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 $$\dot{ \mathcal{P} }$$ be a tagged partition of $$[a, b]$$ with $$ \lvert \lvert \dot{ \mathcal{Q} } \lvert \lvert \ \lt \delta $$. ... ... "I am somewhat puzzled by B&S's use of $$\dot{ \mathcal{P} }$$ and $$\dot{ \mathcal{Q} }$$ 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 $$\dot{ \mathcal{P} }$$ and $$\dot{ \mathcal{Q} }$$ 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 $$\dot{ \mathcal{P} }$$ be a tagged partition of $$[a, b]$$ with $$ \lvert \lvert \dot{ \mathcal{Q} } \lvert \lvert \ \lt \delta $$. ... ... "I am somewhat puzzled by B&S's use of $$\dot{ \mathcal{P} }$$ and $$\dot{ \mathcal{Q} }$$ 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 $$\color{red}\dot{ \mathcal{Q} }$$ be a tagged partition of $$[a, b]$$ with $$ \lvert \lvert \dot{ \mathcal{Q} } \lvert \lvert \ \lt \delta $$. ... ... "
It is particularly unfortunate that there should be an error in what the authors admit is a tricky and delicate proof. The fact that $$\int_a^bf = \int_a^cf + \int_c^bf$$ 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)