# Thread: set of positive outer measure

1. ## set of positive outer measure

Show that if a set X in R has positive outer measure, then there is a bounded subset of X also having positive outer measure.
Here's my unoriginal attempt:

Let X be any set in R, X measurable, $0 < m*(X) < \infty$, so X is bounded
Let $Y = X \cap Q$ and $Y' = X \sim Y$
So, Y, Y' bounded
$m*(Y) = 0$ (countable set)
$m*(X) = m*(Y \cup Y') = m*(Y) + m*(Y') = 0 + m*(Y')$
So, $m*(Y') = m*(X) > 0$

Does this make sense?

Another way I wanted to show this was by representing X as a disjoint union of (finite number of) measurable, bounded sets. Then one of these sets would satisfy the criteria, I think.
If X open, I know that X is the disjoint union of a countable collection of open intervals.
If X closed, or X neither open or closed - what is the disjoint union?

Thanks for any tips.

2. ## Re: set of positive outer measure

I figured out the answer to this question. I'm all set.
And the silly thing with the Y,Y' subsets is not quite correct but it doesn't matter.