Transformations of Moment Functionals

In measure theory several results are known how measure spaces are transformed into each other. But since moment functionals are represented by a measure we investigate in this study the effects and implications of these measure transformations to moment funcationals, especially with dimensionality reduction. We gain characterizations of moment functionals. Among other things we show that for a compact and path connected set K⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\subset \mathbb {R}^n$$\end{document} there exists a measurable function g:K→[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g:K\rightarrow [0,1]$$\end{document} such that any linear functional L:R[x1,⋯,xn]→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L:\mathbb {R}[x_1,\dots ,x_n]\rightarrow \mathbb {R}$$\end{document} is a K-moment functional if and only if it has a continuous extension to some L¯:R[x1,⋯,xn]+R[g]→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{L}:\mathbb {R}[x_1,\dots ,x_n]+\mathbb {R}[g]\rightarrow \mathbb {R}$$\end{document} such that L~:R[t]→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{L}:\mathbb {R}[t]\rightarrow \mathbb {R}$$\end{document} defined by L~(td):=L¯(gd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{L}(t^d):= \overline{L}(g^d)$$\end{document} for all d∈N0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\in \mathbb {N}_0$$\end{document} is a [0, 1]-moment functional (Hausdorff moment problem). Additionally, there exists a continuous function f:[0,1]→K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:[0,1]\rightarrow K$$\end{document} independent on L such that the representing measure μ~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mu }$$\end{document} of L~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{L}$$\end{document} provides the representing measure μ~∘f-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mu }\circ f^{-1}$$\end{document} of L. We also show that every moment functional L:V→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L:\mathcal {V}\rightarrow \mathbb {R}$$\end{document} is represented by λ∘f-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \circ f^{-1}$$\end{document} for some measurable function f:[0,1]→Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:[0,1]\rightarrow \mathbb {R}^n$$\end{document} where λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is the Lebesgue measure on [0, 1].


Introduction
Linear functionals L : V → K with K = R or C belong to the most important structures in mathematics, e.g. for separation arguments. If V is a vector space of functions v : X → K then L is called a moment functional if it is represented by a (non-negative) measure μ on X : If supp μ ⊆ K ⊆ X , then L is called a K-moment functional. Among the moment functionals the most important ones act on polynomials V = R[x 1 , . . . , x n ] on some K ⊆ R n , n ∈ N. The origin of the name moment comes from physics, especially the moment of inertia which is calculated by for a rotation around the x 3 -axis of a body with density distribution ρ. If K is closed then Haviland's Theorem [14,15] states that a linear functional L : R[x 1 , . . . , x n ] → R is a K-moment functional if and only if L(p) ≥ 0 for all p ∈ R[x 1 , . . . , x n ] with p ≥ 0. On the other side p ≥ 0 on K if and only if L(p) ≥ 0 for all K-moment functionals L since every point evaluation is a moment functional. These are the two directions in the duality theorem and the many connections between the moment problem (deciding when a linear functional is a moment functional) and non-negative polynomials (and therefore optimization and many other applications) only start here. See e.g. [1,2,5,8,10,[19][20][21]24,36] and references therein for more on the moment problem, the connection to non-negative polynomials, and applications. Besides the one-point evaluation L(f ) = f (x) the following is probably the simplest moment functional.  Hausdorff Moment Problem 1.2. (see [13] or [19,Thm Our approach in this paper is similar to disintegration. To understand moment functionals better and to simplify them we investigate in this article the possibility of transforming a linear (moment) functional into another linear (moment) functional based on isomorphism and transformation results between measure spaces. While (real) algebraic and functional analytic/operator theoretic results have been applied intensively, deeper measure theoretic aspects have not been studied.
In the theory of moments representing a linear functional L : V → R by an integral (i.e., proving the existence of a representing measure) is done by the use of the Riesz (Riesz-Markov-Kakutani) Theorem [17,23,28]. But other classical results in measure theory [12,16,25,30,31,37,38] have not been applied mainly because these deal with general measurable functions and the moment problem is dominated by the use of polynomials [20,21,36]. But in the operator theoretic approach to the moment problem we work in the general Hilbert space setting and allowing measurable functions instead of using only polynomials is the natural framework. Also recent developments such as [18] (see Theorem 2. 19) have not been considered in the theory of moments so far. The measure theoretic results we use in this paper are distributed over almost a century and hence the mathematical language and definitions used in the original literature can differ enormously from our mathematical 2 Page 4 of 29 Philipp J. di Dio IEOT language today. We therefore give the original references but mainly refer for unified and up to date formulations and definitions to [4]. Let us have a look at the following theorem to see what kind of results we are looking for.
