On a gap series of Mark Kac

Mark Kac gave an example of a function f on the unit interval such that f cannot be written as f(t)=g(2t)-g(t) with an integrable function g, but the limiting variance of ${\displaystyle n^{-\frac{1}{2}}\sum _{k=0}^{n-1}f\left(2^{k}t\right)}$ vanishes. It is proved that there is no measurable g such that f(t)=g(2t)-g(t). It is also proved that there is a non-measurable g which satisfies this equality.