Functor of extension in Hilbert cube and Hilbert space

(1)

Abstract: It is shown that if Ω = Q or Ω = ‘ , then there exists a functor of extension of maps between Z -sets in Ω to mappings of Ω into itself. This functor transforms homeomorphisms into homeomorphisms, thus giving a functorial

setting to a well-known theorem of Anderson [Anderson R.D., On topological infinite deficiency, Michigan Math. J.,

2

1967, 14, 365–383]. It also preserves convergence of sequences of mappings, both pointwise and uniform on

compact sets, and supremum distances as well as uniform continuity, Lipschitz property, nonexpansiveness of maps in appropriate metrics.

MSC: 57N20, 54C20, 18B30

Keywords: Z -set • Functor of extension • Hilbert cube • Fréchet space

Cent. Eur. J. Math. • 12(6) • 2014 • 887-895 DOI: 10.2478/s11533-013- 0386-6

Central European Journal of Mathematics

Functor of extension in Hilbert cube and Hilbert space

Piotr Niemiec^1∗

Research Article

1 Institute of Mathematics, Department of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

Received 28 January 2013; accepted 10 September 2013

1. Introduction

Anderson in his celebrated paper [ 2 ] showed that if Ω = Q or Ω = ‘2 , t h en every homeomorphism between two Z -set s et een Z - ets was generalized [3 , 9] and se tled in an m nif ld modell d an nfi ite- im nsi nal réc et s a e 9]

thpeorloegm in( aofrumncotroeriianlfomrmananteor. Toon fZo-rmseutlatceo osuurl reesult[s, lCet aupstefirx Vth].e Tnhotaationo. f this paper i to strengthen Anderson’s

∗ E-mail: piotr .niemiec@uj.edu.pl

887

in Ω can be extended to an autohomeomorphism of Ω (see also [1] or [8]). The theorem on extending

homeomorphismsb w s t y a o e on i n d e o F h p c

[(which is, in fact, homeomorphic to a Hilbert space, see [ 16 , 17 ] ), and is one of the deepest results in infinite-

dimensionalto o y. F i n s n t .g. 8, h ) e im s

(2)

Functor of extension in Hilbert cube and Hilbert space

Notation and terminology

Below, Ω continues to denote the Hilbert cube Q = [−1; 1]^ωor the Hilbert space ‘2 , and for metrizable spaces X and Y

by an overline; in particular, im(’) denotes the closure of the image of a function ’. For a topological space X , Z(X ) uniform convergence topology). A embeddi g u : (X ; d) → (Y %) bet een met ic pa is called u iform f bo mean a y me ric wh h nduce t t polo of the spac T e olle tion of all om ib bou d me i s

topodlsoupg.icTahlespcaacteegs,orTyopoIf Kcodnetinnoutoeussthfuenccatitoengsory owf

e(eanll)tompaoplosgbiceatwl espenacmesemisbderesnootfeKd b(tyhus K. Wishtehneecvlearss oifsall colabsjects a map of Ω into itself for each K ∈ Z(Ω) and every map ’ between two Z -sets in Ω. Our main result is

There exists a functor E: Z → C such that for any ’ ∈ C(K; L) with K; L ∈ Z(Ω),

E E

(E3)

c

fo

o

r

m

a

p

n

ac

a

t

r

s

b

e

i

t

s

ra

)

r

i

y

ff t

s

h

’

L

.

), the maps E(’n) converge to E(’) pointwise (resp. uniformly on

Ω

,

; and im(’) is completely metrizable iff so is im(E(’)), iff im(E(’)) is homeomorphic

(E6) the image of E(’) and its closure in Ω intersect L along im(’) and im(’), respectively.

Our method also enables extending metrics in a way that the extensor for metrics harmonize with the functor E discussed

Theorem 1.3.