Theorem 1.4 can be seen as a complete characterization of (S-)moment functionals, i.e., every moment functional L : V → R has the form (2) for some f : [0, 1] → S. Additionally, Theorem 1.4 also shows that every moment functional L is represented by λ • f −1 for a measurable function f : Hence, the aim of this paper is to characterize and represent moment functionals in the form of (2) and especially to find additional properties of f : [0, 1] → S.
The notation and result in Theorem 1.4 stimulate the notation of a transformation of a linear (moment) functional. We introduce the following definitions. Definition 1.5. Let X and Y be two Souslin spaces, U and V two vector spaces of real measurable functions on X resp. Y, and K : U → R and L : V → R be two linear functionals. We say L (continuously) transforms into K, symbolized by L K resp. L c K, if there exists a Borel (resp.
We say L strongly (and continuously) transforms into K, symbolized by L s K resp. L sc K, if there exists a surjective Borel (resp. surjective and If in this definition of a transformation a function f : X → Y is fixed because it has special properties, then we denote that in the transformation by f . Of course, we have the implications With this definition Theorem 1.4 can be reformulated to the following statement. The paper is structured as follows. In Sect. 2 we will give the preliminaries on measure theory and integration. Since most of the measure theoretic terminology and results in Sect. 2 (Souslin sets, Lebesgue-Rohlin spaces, isomorphisms between measure spaces etc.) have to our knowledge never been used in connection with the moment problem before, we give the complete definitions, results, and important examples which are essential for this paper (but without proofs).
In Sect. 3 we present basic properties of transformations (Definition 1.5). E.g. in Theorem 3.3 we show that if there exists a transformation L K and K is a moment functional, then also L is a moment functional. So, the transformation (literally and symbolically) aims at moment functionals K to determine whether already L was a moment functional. Section 4 contains then the main results where several non-trivial transformations to [0, 1]-or I k -moment functionals are presented, I k finite union of compact intervals in R. We show, which might already be apparent from Theorem 1.4, that the structure of possible moment functionals K are quite simple. These are always [0, 1]-or I k -moment functionals. However, this simplicity of K has the price that f : [0, 1] → S has little properties. In the worst case as in Theorem 1.4 we only have that f is measurable and each component f i is in L([0, 1], λ). We therefore also present results where f is at least continuous and can therefore approximated by polynomials on [0, 1] in the supremum norm.
In Sect. 5 we give the conclusions and open problems. Additionally, we give and discuss several open questions, especially the restriction that f is a rational or a polynomial map.

Preliminaries: Measure Theory and the Lebesgue Integral
We give here the measure theoretic results used in our paper. Of course, it is possible to go directly to Sect. 3 and the main results in Sect. 4 and consult this Sect. 2 if necessary while reading the results and proofs.
In this article we follow the monographs [9,22], and especially [4] for the measure theory and Lebesgue integral. We denote by P(X ) the power set of a set X , i.e., the set of all subsets of X . Let A ⊆ P(X ) be a σ-algebra on a set X , then we call ( Given F ⊆ P(X ), then by σ(F) we denote the σ-algebra generated by F, i.e., the smallest σ-algebra containing F. The Borel σ-algebra B(X ) of a topological (e.g. Hausdorff) space X is generated by all open sets in X .
Given a measurable space (X , A), a measure μ on (X , A) is a countably additive function μ : A → [0, ∞]. I.e., dissident from [4] for us all measures are non-negative if not otherwise explicitly stated as signed.
Let X be a topological (e.g. locally compact Hausdorff) space. A measure on (X , B(X )) is called Borel measure. A Radon measure μ is a measure over (X , B(X )) such that μ(K) < ∞ for all compact K ⊆ X and μ(V ) = sup{μ(K) | K is compact, K ⊆ V }. By λ n we denote the n-dimensional Lebegue measure on (R n , B(R n )).
We have the following transformation formula.
Proof. It is sufficient to show (3) for f ≥ 0: We have the first result from measure theory. We apply it in Proposition 4.1. The following is a central definition.  The empty set is a Souslin set. Souslin sets are fully characterized. A full answer gives the following theorem.
