Question on convergence in measure and uniform integrability

Question

Suppose $\{f_n\}$ is a sequence of function on $E$, $f_n$ and $f$ are Lebesgue integrable (or absolutely integrable) on $E$, with $\{f_n\} \rightarrow f$ in measure. I want to show that $\lim\limits_{n\rightarrow\infty}{\int_E|f_n-f|=0}$ $\Leftrightarrow$ all the following 3 conditions are satisfied:
(1) $\sup\limits_{n}\int_E{|f_n|}<+\infty$
(2) $\lim\limits_{\mu\rightarrow \infty}\sup\limits_{n} \int_{|f_{n}|\geq\mu}|f_n|=0$
(3) $\lim\limits_{\delta\rightarrow 0}\sup\limits_{n} \int_{|f_{n}|\leq\delta}|f_n|=0$.
For the "$\Rightarrow$" part, I think the 2nd condition probably need the use of Chebyshev's inequality $m( |f_n| \geq \mu) \leq \frac{1}{\mu} \int_E |f_n|$, and similarly the 3rd condition.
For the "$\Leftarrow$" part, maybe we need to prove the uniform integrability and tightness (using the definition on Royden Fitzpatrick Real Analysis 4ed) of $\{f_n-f\}$ , which I don't quite know how to.
Thanks for any corrections or help!