M

Un

e

d

tr

e

(

r

Ω

t

)

he

su

n

c

o

h

ta

th

ti

a

o

t

n

:

of Theorem 1.1, to each (complete) metric d ∈ Metr(L) one may assign a [complete] metric E(d) ∈

E E E

(E9)

(

if

bi

K

-)L

r

p

es

’

pec

∈

t t

C

o

(K

%

;

a

L

n

)

d

is

d

u

iff

nifo

(

r

’

m

)

l

i

y

s

c

s

d)

b

.

edding,

O(muerntpiroonoefds aartethsuerpbreigsiinngnliyngea o sfythaendseucsteions)imapnlde itdheeasw.elHl-okwnoewvenr, tthheeormemainoftoKoelslleorf [t1 h 1 i s ] (pfoarpeΩr a=reQA)ndaenrdsotnh’es rreessuulltt

above, as shown by

(E0) E(’) extends

’;

(E7) E(d) extends d;

n

Theorem 1.1.

Proposition 1.2.

in TopIK). The identity map on X is denoted by idX .

(E5) im

o

(

Ω

E

;

(’)) either is a Z-set in Ω or coincides with Ω;

( 1) (’) is an injection (resp. a surjection or an embedding) iff ’ is so;

( 8) for any K ∈ Z(Ω), the map (C(K; L); dsup) 3 ξ 7→ (ξ) ∈ (C(Ω; Ω); (d)sup) is isometric;

Let = Z(Ω) and = {Ω} . Notice that whenever : → is a functor, then necessarily (K ) = Ω and (’)

is

Z TopI C TopI E Z C E E

( 2) the image of (’) is closed in Ω iff the image of ’ is closed in L, and similarly with “dense” in place of “closed”;

we denote by C(X ; Y ) the set of all maps (that is, continuous functions) from X to Y . The closure operation is marked

stands for the collection of all Z -sets in X ; that is, K ∈ Z(X ) if K is closed and C(Q; X \ K ) is dense in C(Q; X )

(in

the− n n ; w r s ces n i th

u

and u ¹are uniformly continuous (with respect to the metrics d and %). By a compatible metric on a metrizable

spacewe n t ic i s he o gy e. h c c c pat le n ed tr c on

ametrizable space X is denoted by Metr(X ). For d ∈ Metr(Y ) the supremum metric on C(X ; Y ) induced by d is

denotedby bet Top K a s

of

E E E

(3)

888

P.

Niemiec

of Bessaga and Pełczyński [7] on spaces of measurable functions (for Ω = ‘2 ). What is more, the main proof is

(4)

889

nliomnictaotnisotnrutcotpivoeloagnieds (seine [c o 61 n ]tfroarstthteo dheofimneiotimonor)pihniscmaseexΩten=si‘o2n.

theorems  the extensor will not be continuous in the Isnettthinegsn.ext section we shall prove the above results in case of Ω = Q, while Section 3 deals with the Hilbert space

2. Hilbert cube

Throughout this section we assume that Ω = Q. In this case, the main tool to build the functor E will be the well-known theorem of Keller [11] (the proof may also be found in [8, Chapter III, §

3]).

E

ph

v

i

e

c

ry

to

co

Q

m

.

pact, metrizable, infinite-dimensional convex subset of a locally convex topological vector space is homeomor-

E I M

topological disjoint union (or the direct sum) of topological spaces A and B. For simplicity, we shall write A = B if A

In order to define I, let us fix a homeomorphic copy Ω⁰of Q with a metric d⁰∈ Metr(Ω⁰) such that diam(Ω⁰; d⁰) = 1. For defined as follows: I(d) coincides with d on K × K , with d⁰on Ω⁰× Ω⁰and I(d)(x; y) = max (diam(K ; d); 1) if one of x

I(K ) is a compact metrizable space having infinitely many points (1)

(this fact shall be used later), K is a closed subset of I(K ), and I(’) and I(d) extend ’ and d, respectively. Now we

sBtaannadcahrdspwaecaekotfoCp(oKlo;gRy))(,ini.hee. rtihteedt,otphoalnokgsy

twoitthhethReiebsazsicshacorancstiesrtiinzagtioofnfitnhieteorienmte,rfsreocmtiothnes owfesaekt-s^∗otfopthoelofgoyrmof the dual

B(µ; f; ε) = λ ∈ M(K ) :

ZK

f dµ −

ZK

f dλ < ε

1

;

where µ ∈ M(K ), f ∈ C(K; R) and ε > 0. The space M(K ) is compact, convex and metrizable. What is more, M(K ) is

card(K ) � ℵ0 =⇒ M(K ) ∼= Q: (2)

