**Definition:** Let $(X,M,\mu)$ be a measure space and $\{f\_n\}$ a sequence of measurable functions on $x$ that are integrable.  

Then $\{f\_n\}$ is *uniformly integrable* if for every $\epsilon >0$, there is a $\delta >0$ such that if $E$ is a measurable subset of $X$ such that $\mu(E) < \delta$, then $$ \int\_E |f\_n|~d\mu < \epsilon\qquad\text{for every} ~n.$$ 

$\{f\_n\}$ is said to be *tight* if for each $\epsilon >0$, there is a subset $X\_0$ of $X$ such that $\mu(X\_0)< \infty$ and $$\int\_{X\setminus X\_0} |f\_n|~d\mu < \epsilon\qquad\text{for every} ~n.$$ 

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**Theorem:(Vitali Convergence)** Let $(X,M,\mu)$ be a measure space. Let $\{f\_n\}$ be a sequence of uniformly integrable functions that also forms a tight sequence. Suppose $f\_n(x) \to f(x)$ a.e. on $X$. Then, $f$ is integrable and, $$ \lim\_{n\to \infty} \int\_X f\_n~d\mu = \int\_X f~ d\mu.$$ 

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I wish to prove the following: 

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> Let $\{f\_n\}$ be a sequence of non-negative integrable functions on $X$. Suppose that $\{f\_n(x)\} \to 0$ for almost all $x\in X.$. Then $$ \lim\_{n\to\infty} \int f\_n~d\mu =0 \Leftrightarrow \{f\_n\}~\text{is uniformly integrable and tight.}$$
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This is my Attempt:

$(\Leftarrow)$ Suppose $f\_n \to 0$. If $\{f\_n\}$ is uniformly integrable and tight, then by Vitali's Convergence theorem, $\lim\_{n\to \infty} \int f\_n~d\mu = 0$.

$(\Rightarrow)$ Let $\lim\_{n\to \infty} \int f\_n~d\mu = 0$. Let $\epsilon >0$. Then $\exists$ an $N$ such that $\int\_X f\_n ~d\mu< \epsilon$ whenever $n\geq N.$
Also, since $f\_n \geq 0$, if $E$ is a measurable subset of $X$ and $n\geq N$, then $\int \_E f\_n~d\mu < \epsilon.$ 

I know that if I have a finite sequence $\{f\_k\}\_{n=1}^N$ of non-negative integrable functions over $X$, then $\{f\_k\}\_{n=1}^N$ is uniformly integrable, since if $E\subset X$ and $\mu(E)<\delta\_k>0$ then $\int\_E
 f\_k~d\mu < \epsilon$. I can take $\delta=\min (\delta\_1,\ldots, \delta\_k)$ so that $\mu(E)< \delta$ and $$\int\_E f\_k~d\mu < \epsilon.$$ 

I'm afraid this is where I'm stuck and I don't know how to proceed. Any form of help will be very much appreciated. Thanks.