Spaces of measurable functions

SP A CES OF MEASURA BLE FUNCTION S PIOTR NIEMIEC Abstract. F o r a metr izable spa ce X and a finite mea sure space (Ω , M , µ ) let M µ ( X ) and M f µ ( X ) b e the spaces o f a ll equiv alence classes (under the relation of equality almost everywhere mo d µ ) of M -measurable functions from Ω to X whose images are separable and finite, respectively , equipp ed with the top ology o f conv ergence in measure. The main a im of the pap er is to prove the following result: if µ is (nonzero and) nonatomic and X ha s more than one po int , then the space M µ ( X ) is a noncompact absolute retract and M f µ ( A ) is homoto p y dense in M µ ( X ) fo r each dense subset A of X . In particular , if X is c ompletely metrizable, t hen M µ ( X ) is homeomorphic to an infinite-dimensional Hilbert space. 2000 MSC: 54C35, 54C5 5, 54H05, 57N20, 58D15. Key w ords: measurable functions, absolute retracts, infinite-dimen- sional manifolds, reflective isotopy prop erty , Z -sets. In [8] Bessaga and P e lczy´ nski hav e pro v ed that whenev er X is a sep- arable comple tely metrizable top ological s pace ha ving more than one p oin t, then the space M X of Borel functions from [0 , 1] to X with the top ology of con v ergence in measure is homeomorphic to l 2 . Later it turned out that the top ology of l 2 can b e we ll c haracterized. This w a s done b y T oru ´ nczyk[20, 21 ]. After publication of the latter pap ers the n um b er of results on spaces homeomorphic to t he separable infinite- dimensional Hilb ert space has highly rised. F or example, D obro w olski and T oru ´ nczyk[11] hav e shown that ev ery separable completely metriz- able non-lo cally compact top ological group whic h is an AR is home- omorphic to a Hilbert space. Ho w ev er, the problem whether the as- sumption of separability in the latter may b e omitted is still op en ( see [7]). In this pap er w e shall in tro duce a clas s of nonsep arable completely metrizable top ological groups whic h are homeomorphic to Hilb ert spa- ces. Namely , if G is any (nonzero) completely metrizable top ological group and µ is a (nonzero) finite nonato mic measure, then the space M µ ( G ) (defined in Abstract) has a natural structure (induced by the one of G ) of a topo logical group and is homeomorphic to a Hilb ert space. In fact w e shall prov e the follow ing, quite more general, result: if X is a no nempt y metrizable space, µ is a finite no natomic measure and Y = M r µ ( X ) is the subspace of M µ ( X ) consisting of all (equiv a- lence classes of ) functions whose images are con tained in σ -compact subsets of X , t hen Y is an absolute retract suc h that Y ω ∼ = Y . Since infinite-dimensional Hilb ert spaces are the only completely metrizable 1 2 P . NIEMIEC noncompact AR’s homeomorphic to their o wn coun table infinite Carte- sian p ow ers ([20]), the latter men tio ned result may b e seen as a gen- eralization of ear lier results o f Bessaga a nd Pe lczy ´ nski[8 ] as well a s of T oru ´ nczyk[19]. Other purp ose of the pap er is to presen t the idea of extending maps b et w een metrizable spaces to maps b etw een AR’s via functors. Namely , whenev er µ is a finite (nonzero) nonatomic measure, ev ery map f : X → Y has a natural extension M µ ( f ) : M µ ( X ) → M µ ( Y ). What is more, the corresp ondence f ↔ M µ ( f ) preserv es many prop erties (suc h as: b eing an injection, an em b edding, a map with dense image). W e shall sho w that if m is the Leb esgue measure on [0 , 1], the space W = M m ( X ) is alwa ys an AR satisfying the following conditio ns: X is a Z -set in W (pro vided X ha s more than one p oint), W ω ∼ = W , W ha s RIP and is an S- space (in the sense of Schori[16]). As an immediate consequence of this, w e shall obta in t hat if U is a metrizable manifold mo delled on W , then U is W -stable, i.e. U × W ∼ = U . Another issue we shall discuss here concerns the question of whether M m ( M m ( X )) is homeomorphic to M m ( X ). W e shall see that the answ er is affirmativ e for a h uge class of metrizable spaces (namely , for spaces in whic h ev ery closed separable subs et is absolutely measurable), whic h con tains lo cally absolutely Borel spaces and (separable) Souslin ones. Ho w ev er, in general we lea v e this question a s a n op en problem. The article is o rganized as follows. In the first section w e establish notation and terminology , define general spaces of measurable f unc- tions and collect sev eral r esults on them. Section 2 deals with spaces M r µ ( X ), defined in this introduction. W e show there tha t if µ and ν are tw o homog eneous (nonatomic) measures of the same w eight, then the spaces M r µ ( X ) and M r ν ( X ) are naturally homeomorphic, whatev er X is. The third part is dev oted to spaces of measurable functions o v er metrizable AM-spaces (i.e. in whic h eve ry closed separable subset is absolutely measurable). W e pro ve there tha t if X is an AM -space, then M µ ( X ) = M r µ ( X ) for each finite measure µ . In Section 4 w e state a nd pro v e the main result of the pap er, whic h includes the claim that space s of measurable functions are absolute retracts. W e conclude f rom this that suc h spaces o v er completely metrizable ones are homeomorphic to Hilb ert spaces. In the last part we generalize our results of [15] to nonseparable case. Also the idea of extending maps to AR’s via the functors M µ is presen ted. 1. Preliminaries In this pap er R + and N denote the sets of nonnegative reals and in tegers, resp ectiv ely , I = [0 , 1] and m stands for the Lebesgue measure on I . If g is any function, im g stands for the imag e of g . If, in addition, g tak es v alues in a top ological space, im g denotes the closure of im g in the whole space. The we igh t of a top ological space X is SP AC ES OF MEASU RABLE FUNCTIONS 3 denoted by w ( X ) and is understoo d as an infinite cardinal n um b er (i.e. w ( X ) = ℵ 0 for finite X ). All top olog ical spaces which app ear in the pap er are metrizable and all measures are nonnegativ e, finite and nonzero. F or top o logical spaces Y and Z w e shall write Y ∼ = Z iff Y and Z are homeomorphic. By a ma p w e mean a con tin uous function. If X is a metrizable space, X ω stands for the coun table infinite Cartesian p ow er of X , equipp ed with the Tic ho no v top ology , and Metr( X ) denotes the family of all b ounded metrics on X whic h induce the giv en top olog y of X . B ( X ) stands for the σ -algebra of all Borel subsets of X , that is, B ( X ) is the smallest σ -algebra containing all op en su bsets of X . If (Ω 1 × Ω 2 , M , µ ) is the pro duct space of measure spaces (Ω 1 , M 1 , µ 1 ) and (Ω 2 , M 2 , µ 2 ), then we shall write M 1 ⊗ M 2 and µ 1 ⊗ µ 2 for M and µ , resp ectiv ely . Whenev er (Ω , M ) is a measurable space and X is a metrizable space, a function f : Ω → X is M -me asur able , if f − 1 ( U ) ∈ M f or eac h op en subset U of X . Sets w hic h are mem b ers of M are s aid to b e me asur able . By a µ -p artition of B ∈ M w e mean an y family { B j } j ∈ J (with J ⊂ N ) of measurable pairwise disjoint sets suc h that µ ( B j ) > 0 fo r eac h j ∈ J and B = S j ∈ J B j . If the images of M -measurable functions f j : Ω → X j , where j ∈ J ⊂ N , are separable, then also the function Ω ∋ ω 7→ ( f j ( ω )) j ∈ J ∈ Q j ∈ J X j is M -measurable. Therefore, if J = { 1 , 2 } and X 2 = X 1 = X , the set { ω ∈ Ω : f 1 ( ω ) 6 = f 2 ( ω ) } is measurable. W e use standard terminolog y and ideas of measure theory . F or de- tails t he Reader is r eferred e.g. to [1 2]. F or example, ev ery measurable function f : Ω → X with separable image defined on a measure space (Ω , M , µ ) will b e iden tified with its equiv alence class (in the set of all measurable functions Ω → X with separable ima ges) with resp ect to the relation of almost