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SURJECTIVE LE´VY–PROKHOROV ISOMETRIES 7 GYO¨RGYPA´LGEHE´RANDTAMA´STITKOS 1 0 2 n Abstract. AccordingtothefundamentalworkofYu.V.Prokhorov,thegen- a J eraltheoryofstochasticprocessescanberegardedasthetheoryofprobability measures in complete separable metric spaces. Since stochastic processes de- 6 pending upon a continuous parameter are basically probability measures on 1 certainsubspacesofthespaceofallfunctionsofarealvariable,aparticularly importantcaseofthistheoryiswhentheunderlyingmetricspacehasalinear ] A structure. Prokhorov also provided a concrete metrisation of the topology of weak convergence today known as the Lévy–Prokhorov distance. Motivated F by these facts and some recent works related to the characterisation of onto h. isometries of spaces ofBorel probabilitymeasures, herewe givethe complete t description of the structure of surjective Lévy–Prokhorov isometries on the a spaceofallBorelprobabilitymeasuresonanarbitraryseparablerealBanach m space. Our result can be considered as a generalisation of L. Molnár’s ear- [ lier result which characterises surjective Lévy isometries of the space of all probabilitydistributionfunctionsontherealline. However,thepresentmore 1 general settingrequiresthedevelopment ofanessentiallynewtechnique. v 7 6 1. Introduction 2 4 There is a long history and vast literature of isometries (distance preserving 0 maps) on different kind of metric spaces. Two classical results in the case of . normed linear spaces are the Mazur–Ulam theorem which states that every sur- 1 0 jective isometry between real normed spaces is automatically affine (i.e. linear up 7 totranslation),andtheBanach–Stonetheoremwhichprovidesthestructureofonto 1 linear isometries between Banach spaces of continuous scalar-valued functions on : v compact Hausdorff spaces. Since then severalproperties of surjective linear isome- i tries on different types of normed spaces have been explored, see for instance the X papers [2–5,10,13,24]andthe extensive books [16,17]. The reader can find similar r a results on non-linear spaces for example in [6,11,18,21,27,28]. ThestartingpointofourinvestigationisMolna´r’spaper[25]wherethecomplete description of surjective Lévy isometries of the non-linear space D(R) of all cumu- lative distribution functions was given. If F,G∈D(R), then their Lévy distance is 2000MathematicsSubjectClassification. Primary: 46B04,46E27,47B49,54E40,60B10;Sec- ondary: 28A33,60A10, 60B05. Keywordsandphrases. Borelprobabilitymeasures;Weakconvergence;Lévy–Prokhorovmet- ric;Isometries. Gyo¨rgy Pál Gehér was also supported by the “Lendu¨let” Program (LP2012-46/2012) of the Hungarian Academy of Sciences, and by the Hungarian National Research, Development and InnovationOffice–NKFIH(grantno.K115383). Tamás Titkos was also supported by the “Lendu¨let” Program (LP2012-46/2012) of the Hun- garianAcademyofSciences,andbytheNationalResearch,DevelopmentandInnovationOffice– NKFIH(grantno.K104206). 