On uniform distribution for invariant extensions of the linear Lebesgue measure

TL;DR

This paper extends the concept of uniform distribution to invariant measure extensions of Lebesgue measure on [0,1], showing that almost all sequences are uniformly distributed under these measures, analogous to classical theorems.

Contribution

It introduces a new framework for uniform distribution with invariant measure extensions and proves that the set of uniformly distributed sequences has full measure in this context.

Findings

The measure of uniformly distributed sequences is 1 under the extended measures.

An analogue of Hlawka's theorem is established for these measures.

An analogue of Weyl's theorem is also derived.

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.