Hahn-Mazurkiewicz' Theorem 2.6. (see [12,25] The reason why we work with Souslin spaces is revealed in the following theorem. The existence of an isomorphism can be weakened. For Borel measurable function f : X → Y between two Souslin spaces X and Y with f (X ) = Y one always finds nice (i.e., Borel measurable) one-sided inverse functions.
Jankoff 's Theorem 2.12. (see [16] or e.g. [4, Thm. 6.9.1 and 9.1.3]) Let X and Y be two Souslin spaces and let f : X → Y be a surjective Borel mapping. Then there exists a Borel measurable function g : Y → X such that f (g(y)) = y for all y ∈ Y.
In other words, restricting f to some i.e., g is injective, f is surjective, and with X 0 = im g := g(Y) we havẽ Like Theorem 2.11 also the next result shows the importance of working on Souslin sets.  For both results note the difference to Proposition 2.2. In Proposition 2.2 we find for any measurable space X and measure μ a map But for Souslin spaces X in Corollary 2.15 we find a map Theorem 2.14 restricts f : [0, 1] → X to isomorphisms and hence not all measures can be transformed into λ. Atoms in the measure μ prevent it from being isomorphic to λ. In fact, as explained in [4, Rem. 9.7.4], Corollary 2.15 follows from Theorem 2.14 by introducing atoms into f : [0, 1] → X by introducing constant functions into f . But Theorem 2.14 provides that if μ has atoms, it can still be isomorphic mod0 transformed into a measure ν on [0, 1]. Without atoms we could chose ν = λ. So is it possible to transform the non-atomic part of μ to λ and then add the atoms from μ to λ? Yes, we can. This is done on the following spaces.
where M is a Borel set of a complete separable metric space X and μ is a Borel measure on M , is a Lebesgue-Rohlin space. Especially X = R n or PR n are complete metric spaces and therefore any Borel measure on a Borel subset M ∈ B(R n ) gives a Lebesgue-Rohlin space. • We can now transform any measure by an isomorphism mod0 to the Lebesgue measure λ plus atoms.
We will apply Theorem 2.19 especially in connection with the Theorem 2.6. The advantage is here that f on [0, 1] is continuous and can therefore be approximated by polynomials up to any precision ε > 0 in the sup-norm.

Transformations of Linear Functionals: Basic Properties
For the transformation between two linear functionals in Definition 1.5 we get the following technical result. Lemma 3.1. Let X , Y, and Z be Souslin spaces; U, V, and W be vector spaces of real measurable functions on X , Y, and Z respectively; and M : W → R, L : V → R, and K : U → R be linear functionals. The following hold: K. (ii)-(iv) follow in the same way as (i).
Lemma 3.1 can be seen as shortening the sequence: The next lemma shows, that a strong transformation L s K implies the reverse transformation K L.
Lemma 3.2. Let X and Y be Souslin sets, U and V vector spaces of real functions on X resp. Y, and L : V → R and K : U → R be linear functionals.
Since f is surjective by Theorem 2.12 there exists a Borel function g : Y → X such that f (g(y)) = y for all y ∈ Y.
Hence, U • g = V and for all u ∈ U we have While we have so far only transformed linear functionals, the importance of the transformation is revealed in the following result. It shows that the property of being a moment functional is preserved in one or both directions. (ii) L is a moment functional.
(i)→(ii): Let K be a moment functional with representing measure ν on X , then i.e., ν•f −1 is a representing measure of L and hence L is a moment functional.
(ii)→(i): When L s K, then Lemma 3.2 implies K L.
The importance of the transformation and hence Theorem 3.3 can be seen in If K is a moment functional, then all L 1 , . . . , L 8 are moment funtionals. Assume in (4) all transformations are strong transformations s . Then: If one L i or K is a moment functional, then all K, L 1 , . . . , L 8 are moment functionals. Note, the transformation in Definition 1.5 also covers extensions and restrictions of functionals. Let f = id X and let V be a vector space of measurable functions on X , V 0 ⊆ V be a linear subspace, and L : V → R a linear functional. Then shows that if L k is a moment functional, then all L i and L are moment functionals. So far we introduced the transformation of a linear functional and gained basic properties. But as seen from Theorem 1.4 and Corollary 1.6, there are non-trivial results for the transformations. The next section is devoted to these non-trivial transformation results.

