proof of Vitali convergence theorem
Theorem.
Let ${f}_{\mathrm{1}}\mathrm{,}{f}_{\mathrm{2}}\mathrm{,}\mathrm{\dots}$ be ${\mathrm{L}}^{p}$integrable functions on a measure space^{} $\mathrm{(}X\mathrm{,}\mu \mathrm{)}$, for $$. The following conditions are necessary and sufficient for ${f}_{n}$ to be a Cauchy sequence^{} in the ${\mathrm{L}}^{p}\mathit{}\mathrm{(}X\mathrm{,}\mu \mathrm{)}$ norm:
 (i)

(ii)
the functions $\{{{f}_{n}}^{p}\}$ are uniformly integrable; and

(iii)
for each $\u03f5>0$, there is a set $A$ of finite measure, with $$ for all $n$.
Proof.
We abbreviate ${f}_{n}{f}_{m}$ by ${f}_{mn}$.
 Necessity of (i).

Fix $t>0$, and let ${E}_{mn}=\{{f}_{mn}\ge t\}$. Then
$$\mu {({E}_{mn})}^{1/p}=\frac{1}{t}\parallel t\mathbf{\hspace{0.17em}1}({E}_{mn})\parallel \le \frac{1}{t}\parallel {f}_{mn}\parallel \to 0,\text{as}m,n\to \mathrm{\infty}\text{.}$$  Necessity of (ii).

Select $N$ such that $$ when $n\ge N$. The family $\{{{f}_{1}}^{p},\mathrm{\dots},{{f}_{N1}}^{p},{{f}_{N}}^{p}\}$ is uniformly integrable because it consists of only finitely many integrable functions.
So for every $\u03f5>0$, there is $\delta >0$ such that $$ implies $$ for $n\le N$. On the other hand, for $n>N$,
$$ for the same sets $E$, and thus the entire infinite^{} sequence $\{{{f}_{n}}^{p}\}$ is uniformly integrable too.
 Necessity of (iii).

Select $N$ such that $$ for all $n\ge N$. Let $\phi $ be a simple function^{} approximating ${f}_{N}$ in ${\mathbf{L}}^{p}$ norm up to $\u03f5$. Then $$ for all $n\ge N$. Let ${A}_{N}=\{\phi \ne 0\}$ be the support^{} of $\phi $, which must have finite measure. It follows that
$\parallel {f}_{n}\mathrm{\U0001d7cf}(X\setminus {A}_{N})\parallel =\parallel {f}_{n}{f}_{n}\mathrm{\U0001d7cf}({A}_{N})\parallel $ $\le \parallel {f}_{n}\phi \parallel +\parallel \phi {f}_{n}\mathrm{\U0001d7cf}({A}_{N})\parallel $ $=\parallel {f}_{n}\phi \parallel +\parallel (\phi {f}_{n})\mathrm{\U0001d7cf}({A}_{N})\parallel $ $$ For each $$, we can similarly construct sets ${A}_{n}$ of finite measure, such that $$. If we set $A={A}_{1}\cup \mathrm{\cdots}\cup {A}_{N1}\cup {A}_{N}$, a finite union, then $A$ has finite measure, and clearly $$ for any $n$.
 Sufficiency.

We show ${f}_{mn}$ to be small for large $m,n$ by a multistep estimate:
$\parallel {f}_{mn}\parallel $ $\le \parallel {f}_{mn}\mathrm{\U0001d7cf}(A\setminus {E}_{mn})\parallel +\parallel {f}_{mn}\mathrm{\U0001d7cf}({E}_{mn})\parallel +\parallel {f}_{mn}\mathrm{\U0001d7cf}(X\setminus A)\parallel .$ Use condition (iii) to choose $A$ of finite measure such that $$ for every $n$. Then $$.
Let $t=\u03f5/\mu {(A)}^{1/p}>0$, and ${E}_{mn}=\{{f}_{mn}\ge t\}$. By condition (ii) choose $\delta >0$ so that $$ whenever $$. By condition (i), take $N$ such that if $m,n\ge N$, then $$; it follows immediately that $$.
Finally, $\parallel {f}_{mn}\mathrm{\U0001d7cf}(A\setminus {E}_{mn})\parallel \le t\mu {(A)}^{1/p}=\u03f5$, since $$ on the complement of ${E}_{mn}$. Hence $$ for $m,n\ge N$. ∎
Remark. In the statement of the theorem, instead of dealing with Cauchy sequences, we can directly speak of convergence of ${f}_{n}$ to $f$ in ${\mathbf{L}}^{p}$ and in measure. This variation of the theorem is easily proved, for:

•
a sequence converges^{} in ${\mathbf{L}}^{p}$ if and only if it is Cauchy in ${\mathbf{L}}^{p}$;

•
a sequence that converges in measure is automatically Cauchy in measure;

•
a simple adaptation of the argument^{} shows that ${f}_{n}\to f$ in ${\mathbf{L}}^{p}$ implies ${f}_{n}\to f$ in measure; and

•
the limit in measure is unique.
Title  proof of Vitali convergence theorem 

Canonical name  ProofOfVitaliConvergenceTheorem 
Date of creation  20130322 17:31:04 
Last modified on  20130322 17:31:04 
Owner  stevecheng (10074) 
Last modified by  stevecheng (10074) 
Numerical id  5 
Author  stevecheng (10074) 
Entry type  Proof 
Classification  msc 28A20 