Successive Gaussian shell integration composes

On $({\mathbb{R}}^d,{\gamma }_{C_1})\times ({\mathbb{R}}^d,{\gamma }_{C_2})$, the sum map $(y,z)\mapsto y+z$ has characteristic function $\int e^{iu\cdot (y+z)}d({\gamma }_{C_1}\times {\gamma }_{C_2})=\exp (-u^T{C}_{1}u\mathrm{/}2)\exp (-u^T{C}_{2}u\mathrm{/}2)=\exp [-u^T(C_1+C_2)u\mathrm{/}2]\mathrm{.}$ The product measure exists by [F3], the integrand is bounded and therefore integrable, and [F4] justifies the iterated integral/product factorization. Since $C_1+C_2$ is positive semidefinite, [F2] and [F5] identify the pushforward under addition as ${\gamma }_{C_1+C_2}$. This includes singular and zero covariances.

The function $(x,y,z)\mapsto f(x+y+z)$ is nonnegative and Borel. Apply [F6] to integrate first in $y$ and then in $z$; the resulting iterated integral equals the integral against the product law of $(y,z)$. By step 1.1 its pushforward under addition is ${\gamma }_{C_1+C_2}$. Therefore $\int (\int f(x+z+y)d{\gamma }_{C_1}(y))d{\gamma }_{C_2}(z)=\int f(x+w)d{\gamma }_{C_1+C_2}(w),$ including the value $+\mathrm{\infty }$. By [F1] this is exactly $T_{C_2}(T_{C_1}f)(x)=T_{C_1+C_2}f(x)$.

For finite nonnegative Borel $V$, [F1] gives $\exp [-V_{C_1}(x)]=T_{C_1}(e^{-V})(x)$ and the corresponding identity after applying $C_2$. Substituting $f=e^{-V}$ into step 2.1 proves the stated potential-weight identity. Since the normalized Gaussian integrals are expectations under probability laws, no determinant factor occurs. No differentiation or covariance-path limit was used.