Infinite Determinantal Measures

Infinite determinantal measures introduced in this note are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of a determinantal process and a convergent, but not integrable, multiplicative functional. Theorem 2, the main result announced in this note, gives an explicit description for the ergodic decomposition of infinite Pickrell measures on the spaces of infinite complex matrices in terms of infinite determinantal measures obtained by finite-rank perturbations of Bessel point processes.


INTRODUCTION
1.1. Outline of the main results. In this section, our aim is to construct sigma-finite analogues of determinantal measures on spaces of configurations. In Theorem 2 of Section 4, infinite determinantal measures will be seen to arise in the ergodic decomposition of infinite unitarily-invariant measures on spaces of infinite complex matrices.
Informally, a configuration on the phase space E is an unordered collection of points (called particles) of E, possibly with multiplicities; the main assumption is that a bounded subset of E contain only finitely many particles of a given configuration.
To a function g on E assign its multiplicative functional Ψ g on the space of configurations: the functional Ψ g is obtained by multiplying the values of g over all particles of a configuration (see (5)). A probability measure on the space of configurations on E is uniquely characterized by prescribing the expectations of multiplicative functionals; for determinantal probability measures these expectations are given by special Fredholm determinants, see e.g. [30]; the definition is also recalled in (8) below.
Given a subset E ′ ⊂ E, consider the subset Conf(E, E ′ ) of those configurations whose all particles lie in E ′ ; in Proposition 2 below, we shall see that under some additional asumptions the restriction of a determinantal point process onto Conf(E, E ′ ) is again determinantal.
Our main example, the measure B (s) of (24), is defined on the space of configurations on (0, +∞). Almost every configuration is infinite and 1 bounded according to B (s) ; the particles accumulate at zero. If one takes R > 0 and requires all particles to lie in (0, R), then the induced measure of B (s) on the resulting subset of configurations is finite, and, after normalization, determinantal. As R goes to infinity, the measure of the subset Conf((0, +∞); (0, R)) grows, and the measure of the space of all configurations is infinite.
Our general construction will similarly exhaust E by subsets E n in such a way that the weight of Conf(E; E n ) is positive and finite, and the normalized restriction of our infinite determinantal measure onto the subset Conf(E; E n ) is determinantal. A simple example is given by "infinite orthogonal polynomial ensembles", see (3) below. The measure B (s) is a scaling limit of such ensembles. We proceed to precise formulations.

Construction of infinite determinantal measures.
Let E be a locally compact complete metric space, and let Conf(E) be the space of configurations on E endowed with the natural Borel structure (see, e.g., [11], [30]).
Given a Borel subset E ′ ⊂ E, we let Conf(E, E ′ ) be the subspace of configurations all whose particles lie in E ′ .
Given a measure B on a set X and a measurable subset Y ⊂ X such that 0 < B(Y ) < +∞, we let B | Y stand for the restriction of the measure B onto the subset Y .
An infinite determinantal measure is a σ-finite Borel measure B on Conf(E) admitting a filtration of the space E by Borel subsets E n , n ∈ N: such that for any n ∈ N we have (1) 0 < B (Conf(E, E n )) < +∞; (2) the normalized restriction is a determinantal measure; Let µ be a σ-finite Borel measure on E. By the Macchì-Soshnikov Theorem, under some additional assumptions, a determinantal measure can be assigned to an operator of orthogonal projection, or, in other words, to a closed subspace of L 2 (E, µ). In a similar way, an infinite determinantal measure will be assigned to a subspace H of locally square-integrable functions. For example, for infinite analogues of orthogonal polynomial ensembles, H is the subspace of weighted polynomials, see Subsection 1.3 below.
Let L 2,loc (E, µ) be the space of measurable functions on E, locally square integrable with respect to µ, let I 1 (E, µ) be the space of trace-class operators in L 2 (E, µ) and let I 1,loc (E, µ) be the space of operators on L 2 (E, µ) that are locally of trace class (precise definitions are recalled in Section 2).
Let H ⊂ L 2,loc (E, µ) be a linear subspace. If E ′ ⊂ E is a Borel subset such that χ E ′ H is a closed subspace of L 2 (E, µ), then we denote by Π E ′ the operator of orthogonal projection onto the subspace χ E ′ H ⊂ L 2 (E, µ). We now fix a Borel subset E 0 ⊂ E; informally, E 0 is the set where the particles accumulate. We impose the following assumption on E 0 and H.

Assumption 1.
(1) For any bounded Borel set B ⊂ E, the space The requirements (1) and (2) determine the measure B uniquely up to multiplication by a positive constant.
We denote B(H, E 0 ) the one-dimensional cone of nonzero infinite determinantal measures induced by H and E 0 , and, slightly abusing notation, we write B = B(H, E 0 ) for a representative of the cone.
Remark. If B is a bounded set, then, by definition, we have Remark. If E ′ ⊂ E is a Borel subset such that χ E 0 ∪E ′ is a closed subspace in L 2 (E, µ) and the operator Π E 0 ∪E ′ of orthogonal projection onto the subspace χ E 0 ∪E ′ H satisfies then, exhausting E ′ by bounded sets, from Theorem 1 one easily obtains an infinite analogue of an orthogonal polynomial ensemble.
For any b 1 ∈ [a, b), the induced measure is finite and, after normalization, can be represented in determinantal form where K ρ,b 1 N is the N-th Christoffel-Darboux kernel formed by orthonormal polynomials corresponding to the "induced" weight ρ(x)χ [a,b 1 ] (x).
The infinite measure (3) is thus an infinite determinantal measure corresponding to the subspace H ⊂ L 2,loc ([a, b), Leb) spanned by the functions x k ρ(x), k = 0, . . . , N − 1, and the subset E 0 = [a, b 1 ) for an arbitrary b 1 ∈ (a, b). In the problem of ergodic decomposition of infinite Pickrell measures we shall be especially interested in studying scaling limits of such "infinite orthogonal polynomial ensembles".
1.4. Organization of the paper. In the next subsection it is shown that, under certain additional assumptions, an infinite determinantal measure times a multiplicative functional yields after normalization a determinantal point process; for determinantal probability measures this has been established in [8]. We then proceed to our main example of infinite determinantal measures, namely, those obtained as finite-rank perturbations of determinantal point processes. The ergodic decomposition measures of infinite Pickrell measures will be seen to be of this type. In the following subsection it is established that induced processes of an infinite determinantal measure obtained by finite rank perturbation, converge to the unperturbed process.
In Section 2 we recall the definition of determinantal point processes, study the properties of multiplicative functionals of these processes, thus extending the results of [8], and give a sketch of the proof of Theorem 1.
In Section 3 we recall the construction, due to Pickrell [21], [22], [23] in the finite case (see also Neretin [16]) and to Borodin and Olshanski [4] in the infinite case, of Pickrell measures on the space of infinite matrices. We then recall the Olshanski-Vershik approach (see [33], [20]) to the Pickrell classification of finite ergodic unitarily-invariant measures on spaces of infinite matrices as well as the result of [7] that implies that the ergodic components of infinite Pickrell measures are almost surely finite; only the decomposing measure is infinite.
In Section 4 we start by considering finite Pickrell mesures, for which the ergodic decomposition is given, up to a change of variable, by the Bessel point process of Tracy and Widom [32]. The main result of the paper, Theorem 2 , then says that the ergodic decomposition of infinite Pickrell measures is induced by infinite determinantal measures obtained as an explicitly given finite-rank perturbation of the Bessel point processes occurring in the ergodic decomposition of finite Pickrell measures. The scaling limit argument sketched at the end of the section uses precisely the representation, developed in Section 1, of infinite determinantal measures as products of finite determinantal measures and multiplicative functionals.
1.5. Multiplicative functionals. Let g be a non-negative measurable function on E, and introduce the multiplicative functional Ψ g : Conf(E) → R by the formula If the infinite product x∈X g(x) absolutely converges to 0 or to ∞, then we set, respectively, Ψ g (X) = 0 or Ψ g (X) = ∞. If the product in the righthand side fails to converge absolutely, then the multiplicative functional is not defined. We start with an auxiliary proposition.