1 2 GY.P.GEHE´RANDT.TITKOS defined by the following formula: L(F,G):=inf ε>0 ∀t∈R: F(t−ε)−ε≤G(t)≤F(t+ε)+ε . The importance of thi(cid:8)s metri(cid:12)c lies in the fact that it metrises the topology(cid:9)of weak convergence on D(R). Moln(cid:12)a´r’s result reads as follows (see [25, Theorem 1]): let Φ: D(R)→D(R) be a surjective Lévy isometry, i.e., a bijective map satisfying L(F,G)=L(Φ(F),Φ(G)) (∀F,G∈D(R)). Then there is a constant c∈R such that Φ is one of the following two forms: Φ(F)(t)=F(t+c) (∀t∈R,F ∈D(R)), or Φ(F)(t)=1− lim F(−s+c) (∀t∈R,F ∈D(R)). s→t− In other words, every surjective Lévy isometry is implemented by an isometry of Rwith respectto itsusualnorm(orequivalently,bya compositionofa translation and a reflection on R). The investigation of surjective isometries on spaces of Borel probability measures was continued for example in [15,26] for the Kolmogorov–Smirnov distance which is important in the Kolmogorov–Smirnov statistic and test, and in [7,8,20] with respect to the Wasserstein (or Kantorovich) metric which metrises the weak convergence. Let (X,d) be a complete and separable metric space. We will denote the σ- algebra of Borel sets on X by B and the set of all Borel probability measures X by P . The Lévy distance gives a metrisation of weak convergence on D(R), or X equivalently on PR. In 1956 Prokhorov managed to metrise the weak convergence ofP forgeneralcompleteandseparablemetricspaces(X,d). Theso-calledLévy– X Prokhorov distance which was introduced by him in [30] is defined by (1.1) π(µ,ν):=inf ε>0 ∀A∈B : µ(A)≤ν(Aε)+ε , X where (cid:8) (cid:12) (cid:9) Aε := B (x) and(cid:12) B (x):={z ∈X|d(x,z)<ε}. ε ε x∈A [ Forthe details andelementary propertiessee e.g.[19,p. 27]. Let us point outthat in the special case when X =R this metric differs from the originalLévy distance. Here arises the following very natural question: What is the structure of onto Lévy–Prokhorov isometries on P X if X is a general separable real Banach space? Thispaperisdevotedtogivetheanswertothisquestion. Namely,wewillprovethat every such transformation is implemented by an affine isometry of the underlying space X. There are some particularly important cases in our investigation which we em- phasise now. Namely, since stochastic processes depending upon a continuous parameter are basically probability measures on certain subspaces of the space of all functions of a real variable (see e.g. [1,14]), one particularly interesting case is when the underlying Banach space is C([0,1]), i.e. the space of all continuous real-valued functions on [0,1] endowed with the uniform norm k·k . For details ∞ see [30, Chapter 2] or [9, Chapter 2]. Further two important cases are when X is a Euclidean space because of multivariate random variables, and when X is an SURJECTIVE LE´VY–PROKHOROV ISOMETRIES 3 infinite dimensional, separable real Hilbert space because of the theory of random elements in Hilbert spaces. 