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(Note: I have posted a similar post to this recently, but have had no replies ... I am now posting a similar but simple and more focused post with less scope)

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 ...

Proposition 6.1.2 ... and the start of the proof ... reads as follows:

" ... for \(\displaystyle 1 \leq j \leq n, \text{ let } t_j \in [a_j, b_j] \) be arbitrary.

\(\displaystyle B' = \{ x \in \mathbb{R^n} \ \ | \ \ a_j \leq x_j \leq t_j \text{ and } a_k \leq x_k \leq b_k \text{ for } k \neq j \} \)

... now ... to simplify the situation ... consider the case for \(\displaystyle n = 2 \) ... that is for math] R^2 [/math] ...

Thus we consider :

\(\displaystyle B' = \{ ( x_1, x_2 ) \in \mathbb{R^2} \ \ | \ \ a_j \leq x_j \leq t_j \text{ and } a_k \leq x_k \leq b_k \text{ for } k \neq j \} \)

where \(\displaystyle 1 \leq j \leq 2 \)

The sets involved in determining B' are as follows:

For $ j=1$ we consider the set : $ a_1 \leq x_1\leq t_1 \text{ and } a_2 \leq x_2 \leq b_2 $

For $ j=2 $ we consider the set : $ a_2 \leq x_2 \leq t_2 \text{ and } a_1 \leq x_1 \leq b_1$Now ... with these sets ... do we :

$ \bullet $ Take $j=1$ or take $j=2 $ and just consider one set

$ \bullet $ Consider the case for $j=1$ or $j=2$ ... that is , take the union of both sets

$ \bullet $ Consider the case for $j=1$ and $j=2$ ... that is , take the intersection of both setsI think the correct option is to take the intersection of both sets since in the initial specification of B' both conditions seem to apply ...Is that correct?

If it is not correct can someone please explain the mistakes, shortcomings and deficiencies ...

If the intersection is the valid and correct way to proceed then :

\(\displaystyle B' = \{ ( x_1, x_2 ) \in \mathbb{R^2} \ \ | \ \ a_1 \leq x_1 \leq t_1 \text{ and } a_2 \leq x_2 \leq t_2 \} \)Is that correct?Hoping someone can help .

... any help will be much appreciated ...

Peter

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 ...

Proposition 6.1.2 ... and the start of the proof ... reads as follows:

**Definitions and text preliminary to the Proposition reads as follows:**

At the start of the proof in (i) D&K write :" ... for \(\displaystyle 1 \leq j \leq n, \text{ let } t_j \in [a_j, b_j] \) be arbitrary.

**Consider**\(\displaystyle B' = \{ x \in \mathbb{R^n} \ \ | \ \ a_j \leq x_j \leq t_j \text{ and } a_k \leq x_k \leq b_k \text{ for } k \neq j \} \)

... now ... to simplify the situation ... consider the case for \(\displaystyle n = 2 \) ... that is for math] R^2 [/math] ...

Thus we consider :

\(\displaystyle B' = \{ ( x_1, x_2 ) \in \mathbb{R^2} \ \ | \ \ a_j \leq x_j \leq t_j \text{ and } a_k \leq x_k \leq b_k \text{ for } k \neq j \} \)

where \(\displaystyle 1 \leq j \leq 2 \)

__*** Now my problem is how do we validly and correctly determine B' ...__*****The sets involved in determining B' are as follows:

For $ j=1$ we consider the set : $ a_1 \leq x_1\leq t_1 \text{ and } a_2 \leq x_2 \leq b_2 $

For $ j=2 $ we consider the set : $ a_2 \leq x_2 \leq t_2 \text{ and } a_1 \leq x_1 \leq b_1$Now ... with these sets ... do we :

$ \bullet $ Take $j=1$ or take $j=2 $ and just consider one set

$ \bullet $ Consider the case for $j=1$ or $j=2$ ... that is , take the union of both sets

$ \bullet $ Consider the case for $j=1$ and $j=2$ ... that is , take the intersection of both setsI think the correct option is to take the intersection of both sets since in the initial specification of B' both conditions seem to apply ...Is that correct?

If it is not correct can someone please explain the mistakes, shortcomings and deficiencies ...

If the intersection is the valid and correct way to proceed then :

\(\displaystyle B' = \{ ( x_1, x_2 ) \in \mathbb{R^2} \ \ | \ \ a_1 \leq x_1 \leq t_1 \text{ and } a_2 \leq x_2 \leq t_2 \} \)Is that correct?Hoping someone can help .

... any help will be much appreciated ...

Peter

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