On a theorem of Banach concerning periodic functions (Q1108391)

A classical result of S. Banach states: If f: \(R\to R\) is a measurable periodic function with period 1 then \(\limsup_{n}f(nx)= \sup_{0\leq t\leq 1}f(t)\) almost everywhere on [0,1]. In the reviewed paper a \(\sigma\)-algebra \({\mathcal S}\) of subsets of R and a proper \(\sigma\)-ideal \({\mathcal I}\subset {\mathcal S}\) with certain properties are considered. Instead of Lebesgue measurable an \({\mathcal S}\)-measurable function is considered. The usual essential supremum is substituted by an essential supremum naturally defined with respect to the \(\sigma\)-ideal \({\mathcal I}\). An abstract version for an \({\mathcal S}\)-measurable function of the above mentioned result of S. Banach is obtained. Some related results are also given.