2. The setting and the statment of our main result In this sectionwe state the mainresultofthe paper andcollectsomedefinitions and well-known facts about weak convergence of Borel probability measures. For more details the reader is referred to the textbooks of Billingsley [9], Huber [19] and Parthasarathy[29]. Let (X,d) be a complete, separable metric space and denote by C (X;R) the b Banach space of all real-valued bounded continuous functions. Recall that B is X the smallest σ-algebra with respect to each f ∈ C (X;R) is measurable. We say b that an element of P is a Dirac measure if it is concentrated on one point, and X foranx∈X thesymbolδ standsforthecorrespondingDiracmeasure. Thesetof x all Dirac measures on X is denoted by ∆ . The collection of all finitely supported X measures is F := λ δ #I <ℵ , λ =1, λ >0, x ∈X (∀i∈I) , X i xi 0 i i i ( ) Xi∈I (cid:12) Xi∈I (cid:12) which is actually the conv(cid:12)ex hull of ∆X. The support (or spectrum) of µ ∈ PX is the smallest d-closed set S that satisfies µ(S ) = 1. Moreover, it is not hard to µ µ verify the following equation: S = x∈X ∀r >0: µ(B (x))>0 . µ r The closure of a set H ⊆X(cid:8)will be(cid:12)denoted by H. (cid:9) (cid:12) Wesaythatasequenceofmeasures{µ }∞ ⊂P convergesweakly toaµ∈P n n=1 X X if we have f dµ → f dµ (∀f ∈C (X;R)). n b Z Z This type ofconvergenceis metrisedby the Lévy–Prokhorovmetric givenby (1.1). A map ϕ: P →P is called a Lévy–Prokhorov isometry on P if X X X π(µ,ν)=π(ϕ(µ),ϕ(ν)) (∀µ,ν ∈P ) X is satisfied. Now, we are in the position to state the main result of this paper. MainTheorem. Let(X,k·k)beaseparablerealBanachspaceandϕ: P →P be X X asurjectiveLévy–Prokhorovisometry. Thenthereexistsasurjectiveaffineisometry ψ: X →X which implements ϕ, i.e. we have (2.1) (ϕ(µ))(A)=µ(ψ−1[A]) (∀A∈B ), X where ψ−1[A] denotes the inverse-image set {ψ−1(a)|a∈A}. The converse of the above statement is trivial, namely, every transformation of the form (2.1) is obviously an onto Lévy–Prokhorov isometry. Note that our theorem can be re-phrasedin terms of push-forwardmeasures. Namely, the action of ϕ is just the push-forwardwith respect to the isometry ψ: X →X. As we already mentioned, the Lévy–Prokhorov metric on PR differs from the Lévy distance on PR. Therefore, in the special case when X = R, our hypotheses are different from those given in [25, Theorem 1], although the conclusion is the same. 4 GY.P.GEHE´RANDT.TITKOS Our proofis giveninthe next sectionwherewe will havefour majorsteps. This will be followed by some remarks in the final section, where we will also point out thatourMainTheoremstillholdsifwereplaceP with anarbitraryweaklydense X subset S. 3. Proof The proofis divided into four majorsteps. First,we willexplore the actionofϕ on∆ . Thenforfinitelysupportedmeasuresµwewillinvestigatethebehaviourof X itsimageϕ(µ)neartotheverticesoftheconvexhullofS . Thiswillbefollowedby µ providing a procedure which will allow us to obtain important information about the “rest” of ϕ(µ). Finally, we close this section with the proof of the Main Theo- rem. Note that although our main result deals with Borelprobability measures on separableBanachspaces,westateandprovesomeresultsinthecontextofcomplete and separable metric spaces. 