Real Analysis ( measure theory)

[sbashrawi]

Homework Statement

Let A and B be bounded sets for which there is \alpha > 0 such that |a -b| \geq\alpha
for all a in A and b in B. Prove that outer measure of ( A \bigcup B ) = outer measure of (A) + outer measure of (B)

Homework Equations

We know that outer measure of the union is less than the addition of outer measure of the single sets
we need to show that it is greater than the addition of outer measures of the sets

The Attempt at a Solution

 

[Office_Shredder]

Have you tried anything? The fact that the two sets have a gap between them is obviously important. Try going straight to the definition of outer measure and see if you can spot anything useful

 

[sbashrawi]

I know that they are disjoint but it is not clear for me how to use this in proving that
the outer measure of union is greater than the addition of the ouetr measure of the sets.

but I tried this:

A \bigcupB\subset O for some set O covering A and B
i.e A \bigcupB\subset(\bigcup I\cupJ)
So m(A\bigcupB)< m(A) + m(B)
or m(A\bigcupB) + \epsilon > m(A) + m(B)
and since \epsilon is arbirary. then it is done.

but I think there is a gab in this proof and I don't know why