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I am reading Multidimensional Real Analysis II (Integration) by J.J. Duistermaat and J.A.C. Kolk ... and am focused on Chapter 6: Integration ...
I need some help with the proof of Proposition 6.1.2 ... and for this post I will focus on the first auxiliary result ... see (i) ... at the start of the proof ...Near the start of the proof of Proposition 6.1.2 D&K state that :
" ... ... Because b_j - a_j = (b_j - t_j) + (t_j - a_j), it follows straight away that :
[math] \text{ vol}_n (B) = \text{ vol}_n (B') + \text{ vol}_n (B'') [/math]Readers of this post only need to read the very first part of the proof of Proposition 1 (see scanned text below) ... BUT ... I am providing a full text of the proof together with preliminary definitions so readers can get the context and meaning of the overall proof ... but, as I have said, it is not necessary for readers to read any more than the very first few lines of the proof.
Can someone please help me to rigorously prove that [math] \text{ vol}_n (B) = \text{ vol}_n (B') + \text{ vol}_n (B'') [/math] ...Hope someone can help ...
Help will be much appreciated ...
PeterThe proof of Proposition 6.1.2 together with preliminary notes and definitions reads as follows:
Hope that helps,
Peter
I need some help with the proof of Proposition 6.1.2 ... and for this post I will focus on the first auxiliary result ... see (i) ... at the start of the proof ...Near the start of the proof of Proposition 6.1.2 D&K state that :
" ... ... Because b_j - a_j = (b_j - t_j) + (t_j - a_j), it follows straight away that :
[math] \text{ vol}_n (B) = \text{ vol}_n (B') + \text{ vol}_n (B'') [/math]Readers of this post only need to read the very first part of the proof of Proposition 1 (see scanned text below) ... BUT ... I am providing a full text of the proof together with preliminary definitions so readers can get the context and meaning of the overall proof ... but, as I have said, it is not necessary for readers to read any more than the very first few lines of the proof.
Can someone please help me to rigorously prove that [math] \text{ vol}_n (B) = \text{ vol}_n (B') + \text{ vol}_n (B'') [/math] ...Hope someone can help ...
Help will be much appreciated ...
PeterThe proof of Proposition 6.1.2 together with preliminary notes and definitions reads as follows:
Peter
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