ev erywhere equalit y mo d µ . The set of all suc h (equiv alence classes of ) functions is denoted by M µ ( X ). The sub- families o f M µ ( X ) consisting of all those functions whose images are, resp ectiv ely , finite, (at most) coun ta ble and contained in σ - compact subsets of X are denoted b y M f µ ( X ), M c µ ( X ) and M r µ ( X ). W e clearly ha v e M f µ ( X ) ⊂ M c µ ( X ) ⊂ M r µ ( X ) ⊂ M µ ( X ). Eac h of the latter inclu- sions may b e prop er (the example for the last one is giv en in Section 3 , see Example 3.5). If A is a subset of X , we may and shall naturally iden tif y the members of M µ ( A ) with elemen ts of M µ ( X ). Th us, if N stands for M f , M c , M r or M , then N µ ( A ) ⊂ N µ ( X ). Analogously , if N is a σ -subalgebra of M and ν = µ   N , then for N = M , M f , M c , M r the function N ν ( X ) ∋ f 7→ f ∈ N µ ( X ) is w ell defined (a nd is iso- metric with resp ect to the metrics M ν ( d ) a nd M µ ( d ), defined in the sequel, for ev ery d ∈ Me tr( X ) ). The Bo o lean σ -algebra (equipp ed with the metric induced by the measure) asso ciated with a measure space (Ω , M , µ ) will b e denoted by A ( µ ). The w eigh t of A ( µ ) is called b y us the weigh t o f µ and is denoted by w ( µ ). W e call t he measure µ simple if µ ( B ) ∈ { 0 , µ (Ω) } for eac h B ∈ M a nd µ is nona tom i c if for 4 P . NIEMIEC ev ery B ∈ M of p ositiv e µ -measure there is a subset A ∈ M of B with 0 < µ ( A ) < µ ( B ). Finally , µ is homo gene ous if it is nonatomic and for eac h B ∈ M o f p ositive µ -measure, w ( µ ) = w ( µ   B ), where µ   B = µ   M B is a measure on B a nd M B = { A ∈ M : A ⊂ B } . F rom no w on, w e assume that (Ω , M , µ ) is a measure space with (nonzero) finite measure µ and that X is a (nonempt y) metrizable space. The space M µ ( X ) and all its subsets will alw a ys b e equipp ed with the top o logy of conv ergence in measure. In o ther w o rds, a se- quence ( f n ) n of elemen ts of M µ ( X ) conv erges t o f ∈ M µ ( X ) iff ev- ery its subsequence contains a subsequence ( f ν n ) n suc h that f ν n ( ω ) → f ( ω ) ( n → ∞ ) fo r µ -almost all ω ∈ Ω. It is w ell kno wn that if  ∈ Metr( X ), then M µ (  ) ∈ Metr( M µ ( X )), where M µ (  )( f , g ) = R Ω  ( f ( ω ) , g ( ω )) d µ ( ω ) . It is clear t hat if ( X, · ) is a metrizable group, then M µ ( X ) has a natural top olog ical group structure (that is, with the p o in t wise m ulti- plication) induced b y the one of X . F or eac h x ∈ X denote b y δ µ,x ∈ M µ ( X ) the constant function with the o nly v alue equal to x and let ∆ µ ( X ) = { δ µ,x : x ∈ X } and δ µ,X : X ∋ x 7→ δ µ,x ∈ ∆ µ ( X ) ⊂ M µ ( X ). The following are a kind of fo lklore. Most of them can easily b e pro v ed. (M1) ∆ µ ( X ) is closed in M µ ( X ) and δ µ,X : ( X , d ) → ( ∆ µ ( X ) , M µ ( d )) is an isometry f or each d ∈ Metr( X ). In part icular, ∆ µ ( X ) ∼ = X . If X is a g roup, δ µ,X is a homomo rphism. (M2) If N ⊂ M is an algebra o f subsets of X whic h is dense in A ( µ ) and D is a dense subset of X , then the set M f ( N , D ) consisting of suc h functions f ∈ M f µ ( D ) that f − 1 ( { x } ) ∈ N fo r each x ∈ D is dense in M µ ( X ). In particular, w ( M µ ( X )) = max ( w ( µ ) , w ( X )). (M3) If d ∈ Metr( X ), then M µ ( d ) is complete (in the whole space M µ ( X )) iff d is complete. The space M µ ( X ) is completely metriz- able iff X is so. Moreo v er, if car d X > 1, then M µ ( X ) is non- compact. (M4) F or eac h A ⊂ X , M µ ( A ) = M µ ( ¯ A ) (the first closure is in M µ ( X )). (M5) The measure µ is nonatomic iff there is a family { A t } t ∈ I of mea- surable sets suc h t hat A s ⊂ A t for s 6 t and µ ( A t ) = tµ (Ω). (M6) If µ is nonatomic and { A t } t ∈ I is a family as in (M5), then the map λ : M µ ( X ) × M µ ( X ) × I ∋ ( f , g , t ) 7→ f   Ω \ A t ∪ g   A t ∈ M µ ( X ) is con tin uous. Moreo v er, λ ( f , g , 0) = f , λ ( f , g , 1) = g a nd λ ( N µ ( X ) × N µ ( X ) × I ) = N µ ( X ) for N = M f , M c , M r . In particular, each of t he spaces N µ ( X ) with N = M , M f , M c , M r is con tractible, prov ided X is nonempt y (in fa ct they are equicon- nected). SP AC ES OF MEASU RABLE FUNCTIONS 5 (M7) If { A j } j ∈ J ( J ⊂ N ) is a µ - partition of Ω; λ = µ µ (Ω) and λ j = µ   A j µ ( A j ) , then the map Φ : ( M λ ( X ) , M λ ( d )) ∋ f 7→ ( f   A j ) j ∈ J ∈  Q j ∈ J M λ j ( X ) , ˜ d  is an isometry , w here ˜ d (( f j ) j ∈ J , ( g j ) j ∈ J ) = P j ∈ J µ ( A j ) M λ j ( d )( f j , g j ) for d ∈ Metr( X ). Moreov o er, Φ( N λ ( X )) = Y j ∈ J N λ j ( X ) for N = M c , M r . In particular, N λ ( X ) is homeomorphic to Q j ∈ J N λ j ( X ) for N = M , M c , M r . (M8) Let { ( X j , d j ) } j ∈ J ( J ⊂ N ) b e a collection of metric spaces with metrics upp er b o unded by 1 and let { a j } j ∈ J b e a f amily o f p osi- tiv e n um b ers suc h that P j ∈ J a j < + ∞ . Let X = Q j ∈ J X j b e a metric space with metric d (( x j ) j ∈ J , ( y j ) j ∈ J ) = P j ∈ J a j d j ( x j , y j ). Analogously , let D b e the metric on Q j ∈ J M µ ( X j ) give n by D (( f j ) j ∈ J , ( g j ) j ∈ J ) = X j ∈ J a j M µ ( d j )( f j , g j ) . Then the map Ψ : ( M µ ( X ) , M µ ( d )) ∋ F 7→ ( p j ◦ F ) j ∈ J ∈ ( Y j ∈ J M µ ( X j ) , D ) , where p j : X → X j is the natural pro jection, is an isometry . In particular, M µ ( Q j ∈ J X j ) is homeomorphic to Q j ∈ J M µ ( X j ). If J is finite, then (1-1) Ψ( M r µ ( X )) = Y j ∈ J M r µ ( X j ) and Ψ( M c µ ( X )) = Q j ∈ J M c µ ( X j ). (M9) There is a finite or countable collection { A j } j ∈ J ∪ { B k } k ∈ K (eac h of J and K may b e empty) of measurable sets of p o sitiv e µ - measure suc h that µ   A j is simple for eac h j ∈ J , while the mea- sures µ   B k with k ∈ K are homogeneous and of differen t we igh ts. (M10) If µ is an atom, then M µ ( X ) = M f µ ( X ) = ∆ µ ( X ) and th us M µ ( X ) ∼ = X . (M11) (Mahara m[14]) If (Ω j , M j , µ j ) ( j = 1 , 2) a re probabilistic spaces suc h tha t b oth µ 1 and µ 2 are homogeneous and w ( µ 1 ) = w ( µ 2 ), then the Bo olean σ -a lgebras A ( µ 1 ) and A ( µ 2 ) are isometrically isomorphic. The prop erty (M1) sa ys that X may natura lly b e identified (via the map δ µ,X ) with ∆ µ ( X ). The points (M7) and (M9)–(M11) imply tha t if N = M , M c or M r , then N µ ( X ) ∼ = X p × Q j ∈ J N µ j ( X ), where p = n ∈ N if µ has exactly n atoms and p = ω if µ has infi nitely many a toms (if p = 0, w e omit the f actor X p ); and J ⊂ N (if J is empt y , w e omit the factor 6 P . NIEMIEC Q j ∈ J N µ j ( X )) and the measures µ j are pr obabilistic homogeneous and of differen t w eigh ts. W e shall prov e in Section 2 that M r λ ( X ) is natur al ly homeomorphic to M r ν ( X ) if λ and ν are homogeneous and of the same w eight. W e shall also show that the connec tion (1-1) is fulfilled without assumption of finiteness o f J . Our next aim is to prov e that if (Ω j , N j , ν j ) for j = 1 , 2 are tw o measure spaces, then there is a measure space (Ω , N , ν ) suc h that M ν 1 ( M ν 2 ( X )) is natur al l y homeomorphic to M ν ( X ) for each metriz- able space X . T o do this, let Ω = Ω 1 × Ω 2 and π : Ω → Ω 2 b e the natural pro jection. Let N b e the σ -algebra of all subset A of Ω suc h t hat π ( A ∩ ( { ω 1 } × Ω 2 )) ∈ N 2 for each ω 1 ∈ Ω 1 and the function Ω 1 ∋ ω 1 7→ π ( A ∩ ( { ω 1 } × Ω 2 )) ∈ A ( ν 2 ) is N 1 -measurable and its image is se parable. Finally , let ν : N → R + b e giv en b y ν ( A ) = R Ω 1 ν 2 ( π ( A ∩ ( { ω 1 } × Ω 2 ))) d ν 1 ( ω 1 ). It is easy to see that N is indeed a σ - algebra and that ν is a finite measure on Ω. Note also t hat N 1 ⊗ N 2 ⊂ N and ν extends ν 1 ⊗ ν 2 . W e call ν the dir e cte d pr o duct of ν 1 and ν 2 . It w o uld b e quite more reasonable to define (Ω , N , ν ) a s the pro duct space of (Ω 1 , N 1 , ν 1 ) and (Ω 2 , N 2 , ν 2 ). Ho w ever, as w e will see in Section 3 ( Example 3.5), the pro duct space (a s (Ω , N , ν ) b elo w) do es not satisfy the following claim: (M12) F or ev ery b ounded metric space ( X , d ) the map Λ : ( M ν ( X ) , M ν ( d )) → ( M ν 1 ( M ν 2 ( X )) , M ν 1 ( M ν 2 ( d ))) giv en by the fo rm ula (Λ f ( ω 1 ))( ω 2 ) = f ( ω 1 , ω 2 ) is a w ell defined (bijectiv e) isometry . T o sho w that im Λ ⊂ M ν 1 ( M ν 2 ( X )), use the fact that if f : Ω → X is N -measurable and im f is separable, then there is a sequence of N -measurable functions f n : Ω → X with finite images suc h that lim n →∞ f n ( ω ) = f ( ω ) f or eac h ω ∈ Ω. F ur ther, direct calculation s ho ws that Λ is isometric. T o see the surjectivit y , fix an N 1 -measurable func- tion g : Ω 1 → M ν 2 ( X ) with separable image. L et ¯ X b e the completion of X with resp ect t o d . Since M f ν 2 ( X ) is dense in M ν 2 ( X ), there is a sequence of N 1 -measurable functions g n : Ω 1 → M f ν 2 ( X ) with finite images suc h tha t lim n →∞ g n ( ω 1 ) = g ( ω 1 ) f or ev ery ω 1 ∈ Ω 1 . It is easy to che c k that fo r eac h n there is an N -measurable function f n : Ω → X whose imag e is finite and suc h t hat f n ( ω 1 , · ) and g n ( ω 1 ) concide in M ν 2 ( X ) for ev ery ω 1 ∈ Ω 1 (in fact, each f n is N 1 ⊗ N 2 -measurable). Th us (since Λ is isometric), ( f n ) n is a fundamen tal sequence in M ν ( ¯ X ). This means that there is an N -measurable function f : Ω → ¯ X with separable image whic h is the limit of ( f n ) n in M ν ( ¯ X ). W e conclude from this that ¯ Λ f = g , where ¯ Λ is the suitable map ‘Λ’ f or ¯ X . So, the set A 1 = { ω 1 ∈ Ω 1 : f ( ω 1 , · ) 6 = g ( ω 1 ) in M ν 2 ( X ) } b elongs to N 1 and ν 1 ( A 1 ) = 0. Now fix ω 1 ∈ Ω 1 \ A 1 . Let h : Ω 2 → X b e an N 2 -measurable function with separable image whic h coincides with g ( ω 1 ) in M ν 2 (Ω 2 ). Then the set A ω 1 = { ω 2 ∈ Ω 2 : f ( ω 1 , ω 2 ) 6 = h ( ω 2 ) } b elongs to N 2 and SP AC ES OF MEASU RABLE FUNCTIONS 7 ν 2 ( A ω 1 ) = 0. F inally , put A = ( A 1 × Ω 2 ) ∪ S ω 1 ∈ Ω 1 \ A 1 ( { ω 1 } × A ω 1 ) ⊂ Ω and let f ∗ : Ω → X b e suc h that f ∗   A = f   A and f ∗   Ω \ A ≡ b , where b is a fixed elemen t of X . By the construction, A ∈ N , f ∗ ∈ M ν ( X ) and Λ( f ∗ ) = g . The ab ov e defined σ -algebra N and measure ν will b e denoted by us b y N 1 → ⊗ N 2 and ν 1 → ⊗ ν 2 , resp ectiv ely . Since A ( ν 1 → ⊗ ν 2 ) is naturally isometric to M ν 1 → ⊗ ν 2 ( { 0 , 1 } ), the presen ted pro o f of (M12) (especially N 1 ⊗ N 2 -measurabilit y of the functions f n ) yields tha t (M13) F or eac h A ∈ N 1 → ⊗ N 2 there is A 0 ∈ N 1 ⊗ N 2 suc h that ( ν 1 → ⊗ ν 2 )( A \ A 0 ) = ( ν 1 → ⊗ ν 2 )( A 0 \ A ) = 0. In particular, A ( ν 1 → ⊗ ν 2 ) = A ( ν 1 ⊗ ν 2 ) and if ν 1 and ν 2 are homogeneous, so is ν 1 → ⊗ ν 2 . No w w e shall giv e a sufficien t condition ( on a measure µ ) under which the space Y = M µ ( X ) is homeomorphic to Y ω (for eac h X ). T o fo rm u- late it, we need an additional notion. W e sa y that tw o measure spaces (Ω 1 , M 1 , µ 1 ) and (Ω 2 , M 2 , µ 2 ) are p ointwisely isomorphic if there is a bijection ψ : Ω 1 → Ω 2 suc h that for a n y A ⊂ Ω 1 , ψ ( A ) ∈ M 2 iff A ∈ M 1 and µ 2 ( ψ ( A )) = µ 1 ( A ) fo r eve ry A ∈ M 1 . In suc h a situation ψ is called an isom orphism . These spaces are said to b e almost p ointwisely iso- morphic if there are sets A 1 ∈ M 1 and A 2 ∈ M 2 suc h that µ j (Ω j \ A j ) = 0 ( j = 1 , 2) a nd the spaces ( A 1 , M 1   A 1 , µ 1   A 1 ) and ( A 2 , M 2   A 2 , µ 2   A 2 ) are p o intwisely isom orphic . Basicly , ev ery isomorphism ϕ : Ω 1 → Ω 2 in- duces isometries ( M µ 1 ( X ) , M µ 1 ( d )) ∋ f 7→ f ◦ ϕ − 1 ∈ ( M µ 2 ( X ) , M µ 2 ( d )) for an y X a nd d ∈ Metr( X ) (the same for M f , M c and M r -spaces). W e also ha v e: (M14) If there is a measurable set A suc h that 0 < µ ( A ) < µ ( Ω) and the space s (Ω , M , µ µ (Ω) ) and ( A, M   A , µ | A µ ( A ) ) are a lmost point wisely isomorphic, then M µ ( X ) ∼ = M µ ( X ) ω for eac h metrizable space X . T o see this, first of all observ e that there are measurable sets Ω 0 and A 0 suc h that A 0 ⊂ A ∩ Ω 0 , µ (Ω \ Ω 0 ) = µ ( A \ A 0 ) = 0 a nd the spaces (Ω 0 , M   Ω 0 , µ   Ω 0 ) and ( A 0 , M   A 0 , µ   A 0 ) are p oin t wisely isomor- phic. (Indeed, if τ : Ω 1 → A 1 is an isomorphism, where Ω 1 ⊂ Ω and A 1 ⊂ A a re measurable and µ (Ω \ Ω 1 ) = µ ( A \ A 1 ) = 0, then for n > 2 put A n = A n − 1 ∩ Ω n − 1 and Ω n = τ − 1 ( A n ) and finally A 0 = T ∞ n =1 A n and Ω 0 = T ∞ n =1 Ω n .) Since the maps ( M µ ( X ) , M µ ( d )) ∋ f 7→ f   Ω 0 ∈ ( M µ | Ω 0 ( X ) , M µ | Ω 0 ( d )) and ( M µ | A ( X ) , M µ | A ( d )) ∋ f 7→ f   A 0 ∈ ( M µ | A 0 ( X ) , M µ | A 0 ( d )) a re (bijective ) isometries for eve ry b ounded met- ric space ( X , d ), w e may assume that Ω 0 = Ω and A 0 = A . Let ϕ : Ω → A b e a n isomorphism. F or a momen t we will think of ϕ as o f a function from Ω to Ω. Let B 0 = Ω \ A and B n = ϕ n ( B 0 ) ( n > 1 ), where ϕ n denotes t he n -th iterate of ϕ . Note that { B n } ∞ n =0 is a µ -par tition of B = S ∞ n =0 B n . What is more, ϕ ( Ω \ B ) = Ω \ B . But µ (Ω \ B ) µ (Ω) = µ ( ϕ (Ω \ B )) µ ( A ) and th us µ ( Ω \ B ) = 0 . Therefore, as b efore, w e ma y assume that 8 P . NIEMIEC B = Ω. Since ϕ ( B n ) = B n +1 , all the spaces ( B n , M   B n , µ   B n ) are p oin t wisely isomorphic. T ak e a bijection κ : N × N → N and for eac h n, l ∈ N let ψ l,n : B n → B κ ( l,n ) b e an isomorphism. Finally , for a metriz- able space X put h : M µ ( X ) ∋ f 7→ ( S ∞ n =0 [ f   B κ ( l,n ) ◦ ψ l,n ]) ∞ l =0 ∈ M µ ( X ) ω . W e lea v e this as a simple exercise tha t h is a homeomorphism. The p oint (M14) will b e applied in Section 2. W e shall end the section with the t wo more properties of spaces of measurable functions. Recall that a metrizable space X ha s the r efle ctive isotopy pr op erty (in short: RIP) if there is a n ambie n t inv ertible isotopy H : X × X × I → X × X suc h that H ( x, y , 0) = ( x, y ) and H ( x , y , 1) = ( y , x ) (that is, H needs to b e suc h a homotopy that for eac h t ∈ I , the map h t ( x, y ) = H ( x, y , t ) is a homeomorphism of X × X and the function ( x, y , t ) 7→ h − 1 t ( x, y ) is contin uous) (compare [9, Definition IX.2.1 ]). There are other definitions of RIP (see [23],[22 ]), all ‘in v ertible’ v ersions of it a re how ev er equiv alen t for spaces X suc h that X ∼ = X ω . (M6) implies that: (M15) If µ is nonatomic, then the space N µ ( X ) has RIP for eac h metriz- able X and N = M , M f , M c , M r . Indeed, if { A t } t ∈ I is as in (M5), then the map H ( f , g , t ) = ( f   Ω \ A t ∪ g   A t , g   Ω \ A t ∪ f   A t ) is an isotop y w e searc hed for. F ollowing T or u ´ nczyk[17], w e say that a closed subset K of a metriz- able space X is a Z -set if the set C ( Q, X \ K ), where Q is the Hilb ert cub e, is dense in C ( Q, X ) in the top ology o f uniform c on v ergence. (This definition differs from the origina l one by Anderson[1], but b oth these definitions are equiv alen t in ANR’s.) Coun table unions of Z -sets are called σ - Z -sets . The last prop ert y established in this section, whic h shall b e used in Section 5, is (M16) Let µ b e nonatomic. If X has more than one p oin t, then ∆ µ ( X ) is a Z -se t in M µ ( X ). If X is infinite, the set M f µ ( X ) is a σ - Z -set in M µ ( X ). W e shall prov e only the second claim (the first one has similar pro of ). Let { A t } t ∈ I b e as in (M5). It is easy to see that for eac h n the set of all measurable functions whose images ha v e a t most n elemen ts is closed in M µ ( X ) and th us M f µ ( X ) is of t yp e F σ . What is mor e, there is u ∈ M µ ( X ) suc h that u   A t / ∈ M f µ | A t ( X ) for eac h t ∈ I . No w if F : Q → M µ ( X ) is con tin uous, then the maps F n : Q ∋ x 7→ u   A 1 /n ∪ F ( x )   Ω \ A 1 /n ∈ M µ ( X ) con v erge uniformly to F and ha ve imag es