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Zbl 0795.28004
Kahane, Charles S.
Evaluating Lebesgue integrals as limits of Riemann sums. (English)
[J] Math. Jap. 38, No.6, 1073-1076 (1993). ISSN 0025-5513

The purpose of this paper is to prove that if $f\in L\sp p([0,1])$ with $p>2$ and the points $x\sp{(n)}\sb k$ are chosen independently and at random from the intervals $I\sp{(n)}\sb k=\Bigl[{k-1\over n},{k\over n}\Bigr]$, $k=1,2,\dots,n$, then the Riemann sums $${1\over n} \sum\sp n\sb{k=1} f(x\sp{(n)}\sb k)\to \int\sp 1\sb 0 f(x)dx$$ as $n\to +\infty$ with probability 1. A result of the same type has been obtained by {\it J. C. Kieffer} and {\it Č. V. Stanojević} [Proc. Am. Math. Soc. 85, 389-392 (1982; Zbl 0497.28007)].
[J.M.Ayerbe (Sevilla)]

MSC 2000:: *28A25 Integration with respect to measures and other set functions

Keywords: Lebesgue integrals; $L\sp p$-spaces; Riesz-Thorin convexity theorem; random variables; Riemann sums

Citations: Zbl 0497.28007

Cited in: Zbl 1147.28001

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