How to prove $ \lim\limits_{n\to \infty}\int_{X}f_ng d\mu=0 \text{ for all} \ g\in L^q(X,\mu)$ Let $(X,M,\mu)$ be a measure space , let $p,q\in (1,\infty)$ satisfy ${1\over p}+{1\over q}=1$ and let $(f_n)$ be a sequence in $L^p(x,\mu)$.Then ,$$\displaystyle \lim_{n\to \infty}\int_{X}f_ng d\mu=0 \text{ for all} \  g\in L^q(X,\mu),$$
if and only if 
$(i)\sup\{||f_n||_p:n \in \mathbb N\}<\infty$
$(ii)\displaystyle\lim_{n\to \infty}\int_{E}f_nd\mu=0\text{ for all E $\in$ M}$ with $\mu(E)<\infty.$ 
How can I prove this ? Please give some hints.
 A: $( \Rightarrow)$
From hypothesis we have that $f_n \to^w 0$
Thus for every functional $x^* \in (L_p)^*(=L_q)$ we have that $x^*(f_n) \to 0 \Rightarrow |x^*(f_n)| \to 0 $
So $|x^*(f_n)|$ is bounded,thus $\exists M>0$ such that $|x^*(f_n)| \leq M$
Now it proved, using the Uniform Boundness Principle for functionals and the properties of the canonical  function $J:X \to X^{**}$, this theorem:

Theorem:
If $X$ is a normed space and $\mathcal{F} \subseteq X$ such that $\sup_{x \in \mathcal{F}}|x^*(x)|< +\infty$
then $\sup_{x \in \mathcal{F}}||x||<+\infty$

Let $\mathcal{F}=\{f_1,f_2...\}$ in our hypothesis.
Then using the theorem  we have that $$\sup_{n \in \mathbb{N}}|x^*(f_n)|< +\infty \Rightarrow \sup_{n \in \mathbb{N}}||f_n||_p< + \infty$$
Now let $E \in M$ such that $\mu(E)< +\infty$
Then by taking the function $g(x)=1_{E} \in L_q$ you have from hypothesis:$$\lim_{n\to \infty}\int_{E}f_nd\mu=\lim_{n\to \infty}\int_{X}f_ngd\mu=0$$
$( \Leftarrow)$
Let $g \in L_p^*(=L_q)$ and $\epsilon>0$
Then because of the fact that the indicator functions of sets with finite measure are dense in the $L_p$ spaces,we have that:
$\exists s \in L_q$ a  simple function  such that $||g-s||_q \leq \epsilon$
Also from the definition of a simple function and from the $(ii)$ of hypothesis we have that $\int_Xf_ns \to 0$
Thus $$|\int_{X}f_ng| = \int_Xf_n(g+s-s) \leq |\int_Xf_ns|+|\int_Xf_n(g-s)|$$ $$\leq |\int_Xf_ns|+(\sup_{n \in \mathbb{N}}||f_n||_p)||g-s||_q$$ from Holder's inequality.
We deduce that $$\limsup_{n \to +\infty}|\int_{X}f_ng| \leq 0+ \epsilon=\epsilon \Rightarrow \limsup_{n \to +\infty}|\int_{X}f_ng|=0$$
Therefore $$\int_{X}f_ng \to 0,\forall g \in L_q$$

I recommend for you as an exercise to prove the theorem i used in the $(\Rightarrow)$ implication and to remember it because it is extremely useful in functional analysis.

