Let $\alpha\in (0,\infty)^n$ and $\{e_1,...,e_n\}$ be the standard ordered basis for $\mathbb{R}^n$, and define $\alpha_0:=\sum_{i=1}^n \alpha_i$.

Let $Dir(\alpha+e_k)$ be Dirichlet distributions and $\mu_k$ be its distribution measures.

Then, how do I show that $\sum_{i=1}^n \frac{\alpha_i}{\alpha_0} \mu_k$ is the distribution measure of $Dir(\alpha)$?

Thank you in advance!

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