chapter  14
Absolutely nonmeasurable additive functions
ByGilbert W. Bassett Jr., Roger Koenker
Pages 12

It was shown in Chapter 13 that, assuming Martin’s Axiom (MA), there exists an injective absolutely nonmeasurable function f : R → R $ f : \mathbf{R} \rightarrow \mathbf{R} $ . In other words, it was demonstrated therein that some functions f acting from R into R are extremely bad from the measure-theoretical point of view, i.e., those f are nonmeasurable with respect to any nonzero σ $ \sigma $ -finite diffused measure defined on a σ $ \sigma $ -algebra of subsets of R. In the same chapter it was also pointed out that the existence of absolutely nonmeasurable functions acting from R into R cannot be proved within ZFC set theory, so necessarily needs additional set-theoretical axioms.