3.1. Firstmajorstep: theactiononDiracmeasures. Herewewillinvestigate propertiesoftherestrictedmapϕ| . Namely,wewillprovethatϕmaps∆ onto DX X ∆ , furthermore, there is a surjective affine isometry of X which implements this X restriction. Inorderto do this first, we formulatethe metric phrase “distanceone” by means of the supports of measures. Proposition 3.1. Let (X,d) be a complete, separable metric space and µ,ν ∈P . X Then the following statements are equivalent: (i) π(µ,ν)=1, (ii) d(S ,S ):=inf{d(x,y)|x∈S , y ∈S }≥1, µ ν µ ν (iii) S ∩S1 =∅, ν µ (iv) S ∩S1 =∅. µ ν Proof. Observe that (ii) implies the following inequality for every 0<ε<1: 1=µ(S )>ν(Sε)+ε=ε. µ µ Consequentlywehaveπ(µ,ν)≥1. Butonthe otherhand, π(µ,ν)≤1holdsfor all µ,ν ∈P , and therefore the (ii)⇒(i) part is complete. X To prove (i)⇒(ii) assume that ̺ := d(S ,S ) < 1. In this case one can fix two µ ν points x∗ ∈S and y∗ ∈S , and a positive number r >0 which satisfy both µ ν ̺≤d(x∗,y∗)=:̺′ <1 and ̺′+2r <1. We also set t := min{µ(B (x∗)),ν(B (y∗))} which is clearly positive by the very r r definition of the support. We will show that εˆ := max{1−t,̺′ +2r} < 1 is a suitable choice to guarantee µ(A)≤ν(Aεˆ)+εˆ (∀A∈B ) X Indeed, if A∈B satisfies µ(A)≤1−t, then X µ(A)≤1−t≤ν(A1−t)+1−t≤ν(Aεˆ)+εˆ. On the other hand, if µ(A) > 1− t, then we observe that µ(A ∩B (x∗)) > 0, r and consequently A∩B (x∗) is not empty. Let us fix a point z ∈ A∩B (x∗). r r Using the triangle inequality we infer d(y∗,z) ≤ d(y∗,x∗)+d(x∗,z) < ̺′ +r and Br(y∗)⊆B̺′+2r(z)⊆Aεˆ. Therefore we conclude µ(A)≤1≤t+εˆ≤ν(B (y∗))+εˆ≤ν(Aεˆ)+εˆ, r SURJECTIVE LE´VY–PROKHOROV ISOMETRIES 5 which implies π(µ,ν)≤εˆ<1. The equivalence of (ii), (iii) and (iv) follows from the definitions. (cid:3) Next, let us define the unit distance set of a set of measures A ⊆P by X A = ν ∈P ∀µ∈A : π(µ,ν)=1 . u X (Remark that by definition(cid:8)we have(cid:12)∅u = PX.) The follow(cid:9)ing statement gives a (cid:12) metriccharacterisationofDiracmeasureswhenX isaseparablerealBanachspace. We point out that similar results were also crucial ideas in [15,25,26]. Proposition 3.2. Let (X,d) be a complete, separable metric space and µ∈P be X an arbitrary Borel probability measure on it. Then the following three statements are equivalent: (i) ({µ}u)u ={µ}, (ii) there exists an x∈X such that (ii/a) µ=δ , and x (ii/b) B (y)⊆B (x) implies x=y for every y ∈X, 1 1 (iii) #({µ}u)u =1. Proof. First,letuscharacterisetheelementsof({µ}u)u. ItfollowsfromProposition 3.1 that {µ} = ν ∈P S1∩S =∅ = ν ∈P S ∩S1 =∅ . u X ν µ X ν µ Applying this observ(cid:8)ation tw(cid:12)ice, we easily(cid:9)see(cid:8)that (cid:12) (cid:9) (cid:12) (cid:12) (3.1) ({µ}u)u = {ν}u = ϑ∈PX Sν ∩Sϑ1 =∅ , ν∈\{µ}u ν∈PX,\Sν∩Sµ1=∅(cid:8) (cid:12) (cid:9) (cid:12) and therefore we obtain the following equivalence: (3.2) ϑ∈({µ}u)u ⇐⇒ Sϑ1 ⊆Sµ1. (Note that if {µ} =∅, then {ν} =P in (3.1) by definition.) u ν∈∅ u X Now,sinceµ∈({µ}u)u alwaysholds,theequivalenceof(i)and(iii)isapparent. T We continue with proving (i)⇒(ii). Observe that (3.2) implies δz z ∈Sµ ⊆({µ}u)u, thus (ii/a) follows. On the ot(cid:8)her(cid:12)hand, i(cid:9)f any y ∈X satisfies (cid:12) S1 =B (y)⊆B (x)=S1 , δy 1 1 δx then again by (3.2) we infer x=y. Finally, we show (ii)⇒(i). Assume that ϑ ∈ ({µ}u)u. By (ii/b) we get that S1 ⊆S1 holds if and only if S ⊆{x}, which implies (i). (cid:3) ϑ δx ϑ Remark 3.3. Note that if the diameter of the metric space X is less than 1, i.e. there exists an 0<r <1 such that d(x,y)≤r (∀x,y ∈X), then π(µ,ν)≤r holds for every µ,ν ∈ PX. In particular, {µ}u = ∅ and thus ({µ}u)u = PX for every µ∈P . X The following lemma describes the action of ϕ on Dirac measures. Lemma3.4. Let(X,k·k)beaseparablerealBanachspace,andletϕ: P →P be X X asurjectiveLévy–Prokhorovisometry. Thenthereexistsasurjectiveaffineisometry ψ: X →X such that (3.3) ϕ(δ )=δ (∀x∈X). x ψ(x) 6 GY.P.GEHE´RANDT.TITKOS Proof. Since ϕ is a bijective isometry, we have ϕ ({µ}u)u = {ϕ(µ)}u u (∀µ∈PX). Thus an easy applica(cid:0)tion of th(cid:1)e pr(cid:0)evious pr(cid:1)oposition yields ϕ(∆X) = ∆X. This also means that there exists a bijective map ψ: X → X which implements the restriction ϕ| , i.e. ∆X (3.4) ϕ(δ ):=δ (∀x∈X). x ψ(x) We will show that ψ is an isometry. Observe that π(δ ,δ )=min{1,kx −x k} (∀x ,x ∈X). x1 x2 1 2 1 2 Therefore for all α∈(0,1) we have (3.5) kψ(x )−ψ(x )k=α ⇐⇒ kx −x k=α (∀x ,x ∈X). 1 2 1 2 1 2 IfX isone-dimensional,thenitisrathereasytoseethat(3.5)impliestheisometri- ness of ψ. Now, assume that dimX ≥ 2. After suitable renorming, from a result of T.M. Rassias and P. Sˇemrl [31, Theorem 1] we conclude kψ(x)−ψ(y)k=nα ⇐⇒ kx−yk=nα (∀α∈(0,1),n∈N), andthereforeψisindeedanisometry. Finally,bythefamousMazur–Ulamtheorem we obtain that ψ is affine, which completes the proof. (cid:3) Weremarkthatthelaststepoftheproof(usingtheRassias–Sˇemrltheorem)can be also done by the extension theorem of Mankiewicz [23]. In light of the above lemma, from now on we may and do assume without loss of generality that ϕ acts identically on ∆ , i.e., X (3.6) ϕ(δ )=δ (∀x∈X), x x andouraimwillbetoshowthatϕactsidenticallyonthewholeofP . Afterwedo X so,toobtaintheresultofourMainTheoremforgeneralsurjectiveLévy–Prokhorov isometrieswillbestraightforward. Namely,if (3.3)isfulfilled,thenwecanconsider the following modified transformation: (3.7) ϕ : P →P , (ϕ (µ))(A):=(ϕ(µ))(ψ[A]) (∀µ∈P ,A∈B). ψ X X ψ X By our assumption, ϕ fixes every element of ∆ and thus also of P , which ψ X X implies (2.1). Next, let us define the following continuous function for each µ∈P : X W : X →[0,1], W (x):=π(δ ,µ) µ µ x whichwill be calledthe witness function ofµ. The main advantageofthe assumption (3.6) is that the witness function becomes ϕ-invariant, i.e. (3.8) W (x)=π(δ ,µ)=π(ϕ(δ ),ϕ(µ))=π(δ ,ϕ(µ))=W (x) (∀x∈X). µ x x x ϕ(µ) Itis naturalto expect that the