I am studying the Dirichlet Process and I came across this rule that states that given a partition $A_i's$ over the space $\Theta$, which represents the space in which $G_0$ is drawn from, $$(G(A_1),G(A_2),...G(A_k)) \sim Dir(\alpha G_0(A_1),\alpha G_0(A_2),... \alpha G_0(A_k)$$
$G(A_i)$ indicates the measure of $A_i$, which is equal to the sum of the height of the "pickets", $\pi_k$ present in the space $A_i$. How do we determine the measure of $G_0(A_i)$ ? Is this the integral of the probability distribution of $G_0$ over the partition $A_i$ ? I am not sure.
I am not so familiar with measure theory, so I am just basing it on intuition.