Prove that the union of two sets of measure zero has measure zero.
Now, if , then we have as a cover for . Thus, a union of two sets of measure zero is of measure zero.
I don't understand why you talk about intervals... we don't know if we're in the real line, do we ?
Here is a way... The idea is to write as a union of disjoint sets of measures 0.
In particular, we have
Since for any set A,B, , it follows that
And since , it follows that
The same is applied to and
And we have an axiom that says that the measure of the union of disjoint sets is the sum of their measures.
And you're done