If the measure attains only finitely many values then the space is disjoint union of finitely many atoms Let $(X,\mu)$ be a measure space such that $\mu$ attains only finitely many values. I want to show that $X$ can be written as a disjoint union of finitely many atoms. It isn't necessary that a measure space has atoms. However, in this case each of the sets having the smallest non-zero measure must all be atoms. But there's a problem - they need not be disjoint. And if we try something like $$A_1\cup A_2=A_1\cup(A_2\setminus A_1)$$ there is no guarantee that $A_2\setminus A_1$ will be an atom (it can have zero measure). We cannot discard it either. So how do I go about it?
If I can show the space to be a disjoint union of atoms, it most certainly will be a finite union since $\mu$ attains only finitely many values.
 A: First notice that $X$ is atomic. Suppose $A\subset X$ has positive measure but contains no atoms. Since $A$ is not an atom there exists $A_1\subset A$ with $0<\mu(A_1)<\mu(A)$. But since $A_1\:\:(\mu(A_1)>0)$ is not an atom, there exists $A_2\subset A_1$ with $0<\mu(A_2)<\mu(A_1)$ and so on. Thus there is an infinite chain $$\mu(A)>\mu(A_1)>\mu(A_2)>\dots>0$$ which contradicts the fact that $\mu$ assumes only finitely many values.\
Now let $B_1$ be the atom in $X$ having the largest measure. If $\mu(X\setminus B_1)=0$, stop. Otherwise let $B_2$ be the atom with the largest measure contained in $X\setminus B_1$. If $\mu(X\setminus(B_1\cup B_2))=0$, stop. Otherwise let $B_3$ be the atom with the largest measure contained in $X\setminus(B_1\cup B_2)$. This process will end in finitely many steps. For if not, then $$\mu(X\setminus(B_1\cup\dots B_n))=\mu(X\setminus(B_1\cup\dots B_{n+1}))+\mu(X\setminus(B_1\cup\dots B_n)\cap B_{n+1})=\mu(X\setminus(B_1\cup\dots B_{n+1}))+\mu(B_{n+1})$$ i.e. $$\mu(X\setminus(B_1\cup\dots B_n))>\mu(X\setminus(B_1\cup\dots B_{n+1}))$$ since $\mu(B_{n+1})>0$. This would again result in an infinitely descending chain of positive values. Thus, $X$ can be written as a finite disjoint union of atoms and a null set $$X=B_1\cup\dots\cup B_n\cup Z$$
