Does $f<\infty$ a.s. imply that $f$ is integrable?

Question

Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and $f\colon\Omega\to\overline{\mathbb{R}}$ measurable. Does then $f<\infty$ a.s. imply that $f$ is integrable?

I think no, but cannot find a counterexample.

Answer 1

Of course not. Take $\Omega=\mathbb R$, $\mu$ the Lebesgue measure, and $f(t)=1$.