Geodesic
On approximating Lebesgue integrals by Riemann sums
Glasgow mathematical journal, Tome 33 (1991) no. 2, pp. 129-134

Voir la notice de l'article provenant de la source Cambridge University Press

If f is a real function, periodic with period 1, we defineIn the whole paper we write ∫ for , mE for the Lebesgue measure of E ∩ [0,1], where E ⊂ R is any measurable set of period 1, and we also use XE for the characteristic function of the set E. Consistent with this, the meaning of Lp is Lp [0, 1]. For all real xwe haveif f is Riemann-integrable on [0, 1]. However,∫ f exists for all f ∈ L1 and one would wish to extend the validity of (2). As easy examples show, (cf. [3], [7]), (2) does not hold for f ∈ Lp in general if p < 2. Moreover, Rudin [4] showed that (2) may fail for all x even for the characteristic function of an open set, and so, to get a reasonable extension, it is natural to weaken (2) towhere S ⊂ N is some “good” increasing subsequence of N. Naturally, for different function classes F ⊂ L1 we get different meanings of being good. That is, we introduce the class of F-good sequences as 
Révész, Szilárd GY.; Ruzsa, Imre Z. On approximating Lebesgue integrals by Riemann sums. Glasgow mathematical journal, Tome 33 (1991) no. 2, pp. 129-134. doi: 10.1017/S0017089500008156
@article{10_1017_S0017089500008156,
     author = {R\'ev\'esz, Szil\'ard GY. and Ruzsa, Imre Z.},
     title = {On approximating {Lebesgue} integrals by {Riemann} sums},
     journal = {Glasgow mathematical journal},
     pages = {129--134},
     year = {1991},
     volume = {33},
     number = {2},
     doi = {10.1017/S0017089500008156},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500008156/}
}
TY  - JOUR
AU  - Révész, Szilárd GY.
AU  - Ruzsa, Imre Z.
TI  - On approximating Lebesgue integrals by Riemann sums
JO  - Glasgow mathematical journal
PY  - 1991
SP  - 129
EP  - 134
VL  - 33
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500008156/
DO  - 10.1017/S0017089500008156
ID  - 10_1017_S0017089500008156
ER  - 
%0 Journal Article
%A Révész, Szilárd GY.
%A Ruzsa, Imre Z.
%T On approximating Lebesgue integrals by Riemann sums
%J Glasgow mathematical journal
%D 1991
%P 129-134
%V 33
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500008156/
%R 10.1017/S0017089500008156
%F 10_1017_S0017089500008156

[1] 1.Jessen, B., On the approximation of Lebesgue integrals by Riemann sums, Ann. of Math. 35 (1934), 248–251. Google Scholar | DOI

[2] 2.Jessen, B., The theory of integration in a space of an infinite number of dimensions, Acta Math. 63 (1934), 249–323. Google Scholar | DOI

[3] 3.Marczinkiewicz, J. and Zygmund, A., Mean values of trigonometrical polynomials, Fund. Math. 28 (1937), 131–166. Google Scholar | DOI

[4] 4.Rudin, W., An arithmetic property of Riemann sums, Proc. Amer. Math. Soc. 15 (1964), 321–324. Google Scholar | DOI

[5] 5.Salem, R., Sur les sommes Riemanniennes des fonctions sommables, Mat. Tidsskr. B 1948 (1948), 60–62. Google Scholar

[6] 6.Sauer, N., On the density of families of sets, J. Combin. Theory Ser. A. 13 (1972), 145–147. Google Scholar | DOI

[7] 7.Ursell, H. D., On the behaviour of a certain sequence of functions derived from a given one, J. London Math. Soc. 12 (1937), 229–232. Google Scholar | DOI

Cité par Sources :