Dirichlet density normalization by iterated Beta integrals

On the open simplex, each coordinate $p_i$ for $i<k$ and the residual coordinate $p_k=1-{\sum }_{i<k}{p}_i$ is continuous and strictly positive. By [F7], $p\mapsto {\prod }_{j=1}^k{p}_j^{{\alpha }_j-1}$ is therefore continuous, hence Borel measurable there. Since the simplex is open, extending this nonnegative function by zero outside it gives a Borel measurable function on ${\mathbb{R}}^{k-1}$; Borel sets are Lebesgue measurable by [F7], so the integrand has the Lebesgue measurability required by [F2]. Define the stick-breaking map from $(0,1{)}^{k-1}$ to ${\mathrm{\Delta }}_{k-1}$ by $p_i=z_i{\prod }_{j<i}(1-z_j)$ for $i<k$. Its inverse is $z_i=p_i\mathrm{/}(1-{\sum }_{j<i}{p}_j)$, so it is a $C^1$ diffeomorphism. Its derivative matrix is lower triangular with diagonal entries ${\prod }_{j<i}(1-z_j)$; hence its Jacobian is ${\prod }_{i=1}^{k-2}(1-z_i{)}^{k-i-1}$.

Under this change of variables, each factor $p_j^{{\alpha }_j-1}$ contributes $z_j^{{\alpha }_j-1}$ and a factor $(1-z_i{)}^{{\alpha }_j-1}$ for each $i<j$; the Jacobian adds exponent $k-i-1$ to $1-z_i$. Thus the transformed integrand is ${\prod }_{i=1}^{k-1}{z}_i^{{\alpha }_i-1}(1-z_i{)}^{{\sum }_{j>i}{\alpha }_j-1}\mathrm{.}$ This factorization is valid because all coordinates and residuals are strictly positive on the open cube.

Each one-variable factor in step 2.1 is continuous on $(0,1)$, hence Borel measurable by [F7]. Using the product-measure identification in [F3], repeated Tonelli and [F4] now give the integral as ${\prod }_{i=1}^{k-1}B\text{\hspace{0.17em}⁣}({\alpha }_i,{\sum }_{j>i}{\alpha }_j)\mathrm{.}$ Applying [F5] to each factor cancels the intermediate Gamma terms and leaves ${\prod }_{j=1}^k\mathrm{\Gamma }({\alpha }_j)\mathrm{/}\mathrm{\Gamma }({\alpha }_{+})$.

By [F6] the displayed ratio is finite and positive. Multiplying the integrand by the reciprocal constant in [F1] therefore gives integral one, which proves normalization.