 〃
0(dB)<c.Hence,the well‐ knowFl eXhaustion ar_ent deduces that for every positive
nurnber c,there exists a disiOint sequence{E″
}″
≧
1⊂ 〃(γ ),all
of positive
measure,
μ¨
)=O and supθ c〃 0(σ lβ″
μ(yヽ ∪鷹1拓場
)<c for every%≧ 1.Thus,taking EL=
-1(a)for every π≧1,for every positive number c,we have a disioint Sequence{島
ヵ
}π ≧
⊂ A,a1l of positive卜 measure,such that λ(I＼ ∪鷹12)=O and supθ ∈
″0(′ 。
冽E″ )<
such that
1
c for every%≧ 1.Therefore,the same ar_ent as in the proof of Proposition 3 in[6]
deduces that{g。 力:σ ∈ ″}is equimeasurable in L,whence we complete the proof.
Concerning weak″ ‐
fragmentable compact spaces,we have:
Theorem 4.ι ι
′yみ ιασ
ο
ttQ`′ 肋 郷あグ ψα
″α
%グ 〃 ι
′αι
ο
%π 滋″s%決グ a/C(y).
αι
O%′ yα %グ 〃 α
%θ σ
%グ υ
αル%洗
η
ι
ι
滋
」
￠
%″ み
(a)yな
ル
躊ル婆″ ι
υ
ι
(b)Д 9″ ι
6%π み厖ル%σ 励%/:f→ К ttθ ″ιιassο ″グ ′
λ/た
η :〃 (r)_′ (y)協 ο
"グ
`滋
%0′ α
%flsィ滋π″グ δttc.
υ
ι
6%zb厖 ル%σ 滅)%/:f→ К ″λο
(C)乃 ″θ
JJs′ 滋′
η 〃(f)‐ 〃(y)協 ι
aゎ ι
%滋 θ/aJJa"グ %ば s滋 ″πι
%た
.
2+1(gO/)一 EXgO/)‖ 1〕
縫I{:』 ‖
=磐
{:当 1/gび (S))場 (s)激 (s】 }=0.
%/:f→ Кttι sι ′{gO/:′ ∈ 〃}お
ぅ
ι
(d)nフ ″ι
6%π み
厖ル%σ 励
η 〃(f)‐ 〃(y)協 ι
π滋励 ι
物 σ
Ottσ ′物 ιl.
夕 πο″
Minoru MATSUDA
Praグ The same arttent as in[5]and[7]easily deduces that(d)⇒
(C)⇒
(b)。
(b)⇒ (a).This part is crucial and it follows from Theorem 2.
So we have only to see that(a)=⇒ ld),which follows fronl the bounded convergence
theorelTl,since(a)is equivalent to that ttr is weakly precompact。
Rι ″協″
{ら :ノ ∈
a For everyノ ∈ y,letら denote the umt point measure atノ and let δ(y)=
y}.Then
δ(y)is a weak*_compact subset of C(7)申
and wei― ediately
get
that y is″ _fragmentable(resp.weak〃 ・fra≦要lentable)if and only if δ(y)is an
″‐
Radon‐ Nikodtt set(resp.″ ‐
Pettis
set)。
So we knOw that our Theorems 3 and 4 can
be derived easily from Theorems l and 2(Characterization Theorems of″
¨
Radon¨
Pettis sets)in[7],respectively.Consequently,as stated in§
Nikodtt sets and″ ‐
1,the
,τ ε
′way to
″′
‐
〃‐
″ fraglllentable compact spaces and weak
only thing to be ёmphsized in this paper may be that we can present a
analyze general spaces, such as
frattentable compact spaces,by introducing gttπ
ι
π」
″aa Sierpinski functions.
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Generalized Sierpinski Functions and Frattentable Compact Spaces
15
-般 化 され た シ ェル ピンス キ ー 関 数 と
フ ラ グ メ ン ト可 能 な コンパ ク ト空 間
松 田
稔
静岡大学理学部数学教室
〒422-8529静 岡市大谷836
シェルピンスキー関数の拡張 と考えられる一般化されたシェルピンスキー関数を新たに定
義 し,そ れを構成する.こ の関数を利用することにより,二 つの種類のフラグメン ト可能な
コンパクト空間の測度論的側面での,並 行的,直 接的考察を行なう
.
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