For a ∈ K , let €a ∈ M(K ) denote the Dirac measure at a; that is, €a is the probabilistic measure on K such that

card(K ) > 1 =⇒ im(γK ) ∈ Z(M(K )): (3)

AFultrhtohuegr,hfo(r3)aismeelterimc ednt∈arMy aentrd(Ksi)mlpetleM, w(de)w: iMll (sKee) ×inMth(Ke )se→quRe+l

t(hwahteirteisRa+ c=ru[c0i;a∞l p)r)obpeerdtyefi. ned by

M(d)(µ; ν) = sup

ZK

f dµ −

ZK

f dν : f ∈ Contr(K; R)

1

; turn to the definition of the functor

M.

Theorem 2.1.

any K ; L ∈ Z(Q), d ∈ Metr(L) and ’ ∈ C(K ; L) let I(K ) = K ⊕ Ω⁰, and let I(d) ∈ Metr(I(L)) and I(’) ∈ C (I(K ); I(L)) be

infinite-dimensional provided K is an infinite set, and hence, by Theorem 2.1,

and y belongs to K and the other to Ω⁰; I(’)(x) = ’(x) for x ∈ K and I(’)(x) = x for x ∈ Ω⁰. Notice that

€a({a}) = 1. It is clear that the map γK : K 3 a 7→ €a ∈ im(γK ) ⊂ M(K ) is a homeomorphism. What is more,

We shall build the functor using two additional functors, denoted by and . Below we will write A⊕ B

∼for theand B are homeomorphic, and we follow the convention that A ⊂ A

⊕ B.

Fix a compact metrizable space K . Let (K ) be the set of all probabilistic Borel measures on K equipped with theM

(5)

R R M M

metric M(d) was rediscovered by many mathematicians, e.g., by Kantorovich, Monge, Rubinstein, Wasserstein; in Fractal

M(d)(€a; €b) = d(a; b); a; b ∈ K: (4)

Finally, for a map ’ : K → L between compact metrizable spaces, we define M(’): M(K ) → M(L) by the

formula (M(’)(µ))(B) = µ(’⁻¹(B)); µ ∈ M(K ); B ⊂ L is Borel:

RThus M(’)R(µ) is the transport of the measure µ by the map ’. Observe that if λ = M(’)(µ), then for each g ∈ C (L; R),

L g dλ = _Kg◦ ’ dµ. This implies that M(’) ∈ C (M(K ); M(L)). Moreover, M(’) is affine (so its image is a compact

convex set) and

M(’)(γK (a)) = γL (’(a)); a ∈ K : (5)

For transparency, for each K ∈ Z(Q) let δK : K → γI(K )(K ) ⊂ L(K ) be a map obtained by restricting γI(K ). Below

e

d

e

u

c

t

e

i

d

ve

a

ly

b

,

o

t

v

h

e

, f

f

o

r

ll

a

ow

rb

K

:

(L2) δK is a homeomorphism and im(δK ) ∈ Z(L(K

)); (L4) L(d)(δK (x); δK (y)) = d(x; y) for any x; y ∈

K;

and L in Q, a map ’ : K → L and compatible metrics

(L5) the function (C(K ; L); %sup) 3 ξ 7→ L(ξ) ∈ C(L(K ); L(L)); L(%)sup is isometric;

L L L L

(L7) L

a

(’

d

)

%

i

;

s (bi-)Lipschitz (resp. nonexpansive, isometric) with respect to L(d) and L(%) iff such is ’ with respect to d

(L9) L(’) is injective iff ’ is such.

and (4), respectively. Let us briefly show conditions (L5)–(L8) (point (L9) is left to the reader). (L6) follows from the Kre˘ın–Milman theorem: im(L(’)) is a convex compact subset of L(im(’)) (if we natu r ally identify the latter set with the of I(im(’)), which are precisely the extreme points of L(im(’)). In order to check (L 5 ), take ’; ψ ∈ C(K ; L) and µ ∈ L(K ),

L(%) L(’)(µ); L(ψ)(µ)

= sup

Z

f dµ’ −

Z

f dµψ : f ∈ Contr(I(L); R)

1

Geometry it is known as the Hutchinson metric). Observe that

where Contr(K; ) stands for the family of all d-nonexpansive maps of K into . Then (d) ∈ Metr( (K )) (the

collect most important properties of the functor L.

