Banach limits and a measure on the set of 0-1-sequences (Q2304494)

Banach limits are functionals on the space \(\ell^{\infty}\) of bounded sequences which are positive, shift-invariant, and map the constant sequence \((1,1,1,\dots)\) to 1. Denote the set of all Banach limits by \(\mathfrak{B}\) and the set of extreme points of \(\mathfrak{B}\) by \(\mathrm{ext}(\mathfrak{B})\). By identifying each element of \([0,1]\) with the sequence of its binary digits, the Lebesgue measure on \([0,1]\) defines a measure on the set \(2^{\mathbb{N}}\) of 0-1-sequences. In the same way, one can also apply the Hausdorff measure to subsets of \(2^{\mathbb{N}}\). The authors announce some results on Banach limits in connection with these measures on \(2^{\mathbb{N}}\). For example, the set \(\{x\in 2^{\mathbb{N}}:Bx\in\{0,1\} \text{ for all } B\in \mathrm{ext}(\mathfrak{B})\}\) has Lebesgue measure zero.