A random approach to the Lebesgue integral
2008-11-11
Most people meet integrals for the first time in connection with Riemann sums, just to discover later the power of more general approaches. Naturally, the question of a possible convergence of random Riemann sums for Lebesgue integrable functions arises. Jack Grahl [J. Math. Anal. Appl. 340, No. 1, 358-365 (2008; Zbl 1147.28001)] proves that the random Riemann sums of a sequence of partitions whose sizes tend to 0 converge in probability to ∫f. As Olav Nygaard mentions in his review,
“This is not for the first time that random Riemann sums have been studied, as remarked by the author at the end of the paper. J. C. Kieffer and C. V. Stanojevic [Proc. Am. Math. Soc. 85, 389--392 (1982; Zbl 0497.28007)] studied almost sure convergence of random Riemann sums, but with other demands on how to pick evaluation points. Further path to the history of randomized Riemann integral can be found via the review of that paper in Math Reviews, see [MR0656109 (83h:26015)]. C. S. Kahane [Math. Jap. 38, No. 6, 1073--1076 (1993; Zbl 0795.28004)] and A. R. Pruss [Proc. Am. Math. Soc. 124, No. 3, 919--929 (1996; Zbl 0843.60031)] considered almost sure convergence with uniform partitions and revealed the phenomenon
f ∈ L2↔ (δn) ∈ l1.”
