# Measure of unbounded set

## Main Question or Discussion Point

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 sence for me. Could someone help me???

Related Set Theory, Logic, Probability, Statistics News on Phys.org
jbunniii
Homework Helper
Gold Member
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##.

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.