It is easy to check that both andI Mare functors. Now define a functorLas their composition; that is,L= M ◦ .

(L1) L(K ) ∼=

Q;

(L3) L(’)(δK (x)) = δL (’(x)) for each x ∈ K ;

(L8) im

n

(L(’)) = {µ ∈ L(L) : µ (im(’) ⊕ Ω⁰) = 1};

and put µ’ = L(’)(µ) and µψ = L(ψ)(µ). Note that

definition of the topology of L(L) and Lebesgue’s dominated convergence theorem, while (L8) is a consequence of the

set of all measures on I(L) which are supported on I(im(’))) and contains all Dirac’s measures concentrated on points ( 6) the assignment C(K; L) 3 ξ 7→ (ξ) ∈ C( (K ); (L)) preserves pointwise convergence of

sequences;

( 1) follows from (1) and (2), while ( 2) is a consequence of ( 3 ). Further, ( 3) and ( 4 ) are implied by (5)

Z

I(K )

Z

Lemma

2.2. Proof. L L L L

(6)

= sup ^I(L)

Z

I(L)

f ◦ I(’) dµ −

I(

)

f ◦ I(ψ) dµ : f ∈ Contr(I(L); R)

1

:( sZup _I(K

)

|f ◦ I(’) − f ◦ I(ψ)| dµ : f ∈ Contr(I(L); R)

:(

I(K )

% I(’)(x); I(ψ)(x)

dµ(x) :( %sup(’; ψ):

(7)

This gives L(%)sup(L(’); L(ψ)) :( %sup(’; ψ). Since the reverse inequality is immediate (thanks to (L 3 ) and (L 4 ), we see

It remains to show (L7). This property is actually well known in Fractal Geometry, but for the reader’s conve-

s

y

om

x

e

; y

L

;

y

R

)

.

I I I I

I I ontr I

for f ∈ Contr(I(L); R). This simply yields that L(’) satisfies Lip schitz condition with constant L. Similarly, if

%(’(x); ’(y)) � d(x; y)/L for any x; y ∈ K (where still L � 1), then I(%) I(’)(x); I(’)(y) � I(d)(x; y)/L for all x; y ∈ I(K )

im(I(’)) 3 y 7→ L

1 f I(’)⁻¹(y)

∈ R

I R I R

written in the form f = L · g ◦ I(’) with g ∈ Contr(I(L); R) chosen appropriately. Hence, for any µ1 ; µ2 ∈ L(K ), putting

L(%)L(’)(µ1 ); L(’)(µ2 )

= sup

Z

g dν1 −

Z

g dν2

: g ∈ Contr(I(L); R)

1

I(L )

= sup Z ^I(L)

g◦ (’) dµ1 − g◦ I(’) dµ2 : g ∈ Contr(I(L); R)

1

I(K )

Z ^{I(K )} 1

and we are

done. L

sup ^I(K⁾ f dµ1

− ^I(K₎

f dµ2 : f ∈ Contr(I(K ); R) = L

1 L(d)(µ1 ; µ2 )

N Pr o o w o w f e o a f r T e h re e a o d r y e t m o g

1

iv

. 1

e

for

.

L

for every K ∈ Z(Q) there exists a homeomorphism HK : Q → L(K ) which extends δK . We define E by: (E(K ) = Q

for (’) = HL−1 ◦ (’) ◦ HK ; ’ ∈ C( K ; L); K ; L ∈ Z(Q):

and

thus E(’)(x) = ’(x) by (L 3 ). Finally, observe that conditions (E 1 ) and (E 3 ) immediately follow from (L 8 )–(L 9 ) and

Proof of Proposition 1.2 for .

E L

(E5) and (E6) are consequences of (L8). (Indeed: if X is a proper closed subset of a compact metrizable space Y , then {µ ∈ M(Y ) : µ (X ) = 1} is a Z -set in M(Y ); and if E(’)(x) = y ∈ L for some x ∈ Q, then L(’)(µ) = €^yfor µ = HK

(x),

which yields µ I(’)⁻¹({ y } ) = 1 and therefore y ∈ im(’)).

