Beurling slow and regular variation

We give a new theory of Beurling regular variation (Part II). This includes the previously known theory of Beurling slow variation (Part I) to which we contribute by extending Bloom's theorem. Beurling slow variation arose in the classical theory of Karamata slow and regular variation. We show that the Beurling theory includes the Karamata theory.


Introduction
Beurling slow variation (already known) and Beurling regular variation (new here) are related to the classical Karamata (or ordinary) slow and regular variation. This is the study of limit relations of the form written multiplicatively, or written additively. For background on the extensive theory and applications here, see [9] (BGT below). As we show (see § 10.3), the Beurling theory here includes the Karamata theory. One early success of Karamata regular variation was in Wiener Tauberian theory. Here the basic result is the following theorem.
Theorem W (Wiener's Tauberian theorem). For K ∈ L 1 (R) with the Fourier transform K of K non-vanishing on R, and H ∈ L ∞ (R): if

K(x − y)H(y) dy −→ c K(y) dy (x −→ ∞),
then, for all G ∈ L 1 (R), There are corresponding multiplicative versions with The roots of Beurling regular variation lie in a variant of Karamata regular variation, Bojanic-Karamata/de Haan theory (BGT, Ch. III, [24]), which goes back to [23]. The Bojanic-Karamata/de Haan Π-class is an important subclass of the class of Karamata slowly varying functions. These functions are increasing: their (rapidly increasing) inverses form the de Haan Γ-class. This in turn motivates our theory of Beurling regular variation developed in Part II.
For further background and more references, see our arXiv papers [17,18] and [19]. Despite the numerous positive results below, the question that motivated this study remains open, at least in its original form. Does Bloom's theorem/UCT extend to measurable/Baire functions, that is, can one omit the Darboux requirement? Does it even extend to Baire-1 functions?
Part I. Beurling slow variation

G x − y ϕ(x) H(y) dy/ϕ(x) −→ c G(y) dy (x −→ ∞).
Note that the arguments of K and G here involve both the additive group operation on the line and the multiplicative group operation on the half-line. Thus Beurling's Tauberian theorem, although closely related to Wiener's (which it contains, as the case ϕ ≡ 1), is structurally different from it. One may also see here the relevance of the affine group, Aff, already well used for regular variation (see, for example, BGT § 8.5.1 and § § 3, 5).
Analogously to Karamata's UCT, the following result was proved by Bloom in 1976 [21] (an extended and simplified version is in BGT, Theorem 2.11.1).
As above, our motivating question here is whether one can extend this to ϕ measurable and/or Baire; see BGT § 2.11, [45,IV.11] for textbook accounts. We give below a number of results in this direction. Our methods involve tools from infinite combinatorics, and replacement of quantitative measure theory by qualitative measure theory. We also prove a representation theorem (Theorem 8), whence any ϕ ∈ SN satisfies ϕ(x) = o(x), so ϕ ∈ BSV.

Monotone functions
We suggest that the reader cast his eye over the proof of Bloom's theorem, in either [21] or BGT § 2.11, it is quite short. Like most proofs of the UCT for Karamata slow variation, it proceeds by contradiction, assuming that the desired uniformity fails, and working with two sequences, t n ∈ [−T, T ] and x n → ∞, witnessing to its failure.
The next result, in which we assume that ϕ monotone (ϕ increasing to infinity is the only case that requires proof) is quite simple. But it is worth stating explicitly, for three reasons.
(1) It is a complement to Bloom's theorem, and to the best of our knowledge the first new result in the area since 1976.
(2) The case of ϕ increasing is by far the most important one for applications. For, taking G the indicator function of an interval in Beurling's Tauberian theorem, the conclusion there has the form of a moving average: Such moving averages are Riesz (typical) means, and here ϕ increasing to ∞ is natural in context. For a textbook account, see [30] and the recent [19]; for applications, in analysis and probability theory, see [8,20]. The prototypical case is ϕ(x) = x α (0 < α < 1); this corresponds to X ∈ L 1/α for the probability law of X.
(3) Theorem 1 is closely akin to results of de Haan on the Gumbel law Λ in extreme-value theory; see BGT § § 3.10, 8.13.
We offer three proofs (two here and a third after Theorem 2M in § 4) of the result, as each is short and illuminating in its own way.
