Non-measurable sets and the equation 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦)

Abstract

1. A set of S real numbers which has inner measure m*(S) different from its outer measure m*(S) is non-measurable. An extreme form, which we shall call saturated non-measurability, occurs when m*(S) = 0 but m*(SM) = m(M) for every measurable set M, m(M) denoting the measure of M. This is equivalent to: both S and its complement have zero inner measure. More generally, if a fixed set B of positive measure is under consideration, a subset S of B will be called s-non-mble. if both S and its complement relative to B have zero inner measure. This implies m*(S) = 0, m*(S) = m(B) but is implied by these conditions only if m(B) is finite. Our object, in part, is to show that if B is either the set of all real numbers or any half-open finite interval, then for every infinite cardinal k_C (the power of the continuum), B can be partitioned into k disjoint subsets which are s-non-mble. and are mutually congruent under translation (modulo the length of B in the case that B is a finite interval). Sierpinski and Lusin1 have partitioned B into continuum many disjoint s-non-mble. subsets but they are not constructed to be congruent under translation. Other well known constructions do partition B into a countable infinity of mutually congruent non-measurable subsets, but the subsets are not constructed to be saturated non-measurable.2