For any measurable space $(X,\mathscr{M})$, are all weighted counting measures purely atomic?

I have been given the following definitions:

A measure $\mu$ is a weighted counting measure on $(X,\mathscr{M})$ if there exists a function $f:X\rightarrow[0,\infty]$ such that $$\forall E\in\mathscr{M}, \mu(E)=\sum_{x\in E} f(x)$$

A non-null set $A\in\mathscr{M}$ is $\mu$-atomic if for each $B\in\mathscr{M}$ with $B\subseteq A$, either $\mu(B)=0$ or $\mu(B)=\mu(A)$. If every non-null $E\in\mathscr{M}$ contains an atom then we say $\mu$ is purely atomic

My professor proposed the following problem as just something fun to think about, not as a real homework question:

For any measurable space $(X,\mathscr{M})$, are all weighted counting measures purely atomic?

After some thought, I believe that this question reduces to the case when $\mu$ is a finite measure, but I am struggling to make much further progress. Can anyone help me out?