On strong uniform distribution IV

Let be a strictly increasing sequence of natural numbers and let be a space of Lebesgue measurable functions defined on . Let denote the fractional part of the real number . We say that is an sequence if for each we set , then , almost everywhere with respect to Lebesgue measure. Let . In this paper, we show that if is an for , then there exists such that if denotes , . We also show that for any sequence and any nonconstant integrable function on the interval , , almost everywhere with respect to Lebesgue measure.