disjoin t from M f µ ( X ). SP AC ES OF MEASU RABLE FUNCTIONS 9 2. M r -sp a ces A t the b eginning w e shall study certain spaces of measurable func- tions. Fix an infinite cardinal nu m b er α . Eac h of the sets I J , where J is coun table (infinite), will b e equipped with t he Tic hono v top ology . Let T b e a set of cardinalit y α . Let Ω α = I T (= I α ) and M α b e the σ - algebra of all subsets B of Ω α for whic h there are a coun table infinite set J ⊂ T and B 0 ∈ B ( I J ) suc h that B = { ( x t ) t ∈ T : ( x j ) j ∈ J ∈ B 0 } . In other words, M α is the pro duct of α copies of B ( I ). (Note also that, when consider Ω α with the Tic honov top ology , not ev ery op en subset of Ω α is a mem b er of M α . Op en sets whic h are measurable are exactly those whic h are F σ .) Finally , let m α : M α → I b e the pro duct measure of α copies of the Leb esgue measure m on I . The fo llo wing is w ell kno wn: (M17) The measure m α is homogeneous and w ( m α ) = α . The measure spaces ( I , B ( I ) , m ) and (Ω ℵ 0 , M ℵ 0 , m ℵ 0 ) are p oint wisely isomor- phic. W e need to kno w a little bit more ab out the space (Ω α , M α , m α ). But first a few necessary definitions. A Polish space is a separable completely metrizable one. A subset B of a Polish space Y is said to b e absolutely me asur able in Y if for ev ery probabilistic Borel measure µ o n Y there are t w o Borel subsets A and C of Y su c h t hat A ⊂ B ⊂ C a nd µ ( C \ A ) = 0. A separable metrizable space X is absolutely me as ur able , if for eve ry em b edding ϕ of X in t o the Hilb ert cub e Q , ϕ ( X ) is absolutely measurable in Q . Equiv alen tly , X is absolutely metrizable if t here is d ∈ Metr( X ) suc h that X is absolutely measurable in the completion of ( X, d ). A (separable) Souslin s p ac e is the empty space or a contin uous image of the space of all irratio nal n um b ers; or, equiv alen tly , it is a contin- uous image of some P olish space. The follow ing a re imp ort an t for us prop erties of Souslin spaces: (So1) the image of a Borel function defined on a Borel subset of a P o lish space is a Souslin space, (So2) eve ry Souslin space is absolutely measurable (compare with [13, Theorem XI I I.4.1]). It is a kind of fo lklore that ev ery finite Bo rel measure on a P olish space is regular, i.e . it is supported on a σ -compact subs et of the whole space. This im plies that e v ery finite Bo rel measure on a (s eparable) absolutely measurable space is also supp orted on a σ - compact set. All the ab ov e facts yield the fo llo wing result. 2.1. Lemma. L et (Ω , M , µ ) b e a finite m e asur e sp ac e a n d let X b e a metrizable sp ac e. 10 P . NIEMIEC (A) If the i m age of an M -me asur able function f : Ω → X is c on tain e d in a sep ar ab l e absolutely me a sur able subse t of X , then f ∈ M r µ ( X ) . (B) If N is a σ -sub algebr a of M , ν = µ   N and a function f ∈ M r µ ( X ) b elongs to the closur e of M ν ( X ) , then f ∈ M r ν ( X ) , i.e. ther e is an N -me asur able function g : Ω → X whose image is sep ar able and which is µ -al m ost everywher e e qual to f . In p articular, M r ν ( X ) is close d in M r µ ( X ) and M r ν ( X ) = M r ¯ ν ( X ) , wher e ¯ ν = µ   ¯ N and ¯ N c onsists of those A ∈ M for which ther e is B ∈ N w ith µ ( A \ B ) = µ ( B \ A ) = 0 . Pr o of. (A): Let A ⊂ X b e a separable absolutely measurable sup erset of im f . Let λ : B ( A ) ∋ B 7→ µ ( f − 1 ( B )) ∈ R + . Since λ is a finite measure, there is a σ -compact subset K of A suc h that λ ( K ) = λ ( A ). So, µ ( U ) = µ (Ω) for U = f − 1 ( K ). Then f coincides with ˆ f ∈ M r µ ( X ) in M µ ( X ), where ˆ f   U = f   U and ˆ f   Ω \ U ≡ b with b tak en from K . (B): W e only need to pro v e t he first claim. W e ma y assume that the image of f is con tained in a σ -compact subset o f X , sa y K 0 . By the assumption, there is a sequence o f N -measurable functions f n : Ω → X with finite imag es w hic h is po in t wisely con v ergent µ -almost ev erywhere to f . Let K = K 0 ∪ S n im f n . Fix d ∈ Metr ( K ) and let ¯ K b e the completion o f ( K , d ). Note that K is σ -compact and therefore it is a Borel subset of ¯ K . Let B b e the set of a ll those ω ∈ Ω suc h that the sequence ( f n ( ω )) n is conv ergen t in ¯ K . Since f n ’s are N -measurable, B ∈ N . What is more, µ (Ω \ B ) = 0. Th us, aft er c hanging eac h f n so that f n   Ω \ B ≡ b , where b ∈ K , there is an N - measurable function ¯ g : Ω → ¯ K suc h that lim n →∞ f n ( ω ) = ¯ g ( ω ) for eac h ω and ¯ g is equal to f in M µ ( ¯ K ). This yields that the set C = ¯ g − 1 ( K ) b elongs to N a nd µ (Ω \ C ) = 0. Therefore, to end the pro of, it suffices to put g = ¯ g   C and g   Ω \ C ≡ b .  As w e shall see in the next section (Example 3.5), all claims of the p oin t (B) of the ab ov e lemma fail when we replace eac h M r b y M . F or a metrizable space X let M ( X ) = M m ( X ) and fo r an infinite cardinal α , let M α ( X ) = M m α ( X ) (analogous not ation for metrics). The second claim of (M17) yields t hat M ℵ 0 ( X ) ∼ = M ( X ). 2.2. Theorem. F or eve ry infinite c a r dinal numb er α and e ach metriz- able sp a c e X , M α ( X ) = M r m α ( X ) . Pr o of. W e assume that Ω α = I T . Let u : Ω α → X b e M α -measurable with separable image. Since u is the p oint wise limit of a sequence of M α -measurable functions with finite images, we conclude from this that there is a coun table infinite set J ⊂ T suc h that u ( x ) = u ( y ) whenev er x and y a re elemen ts of Ω α suc h that p J ( x ) = p J ( y ), where p J : I T → I J is the nat ural pro jection. This means that there is a Borel function v : I J → X suc h that u = v ◦ p J . Let S = im v = im u . By SP AC ES OF MEASU RABLE FUNCTIONS 11 (So1) and (So2), S is absolutely measurable and thus Lemma 2.1 –(A) finishes the pro of.  The argumen t used in the pro of of the ab ov e theorem sho ws also that M ( X ) = M r m ( X ). F ollowing Sc hori[16 ], we say that a space Y is a n S -sp ac e if there are an elemen t θ ∈ Y and a map f : Y × I → Y suc h t hat: (S1) f ( x, 0) = θ , f ( x, 1) = x , f ( θ , t ) = θ for eac h x ∈ Y and t ∈ I , (S2) fo r ev ery neighbourho o d U of θ in Y there is t ∈ (0 , 1] such that f ( Y × [0 , t ]) ⊂ U , (S3) the map Y × (0 , 1 ] ∋ ( x, t ) 7→ ( f ( x, t ) , t ) ∈ Y × (0 , 1] is an em b ed- ding, (S4) f ( f ( x, t ) , s ) = f ( x, ts ) fo r eac h t, s ∈ I and x ∈ Y . 2.3. Theorem. F or every infinite c ar dinal numb er α and e ach non- empty me trizable sp ac e X , M α ( X ) is an S -sp ac e. Pr o of. As usual, we assume that Ω α = I T . Fix ξ ∈ T and a ∈ X and put Ω = I T \{ ξ } , θ = δ m α ,a and Y = M m α ( X ). W e shall iden tify Ω α with Ω × I . F or ev ery t ∈ (0 , 1] let κ t : Ω × [0 , t ] ∋ ( x, s ) 7→ ( x, s/t ) ∈ Ω × I . Finally , let f : Y × I ∋ ( u, t ) 7→ ( u ◦ κ t ) ∪ θ   Ω × ( t, 1] ∈ Y . It is not to o difficult to sho w that f is contin uous. What is more, f satisfies the axioms (S1) –(S4), which finishes the pro o f.  It is easily seen tha t µ = m α satisfies the assumption of (M14) (for example, lo ok at κ 1 / 2 defined in the for egoing pro of ). So, Theorem 2.2, Theorem 2.3, (M14), (M15) and the results of Schori[16] imply 2.4. Corollary . L et Y = M α ( X ) . Then Y ∼ = Y ω and for ev e ry m e triz- able man i f o ld U mo del le d on Y , U × Y is home omorphic to U . Before w e pro v e t he main result of this section, let us show the follo wing 2.5. Prop osition. L et µ b e any me asur e (define d on a σ -algebr a of subsets of Ω ) and X 0 , X 1 , X 2 , . . . b e an in finite se quenc e of m etrizable sp ac es. L et J = N and X and Ψ b e as in (M8) . Then Ψ ( M r µ ( X )) = Q j ∈ J M r µ ( X j ) . In other wor ds, for any se quenc e ( f n ) ∞ n =0 such that f n ∈ M r µ ( X n ) ther e i s g ∈ M r µ ( Q n ∈ N X n ) such that ( f n ( ω )) ∞ n =0 = g ( ω ) for µ -almost al l ω ∈ Ω . Pr o of. Let f : Ω ∋ ω 7→ ( f n ( ω )) ∞ n =0 ∈ Q n ∈ N X n . Let ¯ X n b e the com- pletion of ( X n , d n ), where d n is a fixed metric on X n . Let K n b e a σ -compact subset of X n suc h that im f n ⊂ K n . Then K n ∈ B ( ¯ X n ) and th us K = Q ∞ n =0 K n ∈ B ( Q ∞ n =0 ¯ X n ). So, K is a bsolutely measurable and im f ⊂ K . No w it remains to a pply Lemma 2.1–(A).  