Riemann Integration - The Additivity Theorem .... B&S Theorem 7.2.9 ....

[Math Amateur]

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ...

I am focused on Chapter 7: The Riemann Integral ...

I need help in fully understanding an aspect of the proof of Theorem 7.2.9 ...Theorem 7.2.9 and its proof ... ... read as follows:View attachment 7319
https://www.physicsforums.com/attachments/7320In the above proof (near to the start ...) we read the following:

" ... let \(\displaystyle \dot{ \mathcal{P} }\) be a tagged partition of \(\displaystyle [a, b]\) with \(\displaystyle \lvert \lvert \dot{ \mathcal{Q} } \lvert \lvert \ \lt \delta \). ... ... "I am somewhat puzzled by B&S's use of \(\displaystyle \dot{ \mathcal{P} }\) and \(\displaystyle \dot{ \mathcal{Q} }\) in this proof ... can someone please explain the use of these symbols ... I know they are different partitions ... but why does B&S introduce them ... what is the logic ... why do we need both ... ... [ ... the use of both \(\displaystyle \dot{ \mathcal{P} }\) and \(\displaystyle \dot{ \mathcal{Q} }\) in the particular statement I quoted seems to me to be most peculiar ... ]

PeterIt may help readers of the above post to have reference to the notation of B&S in setting up the Riemann Integral ... so I am providing the text of Section 7.1 up to and including the definition of the Riemann Integral ... as follows ...View attachment 7321
https://www.physicsforums.com/attachments/7322
https://www.physicsforums.com/attachments/7323

 

[Opalg]

In the above proof (near to the start ...) we read the following:

" ... let \(\displaystyle \dot{ \mathcal{P} }\) be a tagged partition of \(\displaystyle [a, b]\) with \(\displaystyle \lvert \lvert \dot{ \mathcal{Q} } \lvert \lvert \ \lt \delta \). ... ... "I am somewhat puzzled by B&S's use of \(\displaystyle \dot{ \mathcal{P} }\) and \(\displaystyle \dot{ \mathcal{Q} }\) in this proof

It looks to me as though you have identified another misprint in Bartle and Sherbert. The only way I can make sense of it is that the above sentence should read
" ... let \(\displaystyle \color{red}\dot{ \mathcal{Q} }\) be a tagged partition of \(\displaystyle [a, b]\) with \(\displaystyle \lvert \lvert \dot{ \mathcal{Q} } \lvert \lvert \ \lt \delta \). ... ... "
It is particularly unfortunate that there should be an error in what the authors admit is a tricky and delicate proof. The fact that \(\displaystyle \int_a^bf = \int_a^cf + \int_c^bf\) ought to be straightforward. But the integrals are defined in terms of approximating partitions, and a partition of $[a,b]$ may or may not include $c$ as one of its partition points. That is the root cause of the technical difficulties that make this such a complicated and obscure proof.

Bartle and Sherbert would have done well to employ you as a proofreader! (Rofl)

 

[math771]

Hi Peter,

I understand your confusion about the notation used in the proof of Theorem 7.2.9. Let me try to explain it to you.

First, let's review the notation used in the definition of the Riemann Integral. In Section 7.1, B&S define the Riemann Integral as follows:

https://www.physicsforums.com/attachments/7322

Here, P is a partition of the interval [a, b], and \dot{P} is a tagged partition of [a, b], which is a partition with additional information about the points in each subinterval. This notation is used to define the Riemann Sum, which is the sum of the areas of rectangles that approximate the area under the curve.

Now, in the proof of Theorem 7.2.9, B&S use the notation \dot{\mathcal{P}} and \dot{\mathcal{Q}}. These are notations for tagged partitions, just like \dot{P}. The reason they introduce both \dot{\mathcal{P}} and \dot{\mathcal{Q}} in the proof is because they need to consider two different partitions in order to prove the theorem. The first partition, \dot{\mathcal{P}}, is chosen such that its norm is less than \delta, which is a small positive number. This is done to ensure that the rectangles in the Riemann Sum are small enough to accurately approximate the area under the curve. The second partition, \dot{\mathcal{Q}}, is chosen such that its norm is less than half of \delta. This is done to ensure that the difference between the Riemann Sum and the Riemann Integral is small.

In summary, the use of both \dot{\mathcal{P}} and \dot{\mathcal{Q}} in the proof is necessary in order to prove the theorem. I hope this helps clear up your confusion. Let me know if you have any further questions.