For the first, recall that if a sequence of monotone functions converges point-wise to a continuous limit, the convergence is uniform on compact sets. See, for example, Pólya [22, § 17, pp. 104-105]. (The proof is a simple compactness argument, complementing the better-known result of Dini, in which it is the convergence, rather than the functions, that is monotone; see, for example, [64, 7.13].) Theorem 1 (Monotone Beurling UCT). If ϕ ∈ BSV is monotone, then ϕ ∈ SN: the convergence in (BSV) is locally uniform.
First proof. As in [21] or BGT § 2.11, we proceed by contradiction. Pick T > 0, and assume that the convergence is not uniform on [−T, 0] (the case [0, T ] is similar). Then there exist ε 0 ∈ (0, 1), t n ∈ [−T, 0] and x n → ∞ such that Then f n is monotone, and tends point-wise to 0 by (BSV). So, by the Pólya-Szegő result above, the convergence is uniform on compact sets. This contradicts |f n (t n )| ε 0 for all n.
The second proof is based on the following result, thematic for the approach followed in § 4. We need some notation that will also be of use later. Below, x > 0 will be a continuous variable, or a sequence x := {x n } diverging to +∞ (briefly, divergent sequence), according to context. We put Lemma 1. For ϕ > 0 monotonic increasing and {x n } a divergent sequence, each set V x n (ε), and so also each set H x k (ε), is an interval containing 0.
and so s ∈ V x n (ε). The remaining assertions now follow, because an intersection of intervals containing 0 is an interval containing 0.
Second proof of Theorem 1. Suppose otherwise; then there are ε 0 > 0 and sequences x n := x(n) → ∞ and u n → u 0 such that Since ϕ is Beurling slowly varying, the increasing sets H x k (ε 0 ) cover R + and so, being increasing intervals (by Lemma 1), their interiors cover the compact set K := {u n : n = 0, 1, 2, . . .}. So, for some integer k, the set H x k (ε 0 ) already covers K, and then so does V x k (ε 0 ). But this implies that contradicting the above at n = k.
Remark. Of course, the uniformity property of ϕ is equivalent to the sets H x k (ε) containing arbitrarily large intervals [0, t] for large enough k (for all divergent {x n }).

Infinite combinatorics
As usual with proofs involving regular variation the nub lies in infinite combinatorics, to which we now turn. We recall that one can handle Baire and measurable cases together by working bi-topologically, using the Euclidean topology in the Baire case (the primary case) and the density topology in the measure case; see § 10.2. The negligible sets are the meagre sets in the Baire case and the null sets in the measure case; we say that a property holds quasi everywhere if it holds off a negligible set.
We work in the affine group Aff acting on (R, +) using the notation where c n → c 0 = c > 0 and z n → 0 as n → ∞. These are to be viewed as (  [43, 17.47].) We recall the following definition and theorem from [12], which we apply taking the space X to be R with one of E or D.
Definition. A sequence of homeomorphisms h n : X → X satisfies the weak category convergence condition (wcc) if: For any non-meagre open set U ⊆ X, there is a non-meagre open set V ⊆ U such that, for each k ∈ N, Theorem CET (Category Embedding Theorem). Let X be a topological space and h n : X → X be homeomorphisms satisfying (wcc). Then, for any Baire set T, for quasi-all t ∈ T there is an infinite set M t ⊆ N such that From here, we deduce the following lemma.
Proof. It is enough to prove the existence of one such point b, as the Generic Dichotomy Principle (for which see [14,Theorem 3.3]) applies here, because we may prove the existence of such a b in any non-negligible G δ -subset B of B, by replacing B below with B . (One checks that the set of bs with the desired property is Baire, and so its complement in B cannot contain a non-negligible G δ .) Writing T := cB and w n = c n c −1 , so that c n = w n c and w n → 1, put h n (t) := w n t + z n .
Then h n converges to the identity in the supremum metric, so (wcc) holds by [13,Theorem 6.2] (First Verification Theorem), and so Theorem CET above applies for the Euclidean case; applicability in the measure case is established as [11,Corollary 4.1]. (This is the basis on which the affine group preserves negligibility.) So there are t ∈ T and an infinite set of integers M with But t = cb for some b ∈ B and so, as w m c = c m , one has

Darboux property
Here we generalize Bloom's Theorem (from continuity to the Darboux property) and simplify his proof.
We recall the Darboux property, also called the intermediate value property, that if a (realvalued) function attains two values in an interval then it must attain all intermediate values in that interval. Bloom uses continuity only through the Darboux property. It is much weaker than continuity, it does not imply measurability, nor the Baire property. For measurability, see the papers of Halperin [39,40]; for the Baire property, see, for example, [63] and also § 10.2. Conversely, neither measurability nor the Baire property implies the Darboux property, as 1 Q shows.