And now the main result of the section. 12 P . NIEMIEC 2.6. Theorem. L et (Ω 1 , M 1 , µ 1 ) and (Ω 2 , M 2 , µ 2 ) b e two nonatomic me asur e sp ac es such that A ( µ 1 ) and A ( µ 2 ) ar e isometric al ly isomor- phic. L et Φ : A ( µ 1 ) → A ( µ 2 ) b e an isometric isomorp hism of Bo ole an algebr as. Then for every metrizab le X ther e is a unique home om or- phism H : M r µ 1 ( X ) → M r µ 2 ( X ) such that for e ach function f ∈ M c µ 1 ( X ) ther e is a function g ∈ M c µ 2 ( X ) such that g = H ( f ) , im g = im f and g − 1 ( { x } ) = Φ( f − 1 ( { x } )) in A ( µ 2 ) for e v e ry x ∈ X . Wha t is mor e, H ( δ µ 1 ,x ) = H ( δ µ 2 ,x ) for e ach x ∈ X ; and for any d ∈ Metr( X ) , H is an isome try with r esp e ct to the metrics M µ 1 ( d ) and M µ 2 ( d ) . Pr o of. It is clear that the connections b etw een f ∈ M c µ 1 ( X ) and g ∈ M c µ 2 ( X ) described in t he statemen t of t he theorem w ell ( and uniquely) define H on M c µ 1 ( X ). Moreov er, in this step H is a bijection b etw een M c -spaces. It is also clear that H is isometric with resp ect to the suitable metrics (describ ed in the statemen t). Fix d ∈ Metr ( X ) and let ( ¯ X , ¯ d ) be the completion of ( X, d ). Since the spaces ( M µ j ( ¯ X ) , M µ j ( ¯ d )) ( j = 1 , 2) are complete, there is a unique con tin uous extension ¯ H : M µ 1 ( ¯ X ) → M µ 2 ( ¯ X ) , whic h is sim ulta neously a (bijectiv e) isometry . It is enough to c hec k that ¯ H ( M r µ 1 ( X )) ⊂ M r µ 2 ( X ) (b ecause then w e infer analogous inclus ion for ¯ H − 1 ). T ak e an M 1 -measurable function f : Ω 1 → X whose image is con tained in a σ -compact subset of X . This implies that there is a µ 1 -partition { A n } ∞ n =1 of Ω 1 suc h that K n = f ( A n ) (the closure take n in X ) is compact for eac h n > 1. There is a µ 2 -partition { B n } ∞ n =1 of Ω 2 suc h that B n = Φ( A n ) in A ( µ 2 ) f or any n . F or each l > 1 tak e a sequenc e ( f ( l ) n : A l → K l ) ∞ n =1 of M 1 -measurable functions with finite images whic h conv erges p oin t wisely to f   A l . F or ev ery n and l let im f ( l ) n = { x ( l,n ) 1 , . . . , x ( l,n ) p l,n } and let B ( l,n ) 1 , . . . , B ( l,n ) p l,n b e a µ 2 -partition of B l suc h that Φ(( f ( l ) n ) − 1 ( { x ( l,n ) j } )) = B ( l,n ) j in A ( µ 2 ). Define g ( l ) n : B l → K l in the follo wing wa y: g ( l ) n   B ( l,n ) j ≡ x ( l,n ) j . Of course H ( S ∞ l =1 f ( l ) n ) = S ∞ l =1 g ( l ) n ( n > 1) . So — since f n = S ∞ l =1 f ( l ) n tends to f in M µ 1 ( ¯ X ) and ¯ H is isometric — g n = S ∞ l =1 g ( l ) n is a fundamental sequence in M µ 2 ( ¯ X ) and th us also the sequence ( g ( l ) n ) n is fundamen tal in M µ 2 | B l ( ¯ X ). But g ( l ) n is a member of M µ 2 | B l ( K l ), whic h is closed in M µ 2 | B l ( ¯ X ). This implies that there is g ( l ) ∈ M µ 2 | B l ( K l ) whic h is the limit of ( g ( l ) n ) n . The n the function g = S ∞ l =1 g ( l ) is the limit of ( g n ) n in M µ 2 ( ¯ X ). Finally w e conclude that g ∈ M r µ 2 ( X ) and H ( f ) = g .  W e shall denote the unique homeomorphism H corr espo nding to an isometric isomorphism Φ b et w een Bo olean measure algebras, describ ed in Theorem 2 .6, by b Φ. The ab o v e result and (M11) giv e SP AC ES OF MEASU RABLE FUNCTIONS 13 2.7. Corollary . If µ is homo gene ous and of weight α , then M r µ ( X ) ∼ = M α ( X ) . The no te following (M11) com bined with the results of this section leads us to 2.8. Theorem. L e t µ b e nonatomic and let Y = M r µ ( X ) . Then ther e is a finite or c ountable c ol le ction { α j } j ∈ J of differ e nt i n finite c ar dinals such that Y ∼ = Q j ∈ J M α ( X ) . In p articular, Y ω ∼ = Y , Y ha s RIP and is an S-sp ac e and ther e f o r e eve ry me trizable manifold U mo del le d on Y is Y -stable. 3. AM -class 3.1. Definition. A metrizable space is said t o b e an AM-sp ac e [a So- sp ac e ] if ev ery its closed separable subset is absolutely measurable [a Souslin space]. Ev ery S o-space is an AM-space and all lo cally absolutely Borel spaces (in particular, completely metrizable spaces) are So-spaces. It is also w ell kno wn t hat finite or coun table Cartesian pro ducts of AM-spaces [So-spaces] are AM-spaces [So- spaces] as w ell. AM-spaces may b e c haracterized as follo ws: 3.2. Prop osition. F or a metrizable sp ac e X the fol lo w ing c onditions ar e e quivalent: (i) X is an AM-sp ac e, (ii) M r µ ( X ) = M µ ( X ) for ev e ry finite me asur e sp ac e (Ω , M , µ ) , (iii) M r ν ( X ) = M ν ( X ) for any sep ar a b le metric sp ac e Y an d e ac h pr ob- abilistic non a tomic me a s ur e ν define d on B ( Y ) . Pr o of. Thanks to Lemma 2.1–(A), w e only need to pro v e the impli- cation (iii) = ⇒ (i). Let X satisfies the claim of (iii) a nd let A b e a separable closed subset of X . Fix d ∈ Metr( A ) and denote by ˆ A the completion o f ( A, d ). Let µ b e a finite Borel measure on ˆ A . W e may assume tha t µ is nonat omic. Put ν : B ( A ) ∋ B 7→ inf { µ ( C ) : C ∈ B ( ˆ A ) , B ⊂ C } ∈ R + . It is w ell known that ν is a measure. By (iii), there is a Borel function f : A → X whose image is contained in a σ -compact subset of X and suc h tha t f ( a ) = a for ν - almost all a ∈ A . Since A is closed in X , w e may assume that im f is con tained in a σ -compact subset of A , sa y K . Then K ∈ B ( ˆ A ) and ν ( A \ K ) = 0 . Clearly , there is B ∈ B ( ˆ A ) suc h that A ⊂ B and ν ( A ) = µ ( B ) . Then K ⊂ A ⊂ B and µ ( B \ K ) = 0, whic h finishes the pro of.  As an application of the ab ov e c haracterization, thanks to (M12), (M13) and the results of Section 2, we obtain 3.3. Theorem. L et X b e an AM-sp ac e an d d ∈ Metr( X ) . 14 P . NIEMIEC (A) F or any finite me asur e sp ac es (Ω 1 , M 1 , µ 1 ) and (Ω 2 , M 2 , µ 2 ) the map Λ : ( M µ 1 ⊗ µ 2 ( X ) , M µ 1 ⊗ µ 2 ( d )) → ( M µ 1 ( M µ 2 ( X )) , M µ 1 ( M µ 2 ( d ))) given by (Λ f ( ω 1 ))( ω 2 ) = f ( ω 1 , ω 2 ) is a (bije ctive) isometry. (B) F or every two in finite c ar din a l numb ers α and β , M α ( M β ( X )) ∼ = M γ ( X ) , wher e γ = max( α , β ) . In p articular, M ( M ( X )) ∼ = M ( X ) . (C) I f (Ω , M , µ ) is a finite me asur e sp a c e, N is a σ -sub algebr a o f M and ν = µ   N , then M ν ( X ) is clo se d in M µ ( X ) . (D) I f µ is a finite nonatomic me as ur e and Y = M µ ( X ) , then Y ∼ = Y ω , Y has RIP and is an S-sp ac e. It turns out that t he classes of AM-spaces and of So-spaces are in- v arian t under the op erators M µ , as it is sho wn in the following 3.4. Theorem. I f X is an AM-sp ac e [a So-sp ac e], then M µ ( X ) is an AM-sp ac e [a So-sp ac e] as wel l for ev ery fin i te me as ur e sp ac e (Ω , M , µ ) . Pr o of. T ak e a separable a nd closed subset Y of M µ ( X ). Let { f n } ∞ n =1 b e a dense subset of Y . Put A = S ∞ n =1 im f n (the closure take n in X ) and let N b e the smallest σ -subalgebra of M suc h that each of the f unctions f n is N - measurable. Then A is separable and N is a coun tably generated σ - algebra. This means that A ( ν ) is separable, where ν = µ   N . There fore M ν ( A ) is separable as w ell. What is more, b y Theorem 3.3– (C), the space M ν ( A ) is closed in M µ ( X ) and th us Y ⊂ M ν ( X ). Since the classes o f AM-spaces and So-spaces a re closed hereditary , it suffices to sho w that M ν ( X ) is an AM-space [a So-space] if so is X . F urther, thanks to the note follo wing (M11), we ma y assume that ν is nonatomic. But then (see Prop osition 3.2 and Corolla ry 2.7) M ν ( X ) ∼ = M ( X ). So, w e ha v e reduced the pro of to sho wing tha t M ( X ) is an AM-space [a So- space], pro vided X is so and X is s eparable. First w e shall show this for the So-class. Supp ose X is a separable nonempt y Souslin space. Then there is a con tin uous surjection g : R \ Q → X . Put M ( g ) : M ( R \ Q ) ∋ f 7→ g ◦ f ∈ M ( X ) (see the last section). By [15, Theorem 3.3 ], M ( g ) is a contin uous surjection. So, b y the complete metrizabilit y and the separabilit y of M ( R \ Q ), M ( X ) is indeed a Souslin space. No w assume that X is a separable absolutely measurable space. Let S b e a separable metrizable space and let λ b e a probabilistic Borel nonatomic measure on S . It is enough to pro v e that M λ ( M ( X ) ) = M r λ ( M ( X ) ). Let u ∈ M λ ( M ( X ) ). By Theorem 3.3–(A), there is a Borel function v : S × I → X suc h that u ( s ) and v ( s, · ) coincide in M ( X ) for λ - almost all s ∈ S . Since X is absolutely measurable, there is a Borel function w : S × I → X whose image is contained in a σ -compact subset of X (sa y K ) and suc h that v and w coincide in M λ ⊗ m ( X ). Put e u : S ∋ s 7→ w ( s, · ) ∈ M ( K ) ⊂ M ( X ). Then e u and u represen t the same elemen t of M λ ( M ( X ) ). What is more, e u ∈ M λ ( M ( K )) a nd M ( K ) SP AC ES OF MEASU RABLE FUNCTIONS 15 is a Souslin space, whic h yields that e u ∈ M r λ ( M ( K )) ⊂ M r λ ( M ( X ) ). This finishes the pro o f.  W e end the section with the follo wing 3.5. Example. It is w ell known that there exists a subset X of the square I 2 whic h is not Leb esgue measurable, but for eac h t ∈ I the set X t = { s ∈ I : ( t, s ) ∈ X } is a Borel subset of I and m ( X t ) = 1. This implies that X ∈ B ( I ) → ⊗ B ( I ). So, the map f : I 2 → X whic h is the iden tit y on X and constan t on its complemen t is B ( I ) → ⊗ B ( I )-measurable. Ho w ev er, since X is nonmeasurable, there is no g ∈ M m ⊗ m ( X ) whic h coincides with f in M m → ⊗ m ( X ); and f / ∈ M r m → ⊗ m ( X ). Th us w e hav e obtained tha t M r m → ⊗ m ( X ) M m → ⊗ m ( X ) and M m ⊗ m ( X ) M m → ⊗ m ( X ) as w ell. The example sho ws that (M12) is in general not tr ue if w e put there ν = ν 1 ⊗ ν 2 . It also sho ws that if ν is the restriction o f µ to a dense (in A ( µ ) ) σ -subalgebra, then M ν ( X ), in spite of its densit y in M µ ( X ), ma y differ from M µ ( X ). W e do not kno w whether M ( M ( X )) ∼ = M ( X ) if X is as in Exam- ple 3.5 . 4. M ain res ul ts In this section we shall sho w that all considered b y us spaces of measurable functions with resp ect to nonat omic measures are absolute retracts. In our pro of w e shall use the follo wing three results: 4.1. Lem ma ([8 , Theorem 3 .1]) . Ev e ry metrizable sp ac e admits an em- b e dding into the unit spher e of a Hilb ert sp a c e whose image is line arly indep enden t. 4.2. Lemma ([8, the pro o f of Lemma 4.3]) . L et T b e a finite l i n e arly indep enden t subset o f the unit spher e of a Hilb ert sp ac e ( H , h· , −i ) . L et K b e the c onvex hul l of T (in H ) and let D = M ( K ) b e e quipp e d with the top olo gy τ w induc e d by the we ak one of L 2 [lin T ] (= L 2 ( m, lin T )) , i.e. a se q uenc e ( f n ) n of memb ers of D c onver ges to f ∈ D iff Z 1 0 h f n ( t ) , g ( t ) i d t → Z 1 0 h f ( t ) , g ( t ) i d t ( n → ∞ ) for e ach g ∈ D . Then: (BP1) ( D , τ w ) is metrizable c omp act and c onvex, (BP2) the incl usio n map o f M ( T ) into D is an emb e dding, wh en M ( T ) is e quipp e d with the top olo gy of c o n ver genc e in me asur e, (BP3) ther e is a se quenc e of map s fr om D into D who s e image s ar e c ontaine d in M ( T ) which is unif o rmly c onv er gent to the identity map on D . 16 P . NIEMIEC 4.3. Theorem ([18, Theorem 1.1]) . If a metrizable sp ac e X has a b asis (c onsisting of o p en s ets) such that al l finite in terse ctions o f its memb ers ar e homotopic al ly trivial, then X is an ANR. Recall that a top ological space X is homotopic al ly trivial iff for ev ery n > 1 each map o f ∂ I n in to X is extendable to a map of I n in to X . Note also that the empt y space is homotopically trivial. F ollowing [6],[4 ], a subset A of a space X is said to b e homotopy dense (in X ) if there is a homotopy H : X × [0 , 1] → X suc h that H ( x, 0) = x fo r eac h x ∈ X and H ( X × (0 , 1]) ⊂ A . If X is an ANR, then A is homotopy dense in X iff X \ A is lo cally homoto p y negligible in X ( [18]). The main r esult of the pap er has the followin g form: 4.4. Theorem. L et (Ω , M , µ ) b e a finite nonatom i c me asur e s p ac e, X a nonempty metrizable sp ac e and A its d e nse subse t. The n the sp ac e M µ ( X ) is an AR and M f µ ( A ) is hom otopy dense in M µ ( X ) . The pro of of the a b o v e theorem is divided in to a few lemmas. Let us fix a finite nonatomic measure space (Ω , M , µ ), a nonempty metrizable space X a nd its dense subset A . By Lemma 4.1, w e ma y a ssume that X is a linearly indep enden t subset of the unit sphere S o f a Hilb ert space ( H , h· , −i ). F or eac h bounded subse t E of H , the top ology o f con- v ergence in measure in M µ ( E ) coincides with the top olog y induced b y the metric  E ( u, v ) = ( R Ω k u ( ω ) − v ( ω ) k 2 d µ ( ω )) 1 / 2 ( u, v ∈ M µ ( E )). F or eac h Y ⊂ X , w e shall denote b y B  Y ( u, r ) the open ball in ( M µ ( Y ) ,  Y ) with cen ter at u ∈ M µ ( Y ) and of ra dius r > 0 . Our purp ose is to pro v e that if u 1 , . . . , u p ∈ M µ ( X ) and r 1 , . . . , r p > 0, then T p j =1 B  X ( u j , r j ) is homotopically trivial. First w e shall sho w a sp ecial case of this. 4.5. Lemm a. L et T b e a finite subset of X . Then for every u 1 , . . . , u p ∈ M µ ( T ) and e ach r 1 , . . . , r p > 0 , the set G = T p j =1 B  T ( u j , r j ) is homo - topic al ly trivial. Pr o of. Fix k > 1 and tak e a map f : ∂ ( I k ) → G . Let { A t } t ∈ I b e as in (M5) and let E b e an at most coun table dense subset of im f . There is a countably g enerated σ -subalgebra N of M such that eac h member of E is N -measurable and A q ∈ N for q ∈ Q ∩ I . Put ν = µ   N . Then ν is nonatomic a nd A ( ν ) is separable. What is more, since T is clearly an AM-space, M ν ( T ) is closed in M µ ( T ) (Theorem 3.3 –(C)). This implies that im f ⊂ M ν ( T ). No w by Corolla ry 2.7, M ν ( T ) ∼ = M ( T ). Note also that the homeomorphism H (b et w een M ν ( T ) and M ( T ) ) app earing in the statemen t of Theorem 2 .6 is an isometry with resp ect to the metrics  T   M ν ( T ) × M ν ( T ) and d T : M ( T ) × M ( T ) ∋ ( u, v ) 7→ ( R 1 0 k u ( t ) − v ( t ) k 2 d t ) 1 / 2 ∈ R + . So, the in v erse image of G under H coincides with the finite in t ersection of op en d T -balls in M ( T ). This reduces the pro blem to the case when (Ω , M , µ ) = ( I , B ( I ) , m ), whic h w e now assume. Let V = lin T ⊂ H . F ollowing Bessaga and P e lczy´ nski[8 ], SP AC ES OF MEASU RABLE FUNCTIONS 17 consider the Hilbert space L 2 [ V ] of all (equiv alence classes of ) Borel functions w : I → V suc h that R 1 0 k w ( t ) k 2 d t < + ∞ with the scalar pro duct h u, v i V = R 1 0 h u ( t ) , v ( t ) i d t . Put δ j = 1 − r 2 j 2 and U j = { g ∈ L 2 [ V ] : h g , u j i V > δ j } ( j = 1 , . . . , p ). It is easily seen that eac h U j is con v ex and op en in the w eak t op ology of L 2 [ V ]. Let K and ( D , τ w ) b e as in the statemen t o f Lemma 4.2. By (BP2), the top ology of M ( T ) coincides with the one induced by τ w and therefore to the end of t he pro of w e shall deal only with the top ology τ w . Put U = D ∩ T p j =1 U j . Note that U is op en in ( D , τ w ) and U is con v ex. What is more, since T ⊂ S , M ( T ) is contained in the unit sphere of L 2 [ V ] a nd therefore (4-1) U ∩ M ( T ) = G. This implies that f : ∂ ( I k ) → U . Since U is con v ex, there exists a con tin uous extension ˆ f : I k → U of f . F urther, applying Lemma 4.2, tak e a sequenc e of maps ϕ n : D → D whic h is unifo rmly conv ergen t to the iden tit y map on D and suc h that im ϕ n ⊂ M ( T ). Then the sequence f n = ϕ n ◦ ˆ f : I k → D is uniformly conv ergen t to ˆ f . This yields that for infinitely man y n w e ha v e im f n ⊂ U and th us w e ma y assume that the latter inclusion is satisfied for eac h n . But im f n ⊂ im ϕ n ⊂ M ( T ), whic h com bined with (4-1) g iv es im f n ⊂ G . Again b y (BP2), the sequence ( f n   ∂ ( I k ) : ∂ ( I k ) → G ) n tends uniformly to f (with resp ect to the top olog y of M ( T )). Finally , since G is op en in M ( T ) a nd thanks to (M6), G is lo cally equiconnected, whic h implies that fo r some n , f n   ∂ ( I k ) and f are homoto pic in G . So, t he homot op y extension pro p ert y finishes the pro of.  