Let $H$ be the base distribution of a Dirichlet process. How is this process well-defined in case $H(B_1) = 0$? I have read that the parameters of Dirichlet distribution must be strictly positive.

The Dirichlet distribution of order $K \geq 2$ with parameters $\alpha_{1}, \ldots, \alpha_{K} \color{blue}{> 0}$ has a probability density function with respect to Lebesgue measure on the Euclidean space $\mathbf{R}^{K-1}$ given by
$$
f\left(x_{1}, \ldots, x_{K} ; \alpha_{1}, \ldots, \alpha_{K}\right)=\frac{1}{\mathrm{~B}(\boldsymbol{\alpha})} \prod_{i=1}^{K} x_{i}^{\alpha_{i}-1}
$$
where $\left\{x_{k}\right\}_{k=1}^{k=K}$ belong to the standard $(K-1)$-simplex.

Then the Dirichlet distribution is then used in the definition of Dirichlet process.

Given a measurable set $S$, a base probability distribution $H$ and a positive real number $\alpha$, the Dirichlet process $\mathrm{DP}(H, \alpha)$ is a stochastic process whose sample path (or realization, i.e. an infinite sequence of random variates drawn from the process) is a probability distribution over $S$, such that the following holds. For any measurable finite partition of $S$, denoted $\left\{B_{i}\right\}_{i=1}^{n}$, if $X \sim \operatorname{DP}(H, \alpha)$ then
$$
\left(X\left(B_{1}\right), \ldots, X\left(B_{n}\right)\right) \sim \operatorname{Dir}\left(\alpha H\left(B_{1}\right), \ldots, \alpha H\left(B_{n}\right)\right),
$$
where Dir denotes the Dirichlet distribution and the notation $X \sim D$ means that the random variable $X$ has the distribution $D$.

If $H(B_1) = 0$, then $\alpha H(B_1)=0$. How is $\operatorname{Dir}\left(\alpha H\left(B_{1}\right), \ldots, \alpha H\left(B_{n}\right)\right)$ well-defined in this case?
Thank you for your explanation!

The definition of the Dirichlet process, as given on the Wikipedia page, is correct only if we use the definition of the Dirichlet distribution as given in Ferguson, 1973. Ferguson's definition is slightly more general than the one on Wikipedia. Under Ferguson's definition, only one of the $\alpha_j$'s need be positive. With this definition, there is no problem with sets of $H$-measure 0.
