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.