On uniform distribution for invariant extensions of the linear Lebesgue measure

Abstract

The concept of uniform distribution in $[0,1]$ is extended for a certain strictly separated maximal (in the sense of cardinality) family $({\lambda }_t{)}_{t\in [0,1]}$ of invariant extensions of the linear Lebesgue measure $\lambda$ in $[0.1]$, and it is shown that the ${\lambda }_t^{\infty }$ measure of the set of all ${\lambda }_t$-uniformly distributed sequences is equal to $1$, where ${\lambda }_t^{\infty }$ denotes the infinite power of the measure ${\lambda }_t$. This is an analogue of Hlawka's (1956) theorem for ${\lambda }_t$-uniformly distributed sequences. An analogy of Weyl's (1916) theorem is obtained in similar manner.