K-subadditive and K-superadditive set-valued functions bounded on “large” sets

We prove that every K–subadditive set–valued map weakly K–upper bounded on a “large” set (e.g. not null–finite, not Haar–null or not a Haar–meager set), as well as any K–superadditive set–valued map K–lower bounded on a “large” set, is locally K–lower bounded and locally weakly K–upper bounded at every point of its domain.


Introduction and preliminaries
The classical subadditive functions, i.e. functions f : X → R satisfying have many remarkable properties of boundedness discussed, among others, in [11,16,17,19], and recently in [3][4][5][6]. For instance, it is known that if f : R n → R is subadditive and upper bounded on a set T ⊂ R n which is of positive Lebesgue measure or is of the second category with the Baire property, then f is locally bounded at every point of R n (see [16,Theorem 16.2.3]). This classical result was generalized by Bingham et al. in [3] to the case of others "large" sets in abelian Polish groups, e.g. not null-finite, not Haar-meager, not Haar-null sets.
Recall also that a function f is called superadditive, if −f is subadditive.
In this paper we extend the notions of subadditive and superadditive functions to K-subadditive and K-superadditive set-valued maps. Next, we prove theorems which are far-reaching generalizations of the results mentioned above.
For the concept of K-subadditivity and K-superadditivity we refer to the paper [18].
Let X and Y be abelian metric groups (both with invariant metrics). Assume that K is a subsemigroup of Y (i.e. K + K ⊂ K). Denote by n(Y ) the family of all nonempty subsets of Y .
A set-valued map F : for all x 1 , x 2 ∈ X, and K-superadditve, if for all Note that if F is K-subadditive and single-valued and moreover Y is endowed with the relation ≤ K of partial order defined by then conditions (1) and (2) reduce to the following conditions: and respectively. In particular, if Y = R and K = [0, ∞), we obtain the standard definitions of subadditive and superadditive functions.
for all x 1 , x 2 ∈ X, which is the definition of the superadditivity (subadditivity) of set-valued functions introduced and investigated by Smajdor in [20,21]. Note, however, that if F is single-valued, then each of the above inclusions means that F is an additive function (i.e. F (x 1 + x 2 ) = F (x 1 ) + F (x 2 )). Thus subadditive and superadditive set-valued maps are extensions of additive functions, whereas K-subadditive and K-superadditive set-valued maps generalize subadditive and superadditive functions, respectively. Now, let us recall that a subset B of a complete abelian metric group X with an invariant metric is called: • Universally Baire if for each continuous function f : K → X mapping a compact metric space K into X the set f −1 (A+x) has the Baire property for every x ∈ X (see [10]); • Haar-meager if there exist a universally Baire set A ⊃ B, a compact metric space K and a continuous function f : K → X such that f −1 (A + x) is meager in K for each x ∈ X (see [8] and also [2]); • Universally measurable if it is measurable with respect to each complete Borel probability measure on X (see [7]); • Haar-null if there exist a universally measurable set A ⊃ B and a σadditive probability Borel measure μ on X such that μ(A + x) = 0 for each x ∈ X (see [7]). 1 It was proved in [7] and [8] that in each locally compact abelian Polish group the notions of a Haar-meager set and a Haar-null set are equivalent to the notions of a meager set and a set of Haar measure zero, respectively. Moreover, Haar-meager sets and Haar-null sets have many analogous properties (see, e.g. [1,9,14]).
In [2] a new concept of "small" sets was introduced, generalizing (to some extent) the notions of a Haar-meager set and a Haar-null set.

