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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

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    There are a couple of possible meanings for "atom" in this context; how are you defining it? Also, you should explain exactly what $X$ and $F$ are supposed to be.2012-06-01

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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.