Proof of Theorem 1.3 for .

E

L(d) HK (x); HK (y) . Since HK is a homeomorphism between Q and L(K ) and that (L5) is fulfilled.

and therefore for any f ∈ Contr(I(K ); R) the function

νj = L(%)(µj ), j = 1; 2, we obtain

(is well defined and) extends to a function g ∈ Contr( (L); ). This implies that every f ∈ Contr( (K ); ) may be

� Z 1

K ∈ Z(Q) and)

E L

(L5)–(L6), respectively, while (E2) is trivial in the compact

case. Ω = Q Point ( 4) follows from ( 8) and Keller’s theorem (Theorem 2.1) and both

Ω = Q We continue the notation of the section. By ( 2) and Anderson’s theorem [2],

It is readily seen thatEis a functor (sinceL is). Let us check ( 0E ). If ’ ∈ C(K; L) and x ∈ K , then HK ( x) = δK (x )

Ω = Q For d ∈ Metr(K ) (where K ∈ Z(Q)) and x; y ∈ Q we put (d)(x; y) =

I Z

(8)

P.

Niemiec

HK is an isometry of (Q; E(d)) onto (L(K ); L(d)); (6)

ub

∈

gro

A

u

p

th

G

(Q

o

;

f

K

A

)

ut

:

h

(Q

(x

;

)

K

=

) Φ : Auth0 (Q; K ) × G 3 (u; v ) 7→

u◦v ∈ Auth(Q; K ) is a homeomorphism.

e(smebeeed.dgi.n[1 g 2 a ])n. dNaowgroiturpemhoaminosmtoorpnhoitsicme atnhdatboth Auth(K ) and Auth(Q) are completely metrizable, therefore G is closed Φ⁻¹(h) = h◦ [Ψ(hIK )]⁻¹; Ψ(hIK

) :

It is worth mentioning that the functor M introduced above in its full generality was investigated by Banakh in [5, 6].

3. Hilbert space

In this sectioMn we assume that Ω = ‘2 . The Mproof in that case goes similarly. The main differenceI is that we shall change

uons,einobtthaeinssama efuwnactyoraswbitehfoarlel, dtoestihreedfupnrcotpoerrtoifees.x)teMnsoiroeno.veHro,

wtheevelar,ctkheofacuotmhopracistnuensasbolef tthoerespsoalcvee Ωwhmeathkeers itnhethdisetwaailys and Pełczyński [7]. In order to state their result, we have to describe spaces of measurable functions.

M(d): M(X ) × M(X ) 3 (f; g) 7→

Z0 1

d(f(t); g(t)) dt ∈ R+

is a bounded metric on M(X ), (M(X ); M(d)) is separable; and M(d) is complete iff d is so. The topology on M(X ) they converge to f in measure in the sense e.g. of Halmos (see [10, § 2]). Equivalently,

m

t.

etrizable space, then the space M(X ) is homeomorphic to ‘2 iff X is completely metrizable and has

such that the map

[8, Theorem VI.7.1]; for generalizations consult [14]).

into X up to almost everywhere equality. For d ∈ Metr(X ) the function

induced by M(d) is independent of the choice of d ∈ Metr(X ), and functions f1 ; f2 ; : : : ∈ M(X ) converge to f ∈ M(X ) iff

It is enough to put G = { (h) : h ∈ Auth(K )}. Since the map Ψ: Auth(K ) 3 h 7→ (h) ∈ Auth(Q) is an

the functor . (Actually, the same functor as used in Section 2, combined with the functor described below, leads

Let X be a separable nonempty metrizable space and let M(X ) be the space of all Lebesgue measurable functions of [0; 1]

more complicated. Instead of Keller’s theorem, which was used in the previous part, here we need a theorem of Bessaga

Proof.

E E

Theorem

3.1.

