The product-prior posterior factors by coordinates

For each coordinate $j$, the pair $({\theta }_j,X_j)$ is a measurable function of the independent pair $({\xi }_j,Z_j)$. Thus the coordinate pairs are independent: finite rectangle probabilities factor by [F1] and [F2]. Fix finite $J$. For any cylinder event $B$ depending on finitely many observations outside $J$, the random vector $({\theta }_J,X_J)$ is independent of $B$ by that same factorization. The events in $\sigma (X_{J^c})$ for which this independence identity holds form a lambda-system containing the outside-coordinate cylinder pi-system; [F3] extends it to all outside-data events. Hence conditioning on $X_{J^c}$ adds no information about ${\theta }_J$ once $X_J$ is known.

For finite $J$, the joint prior density of ${\theta }_J$ is the product of ${\sigma }_j^{-1}\phi (t_j\mathrm{/}{\sigma }_j)$, and the likelihood of $X_J=x_J$ is proportional to ${\prod }_{j\in J}{e}^{-n(x_j-t_j{)}^2\mathrm{/}2}$. Each one-dimensional normalizer is at most $1$ and is positive: the Gaussian factor is bounded by $1$, is strictly positive everywhere, and the prior density integrates to $1$. Thus the normalized product density is ${\prod }_{j\in J}{p}_{j,n}(t_j\mid x_j)$. Integrating it over rectangles in ${\theta }_J$ and $X_J$ verifies the conditional-law identity for these coordinates. The prior is a Borel probability on $L^2$ by [L1] under [A1], and [A2] supplies the full posterior kernel; step 1.1 extends the finite-coordinate identity to the full observation sequence.

The product of the one-coordinate conditional densities in step 2.1 is exactly the stated finite-dimensional posterior marginal, so the posterior factors by coordinates for every finite index set $J$.