Boundedly finite measures: Separation and convergence by an algebra of functions

We prove general results about separation and weak$^\#$-convergence of boundedly finite measures on separable metric spaces and Souslin spaces. More precisely, we consider an algebra of bounded real-valued, or more generally a $*$-algebra $\mathcal{F}$ of bounded complex-valued functions and give conditions for it to be separating or weak$^\#$-convergence determining for those boundedly finite measures that integrate all functions in $\mathcal{F}$. For separation, it is sufficient if $\mathcal{F}$ separates points, vanishes nowhere, and either consists of only countably many measurable functions, or of arbitrarily many continuous functions. For convergence determining, it is sufficient if $\mathcal{F}$ induces the topology of the underlying space, and every bounded set $A$ admits a function in $\mathcal{F}$ with values bounded away from zero on $A$.


Introduction
Boundedly finite measures play an increasingly important role in probability theory. Classical examples are Itô excursion measures, or Lévy measures, which we will come back to in a moment.
But they also appear in very recent research as speed measures of Brownian motions on Ê-trees [AEW13,ALW15] or sampling measures for spatial Fleming-Viot processes [GSW15]. Convergence of metric measure spaces with boundedly finite measures has been analysed with a view towards probabilistic applications in [ALW16].
For the purpose of illustration, we quickly recall the situation for Lévy measures of infinitely divisible random variables. A real-valued random variable X is called infinitely divisible if, for any n ∈ AE, we can write X d = X 1,n + · · · + X n,n for an i.i.d. sequence of random variables (X i,n ) 1≤i≤n . * Preprint of Electron. Commun. Probab. 21 (2016), no. 60, 1-16. 1 University of Duisburg-Essen, Mathematics, e-Mail: wolfgang.loehr@uni-due.de 2 Institut für Math. Stochastik, Universität Göttingen, e-Mail: trippl@uni-goettingen.de 3 Supported by DFG SPP 1590 1 INTRODUCTION 2 A famous theorem by Lévy and Khintchine states that in that case the Fourier transform of the random variable has a very explicit form: (1) − log exp(−iuX) = iub + c 2 u 2 − Ê\{0} e −iuy − 1 + iuy½ |y|≤1 µ(dy), u ∈ Ê, where b ∈ Ê, c ≥ 0 and µ is a measure on Ê \ {0} satisfying Ê\{0} 1 ∧ |y| 2 µ(dy) < ∞, see Theorem 8.1 of [Sat99]. Let us look at the measure µ, called Lévy measure. The obvious questions are: (a) is the Lévy measure unique? 1 (b) If we have a sequence of infinitely divisible random variables, do their Lévy measures converge and in which sense? Lévy measures are not necessarily finite measures, but are required to be finite on any set which is not close to the origin 0. This motivates to consider measures which are finite on a certain class of sets. Daley and Vere-Jones used Appendix A2.6 in [DVJ03] to present a framework for such questions which they call boundedly finite measures, because the measures are assumed to be finite on bounded sets. The space of these measures is equipped with weak # -convergence, defined as convergence of integrals over bounded continuous functions with bounded support (see Section 2 for definitions). Some extensions are given in [HL06] and [LRR14]. Lévy measures fit into this framework if we change the Euclidean metric on Ê \ {0} such that 0 is sent infinitely far away, an idea also used in [BP06].
How does one prove weak convergence µ n w − → µ of probability measures on a topological space X in situations where it is not feasible to show convergence of f dµ n for all bounded continuous f ∈ C b (X) directly? One possibility is to find a class F ⊆ C b (X) of sufficiently "nice" functions, which is still rich enough to be convergence determining, i.e.