Non-trivial Transformations of Linear Functionals
Let V be a (finite or infinite dimensional) vector space of measurable functions on a Souslin space X . Then by Theorem 2.11 there exist a Souslin set S ⊆ [0, 1] and an isomorphism h : (S, B(S)) → (X , B(X )). This implies that However, by Remark 2.8 S needs not to be a Borel set. So determining whetherL is a moment functional might be as hard as determining whether L is a moment functional. Additionally,L now no longer lives on polynomials but evaluates measurable functions h α = h α1 Allowing general Borel measurable functions on measurable spaces instead of isomorphisms we get Theorem 1.4 in the introduction. There we showed that any moment functional can be expressed as integration with respect to the Lebesgue measure λ on [0, 1].
The next result shows that any moment functional with an atomless representing measure has a "direction" in which it looks like (1)  Proof. Let μ be a representing measure of L. By Proposition 2.2 there exists a measurable f : Then Hence, for any moment functional with an atomless representing measure there exists a function f (a direction) such that it acts on R[f ] ∼ = R[t] as (1), i.e., the Lebesgue measure on [0, 1]. Under some mild conditions every truncated moment functional in the interior of the truncated moment cone has an atomless representing measure. We can even find a linear combination of Gaussian distributions (Gaussian mixture) as a representing measure. This was proven in [7] for the first time. Using the transformation formulation with L Leb from Example 1.1 we can visualize Proposition 4.1 as Note the reverse statement of Proposition 4.1. If a linear functional L can never be (continuously) extended to R[f ] with L(f d ) = L (1) d+1 for some measurable f , then L is not a moment functional with an atomless representing measure. Theorem 1.4 and Proposition 4.1 are very general. Especially Theorem 1.4 works on arbitrary Borel sets of R n (in fact on every Souslin space). For this generality we have to pay the price that f is in general only measurable. Additionally, since we always express L as integration with respect to λ on [0, 1], the chosen f depends on L. If we want additional properties for f to hold, especially continuity and independence from L, then we need to restrict the functionals we want to transform. This can be achieved by restricting the investigation to K-moment functionals on compact and path-connected sets K ⊂ R n . Then from the Theorem 2.6 we get the existence of surjective and continuous functions f : [0, 1] → K. We find the following result. i.e., for any linear functional L : V → R the following are equivalent: Ifμ is a representing measure ofL, thenμ • f −1 is a representing measure of L.
There exists a measurable function g : K → [0, 1] such that f (g(x)) = x for all x ∈ K and if μ is a representing measure of L, then μ • g −1 is a representing measure ofL.
Proof. Since K ⊂ R n is compact and path-connected, by the Hahn-Mazurkiewicz' Theorem 2.6 there exists a continuous and surjective function f : [0, 1] → K. By Example 2.5 or Lemma 2.7 [0, 1] and K are Souslin spaces and f is Borel measurable (since it is continuous). By Theorem 2.12 there exists a measurable function g : K → [0, 1] such that f (g(x)) = x for all x ∈ K.
Let L : V → R be a K-moment functional and μ be a representing measure of L, i.e., supp μ ⊆ K and i.e., μ • g −1 is a representing measure ofL and henceL is a [0, 1]-moment functional.
(ii)→(i): Letμ be a representing measure ofL :Ṽ → R. Then In the previous result the functions f : [0, 1] → K and g : K → [0, 1] do not depend on the functions V or the functional L : V → R. They depend only on K. We can therefore fix such functions f and g and investigate any L resp.L.
If the continuous f can be chosen for each L, then in Theorem 4.2(ii) we can even ensure thatL is represented by the Lebesgue measure λ on [0, 1] if and only if L has a representing measure μ with supp μ = K, see Theorem 4.11 below.
In Theorem 4.2 we required that K consists of one path-connected component. If K consists of more than one component, then we can glue the parts together.

be a vector space of real valued measurable functions on (K, B(K)). There exists a continuous surjective function
such that for any linear functional L : V → R the following are equivalent: Proof. It is sufficient to show the existence of the function f (and g). The rest of the proof is verbatim the same as in the proof of Theorem 4.2.
Since for each i = 1, 2, . . . , k the set K i is compact and path-connected and the translation of the unit interval [0, 1] to [2i−2, 2i−1] is continuous, by the Theorem 2.6 there exists a continuous and surjective for an i ∈ {1, 2, . . . , k}. Then f is continuous and surjective. For we proceed in the same way. By Theorem 2.12 for each Note, that when K consists of countably many compact and pathconnected components (k = ∞), then in Corollary 4.3 f is no longer supported on a bounded (and therefore compact) set: But if e.g. K is a compact and semi-algebraic set, then K has only finitely many path-connected components.