(10)

Fix for a moment a separable metrizable space X . For x ∈ X let €x ≡ x. Put γX : X 3 x 7→ €x ∈ im(γX ) ⊂ M(X ).

card(X ) > 1 =⇒ im(γX ) ∈ Z(M(X )): (8)

Clearly, γX is a homeomorphism. What is more,

(11)

As in Section 2, observe that M

(d)(€x ; €y) = d(x; y); x; y ∈ X; d ∈ Metr(X ): (9) If A is a subset of X , M(A) naturally embeds in M(X ) and therefore we shall consider M(A) as a subset of M(X ). Under

Now let Y be another separable metrizable space and f ∈ C(X; Y ). We define M(f): M(X ) → M(Y ) by the formula metrizable spaces. We collect further properties of the above functor M in the following lemma.

Let X and Y be separable metrizable spaces, f : X → Y be a map and let d and % be any bounded compatible metrics

(M

1

2

.

en (A) is a Z-set in (X ).

(M4) If f1 ; f2 ; f3 ; : : : ∈ C(X; Y ) converge pointw ise or uniformly on com pact sets to f, then M(fn) converge so to M(f).

(M5) The map (C(X; Y ); %sup) 3 u 7→ M(u) ∈ C(M(X ); M(Y )); M(%)sup is isometric.

(M6) M(f) is uniformly continuous (resp. a uniform embedding, (bi-)Lipschitz, nonexpansive, isometric) with respect to

Proof.

To see (M1), take a ∈ X \ A and observe that the maps

Φn : M(X ) 3 f 7→ €aI[0;1/n) ∪ fI[1/n;1] ∈ M(X )

converge uniformly on compact subsets of M(X ) to idM(X ) and their images are disjoint from M(A) (and, of course, M(A)

Items (M2), (M5) and first claims of (M3) and (M4) are quite easy and we leave them to the reader. Also the part

of B(uenl)on∞w=1 wbee ianvsoelvqeuecnricteeroiof nel(e7m).enAtsssuomf eMf(1X; f)2 ;wfh3 ;ic:h: : i∈s

cCo(nXv;eYrg)ecnotntvoeruge∈unMif(oXrm).lyWoen hcaovmeptaoctpsreotvse ttohaft ∈(MC((fXn);(Yu)n.))L∞n=e1t

[t0h;a1t] (:Mu(νfnn()t()u→n))∞nu=(1t)t}enhdass tLoebMes(gf)u(eu)mienatshueretoepqoulaolgytoo1f. MO(bYs)eravnedthwaet

afνrne(udνonn(te).) →In af(usi(mt)i)laforrmta∈nneTr. oBnuet cthheisckms etahnast

bNoouwndaesdsucmonetifnuisouusnimfoornmoltyonceonctoinncuaovues fwunitchtiorenspωe:cRt +to→d Ran+dv%a.nisShiinncge a%t

i0s sbuocuhntdheadt, we conclude that there exists a

%(f(x); f(y)) :( ω(d(x; y)); x; y ∈ X (10)

(csoemebein.ge.d[4w])i.thN(o 1 w 0 ) Jyeinesldesn’sthiante, qfouralaintyy (ua;pvpl∈iedMf(oXr )a, convex function t 7→ M − ω(t) where M is an upper bound of ω)

M(%) M(f)(u); M(f)(v )

=

Z0 1

%(f(u(t)); f(v (t))) dt :(

Z0 1

ω(d(u(t); v (t))) dt :( ω

{Z0 1

d(u(t); v (t)) dt

(

= ω(M(d)(u; v ))

such an agreement one has M(A) = M(A).

(M3) M(f) is an injection or an embedding iff so is f.

on X and Y , respectively.

(M M M

(M(f))(u) = f ◦u. It is easy to verify that M(f) ∈ C(M(X ); M(Y )) and M is a functor in the category of separable

is closed in M( X )).

M(f) is an embedding provided f is so.

M(d) and M(%) iff so is f with respect to d and %.

(M6) concerning (bi-)Lipschitz, nonexpansive and isometric maps is immediate. Let us show the second claim of (M4).

converges to M(f)(u). For an arbitrary subsequence of (un)^∞_n=1take its subsequence (uνn )^∞_n=1such that the set T = {t ∈

and thus (f) is uniformly continuous. Conversely, if (f) is uniformly continuous, then f is so, by ( 2) and (9).

Lemma 3.2.

