# Lebesgue Outer Measure ... Carothers, Proposition 16.1 ...

#### Peter

##### Well-known member
MHB Site Helper
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 $$\displaystyle J_k$$ slightly and shrinking each $$\displaystyle I_n$$ slightly, we may suppose that the $$\displaystyle J_k$$ are open and the $$\displaystyle I_n$$ are closed. ... "

Can someone please explain how Carothers is expecting the $$\displaystyle J_k$$ to be expanded and the $$\displaystyle I_n$$ to be shrunk ... and further, why the proof is still valid after the $$\displaystyle J_k$$ and $$\displaystyle I_n$$ have been altered in this way ... ...

Help will be appreciated ...

Peter