I seem to remember from a measure theory class a long time ago something about the following result. My memory may be totally wrong.

Is it true that on the real line with the ordinary lebesgue measure, any set of positive measure is the union of intervals?

>Is it true that on the real line with the ordinary lebesgue
>measure, any set of positive measure is the union of
>intervals?

Regardless of what you mean by "intervals", I do not believe there is such a statement. If you restrict yourself to open intervals, the counterexample is simple:

the union of a set of positive measure between 1/4 and 3/4 with integers.

For closed intervals, a Cantor set of positive measure is a counterexample.

What you might have in mind is the statement of regularity:

A measurable set is a symetric difference of a Borel set and a set of measure 0. See

http://en.wikipedia.org/wiki/Regularity_theorem_for_Lebesgue_measure