Let be an algebra,

and let be the collection of countable union of sets in .

Let be the collection of countable intersection of sets in .

Let be a premeasure on .

Let be outer measure with [tex] \mu* \mid _

Prove that if , then is -measurable iff with and

Note:

Proof so far.

First, I assume that with and

So I have

Implies that

Implies that

But what I need is , then of course, all I have to show is that since the other inequality follows from the property of outer measure.

Any hints? Thank you.