Probability measures and Milyutin maps between metric spaces

PR OBABILITY MEASURES AND MIL YUTIN MAPS BETWEE N METRIC SP A CES VESKO V ALOV Abstract. W e prov e that the functor ˆ P of Ra do n probability measures transforms any open map betw een completely metrizable spaces in to a soft map. This res ult is applied to esta blish some prop erties of Milyutin maps b etw een completely metrizable s paces. 1. Intr oduction In this pap er we deal with metrizable spaces and con tin uous maps. By a (complete) space w e mean a (completely) metrizable space, a nd b y a measure a proba bilit y Radon measure. Recall that a measure µ on X is said to b e: • pr ob abil i ty if µ ( X ) = 1 ; • R adon if µ ( B ) = sup { µ ( K ) : K ⊂ B and K is compact } for an y Borel set B ⊂ X ; The suppo rt supp µ of a measure µ is the in tersection of all closed subsets A of X with µ ( A ) = µ ( X ). It is w ell kno wn that the support of any measure is non-empt y a nd separable. Ev ery where below ˆ P ( X ) stands for the space of all pro babilit y Radon measures on X equipped with the w eak top o logy with resp ect to C ∗ ( X ). Here, C ∗ ( X ) is the space of b ounded contin uous functions on X with the uniform conv ergence top ology . According t o [2], ˆ P is a functor in the category of metrizable spaces and con tin uous maps. In particular, for an y map f : X → Y there exists a map ˆ P ( f ) : ˆ P ( X ) → ˆ P ( Y ). A systematic study of the functor ˆ P c an b e found in [2 ] a nd [3]. W e also consider the su bspace P β ( X ) ⊂ ˆ P ( X ) consisting of all measures µ suc h that supp µ is compact. This pap er is dev oted to some pr o p erties of Milyutin maps b etw een metrizable space s. W e sa y that f : X → Y is a Milyutin map if there 2000 Mathe matics Subje ct Classific atio n. Prima ry 54C6 0; 60B05 ; Secondary 54C65, 54C1 0. Key wor ds and phr ases. Pro bability measures , supp o r t of measures, Milyutin maps, metric spaces. The a uthor was partially suppo rted by NSERC Grant 26191 4-08. 1 2 exists a map g : Y → ˆ P ( X ) suc h that supp g ( y ) ⊂ f − 1 ( y ) for ev ery y ∈ Y . Such g is called a choice map asso ciated with f . According to [3, Theorem 3.15], fo r an y metrizable X t here exists a barycen tric map b ˆ P ( X ) : ˆ P ( ˆ P ( X )) → ˆ P ( X ) suc h that b ˆ P ( X ) ( ν ) = ν for all ν ∈ ˆ P ( X ). Hence, if g is a choice map asso ciated with f , then the map b ˆ P ( X ) ◦ ˆ P ( g ) : ˆ P ( Y ) → ˆ P ( X ) is a right inv erse of ˆ P ( f ). Consequen tly , f is a Milyutin map if and only if ˆ P ( f ) admits a r ig h t in v erse. Our first principal result concerns t he ques tion when ˆ P ( f ) is a soft map. Recall that a map f : X → Y is soft if for a ny space Z and its closed subset A and an y maps g : Z → Y , h : A → X with ( f ◦ h ) | A = g there exists a map ¯ g : Z → X suc h that ¯ g extends h and f ◦ ¯ g = g . It is easily seen that ev ery soft map is surjectiv e and op en. Theorem 1.1. L et f : X → Y b e a surje ctive op en m ap b etwe en c om- plete sp ac es. Then ˆ P ( f ) : ˆ P ( X ) → ˆ P ( Y ) is a soft map. The pa r ticular cases of Theorem 1.1 when b o t h X and Y are either compact or separable w ere established in [8] a nd [4], resp ective ly . Since any soft map admits a righ t inv erse, a map f satisfying the h yp ot heses of Theorem 1.1 is a Milyutin map. W e apply Theorem 1.1 to obtain some results ab out atomless and exact Milyutin maps in tro- duced in [14]. If f : X → Y is a Milyutin map and there exists a c hoice map g suc h that supp g ( y ) = f − 1 ( y ) (resp., g ( y ) is an atomless mea- sure on f − 1 ( y ) for eac h y ∈ Y , i.e. g ( y )( { x } ) = 0 for all x ∈ f − 1 ( y )), then f is said to b e a n exact (resp., a tom l e ss ) Milyutin map. It w as established in [14] t ha t, in the realm of Polish spaces X and Y , f is exact Milyutin if and only if it is op en. The classes o f atomless ex- act Milyutin maps and atomless Milyutin