Definition 1.
A subset A of an abelian metric group X is called null-finite if there exists a sequence (x n ) n∈N convergent to 0 in X such that the set {n ∈ N : x + x n ∈ A} is finite for every x ∈ X.
The following crucial property of null-finite sets was proved in [2]. Theorem 1. [2][Theorems 5.1 and 6.1] In a complete abelian metric group with an invariant metric: • Each universally Baire null-finite set is Haar-meager, • Each universally measurable null-finite set is Haar-null.
In the same paper [2] the authors applied the above result to show that every real-valued additive (midpoint convex) function upper bounded on a set which is universally measurable non-Haar-null or Borel non-Haar-meager in a complete abelian metric group (linear space) with an invariant metric is continuous. Next, Bingham et al. [3] showed that every subadditive real valued function upper bounded on a set which is "large" in the same sense is locally bounded at each point of the domain.
In this paper we generalize results from [3] to K-subadditive and K-superadditive set-valued maps. Our results are also counterparts of some results from [15] concerning K-midconvex and K-midconcave set-valued maps bounded on "large" sets.

Main results
Let X and Y be abelian metric groups with invariant metrics. Denote by B X (r) and B Y (r) open balls with center 0 and radius r in X and Y , respectively.
This notion generalizes the well-known notion of bounded sets in a real topological vector space. Clearly, if B 1 , B 2 are bounded sets in Y , then the set B 1 + B 2 is also bounded in Y .
Denote by B(Y ) the family of all nonempty bounded subsets of Y . A setvalued map F : X → n(Y ) is called: First we prove a result which generalizes Theorem 2.2 from [3].
First we prove that F is locally weakly K-upper bounded at 0. So, suppose that it is not true and F is not weakly K-upper bounded on any neighborhood of 0. Consequently, for every n ∈ N, U n := B X 1 2 n and B n : there exists x n ∈ U n such that (4) Moreover, by the definition of U n the sequence (x n ) n∈N is convergent to 0 in X. The set A is not null-finite, so there exists a ∈ X such that the set N 0 := {n ∈ N : a + x n ∈ A} is not finite. By (3) we have In view of K-subadditivity, Since Then, in view of (6) we have From (7) and (8) we obtain which contradicts (4). Thus F is locally weakly K-upper bounded at 0, i.e. there are a neighborhood U 0 of 0 and a set B 0 ∈ B(Y ) such that In the second step we prove that F is locally weakly K-upper bounded at every point of the domain. For a proof by contradiction suppose that F is not weakly K-upper bounded on any neighborhood U x0 of some point x 0 . Consequently, for where x 1 − x 0 ∈ U 0 , in view of (9) we can find b ∈ B 0 and k ∈ K such that b − k ∈ F (x 1 − x 0 ). Then, by (11), and consequently, which contradicts (10) and proves that F is locally weakly K-upper bounded at x 0 . Finally, we will show that F is locally K-lower bounded at every point x ∈ X. We have already proved that there exist a neighborhood U 0 of 0 and a set B 0 ∈ B(Y ) such that (9) holds. We may assume that U 0 is symmetric with respect to 0. Fix x ∈ X arbitrarily and take U x := U 0 + x. If y ∈ U x , then x − y ∈ U 0 and by (9) Since F (x) − B ∈ B(Y ), this shows that F is K-lower bounded on U x and finishes the proof.
In Theorem 2 a stronger assumption like K-upper boundedness of F on a "large" set A does not strengthen the statement. More precisely, a set-valued map F : X → B(Y ) which is K-subadditive and K-upper bounded on A need not be locally K-upper bounded at each point of X. Example 1. Let K = [0, ∞) and F : R → B(R) be given by Such a set-valued mapping is K-subadditive and K-upper bounded e.g. on the set [1,2] (it is enough to choose B = [0, 1]). But F is not K-upper bounded at 0. Now, we will prove an analogous result for K-superadditive set-valued maps. Note, however, that this result can not be obtained as a consequence of Theorem 2. Namely, the K-superadditivity of a set-valued map F does not imply the K-subadditivity of −F .

Example 2.
Let Q be the set of all rational numbers. Let F : R → n(R) be given by F (x) = 0, |x| for x ∈ R. Clearly, F is Q-superadditive. Moreover, the set-valued map −F given by −F (x) = − |x|, 0 for x ∈ R is not Qsubadditive (but is Q-superadditive).
Similarly, we can find an example where K is not a group.

