Understanding Riemann Integrable Functions: Interpreting D&K Pages 427-428

In summary, D&K provides a text discussing the concept of Riemann integrable functions with compact support. The text discusses the idea of a function being bounded and zero outside a bounded subset, and then provides a description of a situation in which a function is integrable. If the function is Riemann integrable over a certain set, the number of integrals of the function over the set is independent of the choice of the set. The situation is depicted in two diagrams.
  • #1
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I am not sure of the overall purpose of the concepts developed below regarding Riemann integrable functions with compact support ... nor am I sure of the details ... so I am sketching out the meaning as I understand it in 2 dimensions and depicting the relevant entities in diagrams ... I am hoping that someone can indicate that my interpretation of the ideas is correct and/or point out misinterpretations, shortcomings and errors ...

On D&K pages 427 and 428 we find the following text concerning Riemann integrable functions with compact support ... note that I am assuming the overall purpose of developing the notion of Riemann integrable functions with compact support is to create the linear space of Theorem 6.2.8 together with the other results of the theorem ...
D&K Lower Part of Page 427.png

D&K ... Page 428.png

To try to ensure that I understand the details I am translating the text into the two dimensional case and briefly explaining how I interpret the text ... I also attempt to depict the ideas in diagrams ...

As I said above ... I am hoping that someone can indicate that my interpretation of the ideas is correct and/or point out misinterpretations, shortcomings and errors ...Now ... we let \(\displaystyle f : \mathbb{R^2} \to \mathbb{R} \) be a function such that:

\(\displaystyle f \) is bounded on \(\displaystyle \mathbb{R^2} \) and zero outside a bounded subset \(\displaystyle A \subset \mathbb{R^2} \)

... then there exists a rectangle \(\displaystyle B \subset \mathbb{R^2} \) with \(\displaystyle f(x) = 0 \text{ if } x \notin B \)​
( ... presumably \(\displaystyle f(x) =0 \) for those \(\displaystyle x \) that are inside \(\displaystyle B \) and outside \(\displaystyle A \) ... ... Is that correct? )​
If \(\displaystyle f \) is Riemann integrable over \(\displaystyle B \) , the number \(\displaystyle \int_B f(x) dx \) is independent of the choice of B ... since it only depends on A ... and other rectangles such as B' containing A will give the same result ... that is \(\displaystyle \int_B f(x) dx \) = \(\displaystyle \int_{B'} f(x) dx \) ... since the only non-zero values of f are coming from the set A which is in all rectangles satisfying (6.7) ... ... ... Is that correct?

This situation is depicted in Figure 1 below ...
D&K Pages 427 and 428 ... Figure 1 ... .png
Does Figure 1 correctly and validly depict the situation described in the scanned text above from D&K ...?
Now ... let B' be another rectangle satisfying (6.7) ... ... ... ... ... see Figure 2 below ...
D&K Pages 427 and 428 ... Figure 2 ... .png
In Figure 2 clearly \(\displaystyle B = (B \cap B' ) \cup ( B \text{ \ } B' ) \)

\(\displaystyle B \cap B' \) is a two dimensional rectangle

and

\(\displaystyle B \text{ \ } B' = B_1 \cup B_2 \)Consequently \(\displaystyle \{ B \cap B' \} \cup \{ B_i \ | \ i = 1,2 \} \) is a partition of B ... ...We have \(\displaystyle \int_{ B_i } f(x) dx \text{ for } i = 1, 2 \text{ because } f(x) =0 \text{ for } x = \text{ int}(B_i) \subset \mathbb{R^2} \text{ \ } B' \) ... ...But ... why isn't \(\displaystyle \int_{ B_i } f(x) dx \) zero for the boundary of the \(\displaystyle B_i \) as well ... ?

Is the above a correct interpretation of D&K's text as scanned above?Help will be appreciatedPeter
 
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Hi Peter

Nice job once again looking to $\mathbb{R}^{2}$. Your pictures are also very well done and illustrate the concepts really well.

You asked a number of questions in this post and, as far as I can tell, your answers and intuition are correct. A few additional notes are:
  1. "presumably \(\displaystyle f(x) =0 \) for those \(\displaystyle x \) that are inside \(\displaystyle B \) and outside \(\displaystyle A \) ... ... Is that correct?" Yes, this is correct because if $x\notin A$, then $f(x)=0.$
  2. I have not read Duistermaat & Kolk so I don't know exactly how they plan to utilize bounded functions of compact support. However, the golden rule behind all calculus principles is to use what you know to approximate what you don't know. Bounded functions of compact support are well-behaved and will likely be used to approximate their more unruly counterparts in an attempt to rigorously define a Riemann integral for these more nuanced cases.
  3. "But ... why isn't $\displaystyle\int_{B_{i}}f(x)dx$ zero for the boundary of the $B_{i}$ as well ... ?" It is zero. The boundary of the $B_{i}$ are 1-dimensional "hyperplanes" (though we commonly refer to 1-dimensional hyperplanes as "lines/line segments/intervals/etc") in $\mathbb{R}^{2}$, so they have measure/2-dimensional area = 0. See bottom of page 423 of Duistermaat & Kolk.

As I mentioned, everything in your post looks pretty good. I tried to point out the things that seemed to warrant an additional comment. If there's something you're still looking to dig deeper on, feel free to let me know.
 
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  • #3
Thanks GJA …

… working through your post now …

Looks really helpful …

Thanks again,

Peter
 

1. What is a Riemann integrable function?

A Riemann integrable function is a function that can be integrated using the Riemann integral, which is a method of calculating the area under a curve. It is a fundamental concept in calculus and is used to solve a variety of problems in mathematics and science.

2. How do I determine if a function is Riemann integrable?

To determine if a function is Riemann integrable, you can use the Riemann sum formula to approximate the area under the curve. If the upper and lower Riemann sums converge to the same value, then the function is Riemann integrable. Alternatively, you can also check if the function is continuous on a closed interval, as all continuous functions on a closed interval are Riemann integrable.

3. What is the difference between Riemann integrability and Lebesgue integrability?

Riemann integrability and Lebesgue integrability are two different methods of calculating the area under a curve. Riemann integration is based on dividing the area into smaller rectangles, while Lebesgue integration is based on dividing the area into smaller regions of equal width. Riemann integration is more intuitive and easier to understand, while Lebesgue integration is more general and can be applied to a wider range of functions.

4. Can a function be Riemann integrable but not continuous?

Yes, a function can be Riemann integrable even if it is not continuous. As long as the function is bounded and has a finite number of discontinuities, it can still be Riemann integrable. However, if a function is unbounded or has an infinite number of discontinuities, it is not Riemann integrable.

5. How is Riemann integration used in real-world applications?

Riemann integration is used in many real-world applications, such as calculating the area under a curve to determine the work done by a force, finding the volume of irregular shapes in engineering and physics, and predicting the future value of investments in finance. It is also used in computer science for data analysis and in economics for calculating consumer surplus and producer surplus.

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