Properties of Riemann Integrable Functions with Compact Support .... Proof of D&K Theorem 6.2.8 ....

In summary, the conversation discusses the proof of Theorem 6.2.8 Part (iii) in J. J. Duistermaat and J. A. C. Kolk's book "Multidimensional Analysis Vol.II Chapter 6: Integration". The conversation focuses on the definition of Riemann integrable functions with compact support and the use of supremum and infimum in the proof. The summary also includes a formal proof for the given theorem.
  • #1

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I am reading J. J. Duistermaat and J. A. C. Kolk: Multidimensional Analysis Vol.II Chapter 6: Integration ...

I need help with the proof of Theorem 6.2.8 Part (iii) ...The Definition of Riemann integrable functions with compact support and Theorem 6.2.8 and a brief indication of its proof reads as follows:
D&K ... Defn 6.2.7 and Theorem 6.2.8 .png

D&K ... Defn 6.2.7 and Theorem 6.2.8  ... PART II .png
The definition of supremum and infimum are given in the following text from D&K Vol. I ...
D&K ... Theorem 1.6.1 including Defn of Sup A .png

I cannot locate D&K's definition of sup and inf for functions so I am taking the definition from Joseph L. Taylor's book, "Foundations of Analysis".

Taylor's definition reads as follows:

If \(\displaystyle f: X \to \mathbb{R}\) is a real-valued function and \(\displaystyle A \subset X \) ... ... ...

... then we define ..

\(\displaystyle \text{ sup}_B = \text{sup} \{ f(x) \ | \ x \in B \} \)

and

\(\displaystyle \text{ inf}_B = \text{inf} \{ f(x) \ | \ x \in B \} \)
I need help to formulate a detailed, formal and rigorous proof that \(\displaystyle \text{ sup}_B \ fg - \text{ inf}_B \ fg \leq \text{ sup}_B \ f \text{ sup}_B \ g \ - \ \text{ inf}_B \ f \text{ inf}_B \ g \)I have been unable to make a meaningful start on this proof ...Help will be much appreciated ...

Peter
 
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  • #2
For every $x\in B$, $f(x) \leqslant \sup_B f$ and $g(x) \leqslant \sup_B g$. Therefore $(fg)(x) = f(x)g(x) \leqslant \sup_B f \sup_B g$. Now take the sup over $B$ to get $\sup_B fg \leqslant \sup_B f \sup_B g$. A similar argument shows that $\inf_B fg \geqslant \inf_B f \inf_B g$ and so $-\inf_B fg \leqslant -\inf_B f \inf_B g$.
 
  • #3
Thanks Opalg …

… very much appreciate your help …

Just reflecting on what you have written …

Peter
 

1. What is the D&K Theorem and why is it important in the study of Riemann integrable functions?

The D&K Theorem, also known as the Darboux and Kurzweil theorem, is a fundamental result in the theory of Riemann integration. It states that a function is Riemann integrable if and only if it is bounded and has a set of points of discontinuity of measure zero. This theorem is important because it provides a necessary and sufficient condition for a function to be integrable, which is crucial in applications and further developments in the field.

2. Can you explain the concept of compact support in relation to Riemann integrable functions?

A function with compact support is one that is equal to zero outside of a finite interval. In the context of Riemann integration, this means that the function is bounded and has a finite number of points of discontinuity, making it easier to determine its integrability. The compact support property allows for a simpler proof of the D&K Theorem, as it eliminates the need to consider unbounded functions.

3. What is the significance of the proof of the D&K Theorem 6.2.8 in the study of Riemann integrable functions?

The proof of the D&K Theorem 6.2.8 is significant because it provides a rigorous and systematic approach to determining the integrability of a function. It also showcases the use of various mathematical techniques, such as the Lebesgue covering lemma and the Cauchy criterion, in proving a fundamental result in the theory of Riemann integration.

4. How does the D&K Theorem 6.2.8 relate to other theorems and concepts in Riemann integration?

The D&K Theorem 6.2.8 is closely related to other theorems and concepts in Riemann integration, such as the Lebesgue integrability criterion and the Fundamental Theorem of Calculus. It also serves as a basis for further developments in the field, such as the Lebesgue integral and the theory of integration on higher-dimensional spaces.

5. Are there any practical applications of the D&K Theorem 6.2.8 and its proof in real-world scenarios?

The D&K Theorem 6.2.8 and its proof have numerous practical applications in various fields, such as physics, engineering, and economics. In physics, it is used to determine the work done by a variable force, while in engineering, it is used to calculate the total energy of a system. In economics, it is used to calculate the total utility of a consumer. Additionally, the proof of this theorem has implications in the study of more advanced mathematical concepts, such as measure theory and functional analysis.

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