proof of Vitali convergence theorem

Theorem.

Let $f_{1},f_{2},\ldots$ be $\mathbf{L}^{p}$ -integrable functions on a measure space $(X,\mu)$ , for $1\leq p<\infty$ . The following conditions are necessary and sufficient for $f_{n}$ to be a Cauchy sequence in the $\mathbf{L}^{p}(X,\mu)$ norm:

(i)

the sequence $f_{n}$ is Cauchy in measure;
(ii)

the functions $\{\lvert f_{n}\rvert^{p}\}$ are uniformly integrable; and
(iii)

for each $\epsilon>0$ , there is a set $A$ of finite measure, with $\lVert f_{n}\mathbf{1}(X\setminus A)\rVert<\epsilon$ for all $n$ .

Proof.

We abbreviate $\lvert f_{n}-f_{m}\rvert$ by $f_{mn}$ .

Necessity of (i).

Fix $t>0$ , and let $E_{mn}=\{f_{mn}\geq t\}$ . Then

$\mu(E_{mn})^{1/p}=\frac{1}{t}\lVert t\,\mathbf{1}(E_{mn})\rVert\leq\frac{1}{t}% \lVert f_{mn}\rVert\to 0\,,\quad\text{as $m,n\to\infty$.}$

Necessity of (ii).

Select $N$ such that $\lVert f_{n}-f_{N}\rVert<\epsilon$ when $n\geq N$ . The family $\{\lvert f_{1}\rvert^{p},\ldots,\lvert f_{N-1}\rvert^{p},\lvert f_{N}\rvert^{p}\}$ is uniformly integrable because it consists of only finitely many integrable functions.

So for every $\epsilon>0$ , there is $\delta>0$ such that $\mu(E)<\delta$ implies $\lVert f_{n}\mathbf{1}(E)\rVert<\epsilon$ for $n\leq N$ . On the other hand, for $n>N$ ,

$\lVert f_{n}\mathbf{1}(E)\rVert\leq\lVert(f_{n}-f_{N})\mathbf{1}(E)\rVert+% \lVert f_{N}\mathbf{1}(E)\rVert<2\epsilon$

for the same sets $E$ , and thus the entire infinite sequence $\{\lvert f_{n}\rvert^{p}\}$ is uniformly integrable too.

Necessity of (iii).

Select $N$ such that $\lVert f_{n}-f_{N}\rVert<\epsilon$ for all $n\geq N$ . Let $\varphi$ be a simple function approximating $f_{N}$ in $\mathbf{L}^{p}$ norm up to $\epsilon$ . Then $\lVert f_{n}-\varphi\rVert<2\epsilon$ for all $n\geq N$ . Let $A_{N}=\{\varphi\neq 0\}$ be the support of $\varphi$ , which must have finite measure. It follows that

	$\displaystyle\lVert f_{n}\mathbf{1}(X\setminus A_{N})\rVert=\lVert f_{n}-f_{n}% \mathbf{1}(A_{N})\rVert$	$\displaystyle\leq\lVert f_{n}-\varphi\rVert+\lVert\varphi-f_{n}\mathbf{1}(A_{N% })\rVert$
		$\displaystyle=\lVert f_{n}-\varphi\rVert+\lVert(\varphi-f_{n})\mathbf{1}(A_{N})\rVert$
		$\displaystyle<2\epsilon+2\epsilon\,.$

For each $n<N$ , we can similarly construct sets $A_{n}$ of finite measure, such that $\lVert f_{n}\mathbf{1}(X\setminus A_{n})\rVert<4\epsilon$ . If we set $A=A_{1}\cup\cdots\cup A_{N-1}\cup A_{N}$ , a finite union, then $A$ has finite measure, and clearly $\lVert f_{n}\mathbf{1}(X\setminus A)\rVert<4\epsilon$ for any $n$ .

Sufficiency.

We show $f_{mn}$ to be small for large $m, n$ by a multi-step estimate:

$\displaystyle\lVert f_{mn}\rVert$

$\displaystyle\leq\lVert f_{mn}\mathbf{1}(A\setminus E_{mn})\rVert+\lVert f_{mn% }\mathbf{1}(E_{mn})\rVert+\lVert f_{mn}\mathbf{1}(X\setminus A)\rVert\,.$

Use condition (iii) to choose $A$ of finite measure such that $\lVert f_{n}\mathbf{1}(X\setminus A)\rVert<\epsilon$ for every $n$ . Then $\lVert f_{mn}\mathbf{1}(X\setminus A)\rVert<2\epsilon$ .

Let $t=\epsilon/\mu(A)^{1/p}>0$ , and $E_{mn}=\{f_{mn}\geq t\}$ . By condition (ii) choose $\delta>0$ so that $\lVert f_{n}\mathbf{1}(E)\rVert<\epsilon$ whenever $\mu(E)<\delta$ . By condition (i), take $N$ such that if $m,n\geq N$ , then $\mu(E_{mn})<\delta$ ; it follows immediately that $\lVert f_{mn}\mathbf{1}(E_{mn})\rVert<2\epsilon$ .

Finally, $\lVert f_{mn}\mathbf{1}(A\setminus E_{mn})\rVert\leq t\mu(A)^{1/p}=\epsilon$ , since $f_{mn}<t$ on the complement of $E_{mn}$ . Hence $\lVert f_{mn}\rVert<5\epsilon$ for $m,n\geq N$ . ∎

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	2013-03-22 17:31:04
Last modified on	2013-03-22 17:31:04
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	5
Author	stevecheng (10074)
Entry type	Proof
Classification	msc 28A20