The Hahn–Banach theorem almost everywhere

Roman Badora

doi:10.1007/s00010-015-0365-z

Roman Badora

Open Access

https://doi.org/10.1007/s00010-015-0365-z

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Abstract

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

Highlights

In the year 1960 Erdos [3] raised the following problem: suppose that a function f : R → R satisfies the equation f (x + y) = f (x) + f (y), for almost all (x, y) ∈ R2
Does there exists an additive function a : R → R [i.e. a satisfies a(x + y) = a(x) + a(y), for all (x, y) ∈ R2] such that f (x) = a(x) almost everywhere in R? A positive answer to this question was given by de Bruijn [2]
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 iff there exists a set Z ∈ I such that this condition is satisfied for every x ∈ G\Z

Summary

Introduction

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. If (G, +) and (H, +) are commutative groups, I is a p.l.i. ideal of subsets of G for every Ω(I)-almost additive function f : G → H, i.e. f (x + y) = f (x) + f (y) Ω(I)-a.e. in G × G, there exists a unique homomorphism a : G → H such that f (x) = a(x) I-a.e. in G. Ger [7] generalized de Bruijn’s theorem to the case of non-commutative groups.

Proof of the theorem

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