The Lebesgue syndrome (Q1070368)

This talk is partly historical, partly personal, and partly a looking to the future. The cornerstone in the development on the way of evolution of integrals theories is the Lebesgue's work which is the culmination of all that had gone before. Although this was a giant step forward, soon became clear that it was not good enough to integrate all derivatives, and Denjoy gave his construction - what is now known as the Denjoy-Perron integral. Lebesgue showed that his integral is the limit of Riemann sums, and so did Denjoy, though neither gave anything explicit. The first explicit construction for non-negative functions was given by Beppo Levi. The personal side of this talk explains the author's contribution at this stage: ''I was turned upside down in 1958, throwing away Lebesgue and gresping Riemann''. Returning to Riemann sums but with a more general limit he gets the so called generalized Riemann integral or Riemann- complete integral. This work was independent of J. Kurzweil's paper in 1957. The new integral is more practical and easier than most Lebesgue theories, and, before all, it is a non-absolute theory (the difference being analogous to the difference between all convergent series and the absolutely convergent ones). The Lebesgue syndrome is the ignorance with respect to the work done on non-absolute integration. How large is this syndrome? It may be measured by comparing the number of papers reviewed in one year in M.R.: about 200 on absolute theory against 6 devoted to non-absolute integration. As a consequence, the author throw out a challenge to look at the papers using Lebesgue theory to see whether the proofs can be improved and the contents generalized by generalized Riemann methods.