On Subadditive Functions Bounded Above on a “Large” Set

Abstract

It is well known that boundedness of a subadditive function need not imply its continuity. Here we prove that each subadditive function f : X → R bounded above on a shift-compact (non-Haar-null, non-Haar-meagre) set is locally bounded at each point of the domain. Our results refer to [31, Chapter 16] and papers

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

“…As an immediate consequence of Theorems 2, 3 and 1, we obtain the following generalization of [3][Corollary 2.4].…”

“…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.…”

“…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.…”

“…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.…”

“…The classical subadditive functions, i.e. functions f : X → R satisfying f (x 1 + x 2 ) ≤ f (x 1 ) + f (x 2 ), x 1 , x 2 ∈ X, 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]).…”