# Real Analysis-Monotonicity

1. Sep 18, 2014

### nateHI

**Moderator: Please move my thread to the correct section. Sorry about posting it in the wrong location**

http://press.princeton.edu/chapters/s8008.pdf

In example 2 on page 11 of the text I linked above, the book uses the monotonicity property of the outer measure to conclude
$m_*(Q) \le |Q|$ where Q is a cube.
However, in example 4 on the next page they require a more complicated argument to reach the conclusion
$m_*(R) \le |R|$ where R is a rectangle.
My question is, why couldn't they use monotonicity again?

Last edited: Sep 18, 2014
2. Sep 19, 2014

### mathman

The text is confusing!

3. Sep 23, 2014

### qspeechc

They define the exterior measure using coverings by closed cubes. Example 2 is an closed cube, and Example 4 is a rectangle, and that's what accounts for the difference. In fact, monotonicity has nothing to do with the inequalities you are worried about.

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