We use Lemma 2 to prove Theorem 4B, which implies Bloom's Theorem, as continuous functions are Baire and Darboux. We note a result of Kuratowski and Sierpiński [48] that, for a function of Baire class 1 (for which see below), the Darboux property is equivalent to its graph being connected; so Theorem 4 goes beyond the class of functions considered by Bloom. We begin with some infinite combinatorics associated with a positive function ϕ ∈ BSV.
Definitions. Say that {u n } with limit u is a witness sequence at u (for non-uniformity in ϕ) if there are ε 0 > 0 and a divergent sequence x n such that, for h = log ϕ, Say that {u n } with limit u is a divergent witness sequence if also Thus a divergent witness sequence is a special type of witness sequence, but, as we show, it is these that characterize the absence of uniformity in BSV.
We begin with a lemma that yields simplifications later; it implies a Beurling analogue of the Bounded Equivalence Principle in the Karamata theory, first noted in [10]. As it shifts attention to the origin, we call it the Shift Lemma of BMA. Below uniform near a point u means 'uniformly on sequences converging to u' and is equivalent to local uniformity at u (that is, on compact neighbourhoods of u).
Theorem 2B (Divergence Theorem, Baire version). If ϕ ∈ BSV has the Baire property and u n with limit u is a witness sequence, then u n is a divergent witness sequence.
Proof. As u n is a witness sequence, for some x n → ∞ and ε 0 > 0 one has (1), with h = log ϕ, as always. By the Shift Lemma (Lemma 3), we may assume that u = 0. So (as in the proof of Lemma 2) we will write z n for u n . If z n is not a divergent witness sequence, then {ϕ(x n +z n ϕ(x n ))/ϕ(x n )} contains a bounded subsequence and so a convergent sequence. Without loss of generality, we thus also have Write γ n (s) := c n s + z n and y n : Now take η = ε 0 /3 and amend the notation of § 2 to read These are Baire sets, and as ϕ ∈ BSV. The increasing sequence of sets {H x k (η)} covers R. So for some k, the set H x k (η) is non-negligible. Furthermore, as c > 0, the set c −1 H x k (η) is non-negligible and so, by (4), for some l the set . In particular, for this t and m ∈ M t with m > k, l one has 'absorbing' the affine shift γ m (t) into y. So, by (5), Combining, Theorem 2M (Divergence Theorem, Measure version). If ϕ ∈ BSV is measurable and u n with limit u is a witness sequence, then u n is a divergent witness sequence.
Proof. The argument above applies, with D in place of E.
As an immediate corollary, we have the following.
Third proof of Theorem 1. If not, then there exists a witness sequence u n with limit u. By Lemma 3, without loss of generality so there is N such that both (1/2)ϕ(x n ) < ϕ(x n + wϕ(x n )) and ϕ(x n + vϕ(x n )) < 2ϕ(x n ) for all n > N. By increasing N if necessary, we may take w < u n < v for n > N. But then We also have, as a further corollary, the following theorem.
Proof. Suppose otherwise. As above, by Lemma 3 and Theorem 2, there exists a divergent witness sequence z n → 0 such that, for some x n → ∞ and ε 0 > 0, the inequality (1) holds with u n = z n and y n := x n + z n ϕ(x n ). Without loss of generality, we may assume that y n > 0 and |z n | 1 for all n.
Suppose first that 0 < K < ϕ(x) < L for all x; then 0 < K/L < ϕ(y n )/ϕ(x n ) < L/K for all n, and so the witness sequence is not divergent, which is a contradiction.
Next, suppose that 0 < K < ϕ(x)/x < L for all x (possible as ϕ(x) = o(x)). Without loss of generality, we may now also assume that |z n | < 1/2L for all n and so |z n ϕ(x n )/x n | < 1 2 . Then So, again the witness sequence is not divergent, which is a contradiction.
This gives the following theorem.
Proof. Suppose not; take h = log ϕ. Then there exists a witness sequence v n with limit v, and in particular, for some x n → ∞ and ε 0 > 0, one has inequality (1) modified so that v n replaces u n .
We construct below a convergent sequence u n , with limit u say, such that and also the unmodified (1) holds. This will contradict Theorem 2B.