Proposition 1.
Let a subspace H ⊂ L 2,loc (E, µ) and a Borel subset E 0 ⊂ E satisfy Assumption 1. Let g be a positive bounded measurable function on E such that Then √ gH is a closed subspace in L 2 (E, µ).
Under the assumptions of Proposition 1, let Π g be the operator of orthogonal projection onto the closed subspace √ gH.
Our next aim is to give sufficient conditions for integrability of multiplicative functionals with respect to infinite determinantal measures. We restrict ourselves to the case when the function g only takes values in (0, 1].

Proposition 2.
Let a subspace H ⊂ L 2,loc (E, µ) and a Borel subset E 0 ⊂ E satisfy Assumption 1, and let g : E → (0, 1] be a measurable function such that: We can therefore write B = C · Ψ 1/g · P Π g , where C is a positive constant. Our infinite determinantal measure is thus represented as a product of a determinantal probability measure and a convergent non-integrable multiplicative functional. 1.6. Infinite determinantal measures obtained as finite-rank perturbations of determinantal probability measures. We now consider infinite determinantal measures induced by subspaces H obtained by adding a finite-dimensional subspace V to a closed subspace L ⊂ L 2 (E, µ).
Let, therefore, Q ∈ I 1,loc (E, µ) be the operator of orthogonal projection onto a closed subspace L ⊂ L 2 (E, µ), let V be a finite-dimensional subspace of L 2,loc (E, µ), and set H = L + V . Let E 0 ⊂ E be a Borel subset. We shall need the following assumption on L, V and E 0 .

Assumption 2.
(1) In particular, for any bounded Borel subset B, the subspace χ E 0 ∪B L is closed, as one sees by taking E ′ = E 0 ∪ B in the following clear and that for any function ϕ ∈ L, the equality The subspace H and the Borel subset E 0 therefore define an infinite determinantal measure B = B(H, E 0 ). We now adapt the formulation of Proposition 2 to this particular case.

Proposition 5.
Let L, V , and E 0 satisfy Assumption 2, let B be the corresponding infinite determinantal measure, and let g : where, as before, Π g is the operator of orthogonal projection onto the closed subspace √ gH.
Remark. The subspace √ gH is closed by Proposition 1.

1.7.
Convergence of approximating kernels. Our next aim is to show that, under certain additional assumptions, if a sequence g n of measurable functions converges to 1, then the operators Π gn considered in Proposition 5 converge to Q in I 1,loc (E, µ). Given two closed subspaces H 1 , H 2 in L 2 (E, µ), let α(H 1 , H 2 ) be the angle between H 1 and H 2 , defined as the infimum of angles between all nonzero vectors in H 1 and H 2 ; recall that if one of the subspaces has finite dimension, then the infimum is achieved. Proposition 6. Let L, V , and E 0 satisfy Assumption 2, and assume additionally that we have V ∩ L 2 (E, µ) = 0. Let g n : E → (0, 1] be a sequence of positive measurable functions such that (2) for all n ∈ N we have √ g n V ⊂ L 2 (E, µ); (3) there exists α 0 > 0 such that for all n we have Then, as n → ∞, we have Using the second remark after Theorem 1, one can extend Proposition 6 also to nonnegative functions that admit zero values. Here we restrict ourselves to characteristic functions of the form χ E 0 ∪B with B bounded, in which case we have the following Corollary 1. Let B n be an increasing sequence of bounded Borel sets exhausting E \ E 0 . If there exists α 0 > 0 such that for all n we have Informally, Corollary 1 means that, as n grows, the induced processes of our determinantal measure on subsets Conf(E; E 0 ∪ B n ) converge to the "unperturbed" determinantal point process P Q .

