Let $(\mathbb{R}, \mathcal{B}, \mathbb{P})$ be a probability space. I want to show that $\mathbb{P}$ can be written as $\mathbb{P} = \mu + \nu$, where $\mu$ is a continuous measure (no atoms) and $\nu$ an atomic measure ($\nu = \sum_i \epsilon_i \delta_{x_i}$).
I think first one has to show, that $\mathbb{P}$ has only countable many atoms $x_i$. Then, that $\mathbb{P} - \nu = \mu$ is a measure.
The second part should not be to hard. But how to do the first? If there are uncountable many $x_i \in \mathbb{R}$ with $\mathbb{P}(x_i) >0$, I guess $\mathbb{P}(\mathbb{R})=1$ is not possible anymore, but how to show that?