On uniform distribution for invariant extensions of the linear Lebesgue measure

2016/03/09 by Kirtadze, A., Pantsulaia, G., Rusiashvili, N.
#26A42 #28A05 #28C10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #G.1.0

paper · doi:10.48550/arxiv.1603.04472

Abstract

The concept of uniform distribution in [0,1] is extended for a certain strictly separated maximal (in the sense of cardinality) family (λ_t)_{t ∈ [0,1]} of invariant extensions of the linear Lebesgue measure λ in [0.1], and it is shown that the λ_t^∞ measure of the set of all λ_t-uniformly distributed sequences is equal to 1, where λ_t^∞ denotes the infinite power of the measure λ_t. This is an analogue of Hlawka's (1956) theorem for λ_t-uniformly distributed sequences. An analogy of Weyl's (1916) theorem is obtained in similar manner.

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