Variants on the Berz sublinearity theorem

We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice which Berz used. We consider thinned versions, in which conditions are imposed on only part of the domain of the function -- results of quantifier-weakening type. There are connections with classical results on subadditivity. We close with a discussion of the extensive related literature.


Introduction: sublinearity.
We are concerned here with two questions. The first is to prove, as directly as possible, a linearity result via an appropriate group-homomorphism analogue of the classical Hahn-Banach Extension Theorem HBE [Ban] -see [Bus] for a survey. Much of the HBE literature most naturally elects as its context real Riesz spaces (ordered linear spaces equipped with semigroup action, see §4.8), where some naive analogues can fail -see [BusR]. These do not cover our test-case of the additive reals R, with focus on the fact (e.g. [BinO3]) that for A ⊆ R a dense subgroup, if f : A → R is additive (i.e. a partial homomorphism) and locally bounded, then it is linear: f (x) := ca for some c ∈ R and all a ∈ A. Can this result be deduced by starting with some natural, continuous, subadditive majorant S : R → R (so that, equivalently, S|A ≤f ≤ S|A forS(.) = −S(−.), which is super-additive) and then invoking an (interpolating) additive extension F majorized by S? For then F, automatically being continuous, is linear, because its restriction to the rationals F |Q is so (as in Th. 1 below). Assuming additionally positive-homogeneity, HBE yields an F , but this strategy relies very heavily on powerful selection axioms (formally a weakend version of the Prime Ideal Theorem, itself a weakening of the Axiom of Choice, AC, see §4.7). The alternative is to apply either semigroup results in [Kau], [Fuc], [Kra], or the recent group-theoretic result in [Bad3], but all these again rely on AC (see §4.7 again). We give an answer in Theorem 3 that relies on the much weaker axiom of Dependent Choices, DC (see §4.7 once more). We stress that, throughout the paper, all our results need only DC.
The group analogue (for R) of sublinearity used by [Ber] (cf. [Kau]) requires subadditivity as in Banach's result [Ban,§2.2 Th. 1], but restricts Banach's positive-homogeneity condition to just N-homogeneity: (x ∈ R, n = 0, 1, 2, ...) (with the universal quantifier ∀ on x and n understood here, as is usual in mathematical logic). From here onwards we take this to be our definition of sublinearity. This is of course equivalent to 'positive-rational-homogeneity'. Berz proves and uses a Hahn-Banach theorem in the context of R as a vector space over Q (for which see also [Kuc2,§10.1]) to show that if S : R → R is measurable and sublinear, then S|R + and S|R − are both linear; for generalizations to Baire (i.e. having the Baire property) and universally measurable functions in contexts including Banach spaces, again using only DC, see [BinO2]. Berz's motivation was questions of normability in topological spaces [Ber]. The key result here is Kolmogorov's theorem [Kol]: normability is equivalent to the origin having a bounded convex neighbourhood ( [Rud,Th. 1.39 and p. 400]). Our second, linked, question asks whether the universal quantifier (x ∈ R) above can be weakened to range over an additive subgroup. Since S = 1 R\Q is subadditive and N-homogeneous on Q, but not linear on R ± , the quantifier weakening must be accompanied by an appropriate side-condition. We give in Theorem 5 a necessary and sufficient condition (referring also to a thinnedout domain), by extending the standard asymptotic analysis -as in [HilP] (see Theorem HP below) -of the ratio S(t)/t near 0 and at infinity; this, indeed, permits thinning-out the universal quantifier of N-homogeneity to a dense additive subgroup A.
We come at these questions here employing ideas on quantifier weakening previously applied in [BinO3] to additivity issues in classical regular variation, and in [BinO2] to Jensen-style convexity in Banach spaces. We borrow from [BinO3] two key tools: Theorem 0 below on continuity (exploiting an idea of Goldie), and Theorem 0 + on linear (upper) bounding (exploiting early use by Kingman of the Baire Category Theorem -see [BinO1]), the latter delayed till §3, when we have the preparatory results needed.
