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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 aspects of the proof of Proposition 6.1.2 ...
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 ...
Proposition 6.1.2 reads as follows:
Definitions and text preliminary to the Proposition reads as follows:
I am trying to write a detailed proof of the fact or assertion that
\text{vol}_n (B) = \text{vol}_n (B') + \text{vol}_n (B'') ... ... ... ... (*)
given their definitions as sets ... but have so far been able to formulate a proof ...
I hope someone can help ...
I am especially interested in tying or connecting the proof of (*) to the definitions of B' and B" as sets.To explicitly indicate my concerns I am presenting an example from $$ \mathbb{R^2} $$Let B' = \{ ( x_1, x_2 ) \in \mathbb{R^2} \; \; | \; \; a_1 \leq x_1 \leq t_1 \text{ and } a_2 \leq x_2 \leq b_2 \; \; ; \; \; \text{ and } a_2 \leq x_2 \leq t_2 \text{ and } a_1 \leq x_1 \leq b_1 \}Thus B' is made up of two rectangles, viz.
B'_1 = \{ ( x_1, x_2 ) \in \mathbb{R^2} \; \; | \; \; a_1 \leq x_1 \leq t_1 \text{ and } a_2 \leq x_2 \leq b_2 \}
and
B'_2 = \{ ( x_1, x_2 ) \in \mathbb{R^2} \; \; | \; \; a_2 \leq x_2 \leq t_2 \text{ and } a_1 \leq x_1 \leq b_1 \}
These two rectangles are depicted in Figures 1 and 2 below ...
Now ... let B" = \{ ( x_1, x_2 ) \in \mathbb{R^2} \; \; | \; \; t_1 \leq x_1 \leq b_1 \text{ and } a_2 \leq x_2 \leq b_2 \; \; ; \; \; \text{ and } t_2 \leq x_2 \leq b_2 \text{ and } a_1 \leq x_1 \leq b_1 \}Thus B" is made up of two rectangles, viz.
B"_1 = \{ ( x_1, x_2 ) \in \mathbb{R^2} \; \; | \; \; t_1 \leq x_1 \leq b_1 \text{ and } a_2 \leq x_2 \leq b_2 \}
andB"_2 = \{ ( x_1, x_2 ) \in \mathbb{R^2} \; \; | \; \; t_2 \leq x_2 \leq b_2 \text{ and } a_1 \leq x_1 \leq b_1 \}The above two rectangles B"_1 and B"_2 are depicted in Figures 3 and 4 below ...
Now ... my problem is this: I do not see how
\text vol_n (B) =\text vol_n (B') + \text vol_n (B'') ... ... ... ... (*)
... that is I do not see how the volumes of B' and B" which together comprise the overlapping rectangles B'_1 , B'_2 , B"_1 \text{ and } B"_2 satisfy (*) ...Can someone explain how (*) can be true given my analysis above ... I must be making some errors ...NOTE: I know that (6.2) in the text in defining the conditions for a partition of a rectangle Duistermaat and Kolk write
B_i \cap B_j = \emptyset or \text vol_n ( B_i \cap B_j ) = 0 if i \neq j
... but this is prescriptively defining a partition of a rectangle B... it does not, in my opinion, mean that we can overlook (put to zero) the overlaps or intersections in the rectangles B'_1 , B'_2, B"_1, \text{ and } B"_2 ...
Either way I cannot see how \text vol_n (B) =\text vol_n (B') + \text vol_n (B'') ... ... ... ... (*)
arises from the union of the rectangles
B'_1 , B'_2, B"_1, \text{ and } B"_2 ...Hope someone can explain and clarify this aspect of the proof of Proposition 6.1.2 ...
Peter
Proposition 6.1.2 reads as follows:
I am trying to write a detailed proof of the fact or assertion that
\text{vol}_n (B) = \text{vol}_n (B') + \text{vol}_n (B'') ... ... ... ... (*)
given their definitions as sets ... but have so far been able to formulate a proof ...
I hope someone can help ...
I am especially interested in tying or connecting the proof of (*) to the definitions of B' and B" as sets.To explicitly indicate my concerns I am presenting an example from $$ \mathbb{R^2} $$Let B' = \{ ( x_1, x_2 ) \in \mathbb{R^2} \; \; | \; \; a_1 \leq x_1 \leq t_1 \text{ and } a_2 \leq x_2 \leq b_2 \; \; ; \; \; \text{ and } a_2 \leq x_2 \leq t_2 \text{ and } a_1 \leq x_1 \leq b_1 \}Thus B' is made up of two rectangles, viz.
B'_1 = \{ ( x_1, x_2 ) \in \mathbb{R^2} \; \; | \; \; a_1 \leq x_1 \leq t_1 \text{ and } a_2 \leq x_2 \leq b_2 \}
and
B'_2 = \{ ( x_1, x_2 ) \in \mathbb{R^2} \; \; | \; \; a_2 \leq x_2 \leq t_2 \text{ and } a_1 \leq x_1 \leq b_1 \}
These two rectangles are depicted in Figures 1 and 2 below ...
Now ... let B" = \{ ( x_1, x_2 ) \in \mathbb{R^2} \; \; | \; \; t_1 \leq x_1 \leq b_1 \text{ and } a_2 \leq x_2 \leq b_2 \; \; ; \; \; \text{ and } t_2 \leq x_2 \leq b_2 \text{ and } a_1 \leq x_1 \leq b_1 \}Thus B" is made up of two rectangles, viz.
B"_1 = \{ ( x_1, x_2 ) \in \mathbb{R^2} \; \; | \; \; t_1 \leq x_1 \leq b_1 \text{ and } a_2 \leq x_2 \leq b_2 \}
andB"_2 = \{ ( x_1, x_2 ) \in \mathbb{R^2} \; \; | \; \; t_2 \leq x_2 \leq b_2 \text{ and } a_1 \leq x_1 \leq b_1 \}The above two rectangles B"_1 and B"_2 are depicted in Figures 3 and 4 below ...
Now ... my problem is this: I do not see how
\text vol_n (B) =\text vol_n (B') + \text vol_n (B'') ... ... ... ... (*)
... that is I do not see how the volumes of B' and B" which together comprise the overlapping rectangles B'_1 , B'_2 , B"_1 \text{ and } B"_2 satisfy (*) ...Can someone explain how (*) can be true given my analysis above ... I must be making some errors ...NOTE: I know that (6.2) in the text in defining the conditions for a partition of a rectangle Duistermaat and Kolk write
B_i \cap B_j = \emptyset or \text vol_n ( B_i \cap B_j ) = 0 if i \neq j
... but this is prescriptively defining a partition of a rectangle B... it does not, in my opinion, mean that we can overlook (put to zero) the overlaps or intersections in the rectangles B'_1 , B'_2, B"_1, \text{ and } B"_2 ...
Either way I cannot see how \text vol_n (B) =\text vol_n (B') + \text vol_n (B'') ... ... ... ... (*)
arises from the union of the rectangles
B'_1 , B'_2, B"_1, \text{ and } B"_2 ...Hope someone can explain and clarify this aspect of the proof of Proposition 6.1.2 ...
Peter
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