maps b et w een P olish spaces w ere c hara cterized in [1, Theorem 1.6]. The first class consists of all op en maps p ossessing p erfect fib ers (i.e., without isolated p oints ) [1, Theorem 1 .6], and the second one of a ll maps f : X → Y suc h tha t for some P olish space X 0 ⊂ X the restriction f 0 = f | X 0 : X 0 → Y is a n op en surjection whose fib ers are p erfect [1, Theorem 1.7]. Next theorem is a non-separable analogue of [1, Theorem 1.7]. Theorem 1.2. A c ontinuous surje ction f : X → Y of c omplete sp ac es is an atomless Milyutin map if and only if ther e exists a c o m plete sub- sp ac e X 0 ⊂ X such that f 0 = f | X 0 : X 0 → Y is an op en surje ction and al l fib ers of f 0 ar e p erfe ct sets. Mor e over, for any such f ther e exists a m ap f ∗ : P β ( Y ) → ˆ P ( X ) such that any f ∗ ( µ ) is atomless an d ˆ P ( f )  f ∗ ( µ )  = µ , µ ∈ P β ( Y ) . 3 W e do not kno w whether under the h yp otheses in Theorem 1.2 there exists a map f ∗ : ˆ P ( Y ) → ˆ P ( X ) suc h that eac h f ∗ ( µ ) is atomless and ˆ P ( f )  f ∗ ( µ )  = µ , µ ∈ ˆ P ( Y ). But if we a re in terested in atomless maps defined on Y , w e ha v e the f ollo wing: Theorem 1.3. Every op en surje ction f : X → Y w ith p erfe ct fib ers is a densely a tomless Milyutin map pr ovide d X and Y ar e c omplete sp ac es. Here, a Milyutin map f : X → Y is densely atomless if { g ∈ C h f ( Y , X ) : g ( y ) is at o mless for all y ∈ Y } is a dense G δ -set in the space C h f ( Y , X ) of all c hoice ma ps associat ed with f equipp ed with the source limita tion top ology . A few w ords ab out this top ology . If ( X , d ) is a b ounded (complete) metric space, then there exists a (complete) metric ˆ d on ˆ P ( X ) generating its top olo gy and extending d , see [3]. Then C h f ( Y , X ) is a subspace of the function space C ( Y , ˆ P ( X )) with the source limitation top o logy whose lo cal base at a giv en h ∈ C ( Y , ˆ P ( X )) consists of all sets B ˆ d ( h, α ) = { g ∈ C ( Y , ˆ P ( X )) : ˆ d ( g ( y ) , h ( y )) < α ( y ) for all y ∈ Y } , where α is a con tin uous map fr o m Y in to (0 , ∞ ). It is we ll known that this top ology do es not depend on the metric ˆ d and it has the Baire prop erty in case ˆ P ( X ) is complete. Similar ly , f is said to b e de n sely exact provide d the set { g ∈ C h f ( Y , X ) : supp g ( y ) = f − 1 ( y ) for ev ery y ∈ Y } is a dense and G δ -set in C h f ( Y , X ). When f is b oth densely atomless and densely exact, it is called densely exact atomless. Theorem 1.4. L et f : X → Y b e a n op en surje ction of c omplete sp ac es and π : X → M a map into a sep ar able sp ac e M . Then al l choic e maps h ∈ C h f ( Y , X ) such that π ( supp h ( y )) is dense in π ( f − 1 ( y )) for every y ∈ Y form a dense G δ -set in C h f ( Y , X ) . It is in teresting whethe r in Theorem 1 .4 one can substitute the phrase ” π ( su pp h ( y )) is dense in π ( f − 1 ( y ))” by ” π ( su pp h ( y ) = π ( f − 1 ( y ))”. Next corollary is a pa r ametrization of the P arthasarathy [12] result that p erfect P olish spaces admit atomless measures. It also prov ides a partial answ er of the question [1] whether an y op en surjection f of complete spaces is an exact atomless Milyutin map pro vided all fib ers of f are p erfect P olish spaces. 4 Corollary 1.5. L et f : X → Y b e an op en and close d surje ction of c omplete sp ac es such that al l fi b ers o f f ar e sep ar ab le ( and p erfe ct ) . Then f is densely exact ( atomles s ) Milyutin map. Finally , w e generalize [14, C orollary 1.4 ] and [1, Corollary 1.9 ] as follo ws (b elo w a con tin uous set-v alued map means a ma p whic h is b ot h lo wer and upp er semi-con tinuous): Corollary 1.6. L et X and Y b e c omp lete s p ac es a n d Φ : Y → X a c ontinuous set-value d map such that al l values Φ( y ) ar e close d sep ar a- ble subsets of X . Then ther e exis ts a m ap h : Y → ˆ P ( X ) such that supp h ( y ) = Φ( y ) for every y ∈ Y . If, in addition, al l Φ( y ) ar e p erfe ct sets, the map h c a n b e chosen so that every h ( y ) is atomless. Ac kno wledgemen ts: The author wishes to t ha nk the referee for his/her v aluable remarks and suggestions whic h significan tly impro v ed the pap er. 2. Preliminaries In this section we pro vide some preliminary results and establish the pro of of Theorem 1.1 . Probabilit y Radon measure s on a complete space X can b e describ ed as p ositiv e linear f unctionals µ on C ∗ ( X ) suc h that || µ || = 1 and lim µ ( h α ) = 0 f o r an y decreasing net { h α } ⊂ C ∗ ( X ) whic h p oin t- wisely con v erges to 0, see [15]. Under this in terpretation, supp µ co- incides with the set of all x ∈ X suc h that for ev ery neighborho o d U x of x in X there exists ϕ ∈ C ∗ ( X ) suc h that ϕ ( X \ U x ) = 0 and µ ( ϕ ) 6 = 0. This represen tation of supp µ easily implies that t he set- v alued map supp : ˆ P ( X ) → X (assigning to eac h µ its supp ort) is lo wer semi-contin uous, i.e., { µ ∈ ˆ P ( X ) : supp µ ∩ U 6 = ∅ } is op en in ˆ P ( X ) f o r any op en U ⊂ X . F or eve ry closed F ⊂ X , w e hav e µ ( F ) = inf { µ ( ϕ ) : ϕ ∈ C ( F ) } (see for example [7] in case X is com- pact), where C ( F ) = { ϕ ∈ C ∗ ( X ) : 0 ≤ ϕ ≤ 1 and ϕ ( F ) = 1 } . According to [4], any compatible (complete) metric d on X generates a compatible (complete) metric ˆ d on ˆ P ( X ) suc h that ˆ d  tµ + (1 − t ) µ ′ , tν + (1 − t ) ν ′  ≤ t ˆ d ( µ, µ ′ ) + (1 − t ) ˆ d ( ν, ν ′ ) for all t ∈ [0 , 1] and µ, µ ′ , ν, ν ′ ∈ ˆ P ( X ). It is easily seen that eve ry ball (op en or closed) with resp ect to ˆ d is con vex . Let A ε ( X ) denote the set of all µ ∈ ˆ P ( X ) suc h that µ ( { x } ) ≥ ε for some x ∈ supp µ . F or an y closed K ⊂ X there exists a closed em b edding i : ˆ P ( K ) → ˆ P ( X ) defined b y i ( ν )( h ) = ν ( h | K ) for all ν ∈ 5 ˆ P ( K ) and h ∈ C ∗ ( X ). Eve rywhere b elow w e iden tify ˆ P ( K ) with the set i  ˆ P ( K )  = { µ ∈ ˆ P ( X ) : supp µ ⊂ K } whic h is closed in ˆ P ( X ). Lemma 2.1. L et X b e a c omplete sp ac e, K a p erfe ct close d subset of X and G a c onvex op en subset of ˆ P ( K ) . Then fo r e very ε > 0 we have: (1) A ε ( X ) is a close d subset of ˆ P ( X ) ; (2) A ε ( X ) ∩ G is a nowher e dense set in the closur e G . Pr o of. (1) Since ˆ P ( X ) is metrizable, it suffices to chec k that µ 0 = lim µ n ∈ A ε ( X ) for ev ery con v erg ent sequence { µ n } n ≥ 1 in ˆ P ( X ) with { µ n } ⊂ A ε ( X ). T o this end, let H b e the closure in X of the set S n ≥ 0 supp µ n . Because ev ery µ ∈ ˆ P ( X ) has a separable suppo rt, H is a P olish subset of X . Considering all µ n , n ≥ 0, as elemen ts of ˆ P ( H ), w e ha v e t ha t the sequence { µ n } n ≥ 1 is contained in A ε ( H ) and con ve rges to µ 0 . Therefore, b y [12, Theorem 8.1], µ 0 ∈ A ε ( H ). Conse quen tly , there exists x 0 ∈ H with µ 0 ( { x 0 } ) ≥ ε . Therefore, A ε ( X ) is closed in ˆ P ( X ). (2) Since A ε ( K ) = A ε ( X ) ∩ ˆ P ( K ), it suffices to sho w that A ε ( K ) is no where dense in ˆ P ( K ). Supp ose A ε ( K ) con tains an op en subset W of ˆ P ( K ) and let P ω ( K ) b e the set of all µ ∈ ˆ P ( K ) ha ving a finite supp ort. Since P ω ( K ) is dense in ˆ P ( K ), there exists µ 0 = P i = k i =1 λ i δ x i ∈ P ω ( K ) ∩ W . Here, δ x i denotes Dir a c’s measures at x i and λ i = µ 0 ( { x i } ). Moreo ve r, λ i ≥ ε for at least o ne i . F or eac h i ≤ k and n ≥ 1 c ho ose a neigh b o rho o d V i ⊂ K of x i and n differen t p oints x i (1) , .., x i ( n ) ∈ V i suc h that t he family { V i : 1 ≤ i ≤ k } is disjoint and dist ( x i , x i ( j ) ) ≤ 1 /n for all 1 ≤ j ≤ n . This can b e done b ecause K is p erfect, so ev ery neigh b o rho o d of x i con tains infinitely man y p oin ts. Consider now the measures µ n = i = k X i =1 j = n X j =1 λ i δ x i ( j ) n . Since lim µ n = µ 0 , there exists n 0 suc h that µ n ∈ W for all n ≥ n 0 . Consequen tly , fo r ev ery n ≥ n 0 there exists i ≤ k with λ i /n ≥ ε , a con tradiction.  