F measurable functions & Their atoms Is it true that if X is F measurable, or a Borel function, then it is constant on the atoms of F in all cases?
Thanks 

Consider the $\sigma$-algebra of all subsets of $\mathbb R$ with the measure defined as follows: $\mu(A)=0$ if $A$ is at most countable, $\mu(A)=\infty$ is $A$ is uncountable. Then $\mathbb R$ is an atom for this measure. Any function $f\colon \mathbb R\to\mathbb R$ is measurable, so it does not have to be constant.
Another measure on the same $\sigma$-algebra: $\mu(A)=0$ if $0\notin A$ and $\mu(A)=1$ otherwise. (So, $\mu$ is a unit point mass at $0$). Again, $\mathbb R$ is an atom for this measure. 
