Prove that the union of two sets of measure zero has measure zero.
Let be sets of measure zero. Let for i=1,2 be the sum of lengths of open intervals that cover for , respectively. Since are measure zero, given , we can have , for i=1,2, respectively.
Now, if , then we have as a cover for . Thus, a union of two sets of measure zero is of measure zero.
Hello,
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
where
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
The notion of measure zero does not involve any Lebesgue measure theory in some situations (link). Some analysis books introduce "measure zero" before introducing Lebesgue measure theory.
The definition of measure zero I had used is
"A set of points capable of being enclosed in intervals whose total length is arbitrarily small" (link).