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 $$\int_{X\setminus X_0} |f_n|~d\mu < \epsilon\qquad\text{for every} ~n.$$
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, $$ \lim_{n\to \infty} \int_X f_n~d\mu = \int_X f~ d\mu.$$
I wish to prove the following:
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 $$ \lim_{n\to\infty} \int f_n~d\mu =0 \Leftrightarrow \{f_n\}~\text{is uniformly integrable and tight.}$$
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.
Best Answer
In order to show tightness, fix $\varepsilon>0$. Then you get $N=N(\varepsilon)$ such that $\int_Xf_nd\mu\leq\varepsilon$ if $n\geq N+1$. Now, for all $n\leq N$, you can find a positive $M$ such that $\int_{\{f_n\geq M_n\}}f_nd\mu\leq \varepsilon$, using integrability of $f_n$. (if $f$ is integrable apply the monotone convergence theorem to $f\mathbf 1_{\{|f|\geq n\}})$
Put $A_n:=\{f_n\leq M_n\}$, then $A_n$ is measurable. Take $X_0:=\bigcap_{k=1}^NA_k^c$. Each $A_k^c$ has finite measure (since $\mu(A_k^c)\leq \frac 1{M_k}\int f_kd\mu$) so $X_0$ is of finite measure. Check that we have the wanted inequality.