Lemma 2.2. L et f : X → Y b e an op en surje ction b etwe en c omplete sp ac es such that dimY = 0 . Then ˆ P ( f ) : ˆ P ( X ) → ˆ P ( Y ) is a soft map . Pr o of. According to Theorem 1.3 fro m [4], it suffices to sho w tha t f is ev erywhere lo cally inv ertible. The la st not io n is defined as follows: fo r an y space Z , a p oin t a ∈ Z , a map g : Z → Y and a n op en set U ⊂ X with g ( a ) ∈ f ( U ) there exist a neigh b orho o d V of a in Z and a map h : V → U suc h that f ◦ h = g | V . Obvious ly , f is ev erywhere lo cally in ve rtible pro vided it satisfies the follo wing condition: 6 (*) F or an y op en U ⊂ X and a ∈ f ( U ) there exists a map g : V → U with V b eing a neigh b orho o d of a in Y suc h that f ( g ( y )) = y for all y ∈ V . T o sho w f satisfies ( ∗ ), fix an op en set U ⊂ X and a ∈ f ( U ). Since f is op en, the set V = f ( U ) ⊂ Y is also op en and the set-v alued map Φ : V → U , Φ( y ) = f − 1 ( y ) ∩ U , is low er semi-con tinuous with closed v alues. Moreo v er, U admits a complete metric b ecause X is complete. Then, b y the 0 -dimensional selection theorem of Mic hael [11], Φ has a con tinuous selection g . Obviously , g is as required.  Pro of of Theorem 1.1. First, let us show that ˆ f = ˆ P ( f ) | ˆ P ( f ) − 1 ( Y ) is ev erywhere lo cally inv ertible. It suffices to show that ˆ f satisfies condition ( ∗ ) from Lemma 2.2. Supp ose that U ⊂ ˆ P ( f ) − 1 ( Y ) is op en and y 0 ∈ ˆ f ( U ). W e need to find a map α : V → U , where V is a neigh b o rho o d of y 0 in Y , suc h that ˆ f ( α ( y )) = y for eve ry y ∈ V . T o t his end, ch o ose a 0-dimensional complete space Z and a p erfect Milyutin map g : Z → Y , see [6] (recall that a map is p erfect if it is closed and has compact fib ers). Next, consider t he pull-back T = { ( z , x ) ∈ Z × X : g ( z ) = f ( x ) } of Z and X with resp ect to the maps g a nd f , and let p f : T → Z , p g : T → X b e the corresponding pro jections. Since f is op en, so is p f . F or any y ∈ Y w e ha ve p − 1 f  g − 1 ( y )  = p − 1 g  f − 1 ( y )  = g − 1 ( y ) × f − 1 ( y ). Since g is Milyutin, there exists a map g ∗ : Y → ˆ P ( Z ) such that supp g ∗ ( y ) ⊂ g − 1 ( y ) for all y ∈ Y . Let ˆ p f = ˆ P ( p f ) : ˆ P ( T ) → ˆ P ( Z ) and ˆ p g = ˆ P ( p g ) : ˆ P ( T ) → ˆ P ( X ). T ake an op en set G ⊂ ˆ P ( X ) with G ∩ ˆ P ( f ) − 1 ( Y ) = U and let W = ˆ p g − 1 ( G ). Pic k µ ∗ ∈ G ∩ ˆ P ( f − 1 ( y 0 ) and let ν 0 = µ 0 × µ ∗ b e the pro duct measure, where µ 0 = g ∗ ( y 0 ). Ob viously , ν 0 ∈ ˆ P ( g − 1 ( y 0 ) × f − 1 ( y 0 )) ⊂ ˆ P ( T ). Moreo ve r, ˆ p f ( ν 0 ) = µ 0 and ν 0 ∈ W b ecause ˆ p g ( ν 0 ) = µ ∗ ∈ G . No w w e can complete the pro of that ˆ f is ev erywhere lo cally inv ert- ible. Let g 0 : { y 0 } → ˆ P ( T ) b e the constan t map g 0 ( y 0 ) = ν 0 . Since ˆ p f ( ν 0 ) = g ∗ ( y 0 ) and, b y L emma 2.2, the map ˆ p f is soft, there exists a map θ : Y → ˆ P ( T ) extending g 0 suc h that ˆ p f ◦ θ = g ∗ . Ob viously , V = θ − 1 ( W ) is a neigh b orho o d of y 0 , and define α = ˆ p g ◦ θ . Since for any y ∈ V w e hav e ˆ p f ( θ ( y )) = g ∗ ( y ), p f ( supp θ ( y )) = su pp g ∗ ( y ) ⊂ g − 1 ( y ) and supp θ ( y ) ⊂ g − 1 ( y ) × f − 1 ( y ). So, su pp α ( y ) = p g ( supp θ ( y )) ⊂ f − 1 ( y ). Consequen tly , ˆ f ( α ( y )) = y . Moreov er, α ( y ) ∈ U for all y ∈ V . Since ˆ f is ev erywhere lo cally in ve rtible, by [4 , Theorem 1.3], the map ˆ P ( ˆ f ) : ˆ P ( ˆ Y ) → ˆ P ( Y ) is soft, where ˆ Y = ˆ f − 1 ( Y ). Moreo ve r, ˆ P ( X ) ⊂ ˆ P ( ˆ Y ) ⊂ ˆ P ( ˆ P ( X )) b ecause X ⊂ ˆ Y ⊂ ˆ P ( X ). Therefore the 7 follo wing diagram ˆ P ( ˆ Y ) b ˆ P − − − → ˆ P ( X ) ˆ P ( ˆ f )   y   y ˆ P ( f ) ˆ P ( Y ) i ˆ P ( Y ) − − − → ˆ P ( Y ) is commutativ e. Here, b ˆ P denotes the restriction b ˆ P ( X ) | ˆ P ( ˆ Y ) of the barycen tric map b ˆ P ( X ) : ˆ P ( ˆ P ( X )) → ˆ P ( X ), see [3], and i ˆ P ( Y ) is the iden tity on ˆ P ( Y ). Since b ˆ P retracts eac h ˆ P ( ˆ f ) − 1 ( µ ) on to ˆ P ( f ) − 1 ( µ ), µ ∈ ˆ P ( Y ), and ˆ P ( ˆ f ) is soft, w e finally obtain that ˆ P ( f ) is also soft. The pro of is completed. 