The author considers some properties of sets and functions which are measurable (or nonmeasurable) with respect to certain classes of measures. In the paper, the notion of an absolutely nonmeasurable set (function) is examined. Sierpinski-Zygmund type functions are constructed having additional properties closely connected with the above-mentioned notion. Also, some small subsets of uncountable commutative groups are discussed whose algebraic sum turns out to be an absolutely nonmeasurable set.