Does Everywhere Implies Almost Everywhere

Question

I know the concept of 'almost everywhere' in measure theory. For example, $f:E\to\mathbb{R}$ is bounded almost everywhere means if $B = \{x\in E: f \text{ is not bounded}\}$, then the measure $\mu(B) = 0$. However, I was wondering if $f$ is bounded implies $f$ is bounded almost everywhere. It seems trivially true to me, but I haven't found any discussions related to this. My purpose is to manipulate the measure concept to help with one of my ongoing proofs.

Thanks very much in advance.

Answer 1

(To get it off the unanswered question queue:)

If $f$ is everywhere bounded, $B=\emptyset$ so $\mu(B)=0$ and you're done.