I want to calculate the Lebesgue measure of standard $K-1$ simplex: $\{(x_1,...,x_K)~|~\sum_{i=1}^{K} x_i = 1 \mbox{ and } x_i \ge 0 \mbox{ for all } i \in [1,K]\}$ on $\mathbb{R}^K$. I think the measure is $0$ because the simplex is of dimension $K-1$.
However, the support of Dirichlet distribution is such simplex and Dirichlet distribution has a probability density function with respect to Lebesgue measure on the Euclidean space $\mathbb{R}^K$.
So I am very confused. How Dirichlet distribution could be absolute continuous w.r.t. Lebesgue measure on $\mathbb{R}^K$? How to calculate the volume of simplex by Lebesgue integration?
I found the answer by reading this section in Wikipedia. In fact, the Dirichlet measure is not absolutely continuous w.r.t. the Lebesgue measure on $\mathbb{R}^K$ but the one on on $\mathbb{R}^{K-1}$. The last variable $x_K$ is a bound variable, so the distribution is only w.r.t. $(x_1,...,x_{K-1})$.
I modified the Wikipedia page to remove this error.