On Measurable Sierpiński-zygmund Functions

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 subsets of E. Recall that a set X ⊂ R is universal measure zero if, for every σ-finite diffused Borel measure μ on R, the equality μ∗(X) = 0 is satisfied, where μ∗ denotes, as usual, the outer measure associated with μ. It is well known that there exist uncountable universal measure zero subsets of R (see, e.g., [8]). We have a characterization of absolutely nonmeasurable functions in terms of universal measure zero sets and preimages of singletons. Theorem 1. Let f : E → R be a function. The following two assertions are equivalent: 2000 Mathematics Subject Classification. Primary: 28A05, 28D05.