Theorem 3. Let X and Y be abelian metric groups with invariant metrics.
Assume that A ⊂ X is a set which is not null-finite and K is a subsemigroup of Y . If a set-valued map F : X → B(Y ) is K-superadditive and K-lower bounded on A, then F is locally K-lower bounded and locally weakly K-upper bounded at each point of X.
First we will prove that F is locally K-lower bounded at 0. So, suppose that it is not true and F is not K-lower bounded on any neighborhood of 0. Consequently, for every n ∈ N, U n := B X 1 2 n and B n : there exists x n ∈ U n such that Moreover, by the definition of U n the sequence (x n ) n∈N is convergent to 0 in X. The set A is not null-finite, so there exists a ∈ X such that the set N 0 := {n ∈ N : a + x n ∈ A} is not finite. Then, by (12) we have Since F (−a) ∈ B(Y ), we can find n 0 ∈ N 0 , such that F (−a) ⊂ B Y (n 0 ). Consequently F (−a) ⊂ B Y (n) for every n ≥ n 0 . Fix n ∈ N 0 , n ≥ n 0 . In view of K-superadditivity and (12), which contradicts (13). Thus F is locally K-lower bounded at 0, i.e. there are a neighborhood U 0 of 0 and a set B 0 ∈ B(Y ) such that Now, we will prove that F is locally K-lower bounded at every point of the domain. For a proof by contradiction suppose that F is not K-lower bounded on any neighborhood U x0 of some point x 0 . Consequently, for Since x 1 − x 0 ∈ U 0 , K-superadditivity and (15) implies which contradicts (16) and proves that F is locally K-lower bounded at x 0 . Finally, we will show that F is locally weakly K-upper bounded at each point x ∈ X. Since F is locally K-lower bounded at 0, there are a symmetric neighborhood U 0 of 0 and a set B 0 ∈ B(Y ) such that (15) holds. Fix an arbitrary x ∈ X and take U x := U 0 + x. Take any y ∈ U x . Then y = z + x, where z ∈ U 0 . Since U 0 is symmetric, also −z ∈ U 0 . By the K-superadditivity of F and (15) we have Fix any z 0 ∈ F (x). By (17) Since this condition holds for any y ∈ U x , it proves that F is locally weakly K-upper bounded at x. The proof is finished.
The next example shows that in Theorem 3 a weaker assumption like weakly K-lower boundedness of F : X → B(Y ) on a "large" set A does not imply even a weaker conclusion. More precisely, a set-valued map F which is Ksuperadditive and weakly K-lower bounded on A need not be locally weakly K-lower bounded at each point of X.
Such a set-valued mapping is K-superadditive and weakly K-lower bounded e.g. on the set [1,2]. Moreover, F is not weakly K-lower bounded at 0. Indeed, if F (x) ∩ (B + K) = ∅ for each x ∈ U 0 a neighbourhood U 0 and a bounded set B, then |a(x)| ≤ − inf B for each x ∈ U 0 which contradicts the discontinuity of a.
where a : R → R is an additive discontinuous function, and take K = [0, ∞).
Then F is K-subadditive as well as K-superadditive. Moreover, F is locally weakly K-upper bounded at every point (it is even weakly K-upper bounded on the whole R because F (x) ∩ ([0, 1] − K) = ∅ for every x ∈ R). F is also locally K-lower bounded at every point (it is even K-lower bounded on the whole R because F (x) ⊂ [0, 1] + K for every x ∈ R). However, F is not locally K-upper bounded at any point. Indeed, if for some open set U ⊂ R and some which is impossible because a is discontinuous (by [16][Lemma 9.3.1]).
As an immediate consequence of Theorems 2, 3 and 1, we obtain the following generalization of [3][Corollary 2.4].

Corollary 4.
Let X be a complete abelian metric group and Y be an abelian metric group, both with invariant metrics. Assume that A ⊂ X is a universally measurable non-Haar-null or universally Baire non-Haar-meager set and K is a subsemigroup of Y . If a set-valued map F : X → B(Y ) is K-subadditive and weakly K-upper bounded on A or K-superadditive and K-lower bounded on A, then F is locally K-lower bounded and locally weakly K-upper bounded at each point of X.
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