The proof here splits according as Here we appeal to the Darboux property to replace the sequence {v n } by another sequence {u n } for which the corresponding differences are convergent.
has the Darboux property and f n (0) = 0. Either f n (v n ) ε 0 , and so there exists u n between 0 and v n with f n (u n ) = ε 0 , or −f n (v n ) ε 0 , and so there exists u n with −f n (u n ) = ε 0 . Either way, |f n (u n )| = ε 0 . Without loss of generality, {u n } is convergent with limit u, say, since {v n } is so, and now (6) and (1) hold, the latter as in fact Case (ii). h(x n + v n ϕ(x n )) − h(x n ) versus are bounded. In this case, we can get (2) by passing to a subsequence.
In either case, we contradict Theorem 2B.
Appealing instead to Theorem 2M, the same argument gives the following theorem.
Remarks. (1) The Darboux property in Theorems 4 may be replaced with a weaker local property. It is enough to require that ϕ be locally range-dense, that is, that at each point t there is a bounded open neighbourhood I t such that the range ϕ , or be in the class A 0 of [26, § 2]; cf. also [27].
(2) The proofs of Theorems 4B and 4M begin as Bloom's does, but only in the case (i) of the first step, and even then we appeal to the Darboux property rather than the much stronger assumption of continuity. Thereafter, we are able to use Theorem CET to base the rest of the proof on Baire's category theorem. This enables us to handle Theorems 4B and 4M together, by qualitative measure theory; in contrast, the proofs of Bloom's theorem in [21] and BGT § 2.11 use quantitative measure theory, see § 10.2.
(4) That the Darboux property is natural here will emerge in § 6 on topological dynamics. Part II. Beurling regular variation

The setting
We begin by setting the context of what we call Beurling regular variation, extending the Beurling slow variation of Part I. We use the infinite combinatorics of § 2 to establish (in § 7) a Beurling analogue of the UCT of Karamata theory. In § 6, we discuss the flow issues raised below; there flow rates, time measures and cocycles are introduced. Here we discuss the connection between the orbits of the relevant flows and the Darboux property that plays such a prominent role in Part I. Incidentally, this explains why the Darboux property is quite natural there. These ideas prepare the ground for a Beurling version of the UCT. We deduce a Characterization Theorem in § 8. Armed with these two theorems, we are able in § 9 to establish various Representation Theorems for Beurling regularly varying functions, but only after a review of Bloom's work on the representation of self-neglecting functions, from which we glean Smooth Variation Theorems. We close in § 10 by commenting on the place of Karamata theory, and of de Haan's theory of the Γ-class, relative to the new Beurling theory.
Below the reader should have clearly in mind two isometric topological groups: the real line under addition with (the Euclidean topology and) Haar measure Lebesgue measure dx, and the positive half-line under multiplication, with Haar measure dx/x, and metric d W (x, y) = | log y − log x|. ('W for Weil', as this generates the underlying Weil topology of the Haar measure, for which see [38, § 62; 68].) As usual, we will move back and forth between these two as may be convenient, by using their natural isomorphism exp/log. Again as usual, we work additively in proofs, and multiplicatively in applications; we use the convention h := log f, k := log g.
As in § 1, we need to use both addition and multiplication simultaneously, so Aff is natural here. Recall that on the line the affine group x → ux + v with u > 0 and v real has (right) Haar measure u −1 du dv (or (du/u) dv, as above), see [42,IV,(15.29)]. This explains the presence of the two measure components, the 'du/u' and the 'dv', in (Γ ρ ) of Theorem 10 ( § 9). Generalizations of Karamata's theory of regular variation (BGT; cf. [45]) rely on a group G acting on a space X in circumstances where one can interpret 'limits to infinity' x → ∞ in the following expression: Here an early treatment is [1] followed by [2], but a full topological development dates from our recent papers (for which see [17,18], and for the particular role of group action [55,58]). Recall that a group action A : G × X → X requires two properties: with the maps x → g(x) := A(g, x), also written gx, often being homeomorphisms. An action A defines an A-flow (also referred to as a G-flow), whose orbits are the sets Ax = {gx : g ∈ G}.
In fact, (i) follows from (ii) for surjective A (as A(g, y) = A(1 G , A(g, y))), so we will say that A is a pre-action if just (i) holds, and then continue to use the notation g(x) : = A(g, x); it is helpful here to think of the corresponding sets Ax as orbits of an A-preflow, using the language of flows and topological dynamics [6].
