I am working on the following theorem:
Theorem
Let $(X,\mathscr{A})$ be a measure space with finite measure $\mu$, then $\mu$ can be written uniquely as the sum of a purely atomic measure and a non-atomic measure. Let us call $\mathcal{N}$ is the union of the atoms with respect to $\mu$. We define $$\lambda (A) = \mu(A\setminus \mathcal{N}) \hspace{2cm} \nu (A) = \mu(A\cap \mathcal{N})$$ Where $\lambda$ is non-atomic and $\nu$ is purely atomic, so we have tested that it can be decomposed into said sum, but I don't know how to show that it is unique, at first it occurs to me to give another $\mu_1 + \mu_2 = \mu$ sum representation where $ \mu_1 $ is purely atomic and $ \mu_2 $ is non-atomic, but I don't know what happens when evaluating an atom in a non-atomic measure, that is, if $A$ is an atom, does $ \mu_2 (A) = 0 $?
Can someone help me test the uniqueness?
