Elaborating some points in the proof of Proposition 3 on pg.100 "Royden and Fitzpatrick 4th edition"

Question

The proposition and its proof are given below:

My questions are:

1- why is $|f_{n} - f| < \eta$ and not $<\epsilon$?

2- why is $\{x \in E | |f_{n} - f(x)| > \eta \} \subseteq (E \setminus F)$?

Could anyone help me in finding answers to those questions, please?

Answer 1

I think a confusing thing here is that, technically, there should be written

• $|f_n - f| \color{blue}{\leq} \eta$ on $F$ for all $n \geq N$

if one takes the definition of convergence in measure as given in the book:

For all $\eta > 0$ we have

• $\lim_{n\to\infty}m\left(\{x \in E\, | \, |f_n(x) - f(x)| \color{blue}{>} \eta\}\right) = 0$

Now, fix an arbitrary $\eta>0$. Hence, to show is that for any $\epsilon > 0$ we have $m\left(\{x \in E\, | \, |f_n(x) - f(x)| > \eta\}\right) < \epsilon$ for all $n \geq N$.

So, let $\epsilon >0$. Then Egorov's theorem assures that $f_n \Rightarrow f$ on an $\pmb{F \subseteq E}$ such that $m(E\setminus F) < \epsilon$. So, there is an $N$ such that $|f_n(x) - f(x)| \color{blue}{\leq} \eta$ for $n \geq N$ on $\pmb{F}$. This in turn means, that $\{x \in E\, | \, |f_n(x) - f(x)| > \eta\} \subseteq E\setminus F$ for all $\pmb{n \geq N}$.

Hence $m\left(\{x \in E\, | \, |f_n(x) - f(x)| > \eta\}\right) \leq m(E \setminus F) < \epsilon$ for all $n \geq N$.