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.