On uniform distribution for invariant extensions of the linear Lebesgue measure

The concept of uniform distribution in $[0,1]$ is extended for a certain strictly separated maximal (in the sense of cardinality) family $(\lambda_t)_{t \in [0,1]}$ofinvariantextensionsofthelinearLebesguemeasure$\lambda$in$[0.1]$,anditisshownthatthe$\lambda_t^{\infty}$ measure of the set of all ${\lambda }_t$-uniformly distributed sequences is equal to $1$, where ${\lambda }_t^{\mathrm{\infty }}$ 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.