Let $f \in m(\Omega,\mathcal{F})$, i.e. $f \mapsto [-\infty,\infty]$ and let $A \in \mathcal{F}$ be an atom. Prove that $f$ is almost everywhere constant on A: there exists $k \in [-\infty,\infty]$ such that
$\mu (\{\omega \in A : f(\omega) \neq k \} )=0$.
I was thinking let $k=\frac{1}{\mu(A)}\int \limits_{A} f\,d\mu$.
Then let $B=\{\omega \in A :f(\omega) \neq k \}$. Since A is an atom, and B is a subset of A then $\mu(B)=0$, in which case we're done, or $\mu(B)=\mu(A)$. So I need to show that $\mu(B)<\mu(A)$.