let be a finite measure space ( ) and let , almost everywhere.
Show that for each measurable subset :
My idea for a solution is to use Fatou's lemma:
In one direction:
So we get:
In the other direction, I thought of maybe saying that we know there is an
And an so for all we get that which means
and use Fatou's lemma again on the expression above to get the lim sup smaller or equal to
Is it valid? Am I missing something?
If I do, what can I do to prove the other direction?