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 [5]-see [18] 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 Sect. 4.8), where some naive analogues can fail-see [19]. These do not cover our testcase of the additive reals R, with focus on the fact (e.g. [14]) that for A ⊆ R a dense subgroup, if f : A → R is additive (i.e. a partial homomorphism) and locally bounded (see Theorem R in Sect. 2), then it is linear: f (a) := 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 In Sect. 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 theorem (Sect. 4.9) of probability theory. In Sect. 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 Sect. 4 with a discussion of the extensive background literature.

Subadditivity, sublinearity and theorems of Berz type
We first justify our preferred use of local boundedness. Here and below we write B δ (x) := (x − δ, x + δ) for the open δ-ball around x.
Theorem R. (cf. [50,Th. 16.22]) For A ⊆ R a subgroup and S|A, → R subadditive: if S is bounded above on some interval, say by K on B δ (a) ∩ A, then for any b ∈ A In particular, it is locally bounded on A: so here, local boundedness from above is equivalent to local boundedness.
Proof. Mutatis mutandis, this is [14,Prop. 5(i)], as the proof there relies only on group structure.
The proof in [50] is more involved, and not immediately adaptable to the subgroup setting that we need. The proof offered here we learned from the Referee of [14] (see the Acknowledgements) and in view of that and of some related helpful correspondence with Professor van Rooij we gladly label this Theorem R. As it pinpoints the group dependence, we thank the present Referee for urging us to make explicit reference to local upper boundedness as an alternative assumption (calling to mind the Darboux-Ostrowski-type assumption-see [12]).
We may now begin with a sharpened form of the Berz theorem, with a proof that seems new. Below we write R + := [0, ∞), R − := (−∞, 0], and 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, [13,56]), and so, being subadditive, is locally bounded.
The following is a slightly sharper form of results in [14] 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.
For the proof, we will need the following theorem; for completeness, we show how to make the simple modification needed for the result given in [34,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 [34, 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 : Also S(t)/t itself is bounded on [a, 2a], so β = β(a) < ∞ is well-defined. Suppose first that β > −∞, and let ε > 0. As inf t>0 S(t)/t < β + ε, choose and fix b ≥ a with S(b)/b ≤ (β + ε). For any t ≥ 2b, let n = n(t) ∈ N satisfy (n + 1)b ≤ t < (n + 2)b; then b < t − nb < 2b and This time with K = sup |S([0, 2b])| a bound on S as above, since 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, since also β T < ∞, we have 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-gives (the more convenient) Q +− homogeneity from N-homogeneity, but otherwise a/k ∈ A (a ∈ A , k ∈ N)-gives (the more convenient) Q +− homogeneity from N-homogeneity, but otherwise is innocuous (as any (infinite) subgroup may be extended to a divisible one without change of cardinality-as with the rationals from the integers).

Theorem 3. For A a non-trivial, divisible (so dense) 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 [34, §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 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).
Remark. The conclusions (i)-(iv) of Theorem 3 continue to hold for (just) a dense subgroup A and a locally bounded, sublinear S : A → R, by similar reasoning as follows. As in Th. 1, S is continuous on A, and, since in particular for any x ∈ R and the equalityS A (a) = S(a) for a ∈ A. This equality implies directly (from the same properties of S) that S A (x) is sublinear and locally bounded, so linear on R ± and continuous, by Th. 1. For a more detailed exposition, see the extended arXiv version of the paper.

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 [14], 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 Sect. 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 Sect. 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, [43] studies the lower envelope dominated by S, using instead S(nx)/n-also noted in [29], cf. [3] and [50, 16.2.9]-an approach followed in [31] 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), then, for t ≥ 0, In particular, S(0+) = S * (0+) = 0 and S * is locally bounded; furthermore, Proof. By subadditivity of S, for any n ∈ N and x ∈ R the latter by subadditivity. Combining, 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 → ∞ Finally, by Th. R, S * is locally bounded, since S * is locally bounded 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 [14].
Definition. [13,14,17] 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. (Here again B δ (x) is the open ball about x of radius δ.) See [15, §6.9] or [17, §7] for conditions under which this property is implied by the interiorpoint property of the sets Σ δ x − Σ δ x (cf. 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.
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.

