The Dirichlet normal-mixture density is jointly measurable and normalized

The kernel $K$ in [F2] is a probability kernel because, for fixed $(F,\sigma ,x)$, $A\mapsto F(A)$ is a probability measure, while for fixed Borel $A$, $(F,\sigma ,x)\mapsto F(A)$ is measurable by the evaluation sigma-algebra. The integrand $(\sigma ,x,z)\mapsto (\sigma \sqrt{2\pi }{)}^{-1}\exp (-(x-z{)}^2\mathrm{/}(2{\sigma }^2))$ is continuous on $(0,\mathrm{\infty })\times {\mathbb{R}}^2$ and hence product-measurable. Therefore [F3] makes its integral against $K$ jointly measurable.

The integrand is strictly positive, so its integral against a probability measure is positive. It is bounded above by $(\sigma \sqrt{2\pi }{)}^{-1}$, so the integral is finite and has the stated upper bound.

By [F4], ${\int }_{\mathbb{R}}{p}_{F,\sigma }(x)dx={\int }_{\mathbb{R}}\left({\int }_{\mathbb{R}}\frac{1}{\sigma \sqrt{2\pi }}{e}^{-(x-z{)}^2\mathrm{/}(2{\sigma }^2)}dx\right)F(dz)={\int }_{\mathbb{R}}1F(dz)=1,$ where the inner integral is one by [F1] and $F$ has total mass one. Thus $p_{F,\sigma }$ is a probability density.

For every Borel set $B\subseteq \mathbb{R}$, Tonelli [F4] gives ${\int }_B{p}_{F,\sigma }(x)dx={\int }_{\mathbb{R}}\left({\int }_B\frac{1}{\sigma \sqrt{2\pi }}{e}^{-(x-z{)}^2\mathrm{/}(2{\sigma }^2)}dx\right)F(dz)$. The inner integral is $N(z,{\sigma }^2)(B)$ by [F1], so the right side is the convolution measure $(F\ast N(0,{\sigma }^2))(B)$ by its definition. Hence this density represents the convolution law in the Statement.