If is an outer measure on X and is a sequence of disjointed measurable sets, prove that

Proof so far.

Since is an outer measure on , we know that:

1. if

2.

And since is -measurable, then for each n, we have

Now, since is an outer measure.

So what is left to prove is that

So I have

And I know I will have to use the measurable property, but I don't seem to be making any progress here...

Any hints? Thank you.