Abstract
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.
Highlights
Introduction and preliminariesThe classical subadditive functions, i.e. functions f : X → R satisfying f (x1 + x2) ≤ f (x1) + f (x2), x1, x2 ∈ X, have many remarkable properties of boundedness discussed, among others, in [11, 16, 17, 19], and recently in [3,4,5,6]
It is known that if f : Rn → R is subadditive and upper bounded on a set T ⊂ Rn which is of positive Lebesgue measure or is of the second category with the Baire property, f is locally bounded at every point of Rn. 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 this paper we extend the notions of subadditive and superadditive functions to K–subadditive and K–superadditive set–valued maps
Summary
Introduction and preliminariesThe classical subadditive functions, i.e. functions f : X → R satisfying f (x1 + x2) ≤ f (x1) + f (x2), x1, x2 ∈ X, have many remarkable properties of boundedness discussed, among others, in [11, 16, 17, 19], and recently in [3,4,5,6]. Measurable if it is measurable with respect to each complete Borel probability measure on X (see [7]); K -subadditive and K -superadditive set-valued functions 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.
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