Appendix A: divergence under the conditioning integral

For every natural number $d$, including $d=0$, let $E=\mathbb{R}^{\{0,\ldots,d-1\}}$, with its usual finite-product real normed-space structure (the maximum norm), let $Q$ be any measure on the Borel measurable space $E$, let $F:E\times E\to E$ be any function, and let $x\in E$. For a function $g:E\to E$, write $Dg(y)$ for its real Fréchet derivative, regarded as a continuous linear map, with the convention that this map is zero when $g$ is not differentiable at $y$, and define $\operatorname{div}g(y)=\sum_{i=0}^{d-1}(Dg(y)e_i)_i$, where $e_i$ has coordinate $1$ at $i$ and zero at all other coordinates. Assume that $z\mapsto F(x,z)$ is Bochner integrable with respect to $Q$; that there is a neighborhood of $x$ on which, for every $y$, the function $z\mapsto F(y,z)$ is $Q$-almost everywhere strongly measurable; that $z\mapsto D(F(\cdot,z))(x)$ is $Q$-almost everywhere strongly measurable as a function taking values in the space of continuous real linear maps $E\to E$; and that there exist a set $N\subseteq E$ containing an open neighborhood of $x$ and a real-valued $Q$-integrable function $b:E\to\mathbb{R}$ such that, for $Q$-almost every $z$, simultaneously for every $y\in N$, $\|D(F(\cdot,z))(y)\|_{\mathrm{op}}\le b(z)$, and, for $Q$-almost every $z$, simultaneously for every $y\in N$, the function $F(\cdot,z)$ is real Fréchet differentiable at $y$. The two almost-everywhere conditions may initially have different null exceptional sets, but neither exceptional set depends on $y$. Here almost-everywhere strong measurability means agreement outside a $Q$-null set with a strongly measurable function, and Bochner integrability means almost-everywhere strong measurability together with a finite integral of the norm. Then the function $G:E\to E$ defined by $G(y)=\int_E F(y,z)\,dQ(z)$ is real Fréchet differentiable at $x$, the real-valued function $z\mapsto\operatorname{div}(F(\cdot,z))(x)$ is $Q$-integrable, and $\operatorname{div}G(x)=\int_E\operatorname{div}(F(\cdot,z))(x)\,dQ(z)$. The integral defining $G$ uses the total Bochner-integral convention, which assigns zero to a nonintegrable integrand; the assumptions do not separately require integrability of $F(y,\cdot)$ at every $y\in E$. There is no assumption that $Q$ is a probability measure, finite, or sigma-finite, no joint measurability assumption on $F$, and no global nonnegativity assumption on $b$; its displayed bound forces $b\ge0$ almost everywhere because $x\in N$. The case $Q=0$ is included, in which the almost-everywhere conditions are vacuous and all the displayed integrals are zero. When $d=0$, $E$ is the one-point zero vector space, the defining divergence sum is empty and equals zero, every $E$-valued function is the zero vector, and the asserted equality is $0=0$ for every measure $Q$ on this space. The neighborhood $N$ cannot be empty because it contains $x$.