An In the next theorem we will identify each K-moment functional with a [0, 1]-moment functional, i.e., the Theorem 1.2.
If μ is the representing measure of L, then μ • g −1 representsL. Additionally, there exists a continuous and surjective function f :[0, 1] → K independent on L resp.L such that f (g(x)) = x for all x ∈ K and ifμ is the representing measure ofL, thenμ • f −1 is the representing measure of L.
(i)→(ii): Let L : V → R be a K-moment functional and μ be a representing measure of L with supp μ ⊆ K. g is measurable with |g| ≤ 1 and hence we have that all (8) and hence L extents to R[g]. Let p ∈ R[t], theñ and μ•g −1 is a representing measure ofL, i.e.,L is a [0, 1]-moment functional.
We see that all about L is already known if we know how it acts (viã L) on powers of the fixed (and independent on L) function g.L : R[t] → R is only a Hausdorff moment problem and its representing measureμ provides a representing measure μ =μ • f −1 via a fixed (and independent on L) continuous function f . There shall exists a v ∈ V ⊆ C(K, R) such that v > 0 on K.
By compactness of K and continuity of v this implies 1 ≤ c · v ∈ V for some c > 0, i.e., μ(K) < ∞ in (8). However, since we have to extend L : V → R to L : V + R[g] → R and 1 ∈ R[g] we can assume w.l.o.g. already 1 ∈ V. If 1 ∈ V and L can not be extended to 1, then L can definitely not be extended to R[g] and the statements of Then hold for all f, g ∈ V + R[g] and α ≥ 0. By the Hahn-Banach Theorem there exists an extension L : An extension L in Lemma 4.6 is in general not unique. If V is a point separating algebra on K and L is a K-moment functional, then the extension L is unique (and continuous), since then the representing measure μ of L is unique.
For the extension L it is only necessary that 1 ∈ V to ensure |g| ≤ 1 ∈ V. V ⊆ C(K, I k ) continuous is actually not necessary and hence Lemma 4.6 can be easily weakened.
As in Theorem 4.2 also in Theorem 4.4 the functions f and g do not depend on L orL. They depend only on K. And as in Proposition 4.1 the functionalL is defined in one "direction" R[g] ∼ = R[t] byL(t d ) := L(g d ). But now it no longer needs to be L Leb as in Example 1.1.
The problem of determining whetherL : R[t] → R in Theorem 4.4(ii) is a [0, 1]-moment functional is the Theorem 1.2. This problem is fully solved, analytically as well as numerically. But the function g : K → [0, 1] to establish the equivalence (i) ⇔ (ii) in Theorem 4.4 is a measurable function and not a polynomial. Hence, L(g d ) is not directly accessible unless of course d = 0. Fortunately, since K ⊂ R is compact, R[x 1 , . . . , x n ] is dense in C(K, R). Hence, for any given finite measure μ on K, i.e., μ(K) = L(1) < ∞, we can approximate g by a polynomial g ε ∈ R[x 1 , . . . , x n ] in the L 1 (μ)-norm to any arbitrary precision.
Proof. L is a K-moment functional and therefore has a unique representing measure μ with supp μ ⊆ K. g ≥ 0 and hence there exists a measurable function p : K → [0, 1] such that g = p 2 . Since K is compact and we have p ∈ L 1 (K, μ) and therefore for any ε > 0 there exists a p ε ∈ R[x 1 , . . . , x n ] such that p ε ≤ 1 on K and Set g ε := p 2 ε . Then For d = 0 we have g 0 = g 0 ε = 1, i.e., L(g 0 ) = L(1) = L(g 0 ε ), and for d = 1 we Note, the g ε not only depends on ε > 0 but also on L resp. its representing measure μ. Since g is measurable (but not necessarily continuous) it is not possible to get sup x∈K |g(x) − g ε (x)| ≤ ε. So g ε depends on L. Otherwise assume we find a g ε ∈ R[x 1 , . . . , x n ] such that for any moment functional L (with L(1) = 1), i.e., measure μ on K with μ(K) = 1, we have a contradiction. So the choice of g ε depends on L resp. μ. Additionally, note that in fact we can g ε not only chose to be a square, but in fact any power: g ε = p k ε for a fixed k ∈ N. Just replace p := √ g by p := k √ g in the proof since g ≥ 0 and use the geometric series as in (13) also in (12).