This approach has proven to be particularly fruitful if the topology on X itself is defined in terms of a class of functions. A classical example for such a topology is the weak (weak- * ) topology on (the dual of) a Banach space. A more modern one is the Gromov-weak topology on the space of metric measure spaces, which is induced both by a complete metric and by a class of functions called polynomials (see [GPW09]). That polynomials do not only induce the topology but are even convergence determining was shown with some effort in [DGP11]. But it also follows directly from a general result due to Le Cam, as pointed out in [Löh13]. Le Cam's theorem goes back to [LC57] and states that on a completely regular Hausdorff space X, a set of functions F ⊆ C b (X) is convergence determining for Radon probability measures if it is multiplicatively closed and induces the topology of X. The proof can be found in [HJ77, Proposition 4.1]. A version of Le Cam's theorem for separable metric spaces dropping the "Radon" assumption on the probability measures is given in [BK10]. This version was used extensively for the construction of a tree-valued pruning process in [LVW15]. Our main goal is to extend Le Cam's result to the case of boundedly finite measures and weak # -convergence and, because convergence determining is sometimes too much to ask for, to obtain (weaker) sufficient conditions for F to at least separate boundedly finite measures. A separating class of functions can also be used to prove weak (or weak # ) convergence if tightness is known by other methods. In particular, our results allow to give an answer on the question of uniqueness and convergence of µ in (1) within a general framework. More importantly, they will find applications in future work about spaces of metric measure spaces and Ê-trees such as in the upcoming paper [GGR] which was a driving motivation for the present article. The results will hopefully also facilitate the analysis of spatial population models on unbounded spaces with infinite total population size such as the one in [GSW15], and of other models appearing in modern areas of probability theory.
The rest of the paper is organized as follows. In Section 2, we give our main results about convergence determining (Theorem 2.3) and separating (Theorem 2.7 and Corollary 2.8) classes of functions for boundedly finite measures. In Section 3, we illustrate in four examples how our results can be applied. There, we consider Lévy measures, excursion theory and mass fragmentations.

Separation and convergence of boundedly finite measures
Let (X, d) be a separable metric space, endowed with the Borel σ-field induced by d. By C b (X), we denote the set of bounded continuous functions on the metric space (X, d) with values in .
For real-valued functions we write C b (X; Ê). Note that C b (X; Ê) ⊆ C b (X).
Definition 2.1 (Boundedly finite measures and weak # -convergence). The set of boundedly finite measures M # (X) on X w.r.t. d is given as ♦ Remark 2.2 (Weak # -convergence versus vague convergence). If (X, d) is a Heine-Borel space, i.e. every closed, bounded set is compact, then M # (X) coincides with the set of Radon measures on X, and weak # -convergence with vague convergence. For a general separable metric space, however, M # (X) is a subset of the Radon measures and weak # -convergence is a potentially much stronger convergence than vague convergence. ♣ Consider a set F of measurable, -valued functions on X and define Theorem 2.3 (Convergence determining for boundedly finite measures). Let (X, d) be a separable metric space and F ⊂ C b (X). Assume that (T.1) F is multiplicatively closed and closed under complex conjugation.
(T.2) F induces the topology of X.

(T.3) For every bounded set
Then F is weak # -convergence determining for measures in M # F (X), i.e.
Remark 2.4. 1. (T.1) and (T.2) are classical assumptions for these kind of theorems, see Proposition 4.1 in [HJ77]. Something like (T.3) is necessary to replace the fixed total mass in the weak convergence of probabilities. At least, we have to ensure that F "vanishes nowhere", because if there was x ∈ X with f (x) = 0 for all f ∈ F , then F could not even separate a · δ x for different a ≥ 0. For the purpose of separation of measures, we can do with this weaker requirement S.3 in Theorem 2.7 below. We do not know, however, if it would be enough for Theorem 2.3.
2. For real-valued functions, the part "closed under complex conjugation" is always satisfied. ♣ For the proof, we embed everything in the Hilbert cube H, a technique going back to Urysohn's work on metrisation and also used in [BK10]. Recall that H = [0, 1] AE , with product topology.