In §2 below we discuss subadditivity, sublinearity and theorems of Berz type, proving Theorems 1-3, Th. BM (for Baire/measurable) and Th. HP (for Hille-Phillips). The work of Hille and Phillips is a major ingredient in the Kingman subadditive ergodic theeorem ( §4.9) of probability theory. In §3 we give stronger versions of Berz's theorem by thinning the domain of definition, under appropriate side-conditions, results of quantifier-weakening type (Theorems 4 and 5). We close in §4 with a discussion of the extensive background literature.

Subadditivity, sublinearity and theorems of Berz type.
We begin with a sharpened form of the Berz theorem, with a proof that seems new. Here and below we write B δ (x) := (x − δ, x + δ) for the open δ-ball around x.

This gives as a corollary
Theorem BM ( [Ber], [BinO2], cf. [BinO3]). For S : R → R a sublinear function, if S is Baire/measurable, then both S|R + and S|R − are linear.
Proof. For S Baire/measurable, S is bounded above on a non-negligible set and so, being subadditive, is bounded above on some interval (by the Steinhaus-Weil Theorem, [Oxt], [BinO2]), and so, being subadditive, is locally bounded.
The following is a slightly sharper form of results in [BinO3] with a simpler proof (the subgroup here is initially arbitrary). This extension theorem may be interpreted in Hahn-Banach style as involving a subadditive function S which, relative to a subgroup A, majorizes an additive function G that happens to agree with the restriction S|A.
Theorem 2. If S : R → R is a subadditive locally bounded function and A any non-trivial additive subgroup such that S|A is additive, then S|A is linear.
In particular, for A dense, any additive function G on A has at most one continuous subadditive extension S : R → R.
We will need the following Theorem; for completeness, we show how to make the simple modification needed to the result given in [HilP,Th. 7.6.1]. (We replace their additional blanket condition of measurability of S by local boundedness, and give more of the details, as they are needed later.) Theorem HP. For S : R → R a locally bounded subadditive function so β does not depend on the choice of a > 0. In particular, Proof. Following [HilP,Th.7.5.1], for a > 0 and ma ≤ t < (m + 1)a with m = 2, 3, ..., we note two inequalities, valid according as S(a) ≥ 0 or S(a) < 0 : The case β = −∞ would be similar albeit simpler. In fact it does not arise. Indeed, writing T (t) = S(−t), which is subadditive and locally bounded, yields Remark. In fact Proof of Theorem 2. Put G := S|A, and let β S denote the unique β of Th. HP. For any a ∈ A ∩(0, ∞), we have, by Th. HP, that Now for all a ∈ A, as G(−a) = −G(a) and G(0) = 0 (by additivity), G(a) = β S a. In particular, for S continuous and A dense, The next extension theorem employs majorization and minorization on a subspace. The assumption of subgroup divisibility -a/k ∈ A (a ∈ A , k ∈ N) -is innocuous (as any subgroup may be extended to a divisible one without change of cardinality). Below we write R + := [0, ∞), R − := (−∞, 0], and Theorem 3. For A a dense, divisible subgroup of R and a locally bounded sublinear (in particular, additive) S : A → R, Proof. To lighten the notation, we write S ± for S ± A . Local boundedness of S ± follows immediately from local boundedness of S. Subadditivity is routine, and follows much as in [HilP,§7.8]. As regards sublinearity of S ± , note that if a n → x for a n ∈ A with lim n→∞ S(a n ) = S ± (x), then, as ka n ∈ A for k ∈ N, by sublinearity of S kS + (x) = lim n→∞ kS(a n ) = lim n→∞ S(ka n ) ≤ S + (kx), kS − (x) = lim n→∞ kS(a n ) = lim n→∞ S(ka n ) ≥ S − (kx).
Similarly, for k ∈ N, if a n → kx for a n ∈ A with lim n→∞ S(a n ) = S ± (kx), then a n /k → x with a n /k ∈ A, and so again by sublinearity of S, So kS ± (x) = S ± (kx). By Theorem 1 the four functions S ± A |R ± are linear, and so by dominance the two functions S|A ± are continuous at 0 and so continuous everywhere (as in Th. 1). So if a n → a with a, a n ∈ A, then lim n→∞ S(a n ) = S(a) = S ± (a), proving (iii). So S ± |A + = S|A + ; this implies S|A + is linear and also that S + |R + = S − |R + , since A is dense, proving (i) and (ii) on R + and A + ; similarly on R − and A − . For additive S this means that S + = S − is linear, as is S, proving (iv).

