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I am reading N. L. Carothers' book: "Real Analysis". ... ...

I am focused on Chapter 16: Lebesgue Measure ... ...

I need help with an aspect of the proof of Proposition 16.1 ...

Proposition 16.1 and its proof read as follows:

In the above text from Carothers we read the following:

" ... ... But now, by expanding each $J_k$ slightly and shrinking each $I_n$ slightly, we may suppose that the $J_k$ are open and the $I_n$ are closed. ... "Can someone please explain how Carothers is expecting the $J_k$ to be expanded and the $I_n$ to be shrunk ... and further, why the proof is still valid after the $J_k$ and $I_n$ have been altered in this way ... ...

EDIT: My thoughts ...

We could expand each $J_k$ by altering or replacing intervals of the form $(a, b), [a, b)$ and $(a, b]$ by $[a, b]$ ...

This would expand the $J_k$ by one or two points only leaving the length of the intervals unchanged ...

BUT ... we cannot (as Carothers wishes) then suppose the $J_k$ are open ... indeed they would all be closed ... so we have to find another to expand the $J_k$ slightly

A similar problem arises if we shrink the I_n by replacing intervals of the form $[a, b], [a, b)$ and $(a, b]$ by $(a, b)$ ...

Help will be appreciated ...

Peter

I am focused on Chapter 16: Lebesgue Measure ... ...

I need help with an aspect of the proof of Proposition 16.1 ...

Proposition 16.1 and its proof read as follows:

In the above text from Carothers we read the following:

" ... ... But now, by expanding each $J_k$ slightly and shrinking each $I_n$ slightly, we may suppose that the $J_k$ are open and the $I_n$ are closed. ... "Can someone please explain how Carothers is expecting the $J_k$ to be expanded and the $I_n$ to be shrunk ... and further, why the proof is still valid after the $J_k$ and $I_n$ have been altered in this way ... ...

EDIT: My thoughts ...

We could expand each $J_k$ by altering or replacing intervals of the form $(a, b), [a, b)$ and $(a, b]$ by $[a, b]$ ...

This would expand the $J_k$ by one or two points only leaving the length of the intervals unchanged ...

BUT ... we cannot (as Carothers wishes) then suppose the $J_k$ are open ... indeed they would all be closed ... so we have to find another to expand the $J_k$ slightly

A similar problem arises if we shrink the I_n by replacing intervals of the form $[a, b], [a, b)$ and $(a, b]$ by $(a, b)$ ...

Help will be appreciated ...

Peter

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