Locally integrable functions and locally trace class operators
Choosing an exhausting family B n of bounded sets (for instance, balls of radius tending to infinity) and using (6) with B = B n , we endow the space L 2,loc (E, µ) with a countable family of seminorms which turns it into a complete separable metric space; the topology thus defined does not, of course, depend on the specific choice of the exhausting family.
Let I 1 (E, µ) be the ideal of trace class operators K : L 2 (E, µ) → L 2 (E, µ) (see volume 1 of [26] for the precise definition); the symbol || K|| I 1 will stand for the I 1 -norm of the operator K. Let I 2 (E, µ) be the ideal of Hilbert-Schmidt operators K : L 2 (E, µ) → L 2 (E, µ); the symbol || K|| I 2 will stand for the I 2 -norm of the operator K.
Let I 1,loc (E, µ) be the space of operators K : L 2 (E, µ) → L 2 (E, µ) such that for any bounded Borel subset B ⊂ E we have Again, we endow the space I 1,loc (E, µ) with a countable family of seminorms where, as before, B runs through an exhausting family B n of bounded sets.

Determinantal Point Processes. A Borel probability measure P on
Conf(E) is called determinantal if there exists an operator K ∈ I 1,loc (E, µ) such that for any bounded measurable function g, for which g − 1 is supported in a bounded set B, we have The Fredholm determinant in (8) is well-defined since K ∈ I 1,loc (E, µ). The equation (8) determines the measure P uniquely. If, for a bounded Borel set B ⊂ E, we let # B : Conf(E) → N∪{0} be the function that to a configuration assigns the number of its particles belonging to B, then, for any pairwise disjoint bounded Borel sets B 1 , . . . , B l ⊂ E and any z 1 , . . . , z l ∈ C from (8) we have E P z For further results and background on determinantal point processes, see e.g. [2], [9], [12], [13], [14], [27], [28], [29], [30].
In what follows we suppose that K belongs to I 1,loc (E, µ), and denote the corresponding determinantal measure by P K . Note that P K is uniquely defined by K, but different operators may yield the same measure. By the Macchì-Soshnikov theorem [15], [30], any Hermitian positive contraction that belongs to the class I 1,loc (E, µ) defines a determinantal point process.

Multiplicative functionals.
At the centre of the construction of infinite determinantal measures lies the result of [8] that can informally be summarized as follows: a determinantal measure times a multiplicative functional is again a determinantal measure. In other words, if P K is a determinantal measure on Conf(E) induced by the operator K on L 2 (E, µ), then, under certain additional assumptions, it is shown in [8] that the measure Ψ g P K after normalization yields a determinantal measure.
It is required in [8] that the operator (g − 1)K be of trace class; this assumption is too restrictive for our purposes, and in Propositions 7 and 10 we shall now formulate two more convenient versions of Proposition 1 in [8].
As before, let g be a non-negative measurable function on E. If the operator 1 + (g − 1)K is invertible, then we set By definition, B(g, K),B(g, K) ∈ I 1,loc (E, µ) since K ∈ I 1,loc (E, µ), and, if K is self-adjoint, then so isB(g, K).
In the case when K is self-adjoint, the following proposition generalizes Proposition 1 in [8].
If Q is a projection operator, then the operatorB(g, Q) admits the following description.
Proposition 8. Let L ⊂ L 2 (E, µ) be a closed subspace, and let Q be the operator of orthogonal projection onto L. Let g be a bounded measurable function such that the operator 1 + (g −1)Q is invertible. Then the operator B(g, Q) is the operator of orthogonal projection onto the closure of the subspace √ gL.
We now consider the particular case when g is a characteristic function of a Borel subset. In much the same way as before, if E ′ ⊂ E is a Borel subset such that the subspace χ E ′ L is closed (recall that a sufficient condition for that is provided in Proposition 4), then we set Q E ′ to be the operator of orthogonal projection onto the closed subspace χ E ′ L.
Propositions 10, 7 now yield the following Corollary 2. Let Q ∈ I 1,loc (E, µ) be the operator of orthogonal projection onto a closed subspace L ∈ L 2 (E, µ). Let E ′ ⊂ E be a Borel subset such that χ E ′ Qχ E ′ ∈ I 1 (E, µ). Then Assume, additionally, that for any function ϕ ∈ L, the equality χ E ′ ϕ = 0 implies ϕ = 0. Then the subspace χ E ′ L is closed, and we have The induced measure of a determinantal measure onto the subset of configurations all whose particles lie in E ′ is thus again a determinantal measure. In the case of a discrete phase space, related induced processes were considered by Lyons [12] and by Borodin and Rains [5].