Below we relax the definition of regular variation so that it relies not so much on group action but on asymptotic 'cocycle action' associated with a group G. This will allow us to develop a theory of Beurling regular variation analogous to the Karamata theory, in which the regularly varying functions are those Baire or measurable f which, for some fixed self-neglecting ϕ, possess a non-zero limit function g (not identically zero modulo null/meagre sets) satisfying (so that g(0) = 1). Equivalently, (so that k(0) = 0). This latter equivalence is non-trivial: it follows from Theorem 7 that if g is a non-zero function, then it is in fact positive. Specializing (BRV) to the sequential format one sees that the limit function g is Baire/measurable if f is so. We refer to functions f satisfying (BRV) as (Beurling) ϕ-regularly varying. This takes us beyond the classical development of such a theory restricted to the class Γ of monotonic functions f (BGT § 3.10; de Haan [37]). We prove in Theorem 6 a UCT for Baire/measurable functions f with non-zero limit g, not previously known, and in Theorem 7, the Characterization Theorem, that a Baire/measurable function f is Beurling ϕ-regularly varying with non-zero limit if and only if, for some ρ, one has where ρ is the Beurling ϕ-index of regular variation. Baire and measurable (positive) functions of this type form the class Γ ρ (ϕ) (cf. Omey [57], for f measurable; see also [32] for the analogous power-wise approach to Karamata regular variation).

Topological dynamics: flows, orbits, cocycles
Our approach is to view Beurling regular variation as a generalization of Karamata regular variation obtained by replacing the associativity of group action by a form of asymptotic associativity. To motivate our definition below, take X and G both to be (R, +), ϕ ∈ SN and consider the map Think of t as representing translation. For fixed t put t(x), or just tx := T ϕ t (x) = T ϕ (t, x) = x + tϕ(x), so that 0(x) = x, and so T ϕ is a pre-action. Here we have T ϕ is not a group action, as associativity fails. However, just as in a proper flow context, here too one has a well-defined flow rate, or infinitesimal generator, at x, for which see [6,10,65,13.34] (cf. [2]), There is of course an underlying true flow here, in the measure case, generated † by ϕ > 0 (with 1/ϕ locally integrable) and described by the system of differential equations (writing u x (t) for u(t, x))u (The inverse problem, for t(u) with t(0) = 1, has an explicit increasing integral representation, yielding u x (t) := u(t + t(x)), where u(t(x)) = x, as u and t are inverse.) The 'differential flow' Φ : (t, x) → u x (t) is continuous in t for each x. As such, Φ is termed by Beck a quasi-flow. ‡ In contrast 'translation flow', that is, (t, x) → x + t, being jointly continuous, is a 'continuous flow', briefly a flow. It is interesting to note that, by a general result of Beck (see [ , and f x (0) = 0 for all x. This will be the case when ϕ has the intermediate value property, and so here the Darboux property says simply that orbits embed. Cocycles are thus central to the flow analysis of regular variation. It is this differential flow that the, algebraically much simpler, Beurling preflow circumvents, working not with the continuous translation function f x (t) but tϕ(x), now only measurable, but with the variables separated. Nevertheless, the differential equation above is the source of an immediate interpretation of the integral , † Positivity is key here; x = 0 is a fixed point of the flowu = ϕ(u) when ϕ(x) = |x|. ‡ Beck denotes flows by ϕ(t, x) and uses f where we use ϕ. As we follow the traditional notation of ϕ for self-neglecting functions, the flow here is denoted by Φ. arising in the representation formula (Γ ρ ) of Theorem 10 ( § 9) for a regularly varying function f , as the metric of time measure (in the sense of Beck [6, p. 153]). The metric is the occupationtime measure (cf. BGT § 8.11) of the interval [1, x] under the ϕ-generated flow started at the natural origin of the multiplicative group R + . For ϕ(x) = x, § the ϕ-time measure is Haar measure, and the associated metric is the Weil (multiplicatively invariant) metric with d W (1, x) = | log x|, as in § 5. In general, however, the ϕ-time measure μ ϕ is obtained from Haar measure via the density x/ϕ(x), interpretable as a time-change 'multiplier' w(x) := ϕ(x)/x (cf. [6, 5.41]).
Granted its interpretation, it is only to be expected in (Γ ρ ) that τ x multiplies the index ρ describing the asymptotic behaviour of the function f. The time integral τ x is in fact asymptotically equal to the time taken to reach x from the origin under the Beurling pre-action T ϕ , when ϕ ∈ SN, namely x/ϕ(x). We hope to return to this matter elsewhere.
Actually, T ϕ is even closer to being an action: it is an asymptotic action (that is, asymptotically an action), in view of two properties critical to the development of regular variation. The first (for the second see below Lemma 4) refers to the dual view of the map (t, x) → x(t) = x + tϕ(x) with x fixed (rather than t, as at the beginning of the section). Here we see the affine transformation α x (t) = ϕ(x)t + x. This auxiliary group plays its part through allowing the absorption of a small 'time' variation t + s of t into a small 'space' variation in x. This involves a concatenation formula, earlier identified in [10] as a component in the abstract theory of the index of regular variation.
