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.
Here's my unoriginal attempt: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.
Let X be any set in R, X measurable, , so X is bounded
Let and
So, Y, Y' bounded
(countable set)
So,
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.