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.