Remarks. Actually
, so (in view of the first displayed inequality above, etc.) a less symmetric proof would have fewer steps. Inducing functions (such as S ± from S, above) is a method followed variously, e.g. in [Kuc1,Th. 1], [BinO2,Th. 5].
3. Thinning: a stronger Berz theorem. We now give a stronger version of the Berz theorem by weakening a condition of Heiberg-Seneta type by thinning, as in [BinO3], and requiring the homogeneity assumption to hold on only a dense additive subgroup A of R; all in all, with rather less than sublinearity, we improve on Theorem BM. This comes at the price of assuming more about S. To motivate the next definition, note that for locally bounded subadditive S, the inequalities ( †) of §2 imply that for any a > 0 We note that, with T as in Theorem HP, α S := sup z<0 S(z)/z = −β T is finite (by the remark above), so the definition below fills the gap for sup t>0 S(t)/t, by asking apparently a little less.
Definition. Say that S : R → R satisfies the strong Heiberg-Seneta (SHS) condition if See §4.4 for the origin of this term. For S subadditive, we will see in Proposition 2 that this implies its dual: Proposition 1, to which we now turn, associates to each subadditive function S a sublinear function S * dominating S, here and below to be called the (upper) sublinear envelope of S. (Albeit multiplicatively, [Kau] studies the lower envelope dominated by S, using instead S(nx)/n -also noted in [Fuc], cf. [Bad3] -an approach followed in [GajK] employing the decreasing sequence S(2 n x)/2 n .) However, some assumption on S is needed to ensure that S * is finite-valued: recall that the subadditive function S = 1 R\Q is N-homogeneous on Q, yet S * = (+∞) · 1 R\Q . Proposition 1. For S : R → R locally bounded and subadditive with S(0) = 0, the function defined by is subadditive and sublinear and dominates S. If further S satisfies (SHS), Proof. By subadditivity of S, for any n ∈ N S(x) = S(n.x/n) ≤ nS(x/n) ≤ S * (x).
Suppose now that (SHS) holds. Let ε > 0. Then there is δ > 0 with Fix t > 0. Then for integer n > t/δ and so taking limsup as n → ∞ for t ≥ 0, as S * (0) = 0 (since S(0) = 0). Taking limits as ε ↓ 0 yields Furthermore, for t ≥ 0 In view of the linear bounding of S * (and hence of S) just proved from SHS, we proceed to a weaker property of S in which the domain of the limsup operation is thinned-out. This will nevertheless also yield linear bounding of S (from above), hence finiteness of γ + S , and in turn the bounding of S * . We need a definition and a theorem from [BinO3].
Definition [BinO3,2,5]. Say that Σ ⊆ R is locally Steinhaus-Weil (SW), or has the SW property locally, if for x, y ∈ Σ and, for all δ > 0 sufficiently small, the sets Σ δ z := Σ ∩ B δ (z), for z = x, y, have the interior-point property, that Σ δ x ± Σ δ y has x ± y in its interior. In particular, if furthermore there exists a sequence {z n } n∈N with z n ↑ 0 and S(z n ) → 0, then S is continuous at 0 and so everywhere.
Definition. Say that S : R → R satisfies the weak Heiberg-Seneta (W HS) condition if for some Σ ⊆ (0, ∞), a locally SW set accumulating at 0, Corollary. For S : R → R locally bounded and subadditive with S(0) = 0, if S satisfies W HS, then S is linearly bounded by γ Σ S t for t ≥ 0, and so satisfies SHS with γ + S ≤ γ Σ S .
We now derive in Theorem 5 below a form of Berz's Theorem, in which the weak Heiberg-Seneta condition on S permits a thinned-out assumption of homogeneity; the argument is based on the following result, a corollary of Theorem 1 and Prop 1.
Theorem 4. For S : R → R locally bounded and subadditive with S(0) = 0, if S satisfies W HS, and S * (t n ) → 0 for some sequence t n ↑ 0, then S and its sublinear envelope S * are continuous, and further, by sublinearity, both S * |R + and S * |R − are linear.