The main result (Theorem 4.4) is an easy consequence o f Theorem 4.3 and the fo llo wing 4.6. Lemma. If u 1 , . . . , u p ∈ M µ ( X ) and r 1 , . . . , r p > 0 , then the set W = T p j =1 B  X ( u j , r j ) is homotopic a l ly trivial. Pr o of. W e ma y assume that µ is probabilistic. F or eac h k > 1 let ∆ k = { ( t 0 , . . . , t k ) ∈ I k +1 : P k j =0 t j = 1 } b e the k -dimensional simplex and let ∂ (∆ k ) = { x ∈ ∆ k : x j = 0 for some j } b e its combinatorial b oundary . It is enough to prov e that eac h map of ∂ (∆ k ) in to W is extendable to a map of ∆ k in to W . Fix a map f : ∂ (∆ k ) → W . Since W is op en, im f is compact a nd M f µ ( A ) is dense in M µ ( X ), there are functions u ∗ 1 , . . . , u ∗ p ∈ M f µ ( A ) and num b ers r ∗ 1 , . . . , r ∗ p > 0 suc h that im f ⊂ W ∗ ⊂ W , where W ∗ = T p j =1 B  X ( u ∗ j , r ∗ j ). Th us we may and shall a ssume that (4-2) u 1 , . . . , u p ∈ M f µ ( A ) . As in the pro of o f Lemma 4 .5, ta k e a family { A t } t ∈ I satisfying the claim of (M5). One ma y sho w that for eac h n > 1 the function 18 P . NIEMIEC λ n : M f µ ( A ) n +1 × ∆ n → M f µ ( A ) give n by λ n ( v 0 , . . . , v n ; t 0 , . . . , t n ) = n [ j =0 v j   A t j \ A t j − 1 (with t − 1 = 0 ) is contin uous. F urther, t ak e a p ositiv e num b er ε suc h that (4-3) [ x ∈ ∂ (∆ k ) B  X ( f ( x ) , ε ) ⊂ W and put δ = ε 3 √ k . Fix l > 1. Let K 0 b e the collection o f all f aces of ∆ k and for eac h n > 1 let K n b e the collection of all g eomet- ric simplices obtained by the barycen tric divisions o f all members of K n − 1 . There is N > 1 suc h that diam  X f ( σ ) 6 δ l for every σ ∈ K N . No w fo r eac h ve rtex x of any mem b er of K = K N tak e v x ∈ M f µ ( A ) suc h that  X ( f ( x ) , v x ) 6 δ /l . Let ‘ 4 ’ b e a total order on the set of all v ertices o f all mem b ers of K . T ak e an y σ ∈ K and assume that x 0 ≺ . . . ≺ x k are v ertices of σ . W e define g σ : σ → M f µ ( A ) b y g σ ( P k j =0 t j x j ) = λ k ( v x 0 , . . . , v x k ; t 0 , . . . , t k ) (( t 0 , . . . , t k ) ∈ ∆ k ). Since x 0 , . . . , x k are linearly indep enden t, g σ is contin uous. What is more, if also σ ′ ∈ K , then g σ and g σ ′ coincide on σ ∩ σ ′ . This yields that the union g l of all g σ ’s is a w ell defined contin uous function from ∂ (∆ k ) in to M f µ ( A ). And, what is imp ortan t, there is a finite subset T l of A suc h that im g l ⊂ M µ ( T l ). Moreov er, if x ∈ σ , where σ ∈ K ha s v ertices x 0 ≺ . . . ≺ x k , then (4-4)  X ( g l ( x ) , f ( x )) 2 6 k X j =0  X ( v x j , f ( x )) 2 6 2 k X j =0   X ( v x j , f ( x j )) 2 +  X ( f ( x j ) , f ( x )) 2  6 4 k δ 2 l 2 < ε 2 l 2 and th us, by (4-3) , im g l ⊂ W . No w for t ∈ ( l, l + 1) put g t : ∂ (∆ k ) ∋ x 7→ λ 1 ( g l ( x ) , g l +1 ( x ); l + 1 − t, t − l ) ∈ M f µ ( A ). It is clear that ( g t ) t > 1 is a homoto p y . F urthermore, one c heck s, using (4-4), that for eac h t ∈ [ l, l + 1] with l > 2 and ev ery x ∈ ∂ (∆ k ),  X ( g t ( x ) , f ( x )) 2 < 2 ε 2 l 2 6 ε 2 . So , im g t ⊂ W for t > 2 and the function h : ∂ (∆ k ) × [2 , ∞ ] → W giv en by h ( x, t ) = g t ( x ) for t < + ∞ and h ( x, ∞ ) = f ( x ) is a homotopy connecting g 2 and f . By the homotop y extension prop erty , it suffices to sho w that g 2 is extendable to a map of ∆ k in to W . T o this end, put T = T 2 ∪ S p j =1 im u j ⊂ A . By (4- 2), T is finite. What is mor e, u 1 , . . . , u p ∈ M µ ( T ). Finally , W ∩ M µ ( T ) = T p j =1 B  T ( u j , r j ) a nd g 2 : ∂ (∆ k ) → W ∩ M µ ( T ). So, b y Lemma 4.5, g 2 admits a con tin uo us extension of ∆ k in to W ∩ M µ ( T ), whic h finishes the pro of.  SP AC ES OF MEASU RABLE FUNCTIONS 19 Pr o of of The or em 4 .4 . Let X b e embedded as a linearly indep enden t subset of the unit sphere of a Hilb ert space. By Theorem 4.3 and Lemma 4.6 , M µ ( X ) is a homotopically trivial ANR. This yields that it is an AR. Finally , the last parag raph of the pro o f of Lemma 4.6 shows that for ev ery op en ball B in M µ ( X ) (with resp ect to the metric  X ) the inclusion map B ∩ M f µ ( A ) → B is a (w eak) homotopy equiv alence and hence M µ ( X ) \ M f µ ( A ) is lo cally homo top y negligible in M µ ( X ) ([18]).  4.7. Corollary . If µ is a finite nonatomic me asur e and X is a non- empty metrizable sp ac e, then the sp ac es M f µ ( X ) , M c µ ( X ) , M r µ ( X ) and M µ ( X ) ar e AR’s. 4.8. R em ark. If N , A and M f ( N , A ) ar e as in (M2) and a dditionally N con tains a subfamily { A t } t ∈ I as in (M5), the pro of of Lemma 4.6 show s that M f ( N , A ) is homotop y dense in M µ ( X ) and thus it is a n AR. This implies that the space M s ( X ) ⊂ M ( X ) o f all piecewise constan t functions is a n AR. As a first consequence of Theorem 4.4 w e obtain a generalization of theorems of Bessaga and P e lczy´ nski[8] and of T oru ´ nczyk[19]: 4.9. Theorem. If µ is a finite nonatomic (nonzer o) me asur e and X is a c ompl e tely m etrizable sp ac e which has mor e than one p oin t, then M µ ( X ) i s ho me omorphic to an infinite-dimensi onal Hilb ert sp ac e of dimension α = max( w ( µ ) , w ( X )) . Pr o of. Put Y = M µ ( X ). By Theorem 3.3–(D), Y ω ∼ = Y . But Y is a noncompact AR and th us, by [20, Theorem 5.1], Y is homeomorphic to a Hilb ert space of dimension w ( Y ). So, the observ ation t hat w ( Y ) = α finishes the pro of.  No w rep eating t he pro of s (with M G replaced by M α ( G )) of Theorem 5.1 and Corolla ry 5 .2 of [8] w e g et 4.10. Cor ollary . L et H b e a Hilb ert sp ac e of dimen sion α > ℵ 0 and let G b e a c omp le tely me trizable top o l o gic al gr oup of wei g ht n o gr e ater than α . Then G is (alg e br aic al ly and top olo gic al ly) isomorp h ic to a cl o se d sub gr oup of a gr oup home om o rphic to H and G admits a fr e e action in H . 5. E xtending maps W e b egin this section with 5.1. Definition. Let µ b e a finite measure and let f : X → Y b e a map. Let M µ ( f ) : M µ ( X ) ∋ u 7→ f ◦ u ∈ M µ ( Y ) . M µ ( f ) is said to b e the µ -extension of f . Additionally , let M ( f ) = M m ( f ) and M α ( f ) = M m α ( f ) for eve ry infinite cardinal α . 20 P . NIEMIEC Note that M µ ( f ) is con tin uo us and tha t M µ ( f )( N µ ( X )) ⊂ N µ ( Y ) for N = M f , M c , M r . The connection (5-1) M µ ( f )( δ µ,X ( x )) = δ µ,Y ( f ( x )) ( x ∈ X ) sa ys t hat M µ ( f ) extends f , when w e iden tify the elemen t s of Z with the ones of ∆ µ,Z via δ µ,Z with Z = X , Y , whic h justifies the undertak en terminology . If, in addition, X and Y are top ological groups and f is a gro up ho momorphism, so is M µ ( f ). The Reader will easily c hec k that whenev er µ is a fixed finite measure, the o p erations X 7→ M µ ( X ) and f 7→ M µ ( f ) define a f unctor. This functor has inte resting prop erties, whose pro ofs are left as exercise s (b elo w w e a ssume that g n , g : X → Y are maps): (F1) M µ ( g ) is an injection [em b edding] iff g is so, (F2) im M µ ( g ) = M µ ( im g ), (F3) t he se quence ( M µ ( g n )) n is point wisely [uniformly on compact s ub- sets of M µ ( X )] con v ergent to M µ ( g ) iff the sequenc e ( g n ) n p oin t- wisely [uniformly on compact subsets of X ] con v erges t o g , (F4) f or each  ∈ Metr ( Y ) the map ( C ( X , Y ) ,  sup ) ∋ h 7→ M µ ( h ) ∈ ( C ( M µ ( X ) , M µ ( Y )) , ( M µ (  )) sup ) is isometric ( ‘ C ( A, B )’ denotes the collection of all maps from A to B and ‘ d sup ’ stands for the suprem um metric induced by a b ounded metric d ). It is clear that for eac h f ∈ C ( X , Y ), im M µ ( f ) ⊂ S A M µ ( f ( A )) where A runs o v er all se parable closed subsets of X . W e do not kno w whether the latter inc lusion can alw ay s b e replaced b y the equalit y . W e a re only able to sho w the following result, the pro o f of whic h is similar to the pro of of [1 5, Theorem 3.3]. 