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The concept of uniform distribution in $[0,1]$ is extended for a certainstrictly 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 ofall $\lambda_t$-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 $\lambda_t$-uniformlydistributed sequences. An analogy of Weyl's (1916) theorem is obtained insimilar manner.

On uniform distribution for invariant extensions of the linear Lebesgue measure