Let $\{f_n \}$ be a sequence of measurable functions, $f_n \to f$ $\mu$-a.e. on a measurable set $E$, $\mu(E) < \infty$. Let $\epsilon>0$ be given. Then $\forall \space n \in \mathbb{N} \space \exists A_n \subset E$ with $\mu(A_n) <\frac{\epsilon}{2^n}$ and $\exists N_n$ such that $\forall \space x \notin A_n$ and $k \ge N_n \space |f_k(x) – f(x)| < \epsilon$. That is: if we define $A = \cup_{n=1}^{\infty} A_n$ with $\mu(A) < \epsilon $ then ${f_n}$ converges uniformly on $E \setminus A$.
$\mathbf{Proof}$: (taken from Royden's Real Analysis)
Let $A = \cup_{n=1}^{\infty} A_n \Rightarrow A \subset E$ and $\mu(A) < \sum_{n=1}^{\infty} \frac{\epsilon}{2^n} = \epsilon. \mathbf{Q1}$.
choose $n_0$ such that $\frac{1}{n_0} < \epsilon$. If $x \notin A$ and $k \ge N_{n_0}$ we have $\space |f_k(x) – f(x)| < \frac{1}{n_0} < \epsilon \space$. $ \square$
$\mathbf{Q1}$: First off, I don't see how $\sum_{n=1}^{\infty} \frac{\epsilon}{2^n} = \epsilon$. It's a geometric series: $ \epsilon \sum_{n=1}^{\infty} \frac{1}{2^n} = \epsilon \frac{1}{1-\frac12} = 2 \epsilon$. Am I wrong?
$\mathbf{Q2}$: The idea behind Egoroff is in order to turn almost sure convergence into uniform convergence on $E$ we only need to take away a really small set, right? Interestingly, as $\epsilon \to 0$ (that is $f_n$ is getting closer to $f$), the measure of the set $A$ is getting proportionally smaller ($\mu(A) \to 0$). So are we ultimately taking away a set of zero measure?
Best Answer
A2: You are correct, that for arbitrarily small $\epsilon$ there is a set $A$, such that $\mu(A)<\epsilon$, where uniform convergence fails. So the measure of $A$ can be arbitrarily small. However, also keep in mind that uniform convergence requires a finite $N$ such that given $\delta>0$, for all $k>N$, $|f_k - f|<\delta$. Imagine that as $\epsilon$ gets smaller and smaller, for a fixed $\delta$ this $N$ may get larger and larger. Then in the limit as $\epsilon\to 0$, $N \to\infty$ and uniform convergence would fail.
My favorite text for Egoroff's theorem and related topics is Lieb and Loss's Analysis book.