Definition 2.5 (P, P 0 ). Let P ⊆ C b (H; Ê) be the set of polynomials on H (i.e. functions depending on finitely many coordinates and an algebraic multivariate polynomial in these coordinates). Let P 0 := p ∈ P p(x) = 0 ∀ x = (x n ) n∈AE ∈ H with x 1 = 0 . ♦ Lemma 2.6 (Stone-Weierstrass variant). Let g : H → [0, 1] be continuous with supp g ⊂ H δ for some δ > 0. Then, for every ε > 0 there exists a polynomial p ε ∈ P 0 such that Theng ∈ C b (H; Ê) and we can use the Stone-Weierstrass theorem. So, for ε > 0 we findp ε ∈ P such that Define p ε (x) := x 1pε (x), x ∈ H. Then p ε ∈ P 0 and we get for any a ∈ (0, 1] the estimate sup x∈H, x1=a That is what we needed to show, as the case x 1 = 0 is trivial.
Proof of Theorem 2.3.
Step 1. It is enough to consider functions with values in [0, 1]: First, F may be replaced by where ℜ(f ) and ℑ(f ) are the real and imaginary parts of f , respectively. Because F is closed under complex conjugation due to (T.1), the conditions (T.1), (T.2) and (T.3) are also satisfied for F ′ instead of F . Thus we may assume F to consist of real-valued functions. Second, F may be replaced by which is contained in the vector space generated by F and easily seen to satisfy the prerequisites of the theorem provided that F does. Because F ′ maps to Ê + , we can assume, by normalisation, that the elements of F map into [0, 1].
Step 2. By Assumption (T.2), F induces the topology of X. Because X is a separable, metric space, it has a countable basis, and thus we can choose a countable subfamily of F that still induces the topology of X. Indeed, the family of sets of the form open is a base for the topology, and because X has a countable base, we can select a countable subfamily that is still a base 2 . Therefore there exist f 1 , f 2 , . . . ∈ F with 0 ≤ f m ≤ 1, such that (f m ) m∈AE induces the topology of X. Then ι : X → H, x → (f m (x)) m∈AE is a topological embedding (i.e. a homeomorphism onto its image) of X into H. Identifying X with ι(X), we assume w.l.o.g. X ⊆ H and f n to be the (restriction of the) n th coordinate projection. In particular, being an algebra, the linear span of F contains P 0 (defined in Definition 2.5).
Step 3. Let µ n , µ ∈ M # F (X) with For the claimed weak # -convergence, it is enough to show g dµ n → g dµ for all g : X → [0, 1] which have d-bounded support and are uniformly continuous (by the Portmanteau theorem, Theorem 2.1 in [LRR14]). Because weak convergence depends on the metric only through the induced topology, we may assume g to be uniformly continuous w.r.t. any other metric on X inducing the same topology as d. To this end, we take any metric on H inducing its topology, and assume that g is uniformly continuous w.r.t. its restriction to X (recall that X is a subspace of H by Step 2). By Assumption (T.3), there is f ∈ F and δ > 0 such that for all x ∈ supp(g) we have |f (x)| > δ. We may assume w.l.o.g. that f = f 1 (if not, we define f ′ 1 = f , f ′ m+1 = f m and observe that ι ′ defined with these f ′ m is still an embedding and g uniformly continuous w.r.t. the restriction of the metric on H). Then supp(g) ⊆ H δ . Furthermore, since g is uniformly continuous, it can be extended continuously to the closure of X in H, and by the Tietze extension theorem (e.g. [Mun00, Theorem 35.1]) to a continuous function from H to [0, 1] with support in H δ . We denote the extension again by g. We also identify µ n and µ with their natural extensions to H.
