i hate when i forget math stuff

This is kind of frustrating. I was up last night trying to recall about a proof from grad school, and realize that I can't remember one important point.

Henri Lebesgue

The topic is Lebesque Measure. Specifically, the existence of nonmeasurable sets. It's a pretty fundamental question. There's all sort of stuff built up, starting with the basic definitions and rules. Then there's all sorts of stuff built up around measure, and theory. And, yes, I know the foregoing is a bit hazy. That's because it's been long enough ago that I don't fully remember what are matters of definition and what are results that are proven.

At any rate, with all that stuff around measure, it's good to establish whether there actually are sets that are nonmeasurable. It's not immediately intuitively obvious (at least it wasn't to me when I was a first semester grad student) that there are nonmeasurable sets. All the obvious ways of constructing sets -- take some intervals or single points. Take intersections or unions of them -- don't immediately work.

But I recall the basic construction. Start by splitting the unit interval into equivalence classes where two points are in the same class if their difference is rational. Then take one element from each equivalence class. That set, call it N, is nonmeasurable.

The proof that N is nonmeasurable relies on taking the union of all sets N+a where:
1) N+a is defined as the set of all numbers n+a where n is an element of N; and
2) a is a rational number in the unit interval.
Let's call that union U.

If N is measurable, then it has a measure which must be either zero or positive. Also, because N+a is just a translation of N, its measure is the same as that of N. Finally, N+a and N+b are disjoint for a not equal b.

So, what is the measure of U? We know it has to be finite because U is a subset of [0,2], which has finite measure. But if N has positive measure, then the measure of U, which is the sum of the measures of N+a (for all a in the unit interval) is infinite. Therefore N has measure zero. If N has measure zero, then U has measure zero. This is a contradiction.

And that's where I'm stuck. How do we know that U cannot have measure zero? I think it has to do with the assertion that U contains an interval, and therefore has to have measure greater than or equal to the length of the interval? Maybe it's that U is equal to [0,2], and therefore has to have measure 2? But how do we know that?

Help! Help, help!