On some properties of Hamel bases and their applications to Marczewski measurable functions

Abstract

Abstract:We introduce new properties of Hamel bases. We show that it is consistent with ZFC that such Hamel bases exist.Under the assumption that there exists a Hamel basis with one of these properties we construct a discontinuous and additive function that is Marczewski measurable. Moreover, we show that such a function can additionally have the intermediate value property (and even be an extendable function). Finally, we examine sums and limits of such functions. MSC:

On Subadditive Functions Bounded Above on a “Large” Set

“…Such ideas were also pursued by Z. Kominek on the basis of result by B. Jones (test set there being capable of spanning the reals, as with Hamel bases but not coincidental with these; for recent work on Hamel bases see e.g. [17], [18], [24]). An alternative approach to test sets was developed also in [10] with connections to uniformity results in the theory of regular variation.…”