On uniformly summable functions and a problem of Halmos in ergodic theory

Abstract. Let \(\mathcal L^{*}\) be the \(\|.\|_{1}\)-closure of \(l^{\infty}\), where $$ \|f\|_{1}:=\limsup\limits_{N\rightarrow\infty}N^{-1} \sum\limits_{n\leq N}|f(n)|. $$ The linear functional \(\Lambda\) of the vector space \(\{s\in l^{\infty}: s(n)=const\}\), defined by \(\Lambda(s):=\lim\limits_{N\rightarrow\infty}N^{-1} \sum\limits_{n\leq N}s(n)\), can be extended to a linear functional \(\Lambda^{*}\) on \(\mathcal L^{*}\). We show that, for \(f\in\mathcal L^{*}, \Lambda^{*}(f)\) can be written as an integral, and apply this to a problem formulated
by Halmos in [5].