It is desirable to have a result which separates two boundedly finite measures but requires less than the previous theorem. While "boundedly finite" is essential for the definition of weak #convergence, we can drop this assumption for the purpose of separation and work with M F , the space of measures integrating F as defined in (2), instead of M # F . We can also relax the metrisability assumption on the space X, but do need some topological assumption. Recall that a topological space is Hausdorff if any two distinct points can be separated by open sets, and a Hausdorff topological space X is, by definition, a Souslin space if there exists a Polish space Y and a continuous surjective map from Y onto X. Note that a Souslin space is separable but need not be metrisable. An example is a separable Banach space in its weak topology, which is clearly Souslin but not metrisable. Conversely, not every separable metrisable space is Souslin. In the case of a Souslin space X, and a countable family of functions F , we can drop the topological assumptions on F from the prerequisites of Theorem 2.3 and still obtain the weaker conclusion that F is separating for measures in M F . More precisely, we have Theorem 2.7 (Separation of boundedly finite measures with measurable functions). Let X be a Souslin space (for example a Polish space), and F a countable set of bounded, measurable -valued functions. Assume that (S.1) F is multiplicatively closed and closed under complex conjugation.
(S.2) F separates points of X.
Then F is separating for measures in M F (X), i.e.
Proof. Assume that µ 1 , µ 2 ∈ M F (X) are such that f dµ 1 = f dµ 2 holds for all f ∈ F . We have to show µ 1 = µ 2 . Enumerate F = {f n | n ∈ AE}. Using Step 1 from the proof of Theorem 2.3, we may (and do) assume that f n takes values in [0, 1] for all n ∈ AE. We proceed in two steps: first we show that (4) holds if we assume that instead of (S.3) the following stronger condition holds, In the second step, we reduce the general case to the one where (S.3 † ) holds.
Step 1. Assume that (S.3 † ) holds and define ι : Then ι is measurable and injective by assumption (S.2). Because X is a Souslin space, it is an analytic measurable space ([Coh80, Proposition 8.6.13]) and so is Y := ι(X) ([Coh80, Corollary 8.6.9]). By [Coh80, Proposition 8.6.2], ι is a Borel isomorphism onto Y , i.e. ι −1 : Y → X is measurable. Therefore, Because of (S.3 † ), every x = (x n ) n∈AE ∈ Y satisfies x 1 = 0. We define the metric which induces on Y the topology inhereted as a subspace of H. Let G := {f • ι −1 | f ∈ F }. We show that (Y, r) and G satisfy the prerequisites of Theorem 2.3. G satisfies (T.1) because F does by assumption. By construction of ι, G coincides with the set of restrictions of coordinate projections to Y . Therefore, G induces the topology of Y , i.e. (T.2) is satisfied. An r-bounded set A in Y satisfies δ := inf x∈A |x 1 | > 0, and (T.3) is satisfied with f = f 1 • ι −1 . Thus we can apply Theorem 2.3 and obtain that G is weak # -convergence determining and a fortiori separating for measures in M # G (Y ). If µ 1 , µ 2 ∈ M F (X), then they integrate f 1 , andμ i := µ i • ι −1 , i = 1, 2, are boundedly finite measures on (Y, r). Thus obviouslyμ i ∈ M # G (Y ) and the claim of the theorem follows with (5).
Step 2. Now consider the general case, where (S.3 † ) does not necessarily hold. Define and let F ′ be the set of finite products of elements of F ∪ {f 0 }. Then F ′ is a countable set of measurable functions satisfying (S.1), (S.2) and, because F vanishes nowhere, also (S.3 † ) (with f 1 replaced by f 0 ∈ F ′ ). Hence, by Step 1, F ′ is separating for measures in M F ′ (X). According to the monotone convergence theorem, we have Because every element of F ′ is dominated by an element of F ∪ {f 0 } (recall that elements of F ′ map to [0, 1]), this implies µ 1 , µ 2 ∈ M F ′ (X). Moreover, dominated convergence yields which implies µ 1 = µ 2 by Step 1.
In the case of continuous functions, we can drop the countability of F . Proof. For x ∈ X, let f x be as in (S.3).
