Random sampling in the evaluation of a Lebesgue integral

Random Sampling in the Evaluation of a Lebesgue Integral

The purpose of this note is to show that in the evaluation of a Lebesgue integral a sort of random sampling scheme is permissible. I t is possible, therefore, that the results may have application to questions of probability or statistics. The chief positive result established here is embodied in Theorem 4. With the object of indicating clearly the question at issue, however, we formulate two p...

Rethinking the Lebesgue Integral

a swot analysis of the english program of a bilingual school in iran

با توجه به جایگاه زبان انگلیسی به عنوان زبانی بین المللی و با در نظر گرفتن این واقعیت که دولت ها و مسئولان آموزش و پرورش در سراسر جهان در حال حاضر احساس نیاز به ایجاد موقعیتی برای کودکان جهت یاد گیری زبان انگلیسی درسنین پایین در مدارس دو زبانه می کنند، تحقیق حاضر با استفاده از مدل swot (قوت ها، ضعف ها، فرصتها و تهدیدها) سعی در ارزیابی مدرسه ای دو زبانه در ایران را دارد. جهت انجام این تحقیق در م...

Notes on the Lebesgue Integral

In the definition of the Riemann integral of a function f(x), the x-axis is partitioned and the integral is defined in terms of limits of the Riemann sums ∑n−1 j=0 f(x ∗ j)∆j, where ∆j = xj+1− xj. The basic idea for the Lebesgue integral is to partition the y-axis, which contains the range of f , rather than the x-axis. This seems like a “dumb” idea at first. Shouldn’t the two ways end up givin...

The Sampling Theorem in Variable Lebesgue Spaces

hold. The facts above are well-known as the classical Shannon sampling theorem initially proved by Ogura [10]. Ashino and Mandai [1] generalized the sampling theorem in Lebesgue spaces L0(R) for 1 < p0 < ∞. Their generalized sampling theorem is the following. Theorem 1.1 ([1]). Let r > 0 and 1 < p0 < ∞. Then for all f ∈ L 0(R) with supp f̂ ⊂ [−rπ, rπ], we have the norm inequality C p r ‖f‖Lp0(Rn...