Abstract
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 \( \bar x \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Csörnyei, U. B. Darji, M. J. Evans and P. D. Humke, First-return integrals, J. Math. Anal. Appl., 305 (2005), 546–559.
U. B. Darji and M. J. Evans, A first-return examination of the Lebesgue integral, Real Anal. Exch., 27 (2001–2002), 578–581.
M. J. Evans and P. D. Humke, Almost everywhere first-return recovery, Bull. Polish Acad. Sci.-Math., 52 (2004), 185–195.
J. Grahl, A probabilistic Method for Calculating Lebesgue Integrals, dissertation, University College London (2006), 39pp.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was initiated while the authors were in residence at the Mathematical Institute of St. Andrews University.
Rights and permissions
About this article
Cite this article
Evans, M.J., Humke, P.D. Almost every sequence integrates. Acta Math Hung 117, 35–39 (2007). https://doi.org/10.1007/s10474-007-6046-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-007-6046-1