Almost every sequence integrates

The purpose of this paper is to discuss a first-return integration process which yields the Lebesgue integral of a bounded measurable function f: I → R defined on a compact interval I. The process itself, which has a Riemann flavor, uses the given function f and a sequence of partitions whose norms tend to 0. The “first-return” of a given sequence \( \bar x \) is used to tag the intervals from the partitions. The main result of the paper is that under rather general circumstances this first return integration process yields the Lebesgue integral of the given function f for almost every sequence \( \bar x \).