As for the concatenation formula, one has As for the second property of T ϕ , recall that, for G a group acting on a second group X, a G-cocycle on X is a function σ : G × X → X defined by the condition This definition is already meaningful if a pre-action rather than an action is defined from G × X to X; and so for the purposes of asymptotic analysis one may capture a weak form of associativity as follows using Lemma 4. For a Banach algebra X, we let X −1 denote the set of its invertible elements.
Definition. For X a Banach algebra, given a pre-action T : G × X → X (that is, with 1 G x = x for all x, where, as above, gx : = T (g, x)), an asymptotic G-cocycle on X is a map σ : G × X → X −1 with the property that, for all g, h ∈ G and ε > 0, there is r = r(ε, g, h) such that, for all x with x > r, Say that the cocycle is locally uniform if the inequality holds uniformly on compact (g, h)-sets.
Remark. Baire and measurable cocycles are studied in [58] for their uniform boundedness properties. One sees that the Second and Third Boundedness Theorems proved there hold in the current setting with asymptotic cocycles replacing cocycles. We now verify that replacing X −1 by R + , and taking T = T ϕ and the natural cocycle of regular variation σ f (t, x) := f (tx)/f (x), the above property holds. The case f = ϕ comes first; the general case of ϕ-regularly varying f must wait till after Theorem 7.
regarded as a map into the Banach algebra R, is a locally uniform asymptotic (R, +)-cocycle, that is, for every ε > 0 and compact set K, there is r such that, for all s, t ∈ K and all x with x > r, Proof. Let ε > 0. Given s, t, let I be any open interval with s + t ∈ I. Pick δ > 0 so that the interval J = (1 − δ, 1 + δ) satisfies t + sJ ⊆ I. Next pick r such that, for x > r, both In particular, Noting, as in Lemma 4, that so, for x > r, one has

Uniform convergence theorem
We begin with a lemma that yields simplifications later. As before, we call it a Shift Lemma. It has substantially the same statement concerning local uniformity and proof as Lemma 3 § 4 except that here h = log f, whereas there one has h = log ϕ, so that here the difference h(x n + uϕ(x n )) − h(x n ) tends to k(u) rather than to zero. So we omit the proof.

Lemma 5 (Shift Lemma: uniformity preservation under shift). For any u, convergence in (BRV + ) is uniform near t = 0 if and only if it is uniform near t = u.
Definition. Say that {u n } with limit u is a witness sequence at u (for non-uniformity in h) if there are ε 0 > 0 and a divergent sequence x n such that, for h = log f, Theorem 6 (UCT for ϕ-regular variation). For ϕ ∈ SN, if f has the Baire property (or is measurable) and satisfies (BRV) with limit g strictly positive on a non-negligible set, then f is locally uniformly ϕ-RV .
Proof. Suppose otherwise. We modify a related proof from § 4 (concerned there with the special case of ϕ itself) in two significant details. In the first place, we will need to work relative to the set S := {s > 0 : g(s) > 0} ('S' for support), so that k(s) = log g(s) is well defined on S. Now S is Baire/measurable; as S is non-negligible, by passing to a Baire/measurable subset of S if necessary, we may assume without loss of generality that the restriction k|S is continuous on S, by [47, § 28] in the Baire case and Luzin's Theorem in the measure case [60,Chapter 8]; [43, 17.12]).
Let u n be a witness sequence for the non-uniformity of h and so, for some x n → ∞ and ε 0 > 0, one has (8). By the Shift Lemma (Lemma 5),we may assume that u = 0. So we will write z n for u n . As ϕ is self-neglecting, Write γ n (s) := c n s + z n and y n := x n + z n ϕ(x n ). Then y n = x n (1 + z n ϕ(x n )/x n ) → ∞, (as ϕ(x) = o(x), see Theorem 8) and, as k(0) = 0, Now take η = ε 0 /4 and, for x = {x n }, working in S, put and likewise for y = {y n }. These are Baire sets, and as h ∈ BRV + . The increasing sequence of sets {H x k (η)} covers S. So, for some k, the set H x k (η) is non-negligible. As H x k (η) is non-negligible, by (11), for some l the set B := H x k (η) ∩ H y l (η) is also non-negligible. Taking A := H x k (η), one has B ⊆ H y l (η) and B ⊆ A with A, B nonnegligible. Applying Lemma 2 to the maps γ n (s) = c n s + z n with c = lim n c n = 1, there exist b ∈ B and an infinite set M such that . In particular, for this t and m ∈ M t with m > k, l one has t ∈ V y m (η) and γ m (t) ∈ V x m (η). As t ∈ S and γ m (t) ∈ S (a second critical detail), we have by the continuity of k|S at t, since γ m (t) → t, that, for all m large enough Fix such an m.
