What is the Measure of an Unbounded Set in Lebesgue Outer Measure Theory?

[Funky1981]

Suppose A is not a bounded set and m(A∩B)≤(3/4)m(B) for every B. what is m(A)??

here, m is Lebesgue Outer Measure

My attemption is :

Let An=A∩[-n,n], then m(A)=lim m(An)= lim m(An∩[-n,n]) ≤ lim (3/4)m([-n,n]) = infinite.

is my solution right? I am confusing m(A) < infinite , it doest make sense for me. Could someone help me?

 

[jbunniii]

What you wrote is correct as far as it goes, but $m(A) \leq \infty$ doesn't tell you anything new: this is of course true of the outer measure of any set.

Certainly $m(A) = 0$ is possible: consider $A = \mathbb{Q}$, for example.

Is $m(A) > 0$ possible? Hint: consider $A = B$.

 

[economicsnerd]

There are three possible cases worth thinking about.
- $m(A)=0$, which jbunniii showed is possible.
- $0<m(A)<\infty$, for which jbunniii provided a very useful hint.
- $m(A)=\infty$... Is this possible? Consider the sets $A_n=A\cap[-n,n]$ you defined. If we have to have $0<m(A_n)<\infty$ for some $n\in \mathbb N$ (Is this true?), then maybe the same trick as above can be reused.

It's worth noting that the answer to this question depends on a special property of the Lebesgue measure on $\mathbb R$, which fails for some other infinite measures. Namely, we're using the property that the whole space is a countable union of finite-measure sets.