Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 1147.28001
Grahl, Jack
A random approach to the Lebesgue integral. (English)
[J] J. Math. Anal. Appl. 340, No. 1, 358-365 (2008). ISSN 0022-247X

Let $\mathcal P=(I_k)_{k=1}^n$ be a partition of the unit interval and let $f$ be Lebesgue measurable there. A random Riemann sum is constructed by looking at the evaluation points $t_k\in I_k$ in the usual Riemann sum as random variables over $I_k$ with uniform distribution. One then gets the random sum $S_{\mathcal P}(f)=\sum_{k=1}^n f(t_k)\vert I_k\vert $. It is now easy to see that the expectation of $S_{\mathcal P}(f)$ is the Lebesgue-integral of $f$ over the unit interval.\par Now let the size, $\delta=\max_k \vert I_k\vert $, of $\mathcal P$ tend to zero. It turns out that the following weak law of large numbers is true: Assume $f$ has a Lebesgue integral $\int f$ over the unit interval. Then the random Riemann sums of a sequence of partitions whose sizes tend to 0 converges in probability to $\int f$.\par The corresponding result for almost sure convergence is not true and much harder to analyze. Almost sure convergence of random Riemann sums depends on the way the sizes of the sequence of the partitions tend to 0. The author proves that if $f\in L_p$, $p>1$ and the sizes $\delta_n$ of the sequence ${\mathcal P}_n$ tend to 0 as an $\ell_{p-1}$ sequence, then almost sure convergence of the random Riemann sums towards $\int f$ holds.\par In the other direction the author proves that for every $p>1$ and every $\epsilon>0$ there is an $f\in L_{p-\varepsilon}$ and a sequence $({\mathcal P}_n)$ with $(\delta_n)\in \ell_{p-1}$ such that $P(S_{{\mathcal P}_n}(f)-\int f)=0$!\par 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. {\it J. C. Kieffer} and {\it 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)]. {\it C. S. Kahane} [Math. Jap. 38, No. 6, 1073--1076 (1993; Zbl 0795.28004)] and {\it 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 $f\in L_2\leftrightarrow (\delta_n)\in \ell_1$.
[Olav Nygaard (Kristiansand)]

MSC 2000:: *28A20 Measurable and nonmeasurable functions
28A25 Integration with respect to measures and other set functions
26A42 Ordinary integrals of functions of one real variable

Keywords: Random Riemann integral; Almost sure convergence

Citations: Zbl 0497.28007; Zbl 0795.28004; Zbl 0843.60031

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.
	Elementary number theory. Primes, congruences, and secrets.

Simple Search

Zentralblatt MATH Berlin [Germany]