We have seen in the Beurling UCT how uniformity in the auxiliary function ϕ passes 'out' to ϕ-regularly varying f . For the converse (uniformity passing 'in' from f to ϕ), we note the following.

Characterization theorem
We may now deduce the characterization theorem, which implies in particular that the support set S of the proof of Theorem 6 is in fact all of R.
Theorem 7 (Characterization Theorem). For ϕ ∈ SN, if f > 0 is ϕ-regularly varying and Baire/measurable and satisfies with non-zero limit, that is, g > 0 on a non-negligible set, then, for some ρ (the index of ϕ-regular variation), one has Proof. Proceed as in Theorem 5: writing y := x + sϕ(x), and recalling from Theorem 1 the notation γ = ϕ(x)/ϕ(y), one has Fix s and t ∈ R; passing to limits and using uniformity (by Theorem 6), since γ = ϕ(x)/ϕ(y) → 1. This is the Cauchy functional equation; as is well known, for k Baire/measurable (see Banach [5, Chapter I, § 3, Theorem 4] and Mehdi [51] for the Baire case, [46, 9.4.2] for the measure case, and [15] for an up-to-date discussion) this implies k(x) = ρx for some ρ ∈ R, and so g(x) = e ρx .
Remark. The conclusion that k(x) = ρx (∀x) for some ρ tells us that in fact g > 0 everywhere, which in turn implies the cocycle property below. (If we assumed that g > 0 everywhere, then we could argue more directly, and more nearly as in the Karamata theory, by establishing the cocycle property first and from it deducing the Characterization Theorem.) As an immediate corollary, we now have an extension to Theorem 5.
Corollary 1 (Cocycle property). For ϕ ∈ SN, if f > 0 is ϕ-regularly varying and Baire/measurable and satisfies with non-zero limit, that is, g > 0 on a non-negligible set, then is a locally uniform asymptotic cocycle. Proof. With the notation of Theorem 7, rewrite (14) as where by Theorem 7 both ratios on the right-hand side have non-zero limits g(t) and g(s), as x (and so y) tend to infinity. Given ε > 0, it now follows from (15) using (CF E), and uniformity in compact neighbourhoods of s, t and s + t (by Theorem 6), that, for all large enough x, so that σ f is a locally uniform asymptotic cocycle.

Smooth variation and representation theorems
Before we derive a representation theorem for Baire/measurable Beurling regularly varying functions, we need to link the Baire case to the measure case. Recall the Beck iteration of for which see [6, 1.64] in the context of bounding a flow) and Bloom's result for ϕ ∈ SN concerning the sequence x n+1 = γ n (x 1 ), that is, x n+1 := x n + ϕ(x n ), that, for all x 1 large enough, one has x n → ∞, that is, the sequence gives a Bloom partition of R + (see [21], or BGT § 2.11). We next need to recall a construction due to Bloom in detail as we need a slight amendment.
Proof. Proceed as in [21] or BGT § 2.11; we omit the details.
We now deduce an extension of the Bloom-Shea Representation Theorem in the form of a Smooth Beurling Variation Theorem (for smooth variation, see BGT § 2.1.9, following Balkema et al. [4]). Indeed, the special case ψ = ϕ is included here. Our proof is a variant on Bloom's. It will be convenient to introduce the following definition.
We have just seen that self-neglecting functions are necessarily o(x). We now see that, for ϕ ∈ SN, a ϕ-slowly varying function is also SN if it is o(x). Proof. Since self-neglect is preserved under asymptotic equivalence, without loss of generality we may assume that ψ is smooth. Now ψ(x)/ϕ(x) → 1 (by definition), and so, for fixed u, u[ψ(x)/ϕ(x)] → u. For ψ a ϕ-slowly varying function, by the UCT for ϕ-regular variation ψ(x + tϕ(x))/ϕ(x) −→ 1, locally uniform in u, as ψ is continuous. So, in particular, That is, ψ is BSV, since ψ(x) = o(x). But ψ is continuous, and so, by Bloom's theorem [21], ψ ∈ SN.
