Suppose $1 \leq p \leq \infty$ If $||f_n - f||_p \to 0$ then $f_n \to f $ in measure, and hence some subsequence converges to $f$ a.e. On the other hand, if $f_n \to f$ in measure and $|f_n| \leq g \in L^p$ for all $n$, then $||f_n -f||_p \to 0$
Here is how i prove the first part:
Suppose that $||f_n -f||_p \to 0$ Then $||f_n -f ||_p^p \to 0$, which is equivalent to $\int |f_n -f|^p \to 0$. Assume $1<p<\infty$ and let $\epsilon>0$ be given. Define the sets: $$E_{\epsilon}^n =\{ x | \ |f_n(x) -f(x) | \geq \epsilon\} =\{ x \ | \ |f_n(x) -f(x)|^p \geq \epsilon^p \}$$ then $$0\geq \epsilon^p m(E_{\epsilon}^n) = \int_{E_{\epsilon}^n} \epsilon^p \leq \int_{E_{\epsilon}^n} |f_n -f|^p \leq \int_E |f_n -f|^p \to 0$$ so that $m(E_{\epsilon}^n) \to 0$ Therefore $f_n \to f$ in measure
Can anyone show me how to prove the second part? Thanks