Remark. The burden of proof falls on showing that
Theorem 5 (Quantifier-weakened Berz Theorem). For S : R → R locally bounded and sublinear (in particular for S Baire/measurable and sublinear) with S(0) = 0, if (i) S satisfies W HS and, (ii) A is a dense additive subgroup of R with S|A N-homogeneous -then both S|R + and S|R − are linear: for t ≥ 0 : In particular, for S additive Proof. By the Corollary we may assume that SHS holds. Consider any a ∈ A. Then S * (a) = lim sup n→∞ nS(a/n) = S(a) by Q + -homogeneity of S, and further S * (a/n) = S(a/n) = S(a)/n → 0, taking limits through n ∈ N.
Taking a < 0 gives, via Theorem 3, that S and S * are continuous. Now S = S * , by continuity and density of A, as S * |A =S|A. So S|R + and S|R − are linear, again by Theorem 3. The first formulas comes from Th. HP, and the final one, in the additive case, from S(−t) = −S(t) (and then α S = β S ).
Remark. In the result above, the particular case of S additive includes [BinO3, Th. 1 and Th. 1 ′ b ]. §4. Complements 4.1 Approximate homomorphisms. There are results in which one has a property, such as additivity, which holds only approximately, and deduces that, under suitable restrictions, it holds exactly. For example, in Badora's almosteverywhere version of the Hahn-Banach theorem [Bad4], if the relevant differences are bounded, as in [Bad2], then they vanish. That is, the relevant differences are either identically plus infinity or identically zero. This is a dichotomy, reminiscent of those that occur in probability theory in connection with 0-1 laws (for example, Belyaev's dichotomy [Bel]; [MarR,5.3]). 4.2 Popa (circle) group subadditivity. We recall from [BinO3] that the Popa circle operation on R, introduced in [Pop] (cf. [Jav]), given by a • b = a + bη(a), for η(t) := 1 + ρt with ρ ≥ 0, turns G + := {x ∈ R : 1 + ρx > 0} into a group with R + as a subsemigroup. The latter induces an order on G + which agrees with the usual order (cf. e.g. [FucL]). So a function f : (R + , ×) → (G + , •) satisfying may be viewed as subadditive in the group context. This abstract viewpoint encompasses both the current context of subadditivity (for ρ = 0), and a further significant one arising in the theory of regular variation (the 'Goldie Functional Inequality', for ρ = 1 -cf. [Jab2]); for the latter see [BinO3]. We hope to return to these matters elsewhere. 4.3 Restricted domain. There are results when, as in §3 on quantifier weakening, a property such as additivity or subadditivity holds off some exceptional set (say, almost everywhere), and the conclusion is also similarly restricted. This goes back to work of Hyers and Ulam [CabC], [Bad1]. See also de Bruijn [deB], Ger [Ger]. 4.4 Origin of the Heiberg-Seneta condition. This condition, introduced in regular variation (see [BinGT,Th. 3.2.5], prompting its recent study in [BinO3]), as applied to a subadditive function S : R → R ∪ {−∞, +∞}, took the form lim sup t↓0 S(t) ≤ 0.
