1. ## Liminf to limsup

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!

2. What does "in measure" mean? If i'ts pointwise convergence a.e. with respect to the measure couldn't you apply Fatou two times (to $\displaystyle f_k$ and $\displaystyle -f_k$) to get that $\displaystyle \limsup \int f_k \leq f \leq \liminf \int f_k$

As for your question $\displaystyle \liminf (-f_k)=-\limsup (f_k)$

3. Convergence in measure :

$\displaystyle \forall \epsilon >0, \lim_{n\to\infty} \mu(\{|f_n-f|>\epsilon\})=0$