M M M

(12)

Fcoinnacallvye, iffufncitsio an uτn:ifoRr+m →emRbe+ddvianngis, hwi negmaaty 0respuecaht tthhaetadb(oxv;eya) r:(guτm(e

%n(ft,(xs)t;afr(tyin)g)) ffroormaanyboxu; nyd∈ed Xco, natninduofiunsismhionngotwointhe M(d)(u; v ) :( τ M(%)(M(f)(u); M(f)(v )) for all u; v ∈ M(X ).

(13)

M M

tAWwSo otuhseloilnresmpesadcoef tihsethdeesecmrippttyivetospeotlotghiecoarlysapnadcewoer

haavmeettroizianbtrloedsupcaecethwe hteicrhmiinsoalocgoyn.tinuous imseaegee.ogf. t[1 h 3 e , space R \ Q.

s

o

pa

c

o

e

n

;

d

l

i

e

ti

t

o

X

ns

=

:

∅ be any set and let R be a σ-algebra of subsets of X.

(

ii

i

s

y

et

are two separable metrizable spaces and f ∈ C(X; Y ), then im(M(f)) = M(im(f)), provided X is completely

Proof.

The inclusion “⊂” easily follows from the relation im(M(f)(u)) ⊂ im(f). To prove the reverse one, take a

tof ∈[0;[−101];.1]D. eWfinheatFi:s[0m; o1r] e→, if2^XU biys tohpeenforinmuXla, tFh(etn) ,=byf⁻(¹ S ( { 1 v ),( t f ) ( } U).) CisleaarlSy−o,1uFsl(itn) isspancoeneamnpdtyheanncde,clboyse(dS2 i n ), Xsofoirs atnhye

s{ett∈v [0;(f1(]U:)F) (atn)d∩ tUhe=ref∅or}e ∈it Lis. LNeboewsgTuheeomreemasu 3 r .3 a bglieve(sbyus(Sa3 ) L)e.

beBsugtuev m(efa(Usu)r)ab=le{ftun∈cti[o0n; 1u] :: [0F;(1t]) →∩ UX }t,hsao

t ∅

Proof of Theorem 1.1 for .

d⁰∈ Metr(Ω⁰) such that diam(Ω⁰; d⁰) = 1. Now let I be a functor built in the same way as in the previous part of the

I I I 0 0

• for K ∈ Z(‘2 ) and d ∈ Metr(K ), I(d) ∈ Metr(I(K )) coincides with d on K × K , with d⁰on Ω⁰× Ω⁰and I(d)(x; y) = Now max (diam(K; d); 1)

otherwise. em for Ω = Q. We define a functor L by L = M ◦ I, and for any K ∈ Z(‘2 )

we mimic the p r oIof of the theor I L

based on Theorem 3.1 and (8). Finally, for ’ ∈ C(K ; L) (with K ; L ∈ Z(‘2 )) we put E(’) = H_L⁻¹◦ L(’) ◦ HK . Note that,

0

Our last step is to prove that im( (f)) = (im(f)). It is however not as simple as it looks. To show this, we shall apply

ore[1 s 5 h ,aAl pnpeenditxh])e: following three properties of Souslin spaces (for proofs and more information

Chapter XIII](S1) a inu e e t uslin space;

(S2) the inverse image of a Souslin space under a Borel function (between Borel subsets of sep a rable

completelym i bl ) s a ;

l

t

l

h

o

e

p

re

en

e {x ∈ X

−1

Borel function v : [0; 1] → Y such that v ([0; 1]) ⊂ im(f). Let L denote the σ -algebra of all Lebesgue measurable subsets

u(t) ∈ F (t) for any t ∈ [0 ; 1]. This means that u ∈ M(X ) and (M(f))(u) = v .

such

pape

•

r:

I assigns to each Z -set K in ‘2 the space K

⊕ ‘2 ;

Ω = ‘2 As in Section 2, we fix a homeomorphic copy Ω⁰of ‘2 and a complete metric

• for K ; L ∈ Z(‘2 ) and a map f : K → L, (f) ∈ C( (K ); (L)) coincides with f on K and with idΩ on Ω ;

by Proposition 3.4, im(M(’)) = M(i m (’) ⊕ Ω ) : (11)

denote by δK : K → γ (K )(K ) the restriction of γ (K ) and take a homeomorphism HK : ‘2 → (K ) which extends δK ,

Theorem 3.3.

