# Clarifying Wikipedia's Notation for Definition of a Dirichlet Process

Wikipedia gives the following definition of a Dirichlet process:

1. What does the notation $$H(B_i)$$ and $$X(B_i)$$ mean?

2. What does the notation $$(X(B_1), ..., X(B_n))$$ mean?

My guess to the first question is that $$H(B_i)$$ refers to the probability measure associated with the set $$B_i$$ under the distribution $$H$$'s mass/density function (and similarly for $$X(B_i)$$). Is this correct?

$$H(B_i)$$ is the probability of the set $$B_i$$ under $$H$$, ie, if $$Z\sim H$$, $$H(B_i)\equiv P(Z\in B_i)$$.
$$(X(B_1),\dots, X(B_n)$$ is a random vector. It has a Dirichlet distribution, so it is a set of proportions adding up to 1, which makes sense since the $$B_i$$ are a partition adding up to the whole space.
The means of the $$n$$ proportions are proportional to the $$H(B_i)$$ (but add up to 1), and as $$\alpha$$ increases, the variances of the proportions get smaller. For large $$\alpha$$, $$X(B_i)$$ will be close to $$H(B_i)$$, but for small $$\alpha$$, $$X(B_i)$$ will be vary a lot.