The space I ξ .
To prove Proposition 7, we consider a slightly more general algebra of operators K for which the trace tr K and the Fredholm determinant det(1 + K) can be defined and shown to have the usual properties. The space I ξ (E, µ) is a modification of the space L 1|2 (H) introduced by Borodin, Okounkov and Olshanski [3] and used also by Olshanski in [17]. We proceed to precise formulations. Take a countable partition ξ of our space E into disjoint bounded measurable sets E n , n ∈ N. Introduce the sets (11) {ξ Informally, ξ is considered as a random variable taking integer values. The subspace I ξ (E, µ) ⊂ I 1,loc (E, µ) is now defined as follows: an operator K ∈ I 1,loc (E, µ) belongs to I ξ (E, µ) if (1) K ∈ I 2 (E, µ); (2) ∞ n=1 ||χ En Kχ En || I 1 < +∞.
The space I ξ (E, µ) is normed by the formula By definition, the space I ξ (E, µ) is an algebra. For K ∈ I ξ (E, µ), the Fredholm determinant det(1 + K) is defined by the formula The right-hand side of (12) is well-defined since (1 + K) exp(−K) ∈ I 1 for any K ∈ I 2 .
For K 1 , K 2 ∈ I ξ , we clearly have From the definitions we now immediately obtain Proposition 9. If (g − 1)K ∈ I ξ (E, µ), then Ψ g ∈ L 1 (Conf(E), P K ) and The following Proposition is a generalization of Proposition 1 in [8].

