Measure of unbounded set

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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???
 

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  • #2
jbunniii
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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##.
 
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There are three possible cases worth thinking about.
- [itex]m(A)=0[/itex], which jbunniii showed is possible.
- [itex]0<m(A)<\infty[/itex], for which jbunniii provided a very useful hint.
- [itex]m(A)=\infty[/itex]... Is this possible? Consider the sets [itex]A_n=A\cap[-n,n][/itex] you defined. If we have to have [itex]0<m(A_n)<\infty[/itex] for some [itex]n\in \mathbb N[/itex] (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 [itex]\mathbb R[/itex], 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.
 

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