I want to prove that all probability measures $\mathbb{P}$ on the measurable space $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, where $\mathcal{B}(\mathbb{R})$ σ-algebra Borel, can be decomposed as

$$\mathbb{P}=\mu+\nu$$

where $\mu$ is a discrete measure and $\nu$ a diffuse measure.

Can you provide some guidance or steps that I have to follow in order to prove that ?