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위키데이터

• ID : Q853831

말뭉치

1. Lebesgue integral, way of extending the concept of area inside a curve to include functions that do not have graphs representable pictorially.
2. Lebesgue sums are used to define the Lebesgue integral of a bounded function by partitioning the y-values instead of the x-values as is done with Riemann sums.
3. The Lebesgue integral is defined in terms of upper and lower bounds using the Lebesgue measure of a set.
4. The book is an excellent example how with a clear and understanding exposition of the Lebesgue integral theory the author can achieve two purposes.
5. This is important because the Lebesgue integral theory traditionally creates more missunderstanding when compared to more widely used integrals like the Riemann integral for example.
6. The Lebesgue integral extends the integral to a larger class of functions.
7. The Lebesgue integral plays an important role in probability theory, real analysis, and many other fields in mathematics.
8. The Lebesgue integral is obtained by slicing along the y-axis, using the 1-dimensional Lebesgue measure to measure the "width" of the slices.
9. One approach to constructing the Lebesgue integral is to make use of so-called simple functions: finite real-linear combinations of indicator functions.
10. In essence, the Lebesgue integral is looking at how often a function achieves a certain value rather than the value of a function at a particular point.
11. A new definition of integral-like functionals exploiting the ideas of the Lebesgue integral construction and extending the idea of pan-integrals is given.
12. As a result, a theoretical basis for applications of the generalized Lebesgue integral is provided.
13. Several types of integrals known from the literature are shown to be special cases of generalized Lebesgue integral.
14. We present in this paper several examples of Lebesgue integral calculated directly from its definitions using Mathematica.
15. But it is difficult to find analogical examples for Lebesgue integral in the available literature.
16. As another indication of its superiority, note that none of these convolutions is necessary for the Lebesgue integral.
17. This books starts with a review of the familiar calculus integral and then constructs the Lebesgue integral from the ground up using the same ideas.
18. It turns out that it really can be, via a path to the Lebesgue integral that is different from the one I took as a graduate student.
19. However, the discussion of these topics seems a bit more compact than in a typical real analysis course; they are introduced here primarily as tools for the development of the Lebesgue integral.
20. (**) holds by definition of the Lebesgue integral and the transformed measure.

메타데이터

위키데이터

• ID : Q853831

Spacy 패턴 목록

• [{'LOWER': 'lebesgue'}, {'LEMMA': 'integration'}]
• [{'LOWER': 'lebesgue'}, {'LEMMA': 'integral'}]