3. A tomless Mil yutin maps In this section we provide the pro ofs of Theorems 1.2 a nd 1.3 . Pro of of Theorem 1.2 . Supp ose that f : X → Y is a surjectiv e atomless Milyutin map with X and Y complete spaces. Then there exists a c hoice ma p h : Y → ˆ P ( X ) asso ciat ed with f suc h that h ( y ) is an atomless measure for all y ∈ Y . Let X 0 = S { supp h ( y ) : y ∈ Y } and f 0 = f | X 0 . Since f − 1 0 = supp ◦ h is low er semi-contin uous, f 0 is op en. Hence, by [1, Theorem 3.6], X 0 is complete. Moreo ve r, a ll f − 1 0 ( y ) are p erfect sets b ecause h ( y ) are at omless measures. F or the other implication, assume that f : X → Y is a surjection b et ween complete spaces and there exists a complete subspace X 0 ⊂ X suc h that f 0 = f | X 0 is an op en surjection p ossessing p erfect fib ers. Considering X 0 and f 0 | X 0 , w e may supp ose that f is op en and all of its fib ers f − 1 ( y ), y ∈ Y , are p erfect sets. Then, b y Theorem 1.1, f is Milyutin b ecause ˆ P ( f ) has a right inv erse as a soft map. T o show f is atomless, a s in the pro of of Theorem 1.1 tak e a 0-dimensional complete space Z and a p erfect Milyutin map g : Z → Y . Since g is Milyutin, there exists a map g ∗ : ˆ P ( Y ) → ˆ P ( Z ) suc h that ˆ P ( g )  g ∗ ( µ )  = µ for all µ ∈ ˆ P ( Y ). By Theorem 1.1, ˆ P ( f ) is op en (as a soft map). Hence, ˆ f : ˆ P ( f ) − 1 ( Y ) → Y is also op en (as a restriction of an op en map o n to a preimage-set). So, the set-v alued map Φ : Z → ˆ P ( f ) − 1 ( Y ), Φ( z ) = ˆ f − 1 ( g ( z )), is low er semi-con tin uous. Actually , Φ( z ) = ˆ P ( f − 1 ( g ( z ))) f or ev ery z ∈ Z . Let A n , n ≥ 1, b e the set of all µ ∈ ˆ P ( X ) suc h that µ ( { x } ) ≥ 1 / n f o r some p oint x ∈ supp µ . Since the fibers f − 1 ( y ) are p erfect sets, b y Lemma 2.1, A n are closed in ˆ P ( X ) and all in tersections A n ∩ ˆ P ( f − 1 ( y )) are no where dense in ˆ P ( f − 1 ( y )), y ∈ Y . Then, b y [9, Theorem 1.2], Φ admits a selection θ : Z → ˆ P ( f ) − 1 ( Y ) suc h that 8 θ ( z ) ∈ Φ( z ) \ S ∞ n =1 A n , z ∈ Z . This means that eac h measure θ ( z ) ∈ ˆ P ( f − 1 ( g ( z ))) is atomless. The selection θ g enerates a regular op erator u : C ∗ ( X ) → C ∗ ( Z ) , u ( φ )( z ) = θ ( z )( φ ) for all φ ∈ C ∗ ( X ) and z ∈ Z . Finally , f o r ev ery µ ∈ P β ( Y ) let f ∗ ( µ ) ∈ ˆ P ( X ) b e the measure defined b y f ∗ ( µ )( φ ) = g ∗ ( µ )( u ( φ )), φ ∈ C ∗ ( X ). It is easily seen t hat this definition is correct (i.e., f ∗ ( µ ) ∈ ˆ P ( X )) and f ∗ : P β ( Y ) → ˆ P ( X ) is a con tinuous map. Let us sho w that ˆ P ( f )  f ∗ ( µ )  = µ for ev ery µ ∈ P β ( Y ). It suffices to pro v e that f ∗ ( µ )( α ◦ f ) = µ ( α ) for any α ∈ C ∗ ( Y ). And this is really true b ecause φ = α ◦ f is the constan t α ( y ) on eac h set f − 1 ( y ), y ∈ Y . So, u ( φ )( z ) = θ ( z )( φ ) = α ( y ) for any z ∈ g − 1 ( y ). Thus, u ( φ ) = α ◦ g and f ∗ ( µ )( α ◦ f ) = g ∗ ( µ )( α ◦ g ). Finally , since ˆ P ( g )  g ∗ ( µ )  = µ , we ha ve g ∗ ( µ )( α ◦ g ) = µ ( α ). So, it remains to pro v e only that ev ery f ∗ ( µ ), µ ∈ P β ( Y ), is an atom- less measure. T o this end, fix µ 0 ∈ P β ( Y ), x 0 ∈ sup p f ∗ ( µ 0 ) a nd η > 0. It suffices to find a f unction φ 0 ∈ C ∗ ( X ) with 0 ≤ φ 0 ≤ 1 such that φ 0 ( x 0 ) = 1 and f ∗ ( µ 0 )( φ 0 ) ≤ η . Since θ ( z )( { x 0 } ) = 0, fo r every z ∈ Z there exists φ z ∈ C ∗ ( X ) and a neigh b orho o d U z of z in Z suc h that 0 ≤ φ z ≤ 1, φ z ( x 0 ) = 1 and θ ( z ′ )( φ z ) < η whenev er z ′ ∈ U z . Using the compactness o f g − 1 ( supp µ 0 ) ( r ecall that µ 0 has a compact sup- p ort and g is a p erfect map), w e find neigh b orho