A Large Group of Absolutely Nonmeasurable Additive Functions
Alexander B. Kharazishvili · 2014
The setRR of all real-valued functions defined onR carries several canonical mathematical structures. One of them is a natural commutative group structure on RR. Namely, for any two functions f ∈ RR and g ∈ RR, we have by definition (f + g)(x) = f(x) + g(x) (x ∈ R). In this chapter it is proved, by assuming the Continuum Hypothesis (CH), that there exists a subgroup of RR whose cardinality is strictly greater than c and all nonzero members of which are additive functions absolutely nonmeasurable with respect to the class M(R) of all nonzero σ-finite diffused (i.e., continuous) measures on R.