shape ofthe witness function carriessomeinforma- tion about the measure. The last three major steps of the proof will be devoted to explore this for the ϕ-images of finitely supported measures in the setting of separable Banach spaces. However, as demonstrated by the next example, the witness function usually does not distinguish measures in general complete and separable metric spaces. SURJECTIVE LE´VY–PROKHOROV ISOMETRIES 7 Example 3.5. Consider the complete and separable metric space (X,d) with 1 if i6=j X :={x ,x ,x } and d(x ,x ):= 3 . 1 2 3 i j 0 if i=j (cid:26) Let µ := 1δ + 1δ and ν := 1δ + 1δ . An easy calculation shows that we 2 x1 2 x2 2 x2 2 x3 have π(δ ,µ)= 1 =π(δ ,ν) for all x∈X and hence W ≡W . x 3 x µ ν 3.2. Second major step: isolated atoms on the vertices of the convex hull of the support. Here we will prove that if µ is a finitely supported measure, and xˆ is a vertex of the convex hull of S , then xˆ is an isolated atom of ϕ(µ) and µ µ({xˆ})=(ϕ(µ))({xˆ}). We begin with a technical statement, which will be very useful in the sequel. Proposition 3.6. Let X be a separable real Banach space, and suppose that µ is a finitely supported measure. Then for every ν ∈P , ν 6=µ we have X (3.9) π(µ,ν)=min ε>0|∀A⊆S : µ(A)≤ν(Aε)+ε . µ Proof. First, we observe that(cid:8)in (1.1) it is enough to consider Bor(cid:9)el sets satisfying A⊆S . Furthermore, it is obvious that µ π(µ,ν)=max inf{ε>0|µ(A)≤ν(Aε)+ε} A⊆S . µ Therefore it is enough to sh(cid:8)ow that for each subset A := {(cid:12)a1,...,a(cid:9)k} ⊆ Sµ if the (cid:12) infimum ε :=inf{ε>0|µ(A)≤ν(Aε)+ε} A k =inf ε>0 µ({a ,...,a })≤ν B (a ) +ε  1 k ε j  (cid:12) !  (cid:12) j[=1  (cid:12) ispositive,thenitis actually(cid:12)aminimumifwetaketheclosureofAε insteadofAε.  (cid:12)  The following line of inequalities holds for all 0<h<ε by definition: A k k ν B (a ) +ε −h<µ({a ,...,a })≤ν B (a ) +ε +h. εA−h j A 1 k εA+h j A ! ! j=1 j=1 [ [ Now, taking the limit of the right-hand side as h →0+, and using that 0 < r <s implies B (x)⊆B (x), we obtain r s k k ν B (a ) +ε −δ <µ({a ,...,a })≤ν B (a ) +ε εA−δ j A 1 k εA j A ! ! j=1 j=1 [ [ for every 0<δ <ε , which proves (3.9). (cid:3) A Note that the reasonwhy we excluded the case when µ=ν in (3.9) is that then for every A⊆ S and ε >0 we have µ(A) ≤ ν(Aε)+ε, and for every ∅ =6 A⊆ S µ µ we have µ(A) > 0 = ν(∅)+0 = ν(A0)+0. Of course if we had defined A0 to be A, then (3.9) with ε ≥ 0 instead of ε > 0 would hold for the case µ = ν as well. However,we prefer not to change usual notations. The following propositionplays a key role in the proof. But before stating it we introducesomenotations. Theconvexhulloftwopointsxandy willbedenotedby [x,y], andthe symbol]x,y[ willstand for the set[x,y]\{x,y}. If f is a realvalued 8 GY.P.GEHE´RANDT.TITKOS functiononX andc∈R,thenthe sets{x∈X|f(x)<c}, {x∈X|f(x)=c},and {x∈X|f(x)≤c} will be denoted by {f <c}, {f =c}, and {f ≤c}, respectively. Proposition 3.7. Let (X,k·k)be a separable real Banach space, µ∈F \∆ and X X K be