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!