If Σ is Baire (has the Baire property) and is locally non-meagre, then it is co-meagre (since its quasi interior is everywhere dense). Caveats. 1. Care is needed in identifying locally SW sets: Matoȗsková and Zelený [MatZ] show that in any non-locally compact abelian Polish group there are closed non-Haar null sets A, B such that A + B has empty interior. Recently, Jab lońska [Jab4] has shown that likewise in any non-locally compact abelian Polish group there are closed non-Haar meager sets A, B such that A + B has empty interior. 2. For an example on R of a compact subset S such that S − S contains an interval, but S + S has measure zero and so does not, see [CrnGH] and the recent [BarF]. 3. Here we were concerned with subsets Σ ⊆ R where such 'anomalies' are assumed not to occur. 4.6 Baire/measurable S and S * . Of course if S is Baire/measurable, then so is S * , as the limsup is sequential. Also for A a countable subgroup, the upper and lower limit functions S ± A derived from a subbadditive function S are Baire/measurable, as the image S(A) is countable. 4.7 The Hahn-Banach theorem: variants. There are various theorems of Hahn-Banach type. Text-book accounts, as in e.g. Rudin [Rud,§ 3.2,3.3,3.4], [FucL], deal with dominated extension theorems (without any assumed continuity on the partial function f nor on the dominating function p, HB below), separation theorems for convex sets, and continuous extension theorems. Variations include the assumption that the dominating function p is continuous, e.g. [DodM] (implying continuity of the minorant partial function); another variation -from [FosM], call this 'HB-lite' for our needs in §4.8 below -assumes for given p merely the existence of some linear functional dominated by p. (Here, if the variant axiom is satisfied for all p continuous, then HB follows for all continuous p [FosM,§4]). For a most insightful survey of very many variations in earlier literature see [Bus]. The context also varies, correspondingly, from vector spaces, to topological vector spaces and beyond, so to F-spaces (i.e. topological vector spaces with topology generated by some complete translation-invariant metric, [KalPR]) and Banach spaces. One needs to distinguish between the variants, including the category of space over which the assertions range, when discussing their axiomatic status. Kalton proved ( [Kal], [Dre], cf. [KalPR,Ch. 4]) that an F-space in which the continuous extension theorem (in which f is continuous) holds is necessarily locally convex, a result that is false without metrizability; it is not known whether completeness is necessary.
The dominated extension theorem HB (i.e. without any continuity) is equivalent to a weakened form of PIT, the Prime Ideal Theorem, namely the existence of a non-trivial finitely-additive probability measure (as opposed to a two-valued measure implicit in PIT) on any non-trivial Boolean algebra ( [Lux], [Jec], [TomW], [MycT]) -MB (for 'measure-Boolean') in the terminology of [MycT].
For the relative strengths of HB and the Axiom of Choice AC, see [Pin1,2]; [PinS] provide a model of set theory in which the Axiom of Dependent Choices DC holds but HB fails. Moreover, HB for separable normed spaces is not provable from DC [DodM,Cor. 4]. On the other hand, any separable normed space satisfies the version of HB in which the dominating function p is continuous; indeed the partial function f may first be explicitly extended to the linear span of the union of its domain with the dense countable set -as in the original Banach proof [Ban] by inductive assignment of function-values using the least possible function-value at each stage (as in [DodM,Lemma 9]) -and then to the rest of space, essentially as in Theorem 3, using the continuity conferred by p and our sequential analysis. (Compare [FosM] for various completeness and compactness notions here.) Further to [DodM], we raise, and leave open here, the question as to whether the separable case of Badura's result in [Bad3] can be proved with only DC rather than AC, and the role that completeness (sequential or otherwise) may play here [KalPR].
4.8. The Hahn-Banach Theorem: group analogues. The group analogue of the 'HB-lite' property of §4.7 (mutatis mutandis, with 'additive' replacing 'linear' etc.) delineates a class of groups providing the context for Badora's 'general' Hahn-Banach extension theorem for groups [Bad3,Th. 1], and includes amenable groups; the class is characterized in [Bad3,Th. 3] by the group analogue of HB with a side-condition on p. The more special Hahn-Banach-type extension property for the case of a group G of linear operators g : V → V on a real vector space V is concerned with a p-dominated Ginvariant extension of a G-invariant partial linear operator f (defined on a G-invariant subspace W ) satisfying f (w) ≤ p(w) for w ∈ W, where p is a subadditive and positive-homogeneous functional p : V → R with p(g(v)) ≤ p(v). This as a property of G turns out to be equivalent to G being amenable (Silverman [Sil1,2]) -see [Kle] for a clear albeit early approach. See also [TomW,Th. 12.11]. 4.9 Kingman's Subadditive Ergodic Theorem. Detailed study of subadditivity is partially motivated by links with the Kingman subadditive ergodic theorem, which has been very widely used in probability theory. For background and details, see e.g. [Kin1,2], Steele [Ste].
Postscript. This paper germinated from the constructive and scholarly criticism of successive drafts of [BinO3] by its Referee; it is a pleasure to thank him again here.