There is an open neighbourhood U x of x with f x (y) = 0 for all y ∈ U x . Recall that a topological space is called Lindelöf if every open cover has a countable subcover, and every Polish space has this property (because it has a countable base). Because the property is obviously preserved by continuous maps, every Souslin space is Lindelöf as well. Hence (U x ) x∈X has a countable subcover, and there exists a countable subfamily F 1 of F satisfying (S.3). Similarly, for x, y ∈ X let f xy ∈ F be such that f xy (x) = f xy (y). Then there is an open neighbourhood U xy of (x, y) in X 2 with f xy (u) = f xy (v) for all (u, v) ∈ U xy . Because X 2 is also Souslin and hence Lindelöf, we find a countable subfamily F 2 of F satisfying (S.2). Let F ′ be the closure of F 1 ∪ F 2 under multiplication and complex conjugation. Then F ′ satisfies the prerequisites of Theorem 2.7 and the conclusion follows.
3 Examples 3.1 Example 1: Lévy-Khintchine formula on Ê D Let Z be an infinitely divisible random variable with values in Ê D for D ∈ AE. That means for any n ∈ AE there are i.i.d. random variables Z 1,n , . . . , Z n,n such that Z d = Z 1,n + · · · + Z n,n . Consider It is well-known that there exist b ∈ Ê D , C ∈ S + (Ê D ) a symmetric, positive semidefinite matrix, and µ ∈ M # (X) with (1 ∧ x 2 ∞ ) µ(dx) < ∞ such that Proof. We start with the unique identification of the law.
Step 1. First note that where e k , k = 1, . . . , D are unit vectors in Ê D . Moreover, for k = 1, . . . , D: Hence C and b are unique.
and consider the following two classes of functions, where span denotes the linear span: Then with (6) and the uniqueness of b and C from Step 1, we have G dµ 1 = G dµ 2 for all G ∈ G. Now observe that, using linearity of u → ψ u (x) for every x ∈ Ê D , Hence, F is multiplicatively closed (by the first equality) and F ⊆ G. In particular, (S.1) holds, f dµ 1 = f dµ 2 ∀f ∈ F , and µ 1 , µ 2 ∈ M # F , because functions from G are integrable. Furthermore, F is contained in C b (X) and (S.2) and (S.3) are easily verified. Thus µ 1 = µ 2 follows from Corollary 2.8. Now we show the convergence result.
Step 4. Recall that for the Lévy-triple (b, C, µ) the linear part b depends on the choice of the compensation function ψ u , but C and µ do not. So in order to show µ n w # − − → µ we may choose any (admissible) ψ u we like. Replace ψ u in (7) byψ u (x) = iu t xh(x) for a C 1 -function h : Ê D → Ê with h(0) = 1 and compact support. Then the argument from above still works: F is multiplicatively closed and F ⊂ C b (X), so (T.1) holds.

Moreover, Assumption (T.2) holds by the fact that
for all u ∈ Ê D . The latter is nothing else than the convergence of the characteristic function of the measures δ(x n ) to δ(x). But this implies that x n → x in Ê D , so (T.2) holds.
Step 2. To identify µ we want to use Corollary 2.8. Since (X, d) is a Polish space it is also a Souslin space. Define for The linear span of functions of this kind is defined Then it is easy to see that F ⊂ C b (X).