Remark. The burden of proof falls on showing that
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 N-homogeneity of A, and further, by sublinearity of S * (Prop. 1), S * (a/n) = S * (a)/n = S(a)/n → 0, taking limits through n ∈ N. Taking a < 0 gives, via Theorem 4, that S * and so S is continuous on A. 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 4. The first formulas come from Th. HP, and the final one, in the additive case, from S(−t) = −S(t) (and then α S = β S ).

Remark.
The assumption of divisibility placed on A is innocuous: it may be omitted by the following reasoning. By the Corollary, we may assume that S satisfies SHS, being in fact linearly bounded for t > 0. So, by Theorem 0 + , S(0+ = 0). By the Remarks after Theorem 3, S|A − is linear. So by density of A there is a sequence in A − : a n ↑ 0 say, with S(a n ) → 0. So, by Theorem 0, S is continuous. So by the density of A, S|R ± is linear. In the result above, the particular case of S additive includes [14, Th. 1 and Th. 1 b ].

Approximate homomorphisms
There are results in which one has a property, such as additivity, which holds only approximately, and then deduces that, under suitable restrictions, it holds exactly. For example, in Badora's almost-everywhere version of the Hahn-Banach theorem [4], if the relevant differences are bounded, as in [2], 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 [8]; [53, 5.3.10]).

Popa (circle) group subadditivity
We recall from [14] that the Popa circle operation on R, introduced in [60] (cf. [39]), 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. [30]). So a function f : 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. [36]); for the latter see [13]. We hope to return to these matters elsewhere-cf. [14, §7].

Restricted domain
There are results when, as in Sect. 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 [1,20]. See also de Bruijn [25], Ger [32,33].

Origin of the Heiberg-Seneta condition
This condition, introduced in regular variation (see [10,Th. 3.2.5], prompting its recent study in [14]), as applied to a subadditive function S : For A ⊆ R a dense subgroup, the assumption that S|A is linear together with (HS) guaranteed not only that S is finite-valued with S(0+) = 0, but that in fact S is linear, as in Th. 5, which relates directly to [10, Th. 3.2.5].

Examples of families of locally Steinhaus-Weil sets
The sets listed below are typically, though not always, members of a topology on an underlying set.  [17]. 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).
1. Care is needed in identifying locally SW sets: Matoȗsková and Zelený [54] 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 [38] 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 [23] and the recent [6]. 3. Here we were concerned with subsets Σ ⊆ R where such 'anomalies' are assumed not to occur.

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.

The Hahn-Banach theorem: variants
There are various theorems of Hahn-Banach type. Text-book accounts, as in e.g. Rudin [61, § 3.2, 3.3, 3.4], [30], 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. [26] (implying continuity of the minorant partial function); another variation-from [28], call this 'HB-lite' for our needs in Sect. 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 [28, §4]). For a most insightful survey of very many variations in earlier literature see [18]. 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, [42]) 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 ( [27,41], cf. [42,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 ( [40,52,55,66])-MB (for 'measure-Boolean') in the terminology of [55].
For the relative strengths of HB and the Axiom of Choice AC, see [57,58]; [59] provides 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 [26,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 [5] by inductive assignment of function-values using the least possible function-value at each stage (as in [26,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 [28] for various completeness and compactness notions here.) Further to [26], we raise, and leave open here, the question as to whether the separable case of Badora's result in [3] can be proved with only DC rather than AC, and the role that completeness (sequential or otherwise) may play here [42].

The Hahn-Banach Theorem: group analogues
The group analogue of the 'HB-lite' property of Sect. 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 [3,Th. 1], and includes amenable groups; the class is characterized in [3,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 G-invariant 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 [62,63])-see [46] for a clear albeit early approach. See also [66,Th. 12.11].

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. [44,45], Steele [65].