Example of an Atomic Measure

Definition:

Let $(X,\mathscr{A},\mu)$ be a measurable space, an atom of the measure $\mu$ is a set $A \in\mathscr{A}$ with the property that $\mu(A) > 0$ and for any $B\in \sigma (A)$ either $\mu(B) = 0$, or $\mu(A \setminus B) = 0$. If a measure has atoms it is called atomic; in the opposite case, the measure is called non-atomic (or atomeless). A measure is called purely atomic if $X$ can be written as the union of a finite or countable number of atoms.

From the definition of atoms, we get the following corollary:

Corollary:

Every purely atomic measure is an atomic measure.

I am trying to find an example of an atomic measure that is not purely atomic, can anyone help me?

The sum of the Dirac measure and the Lebesgue measure in $\mathbb R$ is atomic but not purely atomic, because $\{0\}$ is its only atom.

E.g. take $X=[0,1] \cup \{2\} \subseteq \Bbb R$ (as a space), $\lambda$ the standard Lebesgue measure, and use the Borel measure on $X$ defined by: $$\mu(A) = \begin{cases} \lambda(A) & 2 \notin A\\ \lambda(A) + 1 & 2 \in A\\ \end{cases}$$

$\{2\}$ is an atom (of measure $1$) there are no more atoms, so $(X,\mu)$ is not purely atomic.