Remark 3.4. The previous proof also works with Theorem 2.3 and thus allows to deduce a result on the convergence of the characteristics for sequences of infinitely divisible random measures. ♣

Example 3: Excursion measure of Brownian motion
Let P x ∈ M 1 C(Ê + , Ê) be the law of a 1-dimensional Brownian motion started in x ∈ Ê and denote the canonical process by (B t ) t≥0 . Let T 0 := inf{t > 0 : B t = 0} be the first hitting time of the origin 0. It is a folklore fact that the measure µ n := nP 1/n ((B t∧T0 ) t≥0 ∈ ·) converges, as n → ∞, to the Itô excursion measure µ exc of the reflected Brownian motion (|B t |) t≥0 . There are several ways to define µ exc . We use the characterisation given in Theorem XII.4.2 of [RY99] (where µ exc = 2n + for n + used in [RY99]) as definition. Then the σ-finite measure µ exc on X ′ is called Brownian excursion measure. ♦ Since µ exc is obviously not a finite measure, we have to be more precise about what we mean by convergence of µ n to µ exc . In Theorem 1 of [Hut09] it is shown (for a more general class of diffusions) that F dµ n → F dµ exc holds for every F ∈ C b (X ′ ) with the property that there is an ε > 0 with F (e) = 0 whenever e ∞ < ε. This looks very much like weak # -convergence on X ′ \ {0}, where 0 denotes the zero function and is sent infinitely far away. 3 This is, however, not precisely the case, because the map e → e ∞ is not continuous w.r.t. uniform convergence on compacta.
In this subsection we give a setup, where we can apply Theorem 2.3 to obtain a weak # - To this end, we have to modify the topology on (a subspace of) X ′ in two ways. First, we weaken uniform convergence on compacta to convergence in Lebesgue measure, because the latter is induced by "nice" functions and therefore much easier to handle in our framework. This, of course, substantially weakens our result, so that it does not imply the one in [Hut09]. Second, we strengthen the topology (and therefore our result) a bit by additionally requiring convergence of excursion lengths for the convergence of excursions. This allows us to send the zero function infinitely far away by a continuous function, and the result in [Hut09] does not directly include ours.  1. e n → e with respect to d.
In particular, (X, d) is a separable metric space.
Proof. For the "in particular" note that separability of convergence in Lebesgue-measure is wellknown and carries over to d-convergence by the first equivalence.  In order to use Theorem 2.3 to prove Theorem 3.9, we need a set F of continuous functions on X satisfying (T.1) -(T.3). To this end, define for f ∈ L 1 (Ê + ) and g ∈ C b (Ê + ) Denote by C c the continuous functions with compact support and define a set of bounded functions on X by Definition 3.10 (F ). Let F be the multiplicative closure of F ′ . ♦ Lemma 3.11. F is weak # -convergence determining for measures in M # F (X). Proof. F obviously satisfies (T.1) and (T.3). Indeed, F is multiplicatively closed by definition, and A ⊆ X is d-bounded if and only if inf e∈A ζ(e) > 0. Once we have shown (T.2), i.e. that F induces the same topology as d, the claim follows from Theorem 2.3. To this end, note that ζ(e n ) → ζ(e) is necessary for both d-and F -convergence, and recall that under this condition, by Lemma 3.8, d-convergence is equivalent to The set of ϕ f,g of the form ϕ f,g (t, x , and induces the Euclidean topology on Ê 2 + . Thus, by the classical Le Cam theorem, it is convergence determining, and (9) is equivalent to which implies the claim.
On the other hand, recall that T 0 = ζ( B) is the hitting time of 0 of B and observe For fixedt, reordering if necessary, we assume for notational convenience that t 1 ≤ · · · ≤ t n =t and set t 0 = 0. Then, using the Markov property at t n in the first step, where we have used that the sub-Markovian semigroup (Q t ) t≥0 of the killed Brownian motion B and the Markovian semigroup (H t ) t≥0 of the Bessel process ρ are related by H t (x, dy) = x −1 Q t (x, dy)y for x > 0, 2κ(t)y 2 exp(−y 2 /(2t)) dy for x = 0.
Because B is a Feller-process and h ∈ C b , the function x → x h(t n + T 0 ) is also a bounded continuous function. Because also x → x −1 g n (x) is bounded and continuous by assumption, we see that the term inside the outer expectation in (13) is a bounded continuous function in ρ t1 , . . . , ρ tn . Using that ρ is a Feller process, we obtain from (12) and (13) The hitting time T 0 of 0 under P x is known to be Lévy-distributed with scale parameter x, hence Inserting (15) into (14), and applying (11), we obtain the claimed convergence (10).

