  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.
