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arXiv · 1603.04472

On uniform distribution for invariant extensions of the linear Lebesgue measure

A. Kirtadze·G. Pantsulaia·N. Rusiashvili
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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.

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A. Kirtadze, G. Pantsulaia, N. Rusiashvili. 2016-03-09. On uniform distribution for invariant extensions of the linear Lebesgue measure. https://arxiv.org/abs/1603.04472

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