Proposition 10.
Assume that an operator K ∈ I 1,loc (E, µ) induces a determinantal measure P K on Conf(E). Let ξ be a countable measurable partition of E and let g be a nonnegative bounded measurable function on E such that (g − 1)K ∈ I ξ (E, µ) and that the operator 1 + (g − 1)K is invertible. Then the operators B(g, K),B(g, K) induce on Conf(E) a determinantal measure P B(g,K) = PB (g,K) satisfying Indeed, take a bounded measurable function f on E such that We then immediately have K)), and the proposition follows.
Observe now that to a nonnegative function g such that (9) holds, one can easily assign a countable partition ξ such that (g − 1)K ∈ I ξ (E, µ). Proposition 7 is therefore clear from Proposition 10. Let Leb = dz be the Lebesgue measure on Mat(n, C). Following Pickrell [21], take s ∈ R and introduce a measure µ (s) n on Mat(n, C) by the formula The measure µ (s) n is finite if and only if s > −1. For n 1 < n, let π n n 1 : Mat(n, C) → Mat(n 1 , C) be the natural projection map that to a matrix z = (z ij ), i, j = 1, . . . , n, assigns its upper left corner, the matrix π n n 1 (z) = (z ij ), i, j = 1, . . . , n 1 . The measures µ (s) n have the property of consistency with respect to the projections π n n 1 . More precisely, following Borodin and Olshanski [4], p.116, observe that even if the measure µ (s) n is infinite, the fibres of the projection π n n−1 have finite conditional measure as long as n + s > 0. The push-forward (π n n−1 ) * µ (s) n is consequently well-defined, and for any s ∈ R and n > −s we have (14) (π n n−1 ) * µ (s) Now let Mat(N, C) be the space of infinite matrices whose rows and columns are indexed by natural numbers and whose entries are complex: Let π ∞ n : Mat(N, C) → Mat(n, C) be the natural projection map that to an infinite matrix z ∈ Mat(N, C) assigns its upper left n × n-"corner", the matrix (z ij ), i, j = 1, . . . , n.
In this case, (14) implies the relation If s −1, the measures µ (s,λ) are all infinite. In this case, slightly abusing notation, we shall omit the super-script λ and write µ (s) for a measure defined up to a multiplicative constant.
Proposition 11 is obtained from Kakutani's Theorem in the spirit of [4], see also [16].
Let U(∞) be the infinite unitary group: an infinite matrix u = (u ij ) i,j∈N belongs to U(∞) if there exists a natural number n 0 such that the matrix The group U(∞) × U(∞) acts on Mat(N, C) by multiplication on both sides: The Pickrell measures µ (s) are by definition U(∞) × U(∞)-invariant. For the rôle of Pickrell and related mesures in the representation theory of U(∞), see [18], [19], [20].
Theorem 1 and Corollary 1 in [6] imply that the measures µ (s) admit an ergodic decomposition. Furthermore, Theorem 1 in [7] implies that for any s ∈ R the ergodic components of the measure µ (s) are almost surely finite. The main result of this note is an explicit description of the ergodic decomposition of the measures µ (s) for s = −1 − 2k, k ∈ N; in particular, for s < −1 we shall see that the ergodic decomposition is given by an explicitly computed infinite determinantal measure. Mat(N, C). This classification has been obtained by Pickrell [21], [22]; Vershik [33] and Olshanski and Vershik [20] proposed a different approach to this classification in the case of unitarily-invariant measures on the space of infinite Hermitian matrices, and Rabaoui [24], [25] adapted the Olshanski-Vershik approach to the initial problem of Pickrell. In this note, the Olshanski-Vershik approach is followed as well.

