I'm working on the following problem, I'm having trouble with the reverse direction. My question is bolded below. Also could someone check my forward direction?:
Let $(X, \mathcal{M}, \mu)$ be a $\sigma$ finite measure space and $\{f_n\},f \in L^P(X)$. Prove that $f_n \rightharpoonup f$ in $L^p(X)$ iff $\|f_n\|_p \leq c$ for all $n$ and $\int_A f_n\, d\mu \rightarrow \int_A f \, d\mu$ for all $A$ with $\mu(A) < \infty$.
For the reverse direction, we can use the characteristic functions in $L^q$ to build arbitrary functions in $L^q$ and use Monotone Convergence on $A$ equals a ball. Then increase the radius of the ball at each step making error $\epsilon/2^n$. However, I'm having trouble seeing how I use the boundedness of the sequence $f_n$)
(For the forward direction, choosing $\chi_{A}\in L^q(X)$ will get the integral condition and the $\|f_n\|_p$ were bounded because the sequence originally lived in $L^p(X)$.)