Proposition

3.4.

(14)

894

P.

Niemiec

Now in the same way as in Section 2 one checks (E0), (E1)–(E2) (use (M3) and (11)) and (E3) (apply (M4)). The details

are left to the reader.

(15)

Proof of Proposition 1.2 for .

E E from (M 1 ) and (11). Finally, (E 6 ) may briefly be deduced from (11) and the formula for E(’) (cf. the proof of the proposition

P fo r r o Ω o = f o Q f ) T . heorem 1.3

for Ω = ‘2

.

As in Section 2, for d ∈ Metr(K ) (where K ∈ Z(‘2 )) define E( d ) ∈ Metr(‘2 ) by

E(d)(x; y) = L(d) HK (x); HK (y) . Now it suffices to repeat the proof from the previous case, involving (M5) and (M6).

WZ -eseetnsdotfh‘e2 pwahpiecrhwisithcoantniontueotuhsaitnwtehedolimnoittaktinoonwtoifptohleorgeieesx.ists an analogous functor of extension of mappings between

References

[1] Anderson R.D., Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc., [3] A25n(d2e),rs2o8n3–R.DD8 .,, MncCthpaorelongJic.Dal., iOnfinneitxetednedfiincigenhcom,

eMomicohrigpahnismMsattoh.FJr.,é1ch96e7t ,m1a4n, i3fo6ld–s3, 8P3roc. Amer. Math. Soc., 1970, Pacific J. M ., 1956, 6(3), 405–439

[6]] BmaetnraizkahtiTo.nO).,f Ttohpeofluongcytoorfs sPpτacaensd oP

bf ,pMroabta. bSiltiutyd.,m1e9a9s5u, r5e,s8I8I.–

1B0a6ry(cinenRteursssiaofn)Radon probability measures and the BWeasrssaagwa, 1C9z7.,5Pełczyński A., Selected Topics in Infinite- Dimensional Topology,

[10] 1971, 1, 263–272

[11] CHKhaallmpmosanP. .TR..A, .M, DeaesfiucrieenTcyheionryin, fiVnainteN-doismterannsdio, nNalewmaYnoirfko,ld1s9,5G0eneral Topology Appl., Raum, Math. Ann., 1931, 105,

[12] K e Amer. Math. Soc., 1952, 3(3),

u owski K., Mostowski A., Set Theory 2nd ed., PWN, Warsaw, 1 76 saki M., Theory of O erator

gebra . I, Encyclopa dia Ma c .

sets ńacnzdykchHar, aActecroirzraetcioinonoof fl2t-wmanpiafpoldrss”caonndce“CihnagraHctlebreizitng Hilbledrt space topology”, Fund. Math., 1985, 125,

89–93

Ω = ‘2 It is readily seen that ( 4) follows from Theorem 3.1 and (11), while ( 5)

1966, 72(3), 515–519[2] Anderson R. . O o y

5

[5] Banakh T.Oa.,thTopology of spaces of probability measures I. The functors Pτ and Pb , Mat. Stud., 1995, 5, 65–87(in Russian

[4] Aronszajn N2 .,9Panitchpakdi P., Extension of uniformly continuous transformations and hyperconvex metric spaces,

[8] Bessaga Cz.o, Pełczyński A., On spaces of measurable functions, Studia Math., 19IM72P, A4N4, 5M97o–

n6o1g5r. Mat., 58, PWN,

[9]

[13] 48le4–4V8.7L. Jr., Invariant metrics in groups (solution of a problem of

Banach), Proc.[14] K rat , 9

74e8l–e7r5O8 -H., Die Homoiomorphie der kompakten konvexen Mengen im Hilbertschen

[15] Niemiec P., Spaces of measurable functions, Cent. Eur. J. Math., 2013, 111(274),, 1S3p0r4in–

g1e3r1, 6Berlin, 2002[16] Take p Al s e th. S i ,

[17] Toruńczyk H., Characterizing Hilbert space topology, Fund. Math., 1981, 1“1C1o(n3c),er2n4i7n–g26lo2cally homotopy negligibleToru . t o e rn i r manifo s:

(16)

P.

Niemiec

895