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Boundary of a generalized rectangle has Jordan content 0

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0

1

$begingroup$

Show that the boundary of a generalized rectangle has Jordan content
0.

My attempt:
Let $U$ be a generalized rectangle, $U = I_1*...*I_n$, where $I_i = [a_i,b_i]$. I tried to show that $I_1$ has Jordan content 0:
Let $R=[a_1,b_1]*[-fracepsilon4(b_1-a_1),fracepsilon4(b_1-a_1)].$ Then $I_1 subset R$ and $vol(R)=fracepsilon2<epsilon$, so $I_1$ has Jordan content 0. (Is this correct?)

But I don't know how to use this to extend to the generalized rectangle.
Am I thinking in the right direction, and how to extend this to show that boundary of U has Jordan content 0?

real-analysis

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asked Mar 21 at 18:58

dxdydzdxdydz

47610

$endgroup$

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0

1

$begingroup$

Show that the boundary of a generalized rectangle has Jordan content
0.

My attempt:
Let $U$ be a generalized rectangle, $U = I_1*...*I_n$, where $I_i = [a_i,b_i]$. I tried to show that $I_1$ has Jordan content 0:
Let $R=[a_1,b_1]*[-fracepsilon4(b_1-a_1),fracepsilon4(b_1-a_1)].$ Then $I_1 subset R$ and $vol(R)=fracepsilon2<epsilon$, so $I_1$ has Jordan content 0. (Is this correct?)

But I don't know how to use this to extend to the generalized rectangle.
Am I thinking in the right direction, and how to extend this to show that boundary of U has Jordan content 0?

real-analysis

share|cite|improve this question

asked Mar 21 at 18:58

dxdydzdxdydz

47610

$endgroup$

add a comment | 

$begingroup$

Show that the boundary of a generalized rectangle has Jordan content
0.

My attempt:
Let $U$ be a generalized rectangle, $U = I_1*...*I_n$, where $I_i = [a_i,b_i]$. I tried to show that $I_1$ has Jordan content 0:
Let $R=[a_1,b_1]*[-fracepsilon4(b_1-a_1),fracepsilon4(b_1-a_1)].$ Then $I_1 subset R$ and $vol(R)=fracepsilon2<epsilon$, so $I_1$ has Jordan content 0. (Is this correct?)

But I don't know how to use this to extend to the generalized rectangle.
Am I thinking in the right direction, and how to extend this to show that boundary of U has Jordan content 0?

real-analysis

share|cite|improve this question

asked Mar 21 at 18:58

dxdydzdxdydz

47610

$endgroup$

share|cite|improve this question

asked Mar 21 at 18:58

dxdydzdxdydz

47610

share|cite|improve this question

asked Mar 21 at 18:58

dxdydzdxdydz

47610

asked Mar 21 at 18:58

dxdydzdxdydz

47610

asked Mar 21 at 18:58

dxdydzdxdydz

47610