Proof of Egorov's theorem

Question

I was wondering, looking at the proof of Egorove in :http://en.wikipedia.org/wiki/Egorov%27s_theorem, how could we be so sure as to say that each set $E_{n,k}=\bigcup_{m\geq k}\left\{x\in A:|f_m(x)-f(x)| \geq 1/k\right\}$ is measurable? In particular, is it true that for a measurable function $f$, the set $\left\{x:f(x)>c\right\}$ is measurable?

Thanks.

Answer 1

The set $(c,\infty)$ is Borel measurable (it is open), so by definition of measurability, $f^{-1}((c,\infty))$ is measurable. This is precisely $\{x:f(x)>c\}$. Since each function $|f_m-f|$ is measurable, each set $\{x\in A:|f_m(x)-f(x)|\ge\frac1k\}$ is as well, so their union is.