Classification of ergodic measures. First, we recall the classification of ergodic probability U(∞) × U(∞)-invariant measures on
Take z ∈ Mat(N, C), denote z (n) = π ∞ n z, and let (16) λ counted with multiplicities, arranged in non-increasing order. To stress dependence on z, we write λ (1) Let η be an ergodic Borel U(∞)×U(∞)-invariant probability measure on Mat(N, C). Then there exist non-negative real numbers x i , such that for η-almost every z ∈ Mat(N, C) and any i ∈ N we have: (17) x (2) Conversely, given non-negative real numbers γ 0, Introduce the Pickrell set Ω P ⊂ R + × R N + by the formula The set Ω P is, by definition, a closed subset of R + × R N + endowed with the Tychonoff topology.
By Proposition 3 in [6], the subset of ergodic U(∞) × U(∞)-invariant measures is a Borel subset of the space of all Borel probability measures on Mat(N, C) endowed with the natural Borel structure (see, e.g., [1]). Furthermore, if one denotes η ω the Borel ergodic probability measure corresponding to a point ω ∈ Ω P , ω = (γ, x), then the correspondence ω −→ η ω is a Borel isomorphism of the Pickrell set Ω P and the set of U(∞) × U(∞)invariant ergodic probability measures on Mat (N, C).
The Ergodic Decomposition Theorem (Theorem 1 and Corollary 1 of [6]) implies that each Pickrell measure µ (s) , s ∈ R, induces a unique decomposing measure µ (s) on Ω P such that we have (18) µ The integral is understood in the usual weak sense, see [6]. For s > −1, the measure µ (s) is a probability measure on Ω P , while for s −1 the measure µ (s) is infinite.
Set x n }.
The map ω → conf(ω) is bijective in restriction to the subset Ω 0 P . Remark. In the definition of the map conf, the "asymptotic eigenvalues" x n are counted with multiplicities, while, if x n 0 = 0 for some n 0 , then x n 0 and all subsequent terms are discarded, and the resulting configuration is finite. We shall see, however, that the complement Ω P \Ω 0 P is µ (s) -negligible for all s = −1 − 2k, k ∈ N, and, consequently, that, µ (s) -almost surely, all configurations are infinite. It will also develop that, µ (s) -almost surely, all multiplicities are equal to one.
We proceed to the formulation of the main result of this note, an explicit description of the measures µ (s) for s = −1 − 2k, k ∈ N.
(see, e.g., page 295 in Tracy and Widom [32]). The kernel J s induces on L 2 ((0, +∞), Leb) the operator of orthogonal projection onto the subspace of functions whose Hankel transform is supported in [0, 1] (see [32]). Setting x 1 = 4/x, x 2 = 4/y yields a kernel K (s) given by the formula (recall here that a change of variables The kernel K (s) induces on the space L 2 ((0, +∞), Leb) a locally trace class operator of orthogonal projection, for which, slightly abusing notation, we keep the symbol K (s) ; by the Macchì-Soshnikov Theorem, the operator K (s) induces a determinantal measure P K (s) on Conf((0, +∞)). The determinantal measure P K (s) is precisely the decomposing measure for the Pickrell measure µ (s) , as is shown by the following Proposition 12. Let s > −1. Then µ (s) (Ω 0 P ) = 1 and the µ (s) -almost sure bijection ω → conf(ω) identifies the measure µ (s) with the determinantal measure P K (s) .
Sketch of proof of Proposition 12. Take s > −1 and let P Following Pickrell, to a matrix z ∈ Mat(n, C) assign the collection (λ 1 (z), . . . , λ n (z)) of the eigenvalues of the matrix z * z arranged in nonincreasing order ( cf. (16)). The radial part r (n,s) of the Pickrell measure µ (s) n is now defined as the push-forward of the measure µ (s) n under the map z → (λ 1 (z), . . . , λ n (z)) .
The radial part of the Pickrell measure has determinantal form: The change of variables u i = λ i − 1 λ i + 1 , i = 1, . . . , n, reduces K (s) n to the Christoffel-Darboux kernel for the Jacobi orthogonal ensemble with weight Introducing the scaling λ i = n 2 x i , taking n → ∞ and using the classical asymptotics for Jacobi orthogonal polynomials (see, e.g., Szegö [31]), one finds lim convergence being uniform on compact subsets of (0, +∞). To prove that µ (s) (Ω 0 P ) = 1, the method of Section 7 in Borodin and Olshanski [4] is adapted to our situation.

A recurrence relation for Bessel point proceses.
The following observation motivates the construction of the next subsection. Given a finite family of functions f 1 , . . . , f N on the real line, let span(f 1 , . . . , f N ) stand for the vector space these functions span. For any s ∈ R, N ∈ N we clearly have If s > −1, then the informal meaning of (22) is that the space of the first N + 1 normalized Jacobi polynomials with weight (1 − u) s is a rank one perturbation of the space of the first N normalized Jacobi polynomials with weight (1 − u) s+2 .
A similar statement holds true for the Bessel kernel: using the recurrence relation J s+1 (x) = 2s x J s (x) − J s−1 (x) for Bessel functions, one easily obtains the recurrence relation for the Bessel kernels: the Bessel kernel with parameter s is thus a rank one perturbation of the Bessel kernel with parameter s + 2.
For ergodic decomposition measures of infinite Pickrell measures we shall now give a similar description in terms of infinite determinantal measures obtained as finite-rank perturbations of Bessel point processes.
Using Proposition 3, one easily checks that if R > 0 is big enough, then the subspace H (s) ⊂ L 2 ((0, +∞), Leb) and the subset E 0 = (0, R) satisfy Assumption 1. Let    The next step is to take the scaling limit of these infinite determinantal measures. This is achieved, with the use of Propositions 2 and 5, by taking the product with a suitably chosen multiplicative functional and effecting the scaling limit transition for the corresponding determinantal probability measures. The detailed proof of Theorem 2 will be published in the sequel to this note.