o ds U z ( i ) , i = 1 , .., k , co ve ring g − 1 ( supp µ 0 ), and let φ 0 = φ z (1) · φ z (2) · .. · φ z ( k ) . Then φ 0 is as required. Indeed, since ˆ P ( g )  g ∗ ( µ 0 )  = µ 0 , g − 1 ( supp µ 0 ) contains the support of g ∗ ( µ 0 ). Conseque n tly , g ∗ ( µ 0 )( u ( φ 0 )) ≤ max { u ( φ 0 )( z ) : z ∈ g − 1 ( supp µ 0 ) } . So, there exists z 0 ∈ g − 1 ( supp µ 0 ) such that g ∗ ( µ 0 )( u ( φ 0 )) ≤ u ( φ 0 )( z 0 ). Next, c ho ose j with z 0 ∈ U z ( j ) and observ e that φ 0 ≤ φ j implies u ( φ 0 )( z 0 ) ≤ u ( φ j )( z 0 ) = θ ( z 0 )( φ j ). Therefore, f ∗ ( µ 0 )( φ 0 ) ≤ θ ( z 0 )( φ j ) < η b ecause z 0 ∈ U z ( j ) . The pro of is completed. Pro of of Theorem 1.3 . T ak e a 0-dimensional complete space Z , a p erfect Milyutin map g : Z → Y and a map g ∗ : ˆ P ( Y ) → ˆ P ( Z ) whic h is a rig h t in verse of ˆ P ( g ). W e equip ˆ P ( X ) with a conv ex metric ˆ d , and let A n , n ≥ 1, b e the closed subsets of ˆ P ( X ) considered in the pro of of Theorem 1.2. W e need to show that the set A o f all atomless c hoice maps form a dense G δ -subset of C h f ( Y , X ). Since each A n is closed in ˆ P ( X ), it is easily seen that the sets U n = { h ∈ C h f ( Y , X ) : h ( y ) 6∈ A n for all y ∈ Y } are op en in C h f ( Y , X ) and A = T n ≥ 1 U n . T o prov e that A is dense in C h f ( Y , X ), fix h ∈ C h f ( Y , X ) and a function η : Y → (0 , ∞ ). W e 9 are going to find a map h ′ ∈ A suc h that ˆ d ( h ( y ) , h ′ ( y )) ≤ η ( y ) for all y ∈ Y . Denote by B ( h ( g ( z )) , η ( g ( z ))) the op en ball in ˆ P ( X ) (with resp ect to ˆ d ) which is cen tered at h ( g ( z )) and has a radius η ( g ( z )). Define the set- v alued map Φ : Z → ˆ P ( X ), Φ( z ) = ˆ P ( f − 1 ( g ( z ))) ∩ B ( h ( g ( z )) , η ( g ( z ))). This is a conv ex and closed-v alued map b ecause an y ball in ˆ P ( X ) with resp ect to ˆ d is conv ex. Since ˆ f = ˆ P ( f ) |  ˆ P ( f ) − 1 ( Y )  is op en (as a soft map, see Theorem 1.1), the set-v alued map z 7→ ˆ P ( f ) − 1 ( g ( z )) is lo wer semi-con tinu ous. Hence, b y [10, Prop osition 2.5], so is Φ. More- o ver, eac h Φ( z ) is the closure of the conv ex op en set ˆ P ( f − 1 ( g ( z ))) ∩ B ( h ( g ( z )) , η ( g ( z ))) in ˆ P ( f − 1 ( g ( z ))). Hence, according to Lemma 2.1, A n ∩ Φ( z ) , n ≥ 1, are no where dense sets in Φ( z ) for ev ery z ∈ Z . Then, by [9, Theorem 1.2], Φ has a contin uous selection θ : Z → ˆ P ( X ) a voiding the set S ∞ n =1 A n , i.e., with θ ( z ) ∈ Φ( z ) \ S ∞ n =1 A n for ev ery z ∈ Z . F ollo wing the no tations from the pro of of Theorem 1.2, w e ex- tend θ to a map ¯ θ : P β ( Z ) → ˆ P ( X ) b y ¯ θ ( ν )( φ ) = ν ( u ( φ )), φ ∈ C ∗ ( X ). No w let h ′ : Y → ˆ P ( X ) b e the comp osition ¯ θ ◦ g ∗ . It fo llows from the pro of of Theorem 1.2 that h ′ ( y ) is atomless and h ′ ( y ) ∈ ˆ P ( f − 1 ( y )) for all y ∈ Y . So, h ′ ∈ A . It remains to sho w tha t ˆ d ( h ( y ) , h ′ ( y )) ≤ η ( y ), y ∈ Y . T o this end, w e fix y ∈ Y and take a sequence { ν n } ⊂ P β ( g − 1 ( y )) conv erging to g ∗ ( y ) suc h that eac h ν n has a finite supp ort. It is easily seen that if ν = P i = k i =1 t i δ z ( i ) ∈ P β ( g − 1 ( y )) is a measure with a finite supp ort, then ¯ θ ( ν ) = P i = k i =1 t i θ ( z ( i )). Since ˆ d ( θ ( z ( i )) , h ( y )) ≤ η ( y ) for all i and the metric ˆ d is conv ex, w e hav e ˆ d ( ¯ θ ( ν ) , h ( y )) ≤ η ( y ). In pa rticular, ˆ d ( ¯ θ ( ν n ) , h ( y )) ≤ η ( y ) for ev ery n . This implies that ˆ d ( h ′ ( y ) , h ( y )) ≤ η ( y ) b ecause h ′ ( y ) is the limit o f the sequenc e { ¯ θ ( ν n ) } . 