In Corollary 4.3 we extended Theorem 4.2 from a compact and pathconnected K ⊂ R n to an at most countable union of pairwise disjoint, compact, and path-connected K i 's. In Theorem 4.4 we required that K is a compact and path-connected set. Since we needed compactness of [0, 1] in Theorem 4.4 we can at least extend Theorem 4.4 to a finite (disjoint) union of compact and path-connected sets.
Corollary 4.8. Let k, n ∈ N be natural numbers and K ⊂ R n be the union of finitely many compact, path-connected, and pairwise disjoint sets Then there exists a measurable function such that for all linear functionals L : V → R with 1 ∈ V ⊆ C(K, R) the following are equivalent: Then f (g(x)) = x for all x ∈ K and I k ⊂ [0, 1]. We are again facing the problem, that g is measurable but not necessarily a polynomial. But as in Theorem 4.7 we can approximate g by polynomials. Corollary 4.9. Let n, k ∈ N be natural numbers, K ⊂ R n the union of finitely many compact, path-connected, and pairwise disjoint sets K i , K = k i=1 K i , and let g : K → I k be from Corollary 4.8. Then for any ε > 0 and K-moment Proof. Since I k ⊂ [0, 1] it is verbatim the same as the proof of Theorem 4.7.
We have seen in Theorem 4.2 resp. Corollary 4.3 that a linear functional L : V → R is a K-moment functional (K is the countable union of compact and path-connected sets) if and only if it can be transformed by a continuous function f : I → K to a I-moment functional (I is the countable union of intervals [a i , b i ] ∈ R).
If we allow not only continuous functions f , then we can generalize this. If we drop continuity of f but add bijectivity almost everywhere we find that any functional on a Borel set of R n is a moment functional if and only if we can transform it into a moment functional with representing measure "Lebesgue measure on [0, 1] plus countably many point evaluations", see (14). for If we drop bijectivity almost everywhere for f then we get Theorem 1.4, i.e., in (14) we can chose c = L(1) and c i = 0 for all i ∈ N.
In Theorem 1.4 and Theorem 4.10 we can only ensure that f is measurable, but not necessarily continuous or even a polynomial map. The reason is that we can not control the support of a representing measure of L. In Theorem 4.2 we already showed that f can be chosen as continuous and surjective, independent on L. But if we restrict the moment functionals resp. the support of a representing measure and chose f tailor made for each Kmoment functional, then f can be chosen to be continuous and surjective and the representing measure will be the Lebesgue measure λ on [0, 1].  Since f : [0, ∞) → R n is surjective and [0, ∞) and R n are Souslin sets by Lemma 2.7 then by Theorem 2.12 there exists a g ε : R n → [0, ∞) with f ε (g ε (x)) = x for all x ∈ R n . ( * ) implies Similar to Theorem 4.2 we then get the continuous transformation into [0, ∞)-moment functionals.
I.e., L sc L . Ifμ is a representing measure ofL, thenμ • f −1 . There exists a function g : R n → [0, ∞) such that f (g(x)) = x for all x ∈ R n and if μ is a representing measure of L, then μ • g −1 is a representing measure ofL.
Proof. Since R n and [0, ∞) are Souslin sets and f is surjective, by Theorem 2.12 there exists a function g : i.e., μ • g −1 is a representing measure ofL.
Remark 4.14. Similar to Theorem 4.4 we get that for any ε > 0 and g ε from At the end of this section we want to discuss two things that can easily be missed. The first is a crucial technical remark and the second is a historical one.
For most transformations we required that f : X → Y is surjective to apply Theorem 2.12 to get a right-side inverse g : Y → X , i.e., f (g(y)) = y for all y ∈ Y. E.g. in Theorem 4.4 we used this g directly to embed a [0, 1]moment functional into an extension L of L. However, for any f : X → Y of course f : X → f (Y) is surjective. If f is continuous and X Borel, then f (X ) remains even a Borel set. Otherwise f (X ) is at least a Souslin set.
To demonstrate, that f : X → Y needs to be surjective and the re- Note, that in the setting of Theorem 4.4 the multiplication operators are bounded since K is compact. In the setup of K = R n , see Remark 4.14, we have in general unbounded operators and only the easy direction (i)→(ii) was shown. It is open if (ii)→(i) also holds in the unbounded case.