set of positive outer measure

Quote:

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:

Quote:

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.

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.