Note: the earlier question was closed by community due to lack of clarity. So, I have deleted that question and I am asking it again after making changes which I thought were necessary.
While self studying Mathematical Analysis from Tom M Apostol, I am unable to understand proof of theorem 11.23 whose image I am adding.
Due to some problem ( I don't know how) all 4 images are posted at last. For statement and proof of Theorem 11.23 see first 2 images. For Statement of Theorem 10.31 and Theorem 10.38 see image 3 and 4 respectively. Images:
![Theorem 11.31 ]
![Proof of Theorem 11.31]
( Images at last)
Unfortunately , there are a 2 questions in the proof and I have no instructor whose help I can ask for.
These two theorem statements will be useful: ![ Theorem 10.31 ] ! [Theorem 10.38 ( Images at last)
Questions:
1.how did author derived (45) as Theorem 10.31 is not about $ f_{n} $ ie sequence of functions it's about functions , so I can't understand how author used 10.31 while deriving that result.
2.How does continuty of $f_{n} $( I understand why it's continuous )on $\mathbb{R} $ implies that product f $ f_{n} $ is measurable on $\mathbb{R} $ ?
Kindly guide.
(https://i.stack.imgur.com/e2021.jpg)(https://i.stack.imgur.com/MELVV.jpg)(https://i.stack.imgur.com/Wb39K.jpg)(https://i.stack.imgur.com/ZU6YR.jpg)
About your first question, fix a $t$ and then (I am using the notation in Th. 10.31. Change it properly to fit the notation in the proof of 11.23 ) apply Th. 10.31 for $f(x)=e^{-iux}g(x-t)$ and for the intervals $[a_n,b_n]$.
For your second question, (back to the notation in the proof of 11.23) $f$ is measurable and if each $f_n$ continuous then they are also measurable. The product of measurable functions is measurable function.