Example 4: Mass fragmentations
Consider the set S ↓ = s = (s 1 , s 2 , . . . ) ∈ [0, 1] AE i≥1 s i ≤ 1, s 1 ≥ s 2 ≥ s 3 ≥ · · · ≥ 0 of decreasing sequences (s 1 , s 2 , . . . ) with sum less than 1. This set is used to model (mass) fragmentation processes. A perfect introduction to the topic is given in Bertoin's book [Ber06]. In his Section 2.1, the set S ↓ is introduced and endowed with the topology of pointwise convergence. Our goal is to present two sets of real-valued convergence-determining functions on S ↓ , namely Theorem 3.12. The sets G 1 and G 2 are convergence-determining on S ↓ in the sense that for s(n), s ∈ S ↓ , n ∈ AE the following holds for j = 1, 2: Before we give the proof, we relate the set S ↓ to boundedly finite measures. Therefore, let Consider the mapping Φ : Lemma 3.13. The mapping Φ is a homeomorphism from S ↓ to Φ(S ↓ ). Proof.
Step 1. The mapping Φ is injective: For s ∈ S ↓ , all s i , i ≥ 1 can be easily reconstructed.
Step 2. The mapping Φ is continuous: Let s(n), s ∈ S ↓ with s(n) → s. If A is a bounded set in X w.r.t. d with Φ(s)(∂A) = 0, then it is clear that Φ(s(n))(A) → Φ(s)(A) and we can use Proposition A.2.6.II (d) in [DVJ03].
Proof of Theorem 3.12. We only provide the proof for G 1 , since the other proof is similar.
Step 1. Note that we can write G p (s) = x p Φ(s)(dx), p ∈ AE.
We use Theorem 2.3 to show that Φ(s(n)) w # − − → Φ(s) and this suffices for s(n) → s by Lemma 3.13.
Step 2. Of course (X, d) as before Lemma 3.13 is a separable metric space. Moreover, the class F = span{x → x p | p ∈ AE} ⊂ C b (X) of polynomials without constant term on X satisfies the prerequisites of Theorem 2.3, which is easy to check. Finally, Φ(s) ∈ M # F (X) for all s ∈ S ↓ since for all p ∈ AE: Thus, Theorem 2.3 applies and yields the claim.
Corollary 3.15. On the subset S ↓ 1 = {s ∈ S ↓ | i≥1 s i = 1}, the set of functions G 1 generates the topology of pointwise convergence.
Proof. First, the subset is a Polish space with the relative topology since it is a G δ -subset of S ↓ .
Step 1. First suppose that s(n) → s as n → ∞. We can use an approximation argument to get G(s(n)) → G(s) for all G ∈ G 1 : Let ε > 0. Fix z > 0 s.t. i≥1 s i ½ si≤z < ε and s i = z for all i ≥ 1.
Use that in the following equation Once we establish that the last term is small, we see that G p (s(n)) → G p (s). But the last term is bounded by: i≥1 s i (n) p ½ si(n)≤z ≤ i≥1 s i (n)½ si(n)≤z = 1 − i≥1 s i (n)½ si(n)>z ≤ 1 − (1 − ε) = ε.
Note that it was crucial to know that i≥1 s i = 1 in the last proof.
Step 2. The converse direction was established in Theorem 3.12.
Remark 3.16. An application in a similar spirit is given in [GGR], where X is the set of ultrametric measure spaces with diameter in [0, 2h) for certain h > 0. Any ultrametric measure space with diameter in [0, 2h] can be written as a boundedly finite measure on X (in a unique way similar to a prime factorization). This relation can be used for a result analogous to Corollary 3.15. ♣