Proof. With h ρ = log f ρ , one has that We may now establish our main result with f ρ as above.

Any function of this form is ϕ-regularly varying with index ρ.
So f ∼ f ρ φ for some smooth representation φ off .
Proof. By Theorem 8(iii), we may assume that ϕ is smooth. Choose ρ as in Theorem 7 and, referring to the flow rate ϕ(x) > 0 at x, put where h = log f. Soh(x) is Baire/measurable as h is. By Theorem 6 (UCT), locally uniformly in t one has a 'reduction' formula forh : So, substituting u = x + vϕ(x) in the last step, and the convergence under the integral here is locally uniform in t since ϕ ∈ SN. So exp(h) is Beurling ϕ-slowly varying. The converse was established in Lemma 8. The remaining assertion follows from Theorem 8.
As a second corollary of Theorems 6 and 7 and of the de Bruijn-Karamata Representation Theorem (see BGT, Theorems 1.3.1 and 1.3.3), we deduce a Representation Theorem for Beurling regular variation which extends previous results concerned with the class Γ, see BGT, Theorem 3.10.6. We need the following result, which is similar to Bloom's Theorem 4 except that we use regularity of ϕ rather than assume conditions on convergence rates. Lemma 9 (Karamata slow variation). If ϕ ∈ SN with ϕ Baire/measurable eventually bounded away from 0, then Proof. Without loss of generality, suppose that 0 < K < ϕ(x) for all x. Fix v; then 0 |v|/ϕ(x) |v|K −1 for all x. Let ε > 0. Since ϕ ∈ SN, there is X = X(ε, v) such that |ϕ(x + tϕ(x))/ϕ(x) − 1| < ε, (16) for all |t| |v|K −1 and all x X. So, in particular, for x X and t := v/ϕ(x), since |t| |v|K −1 , substitution in (16) yields for x X. This shows that, for each v ∈ R, So log ϕ is Karamata slowly varying in the additive sense; being Baire/measurable, by the UCT of additive Karamata theory, convergence to the limit for log ϕ, and so convergence for ϕ as above, is locally uniform in v.
An alternative 'representation' follows from Lemma 9.
Theorem 10 (Beurling Representation Theorem). For ϕ ∈ SN with ϕ Baire/measurable eventually bounded away from 0, and f measurable and ϕ-regularly varying: there are ρ ∈ R, measurable d(·) → d ∈ (0, ∞) and continuous e(·) → 0 such that f (x) = d(x) exp ρ where h = log f. Here, since 1/ϕ(x) is eventually bounded above as x → ∞ and our analysis is asymptotic, without loss of generality we may assume again by Luzin's Theorem that ϕ here is continuous. By Theorem 6 (UCT), locally uniformly in t one has, as in Theorem 10, a 'reduction' formula forh :h Fix y and let K > 0 be a bound for 1/ϕ, far enough to the right. We will use local uniformity in (17) on the interval |t| |y|K −1 . First, take t = y/ϕ(x), and so |t| |y|/K, so by (17), By Lemma 9, applied to the set {v : |v| |y|}, which corresponds to the w range in the integral above, we haveh since 1/ϕ(x) is bounded. That is,h(x) is slowly varying in the additive Karamata sense (as with log ϕ in the lemma). So, by the Karamata-de Bruijn representation (see BGT, 1.3.3), for some measurable c(·) → c ∈ R and continuous e(·) → 0. Rearranging yields Taking d(x) = e c(x) , we obtain the desired representation. To check this, without loss of generality we now take d(x) = 1, and continue by substituting u = x + sϕ(x) to obtain, from (18) and (19) The proof above remains valid when ρ = 0 for arbitrary ϕ ∈ SN, irrespective of whether ϕ is bounded away from zero or not. Since ϕ is itself ϕ-regularly varying with corresponding index ρ = 0, we have an alternative to the Bloom-Shea representation of ϕ via the de Bruijn-Karamata representation. We record this as the following corollary. We have in the course of the proof of Theorem 8 in fact also shown the following corollary. The result is the key to Beurling's Tauberian theorem [45, IV Theorem 11.1]. Rate-ofconvergence results (Tauberian remainder theorems) are also possible; see, for example, [45,VII.13].
This condition has been extensively studied (see, for example, [44]) and is important in probability theory (work by Szegő, see, for example, [7]).
infinite M t ⊆ N such that {H(t, z m ) : m ∈ M t } ⊆ T, that is, in our context, {t + ϕ(t)z m : m ∈ M t } ⊆ T ; cf. Lemma 2.