Egoroff’s Theorem

Let’s look back at what goes wrong when a sequence of functions doesn’t converge uniformly. Let $X$ be the closed unit interval $\left[0,1\right]$ , and let $f_n(x)=x^n$ . Pointwise, this converges to a function $f$ with $f(x)=0$ for $0\leq x<1$ , and $f(1)=1$ . This convergence can’t be uniform, because the uniform limit of a sequence of continuous functions is continuous.

But things only go wrong at the one point, and the singleton $\{1\}$ has measure zero. That is, the sequence $f_n$ converges almost everywhere to the function with constant value $0$ . The convergence still isn’t uniform, though, because we still have a problem at $\{1\}$ . But if we cut out any open patch and only look at the interval $\left[0,1-\epsilon\right]$ , the convergence is uniform. We might think that this is “uniform a.e.”, but we have to cut out a set of positive measure to make it work. The set can be as small as we want, but we can’t get uniformity by just cutting out $\{1\}$ .

However, what we’ve seen is a general phenomenon expressed in Egoroff’s Theorem: If $E\subseteq X$ is a measurable set of finite measure, and if $\{f_n\}$ is a sequence of a.e. finite-valued measurable functions converging a.e. on $E$ to a finite-valued measurable function $f$ , then for every $\epsilon>0$ there is a measurable subset $F$ with $\mu(F)<\epsilon$ so that $\{f_n\}$ converges uniformly to $f$ on $E\setminus F$ . That is, if we have a.e. convergence we can get to uniform convergence by cutting out an arbitrarily small part of our domain.

First off, we cut out a set of measure zero from $E$ so that $\{f_n\}$ converges pointwise to $f$ . Now we define the measurable sets

$\displaystyle E_n^m=\bigcap\limits_{i=n}^\infty\left\{x\in X\bigg\vert\lvert f_i(x)-f(x)\rvert<\frac{1}{m}\right\}$

As $n$ gets bigger, we’re taking the intersection of fewer and fewer sets, and so $E_1^m\subseteq E_2^m\subseteq\dots$ . Since $\{f_n\}$ converges pointwise to $f$ , eventually the difference $\lvert f_i(x)-f(x)\rvert$ gets down below every $\frac{1}{m}$ , and so $\lim_nE_n^m\supseteq E$ for every $m$ . Thus we conclude that $\lim_n\mu(E\setminus E_n^m)=0$ . And so for every $m$ there is an $N(m)$ so that

$\displaystyle\mu(E\setminus E_{N(m)}^m)<\frac{\epsilon}{2^m}$

Now let’s define

$\displaystyle F=\bigcup\limits_{m=1}^\infty\left(E\setminus E_{N(m)}^n\right)$

This is a measurable set contained in $E$ , and monotonicity tells us that

$\displaystyle\mu(F)=\mu\left(\bigcup\limits_{m=1}^\infty\left(E\setminus E_{N(m)}^n\right)\right)\leq\sum\limits_{m=1}^\infty\mu\left(E\setminus E_{N(m)}^n\right)<\sum\limits_{m=1}^\infty\frac{\epsilon}{2^m}=\epsilon$

We can calculate

$\displaystyle E\setminus F=E\cap\bigcap\limits_{m=1}^\infty E_{N(m)}^m$

And so given any $m$ we take $n\geq N(m)$ . Then for any $x\in E\setminus F$ we have $x\in E_n^m$ , and thus $\lvert f_n(x)-f(x)\rvert<\frac{1}{m}$ . Since we can pick this $n$ independently of $x$ , the convergence on $E\setminus F$ is uniform.

May 17, 2010 - Posted by John Armstrong | Analysis, Measure Theory

1 Comment »

[…] Uniform Convergence From the conclusion of Egoroff’s Theorem we draw a new kind of convergence. We say that a sequence of a.e. finite-valued measurable […]

Pingback by Almost Uniform Convergence « The Unapologetic Mathematician | May 18, 2010 | Reply

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

RSS Feeds

RSS - Posts
RSS - Comments
Feedback

Got something to say? Anonymous questions, comments, and suggestions at Formspring.me!
Subjects

Subjects
Archives

May 2010

M T W T F S S

1 2

3 4 5 6 7 8 9

10 11 12 13 14 15 16

17 18 19 20 21 22 23

24 25 26 27 28 29 30

31

« Apr Jun »
Search for:

The Unapologetic Mathematician

Mathematics for the interested outsider