The Hahn–Banach theorem almost everywhere

The aim of this work is to present an almost everywhere version of the Hahn–Banach extension theorem.


Introduction
In the year 1960 Erdös [3] raised the following problem: suppose that a function f : R → R satisfies the equation for almost all (x, y) ∈ R 2 (in the sense of the planar Lebesque measure). Does there exists an additive function a : R → R [i.e. a satisfies a(x + y) = a(x) + a(y), for all (x, y) ∈ R 2 ] such that almost everywhere in R (in the sense of the linear Lebesque measure)? A positive answer to this question was given by de Bruijn [2] (and, independently, by Jurkat [9]). N. G. de Bruijn has put the Erdös problem into a more general setting.
Let (G, +) be a group. A non-empty family I of subsets of G is called a proper linearly invariant ideal (briefly p.l.i. ideal) iff it satisfies the following conditions (i) G ∈ I; (ii) if U ∈ I and V ⊂ U , then V ∈ I; For a p.l.i. ideal I of subsets of a group G we say that a given condition is satisfied I-almost everywhere in G (written I-a.e.) iff there exists a set Z ∈ I such that this condition is satisfied for every x ∈ G\Z.
The set belonging to the set ideal are regarded as, in a certain sense, small sets (see Kuczma [10]). For example, if G is a second category topological commutative group then the family of all first category subsets of G is a p.l.i. ideal. If G is a commutative locally compact topological group equipped with the Haar measure μ then the family of all subsets of G which have zero measure is a p.l.i. ideal. Moreover, if G is a normed space (dim G ≥ 1) then the family of all bounded subsets of G is a p.l.i. ideal and also, if G is a commutative uncountable group then the family of all countable subsets of G is a p.l.i. ideal.
Let (G, +) be a commutative group. For a p.l.i. ideal I we may define the following family of subsets of G × G (Ger [5,6]): Ger [7] generalized de Bruijn's theorem to the case of non-commutative groups. The notion of p.l.i. ideals and its properties and applications we can find in [10]. One of the most interesting applications is included in the paper of Ger [8] where the author combines the notions of approximately additive and almost additive mappings.
In this paper we proved the following version of the Hahn-Banach extension theorem.
The Hahn-Banach theorem 175 Then for every additive function a : H → R fulfilling there exists an additive function A : G → R such that and

Proof of the theorem
Assume that I is a p.l.i. ideal of subsets of a commutative group (G, +). For a real function f on G we define I f to be the family of all sets Z ∈ I such that f is bounded on the complement of Z. A real function f on G is called I-essentially bounded if and only if the family I f is non-empty. The space of all I-essentially bounded functions on G will be denoted by B I (R, R).
For every element f of the space B I (G, R) the real numbers are correctly defined and are referred to as the I-essential infimum and the I-essential supremum of the function f , respectively. From the Gajda theorem (Gajda [4], see also Badora [1]) we can derive the following. and 2) for all f ∈ B I (G, R) and all z ∈ G, where the function z f : G → R is defined as follows Now we prove our result. From condition (1.1) we infer the existence of the set U 1 ∈ I such that for every x ∈ G\U 1 there exists a set V x ∈ I such that From (1.2) it follows that there exists a set U 0 ∈ I such that for all z ∈ H\(U ∪ U 0 ∪ (x + V x )). From (iv) with x = 0 we get −V x ∈ I. Using again (iv) we infer that x + V x ∈ I. Moreover U ∈ I. Therefore (2.5) means that the function is I-essentially bounded from above. So, we can define the function ϕ : G → R as follows Consequently N ∈ Ω(I).
Let (x, y) ∈ G × G\N be fixed. Then x ∈ U , y ∈ U and x + y ∈ U . For z ∈ G\((y + V −x ) ∪ (x + y + V x )), by (2.3), we have which leads to the following

From this we get
and Vol. 90 (2016) The Hahn-Banach theorem 177 The last inequalities imply that, for x ∈ G\U , the function is bounded which yields that the function belongs to the space B I (G, R). A function α : G → R we define by the formula where M I is a linear functional whose existence guarantees Theorem 2.1 and the subscript y indicates that the functional M I is applied to a function of the variable y.
If we choose (x, y) ∈ G × G\N then, by the linearity of M I and (2.2), we get . The function α is Ω(I)-almost additive and from Theorem 1.1 we obtain the existence of an additive function A : G → R such that (2.7) Next, let x ∈ H\U be fixed and let y ∈ G\(U ∪ (−x + U ) ∪ N [x]). Then Therefore, for x ∈ H\U , using (2.1) we have

Ending comments
Remark 3.1. Note that we can strengthen Theorem 1.2 assuming that the function a is I-almost additive. Then we start the proof from Theorem 1.1.
Remark 3.2. If, in Theorem 1.2, additionally we assume that the functional p is positively homogeneous then we can prove that the function A is linear. Indeed, for a fixed x ∈ G let us observe that condition (1.4) implies that which means that the real additive function is bounded from above, for example, on the interval (0, 1). Whence this function is linear (continuous), i. e.
for some constant c x ∈ R. Putting t = 1 we get c x = A(x) and Hence A is a linear map.
Remark 3.3. We will show that the assumption (iv) imposed on the family I (symmetry with respect to zero) is essential in our theorem. Let G = R and let H = Z. The family I of subsets of R we define as follows: A ∈ I iff A is a countable subset of the interval (c, +∞), for some c ∈ R.
Then H ∈ I, the family I satisfies conditions (i)-(iii) of the definition of a p.l.i. ideal [but the condition (iv) is not fulfilled].
If A : R → R is an additive function satisfying (1.4), then it is bounded from above on some interval (p is bounded from above on each bounded interval). Therefore A is a linear map. So, A(x) = cx, x ∈ R, for some constant c ∈ R. Moreover Z ∈ I and if the function A fulfills (1.3), then A(x) = 2x, for x ∈ R, which is impossible because A, p satisfy (1.4) and (0, +∞) ∈ I. Vol. 90 (2016) The Hahn-Banach theorem 179 Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.