4. Exact Mil yutin maps In this section t he pro of s of Theorem 1.4 a nd Corollaries 1.5 - 1.6 a re established. Lemma 4.1. L et U ⊂ X b e a non-empty op en set i n a sp ac e X . The n the set ˆ U = { ν ∈ ˆ P ( X ) : supp ν ∩ U 6 = ∅ } is op en c onvex and dense in ˆ P ( X ) . Pr o of. Since the support map ν → sup p ν is a lo w er semi-con tin uo us map, ˆ U ⊂ ˆ P ( X ) is op en. T o show it is dense, supp ose there exists an o p en set W = { ν ∈ ˆ P ( X ) : | ν ( φ i ) − ν 0 ( φ i ) | < ε, 1 ≤ i ≤ k } in ˆ P ( X ) with W ⊂ ˆ P ( X ) \ ˆ U , where φ i ∈ C ∗ ( X ) and ε > 0. W e can 10 supp ose that ν 0 has a finite supp ort (recall that t he measures with a finite supp ort f orm a dense set in ˆ P ( X )). Let ν 0 = P j = m j =1 λ j δ x ( j ) suc h that λ j > 0 and P j = m j =1 λ j = 1 . The n supp ν 0 = { x ( j ) : 1 ≤ j ≤ m } ⊂ X \ U . No w, let ν ′ = λ 0 δ x (0) + ( λ 1 − λ 0 ) δ x (1) + P j = m j =2 λ j δ x ( j ) , where x 0 ∈ U and 0 < λ 0 < λ 1 suc h that λ 0 | φ i ( x 0 ) − φ i ( x 1 ) | < ǫ for ev ery i = 1 , 2 , .., k . The c hoice of λ 0 yields that ν ′ ∈ W . Consequen tly , ν ′ 6∈ ˆ U and su pp ν ′ ⊂ X \ U . This contradicts x 0 ∈ U ∩ supp ν ′ . T o sho w ˆ U is con v ex, it suffices to prov e that supp  tν 1 + (1 − t ) ν 2  = supp ν 1 ∪ su pp ν 2 for an y ν 1 , ν 2 ∈ ˆ P ( X ) and any t ∈ (0 , 1). Ob viously , supp ν 1 ∪ supp ν 2 ⊃ supp  tν 1 + (1 − t ) ν 2  . Assume x ∈ supp ν 1 . Then for ev ery neighborho o d V x of x there exists a function φ x ∈ C ∗ ( X ) with φ x ( X \ V x ) = 0 and ν 1 ( φ x ) 6 = 0. Since ν 1 ( φ x ) = ν 1 ( φ + x ) − ν 1 ( φ − x ), where φ + x and φ − x are the p ositiv e and negativ e parts of φ x , w e can supp ose φ x is non-negativ e. Then, ν ( φ x ) ≥ ν 1 ( φ x ) > 0 with ν = ν = tν 1 + (1 − t ) ν 2 . Hence, x ∈ su pp ν whic h completes the pro of.  Pro of of Theorem 1.4. Cho o se a coun table base { V n : n ≥ 1 } for the top olo gy of M , and let B n = { ν ∈ ˆ P ( X ) : su pp ν ∩ π − 1 ( V n ) = ∅ } . By Lemma 4 .1 , each B n is closed in ˆ P ( X ). Let B b e the set of all maps h ∈ C h f ( Y , X ) suc h that π ( supp h ( y )) is dense in π ( f − 1 ( y )) for an y y ∈ Y . Obv iously , B = T n ≥ 1 G n , where G n = { h ∈ C h f ( Y , X ) : h ( y ) 6∈ B n for all y ∈ Y } . It suffices to show that eac h G n is op en and dense in C h f ( Y , X ) with resp ect to the source limitation to p ology . Claim 1 . Each G n is op en in C h f ( Y , X ) . W e can supp ose that each V n is of the form V n = g − 1 n (0 , ∞ ) fo r some non-negativ e function g n ∈ C ∗ ( M ). Then ν ∈ B n if and only if ν ( g n ◦ π ) = 0, n ≥ 1. Ob viously the equality D n ( µ, µ ′ ) = ˆ d ( µ, µ ′ ) + | µ ( g n ◦ π ) − µ ′ ( g n ◦ π ) | , where µ, µ ′ ∈ ˆ P ( X ) and ˆ d is a compatible metric on ˆ P ( X ), defines a compatible metric on ˆ P ( X ) for ev ery n ≥ 1. Giv en h ∈ G n w e consider the con tin uous function α : Y → (0 , ∞ ), α ( y ) = h ( y )( g n ◦ π ) / 2 . W e hav e B D n ( h, α ) ⊂ G n . Indeed, if h ′ ∈ B D n ( h, α ), then | h ′ ( y )( g n ◦ π ) − h ( y )( g n ◦ π ) | ≤ D n ( h ( y ) , h ′ ( y )) < α ( y ) for all y ∈ Y . The last inequalit y implies h ′ ( y )( g n ◦ π ) > α ( y ) > 0, y ∈ Y . Hence, h ′ ( y ) 6∈ B n for all y ∈ Y . So, h ′ ∈ G n whic h completes t he pro of of Claim 1. T o sho w that any G n is dense in C h f ( Y , X ), we fix m ≥ 1 , h ∈ C h f ( Y , X ) and a function η : Y → (0 , ∞ ). W e are going to find a map h ′ ∈ G m with ˆ d ( h ′ ( y ) , h ( y )) ≤ η ( y ) for a ll y ∈ Y . T o this end, follo wing the pr o of of Theorems 1.2 and 1.3, tak e a complete 0-dimensional space Z and a p erfect Milyutin map g : Z → Y with 11 a right in vers e g ∗ : Y → P β ( Z ) . W e also consider the low er semi- con tinuous con v ex and closed-v a lued map Φ : Z → ˆ P ( X ), Φ( z ) = ˆ P ( f − 1 ( g ( z ))) ∩ B ( h ( g ( z )) , η ( g ( z ))). According to Lemma 4.1 , B m ∩ ˆ P ( f − 1 ( g ( z ))) is a closed now here dense subsets of ˆ P ( f − 1 ( g ( z ))) for ev- ery z ∈ Z . Hence, all B m ∩ Φ( z ) are closed a nd no where dense in Φ( z ). Then, by [9, Theorem 1.2], Φ has a contin uous selection θ : Z → ˆ P ( X ) suc h that θ ( z ) ∈ Φ( z ) \ B m , z ∈ Z . As in the pr o of of Theorem 1.3, let h ′ : Y → ˆ P ( X ) b e the comp osition ¯ θ ◦ g ∗ , where ¯ θ : P β ( Z ) → ˆ P ( X ) is an extension of θ defined b y ¯ θ ( ν )( φ ) = ν ( u ( φ )), φ ∈ C ∗ ( X ). F ollo wing the argumen ts from Theorem 1.3 , w e can show that ˆ d ( h ′ ( y ) , h ( y )) ≤ η ( y ) for all y ∈ Y . Next claim completes the pr o of of Theorem 1.4. Claim 2 . h ′ ( y ) 6∈ B m for any y ∈ Y . The pro of of this claim is reduced to find a function φ y ∈ C ∗ ( X ) suc h that φ y  X \ π − 1 ( V m )  = 0 and h ( y )( φ y ) 6 = 0. Inde ed, in suc h a case supp h ( y ) ∩ π − 1 ( V m ) 6 = ∅ . Since θ ( z ) 6∈ B m for all z ∈ g − 1 ( y ), supp θ ( z ) ∩ π − 1 ( V m ) 6 = ∅ . Consequen tly , for an y z ∈ g − 1 ( y ) there ex- ists a function φ z ∈ C ∗ ( X ) with φ z  X \ π − 1 ( V m )  = 0 and θ ( z )( φ z ) 6 = 0. Considering the p ositiv e or nega t ive parts of φ z , w e ma y assume each φ z ≥ 0. Next, use the con tin uity of θ and the compactness of g − 1 ( y ) to find finitely many p oin ts z ( i ) ∈ g − 1 ( y ), i = 1 , 2 , .., k , and neigh b or- ho o ds U z ( i ) suc h that θ ( z )( φ z ( i ) ) > 0 provide d z ∈ U z ( i ) . Finally , let φ y = P i = k i =1 φ z ( i ) . Then φ y  X \ π − 1 ( V m )  = 0 and u ( φ y )( z ) = θ ( z )( φ y ) > 0 for an y z ∈ g − 1 ( y ). So, h ( y )( φ y ) ≥ min { u ( φ y )( z ) : z ∈ g − 1 ( y ) } > 0 b ecause g − 1 ( y ) is compact. This completes the pro of of the claim. Pro of of Co rollary 1.5. Since f is closed with separable fib ers, there exists a map π : X → Q suc h that all restrictions π | f − 1 ( y ), y ∈ Y , are em b eddings, see [13]. Here, Q is the Hilb ert cub e. Then, b y Theorem 1.4 (with M replaced b y Q ), f is densely exact. If, in additio n, t he fib ers o f f are p erfect, b oth Theorems 1.3 and 1.4 imply that f is densely exact atomless. Pro of of Cor ollary 1.6. Consider the graph G (Φ) = ∪{{ y } × Φ( y ) : y ∈ Y } ⊂ Y × X of Φ and the pro jection f : G (Φ) → Y . Since Φ is con tinuous, G (Φ) is closed in Y × X and f is b oth op en and closed. Then G (Φ) is a complete space. No w, by Corollary 1.5, there exists a map h ′ : Y → ˆ P ( G (Φ)) with eac h h ′ ( y ) ∈ ˆ P ( f − 1 ( y ) b eing exact measure. Therefore, supp h ′ ( y ) = f − 1 ( y ). Let h = ˆ P ( π ) ◦ h ′ , where π : G (Φ) → X is the pro jection in to X . Since π em b eds eac h f − 1 ( y ) on to Φ( y ), h is a map from Y into ˆ P ( X ) suc h that supp h ( y ) = Φ( y ) for ev ery y ∈ Y . If Φ( y ) are p erfect sets, so are the fib ers f − 1 ( y ), and 12 h ′ can b e chosen to b e atomless and exact. In suc h a case h is also atomless. Note added in pro of. Recen tly T. Ba nakh informed the a ut ho r that V. Bogac hev a nd A. Kolesnik o v [5] pro ved the follo wing result: The ma p ˆ P ( f ) from Theorem 1.1 is op en. This, in com bination with Mic hael’s con v ex-v alued selection theorem [10], pro vides anot her pro of of Theorem 1.1. Reference s [1] S. Ageev a nd E. Tymc hatyn, On exact atomless Mil utin maps , T opolo gy Appl. 153, 2-3 (2005)), 227 -238 . [2] T. B anakh, T op olo gy of pr ob ability me asur e sp ac es I , Ma t. Studii 5 (19 95), 65-87 (in Russian). [3] T. Banakh, T op olo gy of pr ob ability me asur e sp ac es II , Ma t. Studii 5 (1995), 88-10 6 (in Russian). [4] T. Banach and T. Radul, Ge ometry of mappings of pr ob ability me asur e sp ac es , Mat. Studii 11 (1999), 17 -30 (in Rus s ian). [5] V. Bogachev and A. 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Pasynko v, On ge ometry of c ont inu ous maps of c ount able functional weight , F undam. Prikl. Ma tematik a 4, 1 (199 8), 155– 164 (in Rus s ian). [14] D. Rep ov ˇ s, P . Se menov a nd E. ˇ Sˇ cepin, On exact Milyutin mappings , T o po logy Appl. 81 , (1997 ), 197-20 5. [15] V. V aradara jan, Me asur es on top olo gic al sp ac es , Mat. Sb. 55, 1 (1961), 35- 100 (in Russia n). Dep ar tment of Computer Science and Ma thema tics, Nipissing Uni- versity, 100 College Drive, P.O. Box 500 2, Nor th Ba y, ON, P1B 8L7, Canada E-mail addr ess : vesk ov@ni pissin gu.ca