On measurable functions with vanishing differences - Kuczma, Marek (Annales Mathematicae Silesianae, 1992)
It is shown (under suitable conditions on H⊂R) that if f: R→R is a measurable function such that for an n∈N0 and every h∈H we have Δhn+1f(x) = 0 almost everywhere on R, then f is equal almost everywhere on R to a polynomial of degree at most n. In particular, every measurable polynomial function f: R→R is a polynomial. In fact, these (essentially known) results are here proved in a more general and more abstract form. The paper contains also a version of the Łomnicki-type theorem on measurable microperiodic functions.