5.2. Pr op osition. Whene v er µ is a finite me asur e and f : X → Y is a map, im M r µ ( f ) = S K M r µ ( f ( K )) wher e K runs o ver al l σ -c omp act sub- sets of X and M r µ ( f ) = M µ ( f )   M r µ ( X ) . What is mor e, if v ∈ M r µ ( f ( A )) , wher e A is a (s e p ar able) So uslin subset of X , then v ∈ im M r µ ( f ) . Pr o of. W e only need to pro v e the second claim. Put C = f ( A ) and let L b e a σ - compact subset of C suc h that im v ⊂ L . Let ν : B ( L ) ∋ B 7→ µ ( v − 1 ( B )) ∈ R + . Put K = A ∩ f − 1 ( L ). Then K ∈ B ( A ) and thus K is a Souslin space. Now it suffices t o apply [13 , Theorem XIV.3.1] to obtain a function h : L → K suc h that f ◦ h is the identit y map on L and f or ev ery op en in K set U ⊂ K , h − 1 ( U ) is a mem b er of the σ - algebra generated b y the family of all Souslin s ubsets of L . This implies that for eve ry Borel subset B of K , h − 1 ( B ) is absolutely measurable and therefore there is a Borel function w : L → K and a set B 0 ∈ B ( L ) suc h that ν ( B 0 ) = 0 and w = h on L \ B 0 . No w put u = w ◦ v . By Lemma 2.1–(A), u ∈ M r µ ( X ). What is more, f ◦ u is µ -almost ev erywhere equal t o f ◦ h ◦ v = v , whic h finishes the pro of.  SP AC ES OF MEASU RABLE FUNCTIONS 21 Under the notatio n of Prop osition 5.2 w e get 5.3. Cor ollary . (i) If X is a So-s p ac e, then im M r µ ( f ) = [ A M r µ ( f ( A )) wher e A runs over al l sep ar able clo s e d subsets of X . (ii) If for eve ry c omp act subset L of im f ther e is a (sep ar able) So usli n subset K of X such that L ⊂ f ( K ) , then im M r µ ( f ) = M r µ (im f ) . The ab o v e result leads to the follo wing 5.4. Definition. A ma p f : X → Y is said to b e an s-map if f satisfies the a ssumption of the p oint (ii) of Coro llary 5.3 . Basic examples of s-maps are closed maps whose domains are So- spaces and prop er maps. No w a pplying the G eneral Sc heme and main ideas o f Se ction 3 of [1 5] (with the same f unctor M ), thanks to the homeomorphism extension theorem prov ed in [10], w e easily obtain 5.5. Theorem. L et Ω b e a top ol o gic al sp ac e home omorphic to a non- sep ar able Hilb ert sp ac e. L et Z b e the family (c ate gory) o f maps b etwe en Z -sets of Ω c onsisting of al l p airs ( ϕ, L ) , wher e dom ϕ , i.e. the dom ain of ϕ , and L ar e Z -sets of Ω an d ϕ is an L -val ue d c ontinuous function. Ther e is a functor Z ∋ ( ϕ, L ) 7→ b ϕ L ∈ C (Ω , Ω) of extension which sat- isfies al l the claims of the p oints (a) , (b) , (h) , (i) s tate d on p a ges 1–2 of [15] and the claims of the p oints (d) , (f ) and (g) (o f [1 5] ) c onc erning closur es of im a ges. The functor pr eserves the pr op erties of b eing an inje ction, an emb e dding o r a map with den se image; an d satisfies al l the cla ims of the p oints (c)–(g) of [15 ] for any s-map ϕ . 5.6. R em a rk. Analogous functor as in Theorem 5.5 can b e built using the functor b P studied by Banakh[2, 3] and Banakh and Radul[5]. (F or a metrizable space X , b P ( X ) is the space of all Borel probabilistic mea- sures supp o rted on σ -compact subsets of X a nd f or a map f : X → Y and µ ∈ b P ( X ), b P ( f )( µ ) is the transp ort o f µ under f .) Theorem 2.11 of [5] sa ys t hat b P ( X ) is homeomorphic to an infinite-dimensional Hilb ert space, pro vided X is completely metrizable and noncompact. Th us it is enough to apply G eneral Schem e of [15] and results of Ba nakh[2, 3] on extending maps and b ounded metrics via the functor b P . W e end the pap er with the follow ing t w o questions. Question 1. Is M ( M ( X )) ho meomorphic to M ( X ) fo r an arbitrary metrizable space X ? Question 2. Is M µ ( X ) homeomorphic to M r µ ( X ) for an a rbitrary metrizable space X and any finite measure space (Ω , M , µ ) ? Note that the affirmativ e a nsw er for Question 2 implies the affirma- tiv e one for Question 1. 22 P . NIEMIEC Reference s [1] R.D. Anderson, On t op olo gic al infinite deficiency , Mich. Math. J. 14 (1967), 365–3 83. [2] T.O. Banakh, T op olo gy of sp ac es of pr ob ability me asur es, I , Mat. Stud. 5 (1995), 6 5–87 (Russian). [3] T.O. Ba nakh, T op olo gy of sp ac es of pr ob abili ty me asur es, II , Mat. Stud. 5 (1995), 8 8–106 (Russia n). [4] T. Ba nakh, Char acterization of sp ac es admitting a homotopy dense emb e dding into a Hilb ert manifold , T opolo gy Appl. 86 (199 8), 123– 131. [5] T.O. Banakh and T.N. Radul, T op olo gy of sp ac es of pr ob ability me asure s , Sb. Math. 188 (199 7), 97 3–99 5. [6] T. Bana kh, T. Radul, M. Zarichn yi, Absorbing sets in infinite-dimensional manifolds , VNTL Publisher s, Lviv , 19 96. [7] T. Banakh and I. Zarich nyy , T op olo gic al gr oups and c onvex sets home omorphi c to non-sep ar able Hilb ert sp ac es , Ce n t. Eur. J. Ma th. 6 (200 8), 77 –86. [8] Cz. Bessaga a nd A. Pe lczy ´ nski, On sp ac es of me asur able functions , Studia Math. 44 (1972 ), 59 7–615 . [9] Cz. Bessaga and A. P e lczy ´ nski, Sele cte d t opics in infi n ite-dimensional top olo gy , PWN – Polish Scientific Publisher s, W arszaw a, 1975. [10] T.A. Cha pman, Defi ciency in infin ite-dimensional m anifolds , Gene ral T o p ol. Appl. 1 (197 1), 26 3–27 2. [11] T. Dobrowolski a nd H. T or u´ nczyk , Sep ar able c omplete A NR’s admitting a gr oup structure ar e Hilb ert manifolds , T op ology Appl. 12 (1981), 2 29–2 35. [12] P .R. Halmos, Me asur e t he ory , V a n Nostra nd, New Y ork, 1 950. [13] K. Kuratowski and A. Mostowski, Set The ory with an Intr o du ct ion to Descrip- tive Set The ory , PWN – Polish Scient ific Publishers , W arszaw a, 1976. [14] D. Ma haram, On homo gene ous me asur e algebr as , Pro c. Natl. Acad. Sci. USA 28 (19 42), 10 8–111 . [15] P . Niemiec, F un ctor of c ont inuation in Hilb ert cub e and Hilb ert sp ac e , to app ear in F und. Math. [16] R. Schori, T op olo gic al stability for infin ite-dimensional manifolds , Co mpo s. Math. 23 (1971 ), 87 –100. [17] H. T oru´ nczyk, R emarks on Anderson ’s p ap er “On top olo gic al infinit e defi- ciency” , F und. Ma th. 6 6 (197 0), 3 93–40 1. [18] H. T oru ´ nczyk, C onc erning lo c al ly homotopy ne gligible sets and char acterization of l 2 -manifolds , F und. Math. 101 (1978 ), 93–1 10. [19] H. T oru ´ nczyk, Char acterization of infinite-dimensional manifolds , in: Pr o c e e d- ings of the International Confer enc e on Ge ometric T op olo gy (Warsaw, 1978) , PWN – Polish Scientific Publisher s, W arszaw a, 1980, 43 1–437 . [20] H. T oru ´ nczyk, Char acterizing Hilb ert sp ac e top olo gy , F und. Math. 111 (19 81), 247–2 62. [21] H. T o ru ´ nczyk, A c orr e ct ion of two p ap ers c onc erning H ilb ert manifolds , F und. Math. 125 (198 5), 89 –93. [22] J.E. W es t, Fixe d-p oint sets of tr ansformation gr oups on infin ite-pr o duct sp ac es , Pro c. Amer. Ma th. So c. 21 (1969 ), 575– 582. [23] R.Y.T. W ong, On home omorphi sms of c ert ain infin ite dimensional s p ac es , T r ans. Amer. Math. So c. 128 (1967 ), 148–1 54. Piotr Niemiec, Jagiellonian University, Institute of Ma thema tics, ul. Lojasiewicza 6, 30-348 Kra k ´ ow, Poland E-mail addr ess : pi otr.n iemie c@uj.edu.pl