the convex hull of S . Assume that xˆ is a vertex of K (which is a polytope) µ and set λˆ := µ({xˆ}) (for which we obviously have 0 < λˆ < 1). Then for every ϑ∈P with S ⊆K the following two conditions are equivalent: X ϑ (i) ϑ=λˆδ +(1−λˆ)ϑ where ϑ∈P with S ⊆K\B (xˆ) for some r >0, xˆ X ϑ r (ii) there exist a number 0 < ρ ≤ 1−λˆ and ae half-line e starting from xˆ such that the restrictioneWϑ|e iseof the following form: 1 if kx−xˆk≥1, (3.10) Wϑ|e(x)= kx−xˆk if 1−λˆ <kx−xˆk<1,   1−λˆ if 1−λˆ−ρ≤kx−xˆk≤1−λˆ. Moreover, Sϕ(µ) ⊆K andxˆ is an isolated atom of ϕ(µ) with (ϕ(µ))({xˆ})=λˆ. {f =cˆ} {f =c} e xˇ 1 xˆ S ϑ e Y K Figure 1. Illustration for Proposition 3.7 on the finite dimensional subspace Y. The support S consists of the set of black µ points in K. Proof. First, we construct a half-line e which starts from xˆ and satisfies (3.11) d({x},K)=kx−xˆk<kx−kk (∀x∈e,k ∈K\{xˆ}). Beingtheconvexhullofafinite set,eachvertexofK isstronglyexposed,i.e. there exists a continuous linear functional f ∈X∗ with cˆ:=max{f(y)|y ∈K}=f(xˆ) and K\{xˆ}⊂{f <cˆ}. Let Y be the subspace generated by K. We fix an xˇ∈Y such that xˆ∈B (xˇ) and 1 B (xˇ)∩{f ≤cˆ}∩Y ⊆{f =cˆ}. 1 Note that as Y is finite dimensional, the existence of such an xˇ is guaranteed. Now, we define e to be the half-line starting from xˆ and going through xˇ. It is straightforwardthat e fulfils (3.11). SURJECTIVE LE´VY–PROKHOROV ISOMETRIES 9 Next, we consider an arbitrary ϑ ∈ P which satisfies (i). It is clear form the X compactness of S and S , and the isolatedness of the point xˆ in both S and S , µ ϑ µ ϑ that there is a number c<cˆsuch that (3.12) (S ∪S )\{xˆ}⊆{f <c} µ ϑ holds. Consequently, for every x∈e we have d({x},K)=kx−xˆk>d({x},{f ≤c}∩Y)>d({x},(S ∪S )\{xˆ}). µ ϑ Therefore, if x ∈ e and α := kx−xˆk > 0, then there exists a y ∈ ]x,xˆ[ ⊆ e which satisfies the following equations: (3.13) λˆ =ϑ({xˆ})=ϑ(B (x))=ϑ(B (z)) (∀z ∈[x,y]). α α and (3.14) λˆ =µ({xˆ})=µ(B (x))=µ(B (z)) (∀z ∈[x,y]). α α In fact, y can be chosen to be any point on ]x,xˆ[ such that kx−yk≤d({f =cˆ},{f =c}). We proceed to prove the equivalence of (i) and (ii). Trivially, both (i) and (ii) implies that ϑ∈/ ∆ , and therefore from now on we may and do assume that ϑ is X not a Dirac measure. Recall that according to Proposition 3.6 we have (3.15) W (x)=min ε>0|1≤ϑ(B (x))+ε (∀x∈X). ϑ ε n o If (i) holds, then by combining (3.15) with (3.11) and (3.13) we obtain that Wϑ|e is of the form (3.10) with ρ:=min 1−λˆ,d({f =cˆ},{f =c}) . Conversely,we suppose that ϑ∈(cid:16)PX, Sϑ ⊆K and ϑ satisfies(cid:17)(ii). Let x1 and x2 be the points on e which satisfy kx −xˆk = 1−λˆ and kx −xˆk = 1−λˆ−ρ. By 1 2 (3.10) we have W (x )=W (x )=1−λˆ. Therefore on one hand, we obtain ϑ 1 ϑ 2 1≤ϑ(B (x ))+1−λˆ =ϑ({xˆ})+1−λˆ, 1−λˆ 1 from which λˆ ≤ϑ({xˆ}) follows. On the other hand, 1>ϑ(B (x ))+1−λˆ−δ ≥ϑ({xˆ})+1−λˆ−δ (∀0<δ <ρ) 1−λˆ−δ 2 is satisfied. Hence we infer ϑ({xˆ}) < λˆ +δ for every δ > 0, and thus trivially ϑ({xˆ})=λˆ holds. But we also observe the following: 1>ϑ B1−λˆ−δ(x2) +1−λˆ−δ ≥ϑ B1−λˆ−ρ(x2) +1−λˆ−δ ∀0<δ < ρ2 2 which i(cid:16)mplies (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) λˆ ≥ϑ B1−λˆ−ρ(x2) ≥ϑ({xˆ})=λˆ, 2 whence we conclude that xˆ is(cid:16)indeed an iso(cid:17)lated atom of ϑ. For the last statement first, by Proposition 3.1 we infer π(δ ,µ) = 1 for every x x∈/ K1. The ϕ-invariance of the witness function gives 1=π(δ ,ϕ(µ)) (x∈X \K1), x and hence, again by Proposition 3.1, we conclude S ∩B (x)=∅ (x∈X \K1). ϕ(µ) 1 10 GY.P.GEHE´RANDT.TITKOS Consequently, we obtain S ⊆X\(X \K1)1 ⊆K. ϕ(µ) Finally, an application of the equivalence of (i) and (ii) gives the rest. (cid:3) We have the following consequence. Corollary 3.8. Let (X,k·k) be a separable real Banach space. If µ ∈ F such X that #S ≤ 2, then ϕ(µ) = µ. Moreover, if ν ∈ P with W ≡ W , then µ and ν µ X µ ν coincide. According to the above results, now we know that ϕ fixes every measure which has at most two points in its support. Although we are expecting the same for all µ ∈ F , right now we only have some information about the behaviour of ϕ(µ) X near to the vertices of the convex hull of its support. 3.3. Third major step: the story beyond vertices. Hereweshowaprocedure how the behaviour of ϕ(µ) can be completely explored in case when µ is a finitely supported measure. In order to do so, we need to introduce some technical notations. Let s>0 be a positive parameter and define the s-Lévy–Prokhorov distance π : P ×P →[0,1] by the following formula: s X X (3.16) π (µ,ν):=inf{ε>0|∀A∈B : s·µ(A)≤s·ν(Aε)+ε} (∀µ,ν ∈P ). s X X Note that although we do not know at this point whether π defines a metric on s P , wewillsee this later. The s-witness function (ormodifiedwitness function) of X µ∈P is defined by X W : X →R, W (x):=π (δ ,µ). s,µ s,µ s x Obviously, if we set s = 1, then we get the original Lévy–Prokhorov metric and witness function. In the next two lemmas we collect some properties of the s-Lévy–Prokhorov distance analogous to those provided in the previous major step. Lemma 3.9. Let (X,k·k) be a separable real Banach space and s > 0. Then (P ,π ) is a metric space. Furthermore, for every µ∈F and ν ∈P with ν 6=µ X s X X we have π (µ,ν)=min ε>0 ∀A⊆S : s·µ(A)≤s·ν(Aε)+ε . s µ Proof. For the sake of clar(cid:8)ity, let(cid:12)us use more detailed notations her(cid:9)e. If k·k is a (cid:12) normonX,thendenote byAε,k·k andπ theopenε-neighborhoodofAandthe s,k·k s-Lévy–Prokhorovmetric withrespecttok·k,respectively. Observethatthe Borel σ-algebrasof (X,k·k) and X,1k·k coincide as the norms are equivalent. By an s elementary computation we(cid:0)have Asδ(cid:1),k·k =Aδ,s1k·k, which yields π (µ,ν)=inf ε>0 ∀A∈B : s·µ(A)≤s·ν Aε,k·k +ε s,k·k X =infnsδ >0(cid:12)(cid:12) ∀A∈B : s·µ(A)≤s·ν(cid:0) Asδ,k(cid:1)·k +osδ (cid:12) X n (cid:12) o (3.17) =s·inf δ >(cid:12)0 ∀A∈B : µ(A)≤ν Asδ(cid:0),k·k +(cid:1)δ (cid:12) X =s·infnδ >0(cid:12)(cid:12)(cid:12)∀A∈BX: µ(A)≤ν(cid:0)Aδ,s1k·k(cid:1) +δo =s·π1kn·k(µ,ν)(cid:12)(cid:12) (cid:0) (cid:1) o s (cid:12)