# Outer measures are monotonic

1. Sep 14, 2009

### MrGandalf

I have a sneaky suspicion that this is a fairly simple task, but I just can't seem to break through this particular proof.

The problem statement, all variables and given/known data

The (Lebesgue) outer measure of any set $A\subseteq\mathbb{R}$ is:

$m^*(A) = inf Z_A$

where

$Z_A = \bigg\{\sum_{n=1}^\infty l(I_n)\;:\;I_n\;\text{are intervals},\;A\subseteq\bigcup_{n=1}^\infty I_n\bigg\}$

My problem is to prove that $m^*$ is monotone, i.e

If $A\subset B$ then $m^*(A) \leq m^*(B)$

The hints are to show that $Z_B \subset Z_A$ and then use the definition of the infimum to show that the larger set can't have an infimum greater than the smaller set.

The attempt at a solution

If $I_n$ covers B, then it will also cover A.
$A\subset B\subset \bigcup_n I_n$. Hence $Z_B \subset Z_A$.

And now I am confused! :) I have two questions.

1) I was under the impression that $l(I_n)$ was the length of a certain intervall. But that makes $Z_A$ and $Z_B$ numbers. How can one number be a subset of another number?

2) My other question is regarding the statement about infimums. What does the size of a set have to do with what the infimum can be? For instance.
$A = \{2, 3\}$ and $B = \{1,2,3,4\}$. Here $A\subset B$ but $\text{inf}\; B < \text{inf}\; A$

Hopefully someone can shed some light on this!

2. Sep 14, 2009

### AKG

1) No, Z_A and Z_B are sets of numbers. For each open interval covering (I_n) of A, you get the number $\sum l(I_n)$. As you range over all such coverings, you get a whole bunch of different numbers, and Z_A is the set of those numbers.

2) The size of a set doesn't affect its inf, but if A is a subset of B, then B's inf is less than or equal to A's. Because inf(B) is the greatest lower bound of B, and if A is a subset of B, then every lower bound of B is a lower bound of A, and hence the greatest lower bound of B is a lower bound for A, and hence less than or equal to the greatest lower bound for A.

3. Sep 14, 2009

### MrGandalf

Thank you!

I will charge the proof with newfound courage and motivation. :)