Suppose that $\displaystyle \mid f_n \mid \leq g \in L^1 $ and $\displaystyle f_n \rightarrow f $ in measure.

Then $\displaystyle \int f = \lim \int f_n $.

Proof so far.

Now, $\displaystyle g- f_n \geq 0 $ and $\displaystyle f_n -g \geq 0 $.

So for the first case, then

$\displaystyle \int g - f \leq lim \ inf \int g - f_n = \int g - \ lim \ sup \int f _n $

But how does the lim inf turn into lim sup?

Thank you!