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On approximating Lebesgue integrals by Riemann sums

Published online by Cambridge University Press:  18 May 2009

Szilárd GY. Révész  and
Imre Z. Ruzsa
Szilárd GY. Révész
Affiliation:
Mathematical Research Institute of the Hungarian Academy of Sciences, Budapest, Reáltanoda u. 13–15, POB 127, H-1364Hungary
Imre Z. Ruzsa
Affiliation:
Mathematical Research Institute of the Hungarian Academy of Sciences, Budapest, Reáltanoda u. 13–15, POB 127, H-1364Hungary
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If f is a real function, periodic with period 1, we define

In 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 have

if 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) to

where 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

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 33 , Issue 2 , May 1991 , pp. 129 - 134
Copyright
Copyright © Glasgow Mathematical Journal Trust 1991

References

REFERENCES

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