Sequence in $L^p(X,M,\mu)$ I have two question.
Suppose that {$f_k$} is a sequence in $L^p(X,M,\mu)$ such that
$f(x) = \lim_{k \to \infty} f_k(x)$
exists for $\mu$ -a.e. $x \in X$.
Assume $1\le p<\infty$,
$\liminf_{k\to \infty} ||f_k||_p = a$
is finite.


*

*First one is proving that $f \in L^p$ and $||f||_p \le a$.


And if additionally assume that $||f||_p = \lim_{k \to \infty} ||f_k||_p $. 


*

*Second one is to prove $$\lim_{k \to \infty} ||f-f_k||_p =0 $$


Those are very natural fact, but I want have strict proof of them. How can I approach?

1) is easy using Fatou's lemma
