We investigate some problems of the following type: For which sets H is it true that if f is in a given class F of periodic functions and the difference functions ∆hf(x) = f(x + h) − f(x) are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by H(F ,G), that is, H(F ,G) = {H ⊂ R/Z : (∃f ∈ F \ G) (∀h ∈ H) ∆hf ∈ G}, we try to charact...

It is proved that there exists a Sierpiński-Zygmund function, which is measurable with respect to a certain invariant extension of the Lebesgue measure on the real line R. Let E be a nonempty set and let f : E → R be a function. We say that f is absolutely nonmeasurable if f is nonmeasurable with respect to any nonzero σ-finite diffused (i.e., continuous) measure μ defined on a σ-algebra of sub...