- #1

- 3,988

- 48

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

\(\displaystyle \text vol_n (B) =\text vol_n (B') + \text vol_n (B'') \) ... ... ... ... (*)

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 = \{ 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 \} $

and

$ B"_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

[tex] \text vol_n (B) =\text vol_n (B') + \text vol_n (B'') [/tex] ... ... ... ... (*)

... 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 \(\displaystyle \text vol_n (B) =\text vol_n (B') + \text vol_n (B'') \) ... ... ... ... (*) 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

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

\(\displaystyle \text vol_n (B) =\text vol_n (B') + \text vol_n (B'') \) ... ... ... ... (*)

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 = \{ 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 \} $

and

$ B"_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

[tex] \text vol_n (B) =\text vol_n (B') + \text vol_n (B'') [/tex] ... ... ... ... (*)

... 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 \(\displaystyle \text vol_n (B) =\text vol_n (B') + \text vol_n (B'') \) ... ... ... ... (*) 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

Last edited: