Prove f is measurable on any closed set

[jinsing]

Homework Statement

Prove if $f$ is measurable on R and C is any closed set, f^{-1}(C) is measurable.

Homework Equations

Definition of measurability, closed sets etc.

The Attempt at a Solution

I've been trying for a while to get this proof, but I seem to just end up stuck at the beginning. I think I want to point out that the complement of a closed set is an open set, and open sets are countable unions of open intervals, which are themselves measurable. But I'm not too sure, and I'd sure appreciate a gentle push in the right direction.

Thanks!

 

[micromass]

I think I want to point out that the complement of a closed set is an open set, and open sets are countable unions of open intervals, which are themselves measurable.

Yes, that is correct. So what is bothering you??

 

[jinsing]

I guess I need help formalizing the argument. Would I just assume the hypotheses, point out by definition C' is an open set, and then just mention "open sets are countable unions of open intervals, which are measurable?" Doesn't seem too rigorous..

 

[micromass]

It's rigorous enough for me. (assuming you proved the things like any open set is the countable union of intervals).