Egoroff's theorem in Royden Fitzpatrick (comparison with lemma 10) Hi math stackexchangers,
I have a question about the difference between two math statements (for reference they can be found in Royden Fitzpatrick pages 64-65).

Egoroff's Theorem: Assume $E$ has finite measure. Let $\{f_n\}$ be a sequence of measurable functions on $E$ that converges pointwise on $E$ to the real-valued function $f$. Then for each $\varepsilon > 0$, there is a closed set $F$ contained in $E$ for which
  $$\{f_n\} \to f\text{ uniformly on }F\text{ and }m(E \setminus F) < \varepsilon.$$
Lemma 10: Under the assumptions of Egoroff's Theorem, for each $\eta > 0$ and $\delta > 0$, there is a measurable subset $A$ of $E$ and an index $N$ for which
  $$|f_n - f| < \eta\text{ on }A\text{ for all }n \geq N
\text{ and }m(E \setminus A) < \delta.$$

My question is this. What is the difference between Egoroff's Theorem and Lemma 10? Am I misunderstanding uniform convergence? To me it looks like Lemma 10 provides uniform convergence. Is the difference that Egoroff's Theorem ensures that $F$ is closed but Lemma 10 doesn't ensure that $A$ is closed?
Thanks
 A: A significant difference was not mentioned in the comments:  In the statement of Lemma 10, $A$ depends on $\eta$.  Let us forget about closedness for a minute (I'll come back to that later) and compare the two statements:
Egoroff: $$(\forall \varepsilon>0) (\exists A\subseteq E), m(E\setminus A)<\varepsilon \boldsymbol{\large( \forall \eta>0)}$$ $$(\exists N\in\mathbb N)(\forall x\in A)(\forall n\geq N) |f_n(x)-f(x)|<\eta.$$
Lemma 10: $$(\forall \varepsilon>0) \boldsymbol{\large( \forall \eta>0)} (\exists A\subseteq E), m(E\setminus A)<\varepsilon $$ $$(\exists N\in\mathbb N)(\forall x\in A)(\forall n\geq N) |f_n(x)-f(x)|<\eta.$$
I know this isn't a pretty way to write it, but I hope it makes clear the important difference between the two.  Notice that in Egoroff's theorem, $A$ had to be chosen once and for all to work for all (subsequently chosen) $\eta>0$, whereas in Lemma 10 $A$ only needs to work for a previously fixed $\eta$.  
I remember when I was first learning measure theory from Royden doing an exercise showing how Egoroff's theorem can be proved using Lemma 10 (or whatever it was numbered in my edition).  It was pretty straightforward because the exercise told me what sequences of $\delta$s and $\eta$s to use.

As for closedness, note that Egoroff's theorem without the assumption of closedness applies to all finite measure spaces (where there need not even be a topology in general) and the closedness can be added using inner regularity.  For example, a subset $A$ of $\mathbb R$ is Lebesgue measurable if and only if for all $\varepsilon>0$ there exists a closed set $F\subseteq A$ such that $m^*(A\setminus F)<\varepsilon$ (where $m^*$ denotes Lebesgue outer measure).  Uniform convergence on $A$ implies uniform convergence on F, and $E\setminus F = (E\setminus A)\cup (A\setminus F)$.
