Ways of learning Lebesgue integration

In summary, the conversation discusses the approach to teaching Lebesgue integration in Apostol's Mathematical Analysis, which introduces the concept through different types of functions rather than starting with a theory of measure. The participants agree that this is a valid approach and that there is a simple interplay between sets and functions in which a set is mirrored by its characteristic function. The conversation concludes by stating that the individual is studying by themselves and appreciates the reassurance in their learning process.
  • #1
Castilla
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Good day. I am studying Lebesgue integration in Apostol’s Mathematical Analysis. I have learned already (I believe so) the Dominated Convergence Theorem and the Theorem of Differentiation under the integral sign. But Apostol does not introduce the Lebesgue integration by way of a Theory of Measure. Instead he prefers to define Lebesgue integrals in this order: for step functions, for upper functions, for Lebesgue functions, for measurable functions. (F is a measurable function in an interval I if and only if it is the limit of a sequence of step functions).

Does someone of you have learned Lebesgue integration in this way? Is it equivalent to a study based directly on Measure Theory? Do I am losing something relevant?

Thanks for opinions and excuse the mediocre english.
 
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  • #2
i think you are losing nothing. the integral is treated in the way you describe in several classic, excellent texts including the one you are using.

others include the great Functional Analysis, by Riesz and Nagy, and the fine Analysis II by Lang.

There is a simple interplay between sets and functions ni which a set is mirrored by its characteristic function, the fu ntion whiuch equals 1 on the set and 0 off it. The set is measurable iff the function is measurable.

thus approximating functions by step functions is like aproximating sets by rectangles.

one can begin by defining integrals of step functionsm and taking limits of step functions nd studying when the limits of step functions have integrals, or one can begin by studying sets which are limits of rectangles and when etc etc...

if you have measure theory first then the integral of a positive fucnion can be defined as the measure of the region under the rgaph, and if you have integration first then the measure of a set is the integral of its characteristic function.

comme ci comme ca.

I am not an expert however, indeed a rookie (an old rookie) of sorts in this area.
 
  • #3
Thanks, mathwonk. I am studying by myself and sometimes I need this kind of information to reassure me in what I am doing.
 

1. What is Lebesgue integration?

Lebesgue integration is a mathematical technique used to calculate the area under a curve or the volume of a solid in higher dimensions. It is a more general and powerful method compared to the traditional Riemann integration, allowing for the integration of a wider range of functions.

2. What are the benefits of learning Lebesgue integration?

Learning Lebesgue integration allows for a deeper understanding of mathematical analysis and its applications in various fields such as physics, engineering, economics, and more. It also provides a more rigorous and precise way of solving integration problems compared to other methods.

3. How is Lebesgue integration different from Riemann integration?

Lebesgue integration is based on a different approach to defining and calculating the integral compared to Riemann integration. It uses the concept of measure theory, which allows for the integration of a wider class of functions. Additionally, Lebesgue integration is better suited for handling more complex integrals and can be applied to higher-dimensional spaces.

4. What are some common techniques used in learning Lebesgue integration?

Some common techniques used in learning Lebesgue integration include understanding measure theory, the Lebesgue measure, and the Lebesgue integral. It is also helpful to become familiar with the Lebesgue dominated convergence theorem and the Lebesgue differentiation theorem.

5. How can I improve my understanding of Lebesgue integration?

To improve your understanding of Lebesgue integration, it is essential to practice solving various integration problems, both theoretical and applied. It is also helpful to read textbooks and attend lectures or seminars on the topic. Additionally, seeking guidance from a knowledgeable tutor or mentor can also aid in improving your understanding.

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