I am trying to understand a proof of the following proposition:
Proposition Suppose $G$ is drawn from a dirichlet process $DP(\alpha, H)$, i.e for any arbitrary partition $\{A_1, ..., A_k\}$ of $\Theta$, $ (G(A_1), ... G(A_r)) \sim Dir (\alpha H(A_1), ..., \alpha H(A_k)) $. Then G can be equivalently drawn from the stick breaking scheme.
In the proof the author makes use of the following construction without further elaboration.
"After drawing a sample $\theta$ form $G$, we consider the partition $\{\{\theta\}, \{ \theta^c \} \}$, we get,
$$ (G|\theta(\theta), G| \theta(\theta^c)) \sim Dir((\alpha + 1)\frac{\alpha H + \delta_{\theta}}{\alpha + 1}(\theta), (\alpha + 1) \frac{\alpha H + \delta_{\theta}}{\alpha + 1} (\theta^c)) $$
Hence, $ (G|\theta(\theta), G| \theta(\theta^c)) \sim Dir(1, \alpha) $"
This part is fine. However from here the author concludes that,
$$ G|\theta = \beta \delta_{\theta} + (1 - \beta) G^{'} $$
where $ G^{'}$ is the renormalized measure with the point mass removed and $ \beta \sim Beta